Keywords: microscopie multiphotonique, la génération de seconde harmonique, polarimétrie non linéaire, particules quantiques semi-conducteurs, confinement quantique, hétérostructures hybrides multiphoton microscopy, second-harmonic generation, nonlinear polarimetry, semiconducting quantum dots, quantum confinement, hybrid heterostructures

conditions, le χ (2) de structures à base de semi-conducteurs en mode de confinement quantique peut très largement dépasser sa valeur en milieu massif. De plus, un nouveau type de PQs hybridant des semi-conducteurs en géométries de type "bâtonnet sur sphère" (BS) a été développé sur la base de composantes cristallines de symétries différentes, afin d'augmenter leur non-linéarité quadratique effective, tout en maintenant leur taille dans un régime proche d'un fort confinement quantique. Le nouveau tenseur hybride complexe χ (2) est analysé en terme d'interférence des susceptibilités constitutives, en prenant en compte les différentes formes et symétries associées aux composantes octupolaires et dipolaires. Il en résulte pour de telles structures une exaltation significative du χ (2) , qui excède celle des PQs à constituant unique compte tenu du couplage entre matériaux non-linéaires et d'un temps de décohérence plus long, que nous attribuons à un effet de separation de charge photo-induit.

v

Résumé en français

Les phénomènes de diffusion cohérente non-linéaire ont été récemment proposés en alternatives à la fluorescence comme processus de marquage en microscopie multiphotonique. Les matériaux couramment appliqués dans ce contexte buttent toutefois sur une limite inférieure en taille déterminée par le seuil de détection de signaux faibles en optique non-linéaire. Aucun des efforts récents en détection en génération de secondharmonique (GSH), qui est le processus non-linéaire d'ordre le plus bas, n'a permis de descendre à ce jour au-dessous d'une barrière en taille de 40 nm même en ayant recours aux techniques de détection les plus sensibles telles que le comptage de photons uniques. Les nanoparticules (NPs) restent ainsi dans la famille des nano-diffuseurs de "grande" taille.

Il apparaît toutefois possible de déplacer de façon significative cette limite inférieure vers les plus petites tailles en substituant aux isolants diélectriques ou aux semi-conducteurs à grands gaps des particules quantiques (PQs) à base de semi-conducteurs à gaps directs.

Dans ce travail, un nouveau type de nanosondes hautement non-linéaires a été conçu et développé de façon à franchir cette barrière de taille minimale pour atteindre l'échelle de nanoparticules uniques. Nous considérons ainsi l'excitation résonnante à deux photons de nanoparticules quantiques individuelles à base de CdTe (de la famille des "zinc-blendes") d'un diamètre d'environ 12.5nm, qui fournissent une émission cohérente efficace par GSH jusqu'à hauteur de 10 5 comptages de photons par seconde. Elles présentent de plus l'avantage d'une remarquable sensibilité à l'orientation de leur réseau cristallin octupolaire.

De plus, il a été démontré que les effets de confinement quantique déterminent fortement les caractéristiques de la susceptibilité non-linéaire du second-ordre χ (2) . La caractérisation quantitative du χ (2) des PQs, en particulier leur dispersion spectrale et leur dépendance en taille est menée par spectroscopie de particules uniques ainsi qu'en moyenne d'ensemble par diffusion Hyper-Rayleigh (HRS). Nous fournissons en particulier la preuve que sous certaines

Introduction

Recent achievements in fabrication and characterization of inorganic nanoparticles of high optical quality have inspired new areas of research into the fundamental properties and applications of these nanoscale materials. Basically divided into metals, semiconductors and insulators, morphologically and geometrically tailored nanoscale systems are driven in order to selectively combine desirable properties at the sub-wavelength regime towards optical probing. This approach becomes particularly fascinating and challenging when is put on nonlinear optical effects occurring with small nanoparticle assemblies of typical size below 20 nm. At this scale, the electronic properties of such small nanostructures lie at an intermediate distance between those of a single atom and those of bulk materials. In consequence, this scale provides an inspiring avenue onto the molecular regime for physical properties better known at the bulk level, possibly leading to novel and remarkable responses, depending on size, composition and shape of the nanocrystals.

Applied nanoscale nonlinear optics has reached a certain limit in size dependent detection efficiency of weak signals in nonlinear processes, e.g. in our case the lowest-order nonlinear scattering which is known as second-harmonic generation (SHG). Recent efforts in detection of nonlinear scattering from noncentrosymmetric nanocrystals, deriving from the high nonlinearities of their well known bulk counterparts such as KTiOPO 4 (Le Xuan et al. [1]) or BaTiO 3 (Hsieh et al. [2]), have shown photostable and efficient SHG emission with a high damage threshold for these nanoparticles. None of these efforts, however, has been able to pass a ~40 nm size barrier, thus remaining at the level of large nanoscatterers while resorting to the most sensitive available technologies for SHG detection. As we realize now, this barrier can be overcome when replacing dielectric insulators by direct-gap semiconducting materials. SHG nanocavity-like enhancement in ~80 nm large BaTiO 3 nanoparticles covered by a ~10 nm thick gold shell. In addition, interesting approach of synergistic SHG coupling of gold nanorods into a semiconductor QD cores (Shaviv and Banin [7]) are worth attention. At a more basic level, a novel approach for SHG obtained from purely semiconducting materials can attract a significant interest. As a proof-of-principle experiment, composite rod-on-dot hybrids built from different crystalline symmetry moieties were recently reported (Zielinski et al. [8]), as capable to further improve initially reported results. Their high quadratic nonlinearity was precisely engineered following the blueprint of a pointwise sum of the susceptibilities for each of the components. The nonlinear susceptibility of such hybrid leads to a significant SHG enhancement assigned to slower decoherence and improved spatial charge separation upon excitation, which exceeds that of 'mono-compound' material QDs. This thesis is organized as follows: Chapter I provides an introduction to the downscaling of semiconductors from bulk to the nanoscale. Quantum confinement effects in nanoparticles of different dimensionality are explained using a simple 'spherical box' approximation. These fundamental properties are further investigated towards a better understanding of the electronic band structure as well as optical interactions in QDs. Chapter II deals with fabrication and characterization of suitable materials for second-order nonlinear microscopy, this imaging method being then reviewed in Chapter III. Coherent SHG emission from isolated mono-compound CdTe QDs and hybrid rod-on-dot semiconductor heterostructures is investigated in Chapter IV. These studies focus on spectral features of the second-order nonlinear optical susceptibility of both types of materials. Strong orientational dependence of the SHG tensorial features is pointed out, allowing to infer the spatial orientation of each single nanocrystal. Chapter V addresses single nanoparticle spectroscopy and HRS studies of the size dependent nonlinear susceptibility of CdTe QDs. These studies, together with spectral features reported in the preceding chapter, demonstrate the superiority of QDs over the bulk in the context of second-order nonlinear properties. In the last part of the thesis we provide details about the simulation algorithms used in this work. Finally, this work is summarized and final conclusions and future prospects are drawn. The case of nanoscale semiconductors illustrates the understanding of material properties as a function of size and composition. In this chapter the key features of quantum confined semiconducting structures are briefly introduced, following the literature. Our interest here is to point out the originality of semiconducting nanocrystals (SC NCs) in the context of their optical features. Firstly, we start by explaining the quantum confinement effect, using a simple model, and show how SC properties evolve while crystal dimensionality decreases, starting from the bulk all the way to a single quantum dot. We point out how the NC shape and its material composition translate onto the optical properties, and explain optical excitation mechanisms in relation with absorption and scattering processes. Finally the electronic band structure of bulk CdTe is discussed and compared to that of a single CdTe QDs.

I.1 Shrinking dimensionality of the bulk into the quantum dot scale

The properties of quantum confined SC NCs place these materials between isolated atoms (or small molecules) and a bulk phase of condensed matter. [START_REF] Brus | Quantum crystallites and nonlinear optics[END_REF] Like in any material, fundamental electrical and optical features of SCs vary when the size is reduced and the electronic energy level spacing exceeds different energies characterizing the structural properties, e.g. the temperature of the surrounding environment (kT), the binding energy or the exciton Rydberg energy. Here we show that this size dependence becomes extremely significant when the crystal is shrunk into a quantum confined structure. Semiconductor crystals, where the Fermi level lies between two bands (Figure I-1) exhibit completely different behavior from that of metals or insulators. The band edges dominate the low-energy optical and electrical interactions, so optical excitations across the band gap (BG) depend strongly on cluster size variations (for clusters starting from approximately 10,000 of atoms). This leads also to strong size dependence of electrical carriage transport through the crystalline volume due to the energy variations required for charge appearance or withdrawal on the NC.

I.1.1 Quantum Confinement

The first attempts in understanding the quantum mechanics of confined SC NCs were considered nearly three decades ago, resulting in a description of diffraction by the charges (electrons and holes) of the periodic lattice potential in the semiconductor NC. [START_REF] Brus | A simple model for the ionization potential, electron affinity, and aqueous redox potentials of small semiconductor crystallites[END_REF][START_REF] Brus | Electronic wave functions in semiconductor clusters: experiment and theory[END_REF][START_REF] Alivisatos | Electronic states of semiconductor clusters: Homogeneous and inhomogeneous broadening of the optical spectrum[END_REF] A key feature of SC materials is that charge diffraction leads to a characteristic band structure, Comparison of the energy levels of conduction and valence bands as well as band gaps for different bulk semiconducting materials (II-VI groups SC compounds). [START_REF] Kittel | Introduction to Soild State Physics[END_REF][START_REF] Wei | Calculated natural band offsets of all II-VI and III-V semiconductors: Chemical trends and the role of cation d orbitals[END_REF] where allowed electronic energy states exhibit the form of broad valence and conduction bands (VB and CB, respectively), resulting from a complex set of anisotropic relations between energy and momentum. [START_REF] Kittel | Introduction to Soild State Physics[END_REF] At the macroscopic scale, the bands are continuous, and the energy band gap (E BG ) between them is a fixed parameter, determined by the crystalline material composition (Figure I-2). However, decrease in dimensionality of the SC heterostructure, where at least one of the dimensions (L x(y,z) , see Figure I-3) is under quantum confinement regime results in more confined electronic excitations due to the interaction between the nanoparticle (NP) boundaries (with energies between the orbital spacing) and confined electrons. Decrease in the NC size also increases the energy band gap, as confirmed by the blue-shifted luminescence. In direct-gap size confined SC NCs (such as II-VI SCs), absorption of a photon above the BG usually leads to generation of a single, Coulomb attracted electron-hole pair (an exciton, confined by the material boundaries) and its quick relaxation, with simultaneous reemission of a new photon, which has an energy exactly matching the BG. For example, in CdSe QDs the shift of energy gap varies from 1.2 eV up to 3.0 eV (while its bulk BG energy is equal 1.71 eV). 16 This effect, however, can be controlled also by composition of the material, e.g. different size of the CdSe and CdTe QDs span the visible (VIS) spectral region, [17][18][19][20][21] whereas PbS QDs traverse from VIS to the near infrared (NIR), 22 while materials like ZnSe exhibit efficient luminescence in the ultraviolet (UV). 23 In the first approximation, quantum confinement can be described using a simple 'spherical box' model, [START_REF] Brus | A simple model for the ionization potential, electron affinity, and aqueous redox potentials of small semiconductor crystallites[END_REF][START_REF] Brus | Electronic wave functions in semiconductor clusters: experiment and theory[END_REF]24 where the SC NC is embedded inside an insulating matrix, as illustrated in Figure I-3(d). The exciton resides in the infinite potential well, which leads to a quantized valence and conduction band structures exhibiting ladder-like discrete energy levels for both hole and electron.

In other words, for an ideal SC NC, quantum confinement occurs when the width of the infinite potential approaches the exciton Bohr radius of the material

( ) ( ) ( ) ( ) 2 , , 1 , , , 2 ( )
n n n k n e h U E m z ψ ψ ψ -∇ ∇ + = k r r k r k r ℏ ɶ (I.1)
where the infinite potential ( ) ( ) ( ) and L y directions lead to quantum rods (QRs) and QDs. Therefore the quantum energies of the n-th confined states inside the infinite potential barrier can be expressed as:

U V U z = + r
( )

2 2 2 2 , 2 2 , 1 2 y z y z n n e h y z n n E D m L L π   = +       ℏ (I.5) ( ) 2 2 2 2 2 , , 2 2 2
, 0 2

x y z y x z n n n e h x y z n n n E D m L L L π   = + +       ℏ (I.6)
These expressions lead to a simple common conclusion: the energy of an extra electron in a tiny semiconducting crystallite structure amounts to the conduction band energy (E c ) plus the quantum localization energy of a 'pseudo electron' of effective mass e m (treated as a pseudo particle), which has no direct relationship with the real electron mass, and often displays a smaller value. [START_REF] Brus | Electronic wave functions in semiconductor clusters: experiment and theory[END_REF] The wave function A detailed discussion how the SC NC shape impacts on the optical properties can be found for example in an excellent review by Scholes. 26 Here we explain only how the shape influences the major DOS features, which are directly related to the SC NC optical properties.

In our considerations the most important features are that of QRs and QDs. In the case of the QRs, the DOS is strongly peaked and decays as 1/ E (Figure I-3(c)). Heavy and light hole sub-bands are mixed, leading to coupling between the energy levels of the VB. 27 On the contrary, QDs exhibit atypical Dirac ( ) E δ function like behavior for 3D confined DOS, which allows for accommodation of only two carriers of opposite spins on a single quantized energy level, therefore called "artificial atoms". The first approximation model of the parabolic band, however, does not fully reflect the real complexity of the QD DOS, because there are no SCs exhibiting such simple parabolic conduction and valence bands. [START_REF] Brus | A simple model for the ionization potential, electron affinity, and aqueous redox potentials of small semiconductor crystallites[END_REF] Therefore, more realistic approaches have to be considered, e.g. the multiband effective mass approximation, 16,28 which takes into account degenerated sub-bands, dispersion of light and heavy hole branches for non-zero k, and spin-orbit splitting of the electron sub-bands. The multiband effective mass approximation is more efficient in quantitative description of the Shrinking dimensionality of the bulk into the quantum dot scale | 13 optical properties of SC NCs than the parabolic approximation, and can be used in order to describe the transitions between the quantum size levels (QSLs), and to calculate their energies. 16,28 Figure For tight-bounded semiconducting NCs, the frequency dependent rate of a spontaneous transition from an excited electron-hole state j to the ground state 0 can be determined from Fermi's 'golden rule' namely: 29 where e is the electron charge, 0 ε is the permittivity of vacuum, m is the electron rest mass, . 30 Note, that this dependency applies to both absorption, and scattering processes. The size dependence of j Γ is determined by that of the matrix element modulus o j p and the emission frequency j ω . Assuming, that the exciton states is located near the band extrema, its wave function be expressed as a product of the Bloch function ( ) u r and an 'envelope function' ( ) φ r , and the matrix element of the momentum for an allowed transition is given by CB VB u u p

. [29][30] Therefore the interband transitions are largely determined by crystalline lattice defined Bloch functions, and the CB VB u u p term does not depend on the NC size. The decay rate of an ideal two-level exciton is expected to be proportional to the emission frequency, e.g.

( ) ( ) ( ) ( ) 2 2
, , , 2 (I.15) where: e h exciton m m M + = is the exciton translation mass. The remaining dispersion of the exciton is expressed by its kinetic energy:

X B e h U E m m φ φ φ - ∇ + = + ℏ ̻ ̻ ̻ ̻ ̻ ̻ e h e h e h
2 2 / 2 kin exciton E K M = ℏ
(with K as the excitation vector). A similar rules holds for the QRs, therefore their shape has no impact on the binding energy. For both, 1D and 0D confined structures it is only the strength of the confinement which determines B E .

The strength of the quantum confinement is related to the Coulomb energy of the electronhole interaction, which is of the order of 2 / e a ε . As the size is downscaled, the quantization energy increases as 2 1 / a , therefore the Coulomb energy, which scales as 1/ a becomes only a small correction factor. 16,21 On the contrary, for larger SC NCs the Coulomb electron-hole interaction may surpass the quantization energy. These interactions define the quantum confinement regime, which can be characterized by the ratio between the NC radius, a, and the Bohr radius B a of the bulk exciton. From there on, three different quantum confinement regimes can be distinguished: 24,32 'Weak confinement regime', where B a a ≫ , and the binding energy of an exciton, exc E , is larger than the quantization energy of the electrons and holes. In first approximation, the exciton ground state energy is then expressed as: 32 Coulomb interactions are surpassed by the quantization energy of transitions, so the reduction of the transition energies is relatively small. Interband transitions between quantum-size energy levels of the parabolic VB and CB are governed by selection rules, which limit allowed transitions to levels with the same quantum numbers.

I.1.2 Band structure of CdTe

The band structures of typical II-VI group bulk semiconductors have been well researched using, e.g. 'semi-ab initio' approach. 33 Early efforts on measurements and assignments of the size dependent optical spectrum in QDs were constrained by difficulties in preparation of weakly dispersed, high quality colloidal QDs, with well maintained control over the size dispersion. Successful assignment of the lowest quantized energy levels in many SC QDs could be obtained from a single-band effective mass model. 34 Assignment of the higher excited quantized levels remained difficult due to, e.g. valence-band degeneracy in zinc-blende CdTe QDs, 35 where the conduction band is made of s orbitals of cadmium and its valence band is composed of p orbitals of tellurium.

Following the work of Masumoto et al. [36], the orbital angular momentum in CdTe QDs is mixed with spin angular momentum, and the valence band is split into the topmost 3 2 J = band and the split-off 1 2 J = band. Quantum confined hole envelope function has an additional angular momentum L, therefore possible L -J coupling may lead to a complicated energy level structure for the VB. Nevertheless, CdTe has a less complicated VB than wurtzite CdS and CdSe QDs, and exhibits a larger spin orbit splitting of 0.927 eV and a

smaller BG E = 1.44 eV, therefore the split-off band is expected to mix weakly with the topmost VB.

Since advances in colloidal synthesis techniques allows to obtain NCs of a sufficient quality, e.g. CdSe and CdTe QDs, assignments of multiple numbers of states using sizedependent photoluminescence excitation (PLE) spectroscopy at cryogenic temperatures close to 0 K were carried out with some success. , and (label b')

3/ 2 3 1 e SDD S →
, calculated using the data from

Ref.

[35], assuming that the lowest burned structure correspond to the transition

I.2 Optical interactions

Both linear and nonlinear optical properties in SC QDs result from the transitions between the electron and hole QSLs. In linear process, as recalled, the absorption of an electron from the VB into the CB can be triggered by the absorption of a photon with energy above the BG. In general, direct semiconductors transitions involve electrons and holes of equal momentum, for which the spectral distribution of possible transitions is described by the optical DOS: 37

( ) ( ) ( ) ( ) 2 , DOS e h e h O M E E d ω δ ω = - - ∑ ∫ k k k ℏ ℏ (I.17)
The Dirac δ function ensures energy conservation, with summation over a system dimensionality dependent k vector. The optical matrix element can be calculated for a dipole approximation from the wave functions (I.7) -(I.8), and takes the following form:

( ) ( )

s s j s j e h e h e h m m m m m M u u ψ ψ φ φ = ⋅ = ⋅ ⋅ ∑ p r r p ε ε (I.18)
The oscillator strength is directly proportional to the overlap of the 'envelope functions' of the recombining excitons, which prevents transitions between levels of opposite parity. The second term on the right-hand side of the expression determines the product of the polarization vector ε and the momentum operator p between the Bloch functions (known as the Kane matrix element). It imposes additional selection rules on the polarization of the absorbed/emitted photon in interband transitions, which is determined by the heavy or light character of the hole.

For the first approximation, the absorption spectra (electron-hole interactions) under strong quantum confinement is given by: [START_REF] Brus | A simple model for the ionization potential, electron affinity, and aqueous redox potentials of small semiconductor crystallites[END_REF][START_REF] Brus | Electronic wave functions in semiconductor clusters: experiment and theory[END_REF]16 ( ) ( )

2 2 2 2 2 1.8 1.8 2(
) = ) is given by:

h e BG BG e h e e E E a E a E a m m a a ν ν ν ν ν π ω ε ε = + + - = + - + ℏ ℏ (I.19)
2 B a E a π   ∆ =     R (I.20)
where:

2 2 / 2 r B h m a = R
is the exciton Rydberg energy, dependent on the reduced mass of the exciton.

Radiative relaxation of the exciton exhibits nearly single-exponential character of the emission decay curves. At a given size dependent emission frequency, all the QDs are therefore expected to emit the light at the same rate. The emission decay rates increase with the frequency. Considering the complex structure of CdTe VB, where several hole levels are located close to the band edge, leads to exciton states lying close to the lowest excitation energies. These excitons exhibit much lower probability for spontaneous decay to the ground state, and a high rate of exchange with the lowest energy exciton. The population of the lowest energy exciton is then reduced due to thermal distribution, leading to a lowered rate of the radiative recombination and increased rate for dark excitons. 20,29

I.2.1 Scattering

Non-resonant excitation, where the energy of exciting photon does not match k = 0 transitions, can leads to generation of an exciton, with an excess kinetic energy: Before it recombines such an exciton loses all or part of this kinetic energy, then relaxes towards the band edge. Relaxation of a non-resonant exciton leads to inelastic scattering, where the kinetic energy is not conserved. In zinc-blende CdTe nanocrystals excess energy of the exciton is carried out by longitudinal optical phonons (LO). 38 The energy level structure of CdTe is such that the electrons are completely delocalized in the CB, and holes are localized in the VB, where the level spacing is very small, because an effective mass of the hole is much larger than the effective mass of the electron ( h e m m ≫ ), the relaxation of the hole leads to its coupling with 1 LO phonon. Decreasing of the energy level spacings in the CB may lead to coupling of the electron with several LO phonons, and decrease for higher energy levels down to 1 -2 LO phonons.

I.3 Chapter summary

In conclusion, this chapter contains a theoretical introduction to physical properties of semiconductor NPs, and explains how these properties evolve with the decrease of their size, and how they impact on to the optical features of the nanoscale structures. Quantum confinement effects related to the size, material composition, as well as to the shape of SC NC have been discussed. We point out how Coulomb interactions and the binding energy of the QD structure change with increasing quantum confinement, and explained how these interactions are breaking down into different quantum confinement regimes. Understanding these properties allowed for the introduction of the band structure of the CdTe crystal, where the bulk structure has been compared with that of a single CdTe QD. Finally we have described fundamental optical interactions, which occur in QD structures, showing that both linear and nonlinear optical properties of SC crystals result from transitions between the electron and hole quantized energy levels. Over the last decade, significant advances in the field of NP synthesis enabled fabrication of increasingly complex heterostructures. Composite materials that simultaneously retain quantum confinement are therefore becoming of interest towards the nonlinear optics, mainly due to the enhancement of their nonlinearity. This lead us to propose SHG active CdTe based semiconducting QDs and QD heterostructures as promising candidates for nanolabels in nonlinear microscopy. The following chapter contains structural characterization of dotshaped QDs and rod-on-dot QD hybrids, which have been used throughout this work. We describe their morphology, methods for their synthesis, and give details about sample preparation for microscopic measurements at the sub-wavelength single QD scale.

Chapter II |

II.1 Guidelines for material selections

The contents of this work deal mainly with an investigation of the second-order nonlinear optical properties of NPs exhibiting enhanced nonlinearity, thus opening the way towards decrease of their size and application to two-photon scanning microscopy (TPSM). These NPs have been obtained from semiconductors rather than from insulating inorganic materials, to as to be in a position to benefit from quantum confinement effects. We have chosen to work mainly with CdTe QDs and CdTe/CdS quasi-type-II QDs for the reasons outlined hereafter.

Firstly, CdTe exhibits a relatively strong second-order nonlinearity from a noncentrosymmetric zinc-blende -like crystalline lattice structure. [START_REF] Zielinski | Second-Harmonic Generation from a Single Core/Shell Quantum Dot[END_REF][START_REF] Winter | Characterization of the Second Order Nonlinear Susceptibility of Quantum Confined Semiconductors[END_REF] The energy band gap of CdTe is 1.44 eV, [START_REF] Kittel | Introduction to Soild State Physics[END_REF][START_REF] Wei | Calculated natural band offsets of all II-VI and III-V semiconductors: Chemical trends and the role of cation d orbitals[END_REF] and it can be tuned up to approximately 2.0 eV as a function of QD size. This appears to match the correct range of available frequencies from a Ti-Sapphire femtosecond oscillator. CdTe has the refractive index of about 3.0 at ~1.5eV (295 K), and it is nearly transparent for NIR range. 39 Another important issue is the following: in a strong quantum confinement regime (when QD sizes are about that of the exciton Bohr radius a B or smaller), the transition dipole moment of these NPs or their two-photon cross-section does not follow the same scaling rules as observed for the bulk, but evidence significant enhancement. [START_REF] Winter | Characterization of the Second Order Nonlinear Susceptibility of Quantum Confined Semiconductors[END_REF] This makes SC colloidal QDs promising candidatess for nonlinear optical microscopy.

II.2 Characterization of dot-shaped CdTe/CdS core/shell QDs

Over the last decades, significant advances in the field of semiconductor nanoparticles synthesis enabled fabrication of various core/shell heterostructures. [40][41][42] During this period well developed methodologies of colloidal quantum dot synthesis were defined and implemented, resulting in a high reproducibility and good optical quality of the NPs, to a level making them compatible with commercial exploitation.

In dot-shaped quasi-type-II CdTe/CdS NCs, CdTe has a zinc-blende structure (with 43m point group symmetry). A stereograph showing the symmetry elements of 43m group is presented in Figure II-1(a). It consists of three 4-fold inversion axes perpendicular to four {1,1,1} cube facets, four 3-fold axes along the main diagonals, and six planes of symmetry along the diagonals of facets. Additionally, a set of 24 equivalent points is presented. [START_REF] Kittel | Introduction to Soild State Physics[END_REF][43][44] The single crystalline unit cell of a CdTe zinc-blende like structure is shown in Figure II-1(b).

II.2.1 Chemical synthesis of dot-shaped

II.2.2 Shape and size distribution of dot-shaped CdTe/CdS QDs

Transmission electron microscopy (TEM) characterization shown that particles are somewhat ellipsoidal, with an effective diameter of 10.9 ±1.8 nm (considering the square root of the product of the short and long axes), and a typical ellipticity of 1.17 ±0.13. A typical TEM image (obtained with a Philips CM120 Super Twin transmission microscope) of several NPs appears in Figure III-3(a). Under the prescribed synthesis conditions, some ripening occurs during CdTe NC growth, leading to some relative broadening of the size distribution. The largest particle observed using TEM had an overall (core + shell) effective diameter of 15 nm, while the smallest observed particles had an effective diameter of about 10 nm.

Alternatively, the size distribution was confirmed using calibrating curves proposed by the group of Peng et al.,45 where the emission peak position determines the quantum dots size (neglecting the relatively small Stokes shift of about 8 nm between the emission and first excitonic absorption peaks at these sizes), following the empirical function for CdTe QDs given below: For obtaining CdTe QDs with such control over their size distribution, we have used similar synthesis conditions as described in Paragraph IV. Size determination in case of these three sets of samples (7, 9, 11 Emission and absorption spectra show of 11 nm diameter QDs sample show a very small Stokes shift. In order to obtain the size distribution from the emission spectrum we had to ignore small differences in the absorption cross-sections of different size QDs, and assume a constant quantum yield (QY) for all sizes. The absorption cross-section, however, would blueshift the size distribution since it is larger for larger QDs. Taking into account the QY, size distribution should be red-shifted since the larger QDs are known to exhibit more structural defects and surface traps which lowers the QY. [17][18]20,46 Despite these effects, the extracted size distribution is convenient and more reliable than values obtained from TEM analysis, therefore we chose to use these result in subsequent studies. The mean diameter obtained from the emission spectrum is 11.1 ±2.4 nm. Larger size obtained from the TEM images can be attributed to the appearance of organic materials in the QD solution (which does not affect the emission spectra), as well as to spectral diffusion of individual QDs. To describe their relative lattice orientation at the interface we used the convention described in the literature. [START_REF] Kittel | Introduction to Soild State Physics[END_REF][43][44]47 The epitaxial relationships are therefore characterized by the [01 10] axis of CdS oriented parallel to the This orientation comes from the compatibility of the zinc-blende {111} ± and wurtzite (0001) ± facets at the interface, [47][48][49] whereas the two materials, CdTe and CdS, namely provide minimized mismatch along this interface of about ~9.8% between two lattices. 50

Thus, the relative rotation angle of the CdS rod around its [0001] axis relative to the CdTe core is frozen.

In order to obtain the relative orientation of CdTe and CdS in the RD crystalline structure, we have chosen an arbitrary system of coordinates, which can be found in Appendix C. These coordinates are further applied towards the analysis of the SHG polarization response.

II.4.1 Chemical synthesis of rod-on-dot CdTe/CdS heterostructures

Rod-on-dot NCs were synthesized by the standard colloidal one-pot growth technique.

Briefly, spherical CdTe QDs were grown in a non-coordinating solvent using tetradecyl phosphonic acid as a ligand, following modified procedures suggested by Peng et al. 45 CdS QRs growth was performed by multiple injections of cadmium oleate and sulphur solutions at appropriate temperature and injection rates. The final product can be considered to be a degenerated (one-armed) tetrapod. 48,51 In detail, 60 mg tetradecyl phosphonic acid (PCI synthesis) and 13 mg Cadmium Oxide were added to Size determination is quite important since SHG is expected to scale as the squared volume of the scatterer, this being especially critical when applied to complex structures composed of at least two different types of materials with a strong necessity for characterization of a morphology and dimensions of these heterostructures towards further SHG studies. The size of spherical CdTe cores was determined from the absorption spectrum during growth to 5.9 nm, following the (II.1) empirical function. 45 In the case of geometrically complex heterostructures, such as RD NCs, dimensions are best obtained directly from the TEM images. The size distribution of forty isolated RD NCs, directly obtained from the TEM images is shown in the histograms in Figure II-11(a-c). Size dispersion varies in a range between 5.7 -6.2 nm diameter of the CdTe core (consistent with the optical measurements), and 2.4 -3.6 nm diameter of the CdS rod. The length of the CdS rod is dispersed in a wider range, from 6.0 -16.0 nm, with a mean value of 11.0 nm.

II.5 The role and importance of the shell in type-II QDs

The optical properties of type-II core/shell QDs under the strong confinement regime are very sensitive to the smallest changes in core or shell dimensions. Compared to organically passivated cores, core/shell QD heterostructures can tolerate harsher processing conditions and environments. 53 In general the shell overcoating in QDs increases the chemical stability by protecting the core from the environment, and passivates the core surface trap sites. Shell deposition can, in principle, enable all surface atoms to be passivated, insulating the core from its environment, both physically and electronically. Appearance of the shell simply buries the semiconductor in the potential energy well, and energetically confines the electron and hole, leading to the charge carriers concentrating in the nanocrystal core, away from the surface.

This directly modifies the optical properties of the QD. [START_REF] Alivisatos | Semiconductor Clusters, Nanocrystals, and Quantum Dots[END_REF]40,[54][55] As a result, surface defects states and trap sites have a reduced impact on the luminescence emission efficiency, allowing to come close to ideal conditions (for an ideal QD -without crystalline defects of the surface), whereby for every photon that creates an exciton, one photon is emitted, reaching a 100% luminescence QY. Shell made of the material with a wider band gap than the core increases the probability of the radiative recombination.

The band offset of type-II colloidal QDs can be controlled by careful choice of the material, e.g. quasi-type-II CdSe/CdS or CdTe/CdS (where the electrons are completely delocalized) core/shell heterostructures, or type-II CdTe dots overcoated with a ZnS shell.

Wider band gap of the CdS or ZnS shells (for CdS the band gap is 2.42 eV, ZnS 3.6 eV;

whereas for the CdSe and CdTe cores band gaps are 1.74 eV and 1.44 eV, respectively) [START_REF] Kittel | Introduction to Soild State Physics[END_REF][START_REF] Wei | Calculated natural band offsets of all II-VI and III-V semiconductors: Chemical trends and the role of cation d orbitals[END_REF] not only electronically insulate the cores, but results in a higher threshold to photo-oxidative Sample preparation | 33 degradation and formation of the surface defects, due to a much lower oxidation potential of S 2-than Se 2-or Te 2-anions. 46 The energy band gaps of most popular SC NCs in colloidal synthesis of QDs are illustrated in Figure I-2.

II.6 Sample preparation

Before preparing the samples, QDs had first to be purified after the synthesis, as well as separated from the chemical compounds left in the growth solution. In each case, QDs were purified by dissolving them in toluene and precipitation from the suspension by adding methanol and centrifuging for approximately 5 min. Such a process was repeated usually several times to obtain a high degree of purification. A well performed purification process had a high impact on further optical properties, and reflected in improved signal photostability under two-photon excitation of NPs.

Sample preparation for microscopy measurements required a few conditions. For single NP studies we had to use a polymer host-matrix to embed QDs inside a thin layer and avoid their motion during surface scans. It was important to use a polymer with a similar refractive index to the one of cover slip glass (n ≈ 1.50 in the VIS region), to minimize their refractive index mismatch. Poly(methyl methacrylate) (PMMA) with n ≈ 1.49 (VIS) was found to be a good candidate. After precipitation, QDs were redissolved in toluene or in anisole and mixed with a PMMA-anisole solution (MichroChem 495 MW PMMA 6A, dissolved in anisole with a 2% concentration in mass).

The concentration of the QDs was usually about 2.0 mg of precipitate per 1 mL of the PMMA-solvent solution. This concentration resulted in high enough QDs dispersion on the cover slip glass surface. To avoid aggregation and creation of big multi-crystalline clusters during further processing, QDs solution was sonicated for 1 hour directly before deposition.

The substrates -cover slip glasses (#1) were cleaned in isopropanole, treated with the oxygen plasma and heated up on the hot plate (180 °C).

The prepared colloidal solution of QDs was deposited by spin-coating with 3000 rpm, 1000 acc., through 40 s to allow the solvent to evaporate, which resulted in approximately 40 -50 nm thick polymer layer (measured with the DEKTAK 3ST surface profiler). In some cases (rod-on-dot QDs), when the polymer layer had to be thicker, the deposition was carried out at 4000 rpm, resulting in a ~30 nm thick PMMA host-matrix. After spin-coating, all the samples were held at 180 °C for 1 min to allow QDs to be in contact with the glass surface.

II.7 Chapter summary

We have synthesized and characterized the morphology of three different types of CdTebased QD samples: spherical CdTe QDs, spherical core/shell CdTe/CdS, and rod-on-dot CdTe/CdS QD heterostructures. We have proved a reasonable high degree of control in obtaining these materials with a high reproducibility over their shape and size distributions.

Morphological features were characterized with different microscopy (i.e. TEM, AFM) and single-photon spectroscopy techniques, which allowed determining their respective effective size and shape distributions. These have a high importance towards nonlinear optical properties, which evaluate with the change of the NP shape, size and material composition.

We have also demonstrated a reasonably high degree of control in obtaining these materials with a high reproducibility over their shape and size distributions. In this chapter we briefly introduce the differences between the two-photon excited fluorescence (TPEF) and the second-harmonic generation (SHG) microscopy, putting them in a context of basic underlying processes of different optical properties. Then, we describe a detailed description of the experimental setups and methodologies used for SHG detection from single isolated nanoscatterers, such as quantum dots. Firstly, nonlinear two-photon scanning polarization sensitive microscopy (TPSM) is discussed. We introduce important features of this technique towards further spectral and polarization analysis of the SHG emission from colloidal QDs. Furthermore we discuss a modified version of TPSM experiment based on time correlated single photon counting (TCSPC), which allows for selection of the instantaneously scattered photons arriving subsequent to the laser pulse, excluding the dark counts of the detector. We explain why the TCSPC detection results in an improved signal-to-noise ratio, and allows for the detection of weak SHG signals obtained from small sized QDs.

III.1 Contrast mechanisms for two-photon nonlinear microscopy

Since its first application in nonlinear microcopy at the turn of 70s and 80s, SHG has become a powerful contrast mechanism in nonlinear optical imaging, [56][57] due to the stringent structural conditions in the ability of the specimen to generate second-harmonic of the incident excitation light, which is constrained by the noncentrosymmetry of the medium. 58 Properties of the SHG process are such that it can be applied, e.g. in phase-sensitive interferometric SHG microscopy, 59-60 as a probe of local anisotropies, 61 or to probe surface plasmon enhanced interactions. [62][63] Similar to TPEF microscopy, 64 SHG -based microscopy techniques provide all the benefits of TPEF mechanism, such as superior axial resolution due to the quadratic dependence of SHG on the illumination intensity, entailing reduction of outoff-plane photobleaching and phototoxicity in live cell imaging. [65][66][67] These two processes, however, result from completely different mechanisms and thus exhibit significantly different properties. Here we aim to outline the main features and differences between them, which are of concern for nonlinear microscopy.

III.1.1 Two-photon excited fluorescence

The theory of two-photon absorption and emission process has been developed by Maria Göppert-Mayer during her doctoral research in Göttingen, under the supervision of Max Born. [68][69] TPEF is a resonant absorption process, which involves four energy levels (Figure III-1(a)). Two photons of about the same energy (usually from the same laser source) with frequencies ω are absorbed via a virtual state (with a life time typically shorter than 1 fs) allowing to excite the electron into a real excited molecular vibronic sub-level of the electronically excited state, with identical spectral positioning to the absorption of a single photon possessing twice the energy. Relaxation via up-converted emission of TPEF back to the S 0 ground state follows non-radiative vibrational relaxation into the S 1 energy level (leading to an energy loss). The emitted TPEF photon has an energy slightly lower than the input energy and corresponds to a photon of wavelength slightly longer than half of the excitation wavelength exc λ . TPEF is a second-order nonlinear process resulting in incoherent emission, with a typical radiative life time of about 10 -9 s. It does not comply with the Contrast mechanisms for two-photon nonlinear microscopy | 37 symmetry constraints typical of SHG and may therefore occur for both, centrosymmetric and noncentrosymmetric media or randomly oriented fluorophores.

The efficiency of the TPEF process depends on the probability of two-photon absorption as given by

4 (2 ) 4 p abs abs TPEF P E E ω ω µ σ ≃ ≃
, where: abs µ is the absorption transition dipole and E ω is the incident electric field vector. TPEF σ is the two-photon absorption cross-section in Göppert-Mayer units (GM) ii which provides a quantitative measure of (2 ) p abs P , and E is the incident electric field. Due to difficulties in the direct measurement of the two-photon cross-section, Dashed lines are un-populated virtual energy levels (VS) with life times shorter than 1 fs.

ii 1 GM = 1×10 -50 cm 4 s photon -1 For the two-photon absorption process, the number of TPEF photons detected per unit time and volume can be expressed as:

75 2 1 2 fluo TPEF TPEF N N K I ω φ σ   = × ×     (III.1)
where: N is the number of fluorescent molecules per unit volume (m -3 ); vectors associated respectively with absorption / emission dipoles and the incident field). The factor 1/ 2 reflects the fact that only one photon is emitted, while two photons of the pump beam are absorbed. I ω is a flux of incident photons (photon cm -2 s -1 ), and its power of two is due to the second-order nature of the nonlinear process. For TPEF active nanocrystalline emitters we can roughly assume that number of emitted photons is:

4 3 fluo fluo i emitters i N E I V a ω ∝ ∝ ∝ = ∑ (III.2)
where: fluo i I is a flux of TPEF photons obtained from i emitters within a volume V of average dimension a.

III.1.2 Second-harmonic generation

SHG is a second-order nonlinear scattering process which has a primarily electronic origin.

To be observed SHG requires a high pump energy, therefore it could be discovered only in 1961 by Franken et al.,76 having had to wait development of the first lasers. The three-level SHG process is illustrated in Figure III-1(b). In general two photons of the same frequency ω 'coalesce' to a virtual state to form a single photon of energy exactly twice that of the incident photons. Depending on the excitation wavelength and material features, SHG can be obtained for resonant or in non-resonant energy conditions. The 2ω second-harmonic photon is generated almost instantaneously (within a few fs), therefore it is coherent. The electronic origin of SHG can be explained by looking at the response of an atomic system to a propagating incident electric field E ω , which exerts a dynamical Coulomb force on the internal charge distribution of the material, giving rise to additional components in the spectrum of the incoming field. The nonlinear terms contribute to the polarization only when the high-order terms of the Taylor expansion of E are non-negligible, which is the case for a focused laser beam. The material response is then specified by means of the polarization P (dipole moment per unit volume), which can be explained in a power series of the amplitude E ω of the incident optical electric field: 58

( ) ( ) ( ) ( ) (1) (2) 2 (3) 3 0 ... P t E t E t E t ε χ χ χ   = + + +   ɶ ɶ ɶ ɶ (III.3)
where: 0 ε is the dielectric constant, (1) χ is the linear optical susceptibility, and ( 2,3,...) n χ = are the n-order nonlinear optical susceptibilities (being n+1 rank tensors). Second-harmonic generation is determined by the second-order nonlinear optical susceptibility (2) ijk χ (third-rank tensor), which can be expressed as a cubic matrix as follows:

1 2 3 1 2 2 2 3 1 2 3 ijk ijk i j d d d d d d d d d d d d d d d d d d d d d d d d d d d d k χ = =                 = =                         = (III.4)
where ijk d are the i,j,k nonlinear coefficients, reflecting the symmetry and nonlinear optical properties of the material. Due to the basic parity constraint attached to odd rank tensors ijk d coefficients vanish for materials with inversion symmetry, therefore SHG can be obtained only for noncentrosymmetric media. The specimen has to be also relatively transparent at both fundamental illumination wavelength and the generated harmonic. 58,77 In order to obtain the intensity of the SHG signal from statistically oriented microscopic dipoles with microscopic nonlinearity β , (2) (2) 0

: P E E ε χ
= can be simply related to β via an oriented gas model: 78 ( )

(2) 2 IJK I J K n IJK Nf f f ω ω ω χ β = Ω (III.5)
where N is the molecular density,

( ) 2 ( ) 2 / 3 i i f n ω ω = +
is the Lorentz-Lorenz local field correction factor depending on the refractive index n, ( )

n
β Ω is the statistical tensor of the first hyperpolarizability (dependent on a priori unknown Euler set of angles ( )

, , n n n n θ φ ψ Ω = ).
n Ω can be either a random variable in the case of statistical media (e.g. guest-host polymer), or a well defined unknown quantity for an arbitrary oriented nanocrystal. We take the simplest approximation that SHG is detected along the polarization analysis unit vector I u and is thus proportional to the 2 ( 2)

I P µ ⋅ .
). Therefore, the induced second-order nonlinear polarization SHG intensity can be expressed as: 75

( ) 2 2 (2) 2 , SHG I n J K IJK J K I P N E E β ∝ ∝ Ω ∑ (III.6)
Following equation (III.6) we can determine the number of emitted SHG photons SHG N in a similar manner to that in the case of TPEF, using the second-harmonic scattering crosssection: 79 n ω ω are the refractive indexes at the pump and SHG frequencies, ℏ is a Dirac constant, 0 ε is a dielectric constant of the material, and c is the speed of light. In practice the SHG cross-section is usually much smaller (about four orders of magnitude) than TPEF σ when considered for a single molecule. It may be increased, however, when the molecule is excited near resonance, typically by one to two orders in magnitude, and it scales quadratically with the number of molecules N per volume unit (in contrast to linearly scaling for TPEF σ ). 79 Note, that the molecules are assumed to be uniaxial, with the hyperpolarizability β tensor dominated by a single component in the molecular frame. Such an orientation together with a tightly clustered distribution of the molecules around the SHG focal center accounts for the dipolar nature of the SHG radiation, which is expected to be equally radiating in the forward and backward directions. In other words all the individual molecules coherently scatter the incident field according to their SHG scattering cross-section, which results in the summation of scattered harmonic electric fields in the same manner as for a phase-locked antenna system. 79

The total radiated power SHG P scattered by a single NP can be determined by integrating the SHG intensity over all possible emission angles: 79 where: I ω is the excitation field at the focal spot (for fs laser source the focused pulsed peak intensity will be defined in Subsection III.3), N is the effective number of radiating molecules (which can be assumed as equal to one for a single QD), and ( ) ( )

2 2 0.1 / w λ λ Θ ≈ is a yield
parameter accounting for the reduction of the radiated fields when the molecules are spread over the surface at least of order of exc λ as compared to the situation where they would be concentrated at the beam waist and radiate a perfectly in phase ( ( )

exc

III.2 Second-order nonlinearity of CdTe -from bulk to single QD scale

The high bulk second-order nonlinearity of cadmium telluride arises from its noncentrosymmetric zinc-blende tetrahedral structure (abiding to 43m point group symmetry), and it is associated to a nonlinear electronic polarization mechanism. The form of the d-matrix of nonlinear coefficients for regular 43m symmetry class of the bulk CdTe crystal, assuming the Kleinman symmetry, reduces to the following form: 58 14 25 14 36 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

XYZ d d d d d d     = =     =   (III.9)
For 43m class crystal structures, such as CdTe there are six nonvanishing tensor elements.

Intrinsic permutation symmetries of CdTe crystalline axis further lead to:

xyz xzy d d = = yxz yzx zxy zyx d d d d = = =
. Thus the i component (i = x, y, z) of the second-order nonlinear polarization response can be expressed as:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 14 0 0 0 1 0 0 2 2 0 0 0 0 1 0 2 0 0 0 0 0 1 2 2 x y z i y z x z x y E E E P d E E E E E E ω ω ω ω ω ω ω ω ω ω               =                   (III.10)
Downscaling from bulk crystal size to the nanometric scale -when NP becomes much smaller than the excitation wavelength (QDs may be compared here to a single molecule), enables to describe the nonlinear scatterer as a dielectric sphere, 2,58,[80][81] thus it can be considered as a Rayleigh scatterer. The electric field inside such a particle ( )

E ω induced by
linearly polarized electromagnetic wave is uniform and in-phase, so due to the Landau's electrostatic theory of dielectrics it can be taken as: 80,82 ( ) ( ) ( )

3 2 m m m E E ε ω ω ε ε   =   +     (III.11)
where ( ) m E ω is the fundamental electric field in the surrounding medium (PMMA host- matrix in the case of our experiment), ε and m ε are the linear permittivities of the NP and the surrounding medium respectively.

For SHG, which is simply a frequency doubling process, expression (III.3) transforms into:

( ) ( ) ( ) 2 (2) 0 2 2 XYZ P d E E E ω ε χ ω ω = = ⋅ (III.12)
Since the CdTe cubic zinc-blende -like lattice is an example of a perfectly octupolar symmetry, [START_REF] Zielinski | Second-Harmonic Generation from a Single Core/Shell Quantum Dot[END_REF] one may expect a multipolar character for the emitted SHG field. These properties are, however, related only to the CdTe material features, which cannot sustain a dipole by virtue of its spatial arrangement (nor any other property of a vectorial nature). When addressing the quantum particles, such as single CdTe QDs with a linearly polarized fundamental beam, electrons oscillating at the frequency 2ω have a second harmonic nonlinear polarization which is a vector by the very nature of a polarization. This is not in contradiction with the octupolar character of the CdTe material. Therefore, acting on an 'artificial atom' by linear polarization induces a dipolar polarization of the field.

The ( )

2
P ω polarization is in-phase and uniform inside the NP, however due to our experimental conditions (where the collection sensitivity of the emitted light can be neglected along the Z axis) it implies that only two electric field components along two different crystalline axes of the lattice, such as X and Y can couple to generate a second-harmonic emission dipole moment along the third crystalline axis, such as Z. Thus the nonlinear dipole moment of the CdTe QD can be defined as:

( ) ( ) 3 4 2 2 3 i p P a ω ω π   =     (III.13)
and it radiates like a 'nanoantenna' at the frequency 2ω , where a is the radius of the scatterer, and ( )

2 i
P ω are the elements of the nonlinear polarization (i = X, Y, Z). When these considerations are translated into tensorial properties of the second-order nonlinear susceptibility of CdTe we can limit ourselves to coefficients such as (2) ( ), , X Y j k χ , and in that particular case we discard the (2) , , Z j k χ coefficients. Thus the nonlinear dipole for the CdTe QD can be reduced to the following form:

( ) (2) ( )2 ( )2 , 0 ( ), , , 2 X Y X Y j k j k j k p E E ω ω ω ε χ = ∑ (III.14)
In case of CdTe QDs, excitation with a linearly polarized optical electric field (or with circular polarization, defined further in Subsection V.1), oriented perpendicularly to the z propagation axis of the laboratory frame, allows to generate satisfactory results. We can therefore assume in the following that collection of signal orientating from a (2) , , Z j k χ component is negligible. [START_REF] Zielinski | Second-Harmonic Generation from a Single Core/Shell Quantum Dot[END_REF][START_REF] Winter | Characterization of the Second Order Nonlinear Susceptibility of Quantum Confined Semiconductors[END_REF][START_REF] Zielinski | Nanoegineering the Second Order Susceptibility in Semiconductor Quantum Dot Heterostructures[END_REF] SHG detection from very small NPs is mainly limited by the nonlinearity of the material.

Additionally other constraining factors may appear from a technical point of view, such as due to the numerical aperture of the microscope objective, incident polarization aspects, and quantum efficiency of detectors. 83 Many SHG-active insulating (alternatively wide-gap) inorganic NPs have been recently reported and studied at the single NP scale, e.g. ZnO, [84][85] KNbO 3 , 86-87 Fe(IO 3 ) 3 , 88 BaTiO 3 , 2,89 and KTiOPO 4 . 1,90 Nevertheless, these materials due to their nonlinearity are difficult to detect with standard detection techniques, at least when their size is decreased much below the excitation wavelength (< ~40 nm). To the best of our knowledge, the minimal observable size so far of nanocrystals made of these materials at the single NP scale is larger than 30 nm. The main difficulty arises from a nonlinear scattering nature of SHG, which is expected to scale with the squared volume of the nanoscatterer (

2 SHG scatt I V ∝
). Therefore when we consider real experimental conditions, the number of emitted SHG photons in approximation is proportional to the diameter of the NP (a NP ) to the power of sixth:

( )

2 2 (2) 6 SHG NP rep P N a f χ τ     ∝   ⋅   (III.15)
where P is the power of incident excitation optical electric field, rep f is a repetition rate of the laser pulses (80 MHz for standard Mai-Tai HP, Spectra-Physics oscillator), and τ is a laser pulse duration (typically about 100 fs). Expression (III.15) together with the significant background noise level that goes with nonlinear scattering put a strong limitation for SHG detection from very small NPs. On the other hand, the importance of developing small and photostable NCs, applicable to bio-labeling in cellular or tissue imaging, where the size reduction of nanolabels is a prerequisite, as shown in Ref. [74,96-98] have objected our investigations to a noncentrosymmetric materials, exhibiting much higher second-order nonlinearity than the above mentioned inorganic NPs. Nonlinear coefficients of some inorganic materials and most popular SCs are presented in

III.3 Nonlinear two-photon polarization sensitive microscopy setup

In principle, the nonlinear TPSM setup used for SHG studies is similar to the one described in where the Ti 3+ ions responsible for a lasing are substituted to a small fraction of the Al 3+ host.

This system provides smooth wavelength tuning in a range from 690 -1020 nm (wavelength dependence of the laser output power, as well as characteristics of other optical components can be found in Appendix A). iii One of the main advantages of the Mai-Tai laser system stems from its user-friendly control panel, and the option for automatized continuous scanning over the whole accessible wavelength range, which has been used in this work in order to study the wavelength dependence of the SHG emission, radiated by single QDs. [START_REF] Zielinski | Second-Harmonic Generation from a Single Core/Shell Quantum Dot[END_REF][START_REF] Zielinski | Nanoegineering the Second Order Susceptibility in Semiconductor Quantum Dot Heterostructures[END_REF] Other essential components of the experimental setup are an achromatic half-wave plate . The real value of the signal count rate can be obtained using the correction factor: 1/1 ( )

f d output C t C = -⋅
, where d t is a module dead time (~50 ns), and output C is an output count rate. Finally, the actual photon count rate can be calculated using the following expression: iv In detection of SHG from such small objects as single QDs, providing phase-matching conditions is negligible because for all practical purposes we deal with a scattering by a 'point-like dipole'. Applications of the SHG process in nonlinear microscopy of larger objects, however, demand consideration of high pump electric field intensities in order to obtain efficient enough signals via tight focusing of the incident beam. For tightly focused driving field with an aperture limited angular distribution of half-width 0 α , multi-harmonic radiation patterns are confined into twin-lobed or conical geometries (related to a surface and volume generation, respectively), which are quasi phase-matched and highly peaked at the Gouy phase shift dependent off-axis angles ( 0 / 2 α ± and 0 / 2 α for a surface and volume multi-harmonic generation, respectively). 79,99-100 For both, surface and volume cases multiharmonic scattering propagation is highly limited in the backward direction. Nevertheless, use of the microscope objectives with wide enough angular numerical apertures ( sin )

NA n α = may essentially provide collection of all the scattered harmonic light in the backward direction.

In order to obtain high pulsed peak intensity in a tightly focused excitation volume we used a high numerical aperture oil immersion objective (Nikon, Plan Apo IR, ×100, NA = 1.4). Its high NA value allows for efficient collection of the SHG radiation in the backward direction.

The collected signal was spectrally separated from the NIR excitation beam by the DM.

Additional spectral filtration (short-pass / band-pass filters) was applied in order to select the SHG emission peak in case of potentially appearing TPEL emission. Emitting NPs were localized by scanning surface of the sample carried out with a Piezojena x,y,z positioning piezo stage, with a minimal step of 20 nm. The nature of the emission was confirmed by measuring its spectra, as acquired by Oriel Multispec 7740 spectrometer (160 nm narrow slit) equipped with an Andor Technology DV420-OE CCD camera.

III.3.1 Optical and effective spatial resolution in multi-photon microscopy

The minimal optical resolution of the focused laser beam determines the minimal achievable For coherent point sources, the diffraction limit requires consideration of both the intensity and phase distributions associated with the object. A profile along one of the lateral axis of the image intensity can be expressed in normalized image coordinates as: 102

( ) ( ) ( ) ( ) ( ) 2 1 1 0.61 0.61 2 2 0.61 0.61 j J x J x I x e x x φ π π π π     - +     = + - + (III.18)
where J is a mutual intensity, that is a measure of the spatial coherence of the light at the two coherent point sources, and φ is the relative phase between these two sources. Basically, depending on φ , we can distinguish three situations which are illustrated in

III.5 Chapter summary

Two different contrast mechanisms such, i.e. SHG and TPEF, commonly used in the twophoton microscopy has been reviewed and compared. Properties of both processes were discussed in order to highlight their differences, and point out the importance for development and application of SHG-active nanoprobes, e.g. CdTe QDs, which can give all the features derived from coherent SHG process, alternatively to common fluorescent markers used in bioimaging.

By considering a Rayleigh nanoscatterer model, we compared the physical aspects of the SHG process in CdTe from the bulk, down to the single QD scale. The second-order optical nonlinearity of CdTe was additionally compared with common nonlinear insulators and wide gap semiconducting crystals.

The second part of the chapter contains detailed descriptions of the optical setups used in the experimental part of the thesis. We introduced the diffraction limited TPSM technique used for coherent SHG detection from single QDs (which to our best knowledge are the smallest detectable single SHG-active nanoparticles), and for wavelength dependence of scattered SHG radiation. [START_REF] Zielinski | Second-Harmonic Generation from a Single Core/Shell Quantum Dot[END_REF][START_REF] Zielinski | Nanoegineering the Second Order Susceptibility in Semiconductor Quantum Dot Heterostructures[END_REF] Furthermore combination of TPSM with the time-tagged TCSPC detection setup was proposed for single QD spectroscopy and size dependent measurements of the second-order nonlinear susceptibility. [START_REF] Winter | Characterization of the Second Order Nonlinear Susceptibility of Quantum Confined Semiconductors[END_REF] Chapter IV | Second-harmonic generation from CdTe QDs: single dot studies The choice of CdTe bulk nonlinearity was made to obtain a SHG response which does not strongly depend on the NP shape. Reported values of CdTe nonlinear coefficients range from 64 pm V -1 to 200 pm V -1 (depending on the size and excitation wavelength), [START_REF] Winter | Characterization of the Second Order Nonlinear Susceptibility of Quantum Confined Semiconductors[END_REF]77,[94][95] i.e. about one order of magnitude larger than KTP (see Table IV- emission scans were obtained at an excitation wavelength of λ exc = 980 nm from 100 fs pulsed laser light, and with the incident power of P exc = 0.7 mW (measured at the front of the microscope), which leads to a focused pulsed peak intensity of about 18.2 GW cm -2 at the focal point. The SHG emission was spectrally filtered from eventual contamination by TPEF emission, using a high transmission band-pass filter (480 ±20 nm) and a high quality shortpass filter (transmission ~95%, cut-off at 561 nm). These conditions were enough to obtain a SHG count rate of about 7.5×10 4 cts s -1 , which after correction by the APDs detection efficiency (40% for 490 nm at ambient temperature) yields a signal of about 1.05×10 5 cts s -1 , what is the real SHG photon flux. The observed background appears mainly due to the APDs SHG from single dot-shaped CdTe/CdS core/shell QDs | 57 dark counts at the level of about 950 cts s -1 , leading to a signal-to-noise ratio of about 80, thus providing a high enough contrast. Detection of such a high signal, almost two orders of magnitude higher than the usual limit for bio-imaging, highlights spherical CdTe/CdS core/shell QDs as a promising material for bio-labeling. An important issue of any tested CdTe -based QDs is that the character of the emission has a strong impact on its photostability of the signal. In general our observations show that signals obtained from QDs emitting SHG with a high QY for this efficient process exhibit a higher degree of photo-stability, even at high intensities of the pump light, i.e. ~80 GW cm -2 of the focused pulse peak intensity (see Figure IV-3(a-c)). [START_REF] Zielinski | Second-Harmonic Generation from a Single Core/Shell Quantum Dot[END_REF] In the context of bio-labeling, SHG observed from CdTe/CdS QDs under strong excitation intensities shown reduced photobleaching, which is a main limitation for many organic fluorophores, 72,107 and fully eliminates the problem of emission intermittency such as 'blinking' which is commonly observed for luminescent QDs. [108][109][110][111] Beside SHG, undesirable two-photon excited luminescence (TPEL) can appear as competing nonlinear process. Figure IV-4(b) exemplifies some QDs (obtained with the pulsed peak intensity of about 135 GW cm -2 , and without spectral separation of the SHG emission peak) which, when excited at frequencies far from the one-photon resonance (which appears at 760 nm -see Figure II-3(c)) exhibit TPEL emission with a maximum at about 520 nm, with a decisive influence on the photostability of the NP. Studies of many CdTe/CdS core/shell QDs have shown that even small amounts of luminescence photons observed simultaneously with SHG emission, can dramatically reduce the intensities of both processes, which is a reversed correlation to that which has been observed for a pure and highly photostable SHG signal. We believe that the reduced stability of QDs exhibiting stronger band-edge luminescence is due to induced excited state absorption at the excitation frequency. To analyze the spectral dependence of the SHG process, we considered a single spherical CdTe/CdS core/shell QD, emitting a relatively intense and photostable SHG signal. The spectral response of the SHG emission was first obtained using an Oriel Multispec 77400 spectrometer (160 nm narrow slit), equipped with an Andor Technology DV420-OE CCD camera. For each spectrum the collected SHG emission was obtained from a single QD (Figure IV-4(a)), in presence of a short-pass filter cutting eventual TPEL emission. The excitation wavelength was tuned in a range between 850 nm and 1.0 µm while keeping the incident power at the level of 2.0 mW (measured at the front of the microscope), leading to a mean pulsed peak intensity of about 54.5 GW cm -2 . Spectra were recorded with doubled acquisition time of 10s, including background correction to avoid the appearance of any noise bursts. In this particular case, the SHG QY was not high, however SHG peaks at half the incident wavelength are clearly visible, showing a maximal SHG intensity for the excitation region around 970 nm. This wavelength was further confirmed to be optimal in order to obtain the highest possible SHG intensity, and therefore favorable towards further studies of these QDs. Reliable results for wavelength dependence of SHG emission could be obtained only with a high degree of emission photostability, thus we have chosen to study QDs with a high SHG QY, allowing exciting them at a low incident power. Each dot was firstly investigated using the spectrometer to confirm the absence of TPEL emission. However spectral separation of the SHG peak from an eventual TPEL was ensured setting a good quality short-pass filter in the detection channel during the next experimental step. The wavelength dependence of the second-harmonic (SH) emission has been investigated by tuning the excitation wavelength (using an automatic laser scan with a step of 1 nm per 1 s, controlled by the Mai-Tai control software) and measuring the intensity of the SHG peak observed from CdTe/CdS QDs using two APD detectors (with an acquisition time of 1s). Our femtosecond laser is tunable from quantum efficiency adapted to the green region, thus limiting detection of the SHG signal to wavelengths higher than 400 nm. The SHG emission reflects an effective nonlinear coefficient ( )

2 ( 2) eff χ
corresponding to the squared second-order nonlinear susceptibility.

Therefore we were able to obtain reasonable dependence of the effective (2) χ from 820 nm up to 1000 nm (or from 410 to 500 nm at the doubled frequency), as shown in From one-photon spectroscopy of similar-sized CdSe NCs, 18 it can be seen that when exciting via one-photon transitions up to 1.0 eV above the band gap (as is the case here for 970 nm excitation of QDs with a band gap of 760 nm), the inter-level spacing is of the order of 0.1 eV, roughly corresponding to the observed energy difference in the SHG excitation spectrum. Despite the symmetry difference between one-photon and two-photon transitions, this may suggest that the peaks can be associated with the discrete energy level diagram of the NPs. Our experimental setup does not permit to access the excitation spectrum below 810 nm or above 1020 nm, but it would certainly be interesting to study SHG response when reaching one-photon resonance at 760 nm, as well as two-photon resonance, which is expected at 1520 nm. Although the 970 nm excitation wavelength may not correspond to the highest secondorder nonlinear efficiency it nevertheless leads, to an easily detectable as well as photostable emission for the commonly used wavelength range of the Ti-Sapphire oscilator. averaged value for the bulk CdTe is estimated to be about 140 pm V -1 . 77,94 With the CdTe core diameter of about 12 nm, and thickness of the CdS shell of ~1 nm, the squeezed volume ratio of CdTe over CdS is of about 13.6. The contribution from the shell is expected to be below 10% of the total observed SHG signal. [START_REF] Zielinski | Second-Harmonic Generation from a Single Core/Shell Quantum Dot[END_REF] Thus in further processing of the SHG response from dot-shaped CdTe/CdS QDs we only take into account only a role of the CdTe core. This assumption is very important to further discussed SHG polarization analysis, and it was found to be in a good agreement between experimentally obtained SHG polarization response and the theoretical model. [START_REF] Zielinski | Second-Harmonic Generation from a Single Core/Shell Quantum Dot[END_REF] We note however, that this estimation is very dependent on the actual ratio of the core-shell radius and of the nonlinear coefficients. Clearly a thicker CdS shell would play a major role both on the amplitude and the shape of the SHG signals, since CdTe and CdS belong to different symmetry classes. Finally it has to be pointed out that the epitaxial growth of CdS QDs may lead to two different symmetries of the CdS structure, resulting in a zinc-blende, 113 or wurtzite crystalline lattices. 114 In the case of our QDs, since

CdS shells are grown on the zinc-blende CdTe core, it is possible that it also grows in accordance with a zinc-blende structure, as it is composed of three monolayers. Therefore, in any case, a not well characterized shell structure with possible presence of structural defects, is neglected due to its unknown symmetry and related unknown nonlinearity as well as its weaker volume ratio.

IV.1.3 Orientation dependent SHG polarization response from spherical CdTe/CdS core/shell QDs

One issue in imaging with markers is to separate the signal coming from the nanolabel and the background emission originating from the studied sample. Second-harmonic generation is generally a weaker signal as compared to one or two-photon fluorescence, but it exhibits striking features in contrast with fluorescence signals from an emission dipole (see Sections III.1.1 and III.1.2). One of these distinctive features is coherence, which is being increasingly investigated, 115 and allows to separate the signal coming from the NP, from those obtained from the background scattering. Another one is the symmetry and optical polarization sensitivity of the SHG response, which recently has been explored also at the single NP level. 1,[START_REF] Zielinski | Second-Harmonic Generation from a Single Core/Shell Quantum Dot[END_REF]75,[116][117][118] From this second point of view, CdTe is a peculiar material since it exhibits a tetragonal 43m symmetry, which has been seldom used for other non-centrosymmetric NPs, and which is generally not encountered in SHG-active biological macromolecules. This leads to an uncommon SHG polarization response of four-clover leafs like pattern, which arises due to the octupolar crystalline lattice of a zinc-blende CdTe QDs. This feature could be used as a signature of the NC presence even for a weak SHG intensities due to a small size of the QD, [START_REF] Zielinski | Second-Harmonic Generation from a Single Core/Shell Quantum Dot[END_REF] additionally allowing for the distinction between nanoscatterer and the biological specimen.

IV.1.3.1 Analysis of the SHG polarization dependence

Analysis of the optical polarization of the radiated SHG field, divided into x and y components can be achieved by rotating the angle of the incident electromagnetic field E ω at the fundamental frequency ω. The applied model neglects depolarization and propagation effects associated with the dielectric material. 82 Therefore, the SHG response can be accounted for by the second-order polarization 2 i P ω oscillating at the frequency 2ω: 77

( ) ( ) ( ) ( ) (2)
(2) 0 , 2 2 ; ,

i ijk j k j k P E E ω ε χ ω ω ω ω ω = - ∑ (III.21)
where 0 ε is the vacuum permittivity, and ( )

( ) j k E
ω is the amplitude of the ( ) j k component of the excitation field at excitation frequency ω; i, j and k are the components of the second-order nonlinear optical susceptibility tensor (2) ijk χ , associated with the Cartesian x, y and z axes of the laboratory frame.

( ) (2) 2 i P ω is obtained by rotating the second-order nonlinear susceptibility tensor (2) χ of the CdTe structure in the (X, Y, Z) crystalline axes conventionally attached to 43m symmetry, 77 by the Euler set of angles 1. smoothing the raw data traces using a general form of the SF function, which is directly related to the second-order nonlinear tensor, in order to obtain symmetrical positions of maxima and remove signal noise;

2. the obtained SF function is used as new input data and fitted by a similar function, which is directly related to the (2) χ second-order nonlinear susceptibility of CdTe structure;

3. orientation of the CdTe crystalline lattice is obtained by multiplying the (2) CdTe χ tensor by the Euler set of angles ( )

, , φ θ ψ Ω =
, where the correct position is found during the scan over the full possible space of Ω , using criteria discussed in Appendix B. These criteria decide about a matching degree between both, experimental and calculated SHG polarization patterns. symmetries with respect to the X' crystalline axis, obtained after rotation by the φ angle. Due to these symmetry properties, obtaining the three-dimensional orientation of the crystalline lattice needs an additional technique, which is defocused imaging. 1,90,119 This technique provides an information about the orientation of the emitting dipole in the (x, y) plane, and determines its deviation angle from the z axis of the laboratory frame, as illustrated in These results appear as a limitation in orientational tracking of single NCs, due to the number of possible crystalline orientations resulting in the same SHG polarization response.

These can be obviously narrowed down by defocused imaging, with the inconvenience of longer acquisition times. On the other hand, a possible solution may be found in limiting the freedom of motion of the NC itself, by attaching additional specimen to its surface, e.g. rodlike crystal with a heterojunction between both structures, as in the case of rod-on-dot (RD)

CdTe/CdS QD heterostructures, [START_REF] Zielinski | Nanoegineering the Second Order Susceptibility in Semiconductor Quantum Dot Heterostructures[END_REF] or biological specimen, e.g. complex protein molecule.

IV.2 SHG from single rod-on-dot CdTe/CdS heterostructures

The previous subsection dealt with SHG emission from single dot-shaped CdTe/CdS coreshell QDs using the TPSM technique. The strong second-order nonlinearity of the CdTe core with a large nonlinear coefficient 14 d up to 200 pm V -1 , 5 due to its non-centrosymmetric, tetrahedral zinc-blende crystalline lattice, 77,94-95 allowed for downsizing QDs to a 10-15 nm diameter, while maintaining a strong SH emission, in the range of 7.5×10 4 cts s -1 from a single spherical QD. This was achieved with standard 100 fs pulses of near infrared (NIR) laser light at 80 MHz repetition rate. The observed SH emission is polarization-sensitive, tunable and satisfactorily photostable out of the one-photon resonance. [START_REF] Zielinski | Second-Harmonic Generation from a Single Core/Shell Quantum Dot[END_REF] However, since these spherical CdTe/CdS core-shell QDs are composed of a 7-13 nm diameter CdTe core over-coated by a three monolayer's thick CdS shell (approximately 1 nm thick), the observed SH emission results mainly from the scattering process inside the volume of the octupolar -like CdTe crystalline lattice. The contribution of CdS shell to the different SHG field (when considered with classical additive rules for second harmonic generation from nanoscatterers) is then assumed to account for less than 10%, and can be neglected when simulating the total SHG SHG from single rod-on-dot CdTe/CdS heterostructures | 71 field. Besides, it was also unclear if such a thin layer of CdS, grown on the zinc blende-like CdTe core is well formed, and deprived of crystalline defects, which could possibly influence the bulk wurtzite nonlinearity. Therefore, the total SH emission was considered to originate only from the CdTe core, resulting in a good agreement between the experimental SHG polarization analysis and the theoretical model (see Subsection IV.1.2). [START_REF] Zielinski | Second-Harmonic Generation from a Single Core/Shell Quantum Dot[END_REF] The high potential of SC heterostructures in the field of efficient nanoscale light sources stems from the sensitivity of their optical properties, at both linear and nonlinear levels to modifications of their size, shape and material composition, providing the possibility to control and engineer. This flexibility results from quantum confinement, which strongly determines the density of states and hence the QDs optical properties. [START_REF] Alivisatos | Semiconductor Clusters, Nanocrystals, and Quantum Dots[END_REF]24 Significant advances in the field of nanoparticle synthesis has enabled fabrication of increasingly complex NPs, such as various core/shell heterostructures, [40][41][42]49 anisotropically shaped or branched NPs containing different semiconducting materials, 48,120-123 metal oxidesemiconducting quantum rods, 52 and semiconductor-metal nanocrystal heterostructures. [START_REF] Shaviv | Synergistic Effects on Second Harmonic Generation of Hybrid CdSe-Au Nanoparticles[END_REF]124 State-of-the-art techniques in the development of complex nanoscale objects provide the possibility to engineer the nonlinear properties of these structures. These properties would reflect the combination of tensorial optical responses from different assembled hybrid heterostructure components according to the precise structural engineering guidelines:

(

2) (2) (2) (2) 1 2 3
... where (2) hybrid χ is a third rank tensor of the second-order nonlinear susceptibility of the whole hybrid heterostructure, composed of susceptibilities corresponding to different constituent parts and the related materials with specific symmetries (2) 1(2,3,...) sym χ . This approach is of interest for matching the nonlinear nanometric size sources, e.g. their linear and nonlinear electromagnetic radiation, to any particular application in high resolution optical microscopy. This leads to explore semiconductor-semiconductor hybrid nanostructures in order to tailor their optical nonlinearities. [START_REF] Zielinski | Nanoegineering the Second Order Susceptibility in Semiconductor Quantum Dot Heterostructures[END_REF] In this subsection we report on nonlinear scattering from single hybrid SC QDs where the nonlinear emission is controlled through the choice of QD materials and their geometrical parameters. As a proof-of-principle concept validating experiment we consider RD quantum confined heterostructures made of two SC materials: CdTe and CdS, each having a different crystalline structure (zinc blende corresponding to the 43m point group symmetry for CdTe, and wurtzite with its 6mm point group symmetry for CdS), nonlinear susceptibility tensors and spatial geometry (quantum dot and quantum rod, respectively) leading to: In order to obtain a high SHG signal intensity from photostable RD NCs (where the TPEL yield is close to zero) it is not necessary to use such a high excitation power. For 3.0 mW of exc λ = 890 nm (when polarization of the incident optical electric field is oriented along the dipole moment of a single RD QD) we observed a signal intensity of about 7×10 4 cts s -1 (shown in Figure IV-13(a)), as strong as the typical luminescence level from bright QDs, de- spite the relatively low quantum efficiency of the APDs in the blue range: ~28% at 445 nm (at ambient temperature). Taking into account the transmission of the collection optics as well as the detection efficiencies, this corresponds to the SHG cross-section of the order of 50 GM.

IV.2.1 Special features of SHG emission from rod-on-dot QD heterostructures

Correction by the APDs efficiency would give a corresponding SHG photons count rate of ~9×10 4 cts s -1 , illustrating the real photon flux at the detectors. This surprisingly high value of the count rate was obtained from ~ 18 nm long RD NC hybrid (CdTe core 6 nm diameter and 12 nm long / 3 nm wide CdS rod), having a volume of about 113 nm 3 for the CdTe core and 85 nm 3 for the CdS rod, so that the total volume of about 198 nm 3 is nearly six times smaller than that of spherical CdTe/CdS core/shell QDs (~1150 nm 3 for 13 nm diameter QDs). [START_REF] Zielinski | Second-Harmonic Generation from a Single Core/Shell Quantum Dot[END_REF][START_REF] Zielinski | Nanoegineering the Second Order Susceptibility in Semiconductor Quantum Dot Heterostructures[END_REF] It is however important to mention, that excitation at exc λ = 890 nm for this particular RD NC does Measured in conditions guaranteeing a high level of photostability (without phasematching requirements), and corrected following the same procedure as described in Appendix A, the presented curve provides comprehensive information about spectral regions for optimized two-photon cross-section within the excitation range (currently limited by the setup capabilities). Excitation at exc λ = 925 nm instead of exc λ = 890 nm would then result with a nearly 50% higher count rate, then reaching 1.1×10 5 cts s -1 . SHG obtained with that excitation frequency could be also detected with higher accuracy by Silicon-based APDs, due to their improved detection efficiency at green. -11), can efficiently tune the spectral features thus allowing to optimize the SHG emission intensity. Tunable and high SHG crosssection peaks are then available even from smaller RD QD heterostructures, leading to signal count rates of the order of ~9.8×10 4 cts s -1 , when excited at the optimal frequency of 880 nm, using the same excitation power as for much larger QDs.

Understanding why, for smaller RD QDs the observable signal is still strong despite a reduced volume is a crucial issue, entailing two possible origins for that effect.

IV.2.2 Polarization analysis of the SHG emission of rod-on-dot CdTe/CdS QDs

One of the distinct and foremost advantages arising from the SHG process of nanoscale scatterers is undoubtedly its high sensitivity to the polarization of the incident optical electromagnetic field E ω . This enables to probe the spatial orientation of the scatterer with respect to the laboratory frame. Such experiments have been successfully conducted with both single-material inorganic and organic NCs, where the SHG response originates from a regular crystalline lattice (where it is relatively easy to analyze the experimental polarization response from a theoretical point of view). [1][2][START_REF] Zielinski | Second-Harmonic Generation from a Single Core/Shell Quantum Dot[END_REF]88,116 Nevertheless, the morphological complexity of the RD QD heterostructures, leads to several implications arising from the new geometry and physical features of such newly engineered hybrid NPs. Indeed, analysis of the SHG polarization dependency requires taking into consideration a precise definition of the mutual geometrical relations between the CdTe core and the CdS rod, which have been described in detail in Subsection III.4.

Decomposition of the x and y components of the optical polarization response of the emitted SHG field is achieved as a function of the rotation angle of the incident electromagnetic field E ω at the fundamental frequency ω, using the same method already exposed for dot-shaped CdTe QDs. The RD NC response can thus be specified by a single x y plane) is lower than 45°, which to some extent was forced by the spin coating conditions resulting in ~30 nm thick PMMA host-matrix, where the RD NCs were expected to lie on the surface. However, our model (described in Subsection IV. 3.2.1) can also consider the less frequent case of NPs with nearly vertically oriented rods.

IV.2.2.1 Simulation model for polarization analysis of RD QD heterostructures

In order to obtain a direct expression of the second-harmonic polarization response from a hybrid RD NC consisting of two different materials, we use a linear combination of components of the CdTe and CdS second-order susceptibility tensors, weighing their respective contribution by a fractional volume factor: (2) (2) 1 ( 2)

RD CdTe CdS CdTe CdS V V χ χ χ - ∝ + ⋅ .
The volumes of the spherical CdTe core and of the cylindrical CdS rod are evaluated by considering the shape of each part from TEM-based histograms of the size distribution (see Figure II-11) with the diameter of the CdTe core D CdTe = 6.0 nm and the diameter of the CdS rod D CdS = 3.0 nm. The length of the rod is treated as a variable, and its impact on the total SHG response is analyzed below. The contribution of SH fields generated from both materials to the total SHG field emitted by a single RD scatterer: , resulting from a linear combination of the second-harmonic fields can be obtained from the following expression:

( ) ( ) ( ) (2) (2) 2 (2) (2) (2) (2)
. .

I I I I v v c c E E E E ω ω ω ω χ χ χ χ χ χ * * * * * = + + ∝   ∝ + + +   ∑ (III.26)
where ( )

1 1 2 3 3 2 CdS CdTe CdS CdS CdTe v V V D L D - - = ⋅ =
is the volume ratio between CdS and CdTe, of the order of unity for the RD QD heterostructures reported here.

In intermediary manner, the CdS rod significantly reduces the relative CdTe contribution, leading to approximately equal contributions of both materials when the rod reaches a length of ~8 nm (corresponding to V CdS ≈ 57 nm 3 versus V CdTe ≈ 113 nm 3 ), while longer 11 -13 nm rods (V CdS ≈ 78 -92 nm 3 ) privilege the dipolar character of the SHG polarization response and prevail over the octupolar core contribution. An overgrown 20 nm long rod completely transforms the SHG field into a quasi dipolar-like. Taking into account two facts, primarily that the number of emitted SHG photons is proportional to the squared volume of the emitter and, secondly, the higher zzz nonlinearity of CdTe over that of CdS, it appears that the interference term can have an equally strong constructive or destructive contribution (see

IV.2.2.3 Errors in inferring the orientation of the crystalline lattice from SHG polarization analysis

Any quantitative discussion of the experiment-based simulation results requires a careful inspection of possible error sources. The first significant source undoubtedly comes from optical components inside the experimental setup, especially the dichroism and ellipticity dysfunctions from the dichroic mirror (γ DM (890 nm) = 0.035; φ DM (890 nm) = 2.8 rad) which effect the overall shape of the ( ) (2) 2 i P ω response, and have been indeed corrected.

Another source of error is associated with the signal noise or even with a small decrease of the signal intensity during the measurements (it is clearly visible, e.g. in case of QD1 in Figure IV-17), which deteriorates the simulation and distorts its result. To decrease this type of error and improve the simulation results we have used the raw data smoothing function -SF (defined in Appendix B), which leads to the same intensities for perfectly symmetrical maxima, and become a new set of input data for the simulation program. The SF error is characterized by the relative difference ratios in %, between the raw and smoothed x and y intensity maxima.

On the one hand applying the SF leads to the appearance of relatively small errors with respect to the raw data -in the worst case (QD1), it can be estimated at ±10.7% for the x, and +14.8%, -12.4% for the y component, while for much more regular raw data traces, e.g. QD2, QD3 it appears two times smaller. However, use of the SF significantly decreases the error of simulated fits -both in terms of maxima location and relative maxima intensities, providing well-fitted polarization response. Matching the raw data (or SF for higher efficiency) with the simulation fit -since it cannot be considered quantitatively -is determined from absolute differences, firstly in the maxima location (in °) on polar plots, and then in the relative maxima intensities of x and y components between SF and fit traces

( ) ( ) ( ) ( % % % ) SF fit I I I ∆ = ∆ -∆
. Overall shapes of both polarization responses are additionally characterized by the relative shape difference S xy (defined in Appendix B). The best fit was obtained for SF of QD1, showing its maximal displacements at about 0.7° (x) and 0.8° (y), with ∆I ~2.2% and the relative shape difference S xy = 0.28. A relatively small error was obtained for SF of QD2, with higher maxima displacements of about 7.3° (x) and 3.14° (y), and lower maxima displacement by 2.7° (x) and 0.7° (y), while ∆I was estimated to be of the order of ~1.7% and ~4.1% for lower and higher maxima respectively (with a shape difference S xy = 0.14). The highest errors occurred for QD3, showing its higher maxima displaced by 4.6 ° (x) and 3.2° (y), and lower maxima displaced by 0.1° (x) and 4.9° (y) (∆I ~28.3% for lower and ~0.7% for higher maxima; S xy = 1.12).

A final significant source of error stems from the estimation of the length of the CdS rod used for the simulation process. Best fits (Figure IV-17(b), and the fourth row of Figure IV-20) were obtained for rods ranging from 11 -13 nm, as expected. Simulation of QD2 data with rods varying in the range of 13 ±1 nm leads to ±1.5° of maxima displacements and about 4.0% higher I ∆ . Minimization of that error allows for precise determination of the rod length, with an accuracy of ±0.5 nm.

The error on the out-of-plane orientation of crystalline lattices is estimated to vary within ±5° to ±10° range, which remains acceptable considering that it can be applied with capable accuracy even for RD NCs oriented close to the vertical direction.

IV.3 Accuracy of the polarization analysis from a reduced set of raw data points

Efficient and tunable coherent SHG emission, together with the small size of CdTe -based QD heterostructures studied here open a wide field of applications in bio imaging. Since the SHG response obtained from nanocrystalline materials is highly sensitive to their spatial orientation and morphology, which has been proved for both, spherical and rod-on-dot QD heterostructures, [START_REF] Zielinski | Second-Harmonic Generation from a Single Core/Shell Quantum Dot[END_REF][START_REF] Zielinski | Nanoegineering the Second Order Susceptibility in Semiconductor Quantum Dot Heterostructures[END_REF] a possible application would be three dimensional single NP tracking inside a cellular system. Such an experiment, however, requires fast collection of an efficient enough SHG signal, since the nanolabel is expected to be rotating as a result of its interactions with the cellular environment. Our samples exhibit high SHG signal-to-noise ratio, however, so far all the polarization measurements were performed with a small and therefore time consuming increment of 2° for the incident polarization angle (taking 180 s for a full turn), resulting in high precision for the experiment. Therefore, it is important to determine an optimal increment value, in order to shorten the measurement and acquire a polarization response with intense enough SHG signal level to allow for its processing.

As a proof-of-principle, we consider raw data with a more complicated polarization Our analysis reflects the accuracy and reliability of the applied SF in order to decrease number of raw data points and shorten the measurement time. We found that the optimal value for the increment can be set at 24°, which leads to 15 s for collecting of the raw data and results in a reasonable error of approximately 2.5°. These studies, however, did not consider the possibility of a shorter integration time. With a high SHG signal intensity, which can be guaranteed in the case of currently studied RD QDs, the integration time could be shortened by at least by factor of 3, while the error would not vary significantly. Additionally excitation with optimized SHG cross-section with respect to the NP size, and higher value for the power of the incident beam (the analyzed polarization response has been obtained for 3.0 mW at exc λ = 890 nm, which can be still increased at no expense to the signal photostability) [START_REF] Zielinski | Nanoegineering the Second Order Susceptibility in Semiconductor Quantum Dot Heterostructures[END_REF] should result in a more intense SHG emission, thus allowing for even shorter integration times. Finally, using our algorithm towards three dimensional single NP tracking close to 'real time' data acquisition appears within realistic reach. The only limitation factor is therefore, accurate enough localization of the NP during its motion out of the focal volume.

IV.4 Chapter summary

Concluding our studies on the spherical CdTe/CdS core/shell QDs with ~12.5 nm diameter, it is important to highlight, that choosing these SC NPs allowed for detection of an efficient SHG signal of the order of ~7.5×10 4 cts s -1 . Such a small NPs have been detected for a first time with a standard APD detectors, what designate them as competitive nonlinear nanolabels, an alternative to fluorescent labeling. [START_REF] Zielinski | Second-Harmonic Generation from a Single Core/Shell Quantum Dot[END_REF] Within the standard 800 to 1000 nm spectral range of the femtosecond Ti-Sapphire laser excitation, these QDs show a sizable SHG response with highest SHG cross-section at about 970 nm. However, an extended spectral analysis towards one-photon and two-photon resonances at the conduction band edge would reveal higher SHG cross-section peaks, providing also comprehensive information about the role of the DOS in the nonlinear optical processes, such as coherent SHG. We have shown that the unique features of SHG emission, such as coherence of the emitted field or strong polarization sensitivity may be exploited to increase the signal-to-background ratio or the recognition of these NPs. [START_REF] Zielinski | Second-Harmonic Generation from a Single Core/Shell Quantum Dot[END_REF] Simultaneous measurements of TPEL emission and blinking properties would, moreover, allow us to probe the NP environment, building-up on a wellcontrolled orientation of the nanoparticle crystalline structure. SHG-analysis alone can also be used for orientational imaging, e.g. three-dimensional single NP tracking.

In the second part of the chapter we have shown, that tailored QD heterostructures may lead to even higher effective SHG cross-sections, tunable with the quantum size confinement effect. In this approach we have pointed out guidelines towards the engineering of secondorder nonlinearity of semiconducting hybrid rod-on-dot CdTe/CdS heterostructures at the single NP scale, which provide photostable SHG emission at comparable level like that observed for the spherical dots. These RD hybrid NPs exhibit extremely strong and stable emission although their volume is reduced by at least a factor of four (up to six) from the dotshaped CdTe/CdS QDs. Moreover, it emphasizes the structural relationship between the CdTe core and CdS rod and contributions of their respective (2) χ tensors to the overall SHG field, leading towards a hybrid nonlinear optical susceptibility, which exceeds that of monocompound QDs. The proposed pointwise additive model allows for comprehensive simulation of the polarization response, evidencing quantitative contributions into the total SH emission from both the CdTe core and the CdS rod, as well as from an additional interference term. The high degree of agreement reached between simulation and experimentally observed results cannot be considered as circumstantial but as evidence for the single crystalline nature of the investigated QD samples. It enables precise determination of the spatial orientation of the crystalline structures. It also shows that the hybrid system can be adequately decomposed into a sum of its constituents in the case of semiconductor-semiconductor hybrids. This simple result had to be demonstrated and is even surprising since it does not hold for metalsemiconductor hybrid particles. [START_REF] Shaviv | Synergistic Effects on Second Harmonic Generation of Hybrid CdSe-Au Nanoparticles[END_REF] Still, coupling of the two semiconductor parts may affect the relative magnitude of the (2) χ coefficient. Simulation analysis show that the SHG response strongly depends on the CdS rod length and transforms from the four-leaf clover shape characteristic of the octupolar zinc-blende symmetry of the CdTe core, into a typically dipolar response oriented along the CdS rod and imposed by the main polar axis of its hexagonal symmetry point group. The conceptual and practical possibility to fine tune at will the balance between dipolar and octupolar contributions by adequate stoichiometry based engineering, while maintaining control over this delicate multipolar balance at the nanoscale, and minimizing the size of RD NCs, paves the way towards very efficient and precise orientation monitoring for original bio-imaging applications.

Chapter V | Size dependence of the second-order nonlinearitysingle dot and HRS studies In the following chapter the size dependence of the magnitude of the second-order nonlinear susceptibility (2) χ of CdTe QDs, as well as its dispersion features are investigated. Applied experimental methods are the single particle spectroscopy and hyper-Rayleigh scattering (HRS). In these studies we try to address the question as to how quantum confinement effects can modify the nonlinear optical susceptibility of SC QDs. We show that under adequate conditions, the (2) χ tensor magnitude can significantly surpass that from the corresponding bulk value. These studies prove, most importantly, that careful choice of the material and of the excitation frequencies can result in kilo-counts per second rates for the emission of SHG photons, obtained even from very small appropriately chosen QDs.

V.1 Size dependence of SHG from single CdTe quantum dots

In the previous chapter it was already stated that SHG obtained under nonresonant conditions, as a coherent scattering process, scales as the squared volume of the nanoscatterer. 82 In case of nonlinear properties of nanoscale SC materials, size dependence becomes particularly interesting due to quantum confinement effects and its influence over the energy level spacing. However, until now effects of quantum confinement on coherent nonlinear processes, such as SHG in QDs have not been widely investigated. [3][START_REF] Zielinski | Second-Harmonic Generation from a Single Core/Shell Quantum Dot[END_REF][START_REF] Zielinski | Nanoegineering the Second Order Susceptibility in Semiconductor Quantum Dot Heterostructures[END_REF] Therefore, characterization of the SHG size dependence is particularly important if QDs are to be used as markers in SHG microscopy. [START_REF] Zielinski | Second-Harmonic Generation from a Single Core/Shell Quantum Dot[END_REF] Chemical synthesis and morphological characterization of CdTe spherical QDs used in the following studies have been already described in Subsection III. we have chosen to work under conditions whereby the SHG efficiency is nearly independent of the relative orientation of randomly deposited QDs. In order to obtain such conditions, all studied QDs were excited by a circularly polarized incident excitation beam. The fundamental electric field is expressed in the laboratory frame as:

1 1 2 0 lab E E i     =       (V.1)
Rotating it in respect to the crystalline lattice frame leads to: cos sin 0 cos 0 sin 1 cos cos sin

sin cos 0 0 1 0 cos sin cos 2 2 0 0 1 sin 0 cos 0 sin crystal i E E E i i φ φ φ φ θ φ φ φ φ θ φ φ φ φ θ +                 = - = - +                 - -         (V.2)
where θ is the angle of the z-axis in the crystalline lattice frame relative to the z-axis of the laboratory frame and φ is the angle measuring rotations about the z-axis of the crystal.

Rotations about the z-axis of the laboratory frame need to be considered due to use of the circularly polarized excitation beam. Combining equations (III.10) and (V.2) for the CdTe zinc blende crystalline lattice we obtain:

( )

i P d E i i i θ θ φ θ φ ω θ θ φ θ φ θ φ φ θ φ θ φ φ φ -     = - -     - + - -   (V.3)
Therefore the SHG signal intensity is expressed as:

(

P E d ω φ φ θ φ θ θ θ φ θ φ θ = - + - - - - - - + + + (V.4)
This expression is weakly varying in terms of theta and phi values (up to approximately ±25%).

V.1.1 Detection threshold of the size dependent SHG signal intensity

SHG size dependent statistical measurements were performed using the optical setup equipped with low noise single photon counting detectors and the TCSPC module (Figure III-5). This is crucial for the detection of a low amount of SHG photons from small (~7 nm diameter) QDs, allowing to select only the instantaneously scattered photons concomitant with the pulse, excluding dark counts of the detector (the ~20 dark counts per second of the detector are minimized here to about ~0.03 counts per pixel, which is negligible). An example of the SHG scan obtained with this technique is presented in Figure V-1. The observed background arises from coherent scattering on the PMMA-glass cover slip interface, which arrives into the detector with the pulse. Typical measured coherent background levels range from 1 to 2 counts per 100 ms (10 -20 Hz), which were subtracted from the analyzed peak intensities of the SHG signals corresponding to QDs.

Preliminary results have evidenced the importance of choosing the correct threshold value for identifying peaks, which correspond to SHG emission from QDs. This appears especially important in the case of the smallest QDs with a mean diameter of ~7 nm, which have low SHG yield. Some of the possible SHG peaks featuring the lowest count rates turned out to be simply a background noise, due to the overlap between the SHG distribution and the shot noise distribution. The choice of the threshold is a tradeoff between reduction of the background noise pixels and the desire to retain the maximum number of data points, resulting from actual QDs SHG emission. We assume that the background is spatially homogenous and that it follows a discrete Poissonian distribution, the standard deviation (σ) of the background signal is then its square root. The correct standard deviation σ was determined comparing the results obtained for the larger QDs sample (~11 nm). Using thresholds of 3σ, 4σ, 5σ, 6σ and 7σ above the background, the numbers of detected QDs (and corresponding mean SHG count rates in the brackets) were: 827 (250 Hz), 615 (320 Hz), 393 (420 Hz), 350 (500 Hz) and 327 (530 Hz), respectively. From the shot noise distribution, however, we expect that >230 of these peaks in the 3σ and >40 under the 4σ threshold to result from noise. The mean count rates for 6σ and 7σ thresholds are nearly the same within a few percent due to the eliminated noise pixels. However, raising the threshold to these values leads to significant reduction also the number of real data points, therefore we chose a 5σ threshold as a most reasonable value, keeping in mind that some noise peaks are included in the analysis, whereas symmetrically some real data points have been lost. Exceptionally for the smallest size QDs, we used a lower 4σ threshold, since in this case 5σ eliminates the majority of real data points. * corresponding to spherical CdTe/CdS core shell QDs, studied in Ref. [4] QDs. This is reasonable, since the mean diameter of the CdTe core is about 10.5 nm (excluding the CdS shell, which has been assumed not to contribute to the total SHG field).

The size distribution of 11 nm CdTe dots, determined in Section III. distribution, and correspond to large volumes. This dependence points out the difficulty in detecting second-order nonlinear scattering from QDs, since under conditions used here (800 nm of the excitation wavelength), reasonably high count rates (suitable for standard APD detection) are obtained only from CdTe QDs with diameters larger than 14 nm.

V.1.2 Estimate of the second-order nonlinear susceptibility

In order to obtain the absolute value of the second-order nonlinear optical susceptibility (2) CdTe χ of the investigated CdTe QDs, we had first to calculate the electric excitation field and the second-order nonlinear polarization. Without the possibility of determining the crystalline axis for each single dot, we integrate in these statistical studies the nonlinear polarization response (V.4) over all possible crystal orientations, leading to:

( )

2 4 4 2 2 14 14 2 4 1.2 4.8 P E d E d ω × = ≃ (V.5)
The angle dependent ( )

2 2
P ω values vary by up to ~25% and the effect of the fixed detection angle and corresponding alignment-dependent SHG detection efficiency was numerically found to be small. The nonlinear CdTe coefficients ( 1425 36

d d d = =
) can then be determined by the integrated nonlinear polarization response which expresses as:

( ) ( ) 1/ 2 2 14 2 2 2.2 P d E ω = (V.6)
The electric excitation field can be extracted from the power density of the excitation beam.

In Gaussian units, the power density I is given by:

2 0 2 n c I E π   =     (V.7)
where ( )

1/2 0 0 n ε =
is the refractive index of the surrounding medium (PMMA polymer), c is the speed of light, and E is the electric field.

To obtain the real value of the electric field inside the NP sphere, we take into account its reduction inside the QD due to the higher dielectric constant of CdTe as compared with that of the surrounding PMMA polymer, as from equation (III.11). Accordingly, E should be multiplied by a ( )

/ 2

m m ε ε ε + factor, 82 where ε is the dielectric constant of the scattering nanosphere (CdTe QD), leading to:

2 2 0 3 2 2 m m I E n c ε π ε ε   =   +   (V.8)
The ielectric constant ε of CdTe at 1.5 eV (800 nm) is ~9.0, 39,129 and that of PMMA is

~2.25 m ε . 130
The power density is estimated using the laser power measured at the front of the microscope. In order to obtain the actual value reaching the sample it has to be corrected by the reflectance of the dichroic mirror and transmission of the microscope objective at 800 nm.

The power reaching the sample (~2.3 mW) is then divided by the illumination field size, obtained from the diffraction limited spot size. We have observed some uncertainties in determining the illumination area. The waist of the diffraction limited spot was found by multiplying the measured FWHM of the SHG spot size from 50 individual QDs (~208 nm) by 2 since SHG is a second-order process. Thus, the mean FWHM is found to be 295 ±37 nm, leading to an error of 25% on the area. From this diameter the illumination field size is found to be 7.45×10 4 nm 2 .

To calculate the second-order nonlinear polarization ( )

2

P ω , we need the power (p) of the emitted SHG signal since: Obtained for a 11 nm QD sample, the 420 Hz SHG count rate (using 5σ threshold) leads to a 14 d value of 1.5×10 -7 cm statvolt -1 , what corresponds to 64 pm V -1 (using the convention that, in MKS units system 2 0

P dE ε =
). v This value has been assumed in the model for polarization analysis of RD QD heterostructures described in the previous chapter.

V.1.3 Error sources in determining the size dependence of the second-order nonlinear susceptibility

In our single dot studies, three main sources of error have been identified (all of them are summarized in Table (V-II)). The first one comes from difficulties in determining the properties of the background coherence. Since we do not know its polarization, it is difficult to remove it from the measurement. Therefore we have assumed it is incoherent and subtracted it from the measured count rate, which may have led to an overestimation of the SHG counts. The threshold 5σ used to distinguish QDs from the background counts also introduces error in the SHG count rate, since as a tradeoff threshold means that still some of the included pixels are noise, while using a higher threshold leads to the removal of real data

points. An error of ±20% is estimated for the SHG count rate, which is the difference in the mean count rates obtained when using the 4σ and 6σ standard deviations.

The second significant source of errors arises from the average QD volume. Since values of the nonlinear coefficients d scale with the assumed volume of the scatterer, even relatively v The conversion factor from statvolt cm -1 to pm V -1 is small errors in determining the QD diameter may be significant. We estimate the error in the determination of the QD volumes as ±15%, by taking the difference between the mean volumes as shown in Figure II-6. We note that our analysis assumes that (2) χ can be regarded as dependent only on the NC volume within the size range of our samples. This is not entirely fulfilled but is a reasonable approximation for the relatively narrow experimental size distribution.

The third significant source of error arises from uncertainties in determining the illumination area. The measured FWHM of the spot size of the SHG signal from 50 individual QDs has been found to be 295 ± 37 nm, leading to an error of 25% on the area.

Other small sources of error include errors in the signal collection efficiency (< 10%), the detector quantum efficiency at 400 nm (< 10%), the pulse duration at the sample (~10%), and losses through the optical components (~10%). We also neglect dielectric effects on the dipole emission pattern due to the asymmetry in refractive index between the glass substrate and air. Taking into account that tensor (2) χ follows a square root dependence on the SHG count rate, losses due to the optical components inside the detection channel, and the detector quantum efficiency, a linear dependence on all other quantities, and that the errors are uncorrelated and can be added in quadrature, the (2) χ of CdTe QDs using 800 nm excitation lies within the range 43-85 pm V -1 . 5

V.2 Hyper-Rayleigh Scattering from CdTe QDs

The Hyper-Rayleigh Scattering (HRS) experiment was performed to confirm results obtained from single dot studies, discussed in the previous paragraph. Measurements were carried out for four CdTe QD samples of 5, 7, 9, and 11 nm diameters, each one at several concentrations. QDs were firstly separated from their ligands by centrifugation and then dissolved in toluene. The concentration of each solution was characterized using the Beer-Lambert law:

A CL = ε (V.10)
where A is the absorbance of the first exciton peak in the absorption spectrum, ϵ is the extinction coefficient per mole, C is the molar concentration of the solution, and L is the length of the optical beam path used for recording the spectrum.

The experiment was performed by focusing the linearly polarized output beam of a Spectra-Physics Tsunami Ti-Sapphire laser ( exc λ = 800 nm, 130 fs, 80 MHz; the same as used before for single dot studies) onto a quartz cuvette containing the QD solution and collecting the scattered signal at 90° with respect to the direction of the NIR excitation laser beam (see Our analysis of the HRS signal followed that of conventional HRS experiments. 3,131 A value of the first hyperpolarizability CdTe β per unit cell of CdTe relative to the hyperpolarizability of the solvent s β (toluene in our case) can be extracted from the linear dependence of the HRS signal on the sample concentration, as given by following expression: The number n of unit cells in a QD of volume V is:

( ) ( ) ( ) 2
CdTe A CdTe d n V N M = (V.13)
where CdTe belongs to zinc blende crystalline symmetry (with three 14 25 36

d d d = =
non-zero tensor elements), while for toluene we assume that the only contributing element is zzz β (corresponding to the 33 d matrix element). In order to find the total SHG signal intensity for both CdTe and toluene we follow the method described in Section V.2 taking into account a linearly polarized fundamental field, and assuming that the only collected SHG signal is that one propagating at 90° through a low -NA objective. After summing over all angles we found that in order to satisfy the ratio of total signal intensities / The nonlinear susceptibility il d for the bulk material can be expressed in terms of the firstorder hyperpolarizability ijk β : 132

( ) ( ) 2 0 2 il ijk N f f d ω ω β ε = (V.14)
where N is the number density and ( )

f ω is the local field factor of the material at the incident frequency ω , and 0 ε is a vacuum permittivity. In our experiment the local field factor must include the reduced screening in the composite solution of QDs dissolved in toluene as compared to the screening in a bulk semiconductor material, a corrected local field factor being then: 82 ( ) ( )

2 2 2 CdTe CdTe CdTe n n ε ω ω ≈ ≈ = 9
.0 is the dielectric constant of bulk CdTe. 39,129 The experimental value of β of toluene at 800 nm was found in the Ref.

[134] to be ~0. 23× 10 -30 esu. We take into account the factor of 5.5 of zxy β relative to the hyperpolarizability of toluene, which leads to a CdTe nonlinear coefficient 36 d = 88 pm V - 1 , 135, vi which is in a relatively good agreement with the single QD studies, 5 despite a relatively large experimental error. The result of 36 d = 88 pm V -1 is expected to be subject to an error larger than 25%, arising from that in determining the solution concentrations from the optical densities (~20%), the uncertainty in the hyperpolarizability value of toluene (5%), and the standard deviation of / CdTe s β β obtained from four different QD samples (10%). Another source of error comes from the assumption of the bulk value for the dielectric constant of CdTe, for which the size dependence is expected to be small, 136 which can nevertheless overestimate the d value. A last source of error is the polarity of the solvent, which may lead to an error for the local field factor in expression (V.14), however it is assumed to be small for toluene, 137 and thus negligible in our analysis. * where G is a geometrical factor including the local field factors vi Using the conversion factor of: 1 C 3 m 3 J -2 = 2.7×10 20 esu -1 Relations between spectral and size dependence of CdTe χ(2) features | 105

V.3 Relations between spectral and size dependence of CdTe χ (2) features Both, single dot spectroscopy and HRS investigations of the (2) χ size dependence of CdTe QDs, [START_REF] Winter | Characterization of the Second Order Nonlinear Susceptibility of Quantum Confined Semiconductors[END_REF] have confirmed our earlier observations, reported before for single QD spectral dependence measurements (Subsection IV. Combining these two approaches, the wavelength dependent dispersion curve of (2) χ and numerically obtained values opens the possibility for comparing the nonlinear susceptibility of QDs to that of bulk CdTe, as illustrated in Figure V-6. χ dispersion curve, measured for spherical CdTe QDs (marked with the black line). The grey area is related to the variation of bulk d 36 coefficient, scaled at a similar ratio of the excitation energies between 12 nm CdTe QDs (E BG ~760 nm, excited at 970 nm), and bulk CdTe (E BG ~840 nm, excited at 1.06 µm). [START_REF] Zielinski | Second-Harmonic Generation from a Single Core/Shell Quantum Dot[END_REF][START_REF] Winter | Characterization of the Second Order Nonlinear Susceptibility of Quantum Confined Semiconductors[END_REF]94 The obtained result of 36 d (800 nm) = 64 ±21 pm V -1 is close to the bulk values of 73 pm V -1 (at 1.5 µm) -109 pm V -1 (at 1.06 µm), reported in the literature. 94 Changing the excitation wavelength from 800 nm to resonance with an optimized SHG cross-section at ~970 nm (as shown in Figure IV-5) would lead to four times increase at least for the value of (2) χ , resulting in 36 d ≥ 250 pm V -1 . When comparing the 11-12 nm diameter large CdTe QDs ( BG E ≃ 1.63 eV) illuminated at 970 nm laser light, with the bulk CdTe crystal ( BG E ≃ 1.47 eV) at 1.06 µm, the excitation conditions in terms of the relative excitation energy relative to the band gap appear to be similar. Therefore due to the quantum confinement effects, (2) χ of the single QD can significantly exceed its bulk value, showing completely different behavior when compared to the expected monotonous-like (2) χ dispersion of the bulk crystal. The HRS experiment conducted for smaller 2.4 nm diameter CdSe QDs reports 33 d nonlinear coefficient of 160 pm V -1 , 3 which is much larger than the one obtained for 9 nm diameter QDs and reported for the bulk CdSe. This has pointed out a possibility for the increase of the nonlinear susceptibility by quantum confinement effect, observed while the QDs size has been decreased. As the separation between the band gap and excitation energies shrinks, i.e. as the QD size becomes larger, the resonance enhancement should increase due to the higher density of states further from the band edge, whereas the opposite effect is seen. 3 Therefore, the reported results are attributed to surface SHG, which becomes more significant as the NP volume decreases.

Summarizing our results on the spectral dependence and size dependence experiment of the relative (2) χ , we suggest that resonant effects are important.

General conclusions

This thesis concerns the development of new types of SHG-active probes, which are quantum confined semiconductor nanocrystals, in order to break through the barrier of the smallest detectable size, met for previously researched nanoparticles (>40 nm for insulators or wide gap semiconductors, with SHG count rates of at least 1×10 3 cts s -1 ) towards two-photon nonlinear microscopy. [1][2][84][85][86][87][88]138 The main questions at the starting point of this work were: is the efficiency of nanoscale scatterers really limited by the material bulk properties? If so, how can we deal with such limitation to as to signal, and significantly shrink the size of nanoprobes? As are shown in this work, the most important factors towards the achievement of positive results are a proper choice of materials, well developed chemical synthesis and characterization methods of the samples, and most importantly, optimized conditions for nonlinear optical measurements.

The selected materials are quasi-type-II QDs, based on the CdTe core and CdS overcoating shell, eventually evolved into quantum rod, which belong to the II-VI group of direct gap semiconducting crystals. This choice was made for several reasons (discussed in detail in Chapters I & II). Firstly, CdTe exhibits relatively strong second-order nonlinearity, characteristic of a noncentrosymmetric zinc-blende structure, what is a prerequisite to evenorder terms (corresponding to odd rank tensor) nonlinear scattering processes, as the lowest order one being SHG. 58,77,94 The energy band gap of CdTe QDs can be tuned in a range from 1.44 eV up to about 2.0 eV as a function of QD size, [START_REF] Wei | Calculated natural band offsets of all II-VI and III-V semiconductors: Chemical trends and the role of cation d orbitals[END_REF] and corresponds to the tunable range of frequencies of a Ti-Sapphire fs oscillator. Finally the most important reason, as we realize now, under a strong / intermediate quantum confinement regime is the dipole moment of CdTe QDs and the related two-photon cross-section, which exhibits remarkable enhancement when compared to that of bulk. [START_REF] Winter | Characterization of the Second Order Nonlinear Susceptibility of Quantum Confined Semiconductors[END_REF] In the first report [4], it was shown that efficient and photostable SHG radiation of the order of ~7.5 ×10 4 cts s -1 (nearly 75 times above assumed the lowest detectable value) could be obtained for an individual isolated ~12.5 nm diameter core/shell CdTe/CdS QD under reasonable intensities of the fundamental light. Detailed studies presented within the first part of Chapter IV, shown the assets from the coherent SHG-active QD nanoprobes. Moreover, in contrast to fluorescent QD labels, [110][111]139 SHG active CdTe QDs do not exhibit any emission intermittency such as 'blinking', nor bleaching. Since SHG is a frequency conversion process achievable in non-resonant conditions, its narrow emission peak can be tuned following that of the fundamental light far from the TPEL range, allowing to filter out the latter. Therefore, it may be also easily separated from the auto-fluorescence emission of the biological specimens.

Two-photon resonance conditions, however, can increase TPEL emission, which may then lead to decay of both TPEF and SHG signals. The same contradiction between quantum yield of the TPEL emission and photostability of the SHG signal was observed for different types of samples. [START_REF] Zielinski | Second-Harmonic Generation from a Single Core/Shell Quantum Dot[END_REF][START_REF] Winter | Characterization of the Second Order Nonlinear Susceptibility of Quantum Confined Semiconductors[END_REF][START_REF] Zielinski | Nanoegineering the Second Order Susceptibility in Semiconductor Quantum Dot Heterostructures[END_REF] It was shown that higher TPEL QY may lead to strong fluctuations of the emitted light and finally destroy the dot, while QDs emitting pure SHG signals exhibit remarkable photostability, and endure extremely high pulsed peak intensities up to ~150 GW cm -2 . The effect of reduced stability can be attributed to induced excited state absorption at the excitation frequency of QDs exhibiting stronger band-edge luminescence, while for low luminescence QY the excited state population completely decays nonradiatively within the ~12 ns time interval between excitation pulses. Additional studies are necessary for better understanding of these issues.

Within the standard 800 to 1020 nm spectral tenability range of the fs Ti-Sapphire laser excitation, spherical CdTe/CdS QDs display their highest SHG cross-section around 970 nm. [START_REF] Zielinski | Second-Harmonic Generation from a Single Core/Shell Quantum Dot[END_REF] An extended spectral analysis towards one-and two-photon resonances at the CB edge is expected to reveal even higher SHG cross-section peaks. Such studies would lead to obtaining comprehensive information about the role of DOS in nonlinear optical processes, such as coherent SHG. Therefore, two-photon excitation of the CdTe QDs within a range from 760 nm to 1.52 µm (using an optical parametric oscillator (OPO), and a frequency doubler) retains as an open question and is planned as a continuation to this work.

The new approach for purely semiconducting hybrid QD heterostructures, discussed in the second part of Chapter IV, shows that further engineering of the structural and morphological features at the single nanoparticle level may lead to much higher optical nonlinearities, | General conclusions exceeding that of single-material QDs. The RD QD hybrids built from 6 nm diameter CdTe core, and attached to ~12 nm long / 3 nm wide CdS rod, exhibit comparable SHG signals to that observed from ~12.5 nm diameter spherical CdTe QDs. At the optimal excitation range of 925 nm, where the SHG cross-section reaches a maximum, the obtainable signal count rate resulting from coupling the two materials may reach about 1.1×10 5 cts s -1 , although the overall volume is reduced by at least a factor of four (up to six) from that of spherical QDs. [START_REF] Zielinski | Nanoegineering the Second Order Susceptibility in Semiconductor Quantum Dot Heterostructures[END_REF] Taking into account the transmission of the collection optics as well as the detection efficiencies, this corresponds to a SHG cross-section of the order of 50 GM, as obtained from a standard size RD QD (with a volume of about 200 nm 3 ). This result is remarkable, since reported SHG cross-section value for 100 nm diameter BaTiO 3 (what gives 5.23×10 5 nm 3 ) is equal ~1000 GM. [START_REF] Pu | Nonlinear Optical Properties of Core-Shell Nanocavities for Enhanced Second-Harmonic Generation[END_REF] Obviously, the volume ratio of 1:2600 does not scales with the 1:20 ratio between SHG cross-sections of BaTiO3 and RD QD hybrids. Moreover, the formation of such structures has further contributed to their photostability. This progress has been attributed to the smaller size of the RD QDs, which leads to slower decoherence process. Basically, larger intraband level spacings of small QDs slow down the exciton -LO phonon coupling, which reflects in more intense and stable SHG radiation.

SHG emission from both types of QDs was analyzed in terms of their incident polarization.

The experimental results relate to the crystalline symmetries of the studied nanoparticles. The perfectly octupolar lattice of CdTe QDs features the characteristic four-leaf clover polarization pattern, which is also a proof of monocrystallity. [START_REF] Zielinski | Second-Harmonic Generation from a Single Core/Shell Quantum Dot[END_REF] The same experiment carried out for RD QDs provides substantially different results, evidencing the expected dipolar like polarization response. Analysis of these data, however, due to the complex structure of these hybrid NPs required implementation of a new algorithm. A pointwise tensorial additive model allowed for comprehensive simulation, showing quantitative contributions of the CdTe core and the CdS rod, as well as that of an additional interference term into radiated SHG, which depends in a linear manner on the CdS rod length. [START_REF] Zielinski | Nanoegineering the Second Order Susceptibility in Semiconductor Quantum Dot Heterostructures[END_REF] For both spherical and hybrid structures, a high degree of agreement between simulation and experimental observations was reached, allowing to determine the spatial orientation of the probed crystalline structures. It has also been shown that the hybrid system can be adequately decomposed into a sum of its constituents in the case of purely semiconducting hybrids. [START_REF] Zielinski | Nanoegineering the Second Order Susceptibility in Semiconductor Quantum Dot Heterostructures[END_REF] This simple result is all the more interesting and surprising as it does not hold for metal-semiconductor hybrid particles. [START_REF] Shaviv | Synergistic Effects on Second Harmonic Generation of Hybrid CdSe-Au Nanoparticles[END_REF] These vectorial-like properties of the SHG radiation pave the way towards very efficient and precise orientation monitoring in original bio-imaging applications, e.g. three dimensional tracking of single nanoparticles.

Considering the size and spectral dependence of the second-order nonlinear susceptibility of CdTe QDs, as discussed in Chapter V, allowed to reach its absolute values at different conditions. Using single nanoparticle spectroscopy studies and complementary HRS experiments, it was shown that at 800 nm of the excitation frequency, the nonlinear coefficient d 14 of ~11.5 nm diameter CdTe QDs is found to equal 64 pm V -1 . [START_REF] Winter | Characterization of the Second Order Nonlinear Susceptibility of Quantum Confined Semiconductors[END_REF] Following the spectral analysis, which was carried out in Ref. [8], these may lead to a value of 250 pm V -1 , largely exceeding the bulk value ranging from 73 pm V -1 (at 1.5 µm) to 109 pm V -1 (at 1.06 µm). 77,94 This study illustrates the importance of quantum confinement, and point out its remarkable impact on the resonant enhancement of the SHG nonlinear scattering process.

As a final remark, this work supports the case future applications of semiconducting QD heterostructures as new probes for two-photon polarization sensitive nonlinear microscopy.

The careful choice of material properties such as quantum confined CdTe QDs was justified, allowing to significantly decrease the nanoparticle size much below the earlier ~40 nm barrier. The second-order nonlinear susceptibility features in two-photon resonance conditions are maintained and follow quantum confinement selection rules. Moreover, such previously unexplored results and adapted simulation algorithms based on quadratic tensorial properties discussed within this thesis, path the way towards advanced applications at the subwavelength regime for optical probing. Transmission correction of the raw data was calculated using following expression: . ( Taking into account equations (A-III), (A-IV4) and (A-V), we finally get:

| General conclusions

VIS

( ) ( ) ( ) ( ) )
( ) ( ) ( ) 2 2 4 1 IR SHG photons sample rep sample P N s f χ λ λ τ   ∝ ⋅ ⋅     ⋅   (A.6)
which finally gives the second order nonlinear susceptibility ( ) ( ) 2 χ λ described by the following equation:

( ) ( ) ( ) ( ) 4
χ = =                 = =                         = (B.1)
Applied optical electric field of a linearly polarized excitation beam is expressed as:

2 ( ) cos 1 ( ) ( ) (1 ) sin 1 (1 ) ( ) 0 DM x i y z E E E e E ϕ α α α α γ α γ α         = = -     + -         (B.2)
where: γ is a dichroism and DM ϕ is a phase shift -wavelength dependent properties of the dichroic mirror, between x and y polarization components of the emitted SHG signal (shown in Figure A-10); α is the angle of the linear incident polarization determined in the ( ) , x y plane.

The i -component of second-order nonlinear polarization is equal to:

(2)

2 i ijk j k j k P d E E = ∑∑ (B.3)
We assume that z polarization component is equal zero, and take into consideration the x and y polarization components, which can be written in a following form:

( ) ( ) 111
α γ ϕ α α γ ϕ γ α α γ ϕ α α γ α α α α = + + - +   + -     + + + - +   + + - + - = = + ⋅ + ⋅ + ⋅ 4 4 4 x tan ) cos x SF α α + ⋅ = (B.8)
Which relates to the x polarization component on polarization graphs, where: 2) tan ( 2) tan ( 2) tan ( 2) cos (4 ) 1 tan( 4) tan ( 4) tan ( 4) tan ( 4) cos (360 ) 1 tan(360 ) tan ( 360) tan ( 360) tan ( 360) .20) is the coefficient matrix (written for 2° increment, used for collection of the raw data in experiment), and

2 0 111 2 2 4 x 1 (1 ) d γ = ⋅   + -   (B.9) ( ) 1 112 121 111 2 2 4 (1 ) cos x 2 1 (1 ) MD d d d γ ϕ γ ⋅ - = ⋅ +   + -   (B.10) ( ) 2 2 2 112 121 122 111 2 2 4 (1 ) x 2 cos 2 1 (1 ) DM d d d d γ ϕ γ ⋅ -   = ⋅ + +     + -   (B.11) ( ) 3
B             =                 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ (B

B.1.2 Positions of the maxima

Positions of the SF maxima are determined by calculation of the roots of first-order SF derivative (and by analysis of the sign of its second-order derivative, which in fact has the same form), here shown for I x : x ( 4cos sin ) x (cos 3cos sin )

x (2cos sin 2 cos sin )

x (3cos sin sin ) x (4cos sin )

x cos (2x 4x )cos sin (3x 3x ) cos sin The roots of SF derivatives are equivalent to roots of fourth-degree polynomial: Following the expressions (B.25) and (B.30) we obtain: for exc λ = 970 nm (optimal excitation wavelength for spherical dots, used for polarization analysis), we estimate γ = 0.035 and DM ϕ =1.05 rad. Therefore we obtain:

                =                         (B.
P A B A C D B D E ( , , ) A C D C Q F D F G B D E D F G E G R xyz XYZ d d φ θ ψ                 = Ω ⋅ =                         (B.

B.1.4 Polar graph matching criteria

The difference between calculated and experimental angular positions 1 α , and 2 α of the maxima for x and y polarization components are used to define expressions for different shapes I S :

( )

I 1max, , 1max
M M M M φ φ θ θ ψ ψ φ φ ψ ψ θ θ - -             = =             -       (B.
R α α α α         = - =         -     (C.12)
Thus the overall rotation of CdS matrix is:

CdS CdS,II CdS,I 1 0 0 0 0 1 0 0 1 0 0 1 0 1 0 1 0 0 0 1 0

1 0 0 0 1 0 R R R             = = = -             - - -       (C.
            = = - =             -       (C.

hybrid tensor components

The simulation algorithm of the RD QDs SHG polarization response is similar to the one used for a spherical CdTe dots. Direct simulation, however, needs to take into account the morphological properties of the RD QDs structure, which relates to a direct combination of two (2) ( ) CdTe CdS χ tensors with their contributions proportional to the volume of each material:

(2)

(2) 1 ( 2

P 0 0 0 C 0 0 0 C 0 C 0 C 0 F 0 F 0 0 0 C 0 F 0 C 0 F CdTe XYZ d                 =                 -         (C.15)
and for 14 64 d = pm V -1 (Winter et al. [5]) results with: P 0 = 73.9, C 0 = -36.95, and F 0 = -52.26 pm V -1 .

In case of the CdS rod we rotate following tensor:

                =                         (C.
CdS XYZ d                 =                         (C.17)
The result of this rotation are: K 0 = 15 d = 44.0 pm V -1 , T 0 = 31 d = 40.2 pm V -1 , and S 0 = 33 d = 77.9 pm V -1 , where the nonlinear coefficient values were found in the literature. 77

C.3.1 Linear combination of the SHG radiated fields

The total radiated SHG field is a combination of nonlinear field terms determined by contributions of CdTe core and CdS rod, as well as interference occurring between (see equation (III.26), Section IV.2.2.1). In order to calculate polarization components of the

  Chapter I | Optical properties of semiconducting quantum confined heterostructures I.1 Shrinking dimensionality of the bulk into the quantum dot scale ................................................. 8 I.1.1 Quantum Confinement ........................................................................................................... 8 I.1.1.1 Density of states ............................................................................................................ 12 I.1.1.2 Coulomb interactions and confinement regimes ........................................................... 14 I.1.2 Band structure of CdTe ........................................................................................................ 16 I.2 Optical interactions ...................................................................................................................... 17 I.2.1 Scattering .............................................................................................................................. 19 I.3 Chapter summary ......................................................................................................................... 20

10

 10

Figure I- 1 |

 1 Figure I-1 | Comparison of the changes in the electronic structure accompanying a reduction in size, in metals and semiconductors.

Figure

  Figure I-2 |Comparison of the energy levels of conduction and valence bands as well as band gaps for different bulk semiconducting materials (II-VI groups SC compounds).[START_REF] Kittel | Introduction to Soild State Physics[END_REF][START_REF] Wei | Calculated natural band offsets of all II-VI and III-V semiconductors: Chemical trends and the role of cation d orbitals[END_REF]

  the reduced mass of the exciton, and ε for the dielectric constant of the material. When the width of the infinite potential well is comparable to B a (exactly like in the case of QDs), the VB and CB split into discrete energy surfaces in k-space. In such a case, the Schrödinger equation take the following form: 25

χ

  on the coordinate z of the confined direction due to the bandgap mismatch. We assume Bloch function -like eigenstates subsequently summed over k to describe bonding between the delocalized molecular orbitals (MOs), to be identical throughout the various material shells: is subject to the periodicity of the unit cell (with relatively weak k dependence), the ik r e ⋅ factor is a plane wave-like factor with a wavelength 2 Bloch MO in the case of a single k is infinitely delocalized. Following this assumption, the wave function takes the form: ( ) ( ) ( ) envelope function' of the confined state. Combining the Bloch function behavior with the Schrödinger equation, we obtain: barriers of the potential barrier, the energy of the n-th state confined in a rectangular potential well (2D) of width L z is: the SC NC dimensionality (illustrated in Figure I-3) by confining in L x

I. 1 . 1 . 1

 111 , as a product of an 'envelope function' and a Bloch function can be generalized to determine confined VB and CB states in the wake of progressive reduction of the dimensionality (where coupling between the bands has to be considered). Generalized forms for the CB wave function is the spin of the electron. Density of states As illustrated in Figure I-3 the density of states ρ (DOS) critically depends on the NC dimensionality. The DOS expresses directly the number of possible solutions of the Schrödinger equation (I.1) within a given interval eigenstate energy E and is seen to depend on the structure confinement:

I- 3 |

 3 Figure I-3 | Schematic representation of decreasing dimensionality of a SC crystal from bulk (a), quantum well (QW, (b)), and quantum rod (QR, (c)) into the single QD scale (d), as shown with corresponding densities of states (ρ).

2

 2 Coulomb interactions and confinement regimesIn strongly confined structures, positions of the electrons and holes are constrained.Electrostatic screening is less effective, leading to reduction of the effective dielectric constant ε , which results in an increase of the binding energy ( B E ). For strong enough confinement, Coulomb interactions can be neglected in the confinement direction. In such conditions the single particle 'envelope function' the in-plane confined motion of the electron and hole expressed with respect to respect to in-plane coordinates ̻ e,h . The in-

  confinement regime, Coulomb interactions between the electron and hole are strong and have to be taken into account.'Intermediate confinement regime', which occurs in the SCs exhibiting different effective masses of the electron and hole in an intermediate size ratios e radii of electron and hole, respectively. In this regime, the hole moves in the average potential of a much faster electron (about three orders of magnitude), and stays localized in the center of the NC. The region of its motion around the NC center is much smaller than a, therefore size dependence of the exciton ground state may be determined as a localized donor.'Strong confinement regime' occurring for small NCs, where B a a ≪ . The electron-hole

  Figure I-4(a) illustrates band structures for the case of bulk zinc-blende CdTe crystal, where a single Brillouin zone stems from the Bloch wave of the face-centered cubic crystalline periodic structure of the CdTe crystal. The wave vector k crosses the critical points of the crystalline symmetry, which are as follows: L -a center of a hexagonal facet, Γ -center of the Brillouin zone, X -center of a square facet, and K -middle of an edge joining two hexagonal facets. The center point Γ determines the band edges for the parabolic VB and CB in the first approximation model.

Figure I- 4 |

 4 Figure I-4 | a) The band structure of the bulk, zinc-blende like CdTe crystal. Critical points of the Brillouin zone are explained in the text. Adapted from Huang et al. 33 b) Spacing of the excited states of strongly confined CdTe QDs. Dashed lines (labels from b to f) correspond to the transitions

  5) represent experimentally measured data, with the converging point at the lowest exciton energy transition of bulk CdTe, at 1.596 eV. Figure adapted from Ref. [36].Spacing between the excited states of several samples of strongly confined CdTe QDs is illustrated in FigureI-4(b). The quantized energies of , however the energy separation between these states changes at most by 6nm large dots changes significantly by 68 meV, due to a strongly varying effective mass of the electron. The absorption spectrum of ~5.1 nm large dots exhibits an energy splitting of about 44 meV, showing strong dependence on the QD sizes, however there was no experimental effort towards large-size CdTe QDs, which reach the region close to the intermediate confinement regime of about 10-12 nm diameter.

  of the ν th hole and electron QSLs, respectively. The Coulomb interaction can be calculated from first order perturbation theory. Recombination of the exciton may be assisted by the emission of the photon, which possesses an compared to that of the exciton), where λ the photon wavelength and c the speed of light. The energy of the lowest transition for a known Bohr radius B a , and effective masses of the electron and hole

3

 3 Synthesis and characterization of dot-shaped CdTe QDs .......................................................... 25 II.4 Characterization of rod-on-dot CdTe/CdS heterostructures ....................................................... 27 II.4.1 Chemical synthesis of rod-on-dot CdTe/CdS heterostructures ........................................... 29 II.4.2 Shape determination and effective size distribution of RD CdTe/CdS heterostructures .... 30 II.5 The role and importance of the shell in type-II QDs .................................................................. 32 II.6 Sample preparation ..................................................................................................................... 33 II.7 Chapter summary ....................................................................................................................... 34

  CdTe/CdS core/shell QDs Synthesis of the CdTe/CdS core/shell NPs dedicated to SHG emission required NC sizes of about 10-15 nm diameter. CdTe/CdS dot-shaped QDs used in this work were prepared in a non-coordinating solvent, following the main lines of the procedure given by Peng and coworkers. 45 A scheme of the synthesis and apparatus is shown in Figure II-2(a,b). The synthesis was carried out under argon atmosphere. In short, 0.1 mM CdO was dissolved in 5 ml of octadecene (ODE) and 0.06 g tetradecyl phosphonic acid and heated-up to 280 °C. At this temperature 0.1 mM Te dissolved in trioctylphosphine (TOP) is injected, leading to the formation of approximately 4 nm diameter quantum dots. These were further grown at 290 °C by additional multiple injections of a 0.1 M cadmium oleate solution in ODE and 0.1 M TOP:Te until the emission peak reached about 760 nm (Figure II-3(c)). Once the cores have reached the desired size, the temperature was decreased in order to thermodynamically terminate their growth. The solution was heated again when TOP:Te injections were replaced by TOP:S to initiate shell growth. Injections of sulfur containing reagent were sufficient to form approximately three monolayers of CdS shell, resulting in the formation of an approximately 1 nm thick CdS layer (Figure II-3(b)). Particles were then dissolved in toluene and precipitated by addition of methanol and centrifugation.

  Figure II-1 | a) Stereograph presenting the symmetry elements of unit cell. The lattice constant is a 0 = 0.6482 nm.

Figure II- 2

 2 Figure II-2 | a) Synthesis way of CdTe/CdS dot synthesis apparatus (Ar -argon flow; TC of diluted QDs colloidal solution, carried out in order to control the NPs size

Figure II- 3

 3 Figure II-3 | a) TEM image of monodispersed CdTe/CdS dot single CdTe/CdS core/shell QD. c) Absorption and emission curves of dot from their colloidal mother solution.

II- 4 |

 4 Figure II

  nm dots) was a crucial issue towards measurements of the SHG size dependency. Effective size was thus confirmed by the TEM analysis (see Figure II-6(a)). The effective diameter of the largest NPs obtained with TEM analysis was 12.2 ±1.2 nm, as shown on the histogram in Figure II-6(b). The largest CdTe QDs presented in Figure II-6(a) exhibit some inhomogeneities in crystalline structure orientations, as well as variations of their shapes, which is common for "large" QDs.

Figure II- 5

 5 Figure II-5 | a) Emission and b) absorption curves of 7 nm (green), 9 nm (red) and 11 nm (blue) CdTe QDs.

Figure II- 6

 6 Figure II-6 | a) TEM image of ~11 nm large CdTe QDs. Yellow arrow indicates an example of QD with different crystalline facets orientations. b) Histogram of the size distribution of the largest CdTe QDs based on TEM analysis (with an effective diameter 12.2 ±1.2 nm).The black line delineates a size distribution obtained by using the emission spectrum (using the empirical function (III-1)), with a mean diameter of 11.1 ±2.4 nm. Larger size obtained from the TEM images can be attributed to the occurence of organic materials in the QD solution, which do not affect the emission spectra.

  elementary crystalline lattice of CdS are shown in Figure II-7(a,b). However to show the overall symmetry of the CdS structure a larger repeat unit has to be presented, containing three single unit cells (Figure II-7(c)).

[

  112] axis of CdTe, and the CdS [0001] axis oriented parallel to the [111] axis of CdTe, which means that the 6-fold axis of symmetry (parallel to [0001] ) is oriented along the rods growth direction -shown in Figure II-8 and on the magnified TEM image -Figure II-10(b).

Figure II- 7 |

 7 Figure II-7 | a) Stereograph representing the symmetry elements of the 6mm point group. b) Single unit cell of the CdS wurtzite -like crystal, based on the convention adopted in Refs. [14,43-44]. c) Part of the CdS crystalline structure, showing the symmetry of a right prism with hexagonal base. Coordinates are in nm. The lattice constants are: a 0 = 0.4135 nm and 0

Figure II- 8 |

 8 Figure II-8 | Relative orientation of CdTe and CdS crystalline lattices represented by the unit cell of CdTe and a triple unit cells of CdS. The relations between the [111] and [112] crystalline axis of the CdTe and the [0001] and [01 10] axis of the CdS are marked by green arrows. Rotations of both lattices are discussed in Appendix C.1 and C.2.

Figure II- 9 |

 9 Figure II-9 | Emission (a) and absorption (b) spectra of RD QDs. Black line represents grown CdTe cores, measured just before injecting S:ODE/TOP solution before shell growing process. Red lines represent spectra of the final synthesis product -CdTe/CdS RD QDs.

Figure II- 10

 10 Figure II-10 | a) TEM image showing monodispersion and shape determination of rod-on-dot QDs. b) Magnification of the single RD QD illustrating important structural facets and atomic distances. 52,i c-d) AFM topographical and phase scans of a single and well isolated RD QD embedded into a ~30 nm thick PMMA hostmatrix. e) Cross-section model of the rod-on-dot CdTe/CdS QD.

Figure II- 11

 11 Figure II-11 | a) Determination of the effective diameter of the CdTe core. b) Width dispersion of the CdS rod. c) Length dispersion of the CdS rod.

  Chapter III | Coherent second-harmonic microscopy for detection of single semiconducting QDs III.1 Contrast mechanisms for two-photon nonlinear microscopy .................................................... 36 III.1.1 Two-photon excited fluorescence ...................................................................................... 36 III.1.2 Second-harmonic generation ............................................................................................. 38 III.2 Second-order nonlinearity of CdTe -from bulk to single QD scale......................................... 41 III.3 Nonlinear two-photon polarization sensitive microscopy setup ............................................... 45 III.3.1 Optical and effective spatial resolution in multi-photon microscopy ................................ 48 III.4 Time correlated single photon counting in detection of single nanoparticles ........................... 51 III.5 Chapter summary ...................................................................................................................... 53

  by the two-photon action cross-section, that is the product of the fluorescence quantum yield TPEF φ by the absolute value of the TPEF σ .[70][71][72] In choosing the optimal wavelength for TPEF measurements, doubling the maximum single-photon excitation wavelength is usually a good approximation, however many fluorophores, e.g. Rhodamine B, exhibit different selection rules for single-and two-photon processes, therefore both the wavelength dependence and the absolute value of TPEF TPEF φ σ are important. When excited with a fs pulsed laser, most common fluorescence dyes have TPEF TPEF φ σ values in the range of 1 -300 GM,[70][71]73 however using the SC nanolabels, luminescence core/shell CdSe/ZnS QDs, they may approach extremely large values up to 50,000 GM.74

Figure III- 1 |

 1 Figure III-1 | Perrin-Jablonski energy diagrams of the TPEF (a) versus SHG process (b). Solid black lines correspond to the real energy levels of ground and excited states. The yellow wavy line corresponds to the vibrational relaxation of the electron (with energy loss leading to a small probability of photobleaching) due to the non-radiative transition into the S 1 energy level of the real excited state (RS), followed by TPEF emission. Dashed lines are un-populated virtual energy levels (VS) with life times shorter than 1 fs.

  factor which reflects the collection efficiency, the polarization of the incident electric field E ω and its effectiveness in causing two-photon absorption relative to the fluorophore orientation, and the direction of analysis I u ( ˆabs µ , ˆem µ , Êω are normalized

  further defined in Subsection III.3.1). Since a single QD is treated as a single molecule (because its sizeexc λ ≪), its radiation is assumed to be in phase, the 2 Θ parameter is being then negligible. Expression (III.8) is further used in order to calculate QD SHG crosssections, with known values of the incident focused pulsed peak intensity and experimentally measured SHG count rates (where we assume that SHG

  Refs.[75,90]. However, it has been significantly improved in terms of the laser source, mechanical stability and quality of the optical components. The idea of the experiment is illustrated on the schematic of the optical setup in Figure III-2. The laser source used for excitation is a commercial Spectra-Physics Mai-Tai HP (100 fs pulse duration, 80 MHz repetition rate) mode-locked Ti-Sapphire rod oscillator cavity. In this system the pump laser is a solid state continuous wavelength (CW) Millennia diode-pumped laser, with Nd:YVO 4 lasing medium (Na + ions doped in a yttrium vanadate crystalline matrix). The 1.064 µm emission is converted to the visible through frequency doubling (SHG) in a 90°, noncritically phase-matched, temperature-tuned lithium triborate (LBO) nonlinear crystal. The CW 532 nm output is used to pump the Ti-Sapphire rod in a four level scheme. The Ti-Sapphire rod (with absorption peak at about 510 nm) is composed of Ti 2 O 3 introduced into a melt of Al 2 O 3 ,

  (λ/2), mounted on a compact motorized rotation stage (Newport, PR50) and a polarizing beam splitter (PBS), placed in the detection channel. These components allow for measurements of the incident polarization dependence of the emitted signal for arbitrary orientations of the source NC. In this case, precise rotation of the λ/2 plate turns the incident polarization angle, while the PBS works like an analyzer, which separates the emitted signals (TPEL or SHG) into two different detection channels with different polarization components: vertical P p in a transmission (y or p) and horizontal P s in a reflection mode (x or s). Signal detection was carried out using two single photon counting -avalanche photodiodes (APDs, Perkin-Elmer, Si-based SPCM-AQR-14), both with comparable dark counts ratio ( ) dark counts C of about 100 Hz (at the room temperature). Without illumination, randomly generated dark counts of the APDs follow a Poisson distribution, where the standard deviation is equal to the square root of the counts mean value 1

  dependent APD quantum efficiency, which can be found in Appendix A (Figure A-8).The pump light with known incident polarization and controlled radiation power was directed into the main unit of the setup -an inverted confocal microscope (Nikon, Eclipse T2000-U) by a system of adjustable mirrors. Next, it is being reflected by the dichroic mirror iii Source: User's Manual, Mai-Tai HP, Spectra-Physics. iv Single Photon Counting Module SPCM-AQR Series datasheet, Perkin-Elmer.

Figure III- 2 |

 2 Figure III-2 | Schematic of the two-photon nonlinear scanning microscopy setup: λ/2 -half wave plates, GP -Glan-Taylor prism, L -lenses, M -adjustable silver-protected mirrors, DM -dichroic mirror, SPF -short-pass filter (also additional BG38-39 filters), FM-flip-mounted mirror, PBS -polarizing beam splitter, APD 1-2avalanche photodiodes. Circularly oriented arrow corresponds to the motorized rotation stage.

  diameter of the focal spot, which in practice corresponds to the Airy-disk distribution function (Figure III-3(a)). Considering the Rayleigh criterion, imaging of two incoherent point sources (sum of two known, identical, overlapping component functions) is confined by the diffraction limit. 101-102 Two incoherent point sources are considered distinguishable when the center of the Airy intensity pattern generated by one point source falls exactly on the first zero of the Airy pattern generated by the second. For the VIS region the Rayleigh distance δ is about 250 nm. Theoretically the minimum resolvable distance is characterized by Abbe'incoherent two-photon microscopy, where the excitation probability is proportional to the square of the fundamental incident intensity, the effective spot size is sharpened by a factor of 2 (assuming the Gaussian distribution of the intensity profile), thus the Rayleigh criterion for two-photon excitation (TPE) can be expressed as: 102 0.61 NA λ δ = (III.17) The intensity distribution in the image of two equally bright incoherent point sources separated by the δ distance is illustrated in Figure III-3(b). The central dip is found to fall by a ∆ ratio of about 27% below both peak intensities.

Figure III- 3 |

 3 Figure III-3 | Images of incoherent point-like sources in the optical microscope: a) Airy Disk intensity distribution and its profile (left), in logarithm scale (right). The blue profile shows a Gaussian function related to the IPSF 2 . Profiles were normalized for λ = 690 nm, NA=1.45. Defined distances are: 2 1.22 / Airy D NA δ λ = = ,

   In the first situation ( 0 φ = ) both coherent point sources are totally in phase (this is applied further for SHG imaging of single and isolated QDs). Second case ( both sources are in a quadrature (with identical intensity profile to that of incoherent sources). The last situation (φ π = rad) shows the two sources in phase opposition.102

Figure III- 4 | 4

 44 Figure III-4 | Phase dependent image intensities for two equally bright coherent point sources (marked by the two vertical lines), separated by the Rayleigh resolution distance δ . Figure adapted from Ref. [102].

Figure III- 5 |

 5 Figure III-5 | Schematic of the optical setup for time correlated single photon counting measurements: BSbeam splitter, λ/2 -half wave plate, GP -Glan prism, L -lenses, λ/4 -quarter wave plate, DM -dichroic mirrors, SPF -short-pass filter, F-fluorescence filter, PMT -single photon-counting photomultiplier tube, APD -avalanche photodiode.

Figure III- 6 |

 6 Figure III-6 | Schematic of gated TCSPC single channel system, illustrating the idea for measurements of instantaneously scattered photons, arriving subsequent to the reference laser pulse. Adapted from Ref. [105].

2 . 3

 23 Errors in inferring the orientation of the crystalline lattice from SHG polarization analysis ..................................................................................................................................... 85 IV.3 Accuracy of the polarization analysis from a reduced set of raw data points ........................... 86 IV.4 Chapter summary ...................................................................................................................... 89 In this chapter the SHG emission obtained in the case of individual, well isolated CdTe/CdS SC QDs is discussed. We focus on the second-order nonlinearity of the CdTe material, and comparing its response for the bulk and nanometric scale. We further analyze the spectral response and photostability of observed SHG signals at different conditions of the two-photon excitation. Optimization of the experiment allows to study the spectral dependence of the SHG emission, which is found to be directly connected to effects arising from the quantum confinement in CdTe QDs. Due to the vectorial properties of observed SHG emission, reflecting in a strong optical polarization sensitivity to the properties of CdTe zinc-blende structure, we are able to probe and infer the spatial orientation of crystalline unit cell of individual QDs. Finally we show that the second-order nonlinear response of hybrid CdTe/CdS heterostructures can be engineered based on optimization of material, size and geometrical features.IV.1 SHG from single dot-shaped CdTe/CdS core/shell QDs

  I). With high values of the nonlinear coefficients and a noncentrosymmetric tetrahedral crystalline lattice, we expect to be able to downscale the size of SHG active spherical CdTe -based QDs down to about 10 -15 nm, while maintaining the signal intensity at the level of at least 1×10 3 cts s -1 . Such a size range is also commensurate with the 7.3 nm exciton Bohr radius of CdTe, which marks the onset of size -dependent effects due to quantum confinement. The nonlinear optical response was measured with the TPSM setup illustrated in Figure III-2. Examples of diffraction limited spots corresponding to the SHG emission, from two single dot-shaped CdTe/CdS core/shell QDs (a, b) are shown in Figure IV-1. Scanning conditions were determined by setting the signal acquisition time to 50 ms, and the piezoscanner step to 80 nm per image pixel. Both SHG scans appear somehow ellipsoidal, showing wider cross-sections in vertical directions with a FWHM function of about 350 nm, and a FWHM of about 236 nm for cross-sections in the horizontal direction. Thus the diagonal cross-sections FWHM can be taken as 295 ±56 nm, which is consistent with the diffraction limited experimental system. This anisotropy may be related to vector effects, due to the elliptical excitation region with the ellipse stretched along the direction of the linear polarization. In Figure IV-2 we show smoothed SHG scans with their three-dimensional profiles corresponding to the two SHG spots (a-b), shown in Figure IV-1. The presented SHG

Figure IV- 1 |

 1 Figure IV-1 | (a-b) Scans of SHG spots obtained from two different single CdTe/CdS core/shell QDs.Corresponding cross-sections (organized in columns, where red is a horizontal and blue is a vertical direction) were fitted using Gauss distribution function, and show average FWHM value about 295 ±56 nm.

Figure IV- 2 |

 2 Figure IV-2 | (a-b) Smoothed scans (with large increment) of the SHG Gauss-like signal distribution, obtained from two different single CdTe/CdS core/shell QDs -presented in Figure IV-1. Corresponding threedimensional profiles show signal-to-noise ratio of about 70.

Figure IV- 3 |

 3 Figure IV-3 | Photostability of the SHG emission, obtained from a single CdTe/CdS core/shell QD exhibiting very low luminescence QY at the different excitation wavelengths: a) 800 nm, b) 850nm, c) 900 nm. d) Example of strong signal fluctuation, observed from a single CdTe/CdS QD with a high luminescence yield.

IV. 1 . 1

 11 In QDs exhibiting low luminescence QY the excited state population completely decays nonradiatively within the ~12 ns time interval between excitation pulses, eliminating such possible effect. Longer exposure of QDs exhibiting high luminescence QY under strong intensities results in strong signal fluctuations finally leading QDs to structural damage, and a strong decrease of the emission intensity (Figure IV-3(d)), and sometimes to its complete disappearance. Spectral dependence of the SHG process from spherical CdTe/CdS core/shell QDs

Figure IV- 4 |

 4 Figure IV-4 | a) Set of SHG emission spectra obtained from a single spherical CdTe/CdS QD, showing drift of the SHG peak (FWHM ~3.8 nm) when the excitation wavelength is tuned from 850 nm to 1.0 µm. Intensities of the SHG peaks were not corrected by transmission factors. b) SHG emission spectra with appearance of TPEL, tuned from 830 -940 nm.

  Figure IV-5(a,b),[START_REF] Zielinski | Second-Harmonic Generation from a Single Core/Shell Quantum Dot[END_REF] which provides most relevant data for evaluating the SHG efficiency.112 Obtained raw data was corrected for the Ti-Sapphire laser characteristic -taking into account the incident power dependence on the emission wavelength, transmission and reflection of the optical components of the two-photon microscope setup, as well as the detection efficiency of the Silicon APDs. Detailed description of the data processing can be found in Appendix A.Experimental results show that the SHG signal from four different single QDs exhibit almost the same spectral dependence, implying that the results are reasonably reproducible for QDs of similar size. A maximum of the SHG efficiency appears at 970 nm. The one-photon absorption gap of our NPs has been measured and corresponds to 760 nm (see FigureII-3), in agreement with CdTe NPs with sizes from 10 to 15 nm. Therefore the excitation wavelengths we use are far both from one-photon and two-photon resonances near the band gap which would occur for incident photons respectively at 760 and 1520 nm. Our excitation spectral range actually corresponds to deep resonance two-photon excitation with high-excited transitions, in conditions similar to the probe beam used in Ref.[112]. Low-lying excited levels of CdTe have been discussed in the literature; however it is difficult to assign energy levels for such a deep resonant excitation. Therefore we can only propose a tentative explanation for the excitation spectra shown in FigureIV-5(a,b). The shape of the SHG efficiency about 970 nm may be interpreted as a tradeoff between two competing mechanisms: in a Fermi golden rule picture, the density of states for the two-photon transition is higher for higher energies of the excitation light, which should lead to an increase in the two-photon absorption cross-section. However, exciting a state farther above the band edge implies more rapid decoherence, which leads to the reduction of the SHG QY (and preferential branching to two-photon excited fluorescence). Moreover, at least another secondary maximum can be singled-out in this excitation spectrum at about 910 nm. This corresponds to an energy step of about 0.15 eV between the peaks.

Figure IV- 5 |

 5 Figure IV-5 | a) Excitation wavelength dependence of the SHG intensity, measured for four different and well isolated dot-shaped CdTe/CdS QDs with an average size of about 12.5 nm. b) Averaged excitation wavelength dependence of the SHG intensity over four different spherical QDs. All the curves are given in arbitrary units, and were normalized after their corrections (see Appendix A).

  orientation leading to a new (X', Y', Z') crystalline frame, located inside the fixed (x, y, z) laboratory frame (see FigureIV-6).In theory when the (X, Y, Z) frame of the 43m zinc-blende crystalline lattice of CdTe has exactly the same orientation as the (x, y, z) laboratory frame: be observed, due to the fact that the incident optical electric field is probing a symmetrically oriented structure, illustrated on the top-view in FigureIV-7(d). Note that even a small inclination in the lattice position appears as noncentrosymmetric to the incident field probing its structure, and leads to the appearance of the SHG signal(Figure IV-7(b)).

Figure

  Figure IV-6 | a) Definition of the Euler set of angles

Figure IV- 7 |

 7 Figure IV-7 | Simulation results of the incident polarization dependent response of the SHG when the CdTe crystalline lattice orientation corresponds to the Euler set of angles at: (a)

Figure IV- 8 |

 8 Figure IV-8 | Polarization analysis of the SHG obtained from three different QDs (1-3). a) Experimental results (scattered points) with inserted smoothing function (SF given with continuous lines) curves. b) Simulation results of the experimental polarization response, determining the orientation of the CdTe crystalline lattice (intensities are normalized to 1). c) Three-dimensional orientation of the CdTe single unit cell crystalline lattice, extracted from the simulation results. d) Top-view on the CdTe crystalline single unit cell presented on c) -this view corresponds to the direction probed by the incoming incident beam.

Figure IV- 9 |

 9 Figure IV-9 | Projections of all possible Ω angles, determined by the simulation program using the criteria of S x(y) < 0.3, which correspond to the fit of QD1 experimental data (illustrated in Figure IV-8(a)) for both: x and y polarization components ((a) and (b), respectively). Optimized fits are obtained for smallest possible values of S x(y) (see the blue markers on (c-d)). The full range scans show eight repeated units of different combinations of Ω angles, for which comparable polarization responses can be obtained. Lower graphs (c-d) relate to marked single units areas of both polarization components.

  FigureIV-10(c). Here we did not measure the defocused imaging response experimentally; however, in order to prove the symmetry constraint of the polarization response, we have investigated this analytically for the three pairs of different sets of Ω angles, which basically result in the same polar graphs. Calculated defocused imaging patterns of the SHG emitting dipoles are illustrated in FigureIV-10(c). The results obtained for each pair of polar graphs exhibit mirror-reflected like distribution of the SHG emission patterns, with symmetrical positions of their maxima. These lead to the conclusion, that for the same φ angle, the two θ angles are symmetric with respect to 90° (which is in agreement with the equivalence between θ and 180 θ °-). A similar symmetry effect is observed for the ψ angle of each pair of polar graphs with respect to the 45° (related to the equivalence between ψ and 90 ψ °-).

Figure IV- 10 |

 10 Figure IV-10 | a) Three pairs of simulation results (fits corresponding to QD1, shown in Figure IV-8), with the same values of ϕ angle for each pair. b) Corresponding orientations of the CdTe crystalline lattice, and c) calculated defocused images of the emitting dipoles. All figures are referred to the laboratory frame.

  the morphology of these RD QDs(Chapter III, Subsection III.4) shows obvious geometrical differences compared to previously described dot-shaped CdTe/CdS QDs. Thus, the complexity of the RD NCs leads to several implications on their nonlinear properties, arising from the innovative geometry of such hybrid NPs. The consequences on SHG studies described further, e.g. analysis of the SHG polarization dependency, require knowledge on mutual orientation of the CdTe core with respect to the CdS rod structure.A diluted sample containing isolated CdTe/CdS RD QDs was studied with the same TPSM setup, which has been used before for SHG emitting spherical CdTe/CdS QDs studies.Examples of epi-detected diffraction limited and well-contrasted SHG spots (FWHM ~290 nm), obtained by scanning the sample containing isolated RD NCs are shown in FigureIV-11. Because of their geometrical features (described in details in Paragraph 3.4.2, or in the Ref.[8]) and low thickness of the PMMA host-matrix (~30 nm), RD NCs were expected to lie on the surface of the sample. Thus localization of the single emitters exhibiting dipolar-like emission pattern required scanning the sample with different angles for the incident polarization. To skip this inconvenience, the excitation beam was first circularly polarized in order to minimize the dependence of the SHG intensity on random RD QD orientations. This allowed for efficient localization of all the emitters within the range of the scan. Further studies of single isolated RD QDs required more precise angular characterization of the NC dipole orientation by performing polarization analysis. FigureIV-11(a, b) shows differences in a SHG intensity for the same dot, obtained with the circular incident polarization, in contrast with linear polarization oriented along the dipole -extracted from the polarization analysis (Figure IV-12(c)), and resulting with an approximately 30% higher SHG emission intensity. Both scans were obtained at 890 nm, with the incident excitation power exc P = 3.0 mW, leading to a focused pulse peak intensity of about 94.4 GW cm -2 .

Figure IV- 12

 12 FigureIV-12(d) shows SHG emission signal traces observed with/without spectral filtering of an eventual TPEL contribution. Very small fluctuations observed when the whole spectral range is collected, indicate simultaneous emission of SHG together with a very low amount of TPEL photons, which can be neglected in as much as it does not reduce the SHG intensity. This opens very an interesting perspective on the emission photostability, especially when RD QDs are compared with dot-shaped QDs described before, highlighting the importance of the CdS shell and its good quality, as well as quantum confinement effects. Our findings show that signals obtained from purely SHG emitting RD NCs (Figure IV-13(a)) show a high degree of photostability over a long exposure time, even when doubling the intensity of the excitation in comparison with signals obtained with exc P = 3.0 mW of incident excitation light (at exc λ = 890 nm). We observe here again, an inverse correlation between the intensity of TPEL from these QDs and their photostability. Hence, a high TPEL yield, photostability of the SHG emission is strongly reduced, as shown in Figure IV-13(b). Reduced photostability of some RD QDs exhibiting stronger band-edge luminescence is probably due to induced excited state absorption at the excitation frequency. Due to the CdS monolayer which transforms into a QR on one facet, geometrically well defined RD QDs exhibit improved performance for charge separation, preventing their short times scale radiative recombination, and allowing for longer coherence times of the electron-hole exciton pairs.

Figure IV- 11 |

 11 Figure IV-11 | Scans of SHG spots obtained for the same single rod-on-dot CdTe/CdS heterostructure with: a) circular incident polarization, and b) linear incident polarization oriented along the emission dipole (characterized on the polarization plot -Figure IV-12(d)). Corresponding cross-sections (organized in columns,where red is a horizontal and blue is a vertical direction) were fitted using a gaussian function, and show average FWHM value about 290 nm, close to the diffraction limit.

Figure IV- 12 |

 12 Figure IV-12 | Smoothed scans of the SHG Gaussian-like signal distribution, obtained from the same single rodon-dot CdTe/CdS heterostructure (also presented in Figure IV-9) with: a) circular incident polarization, and b) linear polarization oriented along the dipolar emission pattern. c) Polarization analysis showing dipolar-like SHG emission pattern, oriented at +110 °. d) SHG signal traces (red and blue correspond to x and y polarization components, respectively) showing differences in the intensities and photostability when low amount of TPEL photons is collected, or spectrally removed (by a high transmission > 96% short-pass filter cutting at 561 nm).

  not involve the highest possible SHG cross-section. Therefore, investigation over the SHG wavelength dependency (blue trace in Figure IV-16) for particular RD NC evidences the high potential for this type of material as efficient nonlinear light sources.

Figure IV- 13

 13 Figure IV-13 | a) Signal intensity trace observed for purely SHG emitting RD QD with a high degree of photostability. Observed signal have been obtained with exc λ = 890 nm (3.0 mW). b) Decay of the SHG signal intensity trace (including 20 nm narrow TPEL background) and its fluctuations, observed from a RD QD exhibiting strong TPEL yield. The SHG has been obtained with exc λ = 890 nm and an incident power of 2.0 mW.Spectral separation of the SHG signal from the TPEL was obtained by a 445 ±10 nm band-pass filter.

Figure IV- 14 |

 14 Figure IV-14 | Set of SHG emission spectra obtained from a single rod-on-dot CdTe/CdS heterostructure showing drift of the SHG peak (FWHM ~3.8 nm) when the excitation wavelength is tuned from 850 nm to 950 nm. Intensities of the SHG peaks were not corrected by the transmission factors.

Figure IV- 15 (

 15 Figure IV-15 | a) Signal traces recorded for different incident power from 1-7 mW at 890 nm excitation wavelength (black and red traces correspond to x and y polarization channels respectively, while the blue trace show their averaged values). Strong emission fluctuations appearing at 7 mW are clearly visible. b) Quadratic dependence of the SHG emission intensity on the incident excitation power -data points were extracted from the averaged signal traces (a) by the linear approximation of their 'flat' parts. c) Quadratic dependence of the SHG emission intensity in logarithmic scale.

Figure

  Figure IV-16 | Excitation spectra showing wavelength dependence of the SHG intensity observed from two different size for RD QDs (red and blue traces), using linearly polarized excitation light along their dipoles. The open-black trace represents averaged signal obtained from four different spherical core/shell CdTe/CdS QDs, placed here for comparison.

  For larger SC NPs -like intermediate confined ~12 nm diameter CdTe QDs, the level spacing is tighter, so when excited close to the resonance they exhibit more rapid decoherence via longitudinal optical (LO) phonons, which may lower the SHG QY. Shrinking the size is expected to reduce decoherence, because intraband level spacings are larger for smaller QDs, thus decoherence by exciton-phonon coupling should be slowed down. A second possible origin can arise from the charge separation in the quasi-type-II RD hybrid structure, such as RD CdTe NPs. Reduced decoherence is most probably due to the separation between the hole localized inside the CdTe core and the electron delocalized in the heterostructure. Such an effect has been reported for enhanced spin coherence times in CdSe/CdTe type-II rod-like heterostructures by the group of Scholes, where charge separation has reduced the relaxation rate by one order of magnitude.125

2 P

 2 ω dipole oscillating at the 2ω frequency, as described by the equation (III.21). experimental data obtained for three differently oriented RD NCs (Figure IV-17(a)) evidences very high sensitivity of the polarization response with respect to the orientation of the (2) RD χ tensor, determined by the spatial position of the hybrid heterostructure lattice. The excellent agreement between the fits (Figure IV-17(b)) and the raw data is a most reliable indication of the single dot origin of the harmonic signal, ruling out the occurrence of complex crystalline multiple NC clusters or aggregates, thus opening possibilities for precise determination of the corresponding spatial orientation of both CdTe core and CdS rod crystalline structures. In order to obtain their orientations we have taken into account the mutual relations between both lattices, using the model subsequently leading to expression (III.26). Generated three-dimensional projections (Figure IV-17(c,d)) of hybrid RD NC lattices corresponding to the orientations of experimentally scanned RD NCs, seen from the top view of the x, y laboratory frame, display the mutual positions of the CdTe and CdS lattices. It is clear that the orientation of the dipole is determined by the [ ] 0001 axis of CdS oriented along the rod axis. In each case, the out-of-plane deviation (described by the angle β between the CdS [ ] 0001 axis and the { , }

  and eventual occurrence of the interference effects (either constructive or destructive as presented in Figure IV-18 by means of '+' and '-' symbols respectively) occurring between SHG fields from both sources. Thus the SHG intensity

Figure

  Figure IV-18 and Figure IV-19). Based on our model (equation (III.26)) the interference term follows linearly the growth of the CdS quantum rod.

Figure

  Figure IV-17 | a) Experimentally obtained polarization analysis of the SH emission observed for three RD NCs (QD1-3). Open scattered points represent the raw data, while line traces show smoothed functions (SF). b) Calculated fits of the experimental data smoothed traces. c) 3D projections of the RD crystalline lattice orientations, characterized by the Euler set of angles (in the z-y-z convention) with respect to the laboratory frame (x, y, z). d) Mirror-reflected (with respect to experimental setup configuration) top view of the 3D projections of RD QD crystalline lattices (perpendicularly to the probing incident optical electromagnetic field).

Figure IV- 18 |

 18 Figure IV-18 | Graphical illustration (a) and analytical analysis of the polarization response of decomposed total SHG intensity SHG scatt I , calculated for three differently oriented single RD NCs (b contributions of the CdTe core, CdS rod and experimentally obtained results shown in

Figure

  Figure IV-20 | a) Rod-on-dot NC cross 4 -20 nm on a 6 nm diameter CdTe core. Columns b polarization patterns, obtained for three different RD NCs (with spa IV-17), while growth of their rods is progressing. The fourth row corresponds to the experimentally obtained data.

  response of RD NC, including a noise level and unsymmetrical positions of the maxima. At first sight it may look simpler than typical response of the spherical dots (compare FigureAccuracyof the polarization analysis from a reduced set of raw data points | 87IV-8 and Figure IV-17), however, it is much more demanding for simulation processing due to the larger number of variables and nonlinear interaction terms between two different materials, such as CdTe and CdS. In our studies we determine orientation of the crystalline lattice by polarization analysis, where the experimental raw data is firstly smoothed using an analytical SF function, basing on the form of the SHG polarization response (described in Appendix B), corresponding directly to the SHG phenomenon. In order to eliminate the usually significant experimental noise, the corresponding maxima positions of the SF are determined by a nonlinear least square method, which leads to highly accurate fitting in further processing. Here we analyze how the decrease in number of exploited raw data points (which leads to shortening the probing time of the crystalline orientation) impacts on the shape of calculated SF and its error level, using the same integration time (1s) for each raw data point. An example of such analysis for a few different increments is shown in Figure IV-21. The values of error dependence in the maxima positions of the SF for: 4°, 6°, 8°, 10°, 12°, 20°, 30°, 60° increments were determined with respect to the SF calculated for experimental results obtained with an increment of 2° (first row of Figure IV-21). Final results are displayed in Figure IV-22 and summarized in Table (IV-II). Dependence of the maxima deviation error for each polarization component on the increment value, was estimated as a square root of the sum of squared errors values for both x and y components. Generated results (Figure IV-22) indicate an exponential character. The error value corresponding to the last point (60° increment) was estimated taking into consideration that all maxima have split into two (as shown in Figure IV-21). We have thus averaged the error value by dividing by a factor of two.

Figure IV- 21 |

 21 Figure IV-21 | Experimental results of polarization analysis (a), and corresponding SF curves of x and y components (b), determined by different increments in collecting the raw data: 2°, 10°, 30° and 60°, resulting with: 180, 36, 12, 6 experimental points for the whole turn, respectively.

Figure

  Figure IV-22 | Dependence of maxima deviation error on the increment value. Blue line is an exponential fit obtained for both x and y polarization components.

V. 1

 1 Size dependence of SHG from single CdTe quantum dots ........................................................ 92 V.1.1 Detection threshold of the size dependent SHG signal intensity ........................................ 93 V.1.2 Estimate of the second-order nonlinear susceptibility ........................................................ 97 V.1.3 Error sources in determining the size dependence of the second-order nonlinear susceptibility ................................................................................................................................. 99 V.2 Hyper-Rayleigh Scattering from CdTe QDs ............................................................................ 101 V.3 Relations between spectral and size dependence of CdTe χ (2) features .................................... 105 V.4 Chapter summary ..................................................................................................................... 107

3 .

 3 We have selected three different samples of such QDs with average diameters of approximately 7 nm, 9 nm, and 11 nm (as determined from the optical emission spectrum, Figure II-5 following the empirical function (II.1). The second-order nonlinear polarization response is determined by the third-rank nonlinear susceptibility tensor of CdTe, where the only nonzero and equal elements are 14 d , 25 d and 36 d , as given by equation (III.10). For the statistical studies of SHG obtained from single QDs

Figure V- 1 |

 1 Figure V-1 | Surface scan of 10 by 10 µm area of spin-coated QDs sample, recorded with 60 per 60 pixel resolution and 100 ms integration time. Bright spots of pixels correspond to the SHG emission from single CdTe QDs.

  Recorded background counts with 4σ threshold subtracted peak intensities for each QD size are plotted on the histograms shown below(Figure V-2). All three samples exhibit a large number of QDs with SHG signal intensities between 0 and 100 Hz. These low count rate levels can be explained by the appearance of structural defects of QDs (e.g. see the yellow arrow in FigureII-6), which may drastically reduce the SHG yield, and also by background scattering of some of these signal peaks. Using the 4σ threshold, the mean SHG count rates for each 7, 9, 11 nm QDs size are 97, 234, and 320 Hz. For samples containing 7 and 9 nm diameter QDs these values are higher than they should appear as when going from a 3σ to a 4σ threshold, more than 5% of the data points are lost. Average values of SHG count rates from 7 and 9 nm QDs at the 3σ threshold are 70 and 114 Hz, however these results have to be considered as approximated ones, due to the inclusion of shot noise peaks, as well as discarding many real data peaks below the threshold level. For the clarity all measured count rate results are contained in Table (V-I).

Figure V- 2 |

 2 Figure V-2 | Histograms of SHG signal count rates per 100 ms integration time, obtained using 4σ threshold for: a) 7 nm, b) 9 nm, c) 11 nm large CdTe QDs (semi-log10 scaled). Measured numbers of QDs for each sample are 250, 136 and 615, respectively.

  3 translates into the volume squared distribution of the sample. In Figure V-3 we show the predicted histogram for the 5σ threshold using the volume squared and dipole orientation distribution, corrected by the SHG detection efficiency for each dipole angle (following the correction described below), and plotted along the corresponding histogram. The mean value for the SHG count rate at the 5σ threshold of 420 Hz corresponds to a mean QDs size about 12.3 nm (marked as a dashed vertical line in Figure V-3). QDs exhibiting a count rates higher than 1000 Hz are in the tail of

Figure V- 3 |

 3 Figure V-3 | Normalized histogram of the SHG count rate of the 11 nm QDs using the 5σ threshold (a), and the squared volume distribution of the same sample (b). Dashed vertical lines correspond to averaged values. The x axis of (b) was scaled by fixing the mean value of the squared distribution to the corresponding averaged value of the SHG count rate (420 Hz). Following this, the mean diameter of the QDs corresponding to such a count rate is 12.3 nm.

  the volume of the scatterer. By taking into account the collection efficiency and losses through the optical components of the setup, the number of SHG photons emitted per 100 ms (integration time of the measurement) can be related to the number of measured SHG photons N SHG . In order to obtain its real value the raw data had to be corrected following a similar procedure to the one described in Appendix A, taking into account the transmission of the optical components inside the excitation and detection channels: reflectance (at 800 nm) and transmission (at 400 nm) of the Size dependence of SHG from single CdTe quantum dots | 99 dichroic mirror, microscope objective parameters, as well as quantum efficiency of the detectors (single-photon counting Picoquant PMT, and idQuantique APD). The collection efficiency was calculated by determining the angular distribution of the SHG dipoles and the percentage of signal collected for each dipole orientation by the microscope objective based on its NA.The energy per laser pulse is the number of SHG photons emitted per 100 ms (integration time) divided by the 80 MHz pulse repetition rate and multiplied by the energy of a single SHG photon. Than by dividing the energy per pulse by the ~130 fs pulse duration we get the power of the emitted SHG signal.

Figure V- 4 )

 4 Figure V-4). The signal was collected by a low -NA objective, then passed through a monochromator and detected by PM tube.Figure V-5(a) shows the HRS signal intensity (after correction for the self-absorption of the SH frequency on the detection path) as a function of QDs concentration. A measurement was taken for pure toluene, providing the y-intercept value on the graphs.

Figure V- 4 |

 4 Figure V-4 | Schematic of the hyper-Rayleigh scattering experiment.

  quadratic β tensor components. The common intercept of the HRS signal intensity plots, presented in Figure V-5 as a function of the QDs concentration, is therefore

  and the molecular weight of CdTe, respectively, and A N = 6.02×10 23 mol -1 is Avogadro's number. The slopes of the linear fits to the data for 5, 7, 9, and 11 nm diameter QD samples are 7×10 9 , 7×10 10 , 4×10[START_REF] Brus | A simple model for the ionization potential, electron affinity, and aqueous redox potentials of small semiconductor crystallites[END_REF] , and 1×10 12 counts M -1 , respectively. Dividing equation (V.12) by n results in CdTe g mol-1 . The concentration of toluene was obtained by dividing its density (0.866 g mL -1 ) by its molecular weight (92.14 g mol -1 ), giving s N = 9.4 mol L -1 .Using these inputs, the size dependent values of / to 5 nm, 7 nm, 9 nm, and 11 nm QD sizes, as plotted in Figure V-5(b), and summarized in Table (V-III). An average value of / CdTe s β β = 10.0 is marked by the dashed line in Figure V-5(b). Obtaining the value for single tensor elements which can be then compared with known elements of the (2) CdTe χ tensor is important, since CdTe and toluene have different symmetries.

Figure V- 5 |

 5 Figure V-5 | a) Dependence of the signal intensity on QD concentration in the HRS experiment performed for: (i) 5 nm, (ii) 7 nm, (iii) 9 nm, (iv) 11 nm CdTe QDs. The solid lines are linear fits of the experimental data (the signal obtained from toluene) is separated). b) Orientational mean hyperpolarizability per unit cell, in units of s β , for each QD sizes. The dashed line represents averaged value over all QD sizes.

  to the dashed line in Figure V-5(b), the nonzero coefficients of CdTe have to be 5.5 times higher than the effective β of toluene.

3 . 1 -

 31 photon resonance the efficient SHG process occurs. Results shown in this chapter for single dot studies and HRS experiments show a reversed situation whereby, the distance from the band edge of the two-photon resonance excitation is varied by changing the size and therefore the QDs band gap (BG), while the excitation frequency is set at a constant value of 800 nm.

Figure V- 6 |

 6 Figure V-6 | Wavelength dependence of the ( 2)

Figure A- 2 |

 2 Figure A-2 | The SHG count rate raw data, measured for three different spherical CdTe/CdS core/shell QDs.

Figure A- 3 |

 3 photons in backward direction):

Figure A- 4 |

 4 Figure A-4 | a) Transmission of the dichroic mirror, measured at the spectral range of 400 -510 nm. b)Corrected SHG signal.

Figure A- 5 |

 5 Figure A-5 | a) BG39 filter transmission. b) Corrected SHG signal.

Figure A- 6 |

 6 Figure A-6 | a) BG38 filter transmission. b) Corrected SHG signal.

  of the focal point of the objective at the diffraction limit, and w is the waist of the focus point. At the diffraction limit the lateral (x,y) waist profile

  obtained values of SHG intensities. The least square method is based on transformation of over-determined system of equations into square system by multiplying both sides of equation on the left, by a transposed coefficient matrix: ( ) i coefficients (i = 0, …,4) of the SF function can be found as:

  of coefficients are related to the one-order lower derivatives:

ForB. 1 . 3

 13 conditions are not true, however this relates only to a case of perfectly localized maxima at 90° (or -90°). Direct simulation of the CdTe SHG response Simulation process has three steps: a) Rotation of the second-order susceptibility tensor of the NC over three Euler angles Ω: shown in the Section III.2, the d-matrix of CdTe has a form of: convenience of further simulation we assume 14 d = 1;b) Applying the linearly polarized incident optical electric field: of the SHG intensities for both: x and y components, in order to fit with the experimental measurements:

  29)In order to determine the spatial orientation of the spherical CdTe QDs we use the (z-y-z) convention for the Euler set of angles, for which the rotation matrix is expressed as: (z-y-z) convention are given below in Appendix B.2.

=

  Applying expressions from (B.8) to (B.19), we obtain the coefficients of desired fit function for SHG polarization response of the CdTe NC, spatially characterized by set of Euler angles

  introduced dichroic mirror parameters can be found in FigureB-10, and



  to the rotation steps of the crystalline frame (X, Y, Z), defined by , as shown in FigureIV-6:1. Rotation about the Z axis by φ angle (counterclockwise, from X to Y); 2. Rotation about the new Y axis by θ angle (counterclockwise, from Z to new X); 3. Rotation about the new Z axis by ψ angle (counterclockwise from new X to new Y). axis to be set along y direction, and the [ ] 0001 axis along z direction A) Counterclockwise rotation of the [ ]0001 axis from z to x around y, where α = axis is along y axis, and then it is rotated from y to z (around x), where α =

13 )

 13 Defined orientations of both: CdTe and CdS crystalline lattices in respect to each other are illustrated in FigureII-8.

  0 , C 0 , F 0 , as well as S 0 , T 0 , K 0 are new nonzero components of the CdTe and CdS d matrixes, respectively. For CdTe we rotate tensor: (B.29) by the rotation matrix (C.9), what gives:

  18,36

  Colloidal CdTe-based QD samples: synthesis and characterization II.1 Guidelines for material selections .............................................................................................. 22 II.2 Characterization of dot-shaped CdTe/CdS core/shell QDs ........................................................ 22

II.2.1 Chemical synthesis of dot-shaped CdTe/CdS core/shell QDs ............................................ 23 II.2.2 Shape and size distribution of dot-shaped CdTe/CdS QDs ................................................. 23 II.

II.4 Characterization of rod-on-dot CdTe/CdS heterostructures

	geometrical parameters. As in the case of previous materials, we have used once again
	CdTe/CdS structure. However, RD QDs exhibit expanded CdS shell into a quantum rod (QR)
	structure, grown on one facet of the CdTe core. This has a crucial meaning in further studies
	because SHG is expected to scale as the squared volume of the scatterer size. Another
	important issue is that RD heterostructures are made of two SC materials: CdTe and CdS,
	each one featuring different crystalline lattices (zinc blende and wurtzite, respectively),
	nonlinear susceptibility tensors and spatial geometries. Both components are thus expected to
	contribute differently to the overall SHG field.
	The complexity of RD NCs entails several consequences, arising from the new geometry
	and resulting physical features of such hybrid NPs. This consequently impacts on the SHG
	studies described in Chapter IV, e.g. the analysis of the SHG polarization dependency
	requires a precise definition of the mutual orientation of the CdTe cubic zinc-blende lattice,
	which belongs to the 43m point group symmetry, with respect to the CdS rod structure. In
	this particular case, synthesis conditions (described in the next paragraph) are defined so as to
	impose growth of the CdS rod into the hexagonal wurtzite form, with 6mm point group
	symmetry. The stereograph representing the 6mm symmetry elements of that group and the
	The emission of single
	QDs of the same size can be broadened by ~kT, by about 25 eV, resulting in a total
	inhomogeneous spectral linewidth of about 70 meV.
	SHG active rod-on-dot (RD) heterostructures were developed as a proof-of-principle with the
	intention of obtaining SHG emission from single hybrid SC QDs, where the nonlinearity can
	be controlled through the relative contribution of the participating QD materials and their

Table ( III-I):

 ( Second-order nonlinear optical coefficients of selected inorganic and SC crystals.77

	Material	Symmetry class	λ ω [nm]	n ω	n 2ω	d [pm V -1 ]	Additional Ref.
	Barium titanate	Trigonal 4mm	1,064			d 15 = 17.0
	(BaTiO 3 )					d 31 = 15.7	[91]
	Cadmium selenide (CdSe)	Hexagonal 6mm	10,600	n o = 2.427 n e = 2.445	n o = 2.462 n e = 2.482	d 33 = 6.8 d 15 = 18.0
			800			d 33 = 36.0	[92]
	Cadmium	Hexagonal 6mm	1,064	2.3481	2.6494	d 15 = 44.0
	sulfide (CdS)			2.3316	2.6495	d 31 = 40.2	[93-94]
						d 33 = 77.9
	Cadmium telluride (CdTe)	Tetrahedral 43m	10,600 1,064	2.69	2.71	d 14 = 168 d 14 = 109	[5,94-95]
	Cinnabar (HgS)	Trigonal 32	1,500 10,600	n o = 2.594 n e = 2.845	n o = 2.628 n e = 2.888	d 14 = 73 d 11 = 50.0
	Gallium arsenide (GaAs)	Tetrahedral 43m	10,600	3.27	3.30	d 14 = 368
	Potassium	Rhombic mm2	880			d 15 = 3.92
	titanyl					d 24 = 2.04
	phosphate-KTP					d 31 = 4.74
	(KTiOPO)					d 32 = 2.76
						d 33 = 18.5
	Zinc oxide	Hexagonal 6mm	1,064	1.95	2.05	d 15 = 5.9
	(ZnO)					d 33 = 18.0
						d 35 = 5.4
	Zinc selenide (ZnSe)	Tetrahedral 43m	10,600			d 36 = 33.0
	α-Zinc sulfide	Hexagonal	1,064			d 15 = 7.0
	(ZnS)	6mm				d 33 = 14.0
	Zinc telluride (ZnTe)	Tetrahedral 43m	10,600	2.69	2.70	d 14 = 90

  IV.1 SHG from single dot-shaped CdTe/CdS core/shell QDs .......................................................... 56 IV.1.1 Spectral dependence of the SHG process from spherical CdTe/CdS core/shell QDs ....... 59 IV.1.2 Contribution of the CdS shell to the overall SHG emission process ................................. 62 IV.1.3 Orientation dependent SHG polarization response from spherical CdTe/CdS core/shell QDs ............................................................................................................................................... 63 IV.1.3.1 Analysis of the SHG polarization dependence ........................................................... 64 IV.1.3.2 SHG polarization response with respect to the CdTe symmetry constraint ............... 68 IV.2 SHG from single rod-on-dot CdTe/CdS heterostructures ......................................................... 70

IV.2.1 Special features of SHG emission from rod-on-dot QD heterostructures ......................... 73 IV.2.2 Polarization analysis of the SHG emission of rod-on-dot CdTe/CdS QDs ....................... 78 IV.2.2.1 Simulation model for polarization analysis of RD QD heterostructures .................... 80 IV.2.2.2 The role and impact of the CdS rod length into the total SHG intensity.................... 81 IV.2.

2 Contribution of the CdS shell to the overall SHG emission process

  Dot-shaped CdTe/CdS core shell QDs are built from two noncentrosymmetric materials, both exhibiting nonlinearity (see TableIV-I), thus from this point of view the CdS shell can generate SH emission and also contribute to the total SHG field. Using the simple rules of SHG emission by nanoscatterers (briefly described in Paragraph IV.1), we can estimate the

4

IV.1.

'classical' role of the shell by considering the nonlinearities of the corresponding bulk materials and the square dependence of the volume of the emitter. For bulk CdS crystal reported values can be as high as 78 pm V -1 at 1.064 µm for 33 d nonlinear coefficient, 77 while

Table ( V

 ( -I): SHG count rates dependence on the QD sizes and standard deviation thresholds (

					σ =	(	I	SHG background	) 1/ 2	).
	QDs mean		SHG count rate [Hz] with respect to the standard deviation threshold of:
	diameter [nm]	3σ	4σ	5σ	6σ		7σ
	7	70	97
	9	114	234
	11	250	320	420	500		530
	12.5*	91	102

Table ( V-II):

 ( Summary of the error sources in

		( 2) χ studies
	for a single dot case.
	Source of Error	Percentage Error
	Mean SHG intensity	20%
	Mean QD volume	30%
	Area of illumination	+50%, -0%
	Signal collection efficiency	< 10%
	Detector Quantum efficiency	< 10%
	Optical components	~10%

  77,94 Complementary to our spectroscopic single-dot results HRS studies, comparing to the first hyperpolarizability measurements of the most known CdSe QDs,3 allow for drawing similar conclusions as in the case of single-dot experiments. Converting the values given for

	β	CdSe	/	s β by Jacobson et al. [3] to the form of 33 d coefficient using the same procedure as
	have been used in our CdTe studies (using a CdSe density of 5.91 g cm -1 , M = 191 g mol -1 ,
	and ( ) n ω = 2.7), gives 33 d (800 nm) = 40 pm V -1 (for 9 nm diameter QDs), which is close to
	the reported bulk value of 36 pm V -1 .

92

Comparing the higher hyperpolarizability of CdTe with reported by Jacobson et al. results for CdSe, we expect it to follow the trend for bulk nonlinearities of CdTe and CdSe crystals.

  Both experiments show that

	E	SHG	farther above the band gap leads to a lower (2) χ than SHG E closer to the band gap,
	exhibiting a non-monotonous increase as a function of the	E	SHG	/	BG E ratio. Fourfold increase
	in 36 d observed for a ~12.5 nm diameter CdTe QDs (Figure IV-5) occures when the SHG E
	goes from 1.9 BG E to 1.5 BG E , 4-5 while for 33 d of CdSe NPs, the observed increase is more than
	a factor of two as SHG E changes from 1.6 BG E to 1.2 BG E .

3

Our spectral dependence experiment

  , if only the imaginary part of the roots t are equal to zero and the values of second-order derivative in t are negative.

	0 x' x' 1 t +	+	x'	2	t	2	+	3 x' t	3	+	x'	4	t	4	=	0,	t	=	tan	α	(B.24)
	and the positions of the maxima are equal to	α =	arc	tan	( ) t

  α and 45 factors are in degrees. We have also taken into account the difference in these maxima intensities. Corresponding formulas for different shapes I I S are the following: Note that the second angle α 1max appear in the denominator of (B.44) and (B.45) is not the position of second maximum α 2max , but the position of first maximum shifted by 90º. Functions from (B.42) to (B.45) are taken into account separately for x and y polarization components, to create overall differential shapes: a good agreement with experimentally obtained data. In order to obtain the best fit, simulation program is looking for the smallest possible value of given criteria, which are defined by these functions.

								S	x	I I	=	   	1	-	, ,exp x calc x r r	,exp , 1 x x calc r r + -	   	(B.44)
	where:	x r	,exp	=	I	x	,exp ,exp ( x I α	1max, ,exp 1max, ,exp ( x x α +	) 90)	,	, x calc r	=	I	, x calc ( x calc , I α	1max, , 1max, , ( x calc x calc α +	) 90)	, and:
								S	y	I I	=	   	1	-	, ,exp y calc y r r	1 + -	,exp , y y calc r r	   	(B.45)
	, ,exp x 1max, ,exp y , y calc r α α = To further enhance the fitting process we also minimize the following relative shape different 2 max, , 2 max, ,exp | | | | / 45 x x calc x calc x S α α α = -+ -(B.42) ( ) I 1max, , 2 max, , 2 max, ,exp | | | | / 45 y y calc y calc y S α α α = -+ -(B.43) where: ,exp 1max, ,exp , 1max, , ,exp ,exp 1max, ,exp , 1max, , ( ) ( ) , ( 90) ( 90) y y y calc y calc y y y y calc y calc I I r I I α α α α = + + . I II ( ) ( ) ( ) x y x y x y S S S = + . function ' 1, , 1, ,exp 2, , 2, ,exp 1, ,exp 1, , 2, ,exp 2, , ' 1 1 1 1 xy calc xy xy calc xy xy xy xy calc xy xy calc r r r r S r r r r     = -+ -+ -+ -            (B.46) where: ,exp 1max, ,exp , 1max, , 1, ,exp 1, , ,exp 1max, ,exp , 1max, , ( ) ( ) , ( ) ( ) x x x calc x calc xy xy calc y x y calc x calc I I r r I I α α α α = = , and ,exp 2 max, ,exp 2, ,exp ,exp 2 max, ,exp ( ) , ( ) x x xy y x I r I α α = , 2max, , 2, , , 2max, , ( ) ( ) x calc x calc xy calc y calc x calc I r I α α = . Functions (B.42) -(B.46) significantly improve simulation accuracy, providing results, sin 0 cos 0 sin cos sin 0 0 0 1 sin 0 cos 0 0 1 which are in cos sin cos 0 0 1 0 sin cos 0 xyz xyz xyz xyz

where xy S , which takes into account both x and y components together:

Direct simulation of the SHG response for RD CdTe/CdS heterostructures - linear combination of the RD

	=	    	cos cos sin cos sin α α β -	β β	cos sin sin sin sin sin cos sin α β γ α β γ β γ sin cos cos cos α α -+	γ γ	cos sin cos sin sin cos cos cos α β γ α β γ β + -	sin sin cos sin α γ α γ γ	    
	C.3 χ (	2 )
										14)

  , as described in Section IV.2.2.1. Volumes were determined following the TEM analysis (Figure II-10), with only one variable which is a CdS rod length, for x and y components of the polarization response), used in equation (B.4) relate here into the whole RD QD structure as follows:

	where:		    	11 21 31 a a a		12 22 32 a a a			13 23 33 a a a	    =      	de dbc af dbf ac ae abc df abf dc b ec ef -+ + --	    	,	a	sin ,	b	sin ,	c	sin ,	d	cos ,
	χ	RD	∝		χ	CdTe	+	CdS V V CdTe ⋅	-	χ	) CdS
	and the volume ratio is	CdS v V =	/	CdTe V	=	3	2 CdS CdS D L	/ 2	3 CdTe D	. Sets of: 111 d , 112 d , 121 d , 122 d , and
	211 d , 212 d , 221 d , 222 d tensor components (111 112 121 122 P S, A H, C T, d v d d v d v d = + = = + = +	211	A V, v = +	d	212	=	d	221	C K, v = +	and
	d	222	Q W v = +	. For simplicity we use:
			3 0 11 P P a =	+	0 11 12 3C ( a a	2	+	13 a	2	0 13 ) F (3 a a 12 +	2	-	13 a	2	)
			3 0 21 Q P a =	+	3C	0 21 a a 22 (	2	+	23 a	2	0 23 ) F (3 a a 22 +	2	-	23 a	2	)
			2 0 11 21 A P a a =	+	0 C (2	11 12 22 a a a	+	2	11 13 23 a a a	+	2 a a 12 21	+	2 13 21 a a	)	+
						+	0 F (	2 a a 12 23	+	2	12 13 22 a a a	-	2 13 23 a a	)
			2 0 11 21 C P a a =		+	0 C (2	13 21 23 a a a	+	2	12 21 22 a a a	+	11 22 a a	2	+	11 23 a a	2	)	+
						+	0 F (	13 22 a a	2	+	2	12 22 23 a a a	-	13 23 a a	2	)
			S	=	3 33 11 d a	+	31 11 12 ( d a a	2	+	13 a	2	) 2 +	15 11 12 ( d a a	2	+	13 a	2	)
			T	=		2 33 11 21 d a a	+	31 11 d a a 22 (	2	+	23 a	2	) 2 +	15 21 12 22 ( d a a a	+	13 23 a a	)
			V		=	2 33 11 21 d a a	+	31 21 12 ( d a a	2	+	13 a	2	) 2 +	15 11 12 22 ( d a a a	+	13 23 a a	)
			W	=	3 33 21 d a		+	31 21 d a a (	22	2	+	23 a	2	) 2 +	15 21 d a a (	22	2	+	a	23	2	)
			H		=	2 33 11 21 d a a	+	31 11 12 22 ( d a a a	+	13 23 a a	)	+	15 d a a a 12 11 22 (	+	13 11 23 a a a	+	2 13 21 a a	+	2 a a 12	21	)
			K		=	33 11 21 d a a	2	+	31 21 12 22 ( d a a a	+	13 23 a a	)	+	15 d a a a 13 21 23 (	+	12 21 22 a a a	+	11 23 a a	2	+	11 22 a a	2	)

2

  CdTeEterm we use similar way as shown in Section B.1.3, therefore we express the x

	component as:
	=	(x	0,CdTe	+	x	1,CdTe	tan	α	+	x	2 α ,CdTe 2 tan x 2,CdTe E κ	= +	x I	,CdTe 3,CdTe x	tan =	3 α	+	x	4,CdTe	4 tan ) cos α	4	α	(C.18)
	where:
										x	0,CdTe	=	4 1 (1 ) γ   + -	2	 	2	⋅	P	2	(C.19)
							x	1,CdTe	=	2 4 (1 ) cos 1 (1 ) γ ϕ γ ⋅ -    + -	2 MD	⋅	4AP	(C.20)
					x	2,CdTe	=			2 2 4 (1 ) 1 (1 ) γ γ ⋅ -    + -	2	⋅	 	2 4A 2CP cos 2 ϕ +	DM	 	(C.21)

i CdTe nearest-neighbor distance given for 300 K; source: http://www.semiconductors.co.uk

2 1

* * * * ,CdTe ,CdS ,CdTe ,CdS ,interf ,CdTe ,CdS ,CdTe ,CdS

This pointwise additive model, when applied separately to both x and y polarization components, allows for quantitative analysis of the SHG intensity components described by equation (III.26) accounting for the total SHG intensity emitted by the whole hybrid crystalline structure.

IV.2.2.2 The role and impact of the CdS rod length into the total SHG intensity

Detailed studies of the CdS rod length dependence provides clear evidence of the significant contribution of this feature to the overall SHG response. The polarization response of the SHG emission from the CdTe core shows a characteristic fourfold cloverleaf pattern, [126][127] as expected from its octupolar 43m symmetry (see Paragraph IV.2.3). [START_REF] Zielinski | Second-Harmonic Generation from a Single Core/Shell Quantum Dot[END_REF]128 Growth of the CdS rod changes this response progressively into a dominantly dipolar-like pattern. Extension of the rod leads to a progressive loss of additional maxima for each x and y component. This is imposed by the dipolar 6mm symmetry of the hexagonal form of CdS crystal.

V.4 Chapter summary

Concluding our studies on the second order nonlinear susceptibility of CdTe QDs, we have directly measured the nonlinear coefficient d of single isolated SC nanoscatterer.

Supplementary HRS measurements provide an independent confirmation of our single dot studies and follow the same trend as previously reported HRS experiment on similar SC CdSe NPs. 3 We have determined the d nonlinear coefficient of ~11.5 diameter CdTe QDs excited at 800 nm to reach 64 pm V -1 , which is not far from the bulk value of 73-109 pm V -1 . The importance of resonant enhancement and optimized decoherence is highlighted and shown to play an important role in the SHG emission obtained from QDs under two-photon resonant conditions, leading therefore to an optimized SHG QY. Under proper excitation conditions, the nonlinear susceptibility of SC QDs may considerably surpass the value of the corresponding bulk material, what shows an important role of the quantum confinement effect in nonlinear optics. [START_REF] Winter | Characterization of the Second Order Nonlinear Susceptibility of Quantum Confined Semiconductors[END_REF] Appendix A: Correction of the raw data for SHG wavelength dependence

The excitation spectra of SHG emission have been performed using two APDs in two different polarization planes: X and Y, with an acquisition time of 1 s. Each data series was measured with a constant change of the excitation wavelength with a step 1 nm, in the range 820 -1000 nm, performed automatically with the Ti:Sapphire laser (Mai-Tai HP, Spectra-Physics) control software. For SHG signal below 410 nm we were limited in the detection range by the Silicon APDs, exhibiting quantum efficiency close to zero at 400 nm (at the room temperature). We have also to take into account the wavelength dependence of the transmission / reflection of all optical components, in the excitation / detection channels.

Mirrors, with power variations less than 1%, high quality short-pass filter (Semrock SP01-561RS), and lenses with more than 99% of the transmission, were not taken into consideration. First beam-splitter (BS) was used only to check the emission spectra with the spectrometer, and removed for excitation spectral measurement performed with avalanche photodiodes.

A.1 Correction for the excitation channel

Conversion of the excitation power measured at the input of the microscope to the mean value reaching the sample IR sample P can be obtained from following dependence:

.

where: IR DM R is the reflectivity of the dichroic mirror for IR light;

.

IR Obj

T is the transmission of the objective; IR measured P is the power measured at the front of the microscope. Following equations have been used to extract the second order nonlinear susceptibility tensor

The results of wavelength dependence of the second order nonlinear susceptibility ( ) ( ) ( ) In order to minimize error coming from the noise in the polarization dependence of SHG signal intensity (i.e. non-symmetric positions of the maxima and intensity variations), we have applied SF function, which is directly related to the SHG features understood in general, and can be applied for any type of the crystalline symmetry.

We consider a nonlinear NC, for which the second-order susceptibility tensor is allowed to take any form that can be generally expressed as: 58

Derivations of the full form SF's for both: x and y SHG polarization components are shown below:

( )

SF for the intensity of the y SHG polarization component is obtained in a similar way and given by (κ = 1): using the least square method, where : 1. Rotation about the z axis by ψ angle (counterclockwise, from x to y);

def ac dec af ab def ac aec df bd aef dc dec af bd def ac dec af e def ac bdbc dec af bf bd def ac aec df ab aef dc dec af ab aef dc aec df bd def ac bce d

2. Rotation about the y axis by θ angle (counterclockwise, from z to x);

3. Rotation about the z axis by φ angle (counterclockwise from x to y).

The rotation matrix takes a form of:

, and the rotation axis is oriented along the 1 10     direction, for which the unit vector is:

According to the Rodrigues rotation formula, the rotation matrix is equal to:

where: 1. Rotation of the hybrid crystalline lattice around the Z axis by the angle α (counterclockwise, from X to Y).

2. Rotation of the hybrid crystalline lattice around the new Y axis by the angle β (counterclockwise, from Z to a new X). β determines the out of (x,y) plane angle.

3. Rotation of the structure around the new X axis by γ (counterclockwise, from a new Y to a new Z).

These three steps are associated with the laboratory frame (x, y, z) as follows:

1. Rotation of the hybrid crystalline frame around the x axis by the angle γ (counterclockwise, from y to z).

2. Rotation of the hybrid crystalline frame around the y axis by the angle β (counterclockwise, from z to x).

3. Rotation of the hybrid crystalline frame around the z axis by the angle α (counterclockwise, from x to y).

The rotation matrix is determined by hybrid structure matrix multiplication:

In order to calculate x and y components of the interference term

Appendix D: The list of publications

The list of publications discussed within this thesis:

1. Zielinski, M., Oron, D., Chauvat, D., Zyss, J. Second-harmonic generation from a single core-shell quantum dot. Small, 5, (24), 2835-2840 (2009). 2. Winter, S., Zielinski, M., Chauvat, D., Zyss, J., Oron, D. The Second Order Nonlinear Susceptibility of Quantum Confined Semiconductors -a Single Dot Study. The Journal of Physical Chemistry C, accepted. 3. Zielinski, M., Winter, S., Kolkowski, R., Nogues, C., Oron, D., Zyss, J., Chauvat, D. Nanoengineering the second order susceptibility in semiconductor quantum dot heterostructures. Optics Express, accepted.

The list of my publications not discussed within this manuscript: