Titre : Structures plasmoniques pour le renforcement des effets nonlinaires et la ré alisation de fonctions tout-optiques en photoniques sur silicium Mots clé s : Photonique silicium, plasmonique, optique nonliné aire

Ré sumé :

L'augmentation des flux d'information sur puce conduit l'électronique intégrée à un certain nombre de limitations, lié es en particulier à la saturation des dé bits binaires transmissibles entre blocs et coeurs et au niveau excessif de puissance dissipée. Dans ce contexte, la photonique silicium a été proposé e il y a plusieurs anné es comme une solution inté ressante pour lever certains verrous. Ce domaine, qui a connu un inté rê t marqué depuis, repose sur le dé veloppement de liens optiques sur puce, donc sur le développement de toutes les structures nécessaires pour l'émission, le guidage, la modulation, et la dé tection des signaux optiques. Au stade actuel, les progrè s ont é té spectaculaires mais des difficultés demeurent : d'une part, la puissance consommée par les composants optoé lectroniques, en particulier de modulation, se situe toujours au-dessus des niveaux requis par les applications ; d'autre part, la taille des composants optiques inté gré es classiques ne peut pas ê tre miniaturisé e en-dessous de la limite de diffraction (de l'ordre de 250nm dans les cas usuels de la photonique silicium, dans la fenêtre des longueurs d'onde télécoms λ=1,55µm), ce qui ne permet pas d'envisager une co-intégration poussée de l'optique avec l'électronique CMOS.

Dans cette thèse, nous avons exploré les potentialités de l'utilisation de matériaux organiques nonlinéaires au sein de structures métalliques pour la réalisation de guides d'ondes plasmoniques nonliné aires. Les proprié té s de la plasmonique autorisant la ré alisation de structures sub-longueur d'onde à confinement extrême du champ électromagnétique, les composants qui en découlent sont caracté risé s par un renforcement significatif des effets optiques non-liné aires et leur co-inté gration avec l'électronique devient envisageable en terme de compacité et d'encombrement.

Nous avons dé veloppé une approche basé e sur la thé orie des modes couplé es applicable à des guides à pertes (absorption par les mé taux) et, couplé es à des calculs par é lé ments finis, nous l'avons appliquée à l'exploration des plusieurs effets. Deux types de guides ont été considérés, guides plasmoniques et guides plasmoniques hybrides. Les phé nomè nes de gé né ration de seconde harmonique et de rectification optique (assisté e é lectriquement ou pas) ont é té é tudié s principalement ; les compromis entre pertes de propagation (par absorption) et confinement du champ électromagnétique ont été explorés et l'ensemble a conduit à proposer plusieurs configurations caractérisées par des longueurs d'interaction de quelques dizaines de µm typiquement et des efficacités (de conversion de longueur d'onde, de rectification, etc) se situant au-delà de l'état de l'art actuel.

Ces propositions thé oriques ont é té complé té es par un volet expé rimental, concré tisé par la fabrication de structures plasmoniques, et qui a permis de valider la possibilité d'une injection efficace de la lumiè re depuis une fibre optique vers des guides plasmoniques trè s sub-longueur d'onde.

Acknowledgement

First, I would like to thank my supervisors Prof. Eric Cassan and Prof. Xinliang Zhang for providing me with the opportunity to complete my joint PhD thesis at University Paris-Sud and Huazhong University of Science and Technology. Prof. Eric Cassan is the most easy-going professor I have even seen. He always amazes me with his endless energy, passion, and knowledge. Prof. Xinliang Zhang has been supportive and has given me the freedom to pursue various directions without objection. They were and remains my best role model for a scientist, mentor, and teacher. I would also like to thank my committee members, Prof. Guillaume Bachelier.

Prof. Gilles Renversez, Prof. Alexandre Bouhelier and Prof. Bé atrice Dagens for serving as my committee members. Thank you for letting my defense be an enjoyable moment, and for your brilliant comments and suggestions.

I am also very grateful to all the professors in the two labs Prof. Laurent Vivien, Prof. Delphine Marris-Morini, Prof. Jianji Dong, Prof. Yu Yu and Prof. Lei Shi for their help, scientific advice and insightful discussions. I want to thank present and past group members for the good atmosphere they have built in the lab and good time we have shared together. Thanks to Ping Zhao and Yuan Yu for helping me get used to the graduate life and for sharing their knowledge.

A special thanks to my family for their unconditional support and love. Words cannot express how grateful I am to my mother ShuZhen Zhang, my father Shiyun Zhang and my brother Jixin Zhang for all of the sacrifices that they have made for me. I also want to thank my girlfriend Mengwan Lv for her faith, support and love.

Jihua Zhang December 2015

Orsay, France nonlinearity and existence of free carrier effects in silicon tend to limit the versatility and operating speed of nonlinear optical signal processing, which plays a key role in multiple photonic functionalities.

Plasmonics is a good candidate to overcome the first obstacle due to its ability to confine the optical field into nanoscale beyond the diffraction limit. Moreover, the localized strong field enhancement in plasmonic structures enhances interaction of light and matter, which is promising for enhancing the efficiency and reducing the power consumption in nonlinear applications. For the second limit, organic materials with large nonlinear susceptibility and free of carrier effects have attracted a growing attention in the last years and their integration on silicon have been proposed to be an efficient silicon organic hybrid platform.

In this dissertation, we combine the plasmonic and organic technologies onto the silicon photonics platform to investigate silicon plasmonic organic structures and explore the nonlinear effects induced in them. Silicon plasmonic organic structures, which combine the advantages of silicon with the ultra-compact and strong-field performance of plasmonics and the ultrafast and large-nonlinearity property of organic v materials, have great potentials for nonlinear integrated optics.

A full-vectorial nonlinear coupled-wave equation model valid for lossy plasmonic waveguides is proposed and then used to analyze nonlinear effects in silicon plasmonic waveguides. This dissertation addresses the use of two kinds of plasmonic waveguides: plasmonic slot waveguides (PSW) and hybrid plasmonic waveguides (HPW).

Specifically, an enhanced second harmonic generation (SHG) is proposed in the PSW. A normalized efficiency up to 10 5 W -1 cm -2 is predicted, which is four orders of magnitude higher than those previously reported. Then, by applying two electrodes onto the two sides of PSW, the SHG process can be controlled by the voltage, which is proposed to be a new mechanism for high-speed electro-optic modulation. Another nonlinear effect investigated in PSW is the optical rectification (OR) effect, through which an electrical signal is generated between the two metal slabs when an intensity modulated optical signal is injected into the PSW. This efficient OR process supports flat response for a wide range of wavelength, which can be applied for the realization of high-speed broadband optical detection and demodulation.

The enhanced mid-to-near-infrared SHG in HPW is also explored. The SHG yield is as large as 8.8% for a pumping power of 100 mW and a short length of 120 μm. Then, by utilizing the resonant effect in a microring, the efficiency is further enhanced by two orders of magnitude. This provides a potential route for realizing efficient frequency conversion between mid-infrared and near-infrared wavebands on a chip. Besides, efficient optical parameter amplification (OPA) is also studied in a symmetric HPW.

Based on the OPA, phase regeneration of phase-shift keying signals is then theoretically proposed.

The design, fabrication and measurement of PSWs are also described. The fabricated PSW have good performance in terms of loss. By spin-coating the commercial available second order nonlinear polymer, the nonlinear response of the PSW is under testing at the present writing time.

List of Tables

2.1

Electrooptic coefficients for various second order nonlinear materials ........... 36 is expected to grow to 168 exabytes per month by 2019, up from 59.9 exabytes per month in 2014. This ever-increasing bandwidth requires exponential increases in communication bandwidth at all levels of the system and becomes a big challenge to the conventional electronic circuits due to its intrinsic bandwidth bottleneck and large power consumption at high data rate. At high bit rates of 10 Gbits/s and higher, the circuit's alternating current characteristics begin to dominate the impedance, inductance, and conductance of the electric line, severely distorting the signal.

Meanwhile, electric signal attenuation and power dissipation rise dramatically with higher data rate. To deal with the increasing bandwidth requirement of future data communication networks, optical circuits have been widely accepted as a viable alternative owing to their strengths such as low loss and near infinite optical bandwidth.

Exabytes

Silicon photonics: opportunities and challenges

At present, the fiber based long haul transmission of optical signals is very mature.

Optical connecting between computers with fibers is also commercial. By employing the wavelength division multiplexing and spatial division multiplexing techniques, the gross transmission throughput of 255 Tbit/s has been demonstrated in a single fewmode multicore fibre [1]. This fact of high bandwidth in a single fiber is not enough, the same has to be done at the chip scale. This relates to another aspect of photonics:

the integrated photonics field that deals with on-chip and chip-to-chip optical interconnections and signal processing. In this respect, silicon photonics has drawn a lot of attention in the last decade and has been regarded as the most promising platform in integrated photonics. Motivations to develop silicon photonics include [2][3][4][5][6][7][8]:

 The high-quality of silicon-on-insulator (SOI) wafers, which are an ideal platform for creating planar waveguide circuits. The strong optical confinement offered by the high index contrast between silicon (n = 3.45) and SiO2 (n = 1.45) in the nearinfrared region makes it possible to scale photonic devices to the hundreds of nanometers level;

 Silicon is a transparent material for the wide wavelength band from 1100 nm to 7000 nm approximately, which is far from being limited to the near-infrared communication window of around 1550 nm;

 Silicon has excellent material properties that are important in photonic devices.

These include high thermal conductivity (∼10× higher than GaAs), high optical damage threshold (∼10× higher than GaAs), and high third-order optical nonlinearities;

 Today's mature complementary metal-oxide semiconductor (CMOS) techniques could also allow low-cost, large-scale manufacturing for silicon photonic devices.

Based on silicon photonics, lots of exciting reports and breakthroughs especially on ultrafast nonlinear optical signal processing have been theoretically proposed or experimentally demonstrated [9,10]. For example, R. Salem et al. reported the signal regeneration by using four-wave mixing on silicon chip [11]. Recently, B. Kuyken et al. demonstrated an octave-spanning frequency comb spanning from the telecom wavelength window ~1500 nm to the mid-infrared wavelength range at 3300 nm in a silicon nanophotonic wire waveguide [12]. As shown in Fig. 1.2, by designing the airclad photonic wire with a rectangular cross-section of 1600 nm × 390 nm and slightly over etching by 10nm into the buried oxide, the silicon wire has nonlinear parameter of 38 W -1 m -1 at 2300 nm. The zero-dispersion wavelength is at 2180 nm and the dispersion remains low over a wide spectral band. The spectra at the input from an optical parametric oscillator and output of the waveguide are shown in Fig. 1.2(c) for a pulse energy of 16 pJ. The spectrum of the pulses is significantly broadened in the silicon photonic wire waveguide through self-phase modulation. and the output (black) of the silicon nanowire. Figure taken from ref. [12].

However, in the way to realize chip-scale integrated optical interconnects with low power consumption, advanced functionality and good convergence with electronics, two stumbling blocks emerge in silicon. Firstly, as we know, one of the key advantages of silicon photonics is that photonic and electronic functions can be integrated and fabricated onto the same chip. Nevertheless, the sizes of photonic devices are limited by the diffraction limit, which means that the dimension cannot be smaller than the half wavelength in the dielectric (D ≥ λ/2n). As a result, there exists a size mismatch between photonic and electronic components in their co-integration. Secondly, the lack of second-order optical nonlinearity and the existence of free carrier effect limit the versatility and operating speed of nonlinear silicon photonics, which plays a key role in combining multiple new optical functionalities with electronics on a single chip [9].

As we know in integrated electronics, there is a clear exponential development in the number of transistors per chip, which has been doubling every two years on average during the last four decades. This phenomenon is known as the Moore's law [13].

Similarly, the development of integrated photonics is also considered to obey a photonic Moore's law. The component count on a single chip becomes larger and larger with time [7,14]. For example, Fig. 1.3 shows the complexity development of three mainstream integrated photonic platforms measured as the number of components integrated on a single chip [7]. However, the ultimate miniaturization of optical components, which is of key importance for higher speed and lower power operations, is limited by the diffraction limit. This prevents the further improvement of integration density along with the photonic Moore's law and leads to a size mismatch between electronics and photonics. In the recent years, plasmonics has been explored as a solution to overcome this obstacle due to its ability to confine the optical field into nanoscale beyond the diffraction limit [START_REF] Brongersma | The Case for Plasmonics[END_REF][START_REF] Barnes | Surface plasmon subwavelength optics[END_REF][START_REF] Ozbay | Plasmonics: merging photonics and electronics at nanoscale dimensions[END_REF][START_REF] Gramotnev | Plasmonics beyond the diffraction limit[END_REF][START_REF] Schuller | Plasmonics for extreme light concentration and manipulation[END_REF]. As shown in Fig. 1.4, the dashed lines indicate physical limitations of different technologies. Semiconductor electronics is limited in speed by heat generation and interconnect delay time issues to about 10 GHz. Dielectric photonics is limited in size by the fundamental laws of diffraction. Plasmonics offers the opportunity to combine the size of nanoelectronics and the speed of dielectric photonics, enabling devices that might naturally interface with similar-speed photonic devices and with similar-size electronic components, thus enhancing the synergy between these technologies [START_REF] Brongersma | The Case for Plasmonics[END_REF]. Meanwhile, the localized strong field enhancement in plasmonic structures enhances interaction of light and matter, which is promising for enhancing the efficiency and thus further reducing the power consumption of nonlinear optical operations [START_REF] Kauranen | Nonlinear plasmonics[END_REF]. Due to these distinctive properties, plasmonics is perceived to have a great potential for the development of on-chip optical interconnects. In addition, plasmonics has a lot of other exciting applications beyond integrated photonics. A brief introduction to plasmonics will be given in the next section.

As mentioned before, nonlinear optical signal processing plays a key role especially on ultrafast switching, computing and modern optical communications [START_REF] Langrock | All-optical signal processing using chi((2)) nonlinearities in guided-wave devices[END_REF][START_REF] Ben-Ezra | Applications of All-Optical Signal Processing in Modern Optical Communications[END_REF][START_REF] Zhang | The application of the all-optical signal processing on the optical communication[END_REF][START_REF] Willner | Optically Efficient Nonlinear Signal Processing[END_REF][START_REF] Willner | All-Optical Signal Processing[END_REF]. Silicon offers an abundance of nonlinear optical effects that can be used to generate and process optical signals in low-cost ultra-compact chips at speeds beyond those of today's electronic devices. Many excellent papers have been published to investigate and analyze various nonlinear effects in silicon waveguides [9,[START_REF] Dimitropoulos | Phase-matching and nonlinear optical processes in silicon waveguides[END_REF][START_REF] Espinola | C-band wavelength conversion in silicon photonic wire waveguides[END_REF][START_REF] Koos | Nonlinear silicon-on-insulator waveguides for all-optical signal processing[END_REF][START_REF] Lin | Nonlinear optical phenomena in silicon waveguides: Modeling and applications[END_REF][START_REF] Yin | Impact of two-photon absorption on self-phase modulation in silicon waveguides[END_REF][START_REF] Foster | Nonlinear optics in photonic nanowires[END_REF][START_REF] Tsang | Nonlinear optical properties of silicon waveguides[END_REF][START_REF] Rukhlenko | Nonlinear Silicon Photonics: Analytical Tools[END_REF]. Nevertheless, the centrosymmetry of the silicon crystal prevents second-order nonlinear effects and thus hinders the development of electro-optic silicon modulators and second-order nonlinearity based optical signal processing. Besides, in silicon it exists two photon absorption (TPA), in which an electron makes a transition from the valance band to the conduction band by the simultaneous absorption of two photons. The generated free carriers in the conduction band will subsequently change the refractive index and absorption coefficient of silicon, which are known as the free carrier dispersion (FCD)

and free carrier absorption (FCA), respectively. The long lifetime of the free carriers in the conduction band leads to long-lasting FCD and FCA processes, and ultimately slows down the speed of silicon photonic devices. Meanwhile, the absorption from FCA weakens the optical intensity and in turn decreases the efficiency of nonlinear effects.

To overcome these limitations in silicon nonlinearity, a silicon organic hybrid platform which combines the advantages of silicon with the ultrafast performance of organic materials has attracted a growing attention. It is based on the integration of nonlinear organic polymers with large nonlinear susceptibilities and carrier-effect-free in the silicon photonics platform [START_REF] Baehr-Jones | Polymer Silicon Hybrid Systems: A Platform for Practical Nonlinear Optics †[END_REF][START_REF] Leuthold | Silicon organic hybrid technology-A platform for practical nonlinear optics[END_REF][START_REF] Leuthold | Silicon-Organic Hybrid Electro-Optical Devices[END_REF][START_REF] Xingyu | Polymer-Based Hybrid-Integrated Photonic Devices for Silicon On-Chip Modulation and Board-Level Optical Interconnects[END_REF]. For example, C. Koos et al. demonstrated a silicon-organic hybrid slot waveguide to carry out all-optical high-speed signal processing of all-optical demultiplexing from 170.8 Gbs -1 to 42.7 Gbs -1 [START_REF] Koos | All-optical high-speed signal processing with silicon-organic hybrid slot waveguides[END_REF].

Since plasmonics and organic materials are characterized by several strengths to break the stumbling blocks in silicon for photonic integrated circuits, the next reasonable step is to combine both plasmonics and organics onto the silicon photonics to create the so-called plasmonic organic hybrid (POH) platform [START_REF] Koos | Silicon-Organic Hybrid (SOH) and Plasmonic-Organic Hybrid (POH) Integration[END_REF][START_REF] Melikyan | Plasmonic-organic hybrid (POH) modulators for OOK and BPSK signaling at 40 Gbit/s[END_REF]. The POH optical components are compact, high-speed, low-power-consumption and easily compatible with low-cost CMOS technique, and thus very promising for the future integrated optical signal processing. In this direction, a joint research team from Germany and Switzerland lead by C. Koos and J. Leuthold has reported some pioneer and remarkable works on POH modulators [START_REF] Koos | Silicon-Organic Hybrid (SOH) and Plasmonic-Organic Hybrid (POH) Integration[END_REF][START_REF] Melikyan | Plasmonic-organic hybrid (POH) modulators for OOK and BPSK signaling at 40 Gbit/s[END_REF][START_REF] Haffner | High-speed plasmonic Mach-Zehnder modulator in a waveguide[END_REF][START_REF] Melikyan | High-speed plasmonic phase modulators[END_REF][START_REF] Heni | Plasmonic Mach-Zehnder Modulator with >70 GHz Electrical Bandwidth Demonstrating 90 Gbit/s 4-ASK[END_REF][START_REF] Haffner | All-plasmonic Mach-Zehnder modulator enabling optical high-speed communication at the microscale[END_REF]. For example, a high-speed plasmonic phase modulator with a length of 29 μm and large radio frequency (> 65

GHz) and optical bandwidths (> 120 nm around 1550 nm) was demonstrated in [START_REF] Melikyan | High-speed plasmonic phase modulators[END_REF].

As shown in Fig. 1.5(a), a continuous-wave infrared light guided by the upper-left silicon nanowire was coupled through a silicon taper to the plasmonic slot waveguide.

The slot is filled with a nonlinear organic material, the refractive index n of which can be changed via the Pockels effect by applying a modulating voltage. A second taper transformed the plasmonic mode back to a photonic mode in the lower-right nanowire.

Plasmonics: fundamental and applications

What is plasmonics? Plasmonics is concerned with the study and application of a surprisingly rich set of optical phenomena that can be elicited from metallic nanostructures. At the origin of these phenomena is the excitation of surface plasmons (SPs), which are collective charge oscillations on a metal surface. SPs can take two forms, ranging from freely propagating electron density waves along metal surfaces to localized charge oscillations in metal nanoparticles. In this dissertation we will concentrate on the propagation form of SPs. More specifically we will focus on the surface plasmon polaritons (SPPs), which result from the coupling of propagating SPs with light. SPPs are also defined as electromagnetic surface waves propagating at the interface between a metal and a dielectric material. The SPP mode supported by such an interface is a transverse magnetic (TM) mode, which includes the Ex, Ez and Hy field components. Detailed derivation of SPP modes supported at such a metal-dielectric interface has been reported in many previous review papers and books, for example in [START_REF] Maier | Plasmonics: fundamentals and applications[END_REF], and will not be repeated here. Instead we will present a brief summary of the SPP characteristics. The dispersion relationship of the SPP mode at a metal-dielectric interface shown in Fig. 1.6 is [START_REF] Maier | Plasmonics: fundamentals and applications[END_REF]:

dm 0 dm k ε ε β ε ε   (1.1)
Here β is the propagation constant of the SPP mode propagating in the x direction. k0 is the wave number in vacuum defined by k0 = 2π/λ0 with λ0 being the wavelength in vacuum. In the modeling of SPP, the material dispersion of the dielectric and metal is also to be taken into consideration. Dispersion formulas of the dielectric are often considered in a Sellmeier equation given by:

2 2 ( ) 1 i d i i B λ ε λ λ C    (1.2)
Here λ is the wavelength in vacuum, and Bi and Ci are experimentally determined Sellmeier coefficients. These coefficients are usually quoted for λ in micrometres. The Sellmeier coefficients for many common optical materials can be found in handbooks and papers. For example, for the most used materials in silicon photonics, the Sellmeier coefficients of silicon and silica valid for λ = 1550 nm can be found in [START_REF] Weber | Handbook of optical materials[END_REF] and [START_REF] Kitamura | Optical constants of silica glass from extreme ultraviolet to far infrared at near room temperature[END_REF],

respectively. The relative permittivity of metal is often expressed by a Drude model defined as:

2 p m 2 () i ω ε ω ε ω γω    (1.3)
where ωp is the plasma frequency of the metal and ω is the angular frequency of light. γ = 1/τ where τ is the mean relaxation time of conduction electrons, and ε∞ is a contribution due to interband transitions. For the most used metals in plasmonics, the Drude parameters of gold and silver can be taken as ε∞ = 6.9, ωp = 8.9 eV and γ = 0.07 eV and ε∞ = 3.7, ωp = 9.2 eV and γ = 0.02 eV , respectively [START_REF] West | Searching for better plasmonic materials[END_REF] to match the experimentally measured results in [START_REF] Johnson | Optical Constants of the Noble Metals[END_REF]. Substituting Eqs.

Light line

Dispersion curve of SPP

k β k ε δ k k β k ε δ k       (1.5)
Here kd,m are the components of the wave vector perpendicular to the interface in the two media. δd,m are the evanescent decay lengths of the fields perpendicular to the interface, which quantify the confinement of the wave. As εm is negative and has a large absolute value, the penetration depth of light into the metal is always small. As shown in Fig. 1.7, for small ω values, β is very close to 0 d k ε and the SPP dispersion curve follows the light line very closely. In this case kd is very small and inversely δd is large, thus the SPP wave extends deep into the dielectric. As ω increases, the difference between β and 0 d k ε also increases. Correspondingly, kd becomes larger and δd becomes smaller, which means better confinement of light in the dielectric side.

Especially, when ω approaches the surface plasmon resonance frequency (ωSP), β approaches infinity. ωSP is expressed by

p SP d 1 ω ω ε   (1.6)
Near the surface plasmon resonance frequency, the wave greatly slows down and the extent of the field on either side of the interface becomes vanishingly small. While another important characteristic parameter of the SPP mode: the propagation distance Lp, which is defined as the distance over which the power drops to 1/e of its initial magnitude and given by:

p 1 2 Im[ ] L β  (1.7)
becomes shorter.

To provide here a quantitative idea of the involved orders of magnitude regarding especially the evanescent decay length and propagation distance, we plot them as a function of wavelength in Fig. 1.8 for a silver-air interface. As the wavelength decreases, the SPP mode appears to have a better confinement on the dielectric side (let us notice that d is expressed in µm), while the fraction of the field concentrated in the metal slightly increases (yet, notice that m is given in nm). The propagation distance thus decreases to shorter values due to ohmic absorption in the metal. We can see here the propagation distance of SPP is on the order of millimeters, actually it will become shorter for other metals and dielectrics and roughness of interface in practical demonstrations. The losses have long been the biggest challenge of plasmonics and thus need to be managed carefully. Normally, people tend to use hybrid plasmonic structures with dielectrics, in which the loss penalty is less strong. As a matter of fact, after roughly a decade of research and development, researchers in the field have managed to figure out some potential methods to deal with this biggest challenge [START_REF] Khurgin | How to deal with the loss in plasmonics and metamaterials[END_REF]. These attractive strengths make plasmonics finding applications in many areas of science and technology, including plasmon-assisted catalysis, nanoscale thermal engineering for nanostructure synthesis and phase transformations, heat-assisted magnetic recording and treatment of cancer, optical trapping and sensing, optical imaging and lithography, engineering emission from quantum emitters, thermal radiation engineering, ultrafast and nonlinear plasmonics, plasmonics for solar energy harvesting, and active chipscale plasmonic devices including most notably nanoscale sources, modulators and detectors [START_REF] Brongersma | Introductory lecture: nanoplasmonics[END_REF].

Nonlinear silicon plasmonics

First of all, let us give a brief introduction to nonlinear optics. In nonlinear optics, the total polarization () Pt induced by electric dipoles is not linear in the electric field () Et , but satisfies the more general relation [START_REF] Boyd | Nonlinear Optics[END_REF]:

(1) (2) (3) 0 ( ) [ ( ) : ( ) ( ) ( ) ( ) ( ) ] P t E t E t E t E t E t E t           (1.8)
where χ (1) is known as the linear susceptibility. In the case of linear optics, the induced polarization depends linearly on the electric field strength. χ (2) and χ (3) are known as the second-and third-order nonlinear optical susceptibilities, respectively. In general, χ (i) is a tensor of rank i + 1. In the forms shown, we have also assumed an instantaneous response of the polarization to the electric field strength. The second-order optical nonlinearity vanishes in centrosymmetric crystals, while the third-order nonlinear optical interactions can occur for both centrosymmetric and noncentrosymmetric media.

Common second-order nonlinear effects include second harmonic generation, sum frequency generation, difference frequency generation, Pockels effect (also known as electro-optic effect) and optical rectification effect. Frequently-used third-order nonlinear effects include third harmonic generation, four-wave mixing, self-phase modulation, and cross-phase modulation. These diverse nonlinear effects provide much more degrees of freedom to control the optical field, which make nonlinear optics playing a great impact in today's daily life, such as in the fields of lasers, material interactions and information technology [START_REF] Garmire | Nonlinear optics in daily life[END_REF].

Especially, nonlinear optics has been the hotspot of research in integrated photonics and considerably studied for electro-optic modulation, all-optical signal processing, generation of ultra-short pulses, and ultrafast switching [START_REF] Kauranen | Nonlinear plasmonics[END_REF][START_REF] Zalevsky | Integrated nanophotonic devices[END_REF]. Realizing efficient optical nonlinear effects at nanoscales and within short distances plays a key role in improving the density and reducing the power consumption. Plasmonics opens up an opportunity to simultaneously increase device nonlinear efficiencies, reduce device power consumptions and footprints, speed up the device response time and improve the device flexibility. In principle, the role of plasmonics in nonlinear optics is threefold. Firstly, the strong optical field and related enhanced nonlinearity in plasmonic nanostructures allows nonlinear effects to be utilized with reduced optical power. Secondly, they scale down the sizes of nonlinear components, which is important for the development of integrated photonic devices and, ultimately, of the fully functional nanophotonic circuitry. Thirdly, the response time of plasmonic excitations is ultrafast, allowing optical signals to be manipulated on femtosecond timescales. The detailed principle and applications for the field of nonlinear plasmonics can be found in a good review paper of Ref. [START_REF] Kauranen | Nonlinear plasmonics[END_REF]. Here we just want to highlight the mentioned research status and elicit the next development expectations of the field.

With the continuous development of modern nanofabrication technique, precisely experimental implementations of nonlinear plasmonic devices are possible and demonstrated in plenty of reports. However, with few exceptions the majority of experiments on nonlinear plasmonics have involved free-space beams at least for signal or control light. For certain applications, it is necessary to design systems where all beams are maintained within the integrated plasmonic circuitry [START_REF] Kauranen | Nonlinear plasmonics[END_REF]. In the same time, the compatibility with silicon photonics is also strongly expected. We define the system in which all beams are maintained in the integrated plasmonic circuitry and compatible with silicon photonics as the silicon plasmonic platform. Nonlinear silicon plasmonics is still a young research field, but its basic concepts and great potential have been proposed in a wide variety of ways already. At present, precise experimental fabrication and measurement of the nanoscale silicon plasmonic waveguides is still a big challenge and concepts for more sophisticated waveguide designs and more complex nonlinear effects need to be investigated.

Nevertheless, we envisage the field to become increasingly important in the development of the future integrated nanophotonics, just as traditional nonlinear silicon photonics was in shaping the landscape of modern integrated photonics.

Taking both expectations in mind, in this dissertation we have designed and investigated different kinds of nonlinear silicon plasmonic waveguides and exploited the effective nonlinear effects induced in the nanoscale plasmonic waveguide for optical signal processing.

Organization

The organization of the dissertation is as follows.

In chapter 2, we firstly give a brief review on the plasmonic waveguides.

Especially, we primarily focus on two kinds of waveguides: plasmonic slot waveguides (PSW) and hybrid plasmonic waveguides (HPW). Then, we derive the theoretical model of nonlinear coupled-wave equations which have been used to analyze the nonlinear effects in silicon plasmonic waveguides with losses and for the investigation of the mechanisms proper to enhance optical nonlinear effects.

In chapter 3, we investigate the possibility of enhancing nonlinear effects based on the PSW and the related applications in optical signal processing. Specifically, three kinds of nonlinear effects are studied, including the enhanced second harmonic generation (SHG), a new high-speed electrooptic modulation mechanism based on the electrically controlled SHG and high-speed optical detection based on the optical rectification effect.

In chapter 4, we focus on the HPW-based nonlinear enhancement. Efficient SHG and optical parameter amplification (OPA) are analyzed. The SHG is then further enhanced by using HPW-based ring resonators. The OPA can be applied to regenerate the phase shift keying optical signals.

In chapter 5, we present the experimental realization of PSW. The goal is to demonstrate the optical rectification effect proposed in chapter 3. The chapter includes the design, fabrication and measurement of the structures. By spin-coating an EO polymer onto PSW, the nonlinear property of the coated PSW is investigated.

In chapter 6, we conclude our work and have a look to the future possible steps.

Chapter 2

Theoretical analysis of nonlinear processes in silicon plasmonic waveguides

We aim in this dissertation at investigating the potentials of the combination of silicon photonics and plasmonics for the design of photonic integrated circuits and functions.

We will focus on a particular aspect of this research field, called silicon plasmonics:

the enhancement of nonlinear effects by using plasmonic structures.

In this chapter, we firstly give a brief review of the main plasmonic waveguides proposed in previous works and then focus on two kinds of plasmonic waveguides that can easily be integrated within the silicon photonics platform. In the second part, we derive a set of nonlinear coupled-wave equations (NCEs) which can be exploited to analyze the nonlinear effects induced in the lossy plasmonic waveguides. Note that the NCEs have been widely addressed before in fibers [START_REF]Nonlinear Fiber Optics[END_REF] and silicon waveguides [START_REF] Lin | Nonlinear optical phenomena in silicon waveguides: Modeling and applications[END_REF][START_REF] Osgood | Engineering nonlinearities in nanoscale optical systems: physics and applications in dispersion-engineered silicon nanophotonic wires[END_REF],

in which the losses are negligible. While, in plasmonic waveguides the loss cannot be neglected and thus in the derivation of NCE the normalization condition needs to be considered in term of the unconjugated reciprocity theorem [START_REF] Snyder | Optical waveguide theory[END_REF][START_REF] Ruan | Enhancement of optics-to-THz conversion efficiency by metallic slot waveguides[END_REF]. In Ref. [START_REF] Ruan | Enhancement of optics-to-THz conversion efficiency by metallic slot waveguides[END_REF], the researchers have done similar derivations to model the optics-to-THz conversion in metallic slot waveguides. Here in this manuscript, we obtain a general form of NCE both in the frequency and time domains that can be used to model any kind of nonlinear optical effects induced in a lossy waveguide. Lastly, four general guidelines for the design of integrated waveguides providing strong nonlinear effects are put forward based on the proposed NCEs.

Silicon plasmonic waveguides

Due to the excellent ability to confine light without suffering from the diffraction limit, plasmonic waveguides have attracted a growing interest in the last decade. Many kinds of nanoscale plasmonic waveguides have been proposed theoretically and demonstrated experimentally, such as metal nanowire waveguides [START_REF] Krenn | Non-diffraction-limited light transport by gold nanowires[END_REF][START_REF] Ditlbacher | Silver Nanowires as Surface Plasmon Resonators[END_REF][START_REF] Lal | Noble Metal Nanowires: From Plasmon Waveguides to Passive and Active Devices[END_REF], metal nanoparticle chain array waveguides [START_REF] Quinten | Electromagnetic energy transport via linear chains of silver nanoparticles[END_REF][START_REF] Brongersma | Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit[END_REF][START_REF] Maier | Observation of coupled plasmon-polariton modes in Au nanoparticle chain waveguides of different lengths: Estimation of waveguide loss[END_REF][START_REF] Maier | Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides[END_REF], channel plasmon waveguides being constructed by Vgrooves in metals [START_REF] Bozhevolnyi | Channel plasmon subwavelength waveguide components including interferometers and ring resonators[END_REF], plasmonic slot waveguides (or so called metal-dielectric-metal waveguides in two-dimensional case) consisting of a slot sandwitched between two metallic regions [START_REF] Pile | Two-dimensionally localized modes of a nanoscale gap plasmon waveguide[END_REF][START_REF] Veronis | Guided subwavelength plasmonic mode supported by a slot in a thin metal film[END_REF][START_REF] Liu | Novel surface plasmon waveguide for high integration[END_REF][START_REF] Veronis | Modes of Subwavelength Plasmonic Slot Waveguides[END_REF][START_REF] Han | Experimental realization of subwavelength plasmonic slot waveguides on a silicon platform[END_REF], long range plasmonic waveguides (or so called dielectricmetal-dielectric waveguides in two-dimensional case) comprising a finite-width metal stripe surrounded by dielectric media [START_REF] Berini | Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures[END_REF][START_REF] Berini | Characterization of long-range surfaceplasmon-polariton waveguides[END_REF][START_REF] Boltasseva | Integrated optical components utilizing long-range surface plasmon polaritons[END_REF], hybrid plasmonic waveguides being made up of a high refractive index (RI) dielectric material separated from the metal part by a subwavelength low RI dielectric gap [START_REF] Alam | Super Mode Propagation in Low Index Medium[END_REF][START_REF] Oulton | A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation[END_REF][START_REF] Dai | A silicon-based hybrid plasmonic waveguide with a metal cap for a nano-scale light confinement[END_REF][START_REF] Wu | Conductor-gap-silicon plasmonic waveguides and passive components at subwavelength scale[END_REF], and dielectric loaded plasmonic waveguides made of a dielectric ridge placed on a metal surface [START_REF] Holmgaard | Theoretical analysis of dielectric-loaded surface plasmonpolariton waveguides[END_REF][START_REF] Grandidier | Dielectric-loaded surface plasmon polariton waveguides: Figures of merit and mode characterization by image and Fourier plane leakage microscopy[END_REF][START_REF] Randhawa | Experimental demonstration of dielectric-loaded plasmonic waveguide disk resonators at telecom wavelengths[END_REF].

All these plasmonic waveguides are able to support tight confinement of optical fields in two directions to the nanoscale that dielectric waveguides unable to do.

However, the losses come along with the tight confinement. In plasmonic waveguides, generally there exists a trade-off between confinement and loss. Better confinement is always accompanied by larger losses. This point can be fundamentally understood from In the early stages, the priority was to realize the best confinement of the optical field, while the loss issue was not considered so much. In the initially proposed plasmonic waveguide structures, such as metal nanowire waveguides, nanoparticle array waveguides and channel plasmon waveguides shown in Fig. 2.1, even though ultra-highly localization of the optical field into nanometer scales was achieved, the related structures had large losses, meaning that only short propagation distances (usually ~μm) were possible. Besides, the fabrication and precise control of these waveguide were complex due to their nonrectangular shapes. Additionally, as the related devices were hardly compatible with the silicon photonics platform, we have not considered them in the present manuscript for the design of plasmonic-enhanced nonlinear optical functions on silicon. metal nanoparticle chain waveguides [START_REF] Maier | Observation of coupled plasmon-polariton modes in Au nanoparticle chain waveguides of different lengths: Estimation of waveguide loss[END_REF][START_REF] Maier | Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides[END_REF] and (c) channel plasmon waveguides [START_REF] Bozhevolnyi | Channel plasmon subwavelength waveguide components including interferometers and ring resonators[END_REF].

They can confine the optical field into nanometer scales, while the losses are usually large. In an opposite direction, the long range plasmonic waveguides enable long propagation distances, but their confinement properties show in fact no advantage if compared with conventional dielectric waveguides. Fig. 2.2 shows the schematic and mode profile of a typical long range plasmonic waveguide. By considering an gold stripe with width of 4 μm and thickness of 25 nm, the experimental measured loss is 1.07 dB/mm at λ = 1550 nm, while the mode extends several micrometers away from the metal stripe [START_REF] Berini | Characterization of long-range surfaceplasmon-polariton waveguides[END_REF]. Thus, this kind of waveguides has been disregarded in the investigations performed in this manuscript.

Beyond these two extreme cases, the loss/field-confinement compromise of plasmonic waveguides can fortunately be adjusted by a proper choice of the waveguide material and geometry cross-section. Plasmonic slot waveguides, hybrid plasmonic waveguides and dielectric loaded plasmonic waveguides, as shown in Fig. 2.3, have been proposed in this purpose. These waveguides can also be experimentally realized with the standard CMOS technology, belong to the previously mentioned silicon plasmonic waveguides. The works reported in this manuscript have been mostly based on plasmonic slot waveguides and hybrid plasmonic waveguides. So in the next, we will give a more detailed introduction to these two waveguide families. We will see that due to the difference of the mode confinement mechanism and loss level, they are suitable for different nonlinear applications.

Plasmonic slot waveguides

In view of the strong localization of optical field on a single metal/dielectric interface on the dielectric side due to the large discontinuity difference of the material permittivities, it is intuitive to closely combine two interfaces to form the so-called metal-dielectric-metal waveguide. In a realistic planar scheme, one gets the so-called plasmonic slot waveguide (PSW). One sees that the field is highly localized and greatly enhanced in the nanoscale slot.

Fig. 2.4 shows the typical configuration and mode profile of a PSW. The SPPs on two surfaces that exist in the PSW couple to each other and result in a strong localization of the field in the slot area. The slot width can be chosen down to a few tens of nanometers. Actually, this waveguide geometry is very similar to the silicon slot waveguides, as proposed by Almeida and co-workers [START_REF] Almeida | Guiding and confining light in void nanostructure[END_REF]. Merely the amplitude of the real part of the metal permittivity is even larger than the one of silicon, which makes PSW have an even larger enhancement of optical fields in the slot region than their dielectric counterparts. To have a more quantitative vision, we plot in Fig. The metal, substrate and cladding materials are chosen to be silver, SiO2 and air, respectively. As the slot width decreases, the impact of the height on the mode property becomes weaker. Meanwhile, neff increases, while the loss becomes larger. Especially, as shown in Fig. 2.5(d), PSW is capable of guiding a highly confined mode at λ = 1550 nm into a slot as small as 10 nm × 50 nm. The corresponding mode effective area is as small as 2.04 × 10 -15 m 2 , which is down to λ 2 /1200. However, Lp decreases to 9.35 μm.

Therefore, there usually exists an optimized dimension in PSW-based optoelectronic devices for realizing best device performances. This will be seen in Chapter 3.

The supporting of tight mode confinement, reasonable propagating lengths and intrinsic broadband nature make PSW a promising candidate for chip-scale nanocircuitry. If compared with metal nanowire waveguides, metal nanoparticle array waveguides and channel plasmon waveguides, planar PSWs are easier to fabricate. This waveguide geometry also leads itself appropriate to compact optoelectronic devices as it is convenient to use the same metallic regions to define the optical guide and as electrical contacts to generate a DC/RF electric field across the slot gap filled with a switching medium. For instance, as mentioned in Chapter 1 by Fig. 1.5, a high-speed phase modulator based on this mechanism was recently demonstrated by filling the slot with an electro-optic polymer [START_REF] Melikyan | High-speed plasmonic phase modulators[END_REF]. All these features make PSWs very attractive for the realization of ultra-dense integrated optoelectronic components. For example, PSW-based all-optical logic gates [START_REF] Fu | All-Optical Logic Gates Based on Nanoscale Plasmonic Slot Waveguides[END_REF], subdiffraction directional couplers [START_REF] Kriesch | Functional Plasmonic Nanocircuits with Low Insertion and Propagation Losses[END_REF],

biosensors [START_REF] Osowiecki | Vertically coupled plasmonic slot waveguide cavity for localized biosensing applications[END_REF], high-speed modulators [START_REF] Koos | Silicon-Organic Hybrid (SOH) and Plasmonic-Organic Hybrid (POH) Integration[END_REF][START_REF] Melikyan | Plasmonic-organic hybrid (POH) modulators for OOK and BPSK signaling at 40 Gbit/s[END_REF][START_REF] Haffner | High-speed plasmonic Mach-Zehnder modulator in a waveguide[END_REF][START_REF] Melikyan | High-speed plasmonic phase modulators[END_REF][START_REF] Heni | Plasmonic Mach-Zehnder Modulator with >70 GHz Electrical Bandwidth Demonstrating 90 Gbit/s 4-ASK[END_REF][START_REF] Haffner | All-plasmonic Mach-Zehnder modulator enabling optical high-speed communication at the microscale[END_REF], filters based on plasmon-induced transparency [START_REF] Yang | Tunable ultracompact chip-integrated multichannel filter based on plasmon-induced transparencies[END_REF], fundamental elements such as mirrors, bends and splitters [START_REF] Cai | Elements for Plasmonic Nanocircuits with Three-Dimensional Slot Waveguides[END_REF] , and nonlinear optical devices [START_REF] Rukhlenko | Nonlinear propagation in silicon-based plasmonic waveguides from the standpoint of applications[END_REF][START_REF] Hasan | Second-order nonlinear frequency conversion processes in plasmonic slot waveguides[END_REF][START_REF] Kriesch | Nonlinear switching in a purely plasmonic directional coupler[END_REF][START_REF] Wu | Efficient phase-matched third harmonic generation in an asymmetric plasmonic slot waveguide[END_REF], etc, have been proposed.

When filling the slot gap by nonlinear media, effective nonlinear effects can thus be envisaged at low optical power. For example, spin-coating second order nonlinear polymers in to the slot could give rise to the enhanced second harmonic generation and optical rectification in PSWs [START_REF] Zhang | Highly efficient phase-matched second harmonic generation using an asymmetric plasmonic slot waveguide configuration in hybrid polymer-silicon photonics[END_REF][START_REF] Zhang | Electrically controlled second-harmonic generation in siliconcompatible plasmonic slot waveguides: a new modulation scheme[END_REF][START_REF] Zhang | On-chip high-speed optical detection based on an optical rectification scheme in silicon plasmonic platform[END_REF]. These works will be detailed in Chapter 3.

Hybrid plasmonic waveguide

As mentioned earlier, the main drawback of plasmonic waveguides is the loss issue, particularly for PSW, which losses scale in 0.1-0.5 dB/μm, as shown in Fig. 2.5. This makes them hardly suitable for some linear and nonlinear applications where the loss is a pivotal factor, such as long distance propagation and nonlinear parametric amplification. In Ref. [START_REF] Rukhlenko | Nonlinear propagation in silicon-based plasmonic waveguides from the standpoint of applications[END_REF], Rukhlenko et al. investigated nonlinear metal-silicon-metal (MSM) waveguides and came to the conclusion that even though all nonlinear effects are substantially stronger in MSM plasmonic waveguides than in silicon waveguides of the same thickness, the loss-induced reduced length of MSM waveguides makes the Kerr effect, two-photon absorption, and stimulated Raman scattering too inefficient for device applications. On the contrary, free-carrier effects remain strong enough even in a 10-nm-thick MSM waveguide that is only 500-nm long, and they can dramatically affect the propagation of SPPs. Therefore, it is free-carrier absorption and free-carrier dispersion that should be used to control light in nanosized MSM waveguides [START_REF] Rukhlenko | Nonlinear propagation in silicon-based plasmonic waveguides from the standpoint of applications[END_REF].

To overcome this disadvantage of PSW, people came up more recently with the concept of hybrid plasmonic waveguide (HPW). It has been shown that HPWs have the potential to achieve a nano-scale light confinement as well as relatively long propagation distance simultaneously. In 2007, Alam et al. initially proposed the form of a HPW including a silicon nanowire adjacent to a metal surface [START_REF] Alam | Super Mode Propagation in Low Index Medium[END_REF]. Later in Ref.

[77], the concept of hybrid plasmonic waveguide was proposed formally and a HPW was obtained by putting a dielectric cylinder above a metal surface, for which the mode area was as small as λ 2 /400.

Nonlinear coupled-wave equations

In order to model the nonlinear processes in plasmonic waveguides, such as estimating the efficiency of a nonlinear wavelength conversion process or finding out the phase change of a phase modulation process, the nonlinear coupled mode theory is used in this dissertation. In the following, we will give a detailed derivation for the nonlinear coupled-wave equation (NCE) and then give the key factors to enhance the nonlinear effects in integrated waveguides based on the NCE.

Derivation

For a specific frequency ω in a nonlinear planar waveguide, we consider the nonlinear polarization NL P as a source and the corresponding excited electromagnetic field { , } EH which propagates along the +z direction fulfills the Maxwell's equations in the frequency domain:

NL E i H H i E i P            (2.1)
and can be expanded in terms of all modes at the same frequency:

( ) ( ) ( , ) exp( ) ( ) ( ) ( , ) exp( ) v v v v v vv v v v v v vv E A z E A z E x y ik z H A z H A z H x y ik z     (2.2)
where kν is the propagation constant of ν-th mode. () v Az is the slowly varied complex mode amplitude.

, ,

( , )} v v t v z v v t v z E x y E x y E x y H x y H x y H x y     , , { ( , ) ( , ) ( , ), ( , ) ( , )
are the mode profiles of the v-th guided mode with the subscripts t and z representing the transverse and longitudinal components, respectively. In principle, the expansion of Eq. (2.2) should consider all guided and radiation modes and is continuous over the radiation modes [116]. From a theoretical point of view, the rigorous completeness of the used set of modes is still a question under debate. Involvement of mathematical mode solutions diverging at infinity (thus non normalizable) is in principle necessary [117]. However, in many practical cases, a limited set of interacting waveguide modes is to be considered. In the present manuscript, only the main guided modes have been considered at the different involved frequencies and power exchanges through nonlinear optical mechanisms has been considered between them. This approximation is very common in the study of nonlinear optical waveguides [START_REF] Ruan | Enhancement of optics-to-THz conversion efficiency by metallic slot waveguides[END_REF]. The related guided mode profiles have been normalized as:

1 { ( , ) ( , )} 2 z i j ij e E x y H x y dxdy      (2.3)
Note that this normalization condition is considered here in term of the unconjugated reciprocity theorem, which is applicable to lossy media [START_REF] Snyder | Optical waveguide theory[END_REF]. As a result, the derived NCE is valid for plasmonic waveguides, which are intrinsically lossy due to the absorption loss from the metal.

For a specific guided mode of { , } uu EH , the Maxwell's equations without the nonlinear part are satisfied:

u uu u uu H E i H t E H i E t                   (2.4)
By reversing the sign of z in Maxwell's equations, we can construct a new solution to Eq. (2.4) with the form [START_REF] Ruan | Enhancement of optics-to-THz conversion efficiency by metallic slot waveguides[END_REF]116] (2.1) and Eq. (2.4), we obtain:

: ,, ,, ( , ) ( , ) exp( ) ( , ) exp( ) ( , ) ( , ) exp( ) ( , ) exp( ) u u t u z u u u u u t u z u u u E E x y E x y ik z E x y ik z H H x y H x y ik z H x y ik z                  (2.
( ) ( ) ( ) ( ) ( ) ( )                                             u u u u u NL u u u u u u E H H E E H i H H i E E E H H E E H i H H i E E i P E (2.6) Thus () NL u u u E H E H i P E            (2.7)
The next step is substituting Eq. ( 2.2) into Eq. (2.7) and integrating both sides of Eq.

(2.7) over the cross-section of the waveguide. Applying the orthogonal normalization condition of Eq. ( 2.3) yields the integral of left side to be:

  ( ) ( ) ( ) () () ( ) ( , )                                                                        x u u y u u z u u NL u u u z u u z v v e E H E H e E H E H e E H E H dxdy x y z E H E H dxdy i P E dxdy e E H E H dxdy z e A z E x y z exp( ) ( ) ( , ) exp( ) ( ) exp( ) ( , ) (x, y) ( ) exp( ) ( , ) ( , )                                        vu v z u v v v v z v v u v u v z v v u u v v ik z H dxdy e E A z H x y ik z dxdy z e A z ik z ik z E x y H dxdy z e A z ik z ik z E x y H x y z ( ) exp( ) ( , ) (x, y) ( ) exp( ) ( , ) ( , ) ( ) ( , ) (x, y) ( ) ( , ) ( ,                                      z v v u v u v z v v u u v v u z u u u z u u dxdy e A z ik z ik z E x y H dxdy z e A z ik z ik z E x y H x y dxdy z A z e E x y H dxdy A z e E x y H x zz ) () 4       u y dxdy Az z (2.8)
In the derivation of Eq. (2.8), we have used the following relations:

( , ) (x, y) ( , ) (x, y) 2 ( , ) (x, y) ( , ) (x, y) 2                                   z v u z v u u z u z u u e E x

y H dxdy e E x y H dxdy e E x y H dxdy e E x y H dxdy

(2.9) Thus, we can finally obtain the NCE for guided waves to be:

  4 NL u u A i P E dxdy z       (2.10)
For a lossy waveguide case, the propagation constant can be written into its real and imaginary parts as: ku = βu + iαu/2. Here βu and αu represent the phase propagation constant and attenuation coefficient, respectively. If we set ( )

( )exp( 0.5 )   u u u A z A z z
and substitute it into Eq. (2.10), we obtain the following NCE for lossy waveguides:

  exp( ) ( , ) 24
NL uu u u u A i A i z P E x y dxdy z             (2.11)
in which the relation of (

u u u u u u E E x y ik z E x y i z z          is , ) exp( ) ( , ) exp( 0.5 )
used for the derivation. Now, the power of the lossy u-th mode along the propagation can be expressed as 2 ( ) ( )

uu P z A z  .
In NL P , the part which is irrelevant with the transverse ordinates of (x, y) can be extracted from the integral term. This means that if we set

( , ) (z)exp( )   NL NL NL NL P P x y P i z , we can obtain: (z) exp( ) ( , ) ( , ) 24               NL NL uu uu A i A P i z P x y E x y dxdy z (2.12)
Here ∆β = β NLβu is the phase mismatch. The integral term can be defined as the nonlinear coupling coefficient:

( , ) ( , )       NL u P x y E x y dxdy (2.13)
which only depends on the mode confinement and overlap integral between interacted modes in the nonlinear areas within the waveguide cross-section. Eq. (2.12) is the NCE in the frequency domain. If we want to obtain the time-domain NCE, we need to take the dispersion (i.e. the dependence of ku on ω) into account.

Next, we set ( ) ( ) exp( ) ( ) exp( 0.5 )

u u u u u u A z A z ik z A z i z z     
and substitute it into Eq. (2.10). The resulted new equation is:

    exp( ) exp( ) ( , ) exp( ) 4 ( 0.5 ) ( , ) 4 NL u u u u u u u NL u u u u u A i ik z ik A ik z P E x y ik z dxdy z A i i A P E x y dxdy z                           (2.14)
in which we have used the relations of ku = βu + iαu/2 and ( , ) exp( )

u u u E E x y ik z   .
Then, by considering the dispersion as 0 00 00

1 ( ) ( ) ( ) !! m mm u um u m mm mm                   and 0 00 00 1 ( ) ( ) ( ) !! m mm u um u m mm mm                   (where 0 m u um m        and 0 m u um m       
are the m-th dispersion coefficient and loss dispersion coefficient of the u-th mode at the center frequency of ω0, respectively), we can obtain:

  0 0 0.5 ( ) ( , ) !4 m NL u um um uu m Ai i A P E x y dxdy zm                 (2.15)
Now by implementing an inverse Fourier transformation on both sides of Eq. (2.15), we can get the NCE in the time domain as: for the continuous wave case, while the time domain form given by Eq. (2.16) can be applied in the pulse wave analysis. The only difference between various nonlinear effects lies in the expression of the nonlinear polarization NL P term dependence over the electrical field. For example, when considering a common degenerate four-wave mixing process, where a single pump wave ωp provides amplification for a neighbored signal wave ωs and generates an idler wave on the other side of the pump through a third-order optical nonlinearity (ωi = 2ωp -ωs), the NL P at different frequencies are:

  0 0 0 ( 0.5 ) exp( ) 1 exp( ) ( , ) !4 mm NL ut um um ut tu m m A i i A i t i t P E x y dxdy z m t t                        (2.
* (3) * 0 * (3) * 0 * (3) * 0 3 exp[ (2 ) ] ( , ) ( , ) ( , ) 4 3 exp[ (2 ) ] ( , ) ( , ) ( , ) 4 3 exp[ ( ) ] ( , ) ( , ) ( , ) 2 NL s p p i p i p p i NL i p p s p s p p s NL p s i p s i p p p i P A A A i z E x y E x y E x y P A A A i z E x y E x y E x y P A A A i z E x y E x y E x y                   (2.17)
where χ (3) is the third-order nonlinear susceptibility. The asterisk represents the conjugate term.

Principles to enhance nonlinear effects

In general, nonlinear effects are very weak especially for high-order nonlinear processes due to the small magnitude of nonlinear susceptibilities. As a result, in order to excite efficient nonlinear processes, high pump powers and long propagation lengths are typically needed. For example in the most common used nonlinear silicon nanowire waveguides with dimensions of 220 nm × 450 nm, the nonlinear length is as long as 8

mm for an optical pulse with power of 0.2 W [START_REF] Osgood | Engineering nonlinearities in nanoscale optical systems: physics and applications in dispersion-engineered silicon nanophotonic wires[END_REF]. How to enhance the nonlinear effects is pretty essential for the purpose of reducing the power consumption and footprint of optical devices.

In the next, we will discuss the key factors to enhance the nonlinear efficiency in term of the former derived NCE. Firstly, let us list all the key factors as follows:

 Nonlinear materials with large nonlinear susceptibilities are needed.

 Long distances of interaction between light and matter are expected.

 Phase matching conditions are required in parametric processes.

 Strong optical mode confinement and large overlap integrals between interacted modes are welcome.

Firstly, the most intuitional way is to select nonlinear materials with the highest possible nonlinear susceptibilities. As can be seen in Eq. (2.17 Table 2.1 lists the EO coefficients of various second order nonlinear materials. EO coefficient r and second-order susceptibility χ (2) are interrelated by r = -2χ (2) /n 4 [START_REF] Leuthold | Silicon organic hybrid technology-A platform for practical nonlinear optics[END_REF].

The first material is the most commonly used EO material lithium niobate (LiNbO3), while the other materials are EO polymers. It is clear that the EO coefficients of polymers are one order of magnitude larger than that in LiNbO3. Table 2.2 gives the nonlinear index coefficients of different kinds of third order nonlinear materials which are compatible with the CMOS platform. Nonlinear index coefficient n2 and third-order susceptibility χ (3) are interrelated by n2 = 3Re[χ (3) ]/4ε0cn 2 [START_REF] Leuthold | Silicon organic hybrid technology-A platform for practical nonlinear optics[END_REF]. Similarly, polymers

show the best performance with respect to the reachable nonlinear coefficients. As a matter of fact, there also exists drawbacks for organic polymers. Their integration onto silicon is usually realized by using spin-coating technique. In this technique, most of the material is spun off the wafer during this technological step, which is not suitable for low-cost large-scale manufacturing. While the deposition of SiO2 and Si3N4 is usually realized by plasma enhanced chemical vapor deposition (PECVD) technique. Another drawback of organic polymers is that their short-and long-term stability especially with regard to temperature should be improved. Their performance deteriorates at high temperature. Fortunately, efforts have made to address this problem [136]. In spite of these weaknesses, their excellent nonlinear properties are interesting enough to investigate possible optical functions with low power, which

is precisely what we have done within this manuscript.

In a word, the research on better nonlinear materials is always the most fundamental and effective way to realize high-performance nonlinear devices and progress in this domain is still on-going.

The second key factor is to increase the interaction distance between light and matter. Mathematically speaking, the contribution of the first term on the right side (i.e. loss term) of Eqs. (2.12) and Eq. (2.16) lies in adding an exponential decay into the amplitude solution of the NCE. As nonlinear effects depend on energy density, this means that the strength of nonlinear interaction scales down with the loss level. In nonlinear optics, the effective length is typically expressed as Leff = (1 -e -αL )/α with L and α being the practical length and attenuation coefficient of the nonlinear waveguide, respectively. This means that no matter how long the practical waveguide is, the effective length is always less than a maximum of the propagation length 1/α, i.e.

inversely proportional to the loss. Therefore, in order to have a long light-matter interaction distance, the loss level of the waveguide should be as small as possible.

The loss of a waveguide comes from two aspects: material absorption and scattering loss due to bending and roughness of the sidewalls. Thus the issue can be considered both from the material and fabrication points of view. Firstly, we need to use nonlinear materials with low loss. For example, as already mentioned in Chapter 1, polymers are considered to be better than silicon due to the existence of two photon absorption and concomitant free carrier absorption in silicon for high optical powers in the near-infrared wavelength range. For plasmonic materials, people tend to use noble metals such as silver and gold with relatively low losses [START_REF] West | Searching for better plasmonic materials[END_REF]. In addition, the loss of the plasmonic mode propagating along a metal-dielectric interface decreases for dielectric with smaller refractive index, which further reinforces the advantage of lowindex polymer in plasmonic structures. Secondly, the fabrication procedure should be optimized in order to realize smooth waveguide etched sidewalls. In the same time, as presented in the mode analysis of plasmonic slot and hybrid plasmonic waveguides in Section 2.1, in plasmonic waveguides the loss is also strongly relevant to the geometrical dimension of the waveguide. Besides, an important feature of waveguide size is that it affects the phase matching, as shown hereafter. For these two reasons, the geometrical parameters of plasmonic waveguides need a careful appropriate design.

The third aspect that needs to be considered is the phase matching. In order to satisfy the phase matching condition (PMC), the most popular methods include birefringence, quasi-phase-matching (QPM) and mode phase-matching (MPM)

techniques [137]. Birefringence is the dependence of the refractive index on the direction of polarization of the optical radiation. By making use of this, it is possible to achieve the PMC between the interacting frequencies with different polarizations [START_REF] Boyd | Nonlinear Optics[END_REF].

QPM is one of the most effective techniques for the attainment of phase matching in a material that lacks birefringence and is achieved through periodic modulation of the nonlinear coefficient [138]. However, in silicon plasmonic waveguides, it is not easy to use these two approaches. On the one hand, plasmonic waveguides usually can support only one kind of polarization mode, and most of the silicon-compatible nonlinear materials show little birefringence. On the other hand, periodical poling in a compact plasmonic waveguide is difficult and will increase the cost and complexity of the device.

The MPM method for achieving PMC is to make use of the distinction of modal dispersion properties between different frequencies. In general, waveguides can support several modes, and the mode effective refractive indices of higher-order modes are lower. Therefore it is possible to achieve the PMC if the higher frequency propagates at a higher-order mode while the low one propagates at a lower-order mode. This approach is simple because it does not require any additional technological step after the waveguide fabrication. The MPM is usually realized by tuning the geometrical parameters of the waveguide cross section and has been widely demonstrated as an effective method to achieve PMC in plasmonic waveguides between optical and plasmonic modes [139, 140] or plasmonic modes with different orders [141] or symmetries [142,143].

The last key factor to realize considerable nonlinear effects is the tight optical confinement, or in other words, enlarging the overlap integral item between interacting modes in Eqs. (2.12) and (2.16). This is obvious since nonlinear effects are dependent on optical intensity. Actually, after removing the complex mode amplitude items from the nonlinear polarization, the integral item in Eqs. (2.12) and (2.16) is exactly the effective nonlinear coefficient of a nonlinear waveguide that people have commonly used in nonlinear analysis, such as the interaction factor Γ or the effective index susceptibility γ for the EO effects [START_REF] Leuthold | Silicon-Organic Hybrid Electro-Optical Devices[END_REF]144], nonlinear coefficients for nonlinear silicon waveguides [START_REF] Lin | Nonlinear optical phenomena in silicon waveguides: Modeling and applications[END_REF] or Kerr nonlinear coefficients in subwavelength waveguides [145].

The first step to address this issue is to excite nonlinear effects in miniature waveguides with a cross-sectional area on the micrometre scale and below. This approach has already led to major breakthroughs, for instance signal regeneration using low-power four-wave mixing in a silicon nanowaveguide [11]. The optical confinement can be further increased in silicon slot and photonic crystal waveguides [9,[START_REF] Almeida | Guiding and confining light in void nanostructure[END_REF]. Except from such slot waveguide effects motivated in the cross section of a waveguide, the other way to increase the optical field confinement lies in introducing resonant structures or slow light effects in the propagation direction. As we know it, the optical field is enhanced around the resonant wavelength in a resonant structure such as in microring resonators, gratings or microcavities [146]. The enhancement factor is proportional to the quality factor of the resonator. Thus, resonant structures with high quality factor are often used to enhance nonlinear effects. Slow light is considered to provide another opportunity for further enhancement. When an optical pulse travels in a slow-light waveguide, it is compressed and its energy density is thereby increased extend the interaction time between light and matter within the same waveguide length due to the slower light group velocity, thus can enhance the nonlinear interactions. As mentioned before, for all-dielectric waveguides, the optical field confinement is limited to the diffraction limit. In contrast, plasmonic waveguides can provide a true nano-scale waveguiding and confinement of light, which makes plasmonic waveguides enabling the tightest confinement of light.

In order to have a more visual understanding, we next give a comparison of the third order Kerr nonlinearity between a silicon slot waveguide (SSW) and a plasmonic slot waveguide (PSW) with same slot cross-sectional sizes. For the Kerr nonlinearity in a waveguide, a Kerr nonlinear coefficient γ is often used to evaluate the nonlinear magnitude. According to the full vectorial nonlinear model, γ in a subwavelength waveguide is defined as [145]:

2 eff 2 n A     (2.18)
Here λ is the optical wavelength in free space, 2 n is the nonlinear refractive index averaged over an inhomogeneous cross section weighted with respect to field distribution, and Aeff is the effective mode area. They are expressed as [145]: where z () z S E H e    (ez is the unit vector in z direction). n(x, y) and n2(x, y) are the linear and nonlinear refractive index distribution within the cross-section, respectively.

The comparative results are given in Table 2.3. The cross-sectional size of two slots is 157 nm wide and 220 nm high. The width of two silicon wires in the SSW is 216 nm, which comes from the reported highly nonlinear SSW in [START_REF] Koos | All-optical high-speed signal processing with silicon-organic hybrid slot waveguides[END_REF]. One sees that the PSW has a smaller effective mode area and a larger nonlinear coefficient over one order of magnitude, which means tighter confinement and thus facilitates stronger nonlinearity. Meanwhile, the loss of PSW is much larger than that in SSW due to the existence of metal. In the next, we will yet see that the large nonlinear coefficient and loss in silicon plasmonic waveguides make them applicable for short-distance strong nonlinear effects, while hinders them in long distance nonlinear applications. We take here the case of the degenerate four-wave mixing (FWM) process as an example, whose efficiency in a waveguide is proportional to the factor of η  exp(-αL)(γLeff) 2 [START_REF] Koos | All-optical high-speed signal processing with silicon-organic hybrid slot waveguides[END_REF]. L is the practical length of the waveguide and Leff is the effective length being defined as Leff = [1-exp(-αL)]/α. Fig. 2.8 plots the effective length and FWM efficiency factor as a function of the waveguide length. The parameters of the SSW and PSW are taken from Table 2.3. The black dashed line in Fig. 2.8(a) is the propagation length of PSW divided by the length, which is always larger than the effective length.

After a propagation length of around 100 μm, the effective length nearly reaches its maximum, i.e. the propagation length. But the effective length becomes smaller and smaller compared with the waveguide length for larger waveguide lengths, while for a short waveguide, the effective length is only slightly reduced when compared with the waveguide length (Leff/L = 0.8 when L = 15 μm). In Fig. 2.8(b) the FWM efficiency factor in PSW is larger than that in low-loss SSW when the waveguide is shorter than 96 μm. This can be explained as follows. In the beginning, the loss has little influence on the FWM efficiency due to the short length. Thus the PSW with larger nonlinear coefficient supports larger FWM efficiency, and the efficiency increases as length increases. After a length of around the propagation length, the loss begins to counteract the conversion from the nonlinear process, thus the FWM efficiency decreases. Here the peak efficiency happens at L = 36 μm ~ Lp = 32.85 μm. For this length, the loss of SSW has no influence on the efficiency. These features highlight the potential interest of PSW as compact devices, as the effectiveness of long PSW is limited by linear propagation losses, at least with current technology.

In conclusion, even with larger losses, plasmonic waveguides show superiority over dielectric ones for nonlinear applications when lengths are kept compact, specifically no more than several propagation lengths. The ability to realize efficient nonlinear effects within short distance makes plasmonic waveguides a promising platform for ultra-compact on-chip optical signal processing and communications.

In this dissertation, we have taken the four key factors giving rise to strong nonlinear effects into consideration. Regarding the materials' choices, nonlinear polymers with large nonlinear susceptibilities and low loss will be considered as the active nonlinear materials in the investigations described in next chapters, with silver and gold to realize the metallic layers. With respect to their cross-section geometry, silicon plasmonic waveguides with tight field confinement and nice compatibility with the silicon photonic platform will be considered. Regarding the phase matching conditions, mode phase matching method will be utilized. Based on whole framework, we will proposed enhanced nonlinear effects in plasmonic slot and hybrid plasmonic waveguides, and investigate some of their possible applications for on-chip nonlinear optical signal processing, such as phase regeneration, modulation and detection.

Chapter 3

Enhancement of nonlinear effects using plasmonic slot waveguides

As seen in Chapter 2, plasmonic slot waveguides (PSW) have the ability to squeeze light far below the diffraction limit. The field enhancement at subwavelength scale is really helpful to realize effective nonlinear effects, and based on the supported nanoscale nonlinear effects, compact optical functionalities can thus be envisaged.

Meanwhile, the metallic slabs that define the PSW can also serve as electrodes for the generation of a high electric field with low applied voltages. This feature makes PSW suitable for realizing optoelectronic components with low power consumption. Lastly, PSW has the simplest configuration among all the proposed plasmonic waveguides. It is reasonably easy to fabricate them experimentally with standard nano-fabrication techniques. This chapter will detail numerical investigation of three nonlinear applications exploiting PSW, including enhanced second harmonic generation (SHG), high-speed electro-optic (EO) modulation mechanism based on the electrically controlled SHG, and on-chip high-speed optical detection based on the optical rectification effect.

The rest of the chapter is organized as follows.

In section 3.1, we investigate the most fundamental nonlinear effect, i.e. SHG in PSW. By optimizing the waveguide geometry and exploiting the mode phase matching method, a normalized SHG efficiency more than 1 × 10 5 W -1 cm -2 is predicted within a PSW length shorter than 20 μm.

In section 3.2, we go a step further and take both the second order and third order nonlinearities of the considered polymer into consideration at the same time. By applying an electrical signal between the two metallic slabs, electrically controlled SHG in an integrated waveguide is proposed. The corresponding intensity and phase modulation magnitudes are predicted to be 8 μW/V and 0.12 π/V, respectively. We then consider this process as a possible mechanism of ultra-high speed EO modulation.

In section 3.3, we finally investigate the inverse process of the EO effect, i.e. the optical rectification effect, and apply it for a broadband high-speed optical detection.

The proposed detector characterizes a relatively flat response for a wide optical band from 1.25 μm to 1.7 μm and supports a RF cut-off frequency up to 800 GHz.

3.1 Enhanced second harmonic generation in plasmonic slot waveguides

Introduction

Among all the nonlinear processes, second harmonic generation (SHG) is frequently studied for its simple theoretical principle, reasonable easiness for demonstration in experiments, and various interesting applications [START_REF] Boyd | Nonlinear Optics[END_REF]. In silicon photonics, however, SHG cannot be excited directly in silicon because the second-order susceptibility vanishes in this material owing to crystal centro-symmetry. Exploitation of silicon nitride (Si3N4) has been proposed to circumvent this drawback and induce second-order nonlinear processes in silicon compatible structures [149][150][151][152]. For now, three mechanisms have been proposed in this purpose. Firstly, χ (2) Hence, structures combining the plasmonics and NP have a great potential for nonlinear nano-optics. Here, we propose a second-order nonlinear PSW configuration in hybrid polymer-silicon photonics. To theoretically investigate the potentials of this approach, we consider a NP with high χ (2) as its core nonlinear material. This waveguide is able to tightly confine fields in the subwavelength nonlinear slot both at FF and SHF [START_REF] Veronis | Modes of Subwavelength Plasmonic Slot Waveguides[END_REF], which is crucial to efficiently maximize the nonlinear interaction. In addition, the theoretically considered structures can potentially be fabricated with state-of-the-art silicon compatible fabrication techniques [155]. With a careful choice of the optogeometrical parameters, the phase-matched SHG at the FF wavelength corresponding to λFF = 1550 nm induced in the proposed waveguide is analyzed by numerical simulations. Then, the SHG performance of the structure is modified by enhancing the asymmetry of the PSW. In the new proposed geometry, the nonlinear coupling coefficients are significantly enhanced, leading to prediction of highly efficient SHG with low pump power ( 10 mW) and short waveguide length ( 20 µm).

Waveguide structure and nonlinear modeling approach

.1 shows the cross-section of the proposed waveguide geometry, which is similar to the one reported in [141] for a LiNbO3-based structure, and with a χ (2) polymer infiltrated into a metallic slot. The width and height of the NP region are w and h, respectively. The considered polymer is the doped, crosslinked organic polymer with a refractive index of n = 1.643 and an EO coefficient of r33 = 170 pm/V at the wavelength of 1550 nm [START_REF] Leuthold | Silicon organic hybrid technology-A platform for practical nonlinear optics[END_REF]132]. Based on the conversion equation

(2) 4 1 1 1 /2 lmn l m n lmn nr      [163]
, the corresponding NP second-order nonlinear susceptibility (2) 111  = 619.4 pm/V can be obtained. Metal is defined here as silver due to its relatively low loss in the calculated wavelength range and with a Drude permittivity dispersion given by

  / Ag p i             , with ε∞ = 3.7, ωp = 9.2
eV and γ = 0.02 eV [START_REF] Johnson | Optical Constants of the Noble Metals[END_REF].

Substrate

Silver

Silver Polymer Air h w x y z Fig. 3.1 Schematic cross-section of the proposed plasmonic slot waveguide with χ (2) nonlinear polymer infiltrating into the slot and substrate compatible with SOI platform.

In order to couple light into such a PSW from a standard silicon waveguide, Z.

A A i A A i z z A A i A A i z z                   (3.1)
where Δβ = βSH -2βFF is the phase mismatch and κFF,SH are the nonlinear coupling coefficients (NCC), which are defined by [139]:

    (2) * 0 (2) 0 : ( , ) ( , ) ( , ) : ( , ) ( , ) ( , )
FF SH FF FF SH FF FF SH E x y E x y E x y dxdy E x y E x y E x y dxdy             (3.2)
Note that the χ (2) in two NCC are different in terms of principle. The complete forms are χ (2) (-ω; 2ω, -ω) at FF and χ (2) (-2ω; ω, ω) at SHF, respectively. However, if all the involved optical frequencies are much smaller than the lowest resonance frequency of the nonlinear material system, which is apparently satisfied here, the nonlinear susceptibility would be dispersionless. Thus, we have taken the same value of χ (2) at two frequencies. Besides, the longitudinal components of the modes are neglected in the NCC calculations because the transverse components of both the hybrid modes and the nonlinear susceptibility dominate all over other components.

Then, we give the definition of the normalized conversion efficiency as a factor of merit with which we propose here to quantify the efficiency of SHG conversion process:

  2 2 () (0) ( ) SH p FF p PL P L cm     (3.3)
where PFF(0) is the pump power of the FF, Lp is the length when SHF reaches its maximum (defined as peak position) and PSH(Lp) is the corresponding maximum output power, respectively. This allows better assessment of the conversion process since it takes the conversion efficiency ηp = PSH(Lp)/PFF(0) (to be maximized), the pump power of the FF wave (to be minimized), and the needed device length (to be minimized) simultaneously into account. It should be noted that this definition is slightly different from the standard one in a lossless waveguide, where the waveguide length is fixed. As the optimized device length Lp varies here with the pump power, different losses for the FF and SHF are indeed obtained, meaning a variable normalized conversion efficiency under variations of the pump power.

The key factors to increase the SHG efficiency both in terms of higher ωFF→ωSH conversion yield and smaller device footprint are the reduction of propagation attenuation, the good filling of the phase matching condition (PMC) (i.e. ∆β = 0), and the achievement of large coupling coefficients κFF,SH [START_REF] Ruan | Enhancement of optics-to-THz conversion efficiency by metallic slot waveguides[END_REF]. In the PSW under study here, these three factors can be fulfilled and/or optimized by adjusting the waveguide optogeometrical parameters.

In order to reduce the propagation attenuation, let us first remind that the metal was chosen as Ag due to its relatively low losses and that to further minimize the PSW propagation losses, a polymer with a relatively low index of 1.643 is considered. The choice of the slot width w, however, results from a trade-off between the accepted loss level increasing and the obtained field-induced nonlinear effect enhancement by lowering w. Besides, an important feature of w is that it affects the PMC, as shown hereafter. For these two reasons, this parameter needs a careful appropriate design.

With respect to the PMC, we adopt the mode phase matching method in the present paper, first in the simplest possible configuration before extending it to improve the SHG efficiency as shown hereafter in section 3.1.4.

Enhanced SHG in the proposed plasmonic slot waveguide

Now we calculate the SHG for the case of Fig. 3.2 where the 0-th mode at the FF is converted into the 1-st mode at the SHF. The corresponding NCC is calculated to be by doing a post-processing of "surface integration" after mode calculations in COMSOL: It is shown that a peak power of the generated SHF wave up to 0.021 W is revealed at a propagation length of only 17.9 µm. The corresponding normalized conversion efficiency is 6.55 × 10 3 W -1 cm -2 . This result is larger than the one obtained in LiNbO3based PSW devices by about three orders of magnitude with same pump power at the FF [141]. The improvement of the SHG performance mainly results from two aspects.

* -1 -1/2 71 psm W FF SH   .
On one hand, the considered polymer has a smaller index compared to LiNbO3. This leads to a larger optical power confined into the nonlinear slot region and to a lower waveguide loss level [START_REF] Veronis | Modes of Subwavelength Plasmonic Slot Waveguides[END_REF]. On the other hand, the larger χ (2) in the considered polymer enlarges the coupling coefficients κFF,SH. Starting from this first result, further improvement is investigated hereafter by adjusting the compromise between the in-slot field confinement and the optical propagation losses. To do so, we analyze the influence of the slot width on the SHG process performance. In Fig. 3.4, the slot height to satisfy the PMC, the nonlinear coupling coefficient (κ), the peak efficiency (ηp) and peak position (Lp) of the SHG process, and the corresponding normalized conversion efficiency (η) are plotted versus the slot width for a FF pumping power of 1 W. As can be seen, when the slot width decreases, the height required to ensure the PMC decreases and the coupling coefficient increases correspondingly due to the stronger confinement of the field in a smaller area.

As a result, the maximum power of SHF increases and the propagation length to realize this efficiency becomes shorter, as is shown in Fig. 3.4(b). The normalized efficiency increases as well. It can be concluded that the PSW has better SHG property with smaller width. However, it is necessary to stress also that the fabrication feasibility should be taken into consideration, so that too narrow slots cannot be considered.

Further improvement of SHG efficiency by tailoring the waveguide asymmetry

What limits the interest of the mode phase-matching method to satisfy the PMC is that the spatial overlap integral between modes with different orders or different symmetries is always less than that between modes with same orders or same symmetries. This in turn results to small nonlinear coupling coefficients (NCC), as defined in Eq. (3.2). For example, we can see in Fig. 3.2(b), that the 0-th mode field distribution at the FF is nearly symmetrical in the nonlinear area. However, in the same time, it is shown in Fig.

3.2(d)

that the 1-st mode field distribution at the SHF is nearly anti-symmetrical in the domain of integration. This will inevitably lead to relatively small NCC.

In order to overcome this drawback and enlarge the NCC between the two considered modes, we increase the asymmetry of the PSW by using a Si3N4 substrate In the previous sections, a CW pump power of 1 W was considered to facilitate the direct comparison with the results reported in previous works that considered this pumping condition [109,141,156]. However, it should be noted that lowering the pump power is highly desirable and could enable low-power nonlinear all-optical operations amenable to various practical applications in future generations of integrated photonic circuits. Besides, the input power should be chosen carefully so that the intensity in the sub-wavelength slot remains lower than the damage threshold of the material. Since the modified waveguide has so large NCC, we show that the second harmonic can be efficiently generated under low pump power. . This dramatic improvement makes the proposed plasmonic waveguide geometry very promising and competitive for low-power all-optical signal processing.

Conclusion

In conclusion, we have proposed a silicon compatible nonlinear PSW with a nonlinear polymer infiltrated the metallic slot. For the SHG induced in this PSW, the phase matching condition between the FF and SHF can be satisfied with appropriate designs of the opto-geometrical parameters. Efficient SHG, about three orders of magnitude above the one reported in previous works obtained for LiNbO3-based structures, was demonstrated and the influence of the slot width on SHG efficiency was numerically analyzed. Then, the SHG performance was further improved by one order of magnitude by increasing the spatial asymmetry of the proposed waveguide. In the modified proposed geometry, a normalized SHG conversion efficiency above 1 × 10 5 W -1 cm -2 is predicted for a propagation distance shorter than 20 µm. The output power of the SHF generated in the optimized structure for a low pumping power of 10 mW is comparable to the one obtained in previous works for a 1W pumping power. This obtained dramatic improvement of the nonlinear efficiency allows envisaging practical applications using low-power nonlinear SHG-based all-optical signal processing.

High-speed electro-optical modulator based on electrically

controlled second harmonic generation

Introduction

The versatility of plasmonic slot waveguides (PSW) and their strong ability to confine optical fields at nanoscale voids make them suitable for the enhancement of other nonlinear optical mechanisms apart from the simple SHG. With respect to section 3.1, we now investigate an electrical field induced second harmonic generation (EFISHG) mechanism by making use of an applied electric field.

EFISHG effect, which was first observed in centro-symmetric crystals by applying to them a DC electric field [165], is a frequency-doubling process via four-wave mixing with one of the mixing fields being a DC field or an AC field with a negligible frequency if compared to the optical one. Unlike the conventional second harmonic generation (CSHG) being attributed to the second-order nonlinear susceptibility χ (2) (2𝜔;

𝜔, 𝜔), EFISHG arises from its third-order counterpart χ (3) (2𝜔; 𝜔, 𝜔, 0), which permits the generation of frequency-doubling radiation in either centro-symmetric or noncentro-symmetric materials. EFISHG has been proposed in several materials or structures without χ (2) , such as in glass [START_REF] Koos | All-optical high-speed signal processing with silicon-organic hybrid slot waveguides[END_REF]166], semiconductors [167], optical fibers However, the SH signal generated from EFISHG is usually orders of magnitude smaller than the CSHG contribution. It often requires large voltages (> 100 V) to generate a noticeable EFISHG. Besides, in order to fulfill the phase matching condition (PMC), complex periodic design of electrodes is required [168]. These deficiencies can be modified or simplified by using plasmonic structures. On one hand, the significantly enhanced electromagnetic concentration makes plasmonic structures an ideal platform to support efficient and low-power nonlinear processes [START_REF] Kauranen | Nonlinear plasmonics[END_REF]. On the other hand, it is possible to satisfy the PMC through the modal phase-matching technique [START_REF] Zhang | Highly efficient phase-matched second harmonic generation using an asymmetric plasmonic slot waveguide configuration in hybrid polymer-silicon photonics[END_REF]113].

Lastly, the metallic layers that define the plasmonic structure can also serve as electrodes for the generation of a high DC-like electric field with low applied voltages.

In this direction, Cai et al. experimentally reported an electrically controlled second harmonic generation (ECSHG) in a plasmonic nanocavity filled by a nonlinear polymer (NP) [155], where the authors used electric signals to efficiently and linearly control the SHG in a free-space optics configuration.

In this manuscript, we investigate a planar waveguide integrated counterpart of this approach, which brings several advantages: i) The possible realization of an exact PMC, ii) A good compatibility with the silicon photonics technology. Starting from the work in section 3.1, here we apply an extra control voltage between the two metallic slabs to actively control this efficient SHG. Let us notice that very recently, a highspeed plasmonic phase modulator was experimentally reported based on similar electrically driven PSW [START_REF] Melikyan | High-speed plasmonic phase modulators[END_REF].

To investigate the performances of this planar PSW approach of EISHG and clearly identify the contributions coming from CSHG and EISHG, we firstly consider a centro-symmetric NP with only χ (3) , where a quadratic control of the power of SH by the voltage is predicted. Then by taking both χ (2) and χ (3) into account, the generated SH is almost linearly modulated by the control voltage. At last, we show that this mechanism is able to support small size, efficient intensity and phase modulation for a wide range of voltages with prospectively ultrahigh cut-off electrical frequency.

Waveguide structure and nonlinear modeling approach

Figure 3.9 shows the cross-section of the proposed silicon compatible PSW, which consists of a NP with width w and height h infiltrated into a metallic slot. Additionally, an extra control voltage (defined as Vc) is applied between the two silver electrodes. In a first step, the NP is chosen as DDMEBT (n = 1.8 and χ (3) = 2.310 -19 m 2 /V 2 ), which has been demonstrated as an interesting NP to support high third-order nonlinearity [START_REF] Leuthold | Silicon organic hybrid technology-A platform for practical nonlinear optics[END_REF].

In a second step, the ECSHG is analyzed for a NP having both χ (2) and χ (3) following a similar approach as in [155]. We adopt a typical value of χ (2) = 230 pm/V for the NP

[162] and maintain the original χ (3) value. The material dispersions of Si3N4 and Ag are considered in the simulation. For example, the refractive index of Si3N4 is considered to be 1.99 at 1550 nm and 2.1 at 775 nm, while for Ag the Drude model is employed like in [START_REF] Zhang | Highly efficient phase-matched second harmonic generation using an asymmetric plasmonic slot waveguide configuration in hybrid polymer-silicon photonics[END_REF].

Ag Ag Si 3 N 4

Polymer h w x y z

V c

Fig. 3.9 Schematic cross-section of the proposed active plasmonic slot waveguide.

In this PSW, the PMC can be fulfilled between the fundamental (0-th) waveguide mode at fundamental frequency (FF) and the second-order (1-st) waveguide mode at the second harmonic frequency (SHF). The nonlinear coupling coefficient (NCC), which depends on the overlap integral between the interacted two modes in the NP area, is increased by increasing the waveguide asymmetry [START_REF] Zhang | Highly efficient phase-matched second harmonic generation using an asymmetric plasmonic slot waveguide configuration in hybrid polymer-silicon photonics[END_REF]. Here, the PMC is fulfilled for w = 50 nm, h = 370 nm, where the effective refractive indices of the interacted two modes are nFF = 2.4645+0.0080i and nSH = 2.4642+0.0073i, respectively.

Then, we estimate the tolerable pump power in this PSW by taking into account the damage threshold for the NP layer. In Ref.

[155], the experimental average input power in the considered plasmonic slot nanocavity filled with NP was 34 mW, indicating a peak intensity around 2.4×10 8 W/cm 2 . Setting this intensity as an acceptable value, the tolerable incident power for our PSW is around 44.4 mW for a 50 nm × 370 nm waveguide cross-section size. We thus considered a maximum pump power of 40 mW.

The ECSHG process is analyzed by using the nonlinear coupled equations. When ωFF, the FF, is converted into the SHF ωSH = 2ωFF and an extra electric field of / c c x E V we  (Vc is the applied voltage, ex is the unit vector in x direction) is applied, the nonlinear polarizations are:

(

2) * (3) * 0 0 (2) (3) 0 0 = : 2 3 1 3 = : 2 2 2 NL FF SH FF FF c SH FF c NL SH FF FF SH c FF FF c P E E E E E E E P E E E E E E E                (3.4)
Here,

A i A A A i z z A A i A A A i z z                           (3.
                                    2) (2) 22 0 : ( , ) ( , ) : ( , ) ( , )
E E E             (3.7)
The first term does not include Ec, which is the CSHG part. The last two terms represent the EFISHG contribution. If the NP is centro-symmetric, only the third term then exists.

Effectively controlled SHG by voltage

For the centro-symmetric polymer case, only the EFISHG effect is induced, and the NCC have been calculated to be: * shows the electrically controlled performance of this ECSHG mechanism. The length of the waveguide is set to 16 μm. As expected, the output power of SH is modulated by the control voltage in a quadratic form. For Vc = 20 V, the generated SH is 80 μW. This result shows a promising approach to excite equivalent second-order nonlinearity in materials without χ (2) . when we take both χ (2) and χ (3) into account.

Apart from the χ (3) , some NP also have χ (2) properties [155]. In order to give a theoretical insight of this, next we analyze the ECSHG in the PSW with both χ (3) and χ (2) . The value of χ (2) is chosen to a typical value of χ (2) = 230 pm/V in a NP. In this case, the obtained NCC are: * As shown in Fig. 3.12(a), the output power of SH is not any more quadratically dependent on the control voltage. As the third term is negligible when compared to the second one in Eq. (3.7), the output power of SH is now almost linearly modulated by the control voltage. Another interesting part of this result is that the output phase of the SH can also be linearly modulated by the voltage in the range of 0 < Vc < 17 V, as shown in Fig. 3.12(b). Note that the initial phase of FF is set to π/4 in order to locate the output phase of SH at -π for Vc = 0. This means that both the intensity and phase of the generated SH can be almost linearly modulated by the voltage. This novel property makes this mechanism promising for electric-controlled nonlinear applications. For example, if we apply an AC voltage (still with a negligible frequency if compared to the optical one) instead of a DC one to control the nonlinear SHG process, we may provide a new mechanism for an electro-optical modulator to realize the generation of some optical signals with advanced modulation formats, i.e. both the intensity and phase of the carrier wave carry the information. In addition, it is worth noting that this is a direct intensity modulation instead of translating the phase modulation into an intensity one by using a conventional Mach-Zehnder scheme, which further simplifies the design and complexity of the modulator.

Control Voltage V c (V)

66

The generated SH at Vc = 0 is the CSHG contribution. Setting this power as a reference level, we plot in Fig. 3.13 the normalized change of the SH power as a function of the control voltage. One can see that a nonlinear modulation magnitude of 292% at a bias voltage of 20 V is predicted, while in [155] the corresponding value was about 140%. The resulted efficient SHG and high modulation magnitude are due to the large field enhancement and DC-like field in the slot, large NCC owing to the waveguide vertical asymmetry and the high nonlinear susceptibility of the NP. We also note that further improvement of the SHG performance is possible in terms of the used NP. Here, we have used a very commonly used NP as the active material and taken typical values for the nonlinear susceptibilities. In terms of χ (2) and χ (3) properties, other better options can be envisaged. For example, the doped, crosslinked organic polymer mentioned in [START_REF] Leuthold | Silicon organic hybrid technology-A platform for practical nonlinear optics[END_REF] has a χ (2) = 619 pm/V, and the Poly(3-Hexy1Thiophene) used in [175] has a χ (3) = 2.6×10 -19 m 2 /V 2 .

Discussion and conclusion

In the envisaged modulation scheme, the proposed configuration has to be investigated in terms of electrical bandwidth, driving voltage, and optical extinction ratio. As the nonlinear processes are instantaneous and none of the speed limiting factors in silicon modulators plays a role in this PSW [START_REF] Melikyan | High-speed plasmonic phase modulators[END_REF], ultrahigh speed is expected. The capacitance of the device can be estimated by C = ε0εrhL/w = 2.76 fF, where εr = 3.24 is the relative permittivity of the slot NP. w, h and L = 13 μm are the width, thickness and length of the waveguide, respectively. If we consider a classical load resistance of R = 50 Ω, the maximum cut-off modulation frequency is f0 = 1/2πRC = 1.15 THz. Considering other capacitances coming from substrate and cladding, the realistic bandwidth would be smaller. In spite of this, a large bandwidth is still prospective due to the compact footprint and the ultrafast response of the plasmonic structure. Inversely, this mechanism can be applied to detect ultrafast electric signals. For the driving voltage, it is shown in Fig. 3.12 that the proposed approach supports linear intensity modulation property over a voltage range from 0 V to 20 V, and it is possible to realize a nonlinear phase shift of 2π by altering the voltage from 0 V to 17 V. With respect to the extinction ratio, the average intensity and phase modulation magnitudes are predicted to be 8 μW/V and 0.12 π/V in Fig. 3.12, respectively.

In conclusion, we propose a theoretical insight to the electrically controlled SHG in a silicon compatible hybrid plasmonic slot waveguide with nonlinear polymers infiltrating the slot and a control voltage applied for RF modulation. By considering the NP to be centro-symmetric or noncentro-symmetric, we carry out the dependence of the SHG process on the applied voltage. As a result, efficient SHG and electricallyinduced modulation are proposed. Quantitatively, for the noncentro-symmetric NP case, the average intensity and phase modulation magnitudes are predicted to be 8 μW/V and 0.12 π/V, respectively, within a wide range of voltages. The proposed configuration allows inducing low-power equivalent second-order nonlinear processes in centrosymmetric materials, probing ultrafast electrical signals, and realizing electro-optical modulators with dramatically ultrahigh speed.

Broadband high-speed optical detection based on the optical rectification effect

Introduction

Optical rectification (OR) is a nonlinear process in which a second-order optical nonlinearity converts an optical field into a direct current (DC) or a radio frequency (RF) voltage. It can be seen as the reverse process of the electro-optic (EO) effect [START_REF] Boyd | Nonlinear Optics[END_REF]176] especially for nonlinear applications [START_REF] Kauranen | Nonlinear plasmonics[END_REF]. For example, Ward et al. measured the OR in a sub-wavelength plasmonic nanogap but in a free-space configuration [188]. With respect to using integrated silicon plasmonic waveguides, efficient second order nonlinear processes such as EO modulations [START_REF] Melikyan | High-speed plasmonic phase modulators[END_REF][START_REF] Sun | Design and analysis of a phase modulator based on a metal-polymer-silicon hybrid plasmonic waveguide[END_REF] and frequency conversions [START_REF] Zhang | Highly efficient phase-matched second harmonic generation using an asymmetric plasmonic slot waveguide configuration in hybrid polymer-silicon photonics[END_REF]113,114,156] within short length have been also widely proposed theoretically or demonstrated experimentally due to the large field enhancement and overlap between the interacting frequencies.

In this context, a PSW counterpart of the ORPD proposed in Ref.

[187] is investigated in the present manuscript, bringing several advantages: i) the reduction of the device size and thus the ease of group velocity matching between optical frequency (OF) and RF signals due to the short length;

ii) the simplification of the electrode design because of the fact that in a PSW the metallic slabs which constitute the waveguide also serve as the electrodes directly;

iii) the improvement of the OR efficiency within short lengths (< 23 μm) owing to an increased confinement and overlap between OF and RF modes.

In addition, instead of using an equivalent circuit model where the assumption of uniform OF and RF mode profiles in the slot area was employed, a more general model directly derived from the Maxwell's equations is presented here to calculate the OR voltage. Meanwhile, our model is still able to give similar results as those in Ref. [187] under the same assumption. Specifically, by making use of the optimized PSW structure with a compact footprint of 50nm×90nm×22.4μm, an OR voltage responsivity above 1

V/W and an OR current responsivity about 0.42 mA/W are predicted at the RF of 400

GHz. The corresponding normalized OR efficiency is up to 2.25×10 -4 W -1 . Then the responsivity of the ORPD can be further improved by introducing the electrically induced OR contribution. Moreover, the proposed ORPD supports such an efficient performance over a wide range of light wavelength from 1.25 μm to 1.7 μm, containing all the classical communication bands. The electrical bandwidth of the ORPD is estimated to more than 800 GHz.

Waveguide structure and nonlinear modeling approach

The proposed PSW is shown in Fig. 3.14. It consists of two metallic slabs on a silicon oxide substrate and separated by a narrow gap filled with an EO polymer. The two metallic slabs also serve as the electrodes in the OR process. Through the OR process, an electrical signal is generated between the two electrodes when an intensity modulated optical signal is injected into the PSW. The length of the PSW is l. The bottom inset shows the cross-section of the PSW, in which the width and height of the slot are defined as w and h, respectively. The EO polymer is considered to be the doped, crosslinked organic polymer with a refractive index of np = 1.643 and an EO coefficient of (2) 111  = 619.4 pm/V (r33 = 170 pm/V) at the wavelength of 1550nm [132]. The metal is considered to be silver in forthcoming calculations. In the meantime, results obtained for gold are also given for comparison. It will be shown hereafter that the two metals, which have been chosen due to their better plasmonic properties in the optical domain and higher conductivity in the electrical domain with respect to other ones, present similar voltage responsivities while silver supports larger OR currents and correspondingly larger OR efficiencies. The bottom inset also shows the equivalent circuit at RF for the calculation of the OR current. With respect to the OR mechanism, the PSW is equivalent to a voltage (OR-induced) source VOR and a C capacitance. Z is the external load resistance. The resistance between the electrodes and the slot has been neglected here due to the high conductivity of the considered metal.

To begin, let us set the electric field at the optical frequency (OF) to be with ω being the angular frequency of the OF. Then, by assuming that the OF signal is an intensity modulated signal with a modulation frequency of ωRF, the nonlinear polarization resulting from the OR effect for the RF signal is:

(2) * 0 2: cos( )

NL RF RF t      P E E (3.8)
Now the RF fields {ERF, HRF} fulfill the Maxwell equations with nonlinear polarization:

0 RF RF NL RF RF RF r t tt               H E EP H (3.9)
where εr is the relative permittivity at RF. We can then obtain: When the size of the waveguide is much smaller than the wavelength of the RF signal, i.e. (w, h, l) << λRF, which is satisfied in the proposed PSW, the left part of Eq.

(3.10) is approximately equal to 0. This well-satisfied approximation then leads to:

(2) * 0 2: cos( ) (3.11) This electric field distribution induced from the nonlinear polarization can be expanded over the guided modes, radiation modes, and the evanescent modes at the RF of ωRF [116]. The most interesting part is the conversion efficiency to the fundamental guided mode due to the largest overlap integral between this mode and the nonlinear RF fields, i.e. ERF. Defining the fundamental guided mode distribution at ωRF as {E0, H0}

NL RF RF RF rr t           EE P E
corresponding to 1 V voltage between two electrodes, the generated RF voltage value from the nonlinear OR effect is:

        (2)
                  E H E E H E H E H (3.12)
For simplicity, we will just consider the x components of the field and nonlinear susceptibility, which is a reasonable assumption in PSW [START_REF] Zhang | Highly efficient phase-matched second harmonic generation using an asymmetric plasmonic slot waveguide configuration in hybrid polymer-silicon photonics[END_REF]. Further, if we consider the RF field to be uniform in the cross-section of the gap, i.e. Interestingly, if the optical mode is considered to be uniform in the slot as well, then the generated voltage turns into the same value as the Eq. ( 12) of Ref.

[187], where a lumped-element equivalent circuit model was employed. In addition, it should be noted that even though the length of the PSW is negligible when compared with the RF wavelength λRF, it is usually larger than the optical wavelength λω. As a result, it is needed to take the loss of the optical mode into account. The intensity of the optical mode is attenuated along the propagation in the form of 2 , ( ) exp( )

x I z E z    ,
where α is the attenuation constant of the optical mode. Here, we apply the average value of the intensity along the propagation to calculate the OR-obtained voltage:

(

2) 2 0 , exp( ) 2 [1 exp( )]cos( ) || l xxx RF OR x gap r z dz lt V V E dxdy l hl              (3.14)
The generated current is then:

1 (2) 2 0 , 0 2 [1 exp( )]cos( ) 1 || xxx RF RF OR OR x gap RF RF r lt I Z V E dxdy i C hlZ i w                        (3.15)
where C = ε0εrhl/w is the capacitance and Z is the external load resistance. As a result, the voltage responsivity VR, current responsivity IR and the normalized efficiency of the OR process η in the PSW are: which makes sense since the OR is considered to be the reversed effect of the EO modulation. nm and h = 90 nm, respectively, have been chosen for the purpose of large OR efficiency after an optimization procedure, as shown soon afterwards. As can be seen, both the OF and RF fields are tightly confined in the slot area. Specifically, the electric field at the OF is up to the magnitude of ~10 8 V/m for 1 W, and electric field at the RF arrives to magnitude of ~10 7 V/m for 1 V of drop voltage between the two electrodes.

      2 (2) , 2 (2) , 0 2 2 2 0 2 (2) 0 2 || | | 2 [1 exp( )] 2 || | | 2 [1 exp( )] 2

Effective optical rectification in plasmonic slot waveguides

This matter of fact is likely to facilitate the high efficiency of nonlinear processes involving the filling of the slot. In Fig. 3.16(a) the '5dB length' is defined as the length where the transmittance of the optical mode is -5 dB due to the absorption loss of the metal. The OR results in Fig.

3.16(b)-(d)

are obtained in a PSW with this 5dB length. It is shown that the PSW has larger losses for smaller widths, while the OR shows better performance. This trend can be attributed to the better field confinement at both frequencies for decreasing widths.

With respect to the dependence on the waveguide height, the OR voltage decreases and the current increases when the height increases. This phenomenon can be explained as follows. As shown in Eq. (3.14), the h parameter is in the denominator. Even though the integral term increases for larger heights, the correlation coefficient is smaller than one. In the same time, the impedance of the capacitor goes down as the height goes up, there exists an optimal length as well. As a conclusion, there is an optimized structural dimension for the OR efficiency in the PSW to a certain width. In Fig. 3.18, the black lines mark out the optimized h and l values. In Fig. 3.18(a) the length has been chosen to be the previously introduced 5dB length, while in Fig. 3.18(b) the height has been chosen to be the optimized value for each w obtained in Fig. 3.18(a).

w (nm) h (nm) l (μm) VR (V/W) IR (mA/W) η (10 -4 /W) Table 3.1 lists the optimized sizes and results for a couple of specific widths. As a whole, the OR shows better performance for smaller widths. However, it should be noted that the fabrication of the PSW is difficult and the coupling efficiency between silicon waveguides and PSW becomes low when the width is too narrow. Another issue needs to be accounted is that the poling efficiency of the EO polymer may deteriorate in the ultra-narrow slot. Therefore, we choose a smallest width of 50 nm, as the efficient coupling between the silicon waveguide and PSW with such a gap was demonstrated in a previous paper in an orthogonal junction configuration [189]. The last row in Table 3.1 represents the result for the case of gold. Compared with the silver case, the optimized height is similar and so does the generated voltage, while the optimized length is much shorter due to the larger metallic loss. As a result, the OR current and the efficiency are smaller than that by using silver, which is the reason why we mainly choose silver in this analysis. At last, we note that there still exists rooms for further improvement of the OR efficiency in terms of the used EO polymer. For example, in

Ref.

[187] an EO polymer with r33 = 300 pm/V was considered. In conclusion, the cut-off frequency is estimated to be around 800 GHz in the present configuration. This high speed is believed to be larger than that in the conventional semiconductor photodetectors by one order of magnitude. This capability of generating ultrahigh speed electrical signals may also derive another potential application of the proposed structure, which is the generation of THz radiation.

In the same time, the current responsivity of proposed ORPD is small when compared with the one in the semiconductor detectors, which have typical responsivities of 0.5 to 1A/W. Apart from the way to employ EO polymer with larger nonlinear coefficients, we propose here another method to increase the responsivity of the ORPD by analyzing the electrical induced OR when introducing a DC electrical bias between the two metallic electrodes. For this case, an effective second order nonlinear coefficient of (2) (3) 111 1111

3 c E   is considered when only the x components are taken into account [190]. Here Ec = Vc/w with Vc being the external applied DC voltage.

When a typical value of (3) 1111

 =2.310 -19 m 2 /V 2 is considered [135]
, the OR property in the PSW is shown in Fig. 3.21. It can be seen that the OR process can be linearly controlled and improved by the external applied voltage. The normalized OR efficiency has been further improved more than two times when Vc = 20 V. Note that the total voltage is Vc + VRPωcos(ωRFt) now. Besides, as a common property of nonlinear effect, a high pump power can lead to high OR efficiency. However, it is worth noting that there is a power limitation which comes from two aspects. Firstly, there is an optical damage threshold for the polymer, beyond this value the polymer will get damaged.

Secondly, the loss from the metal will elevate the temperature of device. The temperature cannot exceed the glass transition temperature of the used polymer, or the chromophores of the polymer will reorientate and degrade the second-order nonlinear property of the used polymer. Therefore, the input optical power should be carefully chosen in practical experiments.

Conclusion

Based on an originally proposed and general model, in a PSW with a compact footprint of 50nm90nm22.4μm, an effective OR process with a normalized efficiency of 2.251710 -4 W -1 is predicted for an intensity-modulated optical signal with speed of 400 GHz. The corresponding voltage and current responsibilities are 1.081 V/W and 0.4166 mA/W, respectively. Then, an original approach is additionally proposed for increasing the responsivity of the ORPD by bringing in the electrical induced OR. Apart from the classical semiconductor photodetectors that are based on the carrier-induced photoconductive or photovoltaic effects, the proposed ORPD scheme takes advantage of the transient nonlinear optical effect. This makes ORPD supports ultra-high speed (> 800 GHz) around one order of magnitude larger than the one in semiconductor PD.

Other advantages include compact footprint and no need for external bias. As to our best knowledge, this work is the first investigation of the nonlinear optical rectification effect in an integrated silicon plasmonic waveguide based on standard silicon-oninsulator technologies. Based on this OR scheme, low-power high-speed broadband optical detection and demodulation can be envisaged in silicon chips.

Chapter summary

In this Chapter, we have investigated three kinds of nonlinear processes in the plasmonic slot waveguides (PSW) with compact dimensions.

(1) We theoretically investigated the possible increase of the SHG efficiency in PSW.

The needed phase matching condition is satisfied between the fundamental waveguide mode at the fundamental frequency (FF) and second-order waveguide mode at the second harmonic frequency (SHF) by an appropriate design of the waveguide opto-geometrical parameters. The SHG signal generated in our starting waveguide is three orders of magnitude higher than those previously reported. Then, the SHG performance was further improved by increasing the asymmetry of the structure where nonlinear coupling coefficients as large as 292 psm -1 W -1/2 are predicted. The device length is shorter than 20 µm and the normalized SHG conversion efficiency comes up to more than 1 × 10 5 W -1 cm -2 .

(2) We theoretically investigated the possible realization of an active electro-optical control of the nonlinear SHG mechanism in a PSW. Both the conventional SHG and the electrically induced SHG were taken into account with a moderate pump power of 40 mW at the FF of 1550 nm. The generated SHF power can be modulated by the applied voltage in a quadratic and almost linear form for centrosymmetric and noncentro-symmetric nonlinear polymers integrated in the slot, respectively. Converted power up to 140 µW within a short distance of 16 μm is predicted for a voltage of 10 V. This mechanism may open a new route to realize high-speed advanced modulations or inversely to detect ultrafast electrical signals.

(3) We theoretically investigated the nonlinear optical rectification (OR) process in a

PSW. An electrical signal is generated between the two metal slabs when an intensity-modulated optical signal is injected into the PSW. Optimization of the geometrical parameters is performed to obtain the highest OR efficiency.

Specifically, a voltage responsivity over 1 V/W and a normalized OR efficiency of 2.25×10 -4 W -1 are predicted in a compact PSW with active size of 50nm×90nm×22.4μm for a modulation speed of 400 GHz. The OR efficiency is further increased when introducing the electrically induced OR contribution. The OR characterizes a relatively flat response for a wide range of optical wavelengths from 1.25 μm to 1.7 μm and supports a RF cut-off frequency up to 800 GHz. This efficient OR process paves a new way for realization of high-speed broadband optical detection and demodulation in silicon chips.

All these analysis and results illustrate the key role of PSW in realizing efficient nonlinear effects within short distance and with low power.

Chapter 4 Enhancement of nonlinear effects using hybrid plasmonic waveguides

As mentioned earlier, hybrid plasmonic waveguides (HPW) support a better compromise between confinement and loss when compared with plasmonic slot Recently, the mid-infrared (MIR) silicon photonics has also attracted considerable attention due to the relatively low intrinsic loss of silicon in this waveband range from typically two to eight microns and to the existence of various application areas [2,[191][192][193][194][195], including sensing, medical diagnostics, thermal imaging and free-space communications [196]. It is thus interesting and significant to bridge these two wavelength bands through nonlinear wavelength conversion. Principles and devices based on nonlinear frequency conversion to bridge the MIR-to-NIR gap have been intensively studied [197][198][199]. Another motivation for this lies in the challenges related to the detection of MIR waves. The size of detectors is usually bulky or a cryogenic environment is often needed, which limits the development of the integration level and hinders the low-power consumption of MIR systems. A MIR-to-NIR conversion then allows circumventing these drawbacks by detecting the converted waves using mature telecom-band photodetectors [199,200].

Up to date, MIR to NIR proposed converters in silicon photonics have mostly exploited nonlinear four-wave mixing processes in silicon itself [199,201]. For example, spectral translation of a signal at 2440 nm to the telecom band at 1620 nm was realized based on four-wave mixing in silicon wires [199]. However, the conversion efficiency may be increased by using nonlinear materials with larger nonlinear susceptibilities or lower-order nonlinear processes with larger efficiencies [START_REF] Boyd | Nonlinear Optics[END_REF]. A promising way is to integrate organic materials with large χ (2) into silicon waveguides, which is known as silicon-organic hybrid (SOH) technology [START_REF] Leuthold | Silicon organic hybrid technology-A platform for practical nonlinear optics[END_REF]. For for further improving the performances of nonlinear processes and scaling down their sizes [START_REF] Kauranen | Nonlinear plasmonics[END_REF].

Among the planar plasmonic structures that have been proposed, a well-defined configuration called the hybrid plasmonic waveguide (HPW) was proved to enable subwavelength confinement and long propagation distance in the same time [START_REF] Oulton | A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation[END_REF][START_REF] Dai | A silicon-based hybrid plasmonic waveguide with a metal cap for a nano-scale light confinement[END_REF].

Based on HPW, passive components and nonlinear devices have been studied. For However, the nonlinear HPW they proposed are not compatible with silicon photonics.

Waveguide structure and mode analysis

In this manuscript, we propose a silicon-organic hybrid plasmonic waveguide (SOHPW) easily amenable to fabrication with a nonlinear polymer (NP) in an horizontal slot scheme, as shown in Fig. 4.1. The polymer considered here is the doped, crosslinked organic polymer with a refractive index of n = 1.643 and a nonlinear susceptibility of (2) 111  = 619 pm/V at the wavelength of 1550 nm [START_REF] Leuthold | Silicon organic hybrid technology-A platform for practical nonlinear optics[END_REF]. Material dispersion for modeling the refractive index of SiO2, silicon and Ag are taken from Refs. [START_REF] Kitamura | Optical constants of silica glass from extreme ultraviolet to far infrared at near room temperature[END_REF], [START_REF] Weber | Handbook of optical materials[END_REF] and [START_REF] Johnson | Optical Constants of the Noble Metals[END_REF],

respectively. Based on this SOHPW, we investigate the second-order frequency conversion between a mid-IR wave (3100 nm) and the main optical wavelength of the near-IR window, i.e.  = 1550 nm. Using the mode phase matching (MPM) technique, efficient phase-matched SHG is demonstrated from the fundamental waveguide mode at the fundamental frequency (FF) to the second-order waveguide mode at the second harmonic frequency (SHF). In the SOHPW, the hybrid modes are the results of the coupling between the SPP mode at the Ag-NP interface and the TM polarized dielectric mode with a high-index silicon as its core. At a given wavelength, the SOHPW supports a plasmonic-like mode with greatly enhanced field in the low-index NP slot for a small-size silicon slab.

Conversely, a large-size silicon slab results in a dielectric-like mode with more electromagnetic energy confined into the high-permittivity silicon core and with lower loss [START_REF] Oulton | A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation[END_REF]. Here, for the FF ( = 3100 nm), the supported fundamental mode resulting from the coupling between the SPP and the TM00 dielectric modes belongs to the plasmonic-like kind. With respect to the SHF with smaller wavelength ( = 1550 nm), however, the fundamental mode is dielectric-like. In the same time, the second-order mode at SHF resulting from the coupling between SPP mode and TM01 dielectric mode is plasmonic-like with a strong confinement in the NP layer. In the next, we thus analyze these three modes and demonstrate that the phase matching condition is possible between the two plasmonic-like modes with correct design of the opto-geometrical parameters.

Using a finite-element-based commercial software (COMSOL), we analyze the modes for different waveguide widths. In order to more clearly highlight the mode profiles, we plot in Fig. 4.3 the Ey distributions along a y-cutline at x = -0.5w. For the two modes under interest here, i.e. the 0-th mode at FF and the 1-st mode at SH, the fields are greatly enhanced in the NP slot and there is a change of sign in the silicon part for the 1-st mode at SHF. With respect to the 0-th mode at SHF, the field is mainly in the silicon part and the related loss level of this mode is lower.

Efficient SHG in the proposed hybrid plasmonic waveguide

Then, we investigate the SHG process in this waveguide by solving the nonlinear coupling-wave equations proposed in Chapter 2.

1 ,0 , 0 0* ,0 ,0 , 0 0 ,, 0 , ,0 ,0 0 exp( ) 24 exp(
) 24

FF FF m FF FF FF SH m m m SH m SH m m SH m SH FF FF m A α ω Ai κ A A i β z z A α ω Ai κ A A i β z z                (4.1)
where A are the slowly varying complex amplitudes, Δβm0 = βSH, m -2βFF, 0 (m = 0, 1) are the phase mismatch, β = Re(neff)2π/λ and α = Im(neff)4π/λ represent the phase propagation constant and attenuation coefficient, respectively.

:

m FF SH m FF FF m SH E E E dxdy E E E dxdy             (4.2)
with E being the normalized mode profiles. The calculated NCC values are κ10 = 141 psm -1 W -1/2 and κ00 = 66 psm -1 W -1/2 , respectively. It is interesting to underline that the NCC between the interacting phase-matched modes with different orders is larger than that between the two fundamental modes. It is known indeed that the 0-th mode of a dielectric waveguide is even with respect to the transverse direction and that the 1-st mode is odd, which usually leads to small NCC between these two kinds of modes. In our case, the 1-st mode at the SHF keeps the odd feature as its tail into the silicon layer undergoes a sign change of the field distribution, but the sign change only occurs in the high-index silicon part. In the nonlinear integration area (NP slot), however, the two signs are the same and both considered phase-matched modes are locally enhanced due to their plasmonic-like natures. This results in a large overlap integration in the nonlinear part between the two modes with different orders. We also emphasize the flexibility of the proposed SOHPW geometry for nonlinear conversion with respect to the phase matching condition. The related phase matching tolerance can be written as |Δβ| < Δβc = 2π/L with the considered device length L [START_REF] Boyd | Nonlinear Optics[END_REF].

With L = 120 μm, Δβc is here of 5.236 × 10 4 rad/m. Fig. 4.5(a) and (b) show the phase mismatch between the 0-th mode at FF and 1-st mode at SH as a function of the waveguide width w and the FF wavelength, respectively. It is shown that the condition |Δβ| < Δβc is satisfied for a wide range of width (215 nm -378 nm) and a wide range of wavelength (3050 nm -3150 nm). This result is due to the strong similarity of the dispersion properties between the two interacting modes and to the short needed waveguide length.

Conclusion

In conclusion, we propose a nonlinear SOHPW and investigate the efficient SHG process induced in this waveguide. Due to a specific hybridization approach of the involved modes, the nonlinear coupling coefficient between the two involved plasmonic-like modes with different orders is larger than that between the two fundamental modes. The capability of supporting efficient SHG within short distances, big fabrication-error tolerance and broad bandwidth make it very promising for future highly-integrated nanophotonic circuits.

Further enhancement of second harmonic generation with microring resonators

Microring resonators (MRR) have been proposed to be an attractive component in By setting both the fundamental frequency (FF) and the second harmonic frequency (SHF) at two different resonant wavelengths, a further enhancement factor of SHG efficiency by almost two orders of magnitude is predicted.

Microring structure and design

The proposed silicon-organic hybrid plasmonic microring resonator (SOHPMR) is shown in Fig. 4.6. The straight access waveguide with width wa is coupled with the ring through a gap distance of g. The radius and width of the bended waveguide are R and w, respectively. The cross-section of the HPW is characterized by a NP with a thickness of hp = 70 nm, well suited to find a phase matching point in a bended waveguide, as shown hereafter. The thicknesses of the Ag and Si layers are hm = 100 nm and hSi = 480 nm, respectively. The polymer is chosen as the doped, crosslinked organic polymer with a refractive index of n = 1.643 and a nonlinear susceptibility of (2) 111  = 619 pm/V at the wavelength of 1550 nm [START_REF] Leuthold | Silicon organic hybrid technology-A platform for practical nonlinear optics[END_REF]. The pump light at the FF of λ1 = 3.1 μm is coupled into the SOHPMR with reasonable powers (values are given hereafter). The expected SHF at λ2 = λ1/2 = 1.55 μm is generated and outputs from the through port of the SOHPMR. Such a device can be fabricated by silicon clean room technological processes.

Firstly, the polymer can be spin-coated onto the silicon layer and then by defining a mask on the wafer and etching through, the MRR configuration can be defined.

Following this, metallic layers can be deposited on top through a lift-off process.

Using the modal phase-matching method, the needed phase-matching condition (PMC) to enhance the nonlinear SHG process can be realized between the fundamental (0-th) mode at FF and the second-order (1-st) mode at SHF in the straight HPW [113].

Based on the same mechanism, the new phase-matching points in the bend-waveguide configuration have been searched. The modal properties of bended waveguides have been analyzed by 2D axisymmetric simulation in COMSOL. Material dispersion for modeling the refractive index of SiO2, Si and Ag were taken from Refs. [START_REF] Kitamura | Optical constants of silica glass from extreme ultraviolet to far infrared at near room temperature[END_REF], [START_REF] Weber | Handbook of optical materials[END_REF] and [START_REF] Johnson | Optical Constants of the Noble Metals[END_REF], respectively. Fig. 4.7 shows the difference of the effective refractive index between the interacting two modes as a function of the radius R and width w of the bended waveguide. Note that the fundamental mode of the SHF was not considered in this study. The black line marks out the zero value, i.e. the phase matching line. One can see that the PMC can be fulfilled in the bended HPW as well and that for most of radius values, two phase matching points can be found. in Fig. 4.9(a). We can see that the peak value of the field is not located at the center of the waveguide because of the bended-waveguide property.

The width of the access waveguide was then designed as wa = 358.3 nm  360 nm in order to maximize the fraction of FF field coupled into the ring. Then, nFF = 1.90968+0.00271i and nSH = 1.91268+0.00139i for the access waveguide. At last, in order to set a comparison benchmark for the SHG efficiency in the ring resonator configuration with previous results, the efficiency of the SHG process in a straight HPW was complementary estimated (with w = 369.2 nm  370 nm in this case for PMC consideration).

Nonlinear model approach

In this subsection, we derive the enhancement factor for the SHG in a ring resonator when compared to a single waveguide. In a single lossy waveguide propagating along +z direction, the nonlinear coupled equations (NCE) for the SHG are [START_REF] Zhang | Highly efficient phase-matched second harmonic generation using an asymmetric plasmonic slot waveguide configuration in hybrid polymer-silicon photonics[END_REF]: * exp( ) 24 exp( ) 24

FF FF FF FF FF FF SH SH SH FF SH SH FF FF A α ω Ai κ A A i βz z A α ω Ai κ A A i βz z              (4.3)
where ωFF is the angular frequency of the fundamental frequency (FF). AFF,SH are the slowly varied complex mode amplitudes. Δβ = βSH -2βFF is the phase mismatch. βFF,SH and αFF,SH represent the phase propagation constant and attenuation coefficient, respectively. And κFF,SH are the nonlinear coupling coefficients, which are defined by: are the modes propagating along -z direction. Under the approximation where the pump is not depleted by the nonlinear conversion process, one can solve Eq.

    (2) * 0 (2) 0 : ( , ) ( , ) ( , ) : ( , ) ( , ) ( ,
(4.3) analytically to obtain [START_REF] Ruan | Enhancement of optics-to-THz conversion efficiency by metallic slot waveguides[END_REF]:

2 (0) exp( ) 2 ( ) (0) exp( ) (0) exp( ) 4 2 2 FF FF FF SH SH FF SH SH eff FF SH A A z A i L z A z A z          (4.5)
where the effective length is

1 exp[ ( 2 ) ] () 2 FF SH eff FF SH iz Lz i                (4.6)
So the SHG efficiency in a lossy waveguide with length of L is:

2 2 22 1 2 () ( ) exp( ) (0) 16 (0)
SH FF SH eff SH FF FF AL L L L P A        (4.7)
Note that this single waveguide can be a straight or a bended waveguide.

(



A A i L A i L L A i L A i L             (4.9)
where L is the circumference of the microring given by L = 2πR with R being the radius of the ring. Leff(L) is the effective length defined in Eq. (4.6). The incident condition of the SHG is:

(

, 0

in FF FF SH A A A   (4.10)
Based on Eq. (4.8) through (4.10), we can obtain:

(2) ( )

A t c F i L A A iF A A iF i L A A L L F F i L A t A i L L F F i c                    ( ) 2 (4) 2 ( ) 2 ) ][ ] 2 ( ) exp[( ) ][ ] 42 in SH FF in SH FF SH SH eff FF SH SH FF SH LA A i L L F F i L A c         (4.11)
where the FFF,SH are the field enhancement factors, which can be written as:

          (4.14)
On this circumstance, the SHG shows best performance.

In the analytic solution of Eq. (4.5) and Eq. (4.9) for the NCE of Eq. (4.3), we have made an approximation that the pump is not depleted by the nonlinear conversion. Now we confirm the accuracy of this approximation by comparing the analytic solution with the numerical solution obtained by calculating the NCE directly. the NCE is accurate when the length of the waveguide is smaller than 40 μm. For the final structure we choose, the circumference of the ring is L = 2πR = 14.6 μm. Based on the above analysis, it is reasonable to apply the analytic solution in Eq. (4.5) and Eq. (4.9). Lastly, it should be noted that this analytic solution is only applicable under the condition of low pump power and short waveguide length. Otherwise the direct calculation of the NCE is required.

Enhanced second harmonic generation in microring resonators

In the first benchmarking single straight HPW considered configuration, the nonlinear coupled coefficient was calculated to be κSH = 134 psm -1 W -1/2 . Then by using Eq. (4.7), the SHG efficiency was estimated as η1 = 0.14% for an input pump power In order to further confirm the veracity of the proposed scheme, we have calculated the transmission spectra of the MRR around the SHF and FF in a three dimensional (3D) structure. The software FDTD Lumerical was used to carry out the 3D full-wave simulation. In the simulation, some efforts have been made to make the model convergent. We have modified the perfect matched layer (PML) settings by increasing the "pml kappa" from the original value of 2 to 10, decreasing the "pml sigma" from the original value of 0.25 to 0.15, and changing the "type of pml" from the original type of standard to stabilized. In the time step setting, we have modified the "dt stability factor" from the original value of 0.99 to 0.8. For the mesh setting, we defined an override mesh region near the metal-polymer interface with mesh size of dx = dy = dz = 5 nm. The mesh accuracy for the remaining regions has been chosen to be 3.

The calculated results are shown in Fig. 4.14. Two resonant wavelengths of 1548.4

nm and 3099 nm are presented. The slight difference over the previous mode analysis is probably due to the applied different mesh sizes and different calculation methods in the finite-element mode analysis and 3D FDTD simulation. The MRR shows small extinction ratios as it works deviating from the critical points especially at the SHF. Fig.

Conclusion

To conclude this part, a HPW geometry based on a microring configuration has been proposed to further enhance the second harmonic generation (SHG) process on the that, we explore a potential application, the phase regeneration of degraded phase shift keying (PSK) signals, which is of great significance in optical signal processing.

Waveguide structure and efficient optical parametric amplification

The SHPW studied in this paper has a similar structure as in [212], with a nanometer size metallic layer sandwiched between two NP layers surrounded by two silicon layers, as shown in Fig. 4.16(a). In the waveguide, the refractive index of silicon at the signal wavelength (SW) of λs = 1550 nm and pump wavelength (PW) of λp = 775 nm are 3.48 and 3.71, respectively. Metal is defined as silver with a height hm. The NP has a height hp and is chosen to be the commercially available M1 organic material consisting of chromophores dispersed in a polymer matrix, with refractive index 1.6 and high nonlinear susceptibility (2) 111 The related configuration can also be fabricated with clean room technologies.


Firstly, the polymer can be spin-coated onto a SOI wafer and then by defining a mask on the wafer and etching through, the waveguide configuration can be defined.

Following this, metallic layers can be deposited on top through a lift-off process. Then, the top polymer layer can be realized with another step of spin-coating. Finally, the top silicon layer which made of amorphous silicon can be deposited by plasma-enhanced chemical vapor deposition technique [216]. E being the normalized mode profiles. In the calculation, the two photon absorption (TPA) process in silicon is neglected because most of the field is concentrated in the NP layers and TPA is a third order nonlinear process, being thus weaker than the second order nonlinear exploited process. The losses coming from the amorphous silicon materials, which can be reduced to less than 5 dB/cm in wire waveguides with current technology [216], are also neglected as the needed lengths are very short. The nonlinear coupling coefficient is calculated to be 60 psm -1 W -1/2 . The numerical results of OPA along propagation are displayed in Fig. 4.17 when the incident power Pp(0) and phase φp(0) of the PW are 1 W and 1.5π, and the incident power Ps(0) and phase φs(0) at the SW are 0.01 W and 0, respectively. It is shown that the SW is amplified to a peak power of 0.83 W at 168 µm of propagation length. After the peak point, the pumping gain is overcome by the loss and the SW begins to attenuate. In the next, we analyze the OPA by tuning φs(0), other conditions being unchanged as in Fig. 4.17. Fig. 4.18 shows the phase evolutions of SW and PW along the propagation for φs(0) = 0.2π (a, b) and 0.8π (c, d), respectively. In these two cases, the amplitude evolutions are very similar to the one shown in Fig. 4.17(a) excepted that the distance to reach the SHG power maximum is 181 µm owing to smaller gain. It is shown that the phase at the SW is going to zero and π and stabilizes after 120 µm of propagation when φs(0) = 0.2π and 0.8π, respectively. The length of the waveguide being now set to L = 150 µm, we plot in Fig. 4.19 the gain of OPA and output phase φs(L) at the SW versus φs (0). We can see that the gain changes with the incident phase. In some cases near the notch points, the gain even comes into negative values. More interestingly, if the incident phase at the SW is around π but not differing more than half π, the phase at the SW stabilizes to π. While the input phase is around zero but not differing more than half π, it then stabilizes to zero, as can be seen in Fig. 4. 19(b). This interesting effect is known as 'phase-squeezing' and was measured in the degenerate four-wave mixing phase-sensitive amplifiers based on nonlinear optical fibers [217].

A A i A A i z z A A i A A i z z                   (4.

Conclusion

In conclusion, we propose a nonlinear SHPW and investigate the OPA process induced in this waveguide. Based on the special phase variation property in the OPA, we come up with a potential application in all-optical signal processing, which is phase recovery of PSK signals. The capability of recovery within short distances and tunability makes it promising and competitive in future highly-integrated nanophotonics.

Chapter summary

In this Chapter, we have investigated effective nonlinear processes in the hybrid plasmonic waveguides (HPW).

( All these proposals and results illustrate that HPWs are capable of efficiently favoring nonlinear effects within short distances and low optical powers. Together with plasmonic slot waveguides investigated in Chapter 3, silicon plasmonic waveguides related nonlinear enhancement will play a key role in future photonic functionalities.

However, this field is still in its young age. Most of the related works have been reported

in the way of theoretical calculations, while very few experimental demonstrations have been proposed up to date. Thus, precise experimental fabrication and measurement of nonlinear silicon plasmonic waveguides are strongly expected in the next step of development. In the next chapter, we will present our progress in this regard.

Chapter 5

Progress towards experimental nonlinear plasmonic waveguides

In this chapter, we give all the experimental results that we have obtained by now on the way to demonstrate nonlinear plasmonic waveguides. Among all the integrated plasmonic waveguides, plasmonic slot waveguides (PSWs) have the simplest configuration. We have thus decided to choose them for experimental demonstrations. Yagi-Uda style antennas at λ0 = 1550 nm [START_REF] Kriesch | Functional Plasmonic Nanocircuits with Low Insertion and Propagation Losses[END_REF]. For nonlinear applications, the aforementioned plasmonic modulators realized by filling the slot of PSW with EO polymers are very compact and able to operate ultra-high speed RF signals [START_REF] Melikyan | Plasmonic-organic hybrid (POH) modulators for OOK and BPSK signaling at 40 Gbit/s[END_REF][START_REF] Haffner | High-speed plasmonic Mach-Zehnder modulator in a waveguide[END_REF][START_REF] Melikyan | High-speed plasmonic phase modulators[END_REF][START_REF] Heni | Plasmonic Mach-Zehnder Modulator with >70 GHz Electrical Bandwidth Demonstrating 90 Gbit/s 4-ASK[END_REF][START_REF] Haffner | All-plasmonic Mach-Zehnder modulator enabling optical high-speed communication at the microscale[END_REF].

In the next sections of this chapter, we will detail the whole adopted process for the PSW demonstration, including the design of their dimensions and in/out tapers for efficient light coupling, the detailed fabrication procedures and optical measurement.

The measured results indicate that the fabricated PSWs have good performance in terms of coupling and optical losses. As shown hereafter, a 0.5 dB/µm propagation loss level is experimentally reported at  = 1.55 µm for a PSW with cross-sectional dimension of 66 nm × 150 nm, for which the theoretical propagation loss is 0.434 dB/μm, meaning an agreement of both levels with a moderate error range of 10.6%.

Design and optimization of waveguides and coupling tapers

Our final goal is to experimentally demonstrate the high-speed optical detector The next critical issue is how to efficiently couple light into such a PSW. Many structures have been proposed to do this, such as tapered silicon strip waveguide [START_REF] Melikyan | Plasmonic-organic hybrid (POH) modulators for OOK and BPSK signaling at 40 Gbit/s[END_REF][START_REF] Melikyan | High-speed plasmonic phase modulators[END_REF][START_REF] Han | Experimental realization of subwavelength plasmonic slot waveguides on a silicon platform[END_REF]220], vertical directional coupling between a silicon strip waveguide and a PSW [221,222], nanoantenna couplers [START_REF] Kriesch | Functional Plasmonic Nanocircuits with Low Insertion and Propagation Losses[END_REF][START_REF] Andryieuski | Direct Characterization of Plasmonic Slot Waveguides and Nanocouplers[END_REF] and metallic gratings [START_REF] Fu | All-Optical Logic Gates Based on Nanoscale Plasmonic Slot Waveguides[END_REF]. Among these structures, tapered silicon strip waveguides have properties of high efficiency and broadband coupling. For example, in Ref. In view of the possible fabrication errors, we have designed PSWs with different widths.

PSWs are fabricated in a four-step process with standard nanofabrication techniques.

For the first step, gold markers were deposited onto a SOI wafer with a 220 nm silicon top layer and 3 μm buried oxide layer for the next alignment between different steps.

Specifically, the pattern was defined by using electron beam lithography (EBL) with PMMA as the electron beam resist. After development, 10 nm of titanium and 70 nm of gold were successively deposited onto the wafer by an electron beam evaporation (EBE) system. Then a lift-off process was performed in the acetone to obtain the gold markers. Second, as shown in Fig. 5.5(b), silicon strip waveguides with the taper and grating couplers were defined by EBL with ZEP520 as the e-beam resist. The pattern was then transferred to the silicon layer by inductively coupled plasma reactive ion etching (ICPRIE), resulting in the silicon parts. Third, the metallic slot waveguide was fabricated by another lift-off process with ZEP520 as the e-beam resist. 3nm-thick chromium and 150nm-thick gold layers were evaporated onto the wafer by EBE. The wafer after this procedure is shown in Fig. 5.5(c). Fourth, the M1 polymer was spincoated onto the wafer. The schematic of the final device is shown in Fig. 5.5(d). All these pictures were taken before spin-coating the M1 polymer. After the first measurement of PSWs with air cladding, the polymer was then spin-coated onto the wafer. Using the coating parameters of 500 rpm for 5 seconds and 700 rpm for 40 seconds, a polymer layer with thickness of 2.21 μm was spin-coated onto the wafer.

After coating, the film was first oven dried by ramping the temperature from ambient to 50°C at heating rate of 10 °C/min and held for 2 hours at 50°C. Then the film was further dried at 50°C under vacuum (~ 10 Pa) for 12 hours. After this, the polymer was ready for poling. We recalculated the PSW mode with w = 65.6 nm and h = 150 nm by using COMSOL and got an effective mode index of neff = 1.43491+0.01233i. The corresponding loss is α = 0.434 dB/μm, which is near to the measured value of 0.5 dB/μm. The additional scattering loss from the roughness of sidewall is thus estimated to be 0.066 dB/μm. Another source of this difference comes from the 3 nm chromium adhere layer, which was omitted in the mode calculation. This whole loss may seem to be very large when compared with the one in dielectric waveguides, which can be less than 1 dB/cm in silicon waveguides. However, as mentioned in Chapter 2, one of the advantages of plasmonic waveguides lies in realizing desired function within very short lengths (typically no more than several propagation distances). Thus, the whole losses in plasmonic devices can still be low. A good example is the recently proposed all-plasmonic Mach-Zehnder modulator [START_REF] Haffner | All-plasmonic Mach-Zehnder modulator enabling optical high-speed communication at the microscale[END_REF], in which the insertion loss is 8 ± 1 dB due to a short length of 10 μm. This is not too far away from well-engineered high-speed silicon modulators reporting insertion losses of 4-7 dB with smaller bandwidths. 123 Then, we cover the wafer with the M1 polymer. Using the spin-coating parameter of 500 rpm for 5 seconds and 700 rpm for 40 seconds, a polymer film with thickness of 2.21 μm and good homogeneity is coated onto the wafer. The PSW with polymer cladding is tested with the same setup shown in Fig. 5.11. The results and comparison with air cladding can be seen in Table 5.1. All the values are taken at λ = 1550 nm. We can see that the loss level of the PSW with polymer cladding is similar to the one with air cladding, which means that the loss from the polymer absorption is small. Interestingly, contrary to the air cladding, the PSW with polymer cladding has smaller losses for larger widths. For the width of w = 109.3 nm and length of L = 8.8 μm, the loss contribution from the PSW section is only around 8.785 dB. This matter of fact can be explained as follows. In the optimization design of the taper couplers, the width of the PSW was taken as w = 100 nm. As a result, the closer to 100 nm the width is, the lower the loss. In order to investigate the nonlinear optical rectification effect in the nonlinear PSW, we need to pole the polymer with proper poling condition. According to the datasheet of the M1 polymer, the poling condition we have applied is shown in Fig. 5.13. Firstly, the temperature of the sample was heated from ambient to the polymer transition temperature of 140°C with a ramping rate of 10°C/min. Then the temperature was maintained and a voltage was applied between the two metallic slabs with two electrode probes. The voltage was increased gradually until the electrical field in the slot reached 110 V/μm. The voltage and temperature were kept for 5 seconds. After poling, the poling voltage (used at poling temperature) was maintaind for 3 minutes and the temperature of the sample dropt off to the ambient. In the end, the voltage was taken out. 125

Measurement and results

OLS

Cladding

Note that in the whole poling process, a poling chamber purging with nitrogen continuously is needed to extrude air. Since this poling condition is not so easy to achieve, our experiment gets stuck in this step at present. Right now, we are trying to solve this difficult point. The electrical contact between electrodes of the PSW and outside in the poling process and future measurement are realized by impaling the nonconducting polymer layer with two probe tips. Fig. 5.14 shows the microscope pictures.

In the future, after solving the poling problem of polymer, we will begin to test the nonlinear property of the fabricated PSW.

Chapter summary

To conclude this chapter, we have experientally demonstrated PSWs fabricated in our (2) Efficient second harmonic generation (SHG) from 1550 nm to 775 nm in plasmonic slot waveguides (PSW) was theoretically investigated. The needed phase matching condition was satisfied through the modal phase-matching method by an appropriate design of the waveguide opto-geometrical parameters. The SHG signal generated in the starting waveguide is three orders of magnitude higher than those previously reported. Then, the SHG performance was further improved by increasing the refraction index asymmetry of the structure. The obtained device length is shorter than 20 µm and the normalized SHG conversion efficiency comes up to more than 1 × 10 5 W -1 cm -2 .

( nonlinear organic materials has a relatively low loss of about 0.005 dB/µm and an effective nonlinear OPA coupling coefficient up to 60 psm -1 W -1/2 . Phase recovery process was achieved in this waveguide within a length as short as 150 μm.

(7) Plasmonic slot waveguides were experimentally fabricated. Firstly, the optimized dimensions of PSWs and the taper couplers between the classical dielectric silicon revolution may begin. Actually, for this biggest challenge in the field of plasmonics, researchers have managed to figure out some potential methods to deal with the optical losses in metallic systems [START_REF] Khurgin | How to deal with the loss in plasmonics and metamaterials[END_REF]. For example, many new plasmonic materials beyond gold and silver have been proposed [START_REF] West | Searching for better plasmonic materials[END_REF]229,230]. Or in another way, the field of integrated computation devices in the foreseeable future is to effectively utilize the potential of the three available technologies (electronics, photonics, and plasmonics) so that they may compensate the weaknesses of each other, i.e. the losses in plasmonics with photonics, the size of photonics with plasmonics, and the interface to the outside world with electronics [231].

2 . 2 1 Introduction

 221 Nonlinear coefficients for various third order nonlinear materials .................. 36 2.3 Comparison of Kerr nonlinearity in silicon slot waveguide and plasmonic slot waveguide ......................................................................................................... 41 3.1 The optimized height and length for different widths and the corresponding OR results ................................................................................................................ 77 5.1 Measured loss of PSWs for different top cladding materials and sizes .......... 123 Chapter Today, with the rapid development of information technologies, the capacity of global internet is becoming larger and larger. According to the Cisco Visual Networking Index Global Forecast, global IP traffic has increased fivefold over the past five years, and will increase threefold over the next five years. As shown in Fig. 1.1, overall IP traffic

Fig. 1 . 1

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Fig. 1 .

 1 Fig. 1.2 (a) The simulated dispersion of the photonic wire waveguide. The zero-dispersion

Fig. 1 . 3 Fig. 1 . 4

 1314 Fig. 1.3 Development of chip complexity measured as the number of components per chip

Fig. 1 .Fig. 1 . 5

 115 Fig. 1.5(b) is the scanning electron microscope (SEM) image of the fabricated plasmonic modulator. Fig. 1.5 (c) and (d) show the mode profiles of the plasmonic mode and radio frequency (RF) signals, respectively. Both optical and modulating fields are strongly confined to the slot, resulting in a near-perfect overlap between the optical and RF signals. Consequently, the needed voltage for a phase shift of π at a

Fig. 1 .

 1 Fig. 1.6 illustrates the concept of SPPs at a metal-dielectric interface where the z>0 space is filled by a lossless dielectric material having a εd permittivity, and the material filling the z < 0 space is a metal having a εm permittivity. The intensity and surface charge distributions and electromagnetic field components of the SPP propagating in the x direction are also shown in Fig. 1.6. It propagates along the interface, and is evanescently confined in the perpendicular direction with amplitude exponentially decaying with increasing distances into each medium from the interface.

Fig. 1 . 6

 16 Fig. 1.6 Schematic representation of the intensity distribution and electromagnetic field

  (1.2) and (1.3) into Eq. (1.1), we can get the dispersion relation of SPP. By ignoring the loss and contribution of interband transitions, the permittivity of a lossless metal is

Fig. 1 . 7

 17 Fig. 1.7 Dispersion relation of surface plasmon polariton at a metal-dielectric interface

Fig. 1 .

 1 Fig. 1.7 shows the obtained dispersion relation of the SPP mode at a metaldielectric interface without considering the loss of metal. The black dashed line is the light line in the dielectric. We can see that the propagation constant of the SPP wave is larger than the one of the light line. This leads to two consequences. Firstly, special phase-matching techniques based on grating or prism coupling are required for the excitation of SPP from a free space or fiber optical beam. Secondly and importantly, it results into better confinement of light than in dielectrics. To understand this point, let us have a look to the following equations

Fig. 1 . 8

 18 Fig. 1.8 Penetration depth into the (a) dielectric and (b) metal and (c) propagation distance

  To our best knowledge, few work especially experimental demonstrations have been reported to address these two expectations together. The aforementioned POH modulators belong to one of them. As another example, S. Sederberg et al. recently demonstrated for the first time an efficient third harmonic generation in a silicon-based nanoplasmonic waveguide. As show in Fig. 1.9, the proposed waveguide consisting of a silicon core and a gold cap is simple to fabricate, compatible with modern electronics and photonics technologies, and occupies an ultracompact footprint of 0.40 μm 2 . Femtosecond laser pulses at λ = 1550 nm with input powers in the range of 85 μW -1.52 mW are used to generate strong THG. The conversion efficiencies is up to 2.3 × 10 -5 with waveguide length of 5 μm, which is the highest reported to date in any silicon-based device or nanoplasmonic structure [55].

Fig. 1 . 9

 19 Fig. 1.9 Efficient third harmonic generation in a silicon-based nanoplasmonic waveguide.

Figure

  Figure taken from Ref. [55].

Fig. 1 .

 1 Fig. 1.8 for the SPP mode at a metal-dielectric interface: for smaller wavelengths it supports better confinement in the dielectric (Fig. 1.8(a)), while the fraction of the field concentrated in the metal increases (Fig. 1.8(b)). The propagation distance thus decreases to shorter values due to larger ohmic absorption in the metal (Fig. 1.8(c)).

Fig. 2 . 1

 21 Fig. 2.1 Initially proposed plasmonic waveguides: (a) metal nanowire waveguides [61], (b)

Fig. 2 .

 2 Fig. 2.2 (a) Long range plasmonic waveguide made of a gold stripe with width of 4 μm and thickness of 25 nm. (b) mode intensity contours plotted over a 20 × 20 μm 2 cross-

Fig. 2 . 3

 23 Fig. 2.3 Silicon plasmonic waveguides have better compromise between confinement and

Fig. 2 . 4

 24 Fig. 2.4 Typical configuration and calculated field distribution for the major component Ex

Fig. 2 .

 2 Fig. 2.5 (a) Mode effective index neff, (b) propagation length Lp and (c) mode effective area

Fig. 2 . 6

 26 Fig. 2.6 Typical configuration and calculated field distribution for the major component Ey

Fig. 2 .

 2 Fig. 2.6 shows the typical configuration and mode profile of a HPW. It consists of a high refractive index (RI) region, a metal cap and a low RI spacer between them. This kind of waveguide is also compatible with the silicon on insulator technology. The

Fig. 2 .

 2 Fig. 2.7 (a) Mode effective index neff, (b) propagation length Lp and (c) mode effective area

Fig. 2 .

 2 Fig. 2.7 shows the mode effective index neff, propagation length Lp and mode effective area Aeff of the fundamental mode in HPW as a function of spacer width w for different spacer height h. The thickness of the silicon stripe and the optical wavelength are fixed at 340 nm and λ = 1550 nm, respectively. Similar to the mode property of

       and based on Eq.

  photonics platform. The most used nonlinear materials include silicon itself [120-123], silicon dioxide (in fiber nonlinearity), and newly reported silicon nitride and Hydex[124]. Recently, organic polymers have also attracted a growing attention for nonlinear applications in silicon photonics due to their excellent properties of low linear loss, large nonlinear coefficients and removal of carrier effects, especially for electro-optical (EO) modulation and carrier-free ultrafast optical applications[START_REF] Baehr-Jones | Polymer Silicon Hybrid Systems: A Platform for Practical Nonlinear Optics †[END_REF][START_REF] Leuthold | Silicon organic hybrid technology-A platform for practical nonlinear optics[END_REF][START_REF] Leuthold | Silicon-Organic Hybrid Electro-Optical Devices[END_REF][START_REF] Xingyu | Polymer-Based Hybrid-Integrated Photonic Devices for Silicon On-Chip Modulation and Board-Level Optical Interconnects[END_REF][125][126][127]. More recently, the two dimensional graphene material has been also proposed as a new competitive nonlinear material owing to its large nonlinear coefficient in a broad band and electrical tunability[128][129][130][131].

[

  147]. The second and third order optical nonlinearity enhancement is proportional to the group index squared 2 g n [148]. Understanding slow light from another aspect, it can

Fig. 2 .

 2 Fig. 2.8 (a) Effective length and (b) efficiency factor of third order nonlinear process in

Figure 3 .

 3 2(a) shows the effective indices of the two guided modes as functions of the slot height h for a free space FF wavelength of 1550 nm and a slot width w = 50 nm. The red and blue lines represent the obtained effective indices for modes at the FF (FF = 1550 nm) and the SHF (SH = 775 nm), respectively. We can see that the fundamental mode (labeled 0-th mode) at the FF has a crossing with the second-order mode (labeled 1-st mode) at the SHF indicating a point of PMC. For the considered case of a SiO2 substrate, which has indices of 1.444 and 1.454 for FF and SHF[START_REF] Kitamura | Optical constants of silica glass from extreme ultraviolet to far infrared at near room temperature[END_REF], respectively, the PMC occurs for the slot height h = 489 nm. At this point, the effective indices for the FF and the SHF are 2.2068 + 0.0068i and 2.2069 + 0.0063i, respectively. The corresponding Ex distributions of the different modes are plotted in Fig. 3.2(b), (c) and (d). All the mode profiles shown here have been calculated by using the finite-element-based commercial COMSOL software.

Fig. 3 . 2

 32 Fig. 3.2 Design of the waveguide geometry for satisfying the PMC. (a) Effective indices

Figure 3 .

 3 3 is a plot of the powers of the FF and SHF waves as a function of the propagation distance for an input power of FF source of 1 W. All results presented here and hereafter have been obtained by a numerical simulation of NCE with a self-made Matlab code (please see the detail in the Appendix).

Fig. 3 . 3 .

 33 Fig. 3.3. Optical powers of the FF and SHF waves versus the propagation distance for a

Fig. 3 . 4 .

 34 Fig. 3.4. (a) Slot height h to satisfy the PMC needed for the SHG process and associated

  with larger indices at the FF (nSi3N4 = 1.99) and SHF (nSi3N4 = 2.1). The new structure is still compatible with silicon photonics. The PMC is satisfied for w = 50 nm and h = 490 nm from Fig.3.5(a) and the corresponding mode profiles are shown in Fig.3.5(b), (c) and (d), respectively. At the PMC point, the geometrical size of the waveguide is almost the same as in Fig.3.2. But the mode profiles become quite asymmetrical especially for the 1-st mode at the SHF. In this case, the calculated NCC is up to 292 psm -1 W -1/2 .

Fig. 3 . 5 .Fig. 3 . 6 .

 3536 Fig. 3.5. Design of the waveguide with Si3N4 substrate for satisfying the PMC. (a)

Fig. 3 . 7 .

 37 Fig. 3.7. Normalized Ex distribution at x = 0 for the case of SiO2 substrate (a); Si3N4

Fig. 3 .Fig. 3 . 3 .

 333 Fig.3.6 shows the variations of the optical power of the FF and SHF along propagation in this modified waveguide geometry with the same initial condition as in Fig.3.3. One can see that the maximum efficiency is increased up to 20% and that the propagation length (around 12.5µm) to realize this efficiency is shorter as well. The normalized conversion efficiency is calculated to η = 1.3 × 10 5 W -1 cm -2 , which is over one order of magnitude of further improvement if compared with our starting structure.The principle of this improvement can be seen more clearly in Fig.3.7, where plots of the normalized Ex distributions along the cutline of x = 0 are shown. For the case of a SiO2 substrate shown in Fig.3.7(a), the positive and negative parts of the 1-st mode (blue dotted line) at the SHF have comparable amplitudes, hence making the nonlinear overlap integration with the 0-th mode field small. With respect to the case of Si3N4 substrate shown in Fig.3.7(b), however, the negative part of the 1-st mode becomes negligible in the integration domain. This indicates that the counteraction effect is lower and in turn contributes to the enhancement of the 0-th/1-st NCC. In addition, it should be noted that as the distribution of the generated SHF becomes less symmetric in the modified structure, the in-and-out light coupling efficiency will be easier from/into standard strip silicon waveguides.

Fig. 3 . 8 .

 38 Fig. 3.8. Peak position Lp, peak efficiency ηp, normalized conversion efficiency η and

Figure 3 .

 3 8(a) shows the peak position and efficiency as a function of the pump power with all other parameters and conditions being the same as in Fig.3.6. As expected, it is shown that the conversion efficiency is greatly affected by the pump power, while the peak position only moderately varies, and the change of the calculated normalized conversion efficiency is small as well, as it is shown in the left Y-axis of Fig.3.8(b). We can see that the plasmonic waveguide length remains shorter than 20 µm in all cases, while the normalized conversion efficiency can exceed 1 × 10 5 W -1 cm -2 . Moreover, it should be emphasized that within this nonlinear PSW, the SHG efficiency still remains around 0.4% even if the input power of the FF is as low as to 10 dBm, which represents a continuous wave optical power easily available. This efficiency is over the one in a Si3N4 ring resonator where a pump power of 75 mW was used[151]. Moreover, the optical power of the generated SHF is then still around 40 µW, as shown in the right Y-axis of Fig.3.8(b), i.e. is comparable to values reported in previous work that considered a 1 W pumping power[141]

[

  168], silica waveguides[169] and silicon nitride ring resonators[170]. Applications of EFISHG have also been investigated widely, including the precise measurement of χ(3) tensors [171], the detection of ultrafast electrical signals[172], the characterization of dielectric surface properties[173], and the probing of carrier dynamics in semiconductors[174].

E

  are the electric field for 0-th mode at FF and 1-st mode at SHF, respectively, and can be expressed by βFF,SH are the phase propagation constants. The first to third terms on the right side of Eq. (3.4) stand for the CSHG, χ(2) -induced EO modulation and EFISHG effects, respectively. Note that we have neglected the third-order nonlinear processes of self-phase modulation and cross-phase modulation due to their weak impacts in this study. The nonlinear coupled equations for the considered ECSHG are:

5 )

 5 αFF,SH are the attenuation coefficients, Δβ = βSH -2βFF is the phase mismatch, κii are the NCC defined by[START_REF] Ruan | Enhancement of optics-to-THz conversion efficiency by metallic slot waveguides[END_REF][START_REF] Zhang | Highly efficient phase-matched second harmonic generation using an asymmetric plasmonic slot waveguide configuration in hybrid polymer-silicon photonics[END_REF]:

  take the x-component of the modes and the nonlinear susceptibility into account, which is reasonable in this PSW, the generated power of SHF is [155]:

11 21   6 . 22 ×

 1121622 Vc psm -1 W -1/2 . Fig.3.10(a) shows the power variations along the propagation for a FF of 1550 nm with a control voltage of 10 V and a pump power of 40 mW. For the pump of FF, the power decreases monotonously due to the conversion and intrinsic loss. With respect to the SH, the power increases until a peak position where the nonlinear conversion has equivalent contribution with the loss. After this peak position, the power starts to decrease. Here, the power of SH reaches its peak of 20 μW at a propagation length of 16 μm. Fig.3.10(b)

Fig. 3 . 0 Fig. 3 .

 303 Fig. 3.10 (a) Power variations of FF and SH along propagation for Vc=10 V when the

11 21   104 + 6 . 2 ×

 112162 Vc psm -1 W -1/2 , κ12 = 20.46 × Vc psm -1 W - 1/2 and κ22 =25.78 × Vc psm -1 W -1/2 . Now the nonlinear process includes all the three terms in Eq. (3.7). Similarly, we plot the obtained SHG result for Vc= 10 V in Fig.3.11.The peak distance is similar to the previous one, i.e. around 15.3 µm. However, the generated peak power of SH is enlarged to 140 μW. The added part comes from the first two terms in Eq. (3.7). In Fig.3.11(b), the phases of two signals vary linearly along the propagation due to the χ(2) -induced electro-optical effect.

Fig. 3 .

 3 Fig. 3.12 (a) The generated power and (b) output phase of SH for noncentro-symmetric

Fig. 3 .

 3 Fig. 3.13 The normalized change of the output power of SH as a function of the externally

Fig. 3 .

 3 Fig. 3.14 Schematic of the PSW for investigating the OR effect. The bottom inset shows

Fig. 3 .

 3 Fig. 3.15 x-components of the electric field at (a) optical frequency and (b) RF of 400GHz

Fig. 3 .

 3 Fig. 3.16 (a) 5dB length (where the transmission of the optical mode is -5 dB) of the PSW

Fig. 3 .

 3 Fig. 3.19 The OR property of the PSW as a function of the optical wavelength for w = 50

Fig. 3 .

 3 Fig. 3.20 The OR property of the PSW as a function of the radio frequency for w = 50 nm,

Fig. 3 .

 3 Fig. 3.21 The OR property in the PSW as a function of the control DC voltage for w = 50

4 . 1 Enhanced second harmonic generation in hybrid plasmonic waveguides 4 . 1 . 1 Introduction

 41411 waveguides. The tight confinement and local enhancement in the subwavelength low refractive index (RI) spacer layer yet allows to efficiently enhance nonlinear optical effects in short distances. Meanwhile, their relative low propagation losses further facilitate the effectiveness of the nonlinear processes. In addition, the existence of the high RI region offers another degree of freedom to tune the performance of HPW for fulfilling phase matching conditions (PMC). Reported nonlinear applications based on HPW include second harmonic generation (SHG)[START_REF] Hasan | Second-order nonlinear frequency conversion processes in plasmonic slot waveguides[END_REF] 109, 140], third harmonic generation [112], optical parameter amplification for compromising the loss of SPP mode [139], EO modulators[START_REF] Sun | Design and analysis of a phase modulator based on a metal-polymer-silicon hybrid plasmonic waveguide[END_REF][START_REF] Lou | Design and analysis of ultra-compact EO polymer modulators based on hybrid plasmonic microring resonators[END_REF] and high Kerr nonlinearity[111].These excellent properties have encouraged us to exploit HPW in different configurations, as described hereafter. Specifically, in section 4.1 we theoretically investigate the SHG process from the mid-infrared to the near-infrared band in a HPW based on a nonlinear polymer and then address its further enhancement by applying the HPW into a microring resonator (MRR) configuration in section 4.2. In section 4.3, we similarly study the optical parameter amplification process in a symmetric HPW. Based on the effective phase sensitive amplification process, phase regeneration of the phase shift keying optical signal is finally proposed.Over the last few decades, silicon photonics has been extensively studied to operate near the telecommunication band in the near-infrared (NIR) wavelength region.

  instance, Alloatti et al. proposed a SOH double slot waveguide enabling for the first time efficient second-order nonlinear optical mixing in silicon photonics with standard technology [162]. In addition, plasmonic based nonlinear devices open up opportunities

  example, Lu et al. [109] and Pigozzo et al. [140] studied efficient second harmonic generation (SHG) processes in LiNbO3 and AlGaAs-based HPW, respectively.

Fig. 4 . 1

 41 Fig.4.1 Schematic of the nonlinear hybrid plasmonic waveguide: hSi = 480 nm, hp = 60 nm,

Fig. 4 .

 4 Fig. 4.2(c), (d), and (e) give the distributions of the dominant component Ey for the 0th mode at FF, the 1-st mode at SH, and the 0-th mode at SH, respectively, for w = 329.6 nm. The mode refractive indices (neff) of the three modes are 1.904005 + 0.002648i, 1.904017 + 0.001571i, and 2.730581 + 1.607207 × 10 -4 i, respectively. Here the 0-th mode at the SH has lower loss than the one at the FF due to its dielectric-like property.

SH nFig. 4 .

 4 Fig.4.2 (a), (b) Mode refractive indices as a function of the width. Ey distributions of (c)

Fig. 4 .

 4 Fig.4.3 distributions on the center line along y-direction (powers of modes have been

  nonlinear coupling coefficients (NCC), which are defined by 0

Fig. 4 . 4

 44 Fig.4.4 Intensity variations in the propagation with pump power of 100 mW.

Fig. 4 .LFig. 4 . 5

 445 Fig. 4.4 shows the SHG conversion properties along propagation considering a pump power at FF of only 100 mW (20 dBm). For the 0-th mode at FF shown by the red solid line in Fig. 4.4(a), its power decreases monotonously due to the conversion

  silicon photonics to reinforce optical nonlinearities and realize efficient and compact nonlinear frequency conversions because of the large field enhancements around resonant wavelengths provided by the MRR.[150,[202][203][204][205][206][207][208]. The combination of hybrid plasmonic waveguides (HPW) and MRR has been proposed in several works in which the HPW-based MRR support subwavelength or even submicron radii and large field enhancements in the low-index slot area[START_REF] Lou | Design and analysis of ultra-compact EO polymer modulators based on hybrid plasmonic microring resonators[END_REF][103][104][105] 209]. However, most of these works were focused on the linear response of the MRR. In this work, we apply the same HPW as the one proposed in the previous section into a MRR configuration.

Fig. 4 . 6

 46 Fig. 4.6 Schematic of the silicon-organic hybrid plasmonic microring resonator. Cross-

Fig. 4 . 7

 47 Fig. 4.7 Δn = nSH -nFF as functions of the radius and width of the bended waveguides. The

Fig. 4 .Fig. 4 . 9

 449 Fig. 4.8 (a) neffkFFR and (b) loss of a 90˚ bend at the phase matching line as functions of

Fig. 4 .

 4 11(a) shows the SHG efficiency as a function of the propagation length in the straight HPW when the input pump power of the FF is () in FF P = 20 mW. The dashed line and solid line are the analytic solution and numerical solution, respectively. When the MRR works at the critical coupling point for the FF, i.e. tFF = exp(-παFFR), the field enhancement factor in Eq. (4.13) is calculated to be FFF = 2.276. The power of FF in the MRR is 2 ( ) in FF FF FP = 103.6 mW. Taking this power as the input power of FF, Fig. 4.11(b) shows the comparison for the bend waveguide with radius of R = 2.325 μm. One can see that, for both the straight and bend waveguides we study in this work, the analytic solution for

Fig. 4 .

 4 Fig. 4.11 Analytic solution and numerical solution of the SHG efficiency as functions of

  Fig. 4.12 SHG efficiency η2 as functions of the transmission coefficients for FF tFF and

Fig. 4 .Fig. 4 .

 44 Fig.4.12 shows the final MRR-enhanced SHG efficiency η2 as a function of the two waveguide-to-ring transmission coefficients for the FF (tFF) and SHF (tSH), respectively. As visible, a peak efficiency up to η2 = 21% is realized when the MRR works at the critical coupling point for both the FF and SHF, i.e. when tFF = exp(-παFFR) = 0.898 and tSH = exp(-παSHR) = 0.918. A maximum enhancement factor of η2/η1 = 150 is then expected. However, it is worth noting that it is not possible to let the FF and SH work at the critical coupling points at the same time in practice. Usually, tSH is larger than tFF for a same MRR because the waveguide mode at SH extends less beyond the waveguide boundary due to smaller wavelength. Despite of this, an optimal point can be envisaged in practice by adjusting the geometrical parameters of the MRR.

4. 14 Fig. 4 .

 144 Fig. 4.14 Transmission spectra of the 3D MRR structure around (a) the SHF of 1.55 μm

Fig. 4 .

 4 Fig. 4.15 SHG efficiency η2 as a function of the attenuation coefficients for FF (αFF) and

Fig. 4 .

 4 Fig. 4.16 (a) Schematic diagram of the proposed SHPW. Ey distributions at (b) SW of 1550

Fig. 4 .

 4 Fig. 4.16(b) and (c) show the mode profiles of dominant component Ey at the SW and PW, respectively. The modes are calculated with a finite-element based commercial software (COMSOL Multiphysics) and structural parameters are chosen as: w = 200 nm, hm = 10 nm, hp = 50 nm, hSi = 220 nm. It is shown that most of the fields are concentrated and enhanced within and near the low refractive index NP region both at the SW and PW. This results from the discontinuity of the Ey component at the high index contrast interfaces. The effective refractive indices of the hybrid modes are 1.59 + 8.60 × 10 -5 i and 1.81 + 8.35 × 10 -5 i at SW and PW, respectively. For the SW, a LRSPP-like hybrid mode [77] comes into being from the strong coupling of the LRSPP mode and the fundamental guided mode. With respect to the PW, however, a more like LRSPP hybrid mode is formed due to the weaker coupling of the LRSPP mode and the second-order guided mode at smaller wavelength, which results in more fields confined in the NP part and larger losses.

Fig. 4 .

 4 Fig.4.17(b) and (c) show the phase variations of SW and PW in the propagation, respectively. One can see that the phase of SW is unchanged and the phase of PW jumps to 0.5π at the peak position. This can be explained by the phase coupling equations, as derived in[139]. The initial phase of the SW and PW have an important influence on the OPA process. When the incident phase of SW is not zero, the phase changes while propagating until a stable value is obtained[139].

Fig. 4 .

 4 Fig.4.17 Intensity and phase evolutions in the propagation when φs(0) = 0.

Fig. 4 .

 4 Fig. 4.18 Intensity and phase evolutions in the propagation when (a), (b), φs(0) = 0.2π; (c),

Fig. 4 .

 4 Fig. 4.19 Gain and output phase of SW versus the input phase of SW when L = 150 µm.

4. 3 . 3

 33 Proposal for phase regenerationSimilarly as in[217], we propose to apply this mechanism to the phase recovery of PSK optical signals but in a highly integrated configuration making using of plasmonic waveguides. PSK optical signals result from a digital modulation scheme that conveys data by modulating the phase of a carrier wave, in which a finite number of phase responses under constant intensity are given. Unfortunately, along propagation, the phases usually deviate from their initial values due to amplitude noise and phase modulation effects. These deviations give rise to bit errors in detection, and should thus be removed by a regeneration mechanism[217]. Here, we assume a binary PSK signal with two phases of 0 and π, and the phase deviation of optical signal is smaller than 0.5π. When put the noised PSK optical signal through the nonlinear waveguide we propose, the phase will come back to the right value before detection.The advantages of the phase recovery based on the OPA in SHPW lie in two aspects. On one hand, it can regenerate the phase in variable short distances (tolerant from 120 µm to 200 µm). On the other hand, the trip points and stable points are tunable with the control of φp(0). φs(L) jumps at φs(0)=0.5φp(0)+0.75π and φs(0)=0.5φp(0)-0.25π and stabilizes at 0.5φp(0)-0.75π and 0.5φp(0)+0.25π. Besides, we could also find other applications based on this waveguide nonlinear effect such as phase-sensitive amplification and phase logic processing based on Fig.4.19(a), which are also significant in all-optical signal processing. Additionally, it should be noted that the performance of the OPA and corresponding phase recovery in SHPW may be improved effectively by fulfilling the phase matching condition by an appropriate design of the geometrical parameters[143].

( 3 )

 3 ) We theoretically investigate the quadratic nonlinear property of a silicon-organic HPW with a thin polymer layer deposited on top of a silicon slab and covered by a metal cap. Due to the hybridization property of the waveguide modes, efficient phase-matched SHG from mid-infrared (~ 3.1 μm) to near-infrared wavelength (~ 1.55 μm) is achieved with small fabrication-error sensitivity (225 nm  tolerated waveguide width  378 nm) and large bandwidth ( = 100 nm). The SHG yield is as large as 8.8% for a pumping power of 100 mW and waveguide length of 120 μm.(2) Then, we propose to further enhance the SHG efficiency in the HPW by exploiting the resonant effect of the microring resonator configuration. By making the microring resonator doubly-resonant at the fundamental wavelength of 3.1 μm and second harmonic wavelength of 1.55 μm, the SHG efficiency is enhanced by almost two orders of magnitude when compared to the previous result induced in a straight HPW. A SHG efficiency of 13.71% is predicted for a low pump power of 20 mW in a ring with radius of 2.325 μm. This device provides a potential route for realizing efficient frequency conversion between mid-infrared and near-infrared wavebands on a chip. Phase regeneration of phase-shift keying signals is theoretically proposed for the first time based on the efficient optical parametric amplification (OPA) process in a highly-nonlinear symmetric HPW. This optimized stacked waveguide with nonlinear polymers has a relatively low loss of about 0.005 dB/µm and an effective nonlinear OPA coupling coefficient up to 60 ps/m/W 1/2 . Phase recovery process was achieved in this waveguide within a length as short as 150 μm.

Firstly, let us

  briefly remind that various linear and nonlinear PSWs have been demonstrated. For example, Y. Fu et al. proposed all-optical logic gates based on the interference effect in PSW [85]. A. Kriesch et al. demonstrated subdiffraction directional couplers optically excited with high efficiency from free-space using optical

proposed in Section 3 . 3 .Fig. 5 .Fig. 5 . 1

 33551 Fig. 5.1 shows the schematics of the plasmonic slot waveguide (PSW) and taper couplers that bridge PSW with the in and out silicon waveguides. The whole structure is placed on a silica substrate, and the background top cladding medium material is the M1 polymer. The width and height of the incoming silicon waveguides are wSi = 450 nm and hSi = 220 nm, respectively. The angle and tip width of the taper coupler are defined as θ and d, respectively. The distance between the taper tip and PSW is l.

Fig. 5 .

 5 Fig. 5.2 (a) Voltage responsivity, (b) Current responsivity and (c) Normalized efficiency of

Fig. 5 .

 5 Fig. 5.3 (a) Voltage responsivity, (b) current responsivity and (c) the normalized efficiency

  [220] J. Tian et al. proposed a theoretical coupling efficiencies up to 88% for a wide wavelength range extended from 1460 nm to 1540 nm by tuning the geometrical parameters and experimentally demonstrated a 35% coupling efficiency. More importantly, this coupling scheme provides a good way to bridge the size gap between conventional microscopic silicon photonics and nanoplasmonics, which is a key point for future plasmonic planar integrated optics. For these reasons, we have chosen to use this coupling principle. By sweeping the geometrical parameters (i.e. d, l and θ) in a series of 3D-FDTD simulations, we have obtained an optimized taper size. In the simulation, an override mesh region with mesh size of dx = dy = dz = 5 nm has been added in the metal, slot and taper regions.Note that in the next whole optimization process, we have fixed the following PSW dimensions: w = 100 nm, h = 150 nm and L = 5 μm and considered an operating wavelength of λ = 1550 nm. First, we set the taper as l = 500 nm and θ = 8° and swept d, or set the taper as d = 80 nm and θ = 8° and swept l. Results are shown in Fig. 5.4(a) and (b), respectively. As it can be seen, the transmittance T increases when increasing d or when decreasing l. However, one can notice that the width between the silicon taper and metallic slabs becomes very small for too large d values or too small l ones, which makes the fabrication difficult. So we have chosen the following parameters in experiments: d = 80 nm and l = 500 nm.

Fig. 5 .

 5 Fig. 5.4(c) shows the obtained results for different θ values with fixed d = 80 nm and l = 500 nm. There exists an optimized value around 8 degrees. Therefore, the final geometry of the taper coupler is designed as d = 80 nm, l = 500 nm and θ = 8°, for which the total transmittance is T = 0.301445 (-5.208 dB) for the PSW length of L = 5μm. The total transmission can be expressed as T = t 2 exp(-αL) where t is the coupling efficiency for a single taper and α is the attenuation coefficient for the slot mode. For the PSW with w = 100 nm and h = 150 nm, the effective mode index is calculated to be neff = 1.957911+0.01823654i at λ = 1550 nm, then α = 73925 rad/m. By substituting

Fig. 5 . 4

 54 Fig. 5.4 Transmittance of the plasmonic slot waveguide with two taper couplers at λ = 1550

Fig. 5 . 5

 55 Fig. 5.5 Fabrication procedures of the nonlinear plasmonic slot waveguide.

Fig. 5 .Fig. 5 . 6

 556 Fig. 5.6(a) and (b) show the optical microscope pictures of the structure after the second and third steps, respectively. The inserted pictures are the partially enlarged displays around the PSW part. Note that the stains in the enlarged pictures are located on the high magnification lens of the object, while the wafer is clear. To have a more detailed view of the fabricated waveguide, we used the scanning electron microscopy (SEM) system and we give the related pictures in Figs. 5.7-5.10.

Fig. 5 .Fig. 5 .Fig. 5 .Fig. 5 .Fig. 5 .Fig. 5 .

 555555 Fig. 5.7 shows the reference silicon strip waveguide and grating couplers. The width of the fabricated waveguide is ~412 nm, while the designed width in the L-edit layout was 450 nm. This means that fabrication error around 38 nm exists in this fabrication. Fig. 5.8 shows the SEM pictures of the PSW and taper couplers from silicon

Fig. 5 .Fig. 5 .

 55 Fig. 5.11 Experimental setup to measure the fabricated waveguides.

Fig. 5 .

 5 Fig. 5.13 Poling condition of the M1 polymer.

Fig. 14

 14 Fig. 14 Diagram of the electrodes contact for the plasmonic slot waveguide in the poling

  clean room. The plasmonic modes in fabricated PSWs are successifully excited with a tapered silicon strip waveguide configuration. The measured linear losses are near to the theoretical ones, indicating a good performance of the PSW. After spin-coating the EO polymer M1 onto the wafer, the PSWs kept this good performance.Beyond the linear propagation properties, we now aim at demonstrating the nonlinear response of the fabricated PSWs. At present, we are on the step to efficiently pole the polymer. In the next step, our goal is to demonstrate the optical rectification effect in the PSW and finally apply it to compact high-speed detectors. Meanwhile, the performance of our experimental configuration can be improved in terms of grating couplers and active nonlinear materials. For example, D.Benedikovic et al. recently proposed a grating coupler with a coupling efficiency of -1.3 dB [225]. integrated devices. A conclusion that plasmonic waveguides can support intense nonlinear effects in short distance was obtained.

( 4 )( 6 )

 46 ) The possible realization of an active electro-optical control of the nonlinear SHG mechanism in a PSW was theoretically investigated. Both the conventional SHG and the electrically induced SHG were taken into account with a moderate pump power of 40 mW at the fundamental wavelength of 1550 nm. The generated power of the second harmonic wave can be modulated by the applied voltage in a quadratic and almost linear forms for centro-symmetric and noncentro-symmetric nonlinear polymers integrated in the slot, respectively. Converted power up to 140 µW within a short distance of only 16 μm was predicted for a voltage of 10 V. This mechanism may open a new route to realize high-speed advanced modulations or inversely to detect ultrafast electrical signals. The nonlinear optical rectification (OR) process induced in a PSW with an electrooptic polymer infiltrated into the slot was investigated. An electrical signal was generated between the two metal slabs when an intensity-modulated optical signal was injected into the PSW. Optimization of the geometrical parameters was performed to obtain the highest OR efficiency. Specifically, a voltage responsivity over 1V/W and a normalized OR efficiency of 2.25×10 -4 W -1 were predicted in a compact PSW with active size of 50nm×90nm×22.4μm for a modulation speed of 128 400 GHz. The OR efficiency is further increased when introducing the electrically induced OR contribution. The OR is characterized by a relatively flat response for a wide range of optical wavelengths from 1.25 μm to 1.7 μm and supports a RF cut-off frequency up to 800 GHz. This efficient OR process paves a new way for realization of high-speed broadband optical detection and demodulation schemes in silicon chips.(5) Efficient SHG from mid-infrared (~ 3.1 μm) to near-infrared wavelength (~ 1.55 μm) in the hybrid plasmonic waveguide (HPW) was theoretically investigated.Due to the hybridization property of the waveguide modes, efficient phasematched SHG is achieved with small fabrication-error sensitivity (225 nm < tolerated waveguide width < 378 nm) and large bandwidth (~ 100 nm). The SHG yield is as large as 8.8% for a pumping power of 100 mW and waveguide length of 120 μm. Then the SHG efficiency was further improved by two orders of magnitudes by exploiting the resonant effect of the microring resonator configuration. By making the microring resonator doubly-resonant at the fundamental wavelength of 3.1μm and second harmonic wavelength of 1.55μm, a SHG efficiency of 13.71% is predicted for a low pump power of 20 mW in a ring with radius of 2.325 μm. This device provides a potential route for realizing efficient frequency conversion between mid-infrared and near-infrared wavebands on a chip. Phase regeneration of phase-shift keying signals is theoretically proposed for the first time based on the efficient optical parametric amplification (OPA) process in a highly-nonlinear symmetric HPW. This optimized stacked waveguide with

  2.5 the mode effective index neff, propagation length Lp and mode effective area Aeff of the fundamental mode in PSW as a function of slot width w for different slot height h. Aeff, whose definition can be found in Section 2.2.2 by Eq. (2.19), is a very key parameter in nonlinear enhancement. Note here that these parameters and all other waveguide mode calculations performed by us have been calculated by a finite element method based commercial software (COMSOL Multiphysics) [84].

Thus, we can adjust the compromise between confinement and losses by tuning the dimensional sizes of HPW. In

  PSW, smaller spacer layer leads to larger neff, shorter Lp, and smaller Aeff. another picture, HPW can also be considered as a kind of slot waveguide[START_REF] Almeida | Guiding and confining light in void nanostructure[END_REF], merely it is a vertical slot between a metallic region and a high RI dielectric material (which, in most cases, is silicon). As a result, HPWs compromise the mode properties of both PSWs and silicon slot waveguides.

It shows better confinement properties than silicon slot waveguides and lower optical losses if compared with PSWs. The

	propagation length is on the order of hundreds of
	micrometers, while in PSW is a few tens of micrometers. To have a visual comparison
	with PSW, let us take a look to the related parameters with same size of spacer layer
	(i.e. 10 nm × 50 nm) for two kinds of waveguides. They are Lp = 18.67 μm, Aeff = 2.98
	× 10 -14 m 2 for HPW, and Lp = 9.35 μm, Aeff = 2.04 × 10 -15 m 2 for PSW, respectively.
	Clearly, HPW have lower losses and larger mode size over one order of magnitude, this
	is because some part of the field is confined into the silicon part, as shown in Fig. 2.7(d).
	Another interesting property of HPWs that disappears in PSWs is that the optical
	fields of the TM and TE modes in such a HPW are concentrated in different areas. The
	TM mode is concentrated in the low RI layer, whereas the conventional TE mode is
	concentrated in the high RI region. This makes HPW very interesting in realizing
	polarization components. Because of these attractive features HPWs have attracted a
	growing interest in recent years. Many different applications have been suggested such
	as subwavelength plasmon lasers [97], modulators [98, 99], directional couplers [100-
	102], ring resonators [103-105], polarization devices [106-108], and nonlinear
	applications [109-112] etc.
	When substituting the low RI dielectric with nonlinear materials, nonlinear effects
	can be effectively enhanced. In this manuscript, we analyze the enhanced second
	harmonic generation and phase sensitive amplification based on this kind of
	waveguides without and with the help of micro-ring resonators [113-115], as shown
	hereafter in Chapter 4.

Table 2 .

 2 1 Electrooptic coefficients for various second order nonlinear materials.

	Materials		λ (nm)	n	r33 (pm/V) Ref.
	LiNbO3		1550	2.2	30	[46]
	Doped, crosslinked organic polymer	1550	1.643	170	[132]
	AJ404L		1550	1.6	150	[133]
		M3		1550	1.68	70	[42]
	YLD124/PSLD41 (25:75 wt.%)	1540	1.73	230	[134]
	Silicon	1550	3.48	4.5×10 -18		Yes	[32]
	Silicon dioxide	1550	1.45	2.5×10 -20		No	[56]
	Silicon nitride	1550	1.989	2.5×10 -19		No	[124]
	Hydex	1550	1.5-1.9 1.15×10 -19		No	[124]
	DDMEBT	1500	1.8	1.7×10 -17		No	[135]

Materials

λ (nm) n n2 (m 2 /W) Carrier limited Ref.

Table 2 .

 2

2 Nonlinear coefficients for various third order nonlinear materials.

Table 2 .

 2

		1.6 *	132.2
	Lp (μm)	2714 *	32.85

3 Kerr nonlinearity in silicon slot waveguide and plasmonic slot waveguide.

a Slot is 157 nm wide and formed by two 216 nm × 220 nm silicon wires

[START_REF] Koos | All-optical high-speed signal processing with silicon-organic hybrid slot waveguides[END_REF]

.

b Slot has same size with the one in silicon slot i.e. 157 nm × 220 nm.

* Parameters obtained from

[START_REF] Koos | All-optical high-speed signal processing with silicon-organic hybrid slot waveguides[END_REF]

. Other parameters are calculated using Eqs. (2.18) and (2.19) by a finite element simulation.

  experimentally an electrically controlled SHG in plasmonic slot with NP in the slot [155]. More recently, L. Alloatti et al. reported an impressive secondorder normalized conversion efficiency up to 2900% W -1 cm -2 based on a silicon-

	radius Si3N4 ring resonator by using the electric-field induced SHG process and
	calculated a conversion efficiency of around 3.68 × 10 -3 with a pumping power of 75
	mW [151].
	From Chapter 1, we know that plasmonic based nonlinear devices are among the organic hybrid (SOH) waveguide [162].
	most promising candidates to further increase the efficiency of SHG and reduce the
	sizes of devices due to the ability to allow strong local-enhanced confinement of light
	beyond the limits imposed by the laws of diffraction in dielectric media [20, 153, 154].
	Another advantage of plasmonic structures is that they can respond on the timescale of
	can be induced in silicon
	waveguides by using a stressing Si3N4 overlayer. Based on such a waveguide geometry, relatively small nonlinear susceptibility in LiNbO3, the moderately large nonlinear
	SHG efficiency of about 5 × 10 -8 under a ns pump peak power of around 0.7 W was coupling coefficients (NCC) between different frequencies, and the absorption loss of
	realized for a length of two millimeters [149]. Secondly, SHG was proposed in a Si3N4 the plasmonic modes. Furthermore, LiNbO3 is not easily compatible with silicon
	ring resonator due to the break of centrosymmetry at the interface between the photonics, which represents the best solution for high yield and mass production with
	waveguide and the silica cladding. J. S. Levy et al. observed in a 116 µm ring radius low cost [161]. In the same time, the aforementioned nonlinear polymers (NP) which
	resonator a conversion efficiency of 2.86 × 10 -4 with a 100 µW of second harmonic have nonlinear susceptibilities exceeding those of LiNbO3 and can be integrated into

frequency (SHF) signal generated for a fundamental frequency (FF) pump power of 315 mW [150]. More recently, R. E. P. de Oliveira et al. reported SHG in a 20 µm a few femtoseconds. Many kinds of nonlinear plasmonic structures have been proposed for realizing efficient SHG, such as in plasmonic slot waveguides (PSW) [141, 143, 155], long-range plasmonic waveguides [156], hybrid plasmonic waveguides (HPW) [109], metal surfaces with nanoscale roughness [157], individual metallic nanoaperture [158], plasmonic particle chains [159], and plasmonic core-shell nanowires [160]. To date, the most used nonlinear material in these structures to realize SHG is lithium niobate (LiNbO3) [109, 141, 143, 156]. However, in spite of the generally used continuous wave (CW) pump power of the FF to around 1 W, the peak powers of the generated SHF are usually limited to 10 -5 W [141, 143, 156]. Even if it can go up to 10 -

2 W in HPW, the corresponding waveguide length to realize this efficiency is one millimeter [109], which is probably too long and not suitable for applications into future integrated nanophotonic circuits. The rather small reported efficiencies are due to the silicon photonic structures have drawn more and more attention for nonlinear optics and all-optical high-speed signal processing

[START_REF] Baehr-Jones | Polymer Silicon Hybrid Systems: A Platform for Practical Nonlinear Optics †[END_REF][START_REF] Leuthold | Silicon organic hybrid technology-A platform for practical nonlinear optics[END_REF][START_REF] Koos | All-optical high-speed signal processing with silicon-organic hybrid slot waveguides[END_REF]

. For example, Wenshan

Cai et al. demonstrated

  Han et al. proposed experimentally a Si-plasmonic taper coupler, where a 30% coupling efficiency was reported[START_REF] Han | Experimental realization of subwavelength plasmonic slot waveguides on a silicon platform[END_REF]. More recently, R. Thomas et al. presented theoretically a specially designed coupler with a short taper length to couple light into a PSW, where a maximum coupling efficiency of 72% into a 20 nm slot was predicted[164].Based on the former proposed nonlinear coupled-wave equations (NCE) in chapter 2, we can analyze different kinds of nonlinear processes. In the present case under consideration about the SHG process, the FF ωFF is converted into the SHF ωSH = 2ωFF,

						FF	FF 24 FF FF	FF FF SH	exp(	)
						SH	SH 24 FF SH	SH FF FF	exp(	)
	and the nonlinear polarization terms at FF and SHF are	FF NL P		0 	(2)	:	* SH FF E E	and
	SH NL P		0  0.5	(2)	:	FF FF E E	, respectively. If we consider a single given guided mode both

at FF and SHF, the NCEs describing the SHG process in the nonlinear waveguide are: *

Table 3 .

 3

	50	90	22.4	1.0810	0.4166	2.2517
	80	119	31.5	0.7140	0.3198	1.1417
	110	147	39.6	0.5423	0.2743	0.7438
	140	174	47.1	0.4421	0.2473	0.5467
	50	90	5.4	1.0984	0.1021	0.5607

1. The optimized height and length for different widths and the corresponding voltage responsivity, current responsivity and the optimized normalized efficiency of the OR process. The last row represents the result for the metal of Au.

  )

	with	, FF SH E	( , ) x y and	, FF SH E		( , ) x y	the mode profiles which have been normalized.
	FF E	,	SH		( , ) x y
							 	FF SH	SH FF E x y E x y E x y dxdy FF FF FF SH E x y E x y E x y dxdy          	(4.4)
								96

Table 5 .

 5 1 Measured loss of PSWs for different cladding materials and sizes at λ = 1550 nm

		L (μm)	w (nm)	PSW loss (dB)
	Air	8.8	65.6	11.439
	Air	8.8	76.5	12.933
	Air	8.8	109.3	15.741
	Air	10	65.6	11.877
	Air	10	76.5	13.815
	Air	10	109.3	17.81
	Polymer	8.8	65.6	12.89
	Polymer	8.8	76.5	11.917
	Polymer	8.8	109.3	9.206
	Polymer	10	65.6	13.993
	Polymer	10	76.5	12.82
	Polymer	10	109.3	11.88

W -1/2 for the bended waveguide case.

Bibliography

List of Figures which leads to larger currents. As an interesting result, from Fig. 3.16(d), we can see that there exists an optimal height for a certain width for the OR efficiency. The obtained optimal heights for different widths are given in Table 3.1. Here, the height has been chosen to the optimal value for a certain width coming from the approach giving rise to the results of Fig. 3.16. Next, we set the height to the optimized value and sweep the length of the PSW.

The results are shown in Fig. 3.17. When the length increases, the loss of the optical mode increases, thus reducing the average power along the propagation distance and thus the OR voltage. However, similarly as before, the OR current increases due to the reduction of the capacitor impedance for increasing lengths. As shown in Fig. 3.17(c), the schematic of Fig. 4.10. The interactive waves satisfy the following equations:

( The complex mode amplitudes A are normalized, so that their squared magnitude corresponding to the modal power. c and t are coupling and transmission coefficients with |c| 2 +|t| 2 =1. Besides, based on Eq. (4.5), we can obtain:

silicon photonics platform. By taking the phase matching, resonance and loss level into consideration, the conversion properties are shown to be improved in three aspects: the needed pump power becomes lower, the size of the device is smaller and the efficiency of the SHG is larger.

In the proposed final geometry, a SHG conversion efficiency of around 13.7% is predicted for a pump power as low as 20 mW and a ring radius of 2.325 µm. This obtained dramatic improvement of the nonlinear efficiency with respect to previous results opens avenues to practical applications using low-power nonlinear SHG-based all-optical on-chip signal processing. In SHPW, the loss is even lower while the field confinement is comparable when compared to the conventional HPW. The nonlinear effects in this kind of waveguides may be prominent but have not been studied yet to our best knowledge. In this section, we investigate the nonlinear optical parametric amplification (OPA) process induced in a SHPW with nonlinear polymer (NP) as the active materials. The simulation results reveal that it enables efficient OPA within a short distance around 180 μm. Based on Chapter 6

Conclusion and future steps

With further increase of bandwidth demand and integration density, the expectations for new integrated devices include ultra-compact size, ultrahigh speed, low cost and low power consumption. However, the conventional integrated electronic and dielectric photonic devices can hardly satisfy all these demands. In recent years, plasmonics has come into being a promising platform to meet some of these requirements due to its ability to confine the optical and electrical modes beyond the diffraction limit, support Then, the fabrication steps were introduced. The measured results indicate that the fabricated PSW with size of 66 nm × 150 nm has loss of 0.5 dB/µm, which is near to the theoretical level. Then, the PSWs were spin-coated with EO polymer M1 as the first step to demonstrate the OR effect.

All the above research indicate that silicon plasmonic waveguides open promising solutions to enhance nonlinear effects within short distance while managing the loss issue at an acceptable level, which has great potential for the realization of ultracompact and low power integrated functionalities such as wavelength converters, logical gates, modulators/switches and detectors.

In the same time, the high-quality fabrication and measurement of compact plasmonic waveguides, especially nonlinear plasmonic waveguides, remains a challenge when compared with dielectric photonic waveguides. At present, we have successfully spin-coated the EO polymer onto the PSW wafer. Future steps of our work include:

(1) Poling the EO polymer. This is a difficult point due to the needed harsh poling environment.

(2) Measuring the OR effect after poling in the PSW.

In conclusion, CMOS-compatible plasmonic waveguides are widely considered as a prospective basic element to implement future on-chip subwavelength electronicphotonic integrated circuits. In the future, plasmonic on-chip electronic-photonic integrated circuits and chip-to-chip interconnects are likely to be able to combine optical sources, interconnects, modulators, and detectors on the same chip, within subdiffraction-limited dimensions [226][227][228]. Such an integrated photonic technology could revolutionize the bandwidth, speed, size, cost, and power requirements of modern computational networks, enabling more efficient solutions to increasingly complex problems. As soon as we can overcome the high loss bottleneck, a new plasmonic

Appendix Numerical method to analyze nonlinear effects

Mode calculations at every frequency in COMSOL Nonlinear coupling coefficient calculation by implementing a postprocessing overlap integral in the nonlinear area in COMSOL Numerical solution of the nonlinear coupled-wave equations with a MATLAB code based on the finite difference method All the simulations proposed in this dissertation to analyze various nonlinear effects induced in an integrated waveguide have been based on a numerical processing as shown in Fig. A1. In the first step, we calculate the mode profiles at every frequency of the related nonlinear effect by using the finite-element-based commercial software COMSOL. Then, the nonlinear coupling coefficient as expressed in Eq. (2.13) is calculated through a postprocessing of "Derived Values → Integration → Surface

Integration" in the nonlinear area of the waveguide cross-section in COMSOL. After this, we can substitute this obtained nonlinear coupling coefficient into the nonlinear coupled-wave equation and finally solve it with a self-made MATLAB code based on the finite difference method.

The following is the typical code to obtain the result in Fig. 3.6 for analyzing the second harmonic generation process in the plasmonic slot waveguide.

Enhancement of nonlinear effects using silicon plasmonic structures

With the rapid increasing bandwidth of data transmission and signal processing, integrated electronics encounters strong bottlenecks. Silicon photonics provides a lowcost solution to overcome some of these bottlenecks by introducing on-chip optical links. After a decade of development, silicon photonics is now the most active discipline and most promising platform within the field of integrated optics. However, in the process of further development, new stumbling blocks emerge, among which the fact that the size of photonic devices is limited by the diffraction limit, which results in a large mismatch between photonic and electronic components. Plasmonics seems to be an ideal solution to overcome this obstacle thanks to its ability to confine the optical field into nanoscales beyond the diffraction limit. Meanwhile, the localized strong field enhancement in plasmonic structures enhances interaction of light and matter, which is promising for nonlinear applications.

In this dissertation, we combine the plasmonic and organic technologies onto the silicon photonics platform to create silicon plasmonic organic structures and investigate the nonlinear effects induced in them. Silicon plasmonic organic structures combine the advantages of silicon with ultra-compact performance of plasmonics and ultrafast property of organic materials that have great potentials in nonlinear integrated optics.

A full-vectorial nonlinear coupled-wave equation model which is valid for lossy plasmonic waveguides is proposed and then utilized to analyze the nonlinear effects in silicon plasmonic waveguides. This dissertation addresses the use of two kinds of plasmonic waveguides, plasmonic slot waveguide (PSW) and hybrid plasmonic waveguide (HPW), for nonlinear applications. Specifically, enhanced conventional and electrically induced second harmonic generation and optical rectification effect in PSW and enhanced second harmonic generation in HPW and ring resonators are proposed.

The mode phase matching technique is applied for the phase matching of the nonlinear processes. Based on the effective nonlinear effects within short distances, possible applications in optical signal processing such as phase regeneration, modulation and detection are envisaged.

Design, fabrication and measurement of PSW are also provided. By spin-coating a commercially available second order nonlinear polymer, preliminary results regarding the nonlinear response of the PSW are investigated.