Chapter 1 Introduction

The present work is the result of more than 3 years of research in a doctoral program in optics, materials and nanotechnology. The particular topics discussed and studied are scintillation light and light-matter interactions.

In this introductory chapter a brief description of physical phenomena related to scintillation is provided. The scientic background on which the main ideas discussed in later chapters is presented. As the focus of the work is the control of the light produced in a scintillation event, we rst give a few words on optics and Maxwell's equations and then scintillation light is introduced as a result of highenergy radiation processes. Finally, it is important to mention that this work was carried out in the frame of 2 research projects, something that is explained in more detail below.

A brief overview

The control of the propagation of light has seen tremendous progress over the last centuries. While geometrical optics has allowed mankind to capture our vision of the world since ancient times, in the last two centuries optical technologies have beneted from the eorts of the scientic and engineering communities. Examples of progress in this domain include photographic cameras, optical microscopes, telescopes and related tools that are now ubiquitously used at a global and even extraterrestrial scale.

Beyond ray optics, however, an incredible large amount of phenomena has been synthesized in Maxwell's equations throughout the 19th century. Within this framework, light acts as a wave rather than a ray but naturally leads to geometrical optics within the appropriate conditions. Our insight into the behaviour of light has been signicantly expanded since then. This, in turn, allows us to access otherwise hidden aspects of the world, at least to our perception, such as those revealed by spectroscopic techniques. Furthermore, the rules set by Maxwell's equations have proved to be a robust description of electromagnetic phenomena, not limited to the behaviour of visible light but including all forms of electric and magnetic forces.

Since the end of the 19th century, extensive scientic eorts have led to the understanding of radioactive decay and key concepts in particle physics. In this context, a plethora of analytical tools have found use not only in physics but in 2 1.1. A brief overview science and engineering in general. In this work however the scintillation mechanism is studied from an optics perspective. One of the aims of the work is to provide a description that encompasses the main elements of the optical response related to the scintillation process. To this end, the work is separated in two broad categories: light propagation and light emission. Both categories can be considered as lightmatter interactions.

Of particular interest for this work is scintillation phenomena. The physical description of scintillation has been an active topic of research in physics for some time [START_REF] Schorr | Solid non-crystalline scintillation phosphors[END_REF][START_REF] Heath | Inorganic scintillators: A review of techniques and applications[END_REF][START_REF] Delage | Scintillation of organic materials[END_REF][START_REF] Hajagos | High-z sensitized plastic scintillators: A review[END_REF]. An important number of applications has resulted throughout the 20th century. These have led to sophisticated radiation detectors and complex theoretical tools. Some examples of the processes involved in scintillation include Compton scattering, the photoelectric eect and pair production, to name a few.

A remarkable example of the use of scintillating media lies within medical-physics.

In this regard, the imaging technique called Positron Emission Tomography (PET) makes use of predictions given by one of the most accurate theoretical frameworks available in science nowadays, namely quantum eld theory (see [START_REF] Cherry | Pet: physics, instrumentation, and scanners[END_REF] for an overview).

Nevertheless, the aim of the PhD project is to study the optical response of scintillating media. This means that we borrow the physical picture provided by the nuclear physics community and address the behaviour of the optical elds generated in such materials.

Within the PhD project two research projects have been involved. The projects are related to the subjects studied in this work. The general scope of the projects is related to improvements of scintillating materials. First, the European project TURBOPET [6] was already in an advanced stage by the time the PhD started.

This provided a natural framework to address light propagation issues. Another project named DECISIoN [START_REF] De La Recherche | Nanostructured plastic scintillators for enhanced radiation detection[END_REF] is a currently ongoing collaboration that addresses the issue of generating more light in scintillation crystals through the use of structured media.

The discussions made throughout the chapters reect ideas and insight provided in the frame of the projects. This is specially true for the discussion on light propagation. Within this context it is important to mention that the collaborations leading to this work involved people from dierent domains in physics and engineering. Selected results of the TURBOPET project are presented here and discussed further. This project came to an end in spring 2018.

Concerning the project DECISIoN, the general goal of increasing light emission is pursued. The beginning of this project coincided with the end of the TURBOPET project. A considerably dierent approach is taken within this collaboration. Here, one is interested in microscopic phenomena and the optical response of localized light sources. This project will remain active at least for a couple of years more.

The discussion on light emission given here attempts at providing insight into the mechanisms that could lead to improvements in scintillating materials.

The current understanding: radiative and nonradiative decay

Theoretical methods in physics constitute the fundamental grounds upon which mature technological solutions are built. Most frequently, however, discoveries push theories forward. Over the last few centuries, one could say that correlations between experience and theory have essentially justied the use of formal logical tools found in pure mathematics. In the context of the present work, the theoretical developments needed came into fruition in the late 19th century and throughout the 20th century. After much eort and even at the expense of human lives, what is now called nuclear physics became a mature science. In fact, this eld has proted, and continues to do so, from some of the most sophisticated theoretical frameworks available, namely quantum physics. In particular, we currently consider the standard model of particle physics as bearing fundamental symmetries and notions of the universe. Beyond the fascinating aspects and philosophical implications brought about by the quantum approach, concrete and precise tools for science and technology have resulted from it. Remarkable examples of applying such abstract thinking to real world solutions include medical imaging techniques. Particular emphasis is made on Positron-Emission-Tomography (PET), an imaging technique that relies on predictions given by particle physics. Scintillating media is used to acquire the physical signals emerging from such complex processes.

In the literature one encounters descriptions of the scintillation process in terms of nuclear physics (see, e.g. [START_REF] Hajagos | High-z sensitized plastic scintillators: A review[END_REF] for a modern review). There is no doubt in the fact that the models work and the theory is successful. Indeed, technological solutions rely on accurate descriptions of the physical world. When mature understanding of the physical situation is acquired, one can focus on distinguishing between possible solutions for a given problem. In the present view, the development of theoretical tools and the relation between theory and experience is highly motivated by the success of past work. For example, in the theoretical descriptions provided here, particular emphasis is made on results reported in [START_REF] Vries | Point scatterers for classical waves[END_REF][START_REF] Bouchet | Quantum dipole emitters in structured environments: a scattering approach: tutorial[END_REF][START_REF] Allen | Transmission of isotropic light across a dielectric surface in two and three dimensions[END_REF][START_REF] Walsh | The reection factor of a polished glass surface for diused light[END_REF]. In contrast to previous developments, however, the focus and the context in which these tools are used are considerably dierent.

An unavoidable aspect of the physical situation is the microscopic character of the phenomena involved. Access and use of this microscopic world is now a continuous experience. The use of solid-state electronic devices is an obvious modern example.

Without highlighting further the consequences of looking into very small regions of space, we state that microscopic models of physical phenomena are crucial for studying light emission mechanisms.

Within the current description of the (quantum) microscopic world one encounters the concept of energy levels. This concept is related to physical entities such as electrons or protons for example. One then says that an electron, proton or another particle behave in the following general way: depending on external forces, near-by objects or simply for no reason at all, the particle can migrate to a lower or higher energy level. The result of this transition between energy levels could be a photon.

As mysterious as this mechanism is, it has led to our present understanding of microscopic phenomena [START_REF] Loudon | The quantum theory of light[END_REF][START_REF] Grynberg | Introduction to quantum optics: from the semi-classical approach to quantized light[END_REF]. For example, when the energy transition associated 1.2. The current understanding: radiative and non-radiative decay to one electron results in a photon, one speaks of a radiative transition. On the other hand, if the transition has not resulted in a photon but in another kind of excitation, we say a non-radiative transition has occurred.

From the perspective of nuclear physics, scintillation is a well-established set of mechanisms resulting in light emission. Much eort towards improving the properties of scintillating materials has been made. The approach taken so far towards improving their light-emission response is related to the chemical and mechanical properties of the material. Surely important reasons are behind this choice. For example, a scintillating material must resist high energy radiation. On the other hand, molecules with convenient light-emission properties are known for some time.

From the perspective of electromagnetic theory and in particular, from the perspective of physical optics, the manipulation of the scintillation mechanism is largely unexplored. It can be argued that the scintillation process already involves several complicated mechanisms, at least in its description, where radiation is only described by its spectrum or in terms of rays, for example. In the context of radiation detectors based on scintillation light, one frequently encounters descriptions of the light source based on unpolarized isotropic emission from a point source, for example.

Improvements of the properties of scintillating media have not been addressed, to the best of our knowledge, beyond a ray optics approach. A few exceptions exist, of course, where the transmission properties of scintillating materials are modied with the purpose of extracting as much scintillation light as possible [START_REF] Knapitsch | Photonic crystals: A novel approach to enhance the light output of scintillation based detectors[END_REF].

Scintillating materials can be in the solid and liquid state. Here we will focus only on solid-state scintillating media. Furthermore, in solid-state scintillators one also distinguish two cateogories: inorganic and organic scintillators. Besides the organic/inorganic aspect, they posses physical properties that render their use more convenient for particular applications.

Inorganic scintillators usually have more ecient light emission mechanisms than their organic counterpart. Moreover, they are hard and dense media that is able to better dissipate the energetic radiation being detected, i.e. ionizing radiation.

However, the main drawback of inorganic scintillating materials is the elevated cost of growing this kind of crystals. Technical reports on their properties can be found in [15] for the particular inorganic scintillators used in this work.

Organic scintillators are also a popular choice for the design of radiation detectors. In the solid-state this materials are basically plastics. This type of scintillators benet from rather low cost in their production compared to inorganic crystals.

The main disadvantage in this case is the relative low light yield. In contrast to the inorganic crystals, plastic scintillators are less dense and do not posses the same radiation-resistant characteristics as inorganic scintillators. See [START_REF] Technology | Ej-240 long decay time scintillator[END_REF] for a technical report on their general physical properties for the plastic scintillators of interest in this work.

The properties of light emission in scintillating materials has not been addressed in the past from the perspective of nanophotonics, with the exception of a handful of works in the literature that address enhanced light extraction in such materials.

A plethora of tools related to the microscopic world has resulted from this modern research eld. An electromagnetic approach towards the manipulation of light emitted in scintillating media is the novel aspect of the present work. 1.3 Towards an electromagnetic approach to scintillation phenomena

Electromagnetic theory is now an important reference for all domains of physics. It encompasses a large amount of phenomena and the applications of the theory have resulted in indispensable tools for the modern world. The concept of electromagnetic interactions is of crucial importance in this work. This general term refers to the fact that forces can be exerted upon material objects with the use of electromagnetic elds. In this regard, optical phenomena benets from extensive research.

From an optics perspective, one considers electromagnetic waves that somehow inuence the behaviour of objects. In the context of situations one encounters on a daily basis, one says that shining light onto some material equals to exciting the material. The purpose of an electromagnetic approach to scintillation phenomena is therefore to use optical excitation to our advantage. The concept of excitation immediately implies the notion of interaction. The interactions of light with scintillating media constitute the main subjects of the present work.

1.3. Towards an electromagnetic approach to scintillation phenomena Research on isolated particle scattering or even single-molecule detection has led to mature tools involving the manipulation of light. In this regard, the present work has similarities in the physical systems studied. Towards the end of the thesis we will focus on near-eld eects as well as on the mesoscopic regime. The latter denotes phenomena occurring at characteristic length-scales for which propagating and evanescent components of the light eld interact with near-by objects. A closely related subject is multiple scattering of waves. Although among the theoretical tools we will deal with a quantum approach describing the light source, the interference eects associated to the electromagnetic eld are of a classical nature. The interaction between particles of similar shape and size with localized sources will be also important in the discussion of light emission processes.

The current description of propagation and emission of light in solid media includes wave phenomena. The former addresses mainly the changes in the eld distribution after propagating a very large distance compared to the wavelength of illumination. In this context we think about illumination in terms of an external source of light. At least for the rst part of the work, the intuitive idea of considering a common light bulb or, in modern times, a light-emitting-diode (LED) as illumination is essentially the manner in which we will consider the scintillation light generated inside an inorganic or organic solid medium. Normalized photon count at the at scintillator/air interface obtained by dierent ray tracing software. The angular distribution of photons for hemispherical illumination dI ∼ I 0 cos θ i dΩ for dΩ = sin θ i dθ i dφ i is also shown for comparison. The results of LITRANI and GEANT2 software were obtained by the CERN (Conseil Européen pour la Recherche Nucléaire) in the frame of the TURBOPET project.

New concepts arise when dealing with light distributions at a length scale compared to the wavelength. A general framework must add evanescent elds to the description. While the present discussion on the wave nature of light is strictly on classical physics grounds, it is nonetheless relevant in the context of purely quantum phenomena. The main example of the latter is spontaneous emission of electromagnetic radiation. For example, important excitation mechanisms in light-emitting molecules are currently understood on the basis of microscopic electromagnetic elds 1.3. Towards an electromagnetic approach to scintillation phenomena and the number of electrons or carriers present in the physical system [START_REF] Bouchet | Quantum dipole emitters in structured environments: a scattering approach: tutorial[END_REF]. The concept of electric polarizability and the elds associated to very small distributions of charge is of paramount importance in describing eld-molecule interactions. face. The scintillation process is assumed to yield a diuse light source. With this information, one can estimate the amount of trapped light inside a bulk crystal with a polished exit surface. Although a scintillation event can be considered to yield a localized light source, in this work we focus on average quantities that represent the bulk material. In this way, we avoid describing the light source for a particular scintillation event. For example, one can trace light rays inside a scintillator to simulate the reected and transmitted fraction of light across macroscopic (centimetre-scale) scintillators. Repeating this procedure one can found the distribution of angles with which light rays arrive at some boundary of the crystal. Results of this kind have been obtained in this work and are presented in gure 1.2.

Numerical simulations of two-dimensional gratings are presented and described in terms of their transmission properties. The spatial distribution of reected and transmitted light is also addressed. In this regard, structures with periods smaller than the emission wavelengths (sub-wavelength) are studied. These sub-wavelength structures will not produce diraction orders (only the zeroth or specular order). On the other hand, when the period is large enough, diraction orders appear and the spatial light distribution is modied. The geometry of the particles in the grating has an inuence on the power distribution among diraction orders.

Experimental results of light transmission in at and structured samples are 1.3. Towards an electromagnetic approach to scintillation phenomena presented (see gure 1.3). The transmission of a collimated laser beam through a hemi-cylindrical sample was measured as a function of the angle of incidence of the beam. Here a blue laser is used in order to avoid photoluminescence (PL). The wavelength of operation is 405 nm, close to the maximum emission wavelength of 420 nm. The results show that, above the critical angle, transmission values on the order of 10 percent can be achieved for low-order diracting structures.

Depending on the specic application, distinguishing specular transmission of light from diraction orders may or may not be important. For example, devices relying essentially on detecting overall light levels could benet from strongly diracting structures. Indeed, at oblique incidence, total internal reection is partially avoided by the transmitted orders. On the other hand, if the transmitted light distribution is needed for further processing, diracted orders could result in detrimental eects.

For example, in the context of PET, retrieving the position of scintillation events is crucial for rendering useful images. Here, radionuclides are used to label compounds that will be distributed inside a living body. The distribution of the radionuclides inside the body depends on its specic biochemical properties. The radionuclide is a priori a source of positrons which serve as marks in molecules of interest. Ideally, the interaction of the positron with its surroundings results in positron-electron annihilation, which in turn produces a pair of gamma-photons. The latter must somehow excite a pair scintillators to produce and eventually detect visible light.

Quantitative analysis and mapping of the concentrations of tracers in the body is possible with appropriate congurations of scintillators [START_REF] Cherry | Pet: physics, instrumentation, and scanners[END_REF]. A realistic model should also include the phenomenon of electron capture. This can be avoided in PET by appropriately choosing the radionuclide [START_REF] Qaim | Pet radionuclide production[END_REF]. The scintillators must then detect, in a synchronized fashion, visible photons generated by the interaction of high-energy photons with the material. The fact that coincidence measurements are employed in PET systems stems from the electron-positron annihilation process: when annihilation takes place, two gamma-photons are emitted in opposite directions, 180 degrees apart. In summary, in PET scanners, the detected signal relies ultimately on visible light. The process by which visible light is generated can be traced back to the annihilation of a positron and an electron. The positron, in turn, can be traced to specic atoms attached to molecules of interest. From the perspective of optics, one may attempt at condensing this complicated mechanism into an eective diffuse light source. Evidently, a given annihilation event will produce gamma-photons that can interact with the scintillating medium and produce some amount of visible photons. We do not address such situation. Instead, we focus on average quantities.

For example, here it will be assumed that the light produced from a large number of scintillation events can be described as a diuse wave front.

The second part of this work concerns light emission in scintillation crystals. In this part, the active properties of the material are studied. These properties are related to the internal degrees of freedom of the material. The "active" properties of organic and inorganic scintillators are described. Absorption properties of plastic samples were experimentally determined. A schematic of the experiment and representative results are shown in gure 1.4 and 1.5. Furthermore, the so-called quantum eciency or quantum light yield has been measured, in the UV region, using an integrating sphere and a spectrophotometer. In this second part, the focus is on plastic scintillators, which possess ecient emission mechanisms. A phenomenological approach is provided. The main elements of the approach are related to three-level systems describing the emitters (organic uorophores) and electromagnetic interactions between the emitter and the (possibly structured) dielectric background. The semi-classical approach of the proposed model relies on classical electromagnetic elds and, for the light source, quantized energy levels with their corresponding populations. The interaction between visible radiation and the uorescent molecules is treated in the electric-dipole approximation. Experimental characterization results are presented, whose main goal is to illustrate the theoretical approach describing the main active optical properties of scintillation in polymer-based materials. The characterized samples are bulk plastics with dierent geometries and polished surfaces.

From a theoretical perspective, the well-known case of a (optically) at surface is studied in the context of light emission. The analysis is made in terms of the Local Density of Optical States (LDOS), which essentially counts the number of eigenmodes in a innitesimal frequency range at a given position in space [START_REF] Loudon | The quantum theory of light[END_REF][START_REF] Economou | Green's functions in quantum physics[END_REF].

Classical electrodynamics provides a way to calculate the modication of the LDOS through the electric-dipole and magnetic-dipole elds [START_REF] Novotny | Principles of nano-optics[END_REF][START_REF] Novotny | Single molecule uorescence in inhomogeneous environments[END_REF]. These type of elds are discussed in detail and form the basis of the numerical methods used in this part.

The latter are used to study more complex geometries. The results obtained in this part can be used to estimate uorescence enhancements in structured dielectric environments.

A general approach towards simulations of the radiation-molecule interaction is described in terms of so-called optical Bloch equations [START_REF] Bouchet | Quantum dipole emitters in structured environments: a scattering approach: tutorial[END_REF][START_REF] Loudon | The quantum theory of light[END_REF][START_REF] Grynberg | Introduction to quantum optics: from the semi-classical approach to quantized light[END_REF]. This set of equations describes the temporal evolution of the populations in a three-level system being excited by an electromagnetic eld. The latter could be a scintillation photon or an external excitation. Furthermore, an appropriate laser excitation provides an optical probe of the interaction process, as it could allow simplications in the theoretical description as well as to estimate absorption and emission properties of the material. For example, gures 1.7 and 1.8 are calculations of the eld response for a wire-like particle. The eld in a cross-section of the wire, whose axis extends into the plane of the gure, is shown. This kind of calculations can be considered the starting point for more complex simulations such as those described here for the temporal dynamics of populations in a three-level system. Although the maps show the absolute value of the eld, the real and imaginary parts are required separately.

It may be useful to recall some basic notions allowing for intuitive descriptions of the phenomena studied here: rst, at the title of the work indicates, we emphasize the fact that the optical response will be addressed in detail. A great number of dierent physical phenomena is involved in a scintillation event. This renders its study a dicult task. Indeed, in any practical approach one must choose a particular aspect of the situation to study. Then, while trying to condense physical phenomena of no particular interest into some meaningful quantity, one focus on specic aspects that can be isolated and described in detail. Such approach is taken in this work towards the description of interactions between scintillation light and complex electromagnetic environments. In the following, we assume there is no induced photoluminescence in the physical situations described.

The visible light generated in a scintillation event can be considered a kind of diuse light source, at least in the sense that there is not a xed phase relationship between photons. Furthermore, in order to describe average quantities related to the probability of light to escape the scintillator, we model light incident on the exit surface as coming from an isotropic light source. A well known fact in optics and 14 physics in general, is that the appropriate collection of plane-waves can reconstruct any wave-front. Therefore, as a rst step, we can address the problem of transmission of diuse light by a at surface using the Fresnel equations for reection and transmission by a at interface. In summary, for both planar and non-planar surfaces, the problem reduces to calculating the plane-wave response of the structure.

As light is generated inside the scintillator and detected outside (with a photomultiplier tube, for example), it reaches the interface from the medium with higher refractive index. This corresponds to a conguration of internal illumination. Figure 2.1 shows a schematic representation of internal and external illumination conditions.

For example, reection is commonly studied in external illumination, where light is incident from the less dense medium. On the other hand, the reection of light in internal illumination can lead to total internal reection (TIR). In the latter, special physical considerations come into play, namely accessing purely evanescent waves. In the following, the use of the terms transmission and reection will imply a conguration of internal illumination. The enhancement of the LEE by a structured interface can be dened as the ratio of the transmission by the structure for diuse incident light, and the transmission by a at surface with similar RI. In essence, structured interfaces modify transmission properties. If the purpose is extracting more light (compared to a at surface), structured interfaces should reduce diuse reection as much as possible. Moreover, if the structure employed to modify transmission has negligible absorption properties, decreasing diuse reection should readily yield higher transmitted light levels.

An advantage of surface structures with a thickness H comparable to the wavelength of illumination, H ∼ λ, is the fact that only strongly absorbing media would result in reduced transmission. We can estimate some orders of magnitude from dielectric properties of the scintillators at emission wavelengths. Consider a peak emission wavelength of λ em = 420 nm. The refractive index of an inorganic scintillator such as LYSO is frequently assumed a real quantity. We have measured absorption properties at similar emission wavelengths for plastic scintillators (see chapter on characterization of active optical properties). We found extinction coefcients κ on the order of κ ≈ 10 -7 . This means that the complex refractive index is n c = n + i10 -7 , basically a real quantity. The intensity I(z) of a directional beam traveling through the (homogeneous) scintillator in the z-direction can be written as

I(z) = I 0 e -αz , (2.1)
where α = 4πκ/λ em ≈ 1/40 cm -1 and I 0 is the intensity of the beam at the entrance of the scintillator. For example, a surface structure with a thickness equal to a wavelength, i.e. H = λ will at most reduce the beam's intensity by a factor 1 -e -4πκ ≈ 2 × 10 -3 or 0.2%. The beam is attenuated by 10% after travelling 4 cm (or roughly 95000 wavelengths) inside the scintillator. In order to obtain, for example, a reduction of 10% in the beam's intensity after travelling one wavelength inside the material, the extinction coecient must be κ ≈ 8×10 -3 . Nevertheless, in optical glasses (see, e.g. [START_REF] Inc | Optical glass datasheets[END_REF]), one can easily nd κ ≈ 10 -8 . Therefore, the assumption of a real refractive index for scintillators is appropriate provided the emission wavelength is far from absorption bands. Plastic scintillators may be directly structured on its surface because they are relatively soft media. On the contrary, inorganic scintil-2.1. Light extraction eciency lators may require an external layer on top of which to fabricate the structure. In order to avoid additional reections from this possibly structured layer, the latter should closely match the refractive index of the scintillator. A layer with higher index could result in guided-mode excitation [START_REF] Magnusson | New principle for optical lters[END_REF][START_REF] Wang | Theory and applications of guided-mode resonance lters[END_REF], while lower-index layers, due to low dielectric contrast, are expected to interact less with radiation [24]. Low-index structures, however, provide additional reections induced by the RI mismatch.

Light extraction eciency

In this section we dene a basic gure of merit for light extraction. As the source of light, we will assume a hemispherical wave front towards the exit surface of the scintillator. As we deal only with passive optical properties, we will not calculate η LE directly. Instead, the enhancement of the LEE for a given structure is estimated by comparing it with a known reference. To this end, we denote the probability for a photon emitted inside the scintillator to be transmitted across a at interface as η LE F . The latter will serve as the known reference. F of the incident plane-wave is transmitted by the interface into a medium with RI n > 1. When light is incident from the higher-index side (internal illumination), a fraction T (i) F is transmitted to the opposite side. As it turns out, for a at surface, the diuse transmittance values for external and internal illumination are related through a simple expression (cf. equation 2.4) [START_REF] Walsh | The reection factor of a polished glass surface for diused light[END_REF]25,26]. Therefore, by considering in a rst stage the case of external illumination, we avoid dealing with TIR. In external illumination, the direction of the incident wave is characterized by the angles θ i and φ i . The latter dened in the usual way as the 2.1. Light extraction eciency polar and azimuth angle with respect to a vector normal to the surface. The diusediuse external reection R (e) is dened as the fraction of incident electromagnetic power that is reected per unit surface and includes contributions from all angles of interest:

R (e) = R (e) (θ i )l(θ i )dΩ.

(2.2) Here dΩ = sin θ i dθ i dφ i is an innitesimal solid angle, l(θ i ) is a weight function associated to the change in intensity with the polar emission angle. For our purposes, the integration is carried out over the upper hemisphere. R (e) (θ i ) is the reectance for an unpolarized plane-wave, as given by the Fresnel equations [27] in the case of a at surface. The diuse transmittance in this conguration and for lossless media is

T (e) = 1-R (e) . Isotropic, hemispherical illumination corresponds to l(θ i ) = cos θ i /π [25].
Although in the case of a at surface integration over the azimuth angle φ i only results in a factor of 2π in R F and T F [28], for a surface of arbitrary relief, R (e) represents the φ i -averaged reectance values.

The diuse external reection R (e) has been calculated analytically and its closed-form expression can be found in [START_REF] Walsh | The reection factor of a polished glass surface for diused light[END_REF]25]. It turns out that the diuse reection for external and internal illumination are related as

R (i) F = 1 + (R (e) F -1)/n 2 , (2.3)
which can be rewritten in terms of the transmittance as

T (i) F = T (e)
F /n 2 .

(

With these results, the enhancement of the LEE, η, can be dened as the quotient between the fraction of light extracted from the structure and the fraction extracted from a at surface. We write η as

η = η LE S η LE F = T (i) S T (i) F . (2.5)
We denote the LEE (diuse transmission) of the structured surface as η LE S (T

(i) S )
and that of a at surface as η LE F (T (i) F ). Note that although T (i) S may be computed numerically or analytically (depending on the structure), it is always possible to compute the normalization factor T (i) F analytically. Examples of LEE enhancement are presented for selected congurations in the next sections. As the calculation of η can be computationally expensive for arbitrary surface structures, we present only a few representative cases.

In what follows, we will mainly consider surface structures with periodicity on the plane on the plane parallel to the scintillator surface, perpendicular to the z-axis. The structured surface layer has thickness H and occupies the space between 0 < z < H, matching the RI of the substrate. Periodic structures oer a large number of degrees of freedom which could enable specic mechanisms for light extraction. We distinguish two cases for periodic structures: rstly, those with periods P smaller than a wavelength λ and secondly, structures with periods P greater than λ. In the sub-wavelength case, the considered range of periods includes the 2.1. Light extraction eciency transition from purely specular reection and transmission to low-order diraction. This transition is crucial if anti-reection (or impedance matching) approaches are considered, where P must be smaller than λ and even P λ. In this sense, the limits of these so-called impedance matching approaches are studied. The second case concerns diracting structures. Here, the gratings have larger periods and present more complicated behaviour. The latter stems from the fact that multiple diraction channels depend on the opto-geometrical parameters of the structure in a dierent manner. We describe eects of the geometry of the grating on light transmission. In both cases we provide the formulae required to compute the reected and transmitted angles with which the diracted plane-waves propagate. There is not much emphasis made on rigorous or formal aspects in this chapter. More formal approaches and their relation to Green functions can be found for example in [START_REF] Sirenko | Modern theory of gratings[END_REF] and references therein. Important examples are also briey discussed as alternatives to periodic structures. The latter includes random surfaces and disordered systems, as well as non-diracting resonant structures. For sub-wavelength structures that do not diract light, we expect small changes from the random case, at least for closely-packed structures. For diraction to occur, however, ordered arrays are imperative. • A six-fold increase in the LEE obtained in [START_REF] Boroditsky | Light extraction from optically pumped light-emitting diode by thin-slab photonic crystals[END_REF] for a perforated InGaAs/InP double hetero structure surrounding a 20 nm InGaAs quantum well. By placing a PhC around the edges of the unpatterned active (thin-lm) region, they obtain an optimal external quantum eciency (EQE) [31] of 70%. In this case, the improvement in LEE is achieved thanks to the out-coupling of the guided modes in the active thin lm. When the guided modes reach the nite periodic array of holes, they couple to leaky modes of the periodic structure and eventually can escape into air or into the substrate.

• Holographic lithography was used in [START_REF] Lee | A high-extraction-eciency nanopatterned organic light-emitting diode[END_REF] to dene 2-D arrays of SiO 2 pillars on top of an OLED (layered) structure. The pillars were embedded in a SiN x lm with 800 nm thickness. They obtained experimentally a 50% increase in the LEE for an angular range of ±40 • from the normal.

• In [START_REF] Wierer | Iii-nitride photonic-crystal light-emitting diodes with high extraction eciency[END_REF], a high LEE (73%) was demonstrated in a III-nitride LED, where the PhC is placed on the top of the substrate and a metallic reector is placed on the opposite side of the layered structure. Here, the interplay between Fabry-Perot resonances and diraction by the PhC is characterized by far-eld measurements. • In [START_REF] Thornton | Limit of the moth's eye principle and other impedance-matching corrugations for solar-absorber design[END_REF], arrays of tapered protuberances are studied in order to obtain an optical-impedance (gradual) matching. A reduction of 30 dB was obtained in the reection by the corrugated surface, over a bandwidth 1.4 times the short-wavelength limit of the band.

• In order to nd optimal structures with a graded-index prole, numerical studies of modied-pyramid structures have been done in [START_REF] Southwell | Pyramid-array surface-relief structures producing antireection index matching on optical surfaces[END_REF]. In this work, a polynomial expression for the RI prole is obtained in terms of the dielectric constant of each half-space, yielding a design tool for AR structures made of modied pyramids.

• Several RI proles were studied in [START_REF] Grann | Optimal design for antireective tapered two-dimensional subwavelength grating structures[END_REF] and their AR behaviour compared.

This included the structure proposed by [START_REF] Southwell | Pyramid-array surface-relief structures producing antireection index matching on optical surfaces[END_REF]. They found that a geometry with the so-called "Klopfenstein taper" prole provides better AR properties.

• In [START_REF] Min | Bioinspired self-cleaning antireection coatings[END_REF], bio-inspired and self-cleaning structures were fabricated. Si pillars of high aspect ratio reduced 90% of the reection losses of a bare Si wafer in a broad spectral region. The triangular array of pillars was fabricated by spin-coating silica particles on the Si wafers and then applying an etching process.

• Similar results were obtained in [START_REF] Wang | Biomimetic corrugated silicon nanocone arrays for self-cleaning antireection coatings[END_REF]. In this work, corrugated Si nano-cones patterned the surface of a Si wafer. Their results show reectance values below 1%

for incidence angles up to 70

• at a wavelength of 632.8 nm.

• Results for triangular (2D) geometries were reported in [START_REF] Bose | Dielectric v-ridge gratings: transition from antireection to retroreection[END_REF]. Here, the dependence of the reection properties on the period of the structure is studied. In particular, the transition from anti-reection to retro-reection behaviour was characterized for a xed V-shape.

Light extraction from scintillation crystals

Inorganic scintillators suer from light trapping due to the RI mismatch between the crystal and its surroundings. LEE enhancements are of course useful for scintillation crystals. Although the subject has raised some interest over the past decade, there are only a handful of works in which LEE enhancement is addressed by the use of sub wavelength-or wavelength-scale structures. Much work has been done towards the improvement of the spatial resolution and the photodetector/crystal array conguration. In this regard, representative examples can be found in [START_REF] Anger | Scintillation camera[END_REF][START_REF] Anger | Scintillation camera with multichannel collimators[END_REF][START_REF] Levin | Design of a high-resolution and high-sensitivity scintillation crystal array for pet with nearly complete light collection[END_REF] and references therein. In the context of optics, we could identify the common element in this works as being geometrical optics solutions. Wavelength-scale structures, for which geometrical optics is no longer the appropriate description, also have the potential of enhancing LEE. Some examples include the following works:

• Based on optical and radio-physical simulations, [START_REF] Kronberger | Probing the concepts of photonic crystals on scintillating materials[END_REF] addressed the enhancement in LEE for a cuboid shaped crystal coupled with a high-RI medium to a photo-multiplier tube. The angular distribution of photons reaching the exit surface is obtained for the at and patterned surfaces. The collection of photons reaching the enhancement of the extracted light levels [START_REF] Pignalosa | Giant light extraction enhancement of medical imaging scintillation materials using biologically inspired integrated nanostructures[END_REF].

• An improvement of 1.95 in the light output from a bismuth germanate (BGO) crystal was obtained in [START_REF] Zhu | Enhanced light extraction of scintillator using large-area photonic crystal structures fabricated by soft-x-ray interference lithography[END_REF]. The cuboid crystal was patterned with an array of conical holes by using a combination of x-ray interference lithography and atomic layer deposition.

• More recently, the design, fabrication and characterization of PhCs for light extraction purposes in scintillation crystals was reported [START_REF] Singh | Enhanced scintillation light extraction using nanoimprinted photonic crystals[END_REF]. The performance of the nano-imprinted PhCs on the scintillators resulted in a 50% increase in LE and 20% in energy resolution.

In the following, light extraction (LE) and light trapping are discussed in the context of scintillation crystals. A qualitative description is given in terms of the optical phenomena resulting from a scintillation event.

Surface structures for light extraction

When light is emitted inside a dense (bulk) medium, only a fraction of the total number of emitted photons can escape. The trapped light can bounce inside due to total internal reection (TIR) in the substrate and could be eventually re-absorbed or exit the medium in undesired directions. With the purpose of modelling the visible for the transmission of light by the structure [START_REF] Zhu | Enhanced light extraction of scintillator using large-area photonic crystal structures fabricated by soft-x-ray interference lithography[END_REF].

light generated in a scintillation event, consider an unpolarized isotropic emitter or localized light source inside the scintillator. Consider the latter to be a dense, transparent dielectric of refractive index n ≈ 1.8. In general, the number of photons that escapes the dielectric depends on its bulk shape, but we can expect that the emitted light reaches a particular at boundary as a combination of plane waves impinging at many dierent angles of incidence. Assuming the latter implies the exclusion of the emitters near (r ∼ λ) the surface. If we consider a half-space geometry, only the waves travelling upwards will eventually reach the interface. Equivalently, no reection will occur for waves propagating downwards, as the structure is innite.

Evidently, for any practical application, the macroscopic, bulk shape of the dielectric needs to be taken into account. To this end, one could solve Maxwell's equations for the associated structure of innite dimensions as a rst step and then use this results (reectance and transmittance) to establish the rules of a geometrical optics approach. The price paid would be neglecting edge eects induced by the bulk dielectric corners. For periodic structures without dissipation, boundary conditions can be imposed along the (macroscopic) structured interface. In contrast to the case of continuous dispersion relations (e.g. band-structures), due to Fourier transform properties, the response of piece-wise media with nite macroscopic dimensions of order L mac , possibly L mac λ, can be described with a discrete set of modes [START_REF] Hasan | Finite-size scaling of the density of states in photonic band gap crystals[END_REF].

The latter is represented in frequency space by Dirac's delta functions. In the limit L mac → ∞ the frequency separation between modes vanishes, recovering the continuous case. A description of scaling eects in photonic crystals in terms of the density of electromagnetic states is discussed in [START_REF] Hasan | Finite-size scaling of the density of states in photonic band gap crystals[END_REF]. Here, we restrict ourselves to innite structures, as fabrication of periodic structures is possible over regions with area A λ 2 and with lengths in the same order of magnitude along each dimension.

Nevertheless, as shown in [START_REF] Hasan | Finite-size scaling of the density of states in photonic band gap crystals[END_REF], one can retrieve the nite-L mac correction from the innite structure case.

If the dielectric is surrounded by air, the situation described above implies that for a particular at boundary of the dielectric, only plane waves with incident angles that fulll

θ i < θ c ,
where geometry and a substrate with the refractive index of LYSO, a common scintillator material, which is n = 1.827 at its peak emission wavelength λ = 420 nm [START_REF] Mao | Optical and scintillation properties of inorganic scintillators in high energy physics[END_REF]. In this approximation, we can think of the scintillator material as a dielectric with a large number of emitters placed inside. Then, a scintillation event corresponds to the emission of light in a particular interaction location. In gure 2.7, the transmission by the LYSO/air interface is plotted for internal illumination conditions and both polarizations. One can estimate the reection and transmission of diuse light by a at surface using the Fresnel equations [49]. In this conguration, assuming non-polarized light and hemispherical Lambertian illumination towards the (exit)

θ c = arcsin(1/n),
surface, only 27% of the light can escape. This example is used as a reference in the following sections, where the transmission properties of structured surfaces in this conguration are discussed. In the following sections we focus on periodic arrays of particles placed on top of a dielectric. We will see that two-dimensional gratings play an important role in extracting the light trapped inside the dielectric.

A full electromagnetic treatment of the edge eects, as well as of nite gratings of macroscopic dimensions is beyond the scope of the present work. Indeed, as presented in the bibliographic review, an approach entirely based on impedance matching the exit surface to the surrounding medium has been extensively studied. In the following section we keep the same idea in mind, with some dierences in the application as well as in the specic structure congurations. A key point in this approach is the use of structures with a periodicity smaller than the wavelength. Representative examples are discussed based on parametric numerical studies of light transmission across structured dielectric interfaces. In the numerical calculations that follow, due to the high degree of transparency (at wavelengths of peak emission) of common scintillator material, the extinction coecient is neglected, i.e. we model the system as non-absorbing at emission wavelenghts.

Besides non-diractive structures, the interplay between low-order diraction in reection and the zeroth-order transmission is described. This results in enhanced reection inside the scintillator. A set of formulae is given for the particular case of triangular lattices. This kind of lattices are particularly interesting as they are amenable to fabrication in cm 2 -areas [START_REF] Kadiri | Topology assisted self-organization of colloidal nanoparticles: application to 2d large-scale nanomastering[END_REF][START_REF] Koegler | A novel approach to quantitatively assess the uniformity of binary colloidal crystal assemblies[END_REF][START_REF] Demeter | Fabrication of 2d tio 2 nanopatterns by plasma colloidal lithography[END_REF]. The equations allow the estimation of basic parameters of the structure from simple measurements. They also permit to establish conditions separating the non-diractive from the diractive regime for a given incident eld. If the parameters of the structure are known, the spatial distribution of transmitted and reected light can be predicted. The power distribution among diracted waves, however, is determined only by solving the electromagnetic problem, i.e. by solving Maxwell's equations. Alternatives to periodic structures are of course available. One could also consider for example, rough surfaces, which randomly scatter light and can only be described by their statistical properties. A systematic study of reection of diuse light by rough surfaces is presented in [25] and more generally, prescribed scattering properties through the use of rough surfaces is reported in [START_REF] Maradudin | Designer surfaces. Elsevier[END_REF]. Recently, disordered systems of particles exhibiting transparency eects have been reported [START_REF] Leseur | High-density hyperuniform materials can be transparent[END_REF]. In terms of operating bandwidth, these approaches frequently provide robust responses compared to traditional periodic structures, requiring however complicated numerical simulations in its design. Recall that at some length-scale, all surfaces are rough and need a statistical description. For visible light, however, we consider that the organic and inorganic scintillation materials employed in this work have optically at surfaces (cf. characterization of the at LYSO/air interface).

Results and discussion

In the following we assess the performance of surface structures by computing the transmission of light across them. The incident plane-waves impinge the interface from the dense medium. We start by briey describing the numerical method employed for the calculations of light transmission across the gratings. The electromagnetic elds are computed using Rigorous Coupled-Wave Analysis (RCWA), also known as the Fourier Modal Method (FMM). The package S 4 , an open source software described in detail in [START_REF] Liu | S 4 : A free electromagnetic solver for layered periodic structures[END_REF], was used to compute the transmission values

Results and discussion

presented in this chapter. Here, we describe the main characteristics of the method and highlight particular aspects related to the accuracy of the presented numerical results. In this approach, the (3-D) geometry of the system is built with a number of layers N L . All the layers are parallel to the substrate and of innite extent in this plane. The rst layer corresponds to the scintillator substrate, which occupies the region -∞ < z < 0. The structured layer is placed on top of the substrate, occupying the region dened by 0 < z < H. We will use the terms structured layer to refer to a single or multiple patterned layers. A third layer extending from z > H to z = ∞ is placed above the structured layer, which corresponds to the medium immediately outside the scintillator (see g. 7.

2) and in the present case is air.

The solution of the electromagnetic problem is obtained in terms of propagating and evanescent plane-waves for each layer, where they have a e iqnz dependence for some complex number q n associated to the n-th mode. For example, the α-component of the magnetic eld in the plane of periodicity can be written for the structured layer as

h α (z) = n c n a n e iqnz + b n e -iqn(z-H) , (2.6)
where c n are the Fourier coecients and the pair of coecients {a n , b n } determine the relative weights of the upward-and downward-propagating waves. The values of the latter is determined by the boundary conditions between layers. Each mode is a spatial harmonic of the specularly reected or transmitted beam. In practice, one needs to truncate the innite Fourier expansion to a nite number of harmonics (or diraction orders) N h . By computing the electromagnetic energy present in the incidence and transmission layers for a given incident plane-wave, the transmittance of the structure can be calculated. To this end, the eigenvalue problem is solved with fairly standard numerical schemes [56].

In order to build a lattice of particles, each layer must be appropriately patterned and possess the appropriate size. In the present work we consider cones and cylinders as the constituents of the lattice. Building the cone with layers results in a staircase approximation (see g. 7.2b). Note that, in contrast to a conical geometry, a nite circular cylinder (cf. g 2.13) only requires one layer patterned with a periodic lattice of circles. The computation of a grating of circular pillars is therefore less timeconsuming than the corresponding computation for a grating composed of cones. In fact, the bulk of the computation time has a linear dependence with the number of layers N L and a quadratic dependence with the number of spatial harmonics N h representing the structure, i.e. T (N L , N h ) ∼ N L N 2 h . It is worth commenting on some subtle aspects of the analytical (or numerical) description of the geometry in S

Results and discussion

On the other hand, in RCWA, the Fourier coecients associated to (r) are directly connected to the eigenvalue solution of Maxwell's equations in the spatial-frequency domain. Therefore, discretization artifacts are of a dierent nature in each numerical method mentioned above. In particular, for non-cuboid shapes, RCWA suers from the layer discretization. For conical particles, one could use a large number of thin layers and reduce this eect. The latter improves the spatial resolution with which an ideal cone is constructed but may lead to prohibitive computation costs.

Recall that the solution of the electromagnetic problem is given with N h harmonics, which correspond to propagating and evanescent plane-waves emerging from the structured interface. Therefore, solutions with physical meaning should consider a suciently large number of harmonics N h ensuring convergence of the transmission and reection values, T and R respectively.

For lossless structures one should also monitor energy conservation using the quantity T + R = 1. Both N h and N L should be chosen to ensure a physically reasonable approximation of a realistic structure. Based on convergence studies of the calculations presented hereafter, we precise the reasons behind our choices for N h and N L . For a given structure and incident eld, the criterion for achieving convergence is to obtain variations in transmission of δT ≤ 1% when increasing N h . For small period, non-diracting structures, convergence can be achieved with N h ∼ 50, at least for aspect relations H/D ≤ 5. In the present work, cones were discretized using N L = 20. When diraction takes place, however, the number of harmonics needed for convergence can increase considerably. For example, for a structure of period P = 1 µm and thickness H ∼ λ, one could easily need N h ≥ 300 for convergence, depending on dielectric contrast and aspect relation of the particles in the structure. As T ∼ N L N 2 h , for equal N L , compared to a simulation with N h = 50, one with N h = 300 is a factor of (N h /N h ) 2 = 36 more time-consuming. This means that, while the response of a sub-wavelength structure could be simulated in two hours, a diracting structure with P = 1 µm could take 3 days. The reasons discussed above also justify the choice of pillars as the grating constituents for diracting structures.

Sub-wavelength regime

Light trapping eects are detrimental for devices relying on detecting the light escaping a given medium, e.g. by placing a photodetector or a photo-multiplier outside the dielectric. As a rst approach to enhance light extraction, we address the performance of sub-wavelength cone gratings placed on the surface of a transparent dielectric. This approach is frequently used in anti-reection devices; its aim is to gradually match the impedance between the substrate and the upper medium. As in the case of plane surfaces, the critical angle θ c plays a crucial role in the transmission of light by sub-wavelength gratings. Indeed, conservation of the tangential electric eld components constraints the transmission, as only the zeroth (specular) order can propagate across the grating, at least for relatively small angles of incidence θ.

Based on rigorous numerical calculations of the electromagnetic response of cone gratings, the optical response of dierent geometric congurations are discussed, focusing on applications related to light extraction.

Before presenting numerical results, we start by establishing a condition that allows the separation of diracting and non-diracting regimes. To this end, one 2.2. Results and discussion frequently considers the tangential components of the elds, as rendered by appropriate boundary conditions. The wave vector, for both the incident and diracted wave vectors, are conserved across the interface. For a periodically structured surface, they are related as [START_REF] Loewen | Diraction gratings and applications[END_REF] k ν = k i + (mG 1 + nG 2 ).

(2.7)

Here ν = t (ν = r) for the transmitted (reected) wave vector, while k i is the wave vector of the incident wave. The integers m and n are integers labelling the diracted orders. G 1 and G 2 are the grating's reciprocal lattice vectors.

For a triangular lattice of period P , the lattice vectors are L 1 = P (1, 0) and

L 2 = 1 2 P (1, √ 3
). Dening a matrix M with columns L 1 and L 2 , the reciprocal lattice vectors are given by

M k = 2πM -T , (2.8)
where M -T is the transpose of the inverse of M . The columns of M k are the reciprocal lattice vectors

G 1 = 2π P (1, -1/ √ 3)
and

G 2 = 2π P (0, 2/ √ 3).
(2.9)

Across the interface the tangential wave vector components can be written as

k ν f (θ ν , φ ν ) = k i f (θ i , φ i ) + mG 0 , (2.10a)
k ν g(θ ν , φ ν ) = k i g(θ i , φ i ) + 2n -m √ 3 G 0 . (2.10b)
where ν = r for the reected waves and ν = t for the transmitted waves. In these expressions we dened G 0 = 2π/P and f (θ, φ) = sin θ cos φ, g(θ, φ) = sin θ sin φ.

In order to determine the regime of only zeroth-order transmission, we can write the wave number in the transmitted side as

n 2 t k 2 0 = k 2 x + k 2 y + k 2 z .
(2.11)

Here, k 0 = 2π/λ is the free-space wave number and k α for α = x, y, z are the Cartesian components of the wave vector of the transmitted wave. Only evanescent orders will be transmitted if

n 2 t k 2 0 < (k 2 x + k 2 y ).
(2.12)

The components k x = k i f i + mG 0 and k y = k i g i + qG 0 with q = 2n-m √

3

. Dividing by k 2 0 , we obtain

n 2 t < n i f i + m λ P 2 + n i g i + q λ P 2 .
(2.13)

Developing the squares we get

n 2 t < n 2 i sin 2 θ i + 2n i sin θ i (m cos φ i + q sin φ i ) λ P + m 2 + q 2 λ P 2 . (2.14)
This condition comes from eq. 2.7. Indeed, it determines the emergence of the (m, n) diraction order for a given pair of angles of incidence θ i and φ i . Note that condition 2.13 only depends on the ratio λ/P rather than on both quantities separately.

From sub-wavelength to diracting structures

The geometry of the simulated structured scintillator surfaces is shown in gure 2.8.

The image in the upper row is an illustration of the top of the structure placed on the scintillator. The lower row is a cross-sectional view and the inset shows the staircase approximation inherent to the RCWA approach. In this section, we focus on the general characteristics of an impedance-matching approach to extract light from dense bulk media. The reasons for studying the particular congurations considered here are related to the current centimetre-scale nano-fabrication technology. For an impedance-matching approach to be successful, at least in previously reported work, one considers periods P λ. In this case, because the wavelength of emission is around λ em = 420 nm, the latter condition may render the fabrication unfeasible. It is then relevant from a scientic and technological perspective, to study the limits of the approach in terms of the P/λ ratio. cones. Both the substrate (incidence medium) and the cones have a refractive index of n = 1.81. Plane waves of wavelength λ = 420 nm impinge on the grating from the substrate side. The transmittance maps are a function of the polar angle θ of incidence and the period P of the grating. The base diameter of the cones is D = f P , where f ≤ 1 is dened as the lling fraction. The height H of the cone was set to H = D, corresponding to a cone with apex angle α = 60 • . The intervals for the varied parameters are 72 nm < P < 318 nm, 0 • < θ i < 90 • and the lling fraction was xed at f = 0.8. The azimuth angle was xed at φ i = 0 • . The dramatic change in the transmittance is due to TIR in the substrate. For incident angles below the critical angle, θ < θ c , there are regions where almost perfect transmission can be achieved. This is clearly seen in the red region of the maps, for relatively small 2.2. Results and discussion periods (100 nm < P < 200 nm). For 's' polarization (g. 1a), the decrease in transmission starts a few degrees below θ c ≈ 33 • . In the case of 'p' polarization (g. 1b), there is an increase in the transmittance values at angles near the critical angle θ c . The latter is related to Brewster's angle, in this case θ B ≈ 28.9 • and for which we expect complete transmission of the 'p' polarized component. It is interesting to note that in both polarizations, there is a feature on the transmission maps starting around P ≈ 250 nm at normal incidence. In this region, a grazing order appears in the substrate, reecting back into the dielectric. The spatial distribution of power between reected and transmitted waves is modied and the modication can be seen in the transmission maps. The new reected order is a direct consequence of the phase matching conditions in the substrate/grating interface. For periods P > 200 nm, there is a moderate decrease of the transmission, as a result of the apparition of new orders in the transmitted side. Once again, the new orders change the spatial distribution of power among the waves. In this case, they can be identied with the modes (m, n) = (-1, 0) and (m, n) = (-1, -1) and are shown in gure 2.10. Each pair of harmonics appears in each half-space. As both have the same behaviour for φ = 0 • , only one of them is presented for each half-space.

Critical angle

Diffraction order appears

Although the apparition of grazing orders is completely determined by the phase matching conditions, the power distribution between propagating orders is found only by solving Maxwell's equations and depends on the dielectric properties of the grating constituents and their geometric arrangement, as well as on the surrounding 2.2. Results and discussion (semi-innite) media. The diraction eciency maps in the upper (lower) row correspond to the new reected (transmitted) order. First, it is noted that the zones of non-zero transmission in the maps are bound to regions related to the apparition of the reected and transmitted diracted order. A comparison with gure 2.10 reveals the origin of the observed features. Furthermore, the zones of highest reection and transmission appear close to the critical angle θ c , pointing out the fact that evanescent components and interactions between scatterers are signicant. In this case, f = D/P = 0.8, which implies that each particle is in the near-eld of the adjacent particles, at least for the wavelength and periods considered. We continue the analysis of the sub-wavelength cone gratings in the index matching regime by addressing the dependence of transmission with the size and shape of 2.2. Results and discussion the cone in the lattice. The main results are shown in gure 2.11. The period of the grating was xed at P = 180 nm and the wavelength at λ = 420 nm (P /λ ≈ 0.428).

The period is chosen just below the value for which the rst non-specular orders appear in reection, at least for the chosen incident angles θ i = 0 • and 20 In gure 2.11b, the inuence of cone shape is investigated. In particular, the aspect ratio of the cones is varied in the interval 0.3 < H/D < 5. As expected, in the limit of small height, at H/D ≈ 0.3 in the gure, the transmission values are well approximated by the Fresnel coecients for the corresponding plane surface. For larger aspect ratio, however, the transmission values increase and reach a limiting value close to 99%. This eect can be understood qualitatively as follows. As the height of the cones increases, the spatial change in refractive index, from the substrate/grating interface to the outside medium, is more gradual. Hence, a larger aspect ratio H/D results in a better impedance matching scheme, gradually coupling the dielectric properties of the substrate into the outside medium [START_REF] Thornton | Limit of the moth's eye principle and other impedance-matching corrugations for solar-absorber design[END_REF].

Diracting regime

In order to improve light extraction from bulk crystals, one can also choose diracting structures, i.e. structures with a periodicity larger than or comparable to the wavelength of the emitted light. In the previous section we focus on sub wavelength structures and its transition to diracting structures. In this transition, we see that low-order diraction plays an important role in the extracted light levels. Furthermore, the impedance matching behaviour is lost when diraction takes place.

Nevertheless, diracting structures eectively break TIR and provide an alternative approach to light extraction enhancements.

In general, the spatial distribution of the transmitted and reected photons changes depending on several parameters: the period of the structure, the wavelength of emitted light, the angles of incidence, the RI of the substrate, as well as the constituents of the grating. The diracted light rays propagate with a pair of angles θ ν and φ ν that can be determined from equation 2.7, at least if we know the parameters of the grating and the angles of incidence. The power distribution among the waves, however, can only be determined by solving Maxwell's equations.

Before addressing the power distribution among diracted order, we present angular distributions related to the propagation of transmitted light. Consider a grating with period P = 900 nm. A periodic particle array on top of a substrate, both with RI n = 1.827, is considered. We can use equations 2.15 and 2.16 to deter- γ = λ/P and h i = h(θ i , φ i ) for h = f, g, the angles at which the waves propagate are

ϕ i (°) a) b) θ t (°) ϕ i (°) c) d) θ i (°) θ i (°)
tan φ ν = n i g i + 1 √ 3 (2n -m)γ n i f i + mγ , (2.15)
for the azimuth angle φ ν and

sin θ ν =    n i f i +mγ nν cos φν if |φ t | ≤ π/4 √ 3 n i g i +(2n-m)γ √ 3 nν sin φν if π/4 < |φ t | < π/2 (2.16)
for the polar angles θ ν .

In g. 2.12, the black background in each map corresponds to evanescent waves, i.e. waves that propagate only at angles θ i = 90 In the following we will focus on the diraction properties of wavelength-scale structures and the consequences of using this alternative for light extraction purposes. First, we address the change in the spatial distribution of the transmitted and reected photons, which involves the determination of the angles θ ν and φ ν , where ν = r (reected) or ν = t (transmitted). To this end, we will continue to consider triangular lattices of dielectric scatterers with period P . The grating is composed of pillars whose axis is perpendicular to the surface (see g. 2.13). The power distribution among diracted orders is computed numerically by solving Maxwell's equations in a later section. For now we will consider only assume periodicity at the interface of the scintillator/background interface.

Low-order diraction and light extraction

In gure 2.14, results are presented for normal and oblique incidence angles. Structures with dierent heights, from 100 nm to 400 nm, are considered. For each, the period of the grating is varied between 420 nm and 800 nm. The lling fraction f = D/P , as dened in the previous section, was xed at f = 0.5. The refractive index of the structure and the substrate are equal (n = 1.827). We present transmission values for unpolarized incident light in all calculations, unless stated ). The insets show the diraction pattern for the selected periods of P = 500, 600 and 700 nm at oblique incidence. For normal incidence, besides the absolute scale, the diraction pattern does not change. [START_REF] Rayleigh | The theory of sound[END_REF][START_REF] Rayleigh | On the dynamical theory of gratings[END_REF][START_REF] Rice | Reection of electromagnetic waves from slightly rough surfaces[END_REF]. At some point, however, suciently thick structures will exhibit more complex behaviour. In this case, we might expect particle resonances, near-eld eects and, depending on the refractive indices of the grating's constituents, it may also be possible to excite guided modes [START_REF] Magnusson | New principle for optical lters[END_REF][START_REF] Wang | Theory and applications of guided-mode resonance lters[END_REF] along the structure. A comprehensive review of these eects is beyond the scope of the present work. Nevertheless, it is reasonable to expect that big structures, in relation to the wavelength of illumination, exhibit more competition eects between the eigenmodes of the system (e.g. from the moments of a multipolar expansion [START_REF] Jackson | Classical electrodynamics[END_REF]). The transmission curves will be further described in the following in terms of the diraction eciency of diracted orders. We focus on the dependence of the diraction orders with geometrical parameters of the structure. The goal is to 2.2. Results and discussion describe trends that may help in the design of surface structures for light extraction.

Figure 2.14b shows the simulation results for an incident plane at oblique incidence, at θ inc = 40 • . A at surface would reect light completely in this conguration due to total internal reection (TIR). In this case, however, the diraction grating out-couples light with wave vectors fullling equations 2.7. Furthermore, as the period of the grating is varied and the lling fraction is xed f = 0.5, the size of the pillar also changes. For this reason, resonant eects of the lattice and of the particle could have an inuence on the transmission values [START_REF]It is worth mentioning that, as a general design tool, resonance phenomena can be treated in a framework involving input and output ports. This type of formalisms have been called temporal coupled-mode theories[END_REF].

The transmission of light by a grating can be decomposed in plane waves travelling with a set of well-dened directions. Evanescent components also play a role in the interaction of light with the structure, but here, the measurable signal will only depend on propagating (far-eld) components. In other words, near-eld eects in the structure are not particularly important in this section. The transmission curve show in g. 2.14 contains the contribution of all of these plane waves and therefore contains resonant and non-resonant contributions of each diracted order. Taking into account all the opto-geometrical parameters (dielectric and geometric congurations) of the system at once can be an overwhelming task. For this reason, in what follows we discuss the eect of geometrical parameters on a few diraction orders. The spatial distribution of transmitted light is briey discussed in terms of the diraction pattern for selected grating congurations.

The transmission values presented in g. 2.14a can be decomposed in specular transmission values and the values of the diraction eciency for each order.

More precisely, g. 2.15a shows the specular transmission by the grating, while the diraction eciency is shown for the rst diracted orders in g. 2.15b and 2.15c.

As already mentioned for the transmission curves, at periods P ≈ 485 nm in g.

2.15b and 2.15c, the rst set of diracted orders appears (for normal incidence only).

Sharp features related to the apparition of the orders are also present in the specular transmission curves (g. 2.15a). A comparison of the curves for dierent heights, in each case, illustrates that structures with H λ are not very likely to strongly diract the incoming light. In particular, for H = 100 nm (≈ λ/4), the specular transmission by the grating is fairly constant for the range of periods considered, a behaviour that is reminiscent of a (optically) at surface. The transmission value in this case, T ≈ 75%, is about 16% lower than the corresponding at surface. The latter represents reection losses and possibly the apparition of new diracted orders in transmission. Thicker structures display more complicated behaviour. For example, as the structure thickness H is increased, the zeroth-order transmission is strongly suppressed. For H = 200 nm we can see a dip in the transmission curve for the region around P = 450 nm and a fairly at response for P > 500 nm at T = 50%. For taller structures we can appreciate a less pronounced valley around P = 450 nm and slight variations for P > 600 nm. In these cases (P = 300 nm and P = 400 nm) there are more pronounced variations in the diraction eciencies. Note, for example, that in g. 2.15c the diraction eciency seems to scale with structure thickness. This is particularly true for periods close to the onset of diraction. In g. 2.15c, however, the diraction orders do not present such trend. On the right side of the gure, the diraction pattern at normal incidence is shown for P > 485 nm.

In the transmission of light by a at interface, we encounter total internal reec-

(m,n) = (0,±1) (m,n) = (±1,0) (m,n) = (1,1)
In vertical axis

Out of axis a) b) c) (m,n) = (-1,-1)

Diffraction efficiency

Period (nm) tion in a conguration of internal illumination. This means that light arrives at the interface from the side of the dense medium and is reected from or is transmitted to a less dense medium. In the case of a at LYSO/air interface (cf. g. 2.7), the critical angle θ c = 33.19 The insets in g. 2.16 show the spatial pattern of the diracted (transmitted) light.

At normal incidence, the apparition of new orders depends only on the wavelength and periods of the incident wave, while at oblique incidence it also depends on the particular incidence angles θ i and φ i .

The aim of presenting this set of gures is to illustrate the vast number of degrees of freedom of the problem. For the simulations, we matched the refractive index (RI)

of the structure to that of the scintillator. Indeed, it is possible to fabricate surface dielectric structures with similar RI [START_REF] Kadiri | Topology assisted self-organization of colloidal nanoparticles: application to 2d large-scale nanomastering[END_REF]. For example, at periods P = 600 nm,

Results and discussion

ve diraction orders propagate in transmission. At slightly larger periods, two more orders appear. In general, the diraction eciency can be dierent for each order, as they could be related to distinct physical phenomena. From a design perspective, one could consider that the number of transmitted and/or reected orders is the upper bound for the number of plane-wave channels. Nevertheless, as in the case of the triangular lattice, spatial symmetries of the system reduce the number of independent channels. The term channel is used in the context of temporal coupled-mode theories, where the system is described as input and output ports (with input and output waves or signals) connected by the physical system (the grating in our case). The scattering matrix can be calculated for relatively simple but relevant systems (see, e.g. [START_REF] Fan | Temporal coupled-mode theory for the fano resonance in optical resonators[END_REF][START_REF] Wang | Condition for perfect antireection by optical resonance at material interface[END_REF]). In our situation, the physical system is the grating (or photonic crystal slab on top of a substrate), which we can picture as a black box with input and output ports. The input port represents the interface from the substrate side, the input waves are the incident eld from the same side and, if we do not neglect reections inside the bulk material, the reected specular and diracted orders can also contribute to the input signal. The output channels, in turn, represent the transmitted waves. In this context, only the manipulation of the specular reection has been theoretically addressed [START_REF] Wang | Condition for perfect antireection by optical resonance at material interface[END_REF]. Unfortunately, an analytical description such as the one provided in [START_REF] Wang | Condition for perfect antireection by optical resonance at material interface[END_REF] but for a number of diraction channels N orders > 1 is far from trivial.

We show in gure 2.17 the last results of this section. The transmittance and reectance values for angles of incidence 0 • < θ i < 90 • and 0 • < φ i < 60 • were computed. The polar angle θ i was sampled at angles 1 • apart, while the azimuth angle φ i was sampled every 2 • . A variety of geometrical arrangements for pillar arrays are considered. Furthermore, the gure of merit discussed at the beginning of the chapter is computed with the mentioned calculations (see section 2.1). The results are presented in terms of a light gain. This quantity is simply η -1, where η was dened in section 2.1 as the ratio of diuse transmittance from the structured interface and that of a at substrate. Negative gain values represent suppression of the extracted light compared to a at surface.

The results are divided in three cases: (mostly) sub-wavelength periods 100 nm < P ≤ 500 nm, wavelength-scale periods, 300 nm < P ≤ 800 nm, and periods with values of a few wavelengths, i.e. 1000 nm < P ≤ 1500 nm.

The rst case, the results for smaller periods, shows the maximum LEE enhancement of the overall set of results. In this case the pillar array presents the highest diameter to period ratio or lling fraction. Interestingly, the less thick structures for which the pillar height is H = 100nm provide the best response. Furthermore, these enhancements are present mainly for the smallest periods. A signicant change can be observed for the LEE in the H = 200nm case. Indeed, the particle shape in each of the previous cases is considerably dierent. Consider the latter for P = 100nm in gure 2.17a. In the case H = 100nm, the ratio H/D = 0.7 while for the case H = 200nm it is twice that value. The longer axis of the pillar changes for each case. Therefore, the shape of the particle evidently aects considerably the LEE, even if the particle is a fraction of the wavelength in size.

For the second case a transition between suppression and enhancement of the LEE, as a function of the period P of the grating, can be observed. The negative values of the light gain are attributed here to the apparition of diracted orders in reection. Indeed, this behaviour is reminiscent of the one discussed for gure 2.9.

For each structure thickness H, the light gain is fairly similar. In other words, the height of the pillar for the chosen congurations does not signicantly aect light extraction properties.

The third and nal case of this set of results considers periods P of length equal to a few wavelengths of illumination λ. Evidently, the weak enhancement that results from this grating congurations do not seem very encouraging. There is a slight enhancement in the case of taller pillars. Nevertheless, the percentage is quite low. Therefore, one would not expect to nd much change from a polished exit surface.

Results as the one discussed above illustrate the fact that even if TIR is partially avoided, the manipulation of diraction orders is not trivial. In fact, far from being 2.3. Modeling light propagation at dierent length scales conclusive results on the potentials for LEE in this type of solutions, here we emphasize that a number of parameters could be further explored through numerical simulations. The computations involved are however time consuming. We focus then on providing examples that represent structures well within the capabilities of fabrication.

It is important to remember that this results represent the response of innite surfaces. In order to model the incident diuse light we have used hemispherical Lambertian illumination. In what follows, the eects of nite surfaces and reections from other interfaces of a scintillator/air system will be discussed in terms of ray optics calculations.

Modeling light propagation at dierent length scales

As stated in the beginning, the purpose of this chapter is to provide useful examples and general considerations that could lead to improved solutions for light extraction.

We have considered a diuse wave front as a rst approximation of a scintillation event. As the spatial distribution of light depends on each particular event, a robust solution is required and average quantities of the optical response are of particular interest for this purpose. Light is generated in a region inside the bulk crystal that is related to the high-energy particle trajectory. For a LYSO crystal, the penetration depth is on the order δ γ ≈ 1.2 cm for radiation with energy E = 511

keV [15]. The length-scales involved in modelling the optical phenomena range (for our purposes) from 10 -7 m (hundreds of nm) to 10 -2 m (centimetres). If non-planar surfaces whose features are of linear dimensions D ∼ λ em are considered to enhance light extraction, one must also consider diraction at the structured interface. On some cases approximate theories can simplify the problem [START_REF] Rayleigh | On the dynamical theory of gratings[END_REF][START_REF] Rice | Reection of electromagnetic waves from slightly rough surfaces[END_REF] but the models still describe wave phenomena. On the other hand, the bulk crystal shape can also inuence the collection of light. This inuence however can be thought as a correction to the response of the corresponding innite-size structure. In essence, the eects of the bulk shape are geometrical and can be studied by tracing rays from the source or sources to the exit interface.

Diraction of light resulting from wavelength-scale features on the surface results

in two distinct types of electromagnetic waves: travelling and evanescent waves. The former can propagate indenitely in a transparent medium while the latter decay abruptly near the interface, vanishing for distances R ∼ λ. In fact, evanescent waves only carry energy in a direction perpendicular to the surface. Consider the quantity kR, where k is the wave number in free-space and R is a distance light propagates after being transmitted by the interface. The so-called far-eld limit corresponds to cases where kR 1. Near the surface (R ≤ λ), in general, the eld is composed of travelling and evanescent waves but in the far-eld limit the evanescent components vanish, at least for all practical purposes. Theoretically, in RCWA for example, we retrieve the far-eld limit simply by ignoring evanescent orders. The latter can be determined by the use of condition 2.12. On the other hand, isolated structures embedded in an innite background do not possess periodicity but are of innite size. This is equivalent to the limiting case of a "grating" that has a period P → ∞.

Modeling light propagation at dierent length scales

Then, equation 2.6 becomes an integral. In the far-eld (kR 1), we can write the vector component h α of the scattered wave at R and travelling in direction û as [START_REF] Born | Principles of optics: electromagnetic theory of propagation, interference and diraction of light[END_REF] h

α (Rû) ∼ S sc (k û) e ikR R , (2.17)
with the scattering amplitude S sc given by

S sc (k û) = V F (r )ψ(r )e -ik(û•r ) dV .
(2.18)

In this expressions, the function F (r) is related to the dielectric contrast of scatterer and background materials. For a scintillator surrounded by air

F (r) ∝ [n 2 (r) -1]
. ψ(r) is the total (incident plus scattered) eld inside the volume of the scatterer, i.e. the integration runs only over the volume of the scatterer. The complex amplitude S sc includes contributions from waves scattered by induced sources inside the volume, which propagate in directions û. More precisely, by induced sources we refer to the polarization induced by the incident eld. Note that the shape of the scatterer, due to the û • r exponent in 2.18, also inuences the strength of the scattered eld in direction û. Evidently, there is a strong similarity between the last expressions and a Fourier expansion of the eld. Indeed, one can obtain similar results by taking the spatial Fourier transform of the electric or magnetic eld on a plane, say at z = 0, as a starting point. The latter results in a representation called the angular spectrum (see, e.g. [START_REF] Novotny | Principles of nano-optics[END_REF]). The elds in each of the aforementioned formalisms obey an outgoing wave condition. The problem is that of an open system, at least in the sense that the physical system does is not (spatially) bounded. We thus can interpret equation 2.17 as the limiting behaviour of the waves that will reach distances R λ em . These waves correspond, of course, to the emitted light at the end of a scintillation process. In practice, the actual distance R that light will be able to travel is limited by the absorption of the host medium at wavelengths of emission λ em , in this case the scintillator. Note that in eq. 2.17, S sc (k û) determines a form factor of the spherical wave, which is related to the radiation pattern of the system under illumination. In fact, the asymptotic behaviour of the far-eld, i.e. the radiation pattern, is given by the scattering amplitude S sc (k û). Here, the key point in equations 2.17 and 2.18 point is relating the electromagnetic response of a (piece-wise) body of arbitrary volume V to a macroscopic observable such as the radiation pattern. The corresponding observable for periodic structures is the diraction pattern.

Finite surfaces of macroscopic dimensions

So far we have considered only structures of innite dimensions. The latter can be considered as a starting point in a multi-scale analysis of the light-extraction problem. Figure 2.18 shows, schematically, a comparison of the innite surface case with the nite case. Fig. 2.18a represents a semi-innite medium. A source (denoted with blue arrows) placed at a distance z to the surface is modelled as the diuse hemispherical illumination. All rays within the hemisphere arrive at the interface in the case of an unbounded surface. In g. 2.18b we present the cross-section of a 2.3. Modeling light propagation at dierent length scales nite structure. This geometry is chosen because it corresponds to that of real LYSO crystals used in PET-scanners. The next section focus only on the rays that directly arrive at the interface and neglect every other reection that could eventually escape towards the detector. Afterwards we will consider the contributions from multiple reections inside the crystal. Consider the angular range of the rays that can directly reach the exit surface (see g. 2.18b). This range will depend on the bulk shape of the scintillator. In this case, the symmetry of the crystal and the source position results in the isosceles triangle shown in the cross-section. The angular range is restricted compared to that in the innite case. In gure 2.18c the source is not at a point of such symmetry. The result is that the light output cone gives preference to larger angles of incidence. This is further discussed with the use of numerical calculations.

In order to illustrate the eect of truncation of the exit surface on light extraction, we present results related to the distribution of incident angles with which light rays arrive at the interface. In gure 2.18, a comparison is made between the innite case (g.2.18) For the sake of simplicity we present calculations for planar surfaces. The diuse wave front we have considered previously contains plane-waves propagating in all directions within a hemisphere. This means the angular range of incident plane-waves is -π/2 ≤ θ i ≤ π/2. As the azimuth angle φ i does not play a role in total internal reection, we will focus on the distribution of polar angles of incidence θ i . Due to translation symmetry in the x, y-plane, the amount of light arriving at the interface only depends on z. In the following we estimate this modication by counting the number of light rays transmitted across the surface of the scintillator.

A photograph of the real LYSO crystal as well as the simulated (macroscopic) structure is shown in g. 2.19. On the photograph, except for the exit polished surface, the crystal has roughened surfaces. The latter are meant to scatter light that do not impinge on the exit surface. For the moment we focus our attention on the light rays that arrive at the exit surface directly from the source. The eects of rough surfaces for every other interface of the scintillator are discussed in the next section. In g. 2.19b, the simulated crystal/detector conguration is shown. The crystal, shown in grey and representing the scintillator, is a transparent dielectric with RI n = 1.81. The detector is not placed directly on top of the surface but coupled to a transparent glass window. This conguration has been considered throughout the TURBOPET project [6] and selected results are shown here. The calculations were carried out using an in-house software [66]. For a source in a given position, the number of rays traced was on the order of a few millions. Random sampling of rays within the hemispherical wave front was performed (Monte Carlo approach).

For a scintillator with a nite, at exit surface, one can estimate the inuence of bulk shape by tracing rays from the source to the surface. In order to include potential scintillation events throughout the crystal, one can scan the volume of the material with an isotropic source. While scanning the scintillator's volume, one counts the rays arriving at the detector for each source position. Numerically, the simulations eventually provide enough information to build a histogram of detected rays as a function of the angle with which the ray arrives at the interface. Such a procedure has been pursued using dierent numerical approaches.

The results are shown in gure 2.20. The symbols correspond to numerical calculations. The rst two (green and red squares) were obtained by collaborators at CERN [67] using software intended for particle-physics simulations [START_REF]pour la Recherche Nucléaire, Geant4: A simulation kit[END_REF]. A third set of results corresponds to the in-house ray tracing software (black triangles). The distribution of angles for a hemispherical wave front is also shown (solid blue line).

θ i ' θ i a) b) c) z L top L bottom L depth Light output cone φ
In what concerns the numerical results, each point of the curves was obtained by counting the number of (randomly sampled) rays arriving at the exit interface within an angular range centred about the incidence angle θ i . In order to explain in detail the numerical procedure, consider one isotropic source radiating inside the scintillator at some point r s . First, one needs to choose a distribution of rays that will represent the emitted light propagating inside the scintillator. This distribution of rays is chosen randomly and contains N rays, represented numerically by their angles with respect to the normal to the surface. In practice, N is on the order of a few million. Once the rays emitted by the source at r s are registered together with their angle of incidence, one can make a histogram of the rays as a function of θ i . Now, if we repeat this procedure for a source at r s we would obtain another histogram of the number of rays as a function of θ i . Eventually we can sample inside the scintillator with a large number of isotropic sources and retrieve the corresponding histogram of rays for each case. Finally, inspired by the fact that high-energy radiation decays exponentially when it penetrates the scintillator, we perform an exponentially weighted average of the histograms previously obtained.

We have used the penetration depth for LYSO at E = 511 KeV. In other words, the results for sources located close to the entrance of high-energy radiation, i.e.

Modeling light propagation at dierent length scales

the opposite side of the exit interface, are more important than those obtained with sources closer to the exit interface.

The procedure described above aims at quantifying the average bulk response of a macroscopic scintillator. It is based on ray tracing simulations and provides an estimation of the average performance of the scintillator. It is important to note that it does not corresponds to a particular scintillation event. On the contrary, the curve corresponds to an average over a large number of scintillation events that occur throughout the volume of the scintillator.

In order to further clarify the physical meaning of each point in the curve, we point out the fact that the analysis provided by numerical simulations should be related to the number of rays crossing a given solid angle. For example, consider the rays arriving at the interface at normal or near-normal incidence. At exactly normal incidence we can see there is practically no contribution from these rays. Instead of only stating that the random sampling has not provided the rays, we focus at the fact that a solid angle vanishes near the pole, i.e. for small angles of incidence.

This means that, compared to large number of rays at oblique incidence, the ones travelling in a direction normal to the exit interface are scarce.

The rst curve, obtained with LITRANI, assumed perfectly absorbing walls except for the exit interface. The second one was obtained with GEANT4 and considered absorbing walls in the same manner but with a small probability of reection.

The in-house software considered perfectly absorbing walls except for the exit interface. All the curves match for angles almost at normal incidence. As explained above, this is only a the result of the relation between a solid angle dierential dΩ and the polar angle θ i . For larger angles, the numerical results start to deviate from the case of hemispherical illumination. The overall eect of the nite size structure seems to results in a shift of the angular distribution of photons reaching the interface. Evidently, each numerical approach results in a slightly dierent distribution. The results using LITRANI are in good agreement with the in-house software, with discrepancies towards grazing angles. Compared to these results, the ones obtained using GEANT4 show moderate but more pronounced changes. The dierence could come from considering reections from the partially absorbing interfaces. The preference for larger angles in g. 2.20 can be understood by analyzing the congurations shown in g. 2.18b and 2.18c. Moreover, the shape of the crystal in this case inuences the collection of light through the oset angle ϕ. In this case is ϕ = 39.8 • .

Inuence of the bulk geometry on light extraction

We present now results that illustrate the eect of reective interfaces in the lateral and bottom sides of the crystal. The main results are shown in gure 2.21. First, the case in which no reection occurs at the lateral interfaces is shown in g. 2.21a.

The detected counts are plotted in square pads as a function of the position on the virtual sensor. The width of the resultant peak is related to the critical angle for the scintillator/air interface. Figure 2.21b shows the results for considerably more reecting surfaces with R = 0.5. Here, R is essentially the probability that a ray arriving at the interface is randomly scattered back to the dielectric. In other words, the lateral and bottom interfaces reect light isotropically. Modest changes can be appreciated compared to the ideal non-reecting case. The overall eect of including reections has been the addition of an oset level in the detection count.

a) b) c) R = 0 R = 0.5 R = 1 x (c m ) y ( c m ) x (c m ) y ( c m ) x (c m ) y ( c m )
The extreme case of perfectly (isotropic) reecting lateral and bottom surfaces is shown in g. 2.21c. Here, it is evident that eects at the edges of the detector compete with the signal detected in previous cases. The noisy detected signal at the edges must come from every ray that impinges close to the vertices formed by the exit and lateral surfaces. These results can be considered a rst estimation of the bulk response, as only isotropic radiation and reections have been considered.

The image a virtual detector captures after a large number of rays (on the order of a few millions) have been launched in the simulations is shown in gure 2.22. This image corresponds to the ideal case of 2.21a. The distribution of rays arriving at the detector for this case, a at exit surface and perfectly absorbing side and back walls of the crystal, yields the a relatively narrow spot. Total internal reection only allows for extraction of light rays whose incidence angles are below the critical angle.

In this section we have addressed the reections inside crystal. In this regard, an important distinction can be made between non-diracting, possibly polished structures and diracting ones. The former can be dealt with through the usual Fresnel equations while the latter requires more care as a result of diraction orders. The coupling of a ray-tracing approach with numerical results for the optical response of wavelength-scale structures is not trivial. For each diraction order one must introduce the probability for a ray to be reected or transmitted. In each case, the angles of emergence of diraction orders must be determined with equations 2.7.

Chapter 3 Theoretical description of luminescence in plastic scintillators

The present chapter attempts at providing a basic description of light emission processes in the context of scintillation. A semi-classical approach which addresses light absorption and light emission dynamics is described in terms of a semi-classical polarizability. The latter describes the response of the uorophores in a plastic matrix. Dilute solutions of the light-emitters or plastics with a low-concentration of them can be treated in terms of particles and of the local electromagnetic response of the emitter's surroundings. In the context of scintillating materials, we approach the subject focusing on the processes just before and after the photon is emitted.

The temporal dynamics considered in our discussion of the emission process relate to the relaxation times for higher-energy transitions occurring in scintillation events.

The emission process will be assumed to be, in some sense, independent of earlier transitions. The idea behind this is that as the relaxation process evolves, some carriers encounter states from which they nally decay and emit a photons. The term carrier can refer, in the context of organic uorophores, to electrons involving or being mediated by excitons [START_REF] Davydov | The theory of molecular excitons[END_REF], for example. As we focus essentially on the end of the scintillation process, considering for example a typical model for a uorescent molecule seems to be a reasonable rst step. The purpose of this chapter is then to provide such a description in terms of well-known quantities describing general optical properties of solid-state media. Equivalently, this could be thought of as condensing material degrees of freedom into eective properties.

For practical purposes, one is frequently interested in frameworks in which the models describe reality as accurately as possible. Let us use chapter 2 as an example.

The model used describes the transmission of light from a source inside a scintillating material into air, for example. The scintillating medium has nite but macroscopic dimensions. Realistic, complex optical systems can be simulated in terms of waves and rays if one knows the details of the source. Nevertheless, we have shown that the ideal case, where the scintillator/air interface of interest has innite extension and is perfectly at, already elucidates some aspects of the possible technological solutions (cf. section 2.3). We show the distribution of propagation angles at the interface for a multitude of rays in all directions towards the surface. This distribution is proportional to the probability distribution P (θ i ) and has its maximum at an angle θ i > θ c , where θ c is the critical angle. We could argue the main drawback of the 51 3.1. The solution of Maxwell's equations with integral equations optical model is the innite extent of the surface. The latter, in this model, leads to the prediction of the median at θ i = 45 • , which is not true for nite surfaces. The dierences however but mantain a similar qualitative shape. Another simplication can be made by neglecting reections inside the crystal. For rough surfaces (which are not the exit surface and reect light isotropically) we have seen that the eects are not of paramount importance if one is interested in a rst estimation or even in a reference with which to compare improvements of the eventual solutions.

In the context of the present work we will address the optical modelling of microscopic structures. Contrary to the case in chapter 2, in this chapter we will provide a model for the light source. The manner in which this is done is related to electrical dipoles. We will use this notion to describe the light emission process towards the end of the scintillation process. In particular we will consider scintillators composed of a plastic matrix which contains organic uorophores. In this case, the polymer matrix is responsible for most of the interaction with high-energy excitation such as γ-photons and βor αparticles. At some point in the decay mechanisms, non-radiative transitions can occur between the matrix and embedded uorophores.

Plastic scintillators have been engineered in such a way as to optimize these nonradiative transitions (see, e.g. [START_REF] Hajagos | High-z sensitized plastic scintillators: A review[END_REF] for a modern review). The latter lead to emission in the ultra-violet region of the spectrum. The UV radiation can in turn excite other types of uorophores present in the polymer matrix. The interplay between light emission and light absorption is then of paramount importance in this context. In the following we shall describe and use a formalism that has the basic elements of electromagnetic interactions at the nano-scale.

The solution of Maxwell's equations with integral equations

Classical electrodynamics plays an important role in the description of light emission phenomena. It is therefore reasonable to dedicate this section to briey describe the formalism employed to solve Maxwell's. This set of solutions can eventually result in quantitative estimations. By carefully taking into account elements from a quantum approach, one can establish useful physical notions relating macroscopic and microscopic quantities such as the electric permittivity and electrical susceptibility, for example. Subtle aspects on the denition of this quantities can be found in derivations of the Lorentz-Lorenz formula [START_REF] Born | Principles of optics: electromagnetic theory of propagation, interference and diraction of light[END_REF].

We start from the vector wave equation and state the main results leading to a numerical scheme with which one eventually obtains the unknown eld values.

Beyond the numerical application of the formalism, here it is emphasized that a

Green function formalism provides a convenient theoretical tool. Although it was not mentioned by that name, the f ar-f ield limit discussed in earlier chapters is implicit in this type of framework. This type of formalism is commonly encountered in contexts as diverse as radio wave propagation [START_REF] Sommerfeld | Über die ausbreitung der wellen in der drahtlosen telegraphie[END_REF] and theoretical chemistry [START_REF] Tomasi | Quantum mechanical continuum solvation models[END_REF].

The common aspect in these descriptions is related to the physical notions and models for wave phenomena.

We describe the general form of a Green function and immediately focus on the Helmholtz wave equation. Time-harmonic solutions are considered. The conven-Figure 3.1: Schematic description of a system composed of continuous, piece-wise media.

The dielectric function of polarizable objects (gray) is denoted s , while that of the host medium is h . An external eld, illustrated as a plane wave front (red lines), can impinge on the system. Furthermore, a localized source of radiation can be present in the vicinity of the polarizable objects (red lled circle).

tion, similarly as in previous chapters, is to consider electromagnetic elds with temporal dependence of the form F ∼ e -iωt . Both tensor and vector quantities will be denoted in bold font. General temporal behaviour can be constructed from the appropriate knowledge of single-frequency responses. More precisely, borrowing nomenclature from linear response theory, the formal treatment corresponds to that of a Time-Invariant Linear System. In the context of the present work, it implies that the properties of each material (for example the dielectric constant or the dipole moment) do not change with time.

The physical situation involves particles embedded in a homogeneous background and is represented schematically in gure 3.1. Although we seek a solution of the electromagnetic problem for one particle, the generalization to multiple bodies with dierent (intrinsic) optical properties is straightforward. For each material the model represents a continuum of polarizable entities responding linearly to the eld. The system, composed of polarizable objects in general, could be excited by an external eld. In the gure an incident plane-wave represents an example of such excitation.

Furthermore, in the vinicity of the bodies we could encounter localized sources of radiation. This source or sources could have nite dimensions or could be of a point- Key properties have to be mantained in order for the theory to be consistent. For example, the Kramer-Kronig relations [START_REF] Jackson | Classical electrodynamics[END_REF] indicate that causality is not violated.

Therefore any model describing (ω) must obey the Kramer-Kronig relations. The optical properties of the particle or particles will be also eventually chosen as nonmagnetic, homogeneous and isotropic but they are not restrictions. In order to illustrate dierent aspects in the phenomenology behind the approach we assume a particle of dielectric constant = (ω). The background medium can also be a polarizable, possibly innite, body [START_REF] Martin | Electromagnetic scattering in polarizable backgrounds[END_REF]. Within the scope of the present work we will deal with a background dierent from air. Therefore, we denote the dielectric constant of the background as B . It is important to mention that absorption in the background medium should be strictly avoided, at least if an outgoing-wave condition is pursued. Nevertheless, as justied in the problem of light-extraction from scintillating crystals, one can expect this eects to be negligible for certain spectral regions. More precisely, macroscopic quantities such as the absorbance can provide estimations of the values around which the latter can be considered a sound assumption.

A formalism will be described in order to formally solve Maxwell's equations. We deal with nite medium which will be assumed to be isotropic, homogeneous and non-magnetic. Therefore, the property containing the material response to radiation is condensed in the electric permittivity (ω), which is a function of frequency. In electromagnetic theory, a general solution to the scattering problem can be written with the use of Green function as follows. Here, the term piece-wise media refers to structures that can be well separated in bodies whose physical properties are constant throughout their volumes V j (see, e.g. g. 3.1). For the moment we will only consider one body with volume V and boundary described by Σ, embedded in a homogeneous background. The latter could be air or an isotropic, homogeneous and transparent dielectric with electric permittivity B , which can also be a function of frequency. In practice, the situation of interest will consider a transparent background medium with a weak dependence on frequency, at least at emission wavelengths of interest. Moreover, we will consider structures with dimensions comparable or greater to the wavelength of the interacting radiation. It is assumed then that the macroscopic properties represent each domain, even if it is spatially bounded at micron or sub-micron length-scales.

We start by stating Maxwell's equations, which can be written in dierential form through the following wave equations:

∇ 2 + k 2 H(r, ω) = f h (r, ω), (3.1)
∇ 2 + k 2 E(r, ω) = f e (r, ω), (3.2)
where k 2 = (ω/c). The equations are valid in each subdomain. A number of approaches can be used to solve these equations but we will restrict to solutions in the form of integral equations. The right-hand-side (RHS) of equations 3.1 and 3.2 describe possible sources, except for the trivial cases where f (r, ω) = 0 everywhere.

F(r) = F 0 (r) + V K(r, r )F(r )dV, (3.3)
where, in order the notation to simplify the notation, we dropped the argument for the angular frequency ω. Note eq. 3.3 is an implicit equation for the eld inside volume V . Furthermore, the knowledge of the eld F inside V and F 0 at r are enough to determine the total eld F at that position. In other words, F is determined in all space if the eld is known at all points inside volume V . The tensor K(r, r ) is frequently referred to as a propagator. The latter denotes the fact that it relates the waves coming from points r inside volume V to points r, inside or outside of V . It is described in what follows but we anticipate it is related to the Green function of the system.

In the physical situation, we associate the solution 3.3 to congurations of the system before and after being excited or perturbed by radiation. In this picture, we consider the rst term on the RHS of 3.3 to be the incident eld or an external excitation which is known a priori. The second term, however, includes the selfconsistent solution of the eld inside volume V . Physically it corresponds to the eld scattered by the volume. We can then write eq. 3.3, say for the electric eld

E(r) as E(r) = E inc (r) + E scat (r). (3.4)
The nature of this elds in a physical situation reect the steady-state response of the system. As it was mentioned earlier, we consider the system before and after interaction with radiation. In fact, we assume that initially the system has a relatively simple description, for example a plane wave propagating in free-space. Then, one postulates the existence of a steady-state solution for E some time after the interaction of radiation with the (polarizable) body has taken place. This solution is the sum of the incident and scattered elds. From the previous considerations, we assume the incident eld E inc is known and the general (integral) form of E scat is

E scat (r) = V K(r, r )E(r )dV. (3.5)
It can be shown that the tensor K(r, r ) has the form [START_REF] Martin | Generalized eld propagator for electromagnetic scattering and light connement[END_REF] K(r, r

) = -k 2 0 (r ) -B G(r, r ) (3.6)
and G(r,r') is the Green function of the system, dened as the solution of [START_REF] Economou | Green's functions in quantum physics[END_REF] L{G(r, r )} = δ(r -r )I.

Here, L{} is a dierential operator. The latter is of second order for the Helmholtz (wave equation) operator. δ(r -r ) is Dirac's delta function representing a point source at r and I is the unit dyad. For the electromagnetic problem, the Green function obeys the following equation

∇ × ∇ × G(r, r ) - ω 2 c 2 (r)G(r, r ) = 4πδ(r -r )I.
(3.7)

The solution of Maxwell's equations with integral equations

The solutions of this equation are known for one-, two-and three-dimensional geometries. We will focus on the electric Green function as it is the quantity of interest in the description adopted here for light emission.

At this stage, it is worth commenting on the physical concepts related to the (tensor) Green function G(r, r ). This quantity relates to the notion of electrical dipoles. The latter is the rst-order term in a multipole expansion of the eld [START_REF] Jackson | Classical electrodynamics[END_REF][START_REF] Novotny | Principles of nano-optics[END_REF][START_REF] Stratton | Electromagnetic theory[END_REF]. In physical terms it can be seen as two equal charges of opposite polarity which are innitely close to each other. The mathematical procedure involved, for example in the Green function of a homogeneous medium, can pose some diculties because of a singularity in the case r = r . The latter has nonetheless been extensively studied in dierent contexts (see, e.g. [START_REF] Vries | Point scatterers for classical waves[END_REF][START_REF] Yaghjian | Electric dyadic green's functions in the source region[END_REF][START_REF] Chaumet | Coupled dipole method for scatterers with large permittivity[END_REF][START_REF] Lakhtakia | On two numerical techniques for light scattering by dielectric agglomerated structures[END_REF]] and references therein), which in turn has resulted in several solutions to the problem. The notion of point source associated to a dipole has been used to construct classical models of polarizable media for some time. It renders a model of oscillators responding to the applied elds. One can essentially exploit classical theory to its limits by focusing on aspects at length scales that appropriately represent the problem. This procedure has been shown to yield sound theoretical models in the past [START_REF] Vries | Point scatterers for classical waves[END_REF]. For example, consider a molecule that emits light and has dimensions on the order of a few armstrongs. This light source has physical dimensions l mol , which in turn can be associated to spatial frequencies on the order of k mol ∼ 2π/l mol . For visible and near-UV radiation we are in a regime where the source's wavelength λ l mol or, equivalently, where k mol (ω/c).

Therefore, one can regularize the singularity in G(r, r ) by carefully ltering high frequencies. The ltering procedure can be chosen to ensure the ltered Green function approaches exponentially to the original one. Furthermore, it has been shown it preserves the correct limiting behaviour for r = r . It is then reasonable to assume that such a theoretical framework provides useful information of the optical response at length scales ranging from a few dozens of nanometers to a few microns.

Indeed, this interval will be the range of length scales considered for this part of the work.

The elds of an electrical dipole contain information of central importance in the description of light-matter interactions. This quantity is essentially the eld evaluated at r from a dipole which is "xed" at a position r . The number of electromagnetic modes per unit volume, in the frequency interval between ω and ω + δω, is implicit in G(r, r) [START_REF] Economou | Green's functions in quantum physics[END_REF]. The tensor G(r, r ) corresponds to a 3 × 3 matrix. Each column corresponds to the components of the vector-valued eld, evaluated at r, generated by a dipole in a xed orientation and at position r . Then, by considering three orthogonal orientations one can construct the full tensor. Note that G(r, r ) is independent of any illumination eld. Once the solution G(r, r ) is known for a given system, the interaction of a prescribed radiation eld with the system is given by the scattered eld, as in eq. 3.5.

Important distinctions between macroscopic or average elds and local or microscopic elds can be made in terms of eq. 3.3. In the context of atomic and electronic phenomena these distinctions play a substantial role. Light emission is frequently a result of electronic processes and the electric eld uctuations near the sources (electrons) will evidently have a strong inuence on such processes. Particularly, the strength and uctuations of the elds will aect the population of carriers in dierent energy levels. Therefore, if one attempts at describing the inuence of the 3.1. The solution of Maxwell's equations with integral equations eld on localized sources such as a dipole source, one must consider the local eld [START_REF] Novotny | Single molecule uorescence in inhomogeneous environments[END_REF][START_REF] Giannini | Electrodynamic calculations of spontaneous emission coupled to metal nanostructures of arbitrary shape: nanoantenna-enhanced uorescence[END_REF][START_REF] Kuester | Averaged transition conditions for electromagnetic elds at a metalm[END_REF].

Two-dimensional geometries and boundary values

In this section we focus on the case of geometries invariant in one dimension. Such kind of topology corresponds to innite cylinders for example if their cross-section is a circle. In general, these innite bars or wires can have arbitrary cross-sections and represent the limiting case of real three-dimensional particles. The limit is in this case with respect to the shape of the particle. For example, nite wires which maintain an approximately constant cross-section and are several times longer (along their axes) than another leading order dimension. One would of course nd only an approximate solution with the present approach. We assume that along the y-axis the cross-section is constant (see g. 3.2). This means that the elds will be described in the x, z-plane, or in general at y = const. for our purposes. There is only a phase factor involved in propagating the eld to another plane y = const . We consider localized sources as excitation of a dielectric or conducting body of volume V . In three dimensions, the excitation corresponds to a point source. In two dimensions however, the localized source becomes an oscillating line of charge. This two-dimensional dipole will be considered here as the incident eld. One consequence which turns out to be an advantage for the eld computation is the fact that the problem is of scalar nature. Two uncoupled scalar elds describe the complete vector eld. Recall the Green function is essentially the eld of a dipole. This implies that the Green function is also scalar.

In the following we will take an approach where only boundary values of the eld are needed to determine the electromagnetic elds everywhere in space. Two polarizations, corresponding to the scalar eld solutions, are distinguished in the usual sense as for a plane wave: one with the electric eld parallel to the incidence plane, denoted as scalar eld ψ p and another with the magnetic eld parallel to the incidence plane, denoted as ψ s . For a dipole, there are also a particular polarization orientation which corresponds to the direction of oscillation of the charges. In our coordinate system, we refer to the xand z-polarized dipole as being p-polarized and the y-polarized dipole as being s-polarized. Furthermore we choose the incidence plane as the x, z-plane. This implies that the elds ψ ν for ν = s, p are the only component of the vector elds E = (0, ψ s , 0) and H = (0, ψ p , 0).

The Green function in the present situation is given by a Hankel function [START_REF] Economou | Green's functions in quantum physics[END_REF].

In this section, exclusively, we will make use of a slightly dierent unit system where 0 = µ 0 = 1 (the speed of light remains as c) and there is factor of 4π in the Green function, compared to others sections. The reason is merely convenience and the fact that we will always be interested in quotients of quantities with the same units.

Hence, the choice does not aect comparisons between any of the results.

With this denitions we will seek for solutions in terms of the eld values evaluated precisely at the interface between each domain. The general scheme starts by applying Green's second integral theorem to scalar elds u and v any, and the Green function of the medium, both evaluated at the boundary Σ of the volume V . Evaluating appropriate boundary conditions linking the elds at the background with the eld inside the particle one obtains a system of equations that can be numerically solved. We pose the solution for each polarization separately, as the electric and magnetic elds do not obey the same wave equations.

V u∇ 2 v -v∇ 2 u dV = Σ v ∂u ∂Z -u ∂v ∂Z dΣ. ( 3
We now illustrate the general procedure for the particular case of s polarization. The solution p-polarization follows in a similar manner. We start by applying eq. 3.8 for two "volumes": region I and II (see g. 3.2). Consider region I and its corresponding Green function G I (r, r ). The eld in this region is denoted as ψ I s (r).

With the appropriate identications, the left-hand side of eq. 3.8 can be written as

V ψ I s (r)∇ 2 G I (r, r ) -G I (r, r )∇ 2 ψ I s (r) dV.
(3.9)

3.1. The solution of Maxwell's equations with integral equations Then, we note eq. 3.7 can be written as

∇ 2 + k 2 I G I (r, r ) = -4πδ(r -r ) (3.10)
and recall the scalar form of eq. 3.2 is

∇ 2 + k 2 I ψ I s (r) = f I e (r). (3.11)
In this expressions we denoted the wave number in the background region (or region I) as k I = √ B (ω/c). Substituting equations 3.10 and 3.11 in eq. 3.9 we obtain V 4πψ s (r)δ(r -r ) + G I (r, r )f I e (r ) dV .

(3.12)

Besides leading to a numerical solution, equation 3.12 is a convenient starting point for the discussion of the problem. Note that an identical form can be written in the case of p-polarization by replacing ψ s → ψ p and f e → f h . Furthermore, evaluating the theorem in region II amounts to replacing the superscripts I with II. In the next section we address the problem of describing localized sources in the vicinity of a polarizable object, possibly of linear dimensions comparable to the wavelength of emission λ em . This polarizable object will be referred to as the scatterer as it plays the role of disturbing the source eld that would otherwise propagate indenitely in a transparent background. The body can also absorb radiation and dissipate the absorbed energy through material degrees of freedom. The latter can be related to dierent physical mechanisms.

At some point it is necessary to describe the surface or boundary Σ of volume V in mathematical form. The surface is described by the loci of points where the abrupt changes in material properties occurs. We deal with only one-dimensional surfaces due to the assumed invariance in the y-direction. For now we assume that the function r s is known. The particular numerical approach to obtain this coordinates is described in the chapter presenting the corresponding numerical results. The eld at r is then given by

ψ I s (r) = ψ I,i (r) + 1 4π Σ K I (r, r s )ψ s (r s ) -G I (r, r s )Υ s (r s ) dΣ, (3.13)
where we dened ψ I,i (r) as

ψ I,i s (r) = - 1 4π V G I (r, r )f I e (r )dV (3.14)
and the following quantities:

K I (r, r ) = ∂G I (r, r ) ∂Z , (3.15)
Υ I (r) = ∂ψ I (r) ∂Z .

(3.16)

The integral equation 3.14 will be addressed in what follows. Note that equation 3.13 is surface integral equation corresponding to equation 3.3, at least under our 59 3.1. The solution of Maxwell's equations with integral equations assumptions. Also, note that the volume term ψ I,i s plays the role of an incident eld. This source term will be discussed further in the next sections. For our purposes we emphasize that this incident eld is a known function related to the Green function of a homogeneous medium.

The solution for the case of s-polarization is presented to illustrate the procedure. Consider the presence of localized dipole sources inside and outside the region occupied by volume V or region I and II. As mentioned above, the coordinates r s are the points along the boundary between the particle and the background.

We now write the eld at an arbitrary point r in region I as a sum of an incident eld ψ I,i s (r) and a surface contribution S I s (r)

ψ I s (r) = ψ I,i s (r) + S I s (r).
(3.17)

The incident contribution is identical to eq. 3.14 (with the s subscript). The surface contribution is

S I s (r) = 1 4π Σ K I (r, r s )ψ s (r s ) -G I (r, r s )Υ s (r s ) dΣ. (3.18)
We have yet to specify how to obtain the unkown eld values ψ s and its normal derivative Υ s evaluated at the surface Σ. To this end we evaluate Green's second integral theorem in region II. We assume there are also sources inside region II, so an identical in form of equations 3.17-3.18 would be obtained. In order to obtain the actual result, one replaces the volume of the background with that of the scatterer and takes the point r inside region II. This can be written in a similar form to eq.

3.17:

ψ II s (r s ) = ψ II,i s (r) + S II s (r). (3.19)
Formally, one must take the limit as the point of observation r approaches boundary points r s from region I and, separately from region II, using equations 3.17 and 3.19.

With the results of these evaluations and imposing continuity of the elds across the boundary, one obtains a systems of two coupled integral equations. With this procedure, we pose a system of equations that can be solved for the eld ψ s and its derivative Υ s evaluated at the surface. Taking the limit r → r s , the integral equations are given by

ψ I s (r s ) -ψ I,i s (r s ) = lim υ→0 Σ K I (r s + υĝ, r s )ψ I s (r s ) -G I (r s + υĝ, r s )Υ I s (r s ) dΣ (3.20)
for eq. 3.17 and

ψ II s (r s ) -ψ II,i s (r s ) = lim υ→0 Σ K II (r s + υĝ, r s )ψ II s (r s ) -G II (r s + υĝ, r s )Υ II s (r s ) dΣ (3.21)
for eq. 3.19. Similar denitions to those made for the elds in region I are implied in 3.21. It is worth mentioning that an identical system of equations can be obtained for the orthogonal (p-)polarization. In this case one should consider the appropriate wave equations as ψ p corresponds to the y-component of the magnetic eld. The unit vector ĝ is perpendicular to the surface (see g. 3.2).

The solution of Maxwell's equations with integral equations

The last step in obtaining the coupled equations is to relate tangential eld values across the boundary. This can be achieved by the use of boundary conditions. We assume the usual conditions of continuity for the tangential elds and their normal derivative. For both polarizations and in this notation we can write

ψ I ν (r s ) -ψ II ν (r s ) = 0, (3.22)
for the tangential eld and

1 ν I Υ I ν (r s ) - 1 ν II Υ II ν (r s ) = 0. (3.23)
Hereafter, the notation implies that if ν = s then ν I = ν II = 1. On the other hand, for p-polarization ν I = B and ν II = (ω). From this point forward we will not distinguish between polarization as the equations have similar form. We assume the surface prole r s is a vector-valued function of a parameter t that can be written as r s = ξ(t), η(t) .

(3.24)

Then, we say we evaluate the eld at the surface by writing ψ(r s ) = ψ(t).

Finally we have the elements to write the pair of coupled integral equations in a form amenable to a numerical solution. With this notation we write the coupled integral equations as

W(t) = ψ I (t) - Σ K I (t, t )ψ I (t ) -G I (t, t )Υ I (t ) dΣ and (3.25) Q(t) = Σ K II (t, t )ψ I (t ) - ν II ν I G II (t, t )Υ I (t ) dΣ.
(3.26) Equations 3.25 and 3.26 have two unknown variables. The functions W(t) and Q(t)

play the role of incident elds. The explicit form of the latter will be found by the use of dipole sources. Indeed, in the present work we consider localized light sources in the vicinity of a polarizable object.

The presence of sources in the near-eld

We focus now on the description of the light sources that will represent the scintillation light in our theoretical framework. The model is closely related to eective two-level (quantum) systems [START_REF] Vries | Point scatterers for classical waves[END_REF]. For the moment we will describe the determination of the classical dipole elds needed for the solution of the present (two-dimensional) problem. In a later section we will employ a semi-classical formalism to describe a quantum system with three non-degenerate energy levels. The model will describe the excitation of a molecule. In this part we focus on scattering processes that could lead to modications on the rate of emission of dipole sources.

Coming back to equation 3.14, which is of similar form for region II, we will precise the form of the sources f e,h in the wave equations 3.1 and 3.2. At this stage we make the choice of dipole source which we assume as a point-like source.

The source terms f I,II are related to this dipole source. In symbols we can state f I,II (r) ∝ p 0 δ(r -r 0 ), where p 0 = (p 0x , p 0y , p 0z ) is the dipole moment of the source 3.1. The solution of Maxwell's equations with integral equations at position r 0 . The latter must be taken on the appropriate region to describe f I or f II .

We write the main results for the dipole eld in the case of two orthogonal polarizations. For s polarization we have that [START_REF] Jackson | Classical electrodynamics[END_REF] f I,II e (r) = -4π(ω/c) 2 p 0 δ(r -r 0 ), (3.27) while for p-polarization the source term is given by [START_REF] Giannini | Electrodynamic calculations of spontaneous emission coupled to metal nanostructures of arbitrary shape: nanoantenna-enhanced uorescence[END_REF] f I,II h (r, r 0 ) = i4π(ω/c) 2 ∇ × p 0 δ(r -r 0 ).

(3.28)

Our previous considerations imply that a s-polarized dipole p s (r 0 ) is described by its y-component. p-polarized dipoles p p (r 0 ) are combinations of p 0x and p 0z .

It is now possible to obtain explicit expressions for the incident elds, i.e. to solve eq. 3.14 for both regions. To this end we dene r s -r 0 = R s ûs .

(3.29)

The explicit form of the incident elds then reads

ψ i s (R s ) = iπ(ω/c) 2 p 0y H (1) 0 kR s (3.30)
for s-polarized dipoles and

ψ i p (R s ûs ) = π(ω/c) 2 (x • ûs )p 0z -(ẑ • ûs )p 0x H (1) 1 kR s .
(3.31) for p-polarized dipoles. In writing equations 3.30 and 3.31 we dropped the superscript I, II to simplify notation. We also used the Green function for twodimensional geometries, namely [START_REF] Economou | Green's functions in quantum physics[END_REF][START_REF] Abramowitz | Handbook of mathematical functions: with formulas, graphs, and mathematical tables[END_REF] G I,II (r, r ) = iπH

(1) 0 k I,II |r -r | , (3.32)
where H

n is a Hankel function of the rst kind and order n. The latter is of order 1 in eq. 3.31 as a result of derivatives in eq. 3.28. In general, if there are sources inside and outside the particle, one must consider in each case the localized sources acting as an incident eld on the boundary.

All the elements needed for a numerical solution have been presented. Specication of the precise geometry and material properties will be discussed in the context of numerical results. The matrix elements are known and have similar forms to the incident elds. This is because the matrix elements K I,II (t, t ) and G I,II (t, t ) are themselves source functions relating electromagnetic sources along the boundary of the polarizable object. One eectively condenses the volume scattering contributions, described for example by the integral in eq. 3.3, into sources present precisely at the surf ace. From a practical perspective, this results in ecient computation and rapid convergence, at least compared to a volume integral equation (VIE) method [START_REF] Kern | Surface integral formulation for 3d simulations of plasmonic and high permittivity nanostructures[END_REF].

In the case of light extraction improvements, the criterion for evaluating the performance of a given structure was related to the transmittance of the structure.

Furthermore, this value was normalized to that of a at surface. In this case, as long as we choose some "reasonable" reference, we can estimate quantitatively the inuence of the near-eld scattered by wavelength-scale size structures on the light emitter. To this end we choose to normalize quantities of interest with the corresponding quantity evaluated in vacuum. Another possible normalization can be made using the radiated eld of the dipole in a homogeneous background.

Numerical solution

The numerical solution of such system can be addressed, for example, with linear algebra solutions. This implies discretization of the (continuous) boundary prole.

Truncation of the prole into a nite number of points N p is an issue treated in the section reporting the corresponding numerical results.

Let us dene the following notation for the discrete equations: any eld evaluated at the surface points r s (t j ) will be written as F (t j ) = F j . The discrete version of equations 3.25 and 3.26, allowing for a numerical solution, can be cast in the following form:

W i = Np j=1 M I ij ψ j -N I ij Υ j φ j dt j , (3.33)
Q i = Np j=1 M II ij ψ j - ν II ν I N II ij Υ j φ j dt j . (3.34)
In this expressions we wrote the arc length between consecutive points ds j = φ(t)dt j = φ j |t j+1 -t j |. The latter possess a boundary prole whose parametrization allows an angle t as parameter for r s . In connection to real structures, this boundary represents the interface of the wire and the background in some plane y = const. We will consider only closed surfaces.

The numerical accuracy of the results can be estimated by comparing results of cases for which an analytical description is known. Here, results for circular cylinders obtained with this method were obtained and compared with the Mietype analytical solution for this geometry. Importantly, the accuracy is related to the discretization step ∆s used for the calculations. This quantity is the distance between consecutive sampling points. In a parametric representation in terms of t, such as the one assumed so far, the sampling of t implies a set of values for ∆s.

As it will be later described in detail for particular calculations, we provide only an estimation of ∆s involved in the calculations presented in Chapter 5, which is typically on the order of ∆s ∼ λ/20, where λ is the wavelength of light (inside the particle) used for the simulations. This notion of comparing the wavelength λ with ∆s is illustrated in gure 3.3. We present at last the description of the geometry. The coordinates r s represent the boundary of the wire. At this point we choose some particular points for r s as in gure 3.2. In the present work the discretization will be obtained from Gielis "Superformula" [START_REF] Gielis | A generic geometric transformation that unies a wide range of natural and abstract shapes[END_REF]. The latter is a closed-form equation that can be used to represent a large variety of shapes ranging from circles, squares and triangles, Npeak stars to spirals or ower-like proles. This is a generalization of the equation of an ellipse can be written in polar coordinates (r s , t) as

r s = 1 a cos mt/4 p 1 + 1 b sin mt/4 p 2 -1/q , ( 3.35)
where the set of values {m, p 1 , p 2 , q, a, b} are parameters that determine the shape of the particle. For a rectangular shape, for example, one can set m = 4, q = p 1 = p 2 ∼ 10 , while a and b will determine the aspect relation b/a of the particle. In this example, the resulting prole will not have very sharp corners (see g.

Radiative and non-radiative decay in polymer scintillators

In this section we precise the physical mechanisms that lead to the blue light emission seen in polymer scintillating materials such as those based on polystyrene (PS) or polyviniltoluene (PVT). The aim here is to provide a qualitative description of a scintillation process for the purposes of this work, as extensive research along the past few decades provides detailed descriptions of the processes involved for a number of organic and inorganic materials (see, e.g. [START_REF] Bray | A simple ecient liquid scintillator for counting aqueous solutions in a liquid scintillation counter[END_REF][START_REF] Anger | Scintillation camera[END_REF][START_REF] Anderson | An improved scintillation cocktail of high-solubilizing power[END_REF][START_REF] Hajagos | High-z sensitized plastic scintillators: A review[END_REF][START_REF] Delage | Scintillation of organic materials[END_REF] and references therein).

The latter is also true for the description of light emission in organic uorophores [START_REF] Braem | A femtosecond uorescence study of vibrational relaxation and cooling dynamics of uv dyes[END_REF][START_REF] El-Daly | Photophysical properties, excitation energy transfer and laser activity of 3-(4dimethylaminophenyl)-1-(2-pyridinyl) prop-2-en-1-one (dmapp): a new potential laser dye[END_REF][START_REF] El-Daly | Photophysical parameters, excitation energy transfer, and photoreactivity of 1, 4-bis (5-phenyl-2-oxazolyl) benzene (popop) laser dye[END_REF][START_REF] Delage | Scintillation of organic materials[END_REF], i.e. the description of electronic and vibrational relaxation processes.

Briey, in order to enhance photoluminescence properties of the polymer, organic uorescent dyes are added to the polymer matrix. Energy transfer bewteen the polymer and the added molecules then results in visible light, frequently in the UV and blue region of the spectrum (see g. 3.4).

The goal of a scintillating material is to produce visible light from high-energy excitation such as gamma radiation, fast electrons (β-particles), fast protons, αparticles and heavier nuclei. In general, when ionizing radiation impinges on any material, a part of the radiation's energy is transferred to the material. In the visible range of the spectrum, the interaction between radiation and matter results in excitation of elementary charges such as electrons. A summary of the excitation and relaxation processes occurring in scintillating materials is shown in gure 3.4.

When an excited electron decays to its ground state, it could emit a photon or release energy in some other, non-radiative way. On the other hand, one gamma (γ-) photon, for example, contains roughly a million times the energy of a visible photon. At this energy scale, the interactions are frequently described with particlelike trajectories inside the material. The ionizing particle deposits energy along the trajectory in the material until it stops or exits the material. In a plastic scintillator, this process occurs by exciting molecules along the trajectory of the particle.

Due to the low molar concentrations of added uorophores in the plastic matrix, the main scintillation mechanism along the excited region is related to the excitation of polymer molecules [START_REF] Hajagos | High-z sensitized plastic scintillators: A review[END_REF]. Although uorescent molecules are present in the excited region, we assume their contribution to the overall scintillation process comes after the excitation of the polymer molecules. This is due to the fact that the excited polymer molecules can absorb and emit UV radiation. The UV emission of the matrix is rather inecient but non-radiative transitions can occur by interactions between the matrix and the primary uorophore. In fact, non-radiative transitions are the main energy transfer mechanism between the matrix and PPO. In its (rst singlet) excited state, PPO has a high probability of emission. In turn, this emission excites the secondary uorophore or wavelength-shifter (WS). Afterwards, radiative energy-transfer (violet curly arrow) occurs between the primary and secondary uorophore or wavelength-shifter (WS), which in turn has a high probability of emitting the blue photon. line with a system composed of only paired electrons. Doublet and triplet states are dened by associating two and three spectral lines to systems of one and two unpaired electrons, respectively. Triplet states can be characterized by decaying only in a non-radiative way. Therefore, singlets are responsible for uorescence. There exists also the possibility of triplet-triplet interaction resulting in light emission.

Depending on the particular energetic regime, this process can be more or less signicant. In general, its eciency depends on the excitation energy [START_REF] Hajagos | High-z sensitized plastic scintillators: A review[END_REF].

In the following, using a phenomenological approach, the degrees of freedom are condensed into resonances in order to describe light emission from the organic uorophores. The aim is to consider uorescence as the radiative decay mechanism and 3.3. A semi-classical formalism for a three-level molecule condense every non-radiative relaxation channel into a rate Γ N R . It is stressed that in this description, we do not distinguish between the delayed emission component of real scintillators and the prompt or fast uorescence component. Instead, the emitters are treated as three-level systems close to resonance. Each of the latter is characterized by a resonant frequency ω 0 , a radiative rate Γ and a non-radiative rate Γ N R . The total linewidth of each system is then γ = Γ + Γ N R .

A semi-classical formalism for a three-level molecule

In this section we describe the electromagnetic coupling of a ourescent molecule to a structured medium. The description is based on the electric-dipole approximation and a dynamic polarizability heuristically introduced in the previous section. The model relies on a three-level system for the ourophore. As this model system is described in the literature, we will state the main results for our purposes. We follow the approach reported in [START_REF] Bouchet | Quantum dipole emitters in structured environments: a scattering approach: tutorial[END_REF].

We consider three non-degenerate states |a , |b and |c with the associated spontaneous decay rates Γ bc , Γ ba and Γ ca , respectively. We now recall that in a scintillation event in an organic medium there are radiationless mechanisms with lifetimes much shorter than the radiative lifetime. We associate the non-radiative (fast) transition to Γ bc and the radiative transition to Γ sp ba . Note that Γ ba ≥ Γ sp ba . The dierence in lifetimes implies Γ bc Γ ba .

In order to address the coupling of the eld to the molecule, we assume a quasimonochromatic excitation eld tuned to the transition ab. The system can be described in this case by a density operator ρ that obeys the master equation

d dt ρ = 1 i H 0 + H 1 , ρ + d ρ dt relax , (3.36)
where H 0 is the unperturbed Hamiltonian,

H 0 = ω ab σ+ ab σ- ab + ω ac σ- ac σ- ac , (3.37)
ω ij is the Bohr frequency of the transition ij and σ± ij are the raising (+) and low- ering (-) atomic operators. In the electric-dipole approximation, the interaction In order to compute the populations ρ aa , ρ bb and ρ cc , as well as the coherences ρ ab and ρ ba , we note that ρ aa + ρ bb + ρ cc = 1 and assume the excitation eld to be tuned to transition ab such that |ω -ω ab | ω ab . We further assume that the total decay rate γ ab ω ab . The total rate must satisfy γ ab ≥ Γ ba + Γ bc .

Hamiltonian H 1 is H 1 = -d ab • E(t)(σ + ab + σ- ab ) (3.
(3.39)

Under these assumptions, equation 3.36 leads to three coupled equations commonly known as the optical Bloch equations [START_REF] Loudon | The quantum theory of light[END_REF][START_REF] Grynberg | Introduction to quantum optics: from the semi-classical approach to quantized light[END_REF]. The latter will be discussed in more 67 3.3. A semi-classical formalism for a three-level molecule detail in a later section. The steady state solution of the system can be calculated analytically from the optical Bloch equations for a stationary excitation eld. In the frequency domain, the coherence ρ ba is given by

ρ ba = - Ω + (ω) ω ab -ω -iγ ab /2 1 1 + s , (3.40)
where s is the saturation parameter

s = 2(2Γ ca + Γ bc ) Γ ca (Γ ba + Γ bc ) ∞ -∞ |Ω + (ω )| 2 ω ab -ω -iγ ab /2 dω (3.41)
and Ω + (ω ) is the Fourier transform of the positive-frequency component of the time dependent Rabi frequency

Ω + (t) = ∞ 0 Ω(ω)e -iωt dω. (3.42)
With these results at hand it is possible to dene a dynamic polarizability of the form [START_REF] Bouchet | Quantum dipole emitters in structured environments: a scattering approach: tutorial[END_REF] α ab = 3πc

3 ω 3 ab Γ sp ba ω ab -ω -iγ ab /2 u ⊗ u, (3.43)
where u characterizes the orientation of the transition dipole. This polarizability describes the excitation of the molecule by the electromagnetic eld, at least if the system is far from saturation (s 1). While the polarizability α ab , as dened in 3.43, considers only weak excitation elds, we can extend this notion to the saturated regime by keeping the parameter s in eq. 3.40. Following the notation α ab = α ab u ⊗ u, we can write

(1 + s)α ab = 1 → s = α -1
ab -1

(3.44)

Electromagnetic response of the environment

We have described the excitation of a dipolar emitter using the polarizability α ab .

Nevertheless, this description is only valid for a molecule in free-space or embedded in a homogeneous medium. For a complex background, we must take into account the interaction of the emitted light eld with polarizable objects in its vicinity. The emitter's radiation will also interact with near-by (polarizable) objects, which in turn can scatter the eld back to the source. The fact that the emitted radiation can interact with its own eld through this mechanism is a key point concerning the control of light emission. In order to address the latter, we continue with the description given in [START_REF] Bouchet | Quantum dipole emitters in structured environments: a scattering approach: tutorial[END_REF] and use a scattering formalism to illustrate dierent moleculeeld coupling regimes.

Consider a stationary external eld E inc and an arbitrary inhomogeneous background. The emitter does not interacts with E inc but with an excitation eld E exc resulting from the scattering of E inc by the complex background. We can model these processes using the Green function of the system. The spatial distribution of the electric permittivity, (r), determines the self-consistent eld responsible for the emitter's coupling.

A semi-classical formalism for a three-level molecule

The dyadic Green function G(r, r ) relates the radiated electric eld at r from a dipole with dipole moment d and located at r through

E(r) = k 2 0 G(r, r ) • d, (3.45)
where k 0 = ω/c is the free-space wave number of the incident radiation. By denition, the induced dipole moment d is related to the polarizability of the system, α, by G(r, r , ω) = G b (r, r , ω) + G s (r, r , ω).

d (+) (ω) = α(ω) • E (+) exc (r s , ω).
(3.47)

The rst term on the right-hand side of this equation is associated to a dipole eld in a homogeneous background with dielectric constant B . The second one, G s , is associated to the eld scattered by polarizable inclusions, possibly near the dipole.

Coupling of a dipole emitter to the electromagnetic eld

The key point in describing the interactions of the emitter with the eld is to relate the Green function to the polarizability of the emitter. To this end, the induced dipole moment in the inhomogeneous environment can be written as

d (+) (ω) = α 0 • E (+) exc (r s , ω) + k 2 0 α 0 G s (r s , r s , ω) • d (+) (ω).
α(ω) = α 0 (ω) + 4πk 2 0 α 0 (ω)G s (r s , r s , ω)α(ω), (3.49)
which can be recast in the form

α -1 (ω) = α -1 0 (ω) -k 2 0 G s (r s , r s , ω). (3.50)
This form of the polarizability, sometimes called the dressed or eective polarizability, accounts for the modication of the electromagnetic environment relative to a homogeneous medium. Note that this relation does not depend on the external eld.

It also reveals eigenmodes of the coupled system when α -1 (ω) → 0. Therefore, the modes of the system must satisfy

α -1 0 (ω) = k 2 0 G s (r s , r s , ω). (3.51)
This result shows that the driving mechanisms of the dipole emitter are determined by the bare polarizability α 0 (ω) and the (dipole) eld scattered by the inhomogeneities. In the following we will use this result to describe the coupling of a dipole to a structured environment.

3.4. A at surface as the simplest structured electromagnetic environment 3.4 A at surface as the simplest structured electromagnetic environment

In this section we describe the interaction of a dipolar source with a at substrate.

Although the geometry of the problem is quite simple and a rigorous solution of the electromagnetic problem is known, it is not straightforward to evaluate it numerically. This is because the solution is given in an integral form in reciprocal (k-) space, containing integrands with oscillatory behaviour. Its computation requires some care but is frequently separated into two sums: one involving only propagating components and another involving only evanescent components. The importance of having a rigorous solution and being able to compute it can be justied, among other reasons, with the fact that the half-space geometry represents a quite general system. We have considered a system of innite dimensions but have focused only on the interactions in a small spatial region. Therefore, even for surfaces that are not at at a scale greater to the wavelength, one could expect the trends found for the at surface to be a limiting case. In other words, we could expect a similar behaviour for a similar dipolar source near a locally at surface.

In a half-space geometry, a plane interface separates a medium with refractive index n 1 (z > 0) from another with refractive index n 2 (z < 0). We dene the z-axis to be perpendicular to the interface and the interface to be located at z = 0. For the sake of simplicity, we will consider the case in which the particle is located in the upper medium with refractive index n 1 and assume dipoles located on the x-axis. The Green function associated to the half-space geometry, in the region z > 0, has two contributions: the dipole elds of a homogeneous medium (direct contribution) and those coming from the reection at the interface (reected contribution).

Moreover, in order to relate the Green function to the polarizability, the eld of the dipole needs to be evaluated at its own position. A rigorous solution for the eld scattered by a point dipole in the presence of the plane interface was obtained by 3.4. A at surface as the simplest structured electromagnetic environment Sommerfeld [START_REF] Sommerfeld | Über die ausbreitung der wellen in der drahtlosen telegraphie[END_REF] in the context of radio wave propagation. The general procedure for obtaining such a solution relies on transforming the Green function from direct to reciprocal space. For our purposes, we write the half-space Green function as

G HS (r, r ) = G 0 (r, r ) + G R (r, r ) (3.52)
where G 0 is the Green function of a homogeneous medium with refractive index n 1 and

G R (r, r ) = ∞ 0 2π 0 q dq dφ(q) 2πκ(q) D(q) exp iqR cos φ(q) -κ(q)Z (3.53)
is the reected contribution. The distance Z = z + z , R is the magnitude of the projection of r-r parallel to the surface and κ = q 2 -k 2 1 with k 1 = 2n 1 π/λ and λ the free-space wavelength. The tensor D(q) is related to the dielectric properties of each medium and its expression is given in [START_REF] Panasyuk | Short-distance expansion for the electromagnetic half-space green's tensor: general results and an application to radiative lifetime computations[END_REF]. Unfortunately, an analytic solution for the so-called Sommerfeld integral in the right hand side of equation 3.53) is not known and it is necessary to evaluate it numerically.

The numerical solution of equation 3.53) and the results of section 2 yield the polarizability of a small particle in the presence of the half-space. Furthermore, the polarizability obtained in this way obeys energy conservation for non-absorbing dielectric particles and arbitrary background media. In the following section, we use dierent degrees of approximation for equation 3.53) to obtain the eective polarizability of a particle near a plane surface.

Direct contribution: self-terms

The direct contribution, G 0 (r, r ), is given by

G 0 (r, r ) = - 1 3k 2 1 δ(r -r )I + e ikR 4πR I -rr + 1 -ik 1 R k 2 R 2 (I -3rr) , (3.54)
with the denitions R = |r -r | and r = (r -r )/R. The terms on the right hand side of equation 3.54) are the singular and regular part, respectively, of G 0 (r, r ).

The singular part is an exact result for the depolarization factor of an innitely small cubic or spherical source and exclusion volume [START_REF] Yaghjian | Electric dyadic green's functions in the source region[END_REF]. Indeed, it leads to the electrostatic polarizability when considering point-like particles [START_REF] Chaumet | Coupled dipole method for scatterers with large permittivity[END_REF]. In order to take into account the nite size of the particle, however, the regular part must be properly averaged [START_REF] Chaumet | Coupled dipole method for scatterers with large permittivity[END_REF][START_REF] Lakhtakia | On two numerical techniques for light scattering by dielectric agglomerated structures[END_REF][START_REF] Martin | Electromagnetic scattering in polarizable backgrounds[END_REF]. This average requires some care due to the singularities appearing in the regular part of equation 3.54) when R → 0. In the following, however, we expand the imaginary part in the limit of small k 1 R and write {G 0 (r, r)} ≈ k 1 6π I.

(3.55)

Reected contribution: short-distance expansion

The reected contribution G R can be approximated as [START_REF] Panasyuk | Short-distance expansion for the electromagnetic half-space green's tensor: general results and an application to radiative lifetime computations[END_REF]]

G R (r, r ) ≈ 1 4πk 2 1 L 3 N l=0 (k 1 L) l K (l) (3.56)
3.4. A at surface as the simplest structured electromagnetic environment for small k 1 L. The dyadic K (l) contains the coecients of the lth-order term and L = √ Z 2 + R 2 . The coecients in the expansion are known analytically up to N = 7 for lossless substrates, while for arbitrary substrates the expansion remains valid for N ≤ 3. It is important to notice that equation 3.56) is not a formal power series in k 1 L as K (l) depends on the frequency, as well as Z and R.

In the case of a half-space background, these corrections correspond to improvements of the Image Dipole model (IDM) of a dipole above a surface (see image charge theory in, e.g. [START_REF] Jackson | Classical electrodynamics[END_REF]). Unlike the case of a homogeneous medium, for the half-space geometry, G R (r, r ) is well dened for |r -r | = 0. Using the fact that R = 0 and Z = 2z, we can write

G R (r, r) ≈ 1 32πk 2 1 z 3 N l=1 (2k 1 z) l K (l) .
(3.57)

The explicit expressions for K (l) are given in [START_REF] Panasyuk | Short-distance expansion for the electromagnetic half-space green's tensor: general results and an application to radiative lifetime computations[END_REF]. It is important to mention that this expansion is valid for lossy substrates only for l ≤ 3. Higher orders are only valid for transparent substrates. The zeroth order approximation for G R , associated with the coecients K (0) , corresponds to the well known IDM for a particle above a plane surface. Nevertheless, in the following we will see that higher order corrections provide further insight into the coupling of the dipole to the substrate.

Local density of states near a plane interface

The local electromagnetic density of states (LDOS) is proportional to the imaginary part of the Green tensor. In the following, we will only consider the xx and zz components of G R , which, due to the symmetry of the half-space geometry, are the only necessary components for our purposes. The components of the reected Green tensor lead to the projected LDOS, i.e.

G R αα (z) → ρ α (z) = ρ 0 + 6ω πc 2 {G R αα (z)}. (3.58)
Here, ρ 0 is the LDOS in a homogeneous medium with refractive index n 1 . The normalized LDOS ρα can then be written as

ρα (z) = ρ α (z) ρ 0 = 1 + 6π k 1 {G R αα (z)}. (3.59)
The Image-Dipole model, as mentioned above, corresponds to the lowest-order approximation, i.e., to the case l = 0. The corresponding approximation for the LDOS, ρ(0) , is given by

ρ(0) x (z) =1 + K x λ z 3 and ρ(0) z (z) =1 + K z λ z 3 , (3.60)
where In these last expressions we denote the complex permittivity of the substrate as 2 = 2 + i 2 and 1 = n 2 1 . We can see that, in the IDM, the reected LDOS vanishes for transparent substrates.

K x = 3 64π 3 n 1 2 | 2 + 1 | 2 and K z = 2K x .
The coecients for the order l = 1 are identically zero. However, up to the order l=2, the LDOS can be written as:

ρ(2) x (z) = 1 + L x λ z + K x λ z 3 and (3.62) ρ(2) z (z) = 1 + L z λ z + K z λ z 3 , (3.63)
with

L x = 3 2 16πn 1 3 1 | 2 | 2 + 2 2 1 2 -3 1 | 1 + 2 | 4 and L z = 3 2 8πn 1 1 | 2 | 2 + 6 2 1 2 + 5 3 1 | 1 + 2 | 4 . (3.64)
Note these coecients also vanish for transparent substrates. For absorbing substrates the modication of the LDOS is proportional to 2 . At this level of approximation, the LDOS describes only non-radiative eects associated to the intrinsic losses in the substrate. The terms L α are, however, the rst non-radiative correction to the IDM.

On the other hand, the third other coecient K (3) of the SDE Green function (see eq. 3.57) is purely imaginary for a transparent substrate (see appendix), providing a correction analogous to the one provided by [START_REF] Draine | The discrete-dipole approximation and its application to interstellar graphite grains[END_REF] for a dipole embedded in a homogeneous background. The LDOS can be written in this case as

ρ(3) x (z) = 1 + M x + L x λ z + K x λ z 3 and (3.65) ρ(3) z (z) = 1 + M z + L z λ z + K z λ z 3 , (3.66)
where

M α = 3 2 K (3)
αα .

(3.67)

The explicit expressions of the coecients K

(3) αα are given in the appendix. Note that M α represents a constant modication of the LDOS.

Higher-order terms in the expansion for G R have been published elsewhere [START_REF] Panasyuk | Short-distance expansion for the electromagnetic half-space green's tensor: general results and an application to radiative lifetime computations[END_REF].

In the following section, the short distance expansion is compared to the rigorous solution for the half-space Green function through numerical examples. In particular, the dependence of the LDOS with the distance above the surface is discussed, as well as the accuracy of several orders of approximation. Zone I Zone II

10 -4 10 -3 10 -2 10 -1 d) i = 0.1 Zone I Zone II ρz (z) -1 O(x 3 ) O(x 5 ) O(x 7 ) Exact O(x 0 ) O(x 2 ) O(x 3 ) O(x 7 ) Exact ρz (z) -1 z/λ O(x 0 ) O(x 2 ) O(x 3 ) O(x 7 ) Exact z/λ O(x 0 ) O(x 2 ) O(x 3 ) O(x 7 ) Exact

Image dipole model and higher order corrections

In order to illustrate the inuence of dierent corrections to the IDM, gure 3.6 shows the normalized LDOS above a glass-vacuum interface, in the vacuum region (z > 0). We consider a glass substrate of refractive index n 2 = 1.5 and several degrees of absorption. Figure 3.6a shows the exact (numerical) results for a transparent substrate. The normalized, z-projected LDOS ρ(z) is shown for several orders of approximation as a function of the distance to the substrate. In this case, excellent agreement is found for the seventh order in the chosen z/λ -range. The orders O(x 3 ) and O(x 5 ) provide good agreement in the extreme near-eld (ENF) of the substrate, i.e. in the region z/λ < 10 -3 , but diverge from the exact result at roughly z/λ ≈ 0.005 and z/λ ≈ 0.02, respectively. In gure 3.6b-3.6d we consider lossy substrates. Figure 3.6b shows the results for a weakly lossy substrate with 2 = 2.25 + i10 -5 . The exact integral solution, in this case with absorption, is compared to the orders O(x 0 ), O(x 2 ) and O(x 3 ). The order O(x 7 ) for a lossless substrate is also shown for the sake of comparison. The IDM, corresponding to the zeroth-order approximation, provides good agreement with the exact solution only in 3.4. A at surface as the simplest structured electromagnetic environment the ENF. In this region, one can see that the O(x 2 ) correction is negligibe, implying that the zeroth-order coecient dominates in the ENF. Indeed, this can be seen from the form of LDOS short-distance expansion, where the term proportional to z -3 dominates as z → 0. The next order of approximation, O(x 3 ), corresponding to the rst radiative correction to the half-space polarizability, provides better agreement beyond the ENF and starts to diverge from the exact results at z/λ ≈ 0.01. As the substrate losses increase (g. 3.6c-d), we can observe a similar trend, where the lowest order dominates. Equivalently, there is a characteristic distance for which higher orders start to dominate. This distance is proportional to the imaginary part of 2 as follows. With the use of equation 3.57), for N = 3, we can write

ρz (z) -1 = 3 16k 2 1 K (0) zz z 3 + 4k 2 1 K (2) zz z + 8k 3 1 K (3)
zz .

(3.68)

The rst 2 terms in the right hand side of equation 3.68) are directly proportional to the losses in the substrate. Abbreviating

{K (l)

zz } = K l and imposing the condition

K 0 (z * ) 3 + 4k 2 1 K 2 (z * ) = 8k 3 1 K 3 , (3.69)
we obtain the following equation for k 1 z * :

8K 3 (k 1 z * ) 3 -4K 2 (k 1 z * ) 2 -K 0 = 0.
(z/λ) * = 1 12πn 1 K 3 T 2 1/3 + K 2 2 T 2 -1/3 + K 2 , (3.71)
The explicit expression for T = T (K 0 , K 2 , K 3 ) is given in the appendix. The vertical black lines in gures 3.6b-d indicate (z/λ) * for each case. In gure 3.6, we have denoted "Zone I" as the region where G R,(2) zz dominates the interaction and "Zone II" as the zone where the O(x 3 ) correction dominates. Physically, the rise in G R (r, r)

towards the ENF of the surface (Zone I) can be attributed to the fact that a dipole in this region dissipates energy into the substrate. In other words, in this region the dipole couples to the losses in the substrate. Furthermore, the seventh-order approximation for a transparent substrate provides additional insight into this coupling, since the rise in G R (r, r) towards the ENF is absent in this case. For the O(x 3 ) approximation, however, when the condition 3.69) is fullled, the height at which the coupling to the losses in the substrate are equal to the (radiative) losses of the dipole is z * . Figure 3.7 shows the normalized distance (z/λ) * as a function of the real part of . Each curve on the plot corresponds to a dierent value of i .

The maps show that for i r , the regime where the losses in the substrate dom- inate starts when the dipole approaches the ENF. On the other hand, in the case where i ∼ r , the dipole couples to the losses in the substrate below (z/λ) * ≈ 0.1, consistent with the case shown in gure 3.6d). In gure 3.8 the relative error is shown for the corresponding approximations in gure 3.6. We dene the relative error of the l-th order approximation e (l) as

e (l) = {G R zz (r, r) -G R,(l) zz (r, r)} {G R zz (r, r)} , (3.72)
where G R zz is the zz-component of the rigorous solution and G R,(l) zz is the approximate solution of order l. (3) ∼ 10 -3 ), while the fth-order correction decreases the discrepancy by 5 orders of magnitude, yielding e (5) ∼ 10 -8 . The seventh-order correction, however, has little eect on the accuracy in this region, at least for the considered parameters of the half-space. Nevertheless, there is signicant improvement in G R, [START_REF] De La Recherche | Nanostructured plastic scintillators for enhanced radiation detection[END_REF] zz in the NF region: e (7) starts to monotonically increase at z/λ ≈ 10 -2 , whereas e (5) is already increasing at z/λ = 10 -3 and is nearly 4 orders of magnitude greater than e (7) at z/λ = 10 -2 . For substrates with absorption, gures 3.8b-d show the relative error e (l) for l = 0, 2, 3. As stated above, in gure 3.8b the ENF is clearly the region of reasonable agreement for l = 0 and l = 2. However, for l = 3 the agreement with the exact solution is improved considerably: in the ENF the third-order correction decreases the error by nearly 3 orders of magnitude and it is less than 10% for z/λ < 0.01. On the other hand, e (2) reaches 10% at z/λ ≈ 0.0006. Figure 3.8c shows e (l) for a substrate with 2 = 2.25+i0.001. In this case, the overall accuracy is improved. While e (l) for the second-and zeroth-order approximations remains nearly 2 orders of magnitude greater than the e (3) , e (0) and e (2) reach an er-

d) i = 0.1 Relative error O(x 3 ) O(x 5 ) O(x 7 ) O(x 0 ) O(x 2 ) O(x 3 ) O(x 7 )
Relative error The axes arrangement is similar to that in g. 3.6.

z/λ O(x 0 ) O(x 2 ) O(x 3 ) O(x 7 ) z/λ O(x 0 ) O(x 2 ) O(x 3 ) O(x 7 )
ror of 10% at roughly z/λ ≈ 0.003. As the absorption in the substrate increases, one can observe that G R,(l) zz for l ≤ 3 provide better agreement with the exact solution, implying that low order coecients dominate the reected contribution to G R zz for lossy dielectric substrates. Figure 3.8d) shows the results for e (l) for a substrate with 2 = 2.25 + i0.1. In this case, the fact that the lower orders dominate the reected contribution becomes more evident. Indeed, unlike in gures 3.8b and 3.8c, e (0) and e (2) now have a dierent behaviour.

Chapter 4 Optical characterization of scintillating materials

In this chapter we describe representatives examples illustrating the experimental characterization measurements. The selected results presented here were performed along the period of the present work.

It is worth mentioning that this work was carried out simultaneously within the framework of two research projects: TURBOPET and DECISIoN. The rst, TUR-BOPET [6], is related to the light-propagation aspect of the work. The project came to an end by spring 2018. Structured interfaces were investigated and experiments were carried out towards the characterization of its transmission properties in internal illumination conditions. The main results presented in this section relate to the interaction of light with the at or structured surface of the scintillating material, which is a LYSO crystal with hemi-cylindrical shape (see g. 7.

3). The aim of this part is to show that transmission above the critical angle is possible with the use of wavelength-scale surface structures. This has important consequences for technology such as PET scanners and radiation detectors in general.

In the framework of the project DECISIoN [START_REF] De La Recherche | Nanostructured plastic scintillators for enhanced radiation detection[END_REF], the interest lies mostly on modifying light emission properties. Both descriptions of light-matter interactions involve wave phenomena but in this case one needs to focus on the microscopic aspect of the source of light. In this context it is then important to address the determination of optical constants as a function of the wavelength (or frequency) of light. Spectroscopic information can be retrieved from simple optical measurements using lasers, grating systems and optical detectors.

We present a new approach determining the attenuation coecient, or equivalently the imaginary part of the index of refraction of a homogeneous sample. Furthermore, the measurements of the quantum eciency are described in detail. The internal quantum eciency (IQE) was also measured for a number of commercial plastic scintillator samples.

The purpose of the present chapter is to describe a number of gures relating the concepts and gures of merits presented in earlier chapters. We will present results obtained with relatively simple experimental congurations. Besides the appropriate mechanical hardware, the basic elements needed in the experiments are a continuous-wave laser source, a linear polarizer and a power-meter. The spectra presented was obtained by collecting light from an integrating sphere connected We will not attempt at providing eective medium properties to the structured surface but rather treat it as a diracting dielectric slab whose near-eld is not of particular interest. Details on the fabrication of such kind of surface structures can be found in [START_REF] Kadiri | Topology assisted self-organization of colloidal nanoparticles: application to 2d large-scale nanomastering[END_REF].

There exists a number of factors to take into account in order to obtain reasonable sets of measurements. In particular, for the results presented in this section, the transmission measurements were repeated between 3 and 5 times for a given angle of incidence. This reduces random errors introduced, for example by the laser source uctuations.

Due to the wavelength of excitation at λ = 405 nm we do not obtain strong photoluminescence signals from the LYSO sample. It is worth mentioning at this stage however that a weak photoluminescence (PL) signal is obtained. For excitation of the crystal at this wavelength however the PL is rather be inecient. In average, for each 1000 "pump" photons less than 2 photons are emitted (EQE∼ 1.5%).

Simplications occur for this geometry, as can be deduced from the pictures in We discuss the results in terms of the microscope images presented earlier. Beyond the pictures, the following discussion also stems from results and arguments presented in the framework of the TURBOPET project. The period was estimated at P ≈ 1200 nm for sample A and P ≈ 1000 nm for sample B. First, we note the agreement between the measurements for the polished surface and the predictions for a perfectly at surface assume a value for the refractive index, independently of the transmission measurements. In the computation of the (intensity) Fresnel coecients we have used a value of the RI reported in literature [START_REF] Mao | Optical and scintillation properties of inorganic scintillators in high energy physics[END_REF]. Recall that in chapter 2 we assumed a RI n = 1.81 for the numerical calculations. The reason behind that choice was made in the framework of TURBOPET project and relates to the value of the refractive index at peak emission wavelength.

Measuring absorption in homogeneous media

At the wavelength of the illumination used for the measurements, λ = 405 nm, the light extraction eciency as dened in section 2.1 was found to be -8% for sample A and -25% for sample B. We can compare these values with those from the calculations presented in section 2.2.3 for similar periods. We note that for the perfectly periodic (innite) structures the light gain is far from optimal. Both of the structured scintillators must also possess some level of disorder. The structured samples were fabricated with the use of self-assembly techniques [START_REF] Kadiri | Topology assisted self-organization of colloidal nanoparticles: application to 2d large-scale nanomastering[END_REF]. The resulting surface structures, pillars or cones in this case, can have well-ordered arrays of similar particles. The physical dimensions of ordered arrays is estimated to be at micron scale. The grain size of a typical ordered array was estimated at around 16µm 2 . The regions formed between grains can act like eective boundaries and also scatter light. One could also model each grain as a micron-scale scatterer which is surrounded by other grains. In any case, scattered light will play a role, eectively drawing power from diraction orders. It is not a trivial task to determine if this randomly scattered radiation will contribute more to the reected or transmitted light. Therefore, for the purpose of extracting light through diraction orders, one should consider this eect or reduce it as much as possible. This section presents and describes the characterization of light absorption properties of plastic scintillators. We present one example reecting the wavelength range of optical transparency for a commercial plastic scintillator. The sample has a cylindrical shape. The length along its axes is 1 m. The diameter is 5 cm. Figure 4.4

Measuring absorption in homogeneous media

Measuring absorption in homogeneous media

shows a photograph of the type of scintillators that will be considered. Here we focus on the determination of absorption properties at wavelengths of emission λ em .

The properties considered are related to the imaginary part of the refractive index, i.e. the extinction coecient κ(ω) or κ(λ).

Here a new experimental approach for determining the absorption spectrum of homogeneous media is described. Figure 7.7 presents a schematic representation of the experiment.

For homogeneous media such as polymer-based scintillators, we can use the Beer-Lambert law (see eq. 4.11 in section 4.4). In this case it takes the form

I(z) = I 0 e -α(z-z 0 ) . (4.1)
Note that we wrote a relative distance in the exponent, in contrast to the usual approach of considering a beam traversing all the sample. With this approach one does not determines the quotient I(z)/I 0 . Usually one considers a sample with parallel exit and entrance surfaces and sends a collimated beam through it. The beam impinges at normal incidence. In this work, we generate the sources instead.

At a distance z, a continuous-wave ultra-violet laser with wavelength λ exc = 325 nm excites the sample. The UV laser impinges normally to the interface as shown in gure 4.5. Blue photoluminescence is the result. The absorption of UV light by the sample is so strong that the incident beam is completely stopped at the surface (a few dozen microns actually). Details about the latter will be given in later sections.

The signal that is eventually detected comes from the photoluminescence produced in the plastic, close to the spot where the UV laser impinges the surface. This light has travelled from that region to the exit interface (see g 4.5). The main assumption in this approach is that angles subtended by rays towards the integrating sphere are small, i.e. from around point S to point E in the gure.

We start by presenting the main results of this section.

Phenomenological characterization of an organic uorophore

Let us focus on the case of the commercial plastic scintillators EJ-212 and EJ-240.

Optical characterization of these commercially available samples was performed in terms of absorption properties at the wavelength of emission and of internal and external quantum eciencies in the UV and violet region. It is important to mention that the precise chemical compositions of commercial scintillators are not precisely known. Nevertheless, the uorophores are present at concentrations of the order of ≈ 1% for PPO (primary) and ≈ 0.05% for POPOP (secondary) [START_REF] Hajagos | High-z sensitized plastic scintillators: A review[END_REF]. In this section, we rst review the emission of the constituents in a homogeneous medium (solvent)

and introduce the phenomenological model that will be used describing resonances. Afterwards we will present photoluminescence spectra for the commercial scintillators and further discuss the phenomenological model employed.

The normalized emission and absorption spectra of the scintillator's constituents are presented in g. 4.8. The absorption and emission data was taken from [START_REF] Taniguchi | Database of absorption and uorescence spectra of> 300 common compounds for use in photochem cad[END_REF].

Phenomenological characterization of an organic uorophore

Dissolved in clyclohexane, the quantum yield or internal quantum eciency (IQE) η for PPO is η = 1 [START_REF] Taniguchi | Database of absorption and uorescence spectra of> 300 common compounds for use in photochem cad[END_REF]. In this solvent, reported values for POPOP are η = 1 [START_REF] Taniguchi | Database of absorption and uorescence spectra of> 300 common compounds for use in photochem cad[END_REF] and η = 0.93 [START_REF] El-Daly | Photophysical parameters, excitation energy transfer, and photoreactivity of 1, 4-bis (5-phenyl-2-oxazolyl) benzene (popop) laser dye[END_REF]. Embedded in toluene, however, the IQE has been reported as η = 0.81 [START_REF] El-Daly | Photophysical parameters, excitation energy transfer, and photoreactivity of 1, 4-bis (5-phenyl-2-oxazolyl) benzene (popop) laser dye[END_REF]. First, we note that the polymer matrix posseses an absorption band around λ abs = 260 nm (g. 4.8a). The emission band is shifted towards λ em = 300 nm. For PPO, the main absorption peak is located at λ = 300 nm.

Recall that, although the emission band of the matrix and the absorption band of PPO overlap, the main energy transfer mechanism between the polymer and PPO ) absorption spectra of POPOP (symbols) [START_REF] Taniguchi | Database of absorption and uorescence spectra of> 300 common compounds for use in photochem cad[END_REF]. The solid lines correspond to the best t of the spectra to a sum of Lorentzian functions. The maximum (molar) attenuation coecient for POPOP is 4.71×10 4 cm -1 at a wavelength λ abs = 359 nm. sion). First, energy transfer from the matrix into the primary uorophore (usually of a high quantum-yield or internal quantum eciency) occurs through non-radiative processes and second, the emission from the primary uorophore is the excitation of the wavelength shifter or secondary uorophore (POPOP in this case). The solid lines shown in each case correspond to the t of the spectra to a sum of Lorentzian functions. Indeed, for frequencies close to a transition frequency ω ab , the excitation of a dipole emitter can be characterized by a polarizability of the form [START_REF] Bouchet | Quantum dipole emitters in structured environments: a scattering approach: tutorial[END_REF] α ab (ω

) = 3πc 3 ω 3 ab Γ sp ba ω ab -ω -iγ 0 /2 , (4.4)
with Γ sp ba = ω 3 ab d 2 ab /3π 0 c 3 the spontaneous emission rate (Einstein A coecient) and γ 0 is the total linewidth of the resonance. In this case, the parameters of each resonance were obtained by tting to the equation

S abs (ω) = j L j (ω), (4.5)
where S abs is the molar extinction coecient of POPOP [START_REF] Taniguchi | Database of absorption and uorescence spectra of> 300 common compounds for use in photochem cad[END_REF] and the Lorentzian terms L j (ω) play the role of absorption cross-sections multiplied by molar concentrations. We assume scattering cross sections, as eventually derived from eq. 4.4, can be neglected. By tting the spectra we obtain the linewidths γ 0 , resonance frequencies ω ab and the relative contributions of each resonance. In order to retrieve the radiative decay rate for each resonance we use the following relation [ Here, d ab is the dipole moment of transition ab (in Debye), f is the oscillator strength and E max is the energy of maximum absorption in cm -1 . The integral is carried out for the molar extinction S abs in dm 3 mol -1 cm -1 and ν is the wave number value in cm -1 . Using the total dipole moment d ab , we obtain the contribution of each dipole, w j , and obtain eective spontaneous emission rates

Γ sp j = ω 3 j (w j d ab ) 2 3π 0 c 3 , (4.7)
thus obtaining the set of parameters {ω j , γ j , Γ sp j } representing each internal resonance of the dipole emitter. Normalization ensures that d ab = j w j d j .

The value for the dipole moment d ab obtained with the Lorentzian model diers from the experimental data by ≈ 6%. It is important to emphasize that this value is obtained experimentally and enters the model only to determine the relative weights of the resonances. The results for the absorption (attenuation coecient) spectra of POPOP are summarized in table 4.1.

The characterization of each uorescent molecule in terms of resonant polarizabilities provides information about the decay rates involved towards the end of the scintillation process. We can justify this description heuristically as follows.

Consider the excitation along the trajectory of the ionizing particle. Energy is being deposited in the excited region. The "nal" emitters (POPOP molecules) are present only in low concentrations, which means they are, in average, far (many wavelengths) from each other. It is then reasonable to assume that the excited molecules that could emit blue radiation do not interact with each other. This allows the description of the absorption and emission process as the sum of independent resonant terms. Each resonance is associated to transitions with a particular frequency ω j or energy E ab j and a transition rate Γ sp j .

Measurement of the quantum eciency

The spontaneous emission rates Γ sp j shown in table 4.1 corresponds to radiative decay rates from the rst excited state. The number of radiative transitions from this state is several orders of magnitude smaller than γ 0 , meaning that essentially every transition to the rst excited level decays non-radiatively. As we deal with uorescent molecules, the corresponding model is a three-level atomic system. Radiative transitions from the intermediate states are responsible for the measured photoluminescence signal and can be associated to a radiative decay rate Γ rad . The polarizability, as dened in this section, describes to the excitation of the molecules. Subsequent emission after non-radiative decay to an intermediate level is implied.

Measurement of the quantum eciency

Figure 4.10 shows a schematic representation of the measurements from which emission and absorption spectra are obtained. A detailed description of the experimental methodology followed here can be found in [START_REF] De Mello | An improved experimental determination of external photoluminescence quantum eciency[END_REF]. The measurements involve a linearly polarized laser sent to the interior of an integrating sphere through a small aperture. We consider three situations: the rst (g. 4.10a) corresponds to measuring the laser source without the sample inside the integrating sphere. The laser spectrum obtained in this case will later be used in the computation of the internal quantum eciency (IQE) as a normalization factor. The second measurement (g. The third measurement considers the direct excitation of the sample. In this conguration, the laser impinges directly on the sample. When the beam strikes the sample, a fraction gets reected o the surface. The other fraction penetrates the material and can be partially or completely absorbed by the sample. A fraction of the beam could also be transmitted through the sample. In the direct excitation process, a fraction A is absorbed by the sample. Once again, a fraction µ of the remaining laser light inside the sphere can be absorbed by the sample after bouncing inside the sphere. Therefore, we can write the integrated laser peak for the direct excitation case as

L c = (1 -A)(1 -µ)L a = (1 -A)L b . (4.9)
Clearly, from the knowledge of the laser peaks in each situation, one can obtain the fraction A of light absorbed by the sample. From eq. 4.9 one nds that

A = 1 - L c L b . (4.10)
The fraction absorbed by the sample in direct excitation, A, does not directly determine the absorbance A of the sample. We can estimate A however if we know The internal quantum eciency can be retrieved from a set of 3 dierent measurements: a) the laser source spectrum (without the sample inside the sphere), b) similar to a) but with the sample inside (with the laser striking the sphere's wall rst). In c), the laser beam is impinging directly on the sample; some fraction of the light is reected and another may or not be transmitted.

the refractive index of the sample. In this case, the absorbance A can be related to A through the common logarithm as A = log 10 (1/T 0 ). Here T 0 is the transmittance of the sample at normal incidence, which can only be taken as an approximation using this technique (near-normal incidence is required in order to keep reected light inside the integrating sphere). The transmittance can be estimated from the measurements by considering rst the reected fraction of the light arriving at the surface of the scintillator.

The physical situation from which A could be estimated pertains to the experiment in gure 4.10c. First, note that the fraction of remaining laser light inside the sphere immediately after the direct excitation but before scattering with the wall of the sphere is (1 -A)L a . We know, from the experiment described in gure 4.10b, that a fraction µ will be absorbed after the direct excitation process. This is expressed in equation 4.9. One may then provide an estimation of the transmittance using this information and write

T 0 = (1 -A)(1 -µ) -R 0 , (4.12)
where R 0 is the value for the reectance of the sample at normal incidence.

We now focus on to the emission spectra. Frequently, the spectral range where the pump laser operates is relatively far from the emission wavelengths. Therefore, absorption and emission spectra can be resolved without overlap. In this work, the molecules (PPO and POPOP) are known to have a large Stokes shift [100], on the order of 50 nm, meaning that emission wavelengths are red-shifted by ∆λ ∼ 50 nm.

Optical characterization of polymer based scintillators

In this case, we denote the integrated emission spectra as S a , S b and S c , corresponding to the three situations discussed above. Here, we will use the denition of IQE as [START_REF] De Mello | An improved experimental determination of external photoluminescence quantum eciency[END_REF] that is not absorbed by the sample is scattered by the sphere's wall and excites the sample indirectly, as in g. 4.10b. This can be written as

L c + S c ∝ (1 -A)(L b + S b ). (4.14)
The contribution of luminescence coming from the direct laser excitation of the sample to the overall signal must be proportional to the absorbed number of photons L a A. Furthermore, only a fraction η of the absorbed photons is emitted. With this we can write process is considered in the electric dipole moment approximation. Based on the techniques described in this chapter and borrowing results from photo-physical studies of organic uorophores [START_REF] El-Daly | Photophysical parameters, excitation energy transfer, and photoreactivity of 1, 4-bis (5-phenyl-2-oxazolyl) benzene (popop) laser dye[END_REF][START_REF] El-Daly | Photophysical properties, excitation energy transfer and laser activity of 3-(4dimethylaminophenyl)-1-(2-pyridinyl) prop-2-en-1-one (dmapp): a new potential laser dye[END_REF][START_REF] Braem | A femtosecond uorescence study of vibrational relaxation and cooling dynamics of uv dyes[END_REF], an approach is proposed towards the control and manipulation of light generated by scintillation materials.

L c + S c = ηL a A + (1 -A)(L b + S b ),

Optical characterization of polymer based scintillators

In this chapter we describe the experimental methodology enabling the determination of the absorption and emission properties of plastic scintillation materials. In particular, through the use of an integrating sphere, a spectrophotometer and appropriate laser sources and a few optical components, one can estimate, for example, the radiative and non-radiative contributions in the relaxation processes. Photoluminescence (PL) and absorption spectra are presented. This measurements can be related to the optical properties of the sample through a simple phenomenological model based on resonance phenomena.

The samples used here were commercially available plastic scintillators fabricated by Eljen [START_REF] Technology | Ej-240 long decay time scintillator[END_REF], model EJ-240. Samples with dierent geometries were characterized Measurements were taken for two distinct types of excitation: rst, using an UV laser with central wavelength λ U V = 325 nm (3.81 eV) and linewidth σ U V = 2.5 nm.

In the second case, the pump laser operates at a wavelength of λ Blue = 406 nm (3.05 eV) and has a linewidth σ Blue = 3 nm. In the rst case the UV laser excites both uorophores embedded in the PVT matrix. In the second case, however, the blue laser excites only the secondary uorophore or wavelength-shifter.

UV laser excitation: radiative energy transfer

In order to have a notion of the physical processes following the excitation of the samples, a qualitative description is given in this section. We start from the direct excitation case, which provides useful information concerning the overall energy transfer inside the sample. As the collimated laser beam travels inside the material, it deposits energy and excites molecules along its trajectory. Although we do not know specic information about the proportion of the uorophores in the plastic matrix (commercial samples), it is known [START_REF] Hajagos | High-z sensitized plastic scintillators: A review[END_REF] that, frequently, the primary uorophore is in the order of a 10:1 proportion with the secondary one (or wavelength-shifter).

The nominal density of electrons is n e = 3.33 × 10 23 per cm 3 . This quantity enters classical and quantum-mechanical models of matter. In classical electrodynamics, a Lorentz-oscillator model [START_REF] Stone | Radiation and optics, Radiation and Optics[END_REF] presents some similarities to the models presented here, but in this work we address internal resonances of the emitters. The measurements, however, contain the average (emission and absorption) values corresponding to large numbers of emission events. Nevertheless, keeping in mind that the interpretation has its dierences, we could think of the emitters as a collection of Lorentz sources being excited by the pump laser. Equivalently, we can think of driven oscillators with damping.

If we consider only frequencies close to a resonance (or an emission peak), then we can think of two dierent processes. The rst one is absorption by the primary uorophore and subsequent radiative energy transfer to the secondary uorophore, which ends up in (blue) emission by the latter. The second process is the excitation of the secondary uorophore without interacting with the primary one. We can expect, however, this contribution to be small due to the proportions at which the uorophores are present in the plastic. We also note that absorption of light by the plastic matrix does not play an important role, as the absorption band is located at shorter wavelengths than the laser's wavelength of λ L = 325 nm. In summary, we can expect the main relaxation mechanism present in the measurements to be the rst one, where radiative energy transfer occurs between the primary and secondary uorophores. It is important to keep in mind that saturation and non-linear processes can also be present in the recorded data. The description of such eects it beyond the scopes of the present work. Nevertheless, the present work can be considered a starting point towards more sophisticated models.

At this stage it is worth mentioning some general technical aspects regarding the experimental apparatus. The spectrophotometer used is the model SR-500i made by 4.5. Optical characterization of polymer based scintillators

Andor [START_REF] Andor | [END_REF]. The grating used in the measurements has 84 lines/mm. Together with the Charge-Coupled-Device (CCD) camera employed, this implies that the UV laser peak can be resolved with roughly 25 points. The CCD camera was cooled at -80 • C for every spectrum presented here. The calibration of the integrating sphere was performed with a thermal source and has been extensively used for UV photonics [START_REF] Nomenyo | Photonique UV: structuration top-down du ZnO pour une émission ampliée et un transfert d'énergie ecace[END_REF][START_REF] Zhou | UV-Visible luminescent ZnO based materials: synthesis and emission engineering[END_REF]. The calibration provides corrections terms that weight the detected signal at each wavelength. These corrections attempt at accounting for absorption in the sphere's walls. Unfortunately, as we will see hereafter, some processes in the sphere's walls induced by the incident laser seem to depend on the intensity of the beam. This is most noticeable when the laser strikes directly into the walls of the sphere, which are supposed to uniformly scatter light in all directions inside the sphere.

Instead, in this spectral region, the coating of the walls also seems to absorb the incident laser. Besides absorption by the sphere's walls, weak emission at longer wavelengths, in the blue region, is produced by the wall's coating. This was taken into account as a correction to the values presented hereafter. The absolute value of the correction remained within a level of 0.02, providing a source of systematic error in the measurements. Random errors are also introduced due to the uctuations of the laser source. The signal to noise ratios (SNR) associated to the measurements will be shown and imply deviations from 0.1% to 1% from their mean value. The exposure time in the spectrophotometer was set between 10 and 30 seconds. The long integration times ensures the SNR to be suciently high.

As the measurements approach the UV spectral region from longer wavelengths, the signal to noise ratio (SNR) decreases. This means that the quality of measurements in the UV region suers from higher noise levels than in the blue/visible region of the spectrum. In order to treat this issue, averaging or smoothing techniques need to be used. From a practical perspective, useful signal processing techniques can be employed in order to remove noise from the raw data. In particular, a regularization method named after the Russian physicist Andrey Tikhonov is used. Briey, the technique was developed independently in the context of statistics and more recently, it has been used in applications such as machine learning [START_REF] Krogh | A simple weight decay can improve generalization[END_REF][START_REF] Ng | Feature selection, l 1 vs. l 2 regularization, and rotational invariance[END_REF]. Similarly to a least-squares minimization technique, this regularization scheme considers the minimization of the squared residuals and adds a regularization term. Through an appropriate matrix containing the coecients of the regularization term (frequently a multiple of the identity matrix), one gives preference to solutions with smaller norms. The term regularization is related to the regularization of ill-posed problems. The latter are characterized by the lack of a unique solution, for example, in an usual linear regression approach. This makes the regularization procedure appropriate for noisy data, where one aims at retrieving a meaningful signal from data containing noise [START_REF] Golub | Tikhonov regularization and total least squares[END_REF]. More precisely, this scheme is most valuable where the SNR of the raw data is low, such as in the case of collecting UV light from the integrating sphere. In the following we present results where Tikhonov regularization has been applied. We use the open-source software Octave [START_REF] Eaton | GNU Octave version 4.2.1 manual: a high-level interactive language for numerical computations[END_REF], where we have used the regularization routine rgdtsmcore.

Characterization results

The UV excitation of the plastic scintillator EJ-240 was performed for two distinct bulk geometries: a 150 microns thick disk with a 7 cm diameter and a 10x10x50 We start by presenting the laser spectrum in each situation. Figure 4.11 presents the spectra corresponding to the three measurements described in the previous section. Fig. 4.11a is the measurement of the laser without the sample inside the sphere. Given the sensitivity of the integrating sphere's walls to UV radiation, a Gaussian t to the data is not straightforward. Indeed, the direct evaluation of the IQE using the regularized data in g. sphere. The laser does not strikes the sample directly but rst strikes the sphere's wall. The same issue as before arises, where close to the peak wavelength the measurements are aected by the walls. Finally, in 4.11c the weak signal corresponds to the measurement involving the laser striking the sample directly. In this case the laser does not strikes the wall directly and we can neglect the absorption of the laser light by the coating of the sphere.

In the case of direct excitation, the thickness of the sample is larger than the penetration depth. In other words, the laser beam is completely absorbed within the sample. The laser light remaining inside the sphere comes essentially from the reection at the rst air/scintillator interface. The reection coecient near normal Measured internal quantum eciency η for a variety of samples excited with a continuous-wave laser operating at a wavelength of 325 nm. The irradiance I 0 of the beam and the thickness of the sample are also shown.

incidence is R 0 ≈ 5.1% for a transparent medium (for example for materials a with extinction coecient κ < 10 -3 for our purposes) with RI n. On the assumption of a extinction coecient κ n and n = 1.59 the RI of the scintillator, the penetration depth δ at λ = 325 nm has been estimated at δ = 180 µm. The corresponding extinction coecient turns out to be κ = 1.4 × 10 -4 , consistent with the assumption of weak absorption.

We present the results of the photoluminescence (PL) spectra in gure 4.12. The broad emission peaks are centred at λ em = 430 nm. A smaller peak is also present in the near-UV at λ em = 385nm. This peak must come from radiative decay of the primary uorophore. Recall that a POPOP uorophore (wavelength-shifter) only emits light with wavelengths λ em > 400 nm. Note the noise level increasing towards higher energies in the UV region. Integrating the laser and PL peaks allows the computation of the internal quantum eciency (IQE) dened in the previous section. In this case, we obtain an IQE of η = 0.77 for the 150 microns thick sample. The value of η represents the fraction of the absorbed photons that is subsequently emitted by the sample. Consider the direct excitation case. In this situation, laser light levels are low inside the sphere after the direct excitation of the sample. The overall (wavelength-integrated) absorption of the sample is A ≈ 0.92, meaning that 92% of the incoming light is absorbed. The regularized PL signal reveals that 72% of the beam's power is converted into blue and near-UV light.

A summary of the characterization results for several samples is shown in table 4.2. The emission and absorption properties are summarized in terms of the IQE, the incident beam's irradiance I 0 and the sample thickness. The IQE decreases but a trend is not clear. The results for the characterization of plastic scintillators shown in the table are for samples of dierent bulk geometry (see g. 4.4). The bulk shape will also play a role in light extraction, as discussed in previous sections and in chapter 2.

In g. 4.13 we present a summary of the spectra used to obtain the values in table 4.2. Dierent normalization was used for the laser and emission spectra. For the laser peaks, the (regularized) signal is normalized with the irradiance of the corresponding incident beam and the exposure time. Then, one can interpret the c) the direct sample excitation case. Here, the volume of the sample V (cm 3 ), the power of the beam P 0 (in µW) and the exposure time T exp (sec) were used for normalization.

detected signal as the "Counts" per second and per unit irradiance, i.e.

[Counts] ∝ Events (s • µW/cm 2 ) .

With this normalization, the results for the laser peaks demonstrate the inuence of the incident beam's power on the measurement. As it has been discussed above, the issue of the sphere's walls to UV radiation can be noticed from the shape of the measured laser peaks. In g. 4.13a, the laser spectra without the sample inside is presented. One can clearly see that the least intense beam produces more "counts" per second. Note that, here, counts refer to arbitrary units and not photon counts.

For a perfectly non-interacting sphere, all curves must resemble each other, at least at a degree determined by the laser uctuations (random error source). Here, the 97 4.5. Optical characterization of polymer based scintillators results the for laser peaks can be interpreted as the incident beam interacting with the coating on the sphere's walls. Evidently, the eect is inversely proportional to the incident beam irradiance.

A dierent normalization was applied to the emission spectra. In this case, in order to explain several pumping regimes, the time-integrated signal was normalized by the power P 0 of the beam, the volume of the sample under study V and the exposure time T exp . In this case, the resulting values can be though of as the detected counts per µJ and cm 3 of the sample. On the right in g 4.13a, the background noise and the weak luminescence produced by the coating can be seen. In the right-side of g. 4.13b the PL spectra is shown for the indirect excitation case. In this situation, the normalization represents the eciency of the emission process by absorption in the sample as a whole. Indeed, diuse light excites the sample after the collimated beam impinges on the sphere's walls and is uniformly distributed inside the sphere.

Chapter 5 Dipole scattering by dielectric particles in a scintillating background

In this chapter the last numerical results are presented before giving a summary and some concluding remarks. We will briey describe the interplay between a quantum system that emits light and the classical eld associated to the emission. The semiclassical approach condenses material properties in two distinct manners: through a quantum description of the emitter, a molecule for example, and through classical electrodynamics. The former description allows for quantitative estimation of carrier populations in three-level systems. However, its response to (optical) excitation is determined by classical elds. Furthermore, the interaction of continuous media with the elds also results in (classical) light-matter interactions. The latter are described by Maxwell's equations. Taking all these elements into account have been reported recently in the literature [START_REF] Bouchet | Quantum dipole emitters in structured environments: a scattering approach: tutorial[END_REF] and has been put in the context of the present work in section 3.3.

We will limit the numerical examples to situations involving dipole sources in-

teracting with the near-eld of some dielectric, possibly absorbing structure. It will be stressed that straightforward numerical work can lead to the response of the considered structures in presence of a plane surface. This is possible thanks to the surface Green function discussed in chapter 3. Such situations will not be discussed in this chapter. Nevertheless, general trends discussed previously that apply for a dipole near a at surface remain relevant for the discussion and calculations presented hereafter. Recall that the dipole here is a line of charge. In the considered simulations there is an important number of aspects one has to deal with.

For example, one must consider the incident eld coming from localized sources, its evaluation on the surface prole and further processing to obtain the quantities of interest. In the simulations all the dynamics rely on the excitation eld, i.e. the dipole eld. Propagating and evanescent components are both inuencing the polarization of near-by objects. The latter in turn could at some point inuence the source signicantly. The numerical calculations presented here can be considered quantitative estimations of such inuences.

The formalism described in chapter 3 for a uorescent molecule will be discussed in the context of the present calculations. One of the features of the formalism is that both the amplitude and phase of the electromagnetic eld can be taken into account. This can be seen from the form of the Optical Bloch equations [START_REF] Bouchet | Quantum dipole emitters in structured environments: a scattering approach: tutorial[END_REF][START_REF] Loudon | The quantum theory of light[END_REF][START_REF] Grynberg | Introduction to quantum optics: from the semi-classical approach to quantized light[END_REF].

99

In what follows we separate the results into two categories: spectra for xed dipole congurations and results for quantities involving many congurations of the source or sources. The rst considers aspects related to optical near-eld interactions.

Crucial characteristics of the latter drive the sources through the eld scattered by the dielectric. We will also present a few examples of near-eld maps for a xed wavelength and xed source.

We also present results for a multitude of sources in the neighbourhood of the particle. The theoretical development presented in chapter 3 only considered one source. In order to include number of sources in the simulation, one must add at amplitude level the elds incident in the boundary. Afterwards the determination of the numerical solution is identical.

In essence, the physical model here implies that the incident eld on the surface prole of the wire receives the sum of every dipole at some point in a past, simple and well-known situation. Afterwards, the steady-state solution found numerically represents the conguration of elds after a signicantly large amount of time has passed. The temporal scales involved in this qualitative description are relative to the frequency of temporal oscillations in the emitted light (ω = kc).

For the calculations presented hereafter, we compute the scattered eld in a form identical to that in [START_REF] Valencia | Second-harmonic generation in the scattering of light by two-dimensional particles[END_REF]. The evaluation is done using the numerical solution for the boundary values of the eld. The relation between boundary values of the eld and the eld in an arbitrary point r = Rû in the background region can be written, in the convention used in section 3.1, as

ψ s (r) = (i/4) Σ Z • û k I H (1) 1 (k I R)ψ s (t) -H (1)
0 (k I R)Υ s (t) dt.

(5.1)

In this expression Z is a vector normal to the surface of the particle and the prime indicates a derivative with respect to t.

In order to begin the discussion we will make use of the latter to address some general facts about the population dynamics. Consider the set of coupled equations:

dρ c dt = -(Γ ba + Γ bc )ρ b + 2 {ρ ab Ω (+) (t)}, (5.2a) dρ ca dt = -Γ ca ρ c + Γ bc ρ b and (5.2b) dρ ba dt = -γ ab -iω ab ρ ba + i(ρ b -ρ a )Ω (+) (t) (5.2c)
We use the same notaton as in section 3.3 and t is the time variable. This set of coupled equations is known as the Optical Bloch equations. Recall that the so-called Rabi-frequency is dened as in eq. 3.42, namely

Ω + (t) = ∞ 0 Ω(ω)e -iωt dω.
(5.3) Note this expression only includes the positive-frequency components. In the notation chosen here for the eld, the Rabi frequency Ω(t) may be written as

Ω(t) = -(1/ ) d ab • ψ s (t) , (5.4)
where the dipole moment of transition ab is d ab . This expression can now be related to the presented numerical results. Evidently, eq.5.4 is in the time domain. The set of equations 5.2 describe the temporal uctuations of populations in a three-level system.

First, we note that the driving mechanism is of course the pump eld. The elds are of a classical nature in this model and are completely described by ψ s (t). In a photoluminescence experiment one frequently uses a source with narrow spectral linewidth such as a laser. The irradiance of the beam is proportional to |ψ s (t)| 2 .

Eventually one can relate the incident irradiance and basic characteristics of the source in order to determine the pumping regime. In essence, together with the incident irradiance, one can use the model described in the characterization of organic uorophores, section 4.3, obtain the saturation parameter s and further characterize the onset of the saturation, for example (see eq. 3.44).

Scattering by dielectric nano-walls

The scattering of light by wires is numerically studied in this section. The context of the present work deals with scintillation light, so it is worth mentioning important aspects of the models discussed within the present chapter. First, we stressed the calculations here provide estimations of in-plane behaviour of the elds. This is relevant for structures with wire-like shape. In the simulations, the cross-section of this wire is perfectly constant along its axis of symmetry. One expects this to be a fair approximation for real structures with a cross-section that slightly varies along the axis of the wire, as well as in a smooth fashion compared to the wavelength of illumination.

The discussion presented hereafter relies on the general physical system of a homogeneous background in addition to a two-dimensional particle (or wire). The convenience of studying this type of systems stems from simplications in the theoretical developments and calculations. Beyond the latter, other practical aspects can benet from a geometry such as the one chosen here. For example, in the context of the technology using grating structures, e.g. an important fraction of spectroscopy apparatus, the interfaces are periodically structured along the interface of a solid medium. The in-plane components of the eld are the only elds of interest.

Beyond the general argument made above, here we choose wires of arbitrary cross-section. The role of the wire is expected to be similar to that of an antenna.

This issue of light connement appears frequently in the optical phenomena occurring in metal nanoparticles, i.e. in localized plasmon resonances. For dielectric structures, however, the topic is far from being as studied as metallic nanoparticles.

We have chosen wires with cross-sections of rectangular shape for this part. The background medium will be a model of the plastic matrix of the scintillator. The particle will be a higher-index medium with non-negligible absorption in the UV and blue regions of the spectrum. We take the RI of Titanium Dioxide for the nano-wall (NW). A localized source such as the one described in the theoretical framework, chapter 3, will be considered here as incident eld. NW Width:

ψ s (r 0 ) 2 (a.u.)

Wavelength (nm)

Local eld at the source position 80 nm 140 nm 330 nm Figure 5.2: The magnitude of the eld evaluated at the source position is shown for several NW geometries. The height of the NW was xed at H = 1000 nm and the width was varied. The inset with the black rectangle and the red crossed dot represent the particle and the source, respectively.

The rst set of results is shown in gure 5.2. The situation considered here is that of a dipole situated 70 nm away from the corner of the nano-wall (NW). For example, the coordinates for the case of the NW with width L x = 330 is x 0 = -400 nm and z 0 = 650 nm (see inset in g. 5.2). The height of NW was kept constant at a value of L z = 1000 nm. The results shown are the normalized value of the eld at source position. The source is a s-polarized dipole. Essentially, the latter constitutes an isotropic two-dimensional source. Only this case will be considered as it illustrates the main elements of the interactions, at least for the purpose of this work.

The characteristics of the presented spectra can be qualitatively described as combinations of modes that resonate at certain frequency. A thorough study of the latter is beyond the scope of the present work. Instead we highlight some general aspects concerning the size of the scatterers and its relation to the wavelength of the light source. More precisely we considered the physical situation as follows: a single dipole is placed at point r s , close to a dielectric, absorbing nano-wall (cf. g. 3.2). The dipole is embedded in a homogeneous medium of refractive index n B = 1.59.

Scattering by dielectric nano-walls

For the NW we use the refractive index of Titanium Dioxide (TiO 2 ). In other words, the response of particles composed of such material is simulated. Figure 5.1 presents the complex refractive index for TiO 2 . As mentioned in an earlier chapter, we expect the index contrast to be related to the strength of optical interactions. Furthermore, optical transparency has important implications in terms of potential enhancement of quantities such as the LDOS or the scattering cross-section of the particle. Note, for example, that for wavelengths in the interval 400 < λ < 500 nm the extinction coecient κ, presented in logarithmic scale in the inset at the bottom, is considerably small, more precisely κ ≤ 10 -4 . The background is a transparent dielectric and is considered here as an ideal model of the polymer matrix in a plastic scintillator. The results will be restricted here for the case of transparent background media.

Four distinct NW geometries were chosen as labelled in g. 5.2. The results for the smaller NW, with L x = 80 nm, shows a smooth increase towards the blue region of the spectrum. No sharp peaks dominate for any wavelength of the chosen spectral range. As we increase the value of L x , as in the remaining curves, we see the apparition of well dened peaks and sharp features in the spectra. Beyond describing in detail the features present in the calculations, we discuss the general implications behind the numerical results. First, we point out that the fact the sharper peaks appear for wider NW is related to the resonant character of the modes.

We could argue that this system has some similarities to the case of an innite planar dielectric waveguide (see, e.g. [START_REF] Kasap | Optoelectronics and photonics: principles and practices[END_REF]). The obvious dierence here being the nite boundaries along the vertical axis. Although we limit to a qualitative discussion, we can nonetheless expect the number of modes in the NW increases as in the case of the planar waveguide. Interference phenomena inside the NW results in competition eects between modes. Furthermore, the NW scatters light into the background region, modifying the local eld in its vicinity. As the number of competing modes increases one could expect that the precise conditions for constructive interference to occur at the source position become more restrictive. Therefore, the eld at the source position is signicantly inuenced by radiation escaping from the resonant NW, as abrupt changes in the local eld can occur for wavelengths a few nm apart. In summary, gure 5.2 illustrates the fact that for rectangular structures whose dimensions are on the order of a wavelength, competition between modes can lead to resonances with sharp linewidth. Finally, it is evident from these results that for wavelengths below λ = 350 nm strong suppression of the local eld occurs at the dipole position. This is a direct consequence of absorption by the NW. For example, at this wavelength the imaginary part of the refractive index is κ 350 ∼ 0.06 while at λ = 310 nm it increases by a factor of 10.

Consider the situation for a monochromatic eld with wavelength λ = 426 nm. The near-eld in a region of area A near = 1.6 × 1.6µm 2 was computed for two NW geometries. The dimensions of the nano-walls are the same as those one considered for the results in gure 5.2, namely the cases for which L x = 80 and L x = 330 nm. For the smaller NW, the source is placed at coordinates x 0 = -150 nm and z 0 = -650 nm. For the thicker NW the source is placed at x 0 = -190 nm and z 0 = -650 nm. These gures illustrate some aspects related to the previous discussion on mode competition. Most notably, the dierence can be seen in the eld inside the NW.

For the 80 nm-width NW we see smooth changes in the intensity distribution, while 105 5.2. Structures for dielectric connement the thicker NW presents much more rapid variations compared to the thin one. The intricate eld distribution also hints at an interference pattern resulting from waves bouncing inside dielectric.

Near-eld distributions

Examples of near-eld distributions are shown in this section. The examples are related to the calculations presented above. The wavelength of dipole radiation is λ = 426 nm, which corresponds to the thin vertical line indicated in g. 5.2. For the sake of clarity, the colour scale is shown in a logarithmic scale. We have chosen two NW geometries, namely the smallest and biggest structures considered previously.

For this cases the NW width is L x = 80 and L x = 330 nm. The eld is normalized by the integral of the incident eld over the chosen region, i.e.

|ψ inc | 2 dxdz. (5.5) This normalization is of course arbitrary and its convergence is not guaranteed.

Nevertheless, it has been chosen to give a rst estimation of eld enhancements in the vicinity of the particle. We found that only 22% of a dipole's "power" in the homogeneous background, for the shown spatial region, is accounted for in the scattered eld map for L x = 80 nm. For the case L x = 330 nm, 35% of the dipole's power is present in the scattered eld's magnitude. The eld at localized regions, however, presents modest enhancements on the order of 10. Furthermore, the eld is conned to the particle's volume. We interpret this eect as weak dielectric connement. The chosen congurations presented here are far from optimal systems.

The present examples rather serve as representative cases for which the interplay between the dipole (incident) radiation and the modes of the particle can bee clearly seen.

It is also important to highlight the possible meaning of the eld maps presented in this part. The physical situation involves the eld of a dipole. As has been discussed in previous chapters, this eld contains spectroscopic information about the electromagnetic modes of the system. The quantity describing the latter is the LDOS. It is not a trivial task to determine the relation of the LDOS with the (quantum) population dynamics in a general manner. Some simplications occur however due to the electromagnetic response of the environment. The results are thus useful to describe dierent regimes of the coupling of localized sources with micron or sub-micron size particles.

The results can be interpreted as potential modications on the properties of a virtual source placed at some point in the shown region. Quantitative analysis of such modications can be made using the approach presented in section 3.3.

In short, the frequency and the linewidth of optical resonances are modied. The resonances in these cases are modes of the coupled molecule-eld system [START_REF] Bouchet | Quantum dipole emitters in structured environments: a scattering approach: tutorial[END_REF].

Structures for dielectric connement

The previous sections dealt with wires of rectangular cross-section. Some important trends concerning the eld at the source position were discussed. We continue the 5.2. Structures for dielectric connement discussion by describing near-eld distributions around wires with cross-sections with star-like shape.

We seek to obtain eld enhancement and connement eects, where the source in the neighbourhood of the wire can be inuenced. The precise mechanisms can be precised further by detailed analysis of the theoretical framework describing the excitation of a three-level quantum system. The description takes into account the strength of the incident eld, its frequency content and intrinsic quantum properties of the emitter. Electromagnetic interactions can be understood in this framework as the driving mechanism forcing changes in the populations of carriers in the quantum system. Specifying the electromagnetic response in terms of the squared eld amplitude, proportional to the irradiance, is then a starting point in the discussion of light emission enhancement. Decomposition of the eld into real and imaginary parts is also crucial for the eventual design of experiments or proofs of concept.

Each component of the complex amplitude of the eld plays a role described by the coupled equations describing energy transitions of the carriers.

Consider a multitude of localized sources around the NW. In gures 5.5 and 5.8 we show maps of the scattered eld by the particle in such situation. The dimensions of the NW are L x = 200 nm and L z = 2000 nm. The number of dipoles radiating in all directions is N dip = 32 in this case. The dipoles are randomly distributed and are restricted to the background region. The area over which they are positioned is proportional to the number of dipoles. More precisely, the positions of the dipoles are drawn from a uniform probability distribution and must be outside the particle. The sources are placed over a region of area A dip ∼ (l dip N dip ) 2 . This means the density of emitters remains unchanged in average. For the calculations in gures 5.5 and 5.8 only one realization is shown. In this case there is not strong eld connement but it is superior than the previously reported case in previous sections. The ratio of intensities between incident and scattered radiation is 0.47, i.e. 47% of the free dipole's power is accounted for in the presence of the particle.

We emphasize that the local eld is responsible for inuencing the response of the emitter through the environment. The cross-section of the NW chosen here are meant to illustrate eld enhancement eects inside and outside the dielectric cavity.

In the context of this work we do not directly intend to exploit the eld inside the wire. It is nonetheless an eect that seems to be inherent to resonant phenomena.

Multiple source excitation

We now focus on cases that present eld connement. The purpose of the present work is to provide tools for addressing light management in scintillating materials.

To this end we will now consider the presence of multiple sources in the background surrounding the scatterer.

Besides the presence of multiple dipole sources, the situation remains similar to the one described above. The cross-section of the wire is of a dierent form in this part. We have chosen geometries allowing for stronger enhancement compared to the previous sections. Furthermore, the position of the sources around the wire is chosen randomly. Therefore the simulation is repeated in order to average the eects of disorder. The number of repetitions used for the calculations is N R = 500. Note that there are two distinct average procedures involved in obtaining these results. The rst relates to extracting the average dipole eld from the ensemble of sources. This is done for each realization (or each source conguration). Furthermore we study the eects of increasing the number of particles without increasing the density of emitters. Here we study the inuence of near-eld and intermediate-eld eects, i.e. the eld response at scales for which both propagating and evanescent components cannot be neglected, on dipole sources interacting with their environment. The environment in this case is a homogeneous transparent and innite background and the lossy dielectric particle. No. of sources

ψ s (r 0 ) 2 (a.u.) Wavelength (nm) 1 8 32
Wavelength (nm)

Figure 5.7: Average spectra for a wire with a cross-section like the one shown in the inset. The red dots surrounding the star represent localized sources of radiation. For each wavelength, the value of the mean eld over a total of 500 source congurations is shown.

Results for three type of systems are shown: congurations of 1 source and the particle, as well as 8 and 32 sources in presence of the cavity. The main curves show a clear trend. As we increase the number of sources around the wire, even if in average the density of sources is not changed, the mean eld is decreased considerably in the UV region, while in the blue region it is slightly enhanced. Note the curve on the inset at the top of the gure. The shown values are identical but are plotted in a logarithmic (base 10) scale. Clearly, the results for one isolated source in presence of the particle present sharper and better resolved features. Indeed, in the UV region the average response for one source is roughly one order of magnitude greater than that involving 8 or 32 dipoles. We attribute this decrease on the values of the average eld, at least partially, to interference eects which contribute mainly to absorption by the dielectric cavity.

Scattering properties of the particle (or surface prole) can quantify the absorption properties. In fact, the numerical method used here has been extensively used for this purpose in the past; a few examples can be found in [START_REF] Giannini | Electrodynamic calculations of spontaneous emission coupled to metal nanostructures of arbitrary shape: nanoantenna-enhanced uorescence[END_REF][START_REF] Giannini | Calculations of light scattering from isolated and interacting metallic nanowires of arbitrary cross section by means of green's theorem surface integral equations in parametric form[END_REF][START_REF] Macías | Heuristic optimization for the design of plasmonic nanowires with specic resonant and scattering properties[END_REF]]. Here we found that, at the resonance peak at λ = 426 nm, the scattered eld by the particle due to the dipole excitation is about 11% of a dipole in free-space. We have also found modest local-eld factors, dened as the ratio K = |ψ(r 0 )|/|ψ i (r 0 )| [START_REF] Thomas | Singlemolecule spontaneous emission close to absorbing nanostructures[END_REF], on the order of K = 0.04 at λ = 426 nm.

Finally, one important characteristic that can be seen in gure 5.7 is the evolution of the oscillations in the spectra. The features change with the number of sources.

Towards the UV regions of the spectra, one sees that including more sources result in increased absorption by the particle. For example at λ ∼ 360 nm one can see some peaks in the case of a single source, whereas in the presence of multiple sources the eld is strongly attenuated. Interestingly, not every peak disappears as the number of sources increases around the wire.

Optical response of particle arrays

In this nal section we will discuss the scattering of dipole radiation by a collection of particles. We start from the simplest case, i.e. two particles. Similar results to the ones showed in earlier sections will be presented for this case. For the sake of reasonable computation time, we consider a small number of particles. We come back to the NW geometry.

Dipole scattering by two particles

Consider the presence of multiple dielectric polarizable bodies near localized sources of blue light. One can think of this situation as a rst model approach towards the design of structures whose optical (radiative) properties are useful for detecting scintillation light. In general, the optical properties of the particles will be determined by its chemical constituents as well as the amount of imperfections or defects that any real material has. From this broad perspective we narrow the problem by only considering perfectly homogeneous media. Nevertheless, the situation involving two particles will provide information that is more directly related to potential experiments. The latter does not mean that the descriptions given so far are not useful but rather that they represent general guidelines from which one can choose a particular technological solution.

A near-eld map is shown in g. 5.8. The map illustrates the situation addressed in this section. The localized source is placed at x 0 = -2000 nm and z 0 = -3700 nm. Field enhancements are evidently playing an important role in the vicinity of the wire. Moreover, the distance between particles is 500 nm. This implies that both evanescent and propagating components of the eld are important. Note the position of the source if well within a few wavelengths apart from the structure.

Nevertheless, the pair of particles seem to localize radiation at its vicinity. We discuss with more detail the spectral behaviour of such structure in the following.

Results for the average local eld for a single localized source in presence of two particles with rectangular cross-section are shown in g. 5.9. Each point in the curve is the average response of 300 sources in dierent positions around the particles. We present the values of the elds in more detail. The real and imaginary part of the local eld is shown in gure 5.9. We consider two distinct situations.

Note after a certain number of realizations there is an average area over which the dipole was placed. In practice one sets this value as input for a pseudo-random number generator. These values are indicated in the gure and each corresponds to a dierent colour.

For the average values shown in black, the dipole remains in average closer to As mentioned earlier, the real and imaginary parts determine dierent characteristics of the resonant source. Recall we assume the sources are pumped at nearresonance wavelength. The latter is fairly accurate for the scintillating materials considered in this work (cf. section 4.3). In fact, the "pump" eld in a scintillation event would be the few thousands of photons generated towards the end of relax-5.3. Optical response of particle arrays ation of carriers from high-energy states, e.g. from states more energetic than UV radiation.

Finite ordered arrays of particles

In this section we present maps of the real and imaginary parts of the scattered (local) eld. The response of periodic arrays of wires is simulated. The shown maps are potential modications to optical properties of molecules near the structure.

We present the calculations and briey highlight the main characteristics of each at the bottom of each gure. The latter can be found at the bottom of each gure containing the maps. The details of the calculations remain unchanged. The ratio λ/∆ ∼ 20 for all the calculations presented so far and will continue to be around this value.

We will discuss a few examples of eld distributions around an array of TiO 2 particles with dimensions L x = 300 and L z = 300. A single dipole source is the excitation eld. The scattered elds in several congurations are presented in gures 5.10-5.12. In g. 5.10 we present the squared modulus of the eld around the structures. The top map shows the situation discussed previously, i.e. it represents the same case as g. 5.8 but evaluated in a larger region of space. The eld distribution on the bottom corresponds to a conguration in which the separation of the NW's centre is 1500 nm. It is evident from these maps that interference eects will produce enhancements and suppression of the eld around them.

It is interesting to recall that collective eects of this kind rapidly scale towards a grating-like behaviour. In this sense, phase matching of the scattered elds can build a relative strong signal with appropriate choice of the distance between particles, eventually the period of the grating.

In order to nish this part we present the real and imaginary parts of the local eld in the same situation. Figures 5.11 and 5.12 show the eld distributions in a logarithmic scale (disregarding sign dierences). We present such results to emphasize the wave nature of eventual modications to the source characteristics.

The same considerations apply as for previous discussions on the present chapter.

The eects of multiple sources present simultaneously and averaging eects shown in previous sections are expected to apply also for these cases. Further numerical studies are necessary to obtain optimal results for any particular purpose.

If one attempts at enhancing uorescence in scintillating media, one should rst consider the eective "pumping" resulting from real scintillation events. A starting point could be to consider a single photon being absorbed by a three-level system.

Considering molecules with high quantum eciency, after the absorption process it is likely a photon will be emitted. In essence, quantitative estimation of such dynamics can be studied theoretically and numerically from eqs. 5.2. Eventually in numerical simulations, the absorption process can be regarded as the starting mechanism of a system at rest, which is of course a rst approximation to a real scintillation event. The temporal evolution can then be estimated by solving the optical Bloch equations. From the theoretical perspective, we establish a criterion to evaluate the performance of scintillators in terms of its light extraction properties. We make no reference to specic emission properties of the source. Instead, the approach taken for this part predicts average values associated to the transmission and reection of light by the scintillator. Indeed, it assumes a given distribution of electromagnetic elds at a scintillator/air interface. The only assumption for the source at this point is that its emission its isotropic. Basic considerations are discussed in order to justify the choices of input parameters for the numerical calculations. Quantitatively discussing the notion of transparency in scintillation media is an important aspect of the models in this part and the following chapters as well.

Based on the numerical solution of Maxwell's equations in periodic and lay-118 ered structures, we studied the transmission properties of periodically structured interfaces. As a rst step polished surfaces were discussed. We begin by studying the eld incident on the surface and coming from a large number of localized isotropic sources inside the scintillating medium. Then, transmission properties can be computed. Note that this approach is completely based on classical electrodynamics and that it makes use of well-known results for polished surfaces. Beyond polished interfaces, the potential of increasing light output through diraction and impedance-matching eects is studied. We distinguish two regimes for the trans- Frequently, one assumes that periodic surface structures have innite dimensions. Such is the case of the electromagnetic calculations presented in this section.

In other words, in the calculations the structure itself is innite in the plane of periodicity. This ideal, non-physical but approximate situation has shown to be a robust description if the structure is large enough. In particular, we focused on two general aspects of a nite scintillating medium: the rst is truncation of the exit surface and the second is the reections of light from the side and back walls of the nite medium. Useful trends are found here where a simple model for an inorganic LYSO bulk crystal coupled to a photodetector was simulated.

Light sources in scintillators

We have discussed the characteristics of scintillation light in detail from the perspective of optics and photonics. To this end we began by presenting the formal tools with which the electromagnetic elds are calculated. The main ingredients of such theoretical tools are related to the elds radiated by small sources, compared to the wavelength of radiation, and within the electric-dipole approximation. It is demonstrated that the problem of a localized (dipole) source near polarizable objects can be studied with the use of Green functions. The latter represents a quantity that is crucial for the description of light emission.

The formalism is presented in a way that leads straightforwardly to a numerical solution. In this work, the computation of such solution constitutes our main tool towards quantitative estimation of modications in light emission properties. In this part of the work we present the solution for geometries that are invariant in one direction. The structures associated to such topology have a wire-like shape.

We discussed also how this aects the generality of the work and specify important details of the theoretical framework. In particular, the theoretical treatment of localized sources is discussed. We showed this point-like sources of light produce elds that are essentially a sort of near-eld illumination for polarizable objects in its neighbourhood.

Light emission from plastic scintillating materials are of particular interest in the context of the present work. This type of media allows for nano-and microfabrication of structures in its surface, at least compared to hard inorganic crystals.

Furthermore, there has been much work done in this regard the past few decades.

The advent of nanotechnology and related tools have made possible the manipulation of mechanical, electrical and optical properties of matter, at least down to the nanometer length scale.

In polymer-based scintillators the emitted photons are the result of a nal decay process occurring in organic uorophores. The details of such decay mechanism are described at rst in a qualitative manner. A model allowing for quantitative discussion is presented afterwards. The phenomenological model allows for the determination of eective optical properties related to light emission. The physical system considered in this part is a three-level molecule. In this section we showed the inuence of classical electromagnetic elds on light emitters. The latter can refer to electrons, for example, as they frequently decay to the ground state and emit a photon in the process.

A localized light source in the presence of polarizable objects A well-known situation was studied in this work within the context of light emission.

The situation pertains to the electromagnetic response of a perfectly at, innite surface. This conguration is also called a semi-innite medium or simply a halfspace. The theoretical developments on this part relate to the response of a localized source of radiation in presence of a surface. The problem has been addressed a number of times throughout the 20th century. The agreement between theory and experiment has been shown to be quite accurate. Nevertheless, in the context of modern optics and nanophotonics, such a canonical example can still provide general trends that could be useful for the interpretation of more complex systems.

In parallel to the work on scintillation phenomena addressed here, it is worth mentioning at this stage that related research was developed in the context of a dipole near a at surface. As one could eventually deduce from the formalism presented in this section, there are a number of aspects that can be studied from the set of formulae that has been derived. Indeed, such was the case of the present work. Throughout the course of the research project we studied the interaction of dipoles near at dielectric surfaces. Besides the presented results and discussion, we have found use of the formalism in the description of optical phenomena occurring near Epsilon-Near-Zero (ENZ) materials. The latter are materials whose electric permittivity has a vanishing value. Such case can be encountered, for example, at the plasma frequency of an electron gas (metal). In the context of doped semiconductors, recently, attention has been paid to materials operating in this regime.

The main quantity studied in this context is the Local Density of Optical States or LDOS. The latter is connected to important properties of the medium. In fact, one could say it provides spectroscopic information. The number of electromagnetic modes in an innitesimal frequency interval, the LDOS, can be obtained by the use of Green functions. Therefore, the tools described earlier readily lead to useful results and general trends concerning the theoretical description of light emission.

In the particular application of the formalism, related to ENZ media, we have found that the LDOS is strongly suppressed in a certain region above an ENZ substrate.

Determination of optical constants

The experimental determination of optical properties was reported after the theoretical developments. In particular, measurements of light transmission, light absorption and light emission were performed. In order to measure the transmission of light by plane and structured interfaces, special samples were acquired and characterized. The use of a continuous wave laser provides a source of coherent light that can be used to perform measurements in a narrow spectral region. Furthermore, a spectrophotometer, an integrating sphere and suitable acquisition instrumentation allows for spectral measurements.

The measurements of light transmission by scintillator/air interfaces were performed using a power meter. The experiment also involves directing a collimated In the context of the absorption properties, frequently one encounters the term of active optical properties. The term relates to the conversion of energy resulting from some excitation mechanism, for example a laser.

The measurements of light absorption presented in a rst stage are intended to give useful optical characteristics of bulk materials. The discussion is focused around the concept of optical transparency. In fact, recent work [START_REF] Miller | Fundamental limits to optical response in absorptive systems[END_REF] has theoretically shown that optical transparency and dielectric contrast are crucial quantities for LDOS enhancements, as well as for extinction and scattering cross-section enhancement. Therefore, from a design perspective, it is denitely relevant to address the macroscopic optical properties.

The results in [START_REF] Miller | Fundamental limits to optical response in absorptive systems[END_REF] are obtained starting from a formalism that is essentially identical to the one presented here. It is stressed that, from an electromagnetic perspective, it seems that strong dielectric contrast in a piece wise (inhomogeneous) medium limits global maxima of the electromagnetic response of photonic structures.

Together with a fairly transparent background as well as polarizable inclusions, this provides a general guideline for potential enhancements.

Eective absorption and emission dipoles

Publicly available data of the absorption and emission spectra of well-known organic uorophores was presented. Then we provided a phenomenological description of light emission in such molecules. The concept of resonance is of central importance for the model. Indeed, the validity of the approach is constrained to spectral regions where resonant phenomena dominates the interactions, i.e. close to peak absorption or emission wavelengths. Using this tool, we can estimate the onset of saturation in practical experiments, for example. It was shown that one can also characterize the absorption properties of emitters with eective molecular polarizabilities.

The organic uorophores described are currently used in plastic scintillators.

The specic mechanisms and interactions between them and the polymer matrix are discussed. Here it is emphasized that the model is relatively simple and compatible with a semi-classical approach. The latter involves a quantum formalism that describes the emitting molecule. The elds generated by the emitter are classical electromagnetic waves.

The scintillation process involves a number of relaxation processes. Moreover, the electronic dynamics in organic light-emitting molecules also involve a number of processes, not always allowing for a simple description. Far from providing a comprehensive theory, we aim at illustrating a procedure that relies on eective The precise connections between the LDOS and the Purcell factor depend on the light-matter interaction regime. For example, in the weak-coupling regime and far from saturation, it is well-known that the ratio of the LDOS of an emitter in a cavity to that of the emitter in vacuum (or a reference medium) is proportional to the modication of the radiative rate, i.e.

Γ Γ 0 = ρ ρ 0 ,
where Γ 0 and Γ are the radiative decay rate of the emitter in a homogeneous medium and in the presence of the structure, respectively. The LDOS for ρ and ρ 0 are dened in a similar manner. Saturation and the strong-coupling regime however can be described only with a more general approach.

Consider the model presented in previous chapters dealing with a three-level system as the emitter. The Purcell factor in this case can be computed through the Green function of the system. In this case the Purcell factor represents the strength of the electromagnetic response of the environment [START_REF] Bouchet | Quantum dipole emitters in structured environments: a scattering approach: tutorial[END_REF]. The coupling between the eld and the emitter can then be addressed in terms of the excitation polarizability and the Green function of the electromagnetic environment.

Strong interactions leading to non-negligible Purcell factors are expected for the congurations presented here but with the localized source or sources placed inside a cavity. As the scintillator is represented as the plastic homogeneous and transparent background and the localized sources in it, the cavity should be composed of a scintillating material. This conguration is discussed further in the perspectives.

Validity and applications of the scintillation light models

Both the theoretical developments of the present work and the experimental results dealt with situations involving stationary elds. A main feature of scintillation phenomena is the fact that light is emitted during a short period of time. This is in contrast to considering a stationary eld, i.e. those with average values that do not depend on the particular time interval over which one performs the average.

We recalled that the electromagnetic response due to a harmonic excitation is equivalent to the determination of the spectral response of a single frequency component. By the use of Fourier transforms, one can address the issue of arbitrary time dependence. In practice, this involves the determination of the Rabi frequency associated the molecule excitation. Then, the Optical Bloch equations dictate the temporal evolution provided we have the corresponding values that characterize the intrinsic decay mechanisms of the molecule. The latter can be obtained by the characterization approach discussed earlier for an organic light-emitter.

The numerical results related to light emission were presented and discussed in terms of the local electric eld. The models of interaction between light and the emitter are treated in the electric dipole approximation. Furthermore, the discussion also focuses on near-resonant excitation of the source. These important aspects are a common characteristic of the system under study: resonant energy transfer between pairs of distinct light-emitting molecules occurs in plastic scintillators.

The local electric eld due to a point-like source provides spectroscopic information of the electromagnetic environment in which the source is placed. Indeed, the local eld at the position of the source is related to the electromagnetic density of states or LDOS. In the simulations we considered wire-like particles that are embedded in a homogeneous transparent background. Here we make use of the There is a large number of phenomena we have not mentioned. For example, in the literature concerning scintillation phenomena one cannot avoid discussing topics such as Compton scattering, the photoelectric eect or pair production. This renders the choice of dielectric scatterers a rst step towards the understanding of near-eld interactions that can eventually lead to enhanced light emission in scintillators.

Perspectives

The potential applications of the results presented in this work can be separated in two categories: applications relying on the control of light propagation and those that rely on controlling light emission.

The scintillation process starts with a high-energy excitation of a material (scintillator) and ends in the emission of visible light. The overall control of the scintillation mechanism is beyond the scope of the theoretical and experimental methods presented here. Nevertheless, here we argue that towards the end of the scintillation process there exists the possibility of enhancing the light yield in scintillation.

Two mechanisms have been studied: the rst one is related to the collection of light coming from the scintillation events that occur in the scintillation (homogeneous) medium. The second aspect is closely related to the mechanism of light emission, as mentioned throughout the chapters.

We studied the propagation of scintillation light as it travels inside the scintillator and then reaches the interface, where it is partially reected and transmitted.

With the purpose of improving the extraction of light from scintillators, the response of surface structures placed on top of scintillating materials have been investigated.

Here we presented and discussed numerical and experimental results related to such scintillators with structured surfaces. From a design perspective, the bulk (macroscopic) geometry of the crystal and/or the wavelength-scale geometrical arrangement of the surface structures must be chosen based on numerical schemes. In order to search an optimal conguration, one must focus on the angular distribution of light rays arriving at the scintillator interface.

First we point out that the numerical and theoretical tools allowing for a quantitative estimation of light collection and emission presented here have potential applications that range from scintillating materials and dosimetry to organic lightemitting-diodes (OLEDs). For example, the case of the half-space, which was studied in section 3.4, becomes important in the analysis of the response of supported microand nano-structures.

In the case of improving light extraction from scintillators, the half-space model can be used as the starting point of a more complicated non-planar surface structure.

In general, for arbitrary structure geometry and composition, one can exploit numerical methods such as Finite-Dierence-Time-Domain (FDTD) or those based on nite elements. Nevertheless, for some particular systems such as a nite periodic or aperiodic array of (dielectric or metallic) resonators, the use of Green's function for a half-space could lead to more ecient calculations. Moreover, if the assumption of small scatterers is valid (dipole or Rayleigh scattering), one can benet even more from a Green's function approach. The scattering of waves above and below the structured interface would then lead to the quantication of reection and transmission properties of the interface, which are indispensable gures of merit for light extraction.

The bulk shape of the scintillator is frequently xed in a particular application.

In that case, one focus on the opto-geometrical arrangement of the surface structures. In order to optimize the structure parameters, one could compute the optical response of the interface only for the most relevant components of the (scintillation) wave front arriving at the interface. A rst step in this direction could be to compute the response of the bulk scintillator with polished or at boundaries by tracing a large number of light rays inside the bulk scintillating material. Then, the distribution of angles with which light rays arrive at the interface can be used to optimize the response of surface structures for the most relevant angles in the distribution. Repeating this process in a systematic manner should lead to optimization. The main restriction of such approach is the computational cost in nding the electromagnetic response of the structure.

In the context of light emission in solid media, the response of multilayered structures is also of interest. In particular, this is important for OLED applications, where organic molecules are responsible for the emission of light in such structures.

In this context one nds that absorption, radiation and guided modes become dom- In the last few decades and thanks to developments in micro-and nano-fabrication techniques, a common approach towards enhancing the local electric eld is to include piece wise polarizable inclusions near light emitters. These polarizable objects frequently have dimensions that are comparable or smaller than the wavelength of emission. This is most true for the eld of plasmonics, i.e. where metallic objects are placed near localized sources.

In this work we deal only with (polarizable) dielectric particles that aect the electric eld near by localized sources of light. The reason behind this choice stems from the technological solutions chosen in the frame of the DECISIoN project. In fact, one of the initial goals of the project is to achieve the Purcell eect in low refractive index media.

One may be tempted to use strong eld connement eects provided by plasmonic structures but some diculties quickly arise when making some considerations. Importantly, fabrication issues may eventually limit the performance of such devices. This will not be discussed further in the present text as it pertains to investigations that are out of the scope of this work. Nevertheless we can highlight the importance of intrinsic losses associated to real metals. As commented earlier, the potential enhancements in LDOS values are related to the dielectric contrast.

A large value is expected for the latter in the case of metals. However, as shown in the literature [START_REF] Miller | Fundamental limits to optical response in absorptive systems[END_REF], the maximum LDOS enhancement predicted by theory so far is inversely proportional to the imaginary part of the linear susceptibility (as in the induced polarization discussed in the theoretical framework).

There seems to be much work to do in order to arrive at a practical application of the electromagnetic approach presented here. However, the necessary elements for the determination of the optical response in scintillating materials were described in this work. The results provided here attempt at opening the discussion on the use of optical tools, together with ray optics and diractive elements, illustrating alternatives for light extraction and emission in scintillating media. For example, in the context of light extraction, the use of grating structures has been studied only within the last decade. One can argue there exists a need to explore improvements based on modern solutions from the optics and photonics community that can be equally exploited for technologies related to scintillation.

Chapter 7 Resumé

Un bref aperçu est étudié dans le contexte de l'émission de lumière. L'analyse est faite en termes de densité locale d'états optiques (LDOS), qui compte essentiellement le nombre de modes propres dans une plage de fréquences innitésimale à une position donnée de l'espace [START_REF] Loudon | The quantum theory of light[END_REF][START_REF] Economou | Green's functions in quantum physics[END_REF]. L'électrodynamique classique fournit un moyen de calculer la modication du LDOS à travers les champs dipolaire électrique et magnétique [START_REF] Novotny | Principles of nano-optics[END_REF][START_REF] Novotny | Single molecule uorescence in inhomogeneous environments[END_REF]. Ces types de champs sont discutés en détail et constituent la base des méthodes numériques utilisées dans cette partie. Ces derniers sont utilisés pour étudier des géométries plus complexes. Les résultats obtenus dans cette partie peuvent être utilisés pour estimer les améliorations de la uorescence dans des environnements diélectriques structurés.

Une approche générale des simulations de l'interaction molécule-rayonnement est décrite en termes d'équations Bloch optiques [START_REF] Bouchet | Quantum dipole emitters in structured environments: a scattering approach: tutorial[END_REF][START_REF] Loudon | The quantum theory of light[END_REF][START_REF] Grynberg | Introduction to quantum optics: from the semi-classical approach to quantized light[END_REF] Les mesures d'absorption de la lumière présentées dans une première étape (g.

1. 3 .Figure 1 . 1 :

 311 Figure 1.1: Examples of a) inorganic and organic b) scintillating materials. The photographs are samples whose optical properties were characterized throughout the PhD and that were acquired within the research projects involved. The top picture in a) is the crystal geometry used in PET machines. The bottom picture shows a crystal geometry allowing for optical transmission measurements. The top photograph in b) shows a variety of plastic scintillating materials. A blue/violet laser (reddish color in the picture) impinges on the right-most sample on the photograph. Blue luminescence from the sample is also visible. The bottom picture shows a plastic scintillator's blue luminescence under diuse illumination with light of wavelength λ = 405 nm.

  Figure 1.2: Normalized photon count at the at scintillator/air interface obtained by dierent ray tracing software. The angular distribution of photons for hemispherical illumination dI ∼ I 0 cos θ i dΩ for dΩ = sin θ i dθ i dφ i is also shown for comparison. The results of LITRANI and GEANT2 software were obtained by the CERN (Conseil Européen pour la Recherche Nucléaire) in the frame of the TURBOPET project.

Figure 1 . 3 :

 13 Figure 1.3: Transmittance of LYSO crystals with at and structured exit surfaces. The measurements of the at surface are compared to the Fresnel (intensity) coecients. The structured samples show non-zero transmittance values above the critical angle θ c .

Figure 1 .

 1 6 shows an example of the photoluminescence spectra acquired for the measurements.

1. 3 .Figure 1 . 4 :Figure 1 . 5 :

 31415 Figure 1.4: Schematic representation of the photoluminescence-dependent absorption measurements. Measuring several spectra for dierent positions z source determines the attenuation coecient of the homogeneous sample.

10 1. 3 .Figure 1 . 6 :

 10316 Figure 1.6: Photoluminescence spectra corresponding to 3 experiments needed to determine the quantum eciency.

1. 3 . 2 Figure 1 . 7 :

 3217 Figure 1.7: Near-eld distribution of the squared eld magnitude in the vicinity of a nano-wall with L x = 80 nm. The colour map is shown in a logarithmic scale. a) Field of the dipole source. The inclusion of the particle in this case is for illustration purposes only. b) The scattered eld as a result of the interaction of the localized source with its environment.The brightest colours correspond to values close to 10. The darker tones imply values around 0.1.

1. 3 . 2 Figure 1 . 8 :

 3218 Figure 1.8: Same as in the previous gure but in this case L x = 330 nm.

Figure 2 . 1 :

 21 Figure 2.1: Schematic illustration of a) external and b) internal illumination conditions. Light is represented with blue arrows. A scintillation event will always result in light rays impinging in internal illumination conditions.

2. 1 . 1

 11 Light extraction from light-emitting-diodes In the context of light-emission-diodes (LEDs), extensive research has been conducted towards increasing the amount of light coming from a source inside a HRI medium, quite probably a semiconductor. Some examples are rough surfaces and micro-lens arrays on top of the exit surface. LEE enhancement has also been achieved by placing a photonic crystal (PhC) slab on the substrate. A few representative examples include:

Figure 2 Figure 2

 22 Figure 2.2: a) Representative illustrations of the structures for light extraction and b) angles distribution of the rays impinging on the structured surface [43].

20 2. 1 .Figure 2 . 4 :

 20124 Figure 2.4: a) Representative illustrations of the structures for light extraction and b) enhancement of the extracted light levels [44].

Figure 2 . 5 :

 25 Figure 2.5: a) Sketch of the fabrication process for the structured surfaces and b) results for the transmission of light by the structure [45].

Figure 2 . 6 :

 26 Figure 2.6: a) Schematic of the scintillation system under study b) Results for the transmission values across the structured interface [46].

Figure 2 . 7 :

 27 Figure 2.7: Transmission by a LYSO/air interface as a function of the angle of incidence of the plane wave, two orthogonal linear polarizations and unpolarized light.

Figure 2 . 8 :

 28 Figure 2.8: a) Illustration of the ideal structure geometry. The image shown represents the truncated semi-innite medium whose substrate's structured surface is composed of conical features. b)(Left) Schematic representation of the layered structure for a crosssection of the y = 0 plane. Region (I) is the substrate below the structure, which denoted as region (II). Region (III) is air or a vacuum. The gure on the right is a comparison of the staircase approximation for the simulated prole of the structure (black solid line) with the ideal prole (grey dashed line).

Figure 2 .

 2 Figure 2.9 shows the total transmission by gratings with a triangular lattice of

Figure 2 . 9 :

 29 Figure 2.9: Transmittance as a function of the grating period and the polar angle of incidence for φ = 0 • . a) Polarization of the electric eld perpendicular to the incidence plane. b) Polarization parallel to the incidence plane.

Figure 2 . 10 :

 210 Figure 2.10: Diraction orders in reection (upper row) and transmission (lower row) as a function of the grating period and the polar angle of incidence θ i . The azimuth angle of incidence φ i = 0 • .

Figure 2 . 11 :

 211 Figure 2.11: Dependence of the transmission on a) the lling fraction and b) the height of the cone in the sub wavelength grating. In a) the period is xed at P = 180 nm and the cone's size is varied. In b), the same period is used and a lling fraction of f = 0.8 sets the basal diameter at D = 144 nm. The wavelength λ = 420 nm in both cases.

2. 2 .

 2 Results and discussion mine the directions of propagation of the diracted orders. With the abbreviations θ t (°)

Figure 2 . 12 :

 212 Figure 2.12: Polar transmission angle θ t as a function of the incident angles θ i and φ i for selected orders. The grating is illuminated with a plane wave impinging on the surface at an angle θ i = 40 • from the normal. a-d) Modes (m, n) = (-1, 0), (m, n) = (-2, 0), (m, n) = (-2, -1) and (m, n) = (-3, -1), respectively.

Figure 2 . 13 :

 213 Figure 2.13: Schematic of the simulated structures. The eect of grating thickness is investigated. The inset on the top shows the triangular lattice whose primitive vectors are L 1 = (P, 0) and L 2 = ( 1 2 P, √ 3 2 P ).

Fig. 2 .

 2 Fig.2.14a shows the normal incidence case. Here, it seems that increasing the structure thickness can lead to reduced transmission values for some periods.In this range of periods, between 420 nm and 485 nm, transmission is considerably suppressed. As far as it concerns light extraction, this eect is detrimental and stems

Figure 2 . 14 :

 214 Figure 2.14: Transmission of plane waves by the grating. Light is incident at a) normal incidence and b) oblique incidence (θ inc = 40 •). The insets show the diraction pattern for the selected periods of P = 500, 600 and 700 nm at oblique incidence. For normal incidence, besides the absolute scale, the diraction pattern does not change.

Figure 2 . 15 :

 215 Figure 2.15: Normal incidence diraction by gratings with periods 420 < P < 800 nm. a) shows the specular transmission as a function of grating period for 4 dierent pillar geometries, each one with a dierent height. Similarly, in b) and c), the diraction eciency of the rst diracted orders is presented. The inset on the right shows the diraction pattern at an arbitrary distance behind the exit surface for grating congurations with P > 485 nm. Grey vertical arrows indicate the correlation of increase or decrease in the power of the mode (m, n) with the height H of the structure.

Figure 2 .

 2 Figure 2.16: a-d) Diraction eciencies for a grating illuminated with a plane wave impinging on the surface at an angle θ i = 40 • . Grey vertical arrows indicate the correlation of increase or decrease in the power of the mode (m, n) with the height H of the structure.

Figure 2 .

 2 Figure 2.17: a-c) Enhancement and suppression in the light extraction eciency. The percentage of light gain is simply 100 × (η -1). For each set of curves the Diameter/Period ratio is indicated.

Figure 2 .Figure 2 .Figure 2 . 20 :

 22220 Figure 2.18: a) A semi-innite medium as a model for the scintillator/air interface. A source placed at a distance z from the interface represents scintillation. The latter is denoted with blue arrows. All rays propagating at angles |θ i | < π/2 eventually reach the interface. b) The cross-section of a nite size scintillator crystal. The angular range of light rays at the surface is now |θ i | ≤ ϕ due to the nite size of the crystal. The dimensions of the crystal are L top = 5 cm, L bottom = 4 cm and L depth = 1.2 cm. The upper and lower horizontal interfaces are square surfaces. c) The same cross-section now illustrates the localized light source placed at a point of lower symmetry.

Figure 2 . 1 x

 21 Figure 2.21: Ray-tracing results for the conguration shown in 2.19b. The diuse reectance of the walls is dierent in each case: a) R = 0, b) R = 0.5 and c) R = 1

Figure 2 . 22 :

 222 Figure 2.22: Simulated distribution of rays arriving at the detector. The image is a projection of the ray distribution shown in g. 2.21a.

53 3. 1 .

 531 The solution of Maxwell's equations with integral equations like nature. The red arrow coming out of and into the source represents its own emitted eld backscattered from near-by objects. Thin black, wavy lines represent contribution from scattering due to external excitation.

54 3. 1 .

 541 The solution of Maxwell's equations with integral equations Mathematically, the solution is given in the form

.8) 57 3. 1 .Figure 3 . 2 :

 57132 Figure 3.2: Geometry of the system. The laboratory frame has axes x and ẑ. The local coordinate system indicated in the gure is convenient to evaluate the boundary conditions. The vector-valued function r s depends only on one parameter t. The dipole near the object can have its eld parallel (blue line with arrow) or perpendicular (lled circle) to the incidence plane. Black dots on the surface indicate eventual discretization of the boundary prole for a numerical solution.

  3.3). If the latter is desired one could simply increase the value of q and the related values accordingly. This kind of geometry will be considered in the numerical simulation results presented at a later chapter.

Figure 3 . 3 :

 33 Figure 3.3: Discretization of the wire's surface prole. The wavelength (top) of light λ is compared to the distance between discretization points. The two-dimensional particle will be referred to as a nano-wall in future sections.

Figure 3 . 65 3. 2 .Figure 3

 36523 Figure 3.4 gives a summary of the processes leading to light emission in a plastic scintillator. A number of relaxation processes are presented in terms of a Jablonsky diagram [94, 95]. The microscopic description of the molecule requires a full quantum formalism. In this context, one frequently encounters the concept of singlet and triplet states. These quantum-mechanical states, represent systems of electrons with a given net charge. Singlet states refer to associating one spectral emission

  38) where d ab is the dipole matrix element and E(t) the electric eld at the position of the emitter. The term d ρ dt relax accounts for other relaxation channels such as spontaneous emission.

(3. 46 )

 46 Here, r s is the position of the source. We indicate the ω dependence and the superscript (+) to recall that the corresponding quantities are the Fourier transforms of the positive-frequency component, as in equation 3.42. Now, we write the Green function of the system as the sum of two contributions, i.e.

(3. 48 )

 48 In this expression, α 0 (ω) is the polarizability of the emitter in the homogeneous background. Inserting equation (3.46) into (3.48) we nd that

Figure 3 . 5 :

 35 Figure 3.5: Schematic representation of a small particle above a plane interface. The particle interacts directly with an incident plane wave and with the eld reected at the interface.

( 3

 3 .61) 3.4. A at surface as the simplest structured electromagnetic environment

1 c

 1 ) i = 10 -3

Figure 3 . 6 :

 36 Figure 3.6: Reected contribution of the normalized LDOS for dierent orders of approximation and the exact (numerical) results. a) Transparent substrate with 2 = 2.25, b) substrate with 2 = 2.25 + i10 -5 , c) substrate with 2 = 2.25 + i10 -3 and d) substrate with 2 = 2.25 + i0.1. Note the verical axes in the upper row diers from those in the lower row. The horizontal axes have equal scales.

(3. 70 )

 70 Equation 3.70 has 3 solutions, 2 of which are complex and are discarded. Choosing the real solution for k 1 z * , we obtain the normalized height (z/λ) * at which the losses in the substrate, up to l = 2, are equal to the (radiative) losses for the l = 3 correction:

3. 4 . 75 Figure 3 . 7 :

 47537 Figure 3.7: Normalized distance (in logarithmic scale) above the substrate (z/λ) * where the non-radiative losses in the substrate (the l = 0, 2 terms) are equal to the radiative losses (l = 3).

Figures 3 .

 3 6a-d present the relative error as a function of the distance above the interface for transparent and absorbing substrates. The horizontal thin line in each plot represents an error level of 10%. In gure 3.6a the results for a transparent substrate are shown. We can clearly see that in the ENF of the substrate, the third-order approximation shows relatively small discrepancies with the exact solution (e

Figure 3 . 8 :

 38 Figure 3.8: Comparisons of the relative error in the expansion of the reected contribution in the half-space Green function. a) Transparent substrate, b) substrate with 2 = 2.25 + i10 -5 , c) substrate with 2 = 2.25 + i10 -3 and d) substrate with 2 = 2.25 + i0.1. The axes arrangement is similar to that in g. 3.6.

78 4. 1 .

 781 Measuring internal transmission through a spectrophotometer with an optical ber. In some cases one aperture and one spectral lter are also used with the purpose of ltering spatial frequencies and reducing the beam's intensity. The incident beam used for the shown measurements show is polarized parallel to the plane of incidence.4.1 Measuring internal transmissionIn chapter 2 we addressed the problem of light extraction in terms of the transmission of plane-waves across at and structured interfaces. In this section we come back to this subject from an experimental perspective. In order to probe the interface in a conguration of internal illumination, one can use hemispherical or hemi-cylindrical samples. With the purpose of characterizing gures of merit related to the LEE, measurements were performed on samples with hemi-cylindrical geometry, as shown in gure 4.1a. This means we will measure the transmittance of the structure at the same azimuth angle. On the right side, g. 4.1b, we present a photograph of the real LYSO sample when the collimated laser beam impinges the surface. In the picture the angle of incident is grazing so strong reection inside the crystal can be seen.

Figure 4 .

 4 Figure 4.1: a) Schematic of the experiment designed to probe the interface. b) A photograph of the sample with a collimated laser beam entering the sample and being internally transmitted to an integrating sphere.

gures 7 . 3 .

 73 Inasmuch the incident beam impinges at the centre of exit the surface (in the plane of incidence), only normal reections are produced at the external (polished) interfaces. This provides a convenient platform to probe the interface response.

Figure 4 .

 4 Figure 4.2: a) AFM (top) and SEM (bottom) images of the structured interface of sample A. b) Similar to in a) but for sample B.

Figure 4 .Figure 4 . 3 :

 443 Figure 4.3 presents the main results of this section. We start by describing the results for the sample with polished exit surface. The agreement between the measured transmission values with the theoretical predictions indicate that the roughness of the polished sample contains features much smaller than the illumination wavelength. Further characterization on this sample is needed to quantify the lateral size

Figure 4 . 4 :

 44 Figure 4.4: A photograph of plastic scintillators with dierent bulk shapes. A blue laser (λ = 405 nm) was directed (from left to right) at the 150 microns thick sample at the right side of the picture. Blue luminescence is captured by the camera. The reddish/violet spot reected from the opaque mount is the laser source.

Figure 4 .

 4 6 presents the set of measurements for the absorption properties of the plastic scintillator. The extinction coecient κ and penetration depth δ depth are presented. The latter, related to the attenuation coecient as δ depth = 1/α, represents how far the emitted light can travel before there is a high probability of being absorbed by the homogeneous medium. On the other hand, the extinction coecient κ is the imaginary part of the refractive index. The meaning of measurements A, B and C will be explained in what follows. In gure 4.7 we present the spectra associated to the results shown in gure 4.6 for κ and α. In order to obtain this set of measurements, we have collected light for three source positions. Measurement a,b and c refer to the latter. The results presented in gure 4.6 are (point to point) ratios of the spectra presented in gure 4.7. Consider the positions z a,b of the source (see g. 4.5). In order to retrieve the attenuation coecient α from the measurements, we take two spectra I a (z a , λ) and I b (z b , λ) and evaluate eq. 4.1 for each. This yields I a I b = e -α(za-z b ) , (4.2) which renders the form of α as α = ln I a /I b z b -z a .

Figure 4 . 5 :

 45 Figure 4.5: Schematic representation of the photoluminescence-dependent absorption measurements. Measuring several spectra for dierent positions z source determines the attenuation coecient of the homogeneous sample.

4. 3 .Figure 4 . 6 :Figure 4 . 7 :

 34647 Figure 4.6: a) Measurements of the extinction coecient for a commercial plastic scintillator model EJ-200 from Eljen Technologies. The sample was a rod of 1 meter in length and 6 cm in diameter. b) The penetration depth of emitted light corresponding to the extinction coecient values shown above.

Figure 4 . 8 :

 48 Figure 4.8: Absorption and emission spectra of the main constituents of a comercial plastic scintillator: a) the polymer matrix, b) the primary uorophore and c) the secondary uorophore or wavelength-shifter. The overlap between the emission band of PPO and the absorption band of POPOP results in ecient radiative energy transfer.

Figure 4 .Figure 4

 44 Figure 4.9 shows the normalized emission spectra of PPO (g. 4.9a) and the absorption spectra for POPOP (4.9b). Note the overlap of the emission (PPO) and absorption (POPOP) spectra. The goal of these molecules is to provide energy transfer mechanisms, after ionization, towards a nal radiative transition (light emis-

4 .

 4 10b) considers the sample's indirect excitation. Here, the excitation laser does not directly strikes the sample. Instead, the laser hits the sphere's (rough) walls, which in turn scatter the incident light uniformly inside the sphere. As the sample is placed inside the sphere, we expect a fraction µ of the incoming light to be absorbed after reection of the laser beam o the sphere's walls. If the integrated laser peak has a value L a and that of the indirect excitation is L b (g. 4.10a and 4.10b, respectively), we can write L b = (1 -µ)L a .

4. 4 .Figure 4 . 10 :

 4410 Figure 4.10: Schematic representation of the absorption and emission measurements. The internal quantum eciency can be retrieved from a set of 3 dierent measurements: a) the laser source spectrum (without the sample inside the sphere), b) similar to a) but with the sample inside (with the laser striking the sphere's wall rst). In c), the laser beam is impinging directly on the sample; some fraction of the light is reected and another may or not be transmitted.

  to relate this quantity to the detected signal in the direct and indirect excitation experiments, together with the laser pump spectrum. We start by writing the detected signal in the indirect excitation case as I b = L b + S b . Similarly, the detected signal for the direct excitation case is I c = L c + S c . Then, we note that this signal should be proportional to the indirect excitation by a factor 1 -A. This means that the fraction of the incident laser beam (in direct excitation conguration)

( 4 . 15 )

 415 which can be rearranged to yield the internal quantum eciency ηη = S c -(1 -A)S b L a A .

(4. 16 )

 16 In the following section, absorption and emission spectra are shown and analyzed in terms of a simple model based on a semi-classical formalism. The light-emission

4. 5 .

 5 Optical characterization of polymer based scintillators (see gure 4.4): a 150 microns thick sample with a 15 cm diameter, a 6 mm long cylindrical sample with a 25.4 mm diameter and a 10 × 10 × 50 mm 3 bar. It is worth mentioning that the precise formulations of the scintillators are not publicly available.

Figure 4 .

 4 Figure 4.11: UV laser spectra corresponding to the 3 situations described in gure 4.10: laser spectrum for a) the laser source, b) the indirect and c) the direct excitation case. The incident, linearly polarized laser was set to a power of 20 ± 2µW. The exposure time for the CCD in the spectrophotometer was 30 s. Solid red lines correspond to the best t to a Gaussian function (as discussed in the text). The blue thin lines are the retrieved signals of the raw data after a regularization scheme (Tikhonov regularization).

mm 3

 3 bar. We rst present results for the 150 microns thick sample. The central wavelength of the laser is λ = 325 nm. The power of the incident continuous wave laser was set to 20 µW. The laser spot at the sample was approximately 1 mm in diameter. Here, the laser beam in the direct excitation case hits the sample at angles close to normal incidence (θ inc <5• ), in order to avoid that the reected beam exits the integrating sphere through the aperture. The background signal (schot or 4.5. Optical characterization of polymer based scintillators Poisson noise), where the shutter is closed, corresponds to the baseline level. The data acquisition system is congured to substract the mean value of the noise to the measurement.

1 .

 1 4.11 leads to nonphysical values, such as η > The procedure adopted here is to treat the data as corrupted by the walls at peak intensities. This implies that data points below a certain count level (±2 counts) should not be signicantly aected by the walls. In order to perform the tting, we use the data points matching the wavelengths with lower intensities, i.e. the left-and right-most sides of the peak. The ratio of the peaks is maintained and the central wavelength and linewidth are kept to similar values (≈ 1% relative difference) in each case. In 4.11b the measurement is taken with the sample inside the

Figure 4 . 12 :

 412 Figure 4.12: Photoluminescence spectra corresponding to the 3 situations described in gure 4.10 and to the previous laser spectra.

95 4. 5 .

 955 Optical characterization of polymer based scintillators η A I 0 (mW/cm

96 4. 5 . 1 Figure 4 . 13 :

 9651413 Figure 4.13: Summary of the measurements performed on EJ40 samples with several geometries. The laser peaks (left axis) and PL spectra (right axis) are shown for a) the laser spectra without the sample inside the sphere, b) the indirect sample excitation andc) the direct sample excitation case. Here, the volume of the sample V (cm 3 ), the power of the beam P 0 (in µW) and the exposure time T exp (sec) were used for normalization.

101 5. 1 .Figure 5 . 1 :

 101151 Figure 5.1: The real (top) and imaginary (bottom) part of the complex refractive index for Titanium Dioxide. The inset at the bottom is the extinction coecient κ in logarithmic scale.

Figure 5 . 3 :

 53 Figure 5.3: Near-eld distribution of the squared eld magnitude in the vicinity of a nanowall with L x = 80 nm. The coloured map is in a logarithmic scale. a) Field of the dipole source. The inclusion of the particle in this case is for illustration purposes only (the eld of the dipole in a homogeneous background in is shown for each region). b) The scattered eld as a result of the interaction of the localized source with its environment.The brightest colours correspond to values close to 10. The darker tones imply values around 0.1.

Figure 5 . 4 :

 54 Figure 5.4: Same as in the previous gure but in this case L x = 330 nm.

Figure 5 .

 5 Figure 5.5: a) Incident eld on the background medium. The inclusion of the particle in the map is for illustration purposes only. b) Scattered eld by the particle. Modest eld connement eects are visible inside the wire.

Figure 5 .

 5 Figure 5.7 presents the main results of this section. The star-like shaped particle covers an area of roughly 1 × 1 µm 2 . The inset shows the wire's surface prole in dark grey. Red dots are shown schematically in order to represent the dipole

Figure 5 . 6 :

 56 Figure 5.6: Same situation as in the previous gure but evaluated in a larger region of space.

2 Figure 5 . 8 : 1 -Figure 5 . 9 :

 258159 Figure 5.8: Near-eld distribution for a pair of identical TiO 2 NW. The separation between their centres is 500 nm. The source is placed at x 0 = -2000 nm and z 0 = -3700 nm. The source wavelength is a) λ = 432 nm and b) λ = 437 nm.

Figure 5 . 10 :Figure 5 . 11 :Figure 5 . 12 :

 510511512 Figure 5.10: Near-eld distribution for a pair of identical TiO 2 NW. The separation between their centres is 500 nm. The source is placed at x 0 = -2000 nm and z 0 = -3700 nm. The source wavelength is λ = 432 nm for both cases.

  mission of light by structured interfaces: light transmission by sub wavelength and wavelength-scale structures. Within the description of light transmission by periodically structured surfaces, important distinctions were made in the context of Positron-Emission-Tomography (PET). The rst one relates to the periodicity and to the wavelength of illumination. The latter being scintillation light in this work. The main points discussed in this part were the changes on the spatial distribution of light at the exit interface of a scintillator. The discussed numerical examples elucidate general trends concerning the diraction of light by structured surfaces. Potential applications for scintillating materials were discussed, focusing on PET.

  laser beam towards the sample. Setting the incident angle and the collection of light with the sphere may arguably be the most technically challenging aspects of the transmittance measurements. Technical aspects can be nonetheless considerably simplied by a clever choice of hardware. The latter must ensure that the sample or the laser does not moves and, in a complementary fashion, the other component rotates. A reasonable practical solution is the use of a rail-like mechanical component. The absorption properties of plastic scintillators were also determined by relatively simple experiments. Key facts about the nature of light absorption are discussed. The importance of the discussion relates to the second part of the work, namely light emission in scintillating materials.

Finally, emission and

  absorption spectra is presented in the last section of this chapter. These measurements are concerned with the determination of the internal and external quantum eciency. One can further relate quantities obtained in this experiments to physical processes associated to internal resonances of the light source. Further experimental and theoretical work should provide hints towards the validity of the model.

  quantities representing the light output in a scintillation event. In particular, with this model one could analyze experimental data in order to characterize absorption and emission dipoles of organic light-emitting molecules. The characteristics of the dipole emitters obtained within this formalism can be considered a rst step towards the description of light-matter interactions.Light emission control and the Purcell factorTowards the nal part of this work, numerical simulations related to light emission have been presented and discussed. In this calculations, the quantities of interest are the complex values of the local electric eld. The latter refers to the electromagnetic at microscopic length scales. The presented results have the purpose of compiling a multitude of aspects related to the radiation eld near polarizable structures.Here we discussed the physical situation and the possibility using the calculations to model scintillation phenomena.The Purcell factor F p , a popular gure of merit in the context of nanophotonics and light-matter interactions, represents potential enhancements in the radiative decay rate of a light-emitter. In the present work we do not present results in terms of this quantity. Instead we described the results in terms of the local eld and the number of sources in the neighbourhood of a dielectric cavity. It is nevertheless important to mention the relation of F p to the numerical results related to light emission.

  123 assumptions already discussed in previous chapters, where the transparency of a scintillator at emission wavelengths is of paramount importance. Furthermore, we have focused on aspects related to the presence of multiple localized sources near a dielectric nano-or micro-structure. The dipole sources were placed at random positions around the scatterer. Results were shown for a xed average concentration of sources embedded in the homogeneous (plastic) host.For a multitude of sources surrounding a dielectric cavity, it was shown that absorption increases with the number of localized sources, which are not necessarily in the near-eld of the nano-structure. Furthermore, in regions of low absorption, even if the sources are not inside of the cavity, we found resonances in the average response of the sources in the presence of the cavity.The fact that resonances survive the random interference between localized sources provides some indication of possible changes in uorescence for real scintillators. In fact, the random position of the sources attempts at describing a solid plastic matrix with uorescent molecules randomly distributed inside. Due to the low value of the intrinsic losses in the spectral region of the surviving resonances, we suspect that potential changes in light emission for sources in this conguration should be associated to radiative eects. Further work is needed to assess and quantify potential radiative decay enhancements.

  inant in certain frequency ranges, depending on the properties of each layer. The determination of such regimes is therefore important in the design of experiments as well as devices. With the Green's function formalism used throughout this work, one can study supported structures and extend the formalism of a half-space geometry to that of a multilayered heterostructure.One important step towards modeling more realistic structures is the use of fullwave or 3-dimensional simulations. In this work we studied 2-dimensional sources for the light emission part. Although some similarities are expected to occur in 3dimensional systems, another direction of connement is possible and thus a richer phenomenology would be observed. Quantication of such phenomena in terms of the local eld and the local density of optical states is therefore mandatory in the design of any structures.The semi-classical formalism for light emission presented in this work can of course be rened. Evidently, one could pursue a description in terms of the quantum electromagnetic eld. There are however relevant aspects that can be further investigated using the semi-classical formalism. An example is the coupling of the eld to a classical dipolar light source. Here we have presented results for the local electric eld at the source position. We can consider these values to be a starting point in the description of such coupling. In general, the widths of the resonance peaks which may arise in the scattering of light by dielectric cavities are crucial in estimating the modications in the uorescence spectra of a dipole source in a homogeneous background. Further analysis into the resonant modes excited in dielectric cavities by localized sources should lead to optimization of the dipole-cavity coupling.

Figure 7 . 1 :

 71 Figure 7.1: Exemples de scintillateurs a) inorganiques et b) organiques. Les photographies sont des échantillons dont les propriétés optiques ont été caractérisées tout au long de la thèse et qui ont été acquises dans le cadre des projets de recherche impliqués. La photo du haut en a) représente la géométrie du cristal utilisée dans les machines TEP. La photo du bas montre un cristal permettant des mesures de transmission optique. La photo du haut en b) montre une variété de matières plastiques scintillantes. Un laser bleu / violet (couleur rougeâtre sur la photo) frappe l'échantillon le plus à droite de la photo. La luminescence bleue de l'échantillon est également visible. La photo du bas montre la luminescence d'un scintillateur en plastique sous un éclairage dius avec une lumière de longueur d'onde λ = 405 nm.

Figure 7 . 2 :

 72 Figure 7.2: a) Illustration de la géométrie idéale de la structure. L'image présentée représente le milieu semi-inni tronqué dont la surface structurée du substrat est composée d'éléments coniques. b)(Gauche) Représentation schématique de la structure en couches pour une section transversale du plan y = 0. La région (I) est le substrat situé sous la structure, désigné par région (II). La région (III) est l'air ou le vide. La gure de droite compare l'approximation en escalier du prol simulé de la structure (trait noir continu) avec le prol idéal (trait pointillé gris).

Figure 7 .

 7 Figure 7.3: a) Schéma de l'expérience conçu pour sonder l'interface. b) Une photographie de l'échantillon avec un faisceau laser collimaté pénétrant dans l'échantillon et transmis vers une sphère d'intégration.

Figure 7 . 4 :

 74 Figure 7.4: Description schématique d'un système composé d'objets polarisables. La fonction diélectrique des objets (gris) est notée s , tandis que celle du milieu du fond est h . Un champ externe, illustré par un front d'onde plane (lignes rouges), peut interagir avec le système. De plus, une source de rayonnement localisée peut être présente au voisinage des objets polarisables (cercle rouge).

Figure 7 . 5 :Figure 7 . 6 :

 7576 Figure 7.5: Nombre de photons normalisé à l'interface air / scintillateur plat obtenu par diérents logiciels de traçage de rayons[START_REF]Les résultats du traçage de rayons présentés sont liés à un projet EUROSTARS[END_REF]. Distribution angulaire des photons pour l'éclairage hémisphérique dI ∼ I 0 cos θ i dΩ for dΩ = sin θ i dθ i dφ i est également présenté à des ns de comparaison.

Figure 7 . 7 :

 77 Figure 7.7: Représentation schématique des mesures d'absorption dépendant de la photoluminescence. Mesurer plusieurs spectres pour diérentes positions z source détermine le coecient d'atténuation de l'échantillon homogène.

7. 9 )Figure 7 . 8 :

 978 Figure 7.8: Spectres à partir desquels les propriétés d'absorption des scintillateurs en plastique ont été calculées.

Figure 7 . 9 :Figure 7 . 10 : 2 Figure 7 . 11 : 2 Figure 7 . 12 :

 7971027112712 Figure 7.9: a) Mesures du coecient d'extinction d'un scintillateur en plastique commercial modèle EJ-200 de Eljen Technologies. L'échantillon était une tige de 1 mètre de long et 6 cm de diamètre. b) La profondeur de pénétration de la lumière émise correspondant aux valeurs du coecient d'extinction indiquées ci-dessus.

  2.1. Light extraction eciency 2.1.2 Light extraction and anti-reection structures Anti-reection (AR) structures operating in reection, i.e. not for light extraction purposes, are also interesting for the purpose of LEE enhancement. The key point in this approach is the slow change in refractive index across the interface, leading to an impedance matching of the substrate to the outside medium. Some examples of this approach are

  • and decay exponentially in a direction perpendicular to the surface. Angles θ i and φ i for which θ t < 90 • must

	2.2. Results and discussion
	incident at grazing angles, a considerable part of the transmitted waves propagate
	with angles close to the surface normal (|θ t | < 20 • ).
	be propagating plane-waves. The angles of propagation of transmitted waves cor-
	respond to the regions of bright colours. One can note that, even if the wave is

Table 4 .

 4 1: Resonance characteristics for the internal degrees of fredoom of POPOP. The resonant wavelength is λ 0 .

	93]

, f = 4.32 × 10 -9 S abs (ν)dν.

Table 4 . 2 :

 42 2 ) Signal to Noise Ratio (dB) Sample thickness (cm)

	0.77	0.92	0.28	7 ≤ SNR ≤ 20	0.015
	0.69	0.85	21	13 ≤ SNR ≤ 23	0.015
	0.66	0.93	21	10 ≤ SNR ≤ 23	0.6
	0.56	0.95	21	9 ≤ SNR ≤ 24	1

  cependant, le mécanisme de scintillation est étudié du point de vue de l'optique. L'objectif est de fournir une image englobant les principaux éléments de la réponse optique liée au processus de scintillation. À cette n, le travail est divisé en deux grandes catégories: propagation de la lumière et émission de la lumière. Les deux catégories peuvent être considérées comme des interactions lumière-matière. sources lumineuses localisées. Ce projet restera actif au moins pendant quelques années encore. La discussion sur l'émission de lumière donnée ici tente de donner un aperçu des mécanismes qui pourraient conduire à des améliorations des matériaux scintillants.Dans la description actuelle du monde microscopique (quantique), on rencontre le concept de niveaux d'énergie. Ce concept est lié à des entités physiques telles que des électrons ou des protons, par exemple. On dit alors qu'un électron, un proton ou une autre particule se comportent de la manière générale suivante: en fonction des forces externes, des objets proches ou simplement sans raison, la particule peut migrer vers un niveau d'énergie inférieur ou supérieur. Le résultat de cette transition entre les niveaux d'énergie pourrait être un photon. Aussi mystérieux que soit ce mécanisme, il a conduit à notre compréhension actuelle des phénomènes microscopiques[START_REF] Loudon | The quantum theory of light[END_REF][START_REF] Grynberg | Introduction to quantum optics: from the semi-classical approach to quantized light[END_REF].Les propriétés d'émission de lumière dans ce type de matériaux n'ont pas encore été abordées sous l'angle de la nanophotonique, à l'exception de quelques travaux dans la littérature traitant de l'extraction améliorée de la lumière dans de tels matériaux. Une multitude d'outils liés au monde microscopique est issue de ce domaine de recherche moderne. Une approche de la manipulation de l'émission de lumière dans des milieux scintillants est le nouvel aspect du présent travail.

	Certains des outils les plus importants de la physique moderne reposent sur La compréhension actuelle: désintégration radiative la compréhension de l'interaction de la lumière avec la matière. La comprendre a permis des développements fondamentaux en science. Au c÷ur de l'interaction lumière-matière, parmi de nombreux autres phénomènes, on rencontre l'interaction et non radiative Par exemple, lorsque la transition d'énergie associée à un électron aboutit à un
	entre les équations de Maxwell, les processus atomiques tels que les transitions élec-Les méthodes théoriques en physique constituent les outils fondamentaux sur lesquels photon, on parle de transition radiative. D'autre part, si la transition n'a pas abouti
	troniques, l'émission spontanée, les états de vibration ou la diusion élastique et reposent les solutions technologiques. Au cours des derniers siècles, on pourrait à un photon, on dit qu'une transition non radiative s'est produite.
	inélastique. Ce monde microscopique est expliqué en termes de forces porteuses de dire que les corrélations entre l'expérience et la théorie ont essentiellement justié Du point de vue de la physique nucléaire, la scintillation est un mécanisme
	particules et d'ondes. Selon le phénomène auquel on s'intéresse, une description l'utilisation d'outils logiques formels trouvés en mathématiques pures. Dans le con-bien établi aboutissant à une émission de lumière. Beaucoup d'eorts ont été faits
	particulière peut être plus appropriée. texte du présent travail, les développements théoriques nécessaires se sont concrétisés pour améliorer les propriétés des matériaux scintillants. L'approche adoptée jusqu'à
	Les phénomènes de scintillation sont particulièrement intéressants pour ce travail. présent pour améliorer leur réponse à l'émission de lumière est liée aux propriétés
	chimiques et mécaniques du matériau. Des raisons sûrement importantes sont der-
	Les matériaux scintillants peuvent être à l'état solide et liquide. Ici, nous allons
	nous concentrer uniquement sur les milieux scintillants à l'état solide. En outre,
	dans les scintillateurs à l'état solide, on distingue également deux catégories: les
	scintillateurs inorganiques et organiques. Outre l'aspect organique / inorganique,
	ils possèdent des propriétés physiques qui rendent leur utilisation plus pratique pour
	des applications particulières.
	Les scintillateurs inorganiques ont généralement des mécanismes d'émission de
	Quand on acquiert une compréhension mature de la situation physique, on peut se lumière plus ecaces que leurs homologues organiques. De plus, ce sont des milieux
	concentrer sur la distinction entre les solutions possibles pour un problème donné. durs et denses capables de mieux dissiper le rayonnement énergétique détecté, c'est-
	Même après l'introduction des idées d'espace et de temps liées aux travaux Un autre projet appelé DECISIoN [7] est une collaboration en cours qui s'attaque à la De l'avis actuel, le développement des outils théoriques et la relation entre théorie et à-dire le rayonnement ionisant. Cependant, le principal inconvénient des matériaux
	d'Einstein au début du XXe siècle, les équations de Maxwell sont restées cohérentes. question de générer plus de lumière dans les cristaux de scintillation par l'utilisation expérience sont fortement motivés par le succès des travaux antérieurs. Par exemple, scintillants inorganiques est le coût élevé de la croissance de ce type de cristaux.
	Par contre, si nous examinons susamment le monde microscopique, la description de médias structurés. dans les descriptions théoriques presentées ici, une grande importance est accordée Des rapports techniques sur leurs propriétés peuvent être trouvés dans [15] pour les
	de la matière et des forces donnée par les équations de Maxwell s'eondre. Dans Les discussions menées tout au long des chapitres reètent les idées et les idées aux résultats rapportés dans [8, 9, 10, 11]. Contrairement aux développements scintillateurs inorganiques particuliers utilisés dans ce travail.
	ce cas, ce que l'on appelle maintenant l'électrodynamique quantique fournit des fournies dans le cadre des projets. Cela est particulièrement vrai pour la discussion antérieurs, toutefois, la focalisation et le contexte dans lesquels ces outils sont utilisés Les scintillateurs organiques sont également un choix populaire pour la concep-
	prévisions plus précises. Néanmoins, nous comprenons l'électrodynamique classique sur la propagation de la lumière. Dans ce contexte, il est important de mentionner dièrent considérablement. tion des détecteurs de rayonnement. À l'état solide, ces matériaux sont essentielle-
	comme une approximation puissante. En eet, cette approximation est souvent une que les collaborations menant à ce travail ont impliqué des personnes de diérents Un aspect inévitable de la situation physique est le caractère microscopique des ment des plastiques. Ce type de scintillateur bénécie d'un coût de production
	distinction formelle car sa précision permet, dans de nombreux cas, de trouver une domaines en physique et en ingénierie. Certains résultats du projet TURBOPET phénomènes physiques impliqués. L'accès et l'utilisation de ce monde microscopique relativement faible comparé aux cristaux inorganiques. Le principal inconvénient
	description satisfaisante pour un phénomène donné. sont présentés ici et discutés plus en détail. Ce projet a pris n au printemps 2018. est maintenant une expérience continue. L'utilisation de dispositifs électroniques à dans ce cas est le rendement relativement faible en lumière. Contrairement aux

Le contrôle de la propagation de la lumière a considérablement progressé au cours des derniers siècles. Alors que l'optique géométrique a permis à l'humanité de capturer notre vision du monde depuis l'Antiquité, les technologies optiques ont bénécié au cours des deux derniers siècles des eorts par la communauté des scientiques et des ingénieurs. Des exemples de progrès dans ce domaine comprennent les appareils photographiques, les microscopes optiques, les télescopes et les outils connexes qui sont maintenant utilisés de manière omniprésente à l'échelle mondiale et même extraterrestre.

Au-delà de l'optique des rayons, cependant, une grande quantité de phénomènes a été synthétisée dans les équations de Maxwell tout au long du 19ème siècle. Dans ce cadre, la lumière agit comme une onde plutôt que comme un rayon mais conduit naturellement à une optique géométrique dans les conditions appropriées. Notre compréhension du comportement de la lumière a été considérablement élargie depuis ce temps-là. Cela nous permet d'accéder à des aspects du monde autrement cachés, du moins à notre perception, comme ceux révélés par les techniques spectroscopiques.

En outre, les règles dénies par les équations de Maxwell se sont avérées être une description robuste des phénomènes électromagnétiques, qui ne se limitent pas au comportement de la lumière visible, mais incluent toutes les formes de forces électriques et magnétiques. Depuis la n du 19e siècle, de vastes eorts scientiques ont permis de comprendre la désintégration radioactive et les concepts clés de la physique des particules. Dans ce contexte, une pléthore d'outils analytiques ont trouvé une utilisation non seulement en physique, mais également en science et en ingénierie en général. Dans 127 ce travail La description physique de la scintillation est un sujet de recherche actif en physique depuis un certain temps [1, 2, 3, 4]. Un nombre important d'applications a abouti au cours du 20ème siècle. Celles-ci ont conduit à des détecteurs de rayonnement sophistiqués et à des outils théoriques complexes. Parmi les processus impliqués dans la scintillation, citons la diusion Compton, l'eet photoélectrique et la production de paires, pour ne citer que quelques exemples.

Un exemple remarquable d'utilisation de milieux scintillants est celui de la physique médicale. À cet égard, la technique d'imagerie appelée tomographie par émission de positrons (TEP) utilise les prévisions fournies par l'un des cadres théoriques les plus précis disponibles en science, à savoir la théorie quantique des champs (voir

[START_REF] Cherry | Pet: physics, instrumentation, and scanners[END_REF]

pour un aperçu). L'objectif du projet de thèse est néanmoins d'étudier la réponse optique des milieux scintillants. Cela signie que nous empruntons l'image physique fournie par la communauté de physique nucléaire et abordons le comportement des champs optiques générés dans ces matériaux.

Dans le projet de doctorat, deux projets de recherche ont été impliqués. Les projets sont liés aux sujets étudiés dans ce travail. La portée générale des projets est liée à l'amélioration des matériaux scintillants. Tout d'abord, le projet européen TURBOPET

[6]

était déjà bien avancé au moment où la thèse a commencé. Cela fournissait un cadre naturel pour traiter les problèmes de propagation de la lumière.

Concernant le projet

DECISIoN, l'objectif général d'augmentation de l'émission de lumière est poursuivi. Le début de ce projet a coïncidé avec la n du projet TURBOPET. Une approche considérablement diérente est adoptée dans cette collaboration. Ici, on est intéressé par les phénomènes microscopiques et la réponse optique des à la n du 19e siècle et au cours du 20e siècle. Après de nombreux eorts et même au détriment de vies humaines, ce que l'on appelle maintenant la physique nucléaire est devenu une science mature. En fait, ce domaine a proté, et continue de le faire, de certains des cadres théoriques les plus sophistiqués disponibles, à savoir la physique quantique. En particulier, nous considérons actuellement le modèle standard de la physique des particules comme portant des symétries fondamentales et des notions de l'univers. Au-delà des aspects fascinants et des implications philosophiques apportées par l'approche quantique, des outils concrets et précis pour la science et la technologie en ont résulté. Parmi les exemples remarquables d'application d'une telle pensée abstraite à des solutions du monde réel gurent les techniques d'imagerie médicale. Un accent particulier est mis sur la tomographie par émission de positrons (TEP), une technique d'imagerie qui repose sur des prédictions données par la physique des particules. Des médias scintillant est utilisé pour acquérir les signaux physiques issues de ces processus complexes. Dans la littérature, on rencontre des descriptions du processus de scintillation en termes de physique nucléaire (voir e.g. [4] pour une dicussion moderne). Il ne fait aucun doute que les modèles fonctionnent et que la théorie a du succès. En eet, les solutions technologiques reposent sur des descriptions précises du monde physique. l'état solide est un exemple moderne évident. Sans souligner davantage les conséquences de l'exploration de très petites régions de l'espace, nous armons que les modèles microscopiques de phénomènes physiques sont cruciaux pour l'étude des mécanismes d'émission de lumière. rière ce choix. Par exemple, un matériau scintillant doit résister à une excitation à haute énergie et des molécules possédant des propriétés d'émission de lumière pratiques sont connues depuis un certain temps. Par contre, du point de vue de la théorie électromagnétique et en particulier de l'optique physique, la manipulation du mécanisme de scintillation est largement inexplorée.

Les améliorations des propriétés des milieux scintillants n'ont pas été abordées, à notre connaissance, d'un point de vue électromagnétique. Bien entendu, il existe quelques exceptions où les propriétés de transmission des matériaux scintillants sont modiées an d'extraire le plus de lumière scintillante possible

[START_REF] Knapitsch | Photonic crystals: A novel approach to enhance the light output of scintillation based detectors[END_REF]

. cristaux inorganiques, les scintillateurs plastiques sont moins denses et ne possèdent pas les mêmes caractéristiques de résistance aux radiations que les scintillateurs inorganiques. Voir

[START_REF] Technology | Ej-240 long decay time scintillator[END_REF]

pour un rapport technique sur les propriétés physiques générales des scintillateurs plastiques d'intérêt pour ce travail.

  Le concept d'interactions électromagnétiques est d'une importance cruciale dans ce travail. Ce terme général fait référence au fait que des forces peuvent être exercées sur des objets matériels à l'aide de champs électromagnétiques. À cet égard, les phénomènes optiques bénécient de plus de deux siècles de recherche. exemple principal de ce dernier est l'émission spontanée de rayonnement électromagnétique. Par exemple, on comprend actuellement d'importants mécanismes d'excitation de molécules émettrices de lumière sur la base des champs électromagnétiques microscopiques et du nombre d'électrons ou de porteurs présents dans le système physique[START_REF] Bouchet | Quantum dipole emitters in structured environments: a scattering approach: tutorial[END_REF]. Le concept de polarisabilité électrique et les champs associés à de très petites distributions de charge sont d'une importance primordiale pour la description des interactions champ-molécule.Contrôle de la propagation de lumièreLa première partie du présent travail traite de la propagation de la lumière. Cela signie qu'il aborde les propriétés optiques passives du scintillateur. En particulier, les propriétés de transmission sont étudiées. Sur la base d'études numériques et de la caractérisation expérimentale d'échantillons, les valeurs de mérite et les considérations générales sont discutées en termes de quantité de lumière extraite du scintillateur. Les points de départ sont les coecients de réexion et de transmission (équations de Fresnel) pour une interface plane. Ces structures de sous-longueur d'onde ne produiront pas d'ordres de diraction (uniquement l'ordre de zéro ou spéculaire). En revanche, lorsque la période est susamment grande, des ordres de diraction apparaissent et la distribution spatiale de la lumière est modiée. La géométrie des particules dans le réseau a une inuence sur la distribution de puissance entre les ordres de diraction.

	Du point de vue de l'optique, on ne considère que les ondes de champ électro-
	magnétiques qui inuencent en quelque sorte le comportement des objets. Dans le
	contexte de situations rencontrées au quotidien, on dit que briller la lumière sur cer-
	tains matériaux équivaut à exciter le matériel. Une approche électromagnétique des
	phénomènes de scintillation a donc pour but de tirer parti de l'excitation optique.
	Le concept d'excitation implique immédiatement la notion d'interaction. Les inter-
	actions de la lumière avec des milieux scintillants constituent les sujets principaux
	du présent travail.
	Fréquemment, il est nécessaire de se concentrer sur un aspect particulier de
	l'interaction lumière-matière. Cela signie que les détails de certains des processus
	physiques impliqués sont intégrés dans un paramètre ecace contenant la physique
	La description de la propagation et de l'émission de lumière dans un milieu solide
	comprend des phénomènes d'onde. Le premier aborde principalement les change-
	ments de la distribution du champ après propagation d'une très grande distance
	par rapport à la longueur d'onde d'illumination. Dans ce contexte, on pense à
	l'illumination en termes de source de lumière externe. Au moins pour la première
	partie du travail, l'idée intuitive de considérer une ampoule commune ou, dans les Les résultats expérimentaux de transmission de la lumière dans des échantillons
	plats et structurés sont présentés (voir la gure 7.6). La transmission d'un faisceau temps modernes, une diode émettrice de lumière (LED) comme illumination est es-sentiellement la manière dont on vas considérer la lumière à scintillation générée à laser collimaté à travers un échantillon hémicylindrique a été mesurée en fonction Vers une approche électromagnétique des phénomènes l'intérieur d'un milieu solide inorganique ou organique. de l'angle d'incidence du faisceau. Un laser bleu est utilisé an d'éviter la photo-de scintillation Au-delà de la précédente image intuitive, de nouveaux concepts apparaissent luminescence (PL). La longueur d'onde de fonctionnement est de 405 nm, proche

La théorie électromagnétique est maintenant une référence importante pour tous les domaines de la physique. Il englobe une grande quantité de phénomènes et les applications de la théorie ont permis de créer des outils indispensables au monde moderne.

essentielle. Par exemple, les formalismes "toute matière" tentent de décrire les phénomènes en termes de degrés de liberté atomiques, tandis que les processus radiatifs se condensent en durées de vies et en déplacement de Lamb des niveaux atomiques

[START_REF] Vries | Point scatterers for classical waves[END_REF]

. D'autre part, les représentations "tout lumière" traitent de la réponse électromagnétique du système et d'une description ecace des processus atomiques au moyen d'une polarisabilité atomique. Évidemment, l'utilité de chaque approche dépend des détails expérimentaux et / ou théoriques d'un problème donné.

Les recherches sur la diusion de particules isolées, voire la détection de molécules uniques, ont permis de mettre au point des outils sophistiqués impliquant la manipulation de la lumière. À cet égard, les travaux actuels présentent des similitudes dans les systèmes physiques étudiés. Vers la n de la thèse, nous nous concentrerons sur les eets en champ proche ainsi que sur le régime mésoscopique. Ce dernier désigne des phénomènes se produisant à des échelles de longueurs caractéristiques pour lesquelles des composantes du champ lumineux se propageant et évanescentes interagissent avec des objets proches. Un sujet étroitement lié est la diusion multiple des ondes. Bien que parmi les outils théoriques, nous aborderons une approche quantique décrivant la source de lumière, les eets d'interférence associés au champ électromagnétique sont de nature classique. lorsque l'on traite de la lumière à l'échelle de la longueur d'onde. Un cadre général doit ajouter des champs évanescents à la description. Bien que la discussion actuelle sur la nature d'onde de la lumière soit strictement basée sur des bases de la physique classique, il est néanmoins pertinent dans le contexte de phénomènes purement quantiques. L'Nous considérons que le processus de scintillation fournit une source de lumière diuse. Avec cette information, on peut estimer la quantité de lumière piégée dans un cristal en vrac avec une surface de sortie polie. Bien qu'un événement de scintillation puisse être considéré comme produisant une source de lumière localisée, dans ce travail, on vas concentrés sur les quantités moyennes représentant le matériau en vrac. De cette manière, nous évitons de décrire la source de lumière pour un événement de scintillation particulier. Par exemple, il est possible de suivre les rayons lumineux dans un scintillateur pour simuler la fraction de lumière rééchie et transmise sur des scintillateurs macroscopiques (échelle centimétrique). En répétant cette procédure, on peut trouver la distribution des angles avec lesquels les rayons lumineux atteignent une limite du cristal. Des résultats de ce type ont été obtenus dans ce travail et sont présentés dans la gure 7.5.

Des simulations numériques de réseaux bidimensionnels sont présentées et décrites en fonction de leurs propriétés de transmission. La distribution spatiale de la lumière rééchie et transmise est également abordée. À cet égard, des structures de périodes inférieures aux longueurs d'onde d'émission (sous-longueur d'onde) sont étudiées.

de la longueur d'onde maximale d'émission de 420 nm. Les résultats montrent que, au-dessus de l'angle critique, les valeurs de transmission de l'ordre de 10 pour cent peuvent être obtenus pour des structures de diraction d'ordre inférieur. Selon l'application spécique, la distinction entre la lumière transmise spécu-

  lairement et les ordres de diraction peut être importante ou non. Par exemple, des dispositifs reposant essentiellement sur la détection des niveaux de lumière globaux pourraient bénécier de structures fortement diractantes. Eectivement, à incidence oblique, une réexion interne totale est partiellement évitée par les ordres transmis. D'autre part, si la distribution de la lumière transmise est nécessaire pour un traitement ultérieur, les ordres diractés pourraient avoir des eets néfastes. Par exemple, dans le contexte du PET, la récupération de la position des événements de scintillation est cruciale pour le rendu d'images utiles. Ici, les radionucléides sont utilisés pour marquer les composés qui seront distribués dans un corps vivant. Cette distribution du radionucléide dans le corps dépend de ses propriétés biochimiques spéciques. Le radionucléide est a priori une source de positrons servant de marques aux molécules d'intérêt. Idéalement, l'interaction du positron avec son environnement entraîne une annihilation positron-électron, qui produit à son tour une paire de photons gamma. Ce dernier doit en quelque sorte exciter une paire de scintillateurs pour produire et éventuellement détecter la lumière visible.

	un grand nombre d'événements de scintillation peut être décrite comme un front
	d'onde diuse.
	Contrôle de l'émission de lumière
	La deuxième partie de ce travail concerne l'émission de lumière dans les cristaux de
	scintillation. Dans cette partie, les propriétés actives du matériau sont étudiées. Ces
	propriétés sont liées aux degrés de liberté internes du matériau. Les propriétés ac-
	tives des scintillateurs organiques et inorganiques sont brièvement décrites dans les
	sections suivantes. Dans cette partie, l'accent est mis sur les scintillateurs plastique,

L'analyse quantitative et la cartographie des concentrations de traceurs dans le corps sont alors possibles avec des congurations appropriées de scintillateurs

[START_REF] Cherry | Pet: physics, instrumentation, and scanners[END_REF]

. Un modèle réaliste devrait également inclure le phénomène de capture d'électrons. Cela peut être évité dans le PET en choisissant de manière appropriée le radionucléide

[START_REF] Qaim | Pet radionuclide production[END_REF]

. Les scintillateurs doivent alors détecter, de manière synchronisée, les photons visibles générés par l'interaction de photons de haute énergie avec le matériau. Le fait que les mesures de coïncidence sont utilisés en PET provient du processus d'annihilation électron-positron: quand l'annihilation a lieu, deux photons gamma sont émis dans des directions opposées, 180 degrés d'intervalle. En résumé, dans les scanners PET, le signal détecté repose nalement sur la lumière visible. Le processus par lequel la lumière visible est générée peut être retracé jusqu'à l'annihilation d'un positron et d'un électron. À son tour, le positron peut être attribué à des atomes spéciques liés à des molécules d'intérêt. Du point de vue optique, on peut condenser ce mécanisme complexe en une source de lumière diuse ecace. De toute évidence, un événement d'annihilation donné produira des photons gamma pouvant interagir avec le milieu scintillant et produire une certaine quantité de photons visibles. Nous ne traitons pas une telle situation. Au lieu de cela, on est concentré sur les quantités moyennes. Par exemple, on supposera ici que la lumière produite dans qui possèdent des mécanismes d'émission ecaces. Une approche phénoménologique est proposée. Les principaux éléments de l'approche concernent les systèmes à trois niveaux décrivant les émetteurs (uorophores organiques) et les interactions électromagnétiques entre l'émetteur et le fond diélectrique, éventuellement structuré (voir la gure 7.4). L'approche semi-classique du modèle proposé repose sur les champs électromagnétiques classiques et les niveaux d'énergie et leurs populations quantiées correspondantes. L'interaction entre le rayonnement visible et les molécules uorescentes est traitée dans l'approximation dipolaire. Les résultats de caractérisation expérimentale sont présentés, son objectif principal est d'illustrer l'approche théorique décrivant les principales propriétés optiques actives de la scintillation dans des matériaux à base de polymères. Les échantillons caractérisés sont des plastiques en vrac de diérentes géométries et surfaces polies.

2.1. Light extraction eciency

as well as on the eld solutions. In general, RCWA provides the solution of the electromagnetic problem in a spatial-frequency domain. This implies that the structure is not described by the distribution of the electric permittivity (r), where r = (x, y, z) are coordinates in real space. Instead, it is described with the associated Fourier coecients. The relation of the geometry of the system with the solution is then dierent, for example, from that in a nite-element method or an integral solution. In the latter, the coordinates are specied a priori and eective, localized sources in real space provide the solution.

2.2. Results and discussion

2.3. Modeling light propagation at dierent length scales

3.2. Radiative and non-radiative decay in polymer scintillators

Acknowledgements I would like to acknowledge the people who has been involved in this work and provided useful advice. I must rst acknowledge the kind welcome provided by the people at Université de Technologie de Troyes. The present work would not have been possible without the support, guidance and thrust provided by Dr. Gilles Lerondel and Dr. Alexandre Vial. The making of this PhD thesis took place during periods of time that coincided with active research projects.

I would like to thank all the people involved in the projects and to every person that contributed in some way to the realization of this thesis. This work has beneted from nancial support from the former Champagne-

Ardenne region, in France.

There is of course many people involved in the projects and therefore share the credit for any contribution eventually resulting from this work.

polarisation induite présentée dans le cadre théorique).

Il semble y avoir beaucoup de travail à faire pour parvenir à une application pratique de l'approche électromagnétique présentée ici. Cependant, les éléments nécessaires à la détermination de la réponse optique dans les matériaux scintillants ont été décrits dans ce travail. Les résultats fournis ici tentent d'ouvrir la discussion sur l'utilisation d'outils optiques au-delà de l'optique des rayons. L'utilisation de structures en réseau n'a été étudiée qu'au cours de la dernière décennie. On peut alors soutenir qu'il existe un besoin d'explorer des améliorations basées sur des solutions modernes de la communauté optique et photonique qui peuvent être également exploitées pour les technologies liées à la scintillation.

Appendices

Appendix A Components of the short-distance expansion coecients K (l) The diagonal elements of K (l) for the orders l = 0, 1, 2 are given by

The third order, K (3) , is given by

Appendix B Determination of z *

The only real solution of

is given by

where

Bibliography

[25] A. K. González-Alcalde, E. R. Méndez, E. Terán, F. L. Cuppo, J. Olivares, and A. García-Valenzuela, Reection of diuse light from dielectric onedimensional rough surfaces, JOSA A, vol. 33, no. 3, pp. 373382, 2016.

[26] F. Stern, Transmission of isotropic radiation across an interface between two dielectrics, Applied Optics, vol. 3, no. 1, pp. 111113, 1964.

[27] Here the Fresnel equations refer to the intensity ratio between incident and reected power per unit surface. For unpolarized incident light and external illumination, they can be written as

In this notation Snell's law is given by sin θ i = n sin θ t .

[28] This factor appears in several works dealing with isotropic transmission of light by at surfaces. For example, as one sees from equation ( 3) in [START_REF] Allen | Transmission of isotropic light across a dielectric surface in two and three dimensions[END_REF], dΩ takes the form dΩ = 2π sin θdθ after integration over the azimuth φ angles.