Thanks Feng Tang

Feifei Zhang

Jiyong Wang

Fang DAI Yi Huang

Shijian Wang

Yunlong Zhu

Junze Zhou

Hongshi Chen

Dandan Ge

Quan Liu

Lan Zhou

Xiaolun Xu

Alma K Gonzalez Alcalde

Avril A Meza Olivo

Ali Issa

Claudio Silvestre Castro

Wajdi Chaabani

Keywords: optiques, 5, 4, 3 Performances électriques, fundamental TM mode incidence polaritons plasmonaires de surface light modulators -graphene -plasmons (physics) -nanophotonics -optoelectronics

My warmest gratitude goes to my dear parents, my little sister, and my beloved Shiyu. Thanks for your selfless love, unconditional

i Abstract This thesis work aims to high performances subwavelength graphene-based electro-optic modulator (EOM) integrated on a silicon photonic waveguide by plasmonic enhancement of graphene-light interaction toward optical interconnect application.

We first analyzed theoretically/numerically/experimentally a configuration where a single graphene layer was placed on top of a silicon photonic waveguide. The in-plane selectivity of the absorption derived by the surface conductivity model of graphene and the numerical simulation results allow to confirm the weak interaction of Si photonic mode with graphene. We also conducted the fabrication of this configuration. Near-field scanning optical microscopy (NSOM) measurements confirmed the weak interaction of photonic modes with graphene.

To enhance the interaction between waveguide's mode and graphene, the use of plasmonic waveguides such as nano-strip, nano-slot and chain of metal nanoparticles were numerically analyzed and optimized. In-plane electric field density enhancements into the graphene flake of respectively 10, 6, and 40 times were obtained.

An ultra-compact EOM based on graphene and a periodic chain of metal nanoparticles integrated on a silicon waveguide was finally proposed and numerically demonstrated. The benefit from the in-plane enhancement of hybrid plasmonic modes allows to achieve sub-λ scale footprint without sacrificing the modulation depth. Numerical analysis shows that the EO nanomodulator presents a modulation depth as high as 4 dB/µm at nonresonant wavelength of 1.65 µm, with a transmission loss of 2.6 dB. Due to the sub-λ scale footprint of the proposed EO nanomodulator, the modulation speed is up to 39 GHz and the energy consumption is as low as 21.2 fJ/b.

In parallel to this study, we also made the numerical and experimental demonstration of an extremely compact TM-pass polarizer based on the use of periodic chain of metal nanoparticles in silicon integrated photonics. The experimental results revealed an extinction ratio about 23 dB/µm at 1.592 µm, an insertion loss of 2.4 dB/µm in the 1.54 µm to 1.64 µm wavelength range, and the length as short as 0.63 µm.

In conclusion, this doctoral research project provided a key for the development of plasmonic-enhanced interaction with graphene towards high performance sub-λ EOM. This thesis work brings a contribution to photonic communication interconnects and compact photonic integrated circuit. [START_REF] Davis | Plasmonics: the convergence between optics and electronics[END_REF] The effect of substrate on conductivity of graphene at 1. [START_REF] Michael R Watts | Ultralow power silicon microdisk modulators and switches[END_REF]

Chapter 1 Introduction

Electro-optic modulator (EOM) is a basic component of optical interconnect systems. In this chapter, we firstly present the general concept of optical interconnects and the application of EOMs in optical interconnect systems. Then, on the basis of requirements of optical interconnect applications, we present the state-of-the-art of EOM that includes traditional EOMs, graphenebased EOMs, and plasmonic-enhanced graphene-based EOMs. Finally, by summarizing the problems of recent EOMs toward optical interaction applications, the objective and organization of this thesis are described.

Integrated electro-optical modulators (EOM) as optical interconnects 1.General concept of optical interconnect

Through sixty years of rapid development, integrated circuit (IC) based on electronics has became the basis of modern information and communication technologies. Due to the tremendous amount of data and larger electrical computational systems, data transport of intra/inter-system across short electrical wires (electrical interconnects) became indispensable. In recent years, the bandwidth limitation of electrical integrated circuits can not longer meet the growing demand on data processing and telecommunication. For example, the available bandwidth per compute operation (Mbps/FLOPS) will likely reach its end at 5 nm technology node [START_REF] Sun | The case for hybrid photonic plasmonic interconnects (hyppis): low-latency energy-and-area-efficient on-chip interconnects[END_REF]. Besides the limitation of bandwidth, electrical wires have large energy consumption and signal attenuation.

As the optical equivalent of electrical integrated circuit, photonic integrated circuit (PIC) based on photonics can be divided into photonic signaling and photonic computing. Photonic computing focuses on replacing current computer components with optical equivalents, for instance replacing transistor by EOMs or electro-optic switches, resulting in an optical digital computer system processing binary data [START_REF] Athale | Optical computing: past and future[END_REF][START_REF] Ashok V Krishnamoorthy | Computer systems based on silicon photonic interconnects[END_REF]. Nevertheless, there are doubts about the future capabilities of optical computing: whether or not they may be able to compete with semiconductor-based electronic computers in terms of speed, power consumption, cost, and size is an open question. So far, the electrical circuit is good in calculation and memory, and photonic circuit is more suitable for signal transmission [START_REF] Davis | Plasmonics: the convergence between optics and electronics[END_REF]. So currently, information in communication networks is processed by electronic chips and then transmitted by optical signal. Because of the weak interaction between photons, photonic systems have weak non-linear behaviors, such as switch and modulation and therefore, not a suitable candidate for data computing. Photon has no mass so it cannot be confined stably in small space, which is important for memory, and hence it is not suitable for memory [START_REF] Davis | Plasmonic circuits for manipulating optical information[END_REF]. Nevertheless, PIC is more promising in data transmission due to ultra-fast speed, bandwidth, and low energy consumption. For example, optical fibers have optical loss of only 0.2 dB/km at 1.55 µm [START_REF] Atef | Optoelectronic circuits in nanometer cmos technology[END_REF] and can support huge amounts of data at very high speed with ultra-low power consumption. Optical data transmission based on photonics are considered as the alternative of electronic data transmission since the 1990s [START_REF] Shin | Comparison of electrical and optical interconnect[END_REF] due to less energy consumption and higher bandwidth.

Generally speaking, optical data transmission (photonic signaling) includes optical communications and interconnects [START_REF] Wang | On-chip silicon photonic signaling and processing: a review[END_REF]. Optical communications is the far distance data transmission by optical fibers [START_REF] Welch | The realization of large-scale photonic integrated circuits and the associated impact on fiber-optic communication systems[END_REF], while intra-and interchip data transmission by waveguides can be classified into optical interconnects [START_REF] Joseph W Goodman | Optical interconnections for vlsi systems[END_REF][START_REF] David | Rationale and challenges for optical interconnects to electronic chips[END_REF]. So optical data transmission in the distance from deep-space mission to access network can be classified into optical communications, while data center and high-performance computing applications from rack-to-rack transmission to on-chip data transmission can be classified into optical interconnects (Figure 1.1).

FIGURE 1.1: Optical communications and interconnects from

deep-space to on-chip [START_REF] Wang | On-chip silicon photonic signaling and processing: a review[END_REF].

1.1. Integrated EOM as optical interconnects 3 Figure 1.2 is a typical configuration of optical communication/interconnect system [START_REF] Paniccia | Introducing intel's advances in silicon photonics[END_REF]. There are three parts in this system: The first part is the electronic chip that sends electrical binary signal to a photonic chip. The second part are two photonic chips. The electrical binary signal is received and detected by the two photonic chips. Here, on-chip laser, electro-optic modulator and photodetector are used in the transceiving of optical binary signals. The electrical binary signal is translated to optical binary signal by the modulators, and then received by the photodetector and translated again as electrical binary signal. Finally, the optical binary signal is transmitted by fibers between two photonic chips (optical communication) or waveguides in intra-/inter-chip system (optical interconnect). Note that the optical binary signals in the singlemode waveguides/fibers are combined and demultiplexed by multiplexers that has both large bandwidth and low attenuation. This is why PIC is more suitable than electrical integrated circuits in communication and interconnect. In the past several decades, fiber-optical communication has achieved great success in the telecommunication industry, nevertheless, industrial application of intra-and inter-chip optical interconnect still are a big challenge due to the difficulties of practical and high-efficiency discrete devices and their integration such as lasers, electro-optic modulators and photodetectors.

For instance, the length of field-effect-transistor (FET) is just tens of nanometers whereas its optical counterparts are from several hundreds micrometers to several millimeters. A reason is that active electro-optic devices based on pure photonic elements often rely on optical non-linear physical effects requiring strong light-matter interaction, which is inherently weak in photonic systems, this leads to large footprint and significant power consumption. Diffraction limit breaking by plasmonics allow high field densities in surface plasmon polaritons (SPPs) modes. SPPs represent a huge opportunity to enhance light-matter interaction and to implement sub-wavelength scale footprint discrete optical devices. Recently, a few plasmonic devices have been presented, ranging from plasmonic waveguides [START_REF] Rupert F Oulton | A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation[END_REF][START_REF] Volker | Experimental demonstration of low-loss optical waveguiding at deep sub-wavelength scales[END_REF], plasmonic sources [START_REF] Ma | Plasmon lasers: coherent light source at molecular scales[END_REF], plasmonic modulators [START_REF] Haffner | All-plasmonic machzehnder modulator enabling optical high-speed communication at the microscale[END_REF][START_REF] Dionne | Plasmostor: a metal-oxide-si field effect plasmonic modulator[END_REF] to plasmonic photodetectors [START_REF] Falk | Near-field electrical detection of optical plasmons and single-plasmon sources[END_REF].

Electro-optic modulators and its performance metrics

In order to obtain binary optical information, optical modulators are required to change the amplitude, phase, or polarization state of output continuous wave of light source. Optical modulators are typically based on direct or external modulation. With direct modulation, the light is directly modulated inside the laser. With external modulation, the output light of the laser is operated continuously by an additional optical modulator. For the external optical modulator, there are electro-optical modulator, magneto-optical modulator, acousto-optic modulator, liquid crystal modulators, micromechanical modulators, and so on. They are used in different application areas, such as in optical communications, displays, active Q switching or mode locking of lasers, and in optical metrology.

Electro-optic modulator (EOM) is a basic component of optical communication and interconnect system. It is an optical device based on electro-optic effect (for example: Pockel's effect, Kerr's effect, plasma dispersion effect, electro-absorption effect, Franz-Keldysh effect, Quantum-confined Stark effect, and so on [START_REF] William | Fundamentals of guided-wave optoelectronic devices[END_REF]), which allow us to control the amplitude, phase, or polarization state of an optical beam electrically. For instance, in optical interconnect systems, these modulators translate electrical binary signal onto an optical binary signal. Figure 1. 3 (a) is the schematic of IBM's EOM. It is a three-port device that includes electrical part and optical part. Electrical ports are electrodes that are used to apply an electrical binary signal into the EOM, optical ports is an optical waveguide, active material as a bridge between electrical binary signal and optical binary signal. The basic principle of EOM is shown in Figure 1. 3 (b). The electrical binary signal (V d ) induces a change of free carrier concentration of active material and then leads to a shift in the real part (n) and imaginary part (α) of the propagation constant of light through the device. Finally, ∆n leads to a phase shift and ∆α causes different absorption under different V d .

Performance metrics of EOMs

Modulation speed or 3dB bandwidth is one of the most important parameter of optical interconnect system. For an EOM, the modulation speed can be calculated by [START_REF] Peng | Highly efficient graphene-on-gap modulator by employing the hybrid plasmonic effect[END_REF]:

f 3dB = 1 2πRC , (1.1)
where R and C are respectively total resistance and total capacitance of the EOM. The energy consumption of an EOM can be calculated as [START_REF] Christopher | Graphene electro-optic modulator with 30 GHz bandwidth[END_REF][START_REF] Qiu | Efficient modulation of 1.55 µm radiation with gated graphene on a silicon microring resonator[END_REF]:

E bit = 1 4 C∆V 2 , (1.2)
where ∆V is the difference between the drive voltages V d at on and off states. The energy consumption is a very important metric as current electrical interconnects consume about 80% of a microprocessor's power [START_REF] Graham T Reed | Silicon optical modulators[END_REF]. Optical interconnects consume lower energy than that of their electrical counterparts and therefore, makes them more competitive for future networks.

Chapter 1. Introduction

With the ongoing miniaturization and integration of optoelectronic integrated circuits, high modulation strength and small device footprint are always important figures of merit of discrete devices. Considering the electrical counterpart field-effect-transistor (FET) of EOM is just tens of nanometers short, the objective of EOMs is to fabricate subwavelength devices without sacrifice performance [START_REF] Fratalocchi | Nano-optics gets practical[END_REF]. Hence small footprint is the requirement for dense optical interconnect especially intra-chip optical interconnect that requires subwavelength devices. Small footprint also brings low resistance and capacitance and provides not only high modulation speed but also low energy consumption [START_REF] Sun | Optical modulators with 2D layered materials[END_REF]. Modulation strength is another important performance metric of EOMs that can be defined as the transmission ratio between on and off states. In order to take both the performance (modulation strength) and footprint into account, the modulation depth (MD) is redefined in dB/µm as [START_REF] Graham T Reed | Silicon optical modulators[END_REF]:

MD = 1 L 10 log(T max /T min ) , (1.3)
where L is effective length of the active region of the EOM, T max and T min are respectively the transmission at on state and off state expressed in dB.

Although a high modulation depth (i.e. > 7 dB) is preferred for optical interconnect, 5 dB is often sufficient [START_REF] Graham T Reed | Silicon optical modulators[END_REF].

In order to make a performances comparison between different EOMs, a unified performance benchmark figure of merit (FoM) is defined as [START_REF] Amin | Attojoule-efficient graphene optical modulators[END_REF]:

FoM = f 3dB S • E bit , (1.4)
where S is the area of the active region. Besides MD, footprint, modulation speed and energy consumption, other important parameters of EOM include: insertion loss (IL, which is the intrinsic loss of an EOM in on state), working spectrum (working bandwidth of an EOM), drive voltage V d (including drive voltage of on state and off state, which is more important for traditional EOMs due to higher drive voltage bring stronger light-active material interaction and hence shorter active region), total resistance R, total capacitance C, and so on.

As a practical and high-efficiency EOM, the best possible performances are always purposes of researchers. Nevertheless, for recent application of EOMs in optical interconnect, modulation depth higher than 3 dB/µm, on the scale of 10 fJ/b energy consumption, higher than 10 GHz modulation speed and subwavelength scale footprint are required by 2020 as shown in Table 1.1 [START_REF] Miller | Device requirements for optical interconnects to CMOS silicon chips[END_REF]. which considers MD and IL in its whole length as a device. In this case, the requirements of both MD and IL are 3 dB.

State-of-the-art of photonic EOM 1.2.1 LiNbO 3 -and semiconductor-based EOMs

Early demonstrations of EOMs are mainly based on electro-optic effect such as Pockel's effect (LiNbO 3 , EO polymer), electro-absorption effect (Ge and III-V compound semiconductors), and plasma dispersion effect (Si, ITO). The basic configuration of EOMs includes pin (p-type/intrinsic/n-type layers) diode structure, pn junction or MOS (diode metal-oxide-semiconductor) capacitor are used. Nevertheless, because of weak interaction between light and the electro-optic materials, Mach-Zehnder interferometer (MZI) and microring resonator (MRR) are widely used to enhance the interaction between light and electro-optic materials. Many electro-optic materials have been used in EOMs, such as LiNbO 3 , Ge, GeSi, Ge/SiGe quantum wells (QWs), Si, EO polymer, ITO, and so on (Table 1.2). ) is the first electro-optic material applied in EOMs that began in the 1970s due to its significantly large Pockels coefficients [START_REF] Ip Kaminow | Lithium niobate ridge waveguide modulator[END_REF].

The significantly Pockel's effect of LiNbO 3 leads to a large change of refractive index of LiNbO 3 , which is linearly proportional to the applied voltage (∆n ∝ V d ). The accumulated change in the phase produced by the Pockel's effect is [START_REF] William | Fundamentals of guided-wave optoelectronic devices[END_REF]:

∆φ = ∆n(V d )kl , (1.5)
where l is the length of electrode, k = 2π/λ 0 is the wave vector in vacuum.

Once the phase modulation is achieved, amplitude modulation can be designed with the use of a Mach-Zehnder interferometer [START_REF] Cheng | Mach-zehnder modulators with lithium niobate ridge waveguides fabricated by proton-exchange wet etch and nickel indiffusion[END_REF][START_REF] Howerton | Fully packaged, broad-band LiNbO 3 modulator with low drive voltage[END_REF]. Nevertheless, the length of this kinds of EOMs is in millimeter range because the achievable index change is small and thus, either high voltages or long electrode lengths are needed to obtain sufficient modulation. This limits the ability to integrate multiple optical components on the same chip. To reduce the size of EOMs, micro-ring resonator based on LiNbO 3 EOMs have also been presented, but their size are still several hundreds micrometer and their working bandwidth is narrow [START_REF] Guarino | Electro-optically tunable microring resonators in lithium niobate[END_REF]. Although efforts are done towards the integration of LiNbO 3 in photonic integrated circuits [START_REF] Chen | Hybrid silicon and lithium niobate electro-optical ring modulator[END_REF][START_REF] Rao | Heterogeneous microring and Mach-Zehnder modulators based on lithium niobate and chalcogenide glasses on silicon[END_REF], bottleneck still exists. [START_REF] Rao | Heterogeneous microring and Mach-Zehnder modulators based on lithium niobate and chalcogenide glasses on silicon[END_REF]. Currently, LiNbO 3 -based EOMs are mainly used for fiber optical communications.

Ge and III-V semiconductor compounds based EOMs

Germanium (Ge) and III-V compound semiconductors, such as the compounds of Ga, As, P, In, allow compatibility with standard complementary metal-oxide-semiconductor (CMOS) processing. They are based on a strong electro-absorption (EA) effect, which is intrinsically an ultrafast process suitable for high-speed modulation. The electro-absorption effect is a modulation of the absorption of a well guided mode in Ge and III-V compound semiconductor waveguides because the imaginary part of refractive index (absorption coefficient α) can be changed by applied voltage (∆α = α(V d )).

The electro-absorption modulators are based either on the Franz-Keldysh effect in bulk semiconductors or the Quantum Confined Stark Effect (QCSE) in quantum-well structures [START_REF] Liu | Waveguideintegrated, ultralow-energy gesi electro-absorption modulators[END_REF]. Ge is the most used electro-absorption modulation material because of significant direct-gap transition in bulk Ge. Bulk Ge or SiGe electro-absorption EOMs [START_REF] Liu | Waveguideintegrated, ultralow-energy gesi electro-absorption modulators[END_REF][START_REF] Eu-Jin | Novel evanescent-coupled germanium electro-absorption modulator featuring monolithic integration with germanium pin photodetector[END_REF][START_REF] Feng | High speed GeSi electro-absorption modulator at 1550 nm wavelength on soi waveguide[END_REF] based on the Franz-Keldysh effect have been demonstrated with low energy consumption [START_REF] Liu | Waveguideintegrated, ultralow-energy gesi electro-absorption modulators[END_REF] and high modulation speed [START_REF] Feng | High speed GeSi electro-absorption modulator at 1550 nm wavelength on soi waveguide[END_REF]. Ge/SiGe quantum wells (QWs) [START_REF] Chaisakul | 23 GHz Ge/SiGe multiple quantum well electro-absorption modulator[END_REF] also has demonstrated a strong Quantum-confined Stark Effect. Additionally, other III-V compound semiconductors such as InP-InGaAsP [START_REF] Irmscher | InP-InGaAsP high-speed traveling-wave electroabsorption modulators with integrated termination resistors[END_REF] and InGaAs/InAlAs QW [START_REF] Ido | Ultra-high-speed multiple-quantumwell electro-absorption optical modulators with integrated waveguides[END_REF] have been presented based either on the Franz-Keldysh effect or the Quantum-confined Stark Effect with high modulation speed. Compare with LiNbO 3 and Si based EOMs, Ge and III-V compound semiconductors electro-absorption modulators are one of best approaches to obtain low energy consumption between 10 fJ/b to 100 fJ/b and high speed modulation due to its micrometer scale footprint (and low capacitance) and subpicosecond operation time for a wide spectral range [START_REF] Schmitt-Rink | How fast is excitonic electroabsorption?[END_REF]. Realizing both Ge and Si photonics and Si electronics on a thick SOI substrate is possible. For example, epitaxially growth, wafer bonding, or die bonding of Ge and III-V materials on Si are currently under investigation by several research teams [START_REF] Liang | Recent progress in lasers on silicon[END_REF], but it can lead to a significant degradation of the electronic performance in terms of heat dissipation and device reliability and complicates device designs [START_REF] Chaisakul | Integrated germanium optical interconnects on silicon substrates[END_REF]. Further, in order to achieve a large modulation depth, all Ge-on-Si electro-absorption modulators to date have required voltage levels that are not compatible with CMOS drive levels [START_REF] Timurdogan | An ultralow power athermal silicon modulator[END_REF]. Finally, since electro-absorption devices are based on a band-edge effect, they are inherently narrowband.

Si-based EOMs

Silicon is the most widely investigated material for EOMs because the dominance of Si as the semiconductor of choice for microelectronics leads to potential attraction of EOMs integration with electronics in a cost-effective manner. However, Si-based EOM is inherently restricted as an optical material for two primary reasons. Firstly, the indirect band gap of Si means that light emission from Si is inefficient, and hence small electro-absorption effect like Ge and III-V compound semiconductors. Secondly, the centrosymmetric crystal structure of Si means that it does not exhibit a linear electro-optic effect (Pockels effect) [START_REF] Soref | Electrooptical effects in silicon[END_REF]. Most of Si-based EOMs are designed by plasma dispersion effect. The plasma dispersion effect is related to the density of free carriers in a semiconductor, which changes both the real part and imaginary part of the refractive index. At λ 0 = 1.55 µm, the change of the real part and imaginary part of the refractive index can be expressed as [START_REF] Soref | Electrooptical effects in silicon[END_REF]: where ∆N e and ∆N h are the change of electron and hole densities. The densities of electron and hole can be changed by applied voltage V d [START_REF] Liu | A high-speed silicon optical modulator based on a metal-oxidesemiconductor capacitor[END_REF]. This kind of electrical manipulation of the charge density interacting with the propagating light is achieved through carrier depletion in pn junction or carrier injection in MOS capacitor and pin diode structure formed by doped silicon. Note that the plasma dispersion effect of doped silicon is weaker than electro-absorption effect of Ge and III-V compound semiconductors. Owing to intrinsically low free carrier concentration and weak electro-optical effect of silicon, most demonstrated Si-based EOMs utilize either a Mach-Zehnder interferometer or a micro-ring resonator. The silicon MZI modulator is usually a few millimeters long and hence consumes tens or hundreds of milliwatts, relatively high insertion loss (∼3-5 dB) and drive voltage (∼2-5 V). The silicon micro-ring modulator has low power consumption, but the working wavelength is sensitive to fabrication imperfections and temperature variation [START_REF] Barwicz | Silicon photonics for compact, energy-efficient interconnects[END_REF]. Figure 1.4 shows typical configuration of MZI and MRR based EOM.

∆n = ∆n e + ∆n h = -[8.
The early Si-based EOMs in the 1990s were generally slow (∼MHz) and therefore, not suitable for high-speed optical communications and interconnects [START_REF] Richard | Kramers-Kronig analysis of electro-optical switching in silicon[END_REF][START_REF] Dainesi | CMOS compatible fully integrated Mach-Zehnder interferometer in soi technology[END_REF]. The first Si-based EOM with modulation speed up to 1 GHz, based on MOS capacitor (carrier injection) and MZI, was fabricated by Intel Corporation in 2004 (Figure 1.5 a) [START_REF] Liu | A high-speed silicon optical modulator based on a metal-oxidesemiconductor capacitor[END_REF]. Silicon EOMs based on pn junction (carrier depletion) and MZI were also fabricated with a modulation speed up to 40 GHz (Figure 1.5 b) [START_REF] Liu | High-speed optical modulation based on carrier depletion in a silicon waveguide[END_REF][START_REF] Liao | 40 Gbit/s silicon optical modulator for high-speed applications[END_REF]. To date, above the optical communication and optical interconnect modulation speed requirement (> 10 GHz) silicon EOMs based on pn junction and MZI have been widely investigated [START_REF] Liu | High-speed optical modulation based on carrier depletion in a silicon waveguide[END_REF][START_REF] Liao | 40 Gbit/s silicon optical modulator for high-speed applications[END_REF][START_REF] Samani | Experimental parametric study of 128 Gb/s PAM-4 transmission system using a multi-electrode silicon photonic Mach-Zehnder modulator[END_REF][START_REF] Streshinsky | Low power 50 Gb/s silicon traveling wave Mach-Zehnder modulator near 1300 nm[END_REF]. Nevertheless, the length of MZI structures are usually in the millimeter scale and therefore, with low modulation depth and high energy consumption [START_REF] Streshinsky | Low power 50 Gb/s silicon traveling wave Mach-Zehnder modulator near 1300 nm[END_REF]. High-speed microring resonators were first introduced into Si-based EOM in 2005 [START_REF] Xu | Micrometre-scale silicon electro-optic modulator[END_REF]. The MRR was fabricated with pin doped silicon (carrier injection, Figure 1.5 c) and the footprint of the EOM as low as 12µm × 12 µm because of the MRR structure, and therefore high modulation depth. After this work, MRR-based silicon EOMs using pn structure (carrier depletion) was made with relatively low energy consumption [START_REF] You | 12.5 Gbps optical modulation of silicon racetrack resonator based on carrier-depletion in asymmetric pn diode[END_REF][START_REF] Michael R Watts | Ultralow power silicon microdisk modulators and switches[END_REF][START_REF] Fy Gardes | Highspeed modulation of a compact silicon ring resonator based on a reverse-biased pn diode[END_REF][START_REF] Dong | Low V pp , ultralow-energy, compact, high-speed silicon electro-optic modulator[END_REF]. Lower energy consumption (< 10 fJ/b) Si-based MRR EOMs with high modulation speed (> 10 GHz) are presented since 2011 [START_REF] Michael R Watts | Vertical junction silicon microdisk modulators and switches[END_REF][START_REF] Timurdogan | Vertical junction silicon microdisk modulators at 25Gb/s[END_REF][START_REF] William A Zortman | Low-voltage differentially-signaled modulators[END_REF][START_REF] Li | 25Gb/s 1V-driving CMOS ring modulator with integrated thermal tuning[END_REF][START_REF] Timurdogan | An ultralow power athermal silicon modulator[END_REF]. These EOMs used a vertical junction that maximizes the overlap between the depletion region of pn junction and the optical mode, thereby minimizing size and power consumption [START_REF] Timurdogan | An ultralow power athermal silicon modulator[END_REF].

Chapter 1. Introduction

Moreover, ultra-high modulation speed electro-optical polymer EOMs [START_REF] Chen | Demonstration of 110 GHz electro-optic polymer modulators[END_REF][START_REF] Mallari | 100Gbps EO polymer modulator product and its characterization using a real-time digitizer[END_REF][START_REF] Huang | Broadband modulation performance of 100-GHz EO polymer mzms[END_REF] as well as silicon-polymer hybrid EOM [START_REF] Alloatti | 100 GHz silicon-organic hybrid modulator[END_REF] and plasmonic-polymer hybrid EOM [START_REF] Melikyan | High-speed plasmonic phase modulators[END_REF] have been investigated. Absorption coefficient electrically tuneable indium-tin-oxide (ITO) layer on Si waveguide has been used in EOM and shown excellent performance [START_REF] Volker | Ultra-compact silicon nanophotonic modulator with broadband response[END_REF]. This configuration integrates plasmonic on Si waveguide and therefore efficiently concentrates the electric field of optical modes into a nanometer thin region across the ITO layer. Table 1.3 shows performances of traditional EOMs at telecom frequencies. As shown in Tables 1.1 and 1.3, traditional EOMs have fulfill the requirements for fiber optical communication and inter-chip interconnect in modulation speed and energy consumption nevertheless, the interaction between light and traditional electro-optic materials are still weak and therefore low modulation depth and relatively large footprint are drawbacks for high density subwavelength intra-chip optical interconnect.

Material

Footprint µm [START_REF] Cheng | Mach-zehnder modulators with lithium niobate ridge waveguides fabricated by proton-exchange wet etch and nickel indiffusion[END_REF] 6 000 0.0022 * 0.0014 --LiNbO 3 (MZI) [START_REF] Howerton | Fully packaged, broad-band LiNbO 3 modulator with low drive voltage[END_REF] 41 000 0.0006 * 0.0002 40 -LiNbO 3 (MRR) [START_REF] Guarino | Electro-optically tunable microring resonators in lithium niobate[END_REF] 200×200 0.003

MD dB/µm IL dB/µm f 3dB GHz E bit fJ/b LiNbO 3 (MZI)

Graphene-based photonic EOMs

Graphene is an emerging band-filling effect based electro-optic material due to its electrical tunability. The potential of graphene as an electro-optic material comes from the following features: First, monolayer graphene exhibits a remarkably high light absorption of 2.3% considering graphene is only one atomic layer thick [START_REF] Li | Optical absorption in graphene integrated on silicon waveguides[END_REF], which is important for nonlineal electro-optic effect including modulation. Second, due to its unique linear and gapless band structure, graphene can absorb electromagnetic wave over an ultra-broad spectral ranging from the ultraviolet (UV) to terahertz (THz) region, this provides opportunities for broadband operating especially at telecom windows [START_REF] Stauber | Optical conductivity of graphene in the visible region of the spectrum[END_REF]. Third, graphene has the highest charge carrier mobility of up to 140 000 cm 2 V -1 s -1 [START_REF] Wang | One-dimensional electrical contact to a two-dimensional material[END_REF] at room temperature when graphene is encapsulated on hexagonal boron nitride (hBN), this allows faster modulation speed up to 500 GHz in theory because high mobility of graphene will reduce the RC delay time by a lowered graphene resistance. Fourth, the interband absorption and transmission of graphene can be controlled by tuning the Fermi level (E F ), shifting the applied drive voltage or chemical doping [START_REF] Wang | Gate-variable optical transitions in graphene[END_REF]. Fifth, monolayer graphene couples strongly to light when graphene is located on top of photonic waveguide, this configuration is compatible with silicon optical interconnect system and offers 0.1 dB/µm high modulation depth [START_REF] Liu | A graphene-based broadband optical modulator[END_REF]. Finally, due to its mechanical flexibility and chemical robustness, graphene can be integrated on any substrates and on different photonic devices such as waveguides, fibers, optical micro-cavities, and most importantly, compatible with the mature Si CMOS platform [START_REF] Shiue | Active 2D materials for on-chip nanophotonics and quantum optics[END_REF].

The working principle of graphene-based EOMs rely on the Pauli blocking effect [START_REF] Wang | Gate-variable optical transitions in graphene[END_REF][START_REF] Li | Dirac charge dynamics in graphene by infrared spectroscopy[END_REF] (Figure 1.6). The light absorption of graphene can be caused by inter-band (Figure 1.6 a-c) and intra-band absorption (Figure 1.6 d). Intraband absorption is the transition absorption of free carriers from low energy band to high energy band within conduction band, which is weak at telecommunication wavelength. Stronger inter-band absorption is modulated by Fermi level E F of graphene. In an intrinsic graphene, the Fermi level of graphene is in the middle of conduction band and valence band (Figure 1.6 a). Because of the gapless energy band structure of monolayer graphene, a single photon can be easily absorbed due to the transition of carriers from valence band to conduction band. This kind of inter-band absorption in terms of symmetric inter-band transition can take place over a broad wavelength range from ultraviolet to terahertz. Nevertheless, the inter-band absorption is forbidden when the Fermi level of graphene is high (Figure 1 (Figure 1.6 c) than half of the energy of incidence photon (hω/2). When the Fermi level is higher than hω 0 /2, there are no interband transition since all electron states in the conduction band are filled up. In this case also, the incident photons are not absorbed. When the Fermi level is lower than the transition threshold -hω 0 /2, positive charges accumulates in the valence band and therefore, no electrons are available for interband absorption. In this case, the incident photons propagate through a transparent graphene. For a EOM, the Fermi level is modulated by applied voltage

V d through a capacitor in terms of E F = hv F π 0 d et |V d + V 0 | [77]
, where t and d are respectively the thickness and permittivity of dielectric of capacitor structure, v F = 1.49 × 10 6 m/s [START_REF] Hwang | Fermi velocity engineering in graphene by substrate modification[END_REF] is the Fermi velocity and V 0 is the voltage offset caused by natural doping, which can be set to 0.

The first graphene-based EOM integrated on Si waveguide was presented in 2011 by Zhang's group [START_REF] Liu | A graphene-based broadband optical modulator[END_REF] (Figure 1.7). By integrating graphene on photonic waveguide, graphene strongly couples with the evanescent field of the guided optical mode. This configuration enhances the interaction strength between graphene and light. This strength is only determined by the length of the device and so complete light absorption can be achieved by suitable device length (Figure 1.7 a-b). Although graphene has much drastic change of optical absorption compared to silicon and the length of the EOM was optimized to 40 µm, the modulation depth of such an EOM is still limited to 0.1 dB/µm due to weak overlap between the propagating mode and graphene, which is in same level with Ge and III-V compound semiconductors based EOMs (Table 1.3). Following this concept, similar design are experimentally demonstrated with higher modulation speed and lower energy consumption [START_REF] Hu | Broadband 10 Gb/s operation of graphene electro-absorption modulator on silicon[END_REF]. Light-graphene overlap can be slightly increase by double-layer graphene configuration (Figure 1.8 a), but the modulation depth is still 0.16 dB/µm [START_REF] Liu | Double-layer graphene optical modulator[END_REF]. Later groups reported improved device designs on double layer graphenebased EOMs in both theory [START_REF] Steven | High-speed waveguide-coupled graphene-on-graphene optical modulators[END_REF] and experiment [START_REF] Mohsin | Graphene based low insertion loss electro-absorption modulator on SOI waveguide[END_REF][START_REF] Youngblood | Multifunctional graphene optical modulator and photodetector integrated on silicon waveguides[END_REF][START_REF] Dalir | Athermal broadband graphene optical modulator with 35 GHz speed[END_REF], but only modulation speed was improved [START_REF] Dalir | Athermal broadband graphene optical modulator with 35 GHz speed[END_REF].

The idea to integrate multilayer graphene in horizontal slot waveguide (Figure 1.8 b) have also been widely discussed in theory [START_REF] Kim | A role for graphene in silicon-based semiconductor devices[END_REF][START_REF] Yang | Low-chirp highextinction-ratio modulator based on graphene-silicon waveguide[END_REF][START_REF] Gosciniak | Theoretical investigation of graphene-based photonic modulators[END_REF][START_REF] Shengwei Ye | Electro-absorption optical modulator using dualgraphene-on-graphene configuration[END_REF], nevertheless, this configuration is proved not be efficient in modulating TM modes due to the anisotropic material properties of graphene which the in-plane permittivity ( || ) can be actively tuned by the Fermi level whereas the out-ofplane permittivity ( ⊥ ) does not vary with the external parameters [START_REF] Hao | Ultra-compact optical modulator by graphene induced electro-refraction effect[END_REF][START_REF] Kwon | Discussion of the epsilon-near-zero effect of graphene in a horizontal slot waveguide[END_REF]. Further enhancement of graphene-light interaction is demonstrated by integrating graphene on microring resonator in 2014 by Qiu and their colleagues [START_REF] Qiu | Efficient modulation of 1.55 µm radiation with gated graphene on a silicon microring resonator[END_REF] (Figure 1.8 c). Improved designs are presented with either monolayer graphene [START_REF] Ding | Effective electrooptical modulation with high extinction ratio by a graphene-silicon microring resonator[END_REF] or double-layer graphene [START_REF] Christopher T Phare | Graphene electro-optic modulator with 30 GHz bandwidth[END_REF]. Such configuration present shorter footprints and faster modulation speed. Nevertheless, these devices suffered from narrow working bandwidth (< 0.1 nm [START_REF] Dalir | Athermal broadband graphene optical modulator with 35 GHz speed[END_REF]), high sensitivity to fabrication tolerance and poor thermal stability due to the application of resonator. As LiNbO 3 and Si-based EOMs, Mach-Zehnder interferometer was used in some theoretical designs of graphene-based EOM [START_REF] Hao | Ultra-compact optical modulator by graphene induced electro-refraction effect[END_REF][START_REF] Sorianello | Design optimization of single and double layer graphene phase modulators in SOI[END_REF] (Figure 1.8 d). But several hundreds of micrometers long makes them not be appropriate for graphene-based EOMs.

To date, graphene-based EOMs are more promising than Si-based EOMs. Because of stronger light-active material interaction, the length of graphenebased EOMs have been reduced to the scale of tens micrometer, which are 1.2. State-of-the-art of photonic EOM

Plasmonic-enhanced photonic EOMs

As weak light-active materials interaction, to enhance such interaction is the main challenge of high performance EOMs. Surface plasmon polaritons (SPPs) are coupling between an electromagnetic wave and the collective oscillation of a free electron-plasma in a metal. It is a transverse-magnetic (TM)-polarization surface wave that is excited only at a metal and dielectric interface. The most attractive feature of such SPPs modes supported by plasmonic structures is the ability to strongly confine light well-below the diffraction limit [START_REF] Maier | Plasmonics: fundamentals and applications[END_REF] and therefore, are suitable candidates for the enhancement of the interaction of the mode field with active materials by increasing the overlap of plasmonic modes within active materials. By squeezing the plasmonic modes into the active material region to increase the light-active materials interaction and thus decrease the footprint, several configurations that combine active materials and hybrid plasmonic waveguides have been experimentally demonstrated in nanophotonic and plasmonic scientific communities. For example, the modulation depth of ITO on silicon waveguide Chapter 1. Introduction EOM has been enhanced up to 1 dB/µm by integrating an Au strip plasmonic waveguide on ITO [START_REF] Volker | Ultra-compact silicon nanophotonic modulator with broadband response[END_REF]. In order to decrease the footprint of EOM, plasmonic slot waveguide has been used in a EO polymer-based EOM [START_REF] Melikyan | High-speed plasmonic phase modulators[END_REF]. Only 10 µm long all-plasmonic Mach-Zehnder modulator has been presented [START_REF] Haffner | All-plasmonic machzehnder modulator enabling optical high-speed communication at the microscale[END_REF].

Plasmonic-enhanced graphene-based photonic EOMs

Gold strip and slot plasmonic waveguides are also demonstrated in graphenebased EOMs. Theoretical EOM designs based on Au plasmonic strip integrated on silicon hybrid waveguides have been presented with ultra-high modulation depth about 10 dB/µm [START_REF] Lu | Nanoscale electro-optic modulators based on graphene-slot waveguides[END_REF][START_REF] Gosciniak | Graphene-based waveguide integrated dielectric-loaded plasmonic electro-absorption modulators[END_REF][START_REF] Chen | A broadband optical modulator based on a graphene hybrid plasmonic waveguide[END_REF] (Figure 1.9 a). Nevertheless, modulation depths in these configurations have been proved overestimated by considering graphene as an isotropic 3D material with effective thickness and permittivity. Indeed, the in-plane permittivity of graphene can be adjusted with doping concentration, while the out-of-plane permittivity is a constant with the value of 2.5 based on the dielectric constant of graphite [START_REF] Gao | Excitation of plasmonic waves in graphene by guided-mode resonances[END_REF]. Thus, by simply placing a 2D material on a plasmonic waveguide surface will not lead to increased overlap despite the high field concentration near that metal-dielectric interface. This concept has been discussed in [START_REF] Ansell | Hybrid graphene plasmonic waveguide modulators[END_REF], [START_REF] Shin | Broadband silicon optical modulator using a graphene-integrated hybrid plasmonic waveguide[END_REF], [START_REF] Zhu | Modeling of 2D graphene material for plasmonic hybrid waveguide with enhanced near-infrared modulation[END_REF] and [START_REF] Ma | Two-dimensional material-based mode confinement engineering in electro-optic modulators[END_REF]. In addition, Kwon has theoretically proved that horizontal slot waveguides are not appropriate for graphene-based EOMs for TM modes incidence [START_REF] Kwon | Discussion of the epsilon-near-zero effect of graphene in a horizontal slot waveguide[END_REF] , and Li and his colleagues demonstrated that graphene has a strong electric field components selectivity, which means only the in-plane components, parallel to graphene, strongly interact with graphene [START_REF] Li | Optical absorption in graphene integrated on silicon waveguides[END_REF]. Such selectivity requires a special plasmonic structure design and Au plasmonic strip on silicon hybrid waveguide is no longer suitable for high performances EOMs. Following theoretical designs have shown modulation depths of such a configuration to be from 0.31 dB/µm to 1.23 dB/µm [START_REF] Huang | Waveguide-coupled hybrid plasmonic modulator based on graphene[END_REF][START_REF] Peng | Highly efficient graphene-on-gap modulator by employing the hybrid plasmonic effect[END_REF][START_REF] Sun | Highlyefficient graphene-based silicon hybrid plasmonic waveguide modulator[END_REF][START_REF] Zhu | Modeling of 2D graphene material for plasmonic hybrid waveguide with enhanced near-infrared modulation[END_REF]. Also, it has been demonstrated experimentally a modulation depth of about 0.25 dB/µm [START_REF] Khan | Sub-voltage graphene-plasmon based electro-absorption modulator[END_REF]. By contrast, Au slot plasmonic enhanced EOMs are more promising in high performances EOM (Figure 1.9 b). Nevertheless, theoretical prediction of modulation depth of Au plasmonic slot waveguides integrated in silicon photonics circuits is 0.524 dB/µm [START_REF] Luan | Design and optimization of a graphene modulator based on hybrid plasmonic waveguide with double lowindex slots[END_REF] and the measurement value is 0.13 dB/µm [START_REF] Ding | Efficient electro-optic modulation in low-loss grapheneplasmonic slot waveguides[END_REF] which are still not enough. Moreover, a localized plasmonicenhanced EOM was designed and the ultra-high modulation depth about 10 dB/µm was predicted in theory [START_REF] Huang | Low-energy high-speed plasmonic enhanced modulator using graphene[END_REF], and an Au wedge structure was designed to enhance the transverse SPP fields on the graphene EOM and experimental modulation depth > 0.03 dB/µm was achieved experimentally [START_REF] Ansell | Hybrid graphene plasmonic waveguide modulators[END_REF]. As shown in Table 1.5, high modulation depths in plasmonic enhanced graphene-based EOMs have been confirmed both in theory and experiment, and the length of EOMs has been decreased to several micrometers. These configurations are more competitive than other EOMs (Table 1.3, 1.4). Nevertheless, these results do not meet the requirement of recent optical interconnects (Table 1.1).

Objective of the thesis

Currently, EOMs based on a silicon micro-disk have met the speed and energy requirements of recent optical interconnects, but they still suffer from bulky footprint, reduced thermal stability, and relatively low modulation depth (Table 1.3). The intrinsic low carrier concentration and the weak freecarrier dispersion of silicon lead the improvement of further designs more and more difficult to find. Although graphene-based EOMs promise better performances, footprints are still in tens micrometer scale, modulation depth and modulation speed are still low, and especially the energy consumption is too high (Table 1.4). This is explained by the weak overlap between the light field and the 0.34-nm-thick graphene flake eventhough the waveguide dimensions were optimized. Benefiting from optical mode confinement wellbelow the diffraction limit, surface plasmon-polaritons (SPP s) supported by plasmonic structures are suitable candidates for the enhancement of the interaction of the mode field with graphene. Nevertheless, the experimentally confirmed performances are still far away from the requirements of optical interconnects (Table 1.1, 1.5). The reason is that only the in-plane electric field components of propagating modes strongly interact with graphene. However, electric fields of SPPs supported by metal-dielectric interfaces are predominantly transverse in dielectrics and the perpendicular electric fields do not excite currents into graphene [START_REF] Ansell | Hybrid graphene plasmonic waveguide modulators[END_REF]. Therefore it is difficult to design plasmonic waveguide modes with large in-plane electric field components.

Considering this context, this doctoral research project focused on a improved design of plasmonic enhanced graphene-based EOM that meets all the performance requirements for future optical interconnect applications (Table 1.1). Plasmonic Bloch modes supported by metal nanoparticle chain (MNC) waveguide present high confinement, well below the diffraction limit, and strong in-plane electric field components. These features would leads to subwavelength footprint, high modulation speeds (> 10 GHz ) and low energy consumption (∼ 10 fJ/b). High modulation depth (> 3 dB/µm) and low insertion loss (< 2.5 dB/µm).

Methodology

To this end, following intermediate targets are required:

• Description and electromagnetical modeling of graphene across the telecommunication frequencies.

Organization of the thesis

21

• Simulation of graphene modeled as an anisotropic surface conductivity.

• Understanding mode coupling and modes interaction with graphene.

• Analyses of mode coupling between photonic and plasmonic modes.

• To propose a plasmonic hybrid structure that enhances the interaction of light with graphene.

• Optical simulation and optimization of the designed nanomodulator with the use of FDTD Solution and MODE Solution.

• Electrical simulation of the designed nanomodulator using DEVICE of Lumerical.

• Fabrication and transfer of graphene and other 2D materials on silicon waveguide conducted at Center for Integrated Nanostructure Physics (CINAP), Sungkyunkwan University (SKKU), Korea.

• Patterning of graphene and hBN by electron beam lithography (EBL) and reactive ion etching (RIE).

• Test of graphene quality using Raman spectrometer.

• Integration of well aligned gold nanoparticle chain on silicon waveguide with standard fabrication facilities.

• Fabrication of the electrodes with low graphene contact resistance.

• Far-field transmission spectra measurement of fabricated structure.

• Near-field NSOM measurement.

Organization of the thesis

This manuscript is organized as follows:

In Chapter 1, optical interconnect toward inter and intra-chip communication is presented. As a basic component of optical interconnect system, the parameters and the requirements of EOM for optical interconnect are discussed. We also present the state-of-the-art, which includes traditional EOMs, graphene-based EOMs on Si waveguide, and plasmonic-enhanced graphene-based EOMs. Finally, the problems of current EOMs are summarized and the main objective of this thesis is presented.

Chapter 1. Introduction

In Chapter 2, the electro-optical properties of graphene and its modeling will be firstly presented. Based on the anisotropic surface conductivity model of graphene, we introduce the absorption anisotropy in graphene. With these results, we will numerically discuss the interaction between graphene and photonic modes in order to enhance graphene-light interaction. And then, after introduction of the fabrication, transfer, and patterning of graphene/hBN heterostructures, and its test using Raman spectrometer, we will finally conduct the near-field scanning optical microscopy (NSOM) measurement of graphene/hBN heterostructures integrated on Si waveguide for experimental investigation of the interaction between graphene and photonic modes.

In Chapter 3, we will investigate further enhancement of graphene-light interaction by hybrid plasmonic waveguides. We will firstly introduce the weak in-plane enhancement of standard plasmonic waveguides such as metaldielectric interface, metal-dielectric-metal (MIM) multilayers, nano-strip, and nano-slot plasmonic waveguides. And then we will demonstrate giant inplane enhancement using a metal nanoparticle chain (MNC). A comparative study between all these configurations will be presented. Finally, a secondary application of the in-plane enhancement offered by the MNC plasmonic waveguide towards nanoscale TM-pass polarizer will be numerically and experimentally demonstrated.

In Chapter 4, we will investigate a graphene-based EOM consisting in an MNC hybrid plasmonic waveguide. Plasmonic-enhanced performances including sub-λ footprint, high modulation depth at non-resonant wavelength, relatively small insertion loss, fast modulation speed and low energy consumption due to sub-λ footprint will be introduced. We will also present the impact factors of modulation depth and insertion loss, the equivalent electrical circuits for modulation speed calculation, and current progress on fabrication of the EOM.

Chapter 2

Graphene on silicon photonics as 2D active material for EOM

Introduction

Due to the unique electrical and optical properties, graphene has attracted special attention in electronics (transistors, transparent electrodes due to its ultra-high mobility), photonics (electro-optic modulators, photodetectors, polarizers because of electrical tunability of energy band of graphene), and plasmonics (graphene plasmonic induced by intraband electrons transition).

This chapter focuses on the properties of graphene and the interaction between graphene and light in telecommunication spectrum. We first introduce the properties of graphene including its energy band structure, electric tunable Fermi level, and its ultra-high mobility in different substrates such as SiO 2 and hexagonal boron nitride (hBN). We will then present the modeling of anisotropic surface conductivity of graphene across the telecommunication spectrum. Following this anisotropic concept, we introduce the inplane components selectivity absorption of graphene derived by universal graphene absorption formula, and then we analyze the interaction between graphene and light using in-plane components selectivity when graphene is in free space and on a photonic waveguide. Finally, after the presentation of fabrication and Raman spectroscopy characterization of graphene/hBN heterostructures, the interaction between graphene and photonic modes will be investigated using near-field scanning optical microscopy (NSOM).

Electronic properties of graphene

Graphene, a zero bandgap semimetal, is a flat monolayer of carbon atoms tightly packed into a two-dimensional (2D) hexagonal lattice [START_REF] Ak Geim | The rise of graphene[END_REF] (Figure 24 Chapter 2. Graphene on silicon photonics as 2D active material for EOM 2.1). It is the basic unit of graphite, carbon nanotubes, fullerenes, and other carbon materials. In 2004, Geim and Novoselov produced the first graphene monolayer using micro-mechanical exfoliation and studied the electric field effect of graphene [START_REF] Kostya S Novoselov | Electric field effect in atomically thin carbon films[END_REF]. After that, many devices based on graphene have been demonstrated, for example, optical waveguides [START_REF] Vakil | Transformation optics using graphene[END_REF], electro-optic modulators [START_REF] Liu | A graphene-based broadband optical modulator[END_REF], photodetectors [START_REF] Mueller | Graphene photodetectors for high-speed optical communications[END_REF], polarization controllers [START_REF] Bao | Broadband graphene polarizer[END_REF], ultrafast pulsed lasers [START_REF] Sun | Graphene mode-locked ultrafast laser[END_REF][START_REF] Bao | Atomic-layer graphene as a saturable absorber for ultrafast pulsed lasers[END_REF], light emitter [START_REF] Duck | Bright visible light emission from graphene[END_REF], field effect transistors [START_REF] Schwierz | Graphene transistors[END_REF], ... . These devices are based either on the electrical or on the optical properties of graphene. The unique electrical properties include outstanding ballistic transport properties and longest mean free path at room temperature, distinctive integral and half-integral quantum hall effect, the highest mobility to increase the speed of devices [START_REF] Guo | Graphene doping: a review[END_REF]. Graphene also has interesting optical properties, for instance, tunable interband absorption [START_REF] Wang | Gate-variable optical transitions in graphene[END_REF], saturable absorption at high power levels [START_REF] Bao | Atomic-layer graphene as a saturable absorber for ultrafast pulsed lasers[END_REF], ultra-wide band absorption [START_REF] Rahul Raveendran Nair | Fine structure constant defines visual transparency of graphene[END_REF][START_REF] Mak | Measurement of the optical conductivity of graphene[END_REF], and graphene plasmonics [START_REF] Victor W Brar | Hybrid surface-phonon-plasmon polariton modes in graphene/monolayer h-BN heterostructures[END_REF]. or stacked into 3D graphite [START_REF] Ak Geim | The rise of graphene[END_REF].

Energy band structure and tunability of graphene

The electrical and optical properties of graphene come from its zero bandgap energy band structure. Graphene consists of carbon atoms arranged in hexagonal lattice. The energy band structure of a single hexagonal lattice is shown in Figure 2.2. The electrical and optical properties of graphene is characterized by its linear dispersion relation or Dirac cones at the K and K points of the Brillouin zone. The energy bands in the Brillouin zone can been expressed as [START_REF] Ah Castro Neto | The electronic properties of graphene[END_REF]:

E k (k x , k y ) = ±t 3 + f (k x , k y ) -t f (k x , k y ) (2.1)
where f (k x , k y ) = 2cos( √ 3k y a) + 4cos(

√ 3 2 k y a)cos( 3 2 k x a), a ≈ 1.42
• A is the carbon-carbon distance, t ≈ 2.8 eV is the nearest-neighbor hopping energy (electrons hopping between different sublattices), and t = -0.2 t is the next nearest-neighbor hopping energy (electrons hopping in the same sublattice). In the position near Dirac point, linear energy-momentum dispersion relation given by [START_REF] Ah Castro Neto | The electronic properties of graphene[END_REF]:

E ± (κ) = ±hv f |κ| (2.2)
where κ = k -K, v f is the Fermi velocity. This linear dispersion of graphene energy band is shown in the right inset of The energy band structure of graphene is tunable by doping. For example, adsorbate-induced doping (or surface transfer doping) [START_REF] To Wehling | Molecular doping of graphene[END_REF][START_REF] Wang | N-doping of graphene through electrothermal reactions with ammonia[END_REF], chemical substitutional doping [START_REF] Liu | Chemical doping of graphene[END_REF], and electrical doping [START_REF] Wang | Gate-variable optical transitions in graphene[END_REF]. The adsorbateinduced is achieved by electron exchange between graphene and dopants that adsorb on the surface of graphene. For example, different molecules and atoms such as NO 2 [START_REF] To Wehling | Molecular doping of graphene[END_REF], NH 3 [START_REF] Wang | N-doping of graphene through electrothermal reactions with ammonia[END_REF], Bi [START_REF] Gierz | Atomic hole doping of graphene[END_REF] have been used to induce p-type doping in graphene, while n-type doping can been achieved by using SiC substrate [START_REF] Gierz | Atomic hole doping of graphene[END_REF] (Figure 2.3 a), different metal induced doping has been used in graphene-based pin junction [START_REF] Youngblood | Multifunctional graphene optical modulator and photodetector integrated on silicon waveguides[END_REF]. Chemical substitutional doping is to substitute carbon atoms in the honeycomb lattice of graphene by other atoms with different number of valence electrons such as nitrogen and boron [START_REF] Wei | Synthesis of n-doped graphene by chemical vapor deposition and its electrical properties[END_REF][START_REF] Ls Panchakarla | Synthesis, structure, and properties of boron-and nitrogen-doped graphene[END_REF]. Electrical doping is the most importing doping method due to the application in graphene-based field effect transistors and electro-optic devices such as electro-optic modulators and photodetectors. It is achieved by changing drive voltage applied in graphene [START_REF] Kostya S Novoselov | Electric field effect in atomically thin carbon films[END_REF][START_REF] Das | Monitoring dopants by raman scattering in an electrochemically topgated graphene transistor[END_REF]. By changing drive voltage, the Fermi level of pristine graphene, which depends on carrier concentration, can be finely modified from conduction band to valence band, and then p-typed and n-typed doping can be obtained (Figure 2.3 b). This special behavior is due to its low density of states near the Dirac point and 2D structure that confines electrons in one atomic layer, which causes the Fermi level shift significantly with change of carrier density [START_REF] Ak Geim | The rise of graphene[END_REF]. Electrical doping provides the opportunity for the development of electro-optic devices based on graphene.

Note that adsorbate-induced doping and chemical substitutional doping can be used to tune the Fermi level and open the bandgap of graphene. While electrical doping, the Fermi level of graphene can be tuned by the polarity and value of applied drive voltage, but the bandgap cannot be opened.

Anisotropic surface conductivity model

Both isotropic [START_REF] Lu | Nanoscale electro-optic modulators based on graphene-slot waveguides[END_REF][START_REF] Gosciniak | Graphene-based waveguide integrated dielectric-loaded plasmonic electro-absorption modulators[END_REF][START_REF] Chen | A broadband optical modulator based on a graphene hybrid plasmonic waveguide[END_REF][START_REF] Yang | Low-chirp highextinction-ratio modulator based on graphene-silicon waveguide[END_REF][START_REF] Gosciniak | Theoretical investigation of graphene-based photonic modulators[END_REF] and anisotropic [START_REF] Wang | Graphene, hexagonal boron nitride, and their heterostructures: properties and applications[END_REF][START_REF] Hao | Ultra-compact optical modulator by graphene induced electro-refraction effect[END_REF][START_REF] Huang | Low-energy high-speed plasmonic enhanced modulator using graphene[END_REF] models have been used in nanophotonic and plasmonic communities, this leads to a confusion in the investigation. This confusion have been discussed in [START_REF] Zhu | Modeling of 2D graphene material for plasmonic hybrid waveguide with enhanced near-infrared modulation[END_REF][START_REF] Shin | Broadband silicon optical modulator using a graphene-integrated hybrid plasmonic waveguide[END_REF]. Indeed, as a 2D material, graphene should be modeled as an anisotropic material according to the discussions in [START_REF] Ma | Two-dimensional material-based mode confinement engineering in electro-optic modulators[END_REF] and [START_REF] Ansell | Hybrid graphene plasmonic waveguide modulators[END_REF]. Physically, 2D graphene is characterized by surface conductivity, which consists of interband σ inter and intraband σ intra terms, and can be expressed as a tensor [START_REF] Chatzidimitriou | Rigorous calculation of nonlinear parameters in graphenecomprising waveguides[END_REF]:

σ s =    σ xx σ xy 0 σ yx σ yy 0 0 0 0    , (2.3)
where σ xy = σ yx ≈ 0 in absence of an external magnetic, and σ xx = σ yy = σ. The in-plane conductivity σ can be deduced from Kubo formalism by [START_REF] Hanson | Dyadic green's functions and guided surface waves for a surface conductivity model of graphene[END_REF][START_REF] Falkovsky | Optical properties of graphene[END_REF]:

σ(ω, E f , Γ, T) = σ inter (ω, E f , Γ, T) + σ intra (ω, E f , Γ, T) (2.4) σ inter (ω, E f , Γ, T) = ie 2 (ω + i2Γ) πh 2 ∞ 0 f d (-ξ) -f d (ξ) (ω + i2Γ) 2 -2ξ h 2 dξ (2.5) σ intra (ω, E f , Γ, T) = - ie 2 πh 2 (ω + i2Γ) ∞ 0 ξ ∂ f d (ξ) ∂ξ - ∂ f d (-ξ) ∂ξ dξ (2.6)
where ω is the frequency, E f is Fermi level, h is the reduced Plank constant,

f d (ξ) = {exp [(ξ -E f )/k B T] + 1} -1 is the Fermi-Dirac distribution, k B is
Boltzmann's constant, and temperature T = 300 K. The scattering rate Γ can be calculated by [START_REF] Thongrattanasiri | Complete optical absorption in periodically patterned graphene[END_REF]

Γ = ehv 2 f /(2µE f ) (2.7)
where v f , ranging from 0.85 ×10 6 m/s (graphene on metal) to 3 ×10 6 m/s (suspended graphene) [START_REF] Hwang | Fermi velocity engineering in graphene by substrate modification[END_REF], is the substrate modified Fermi velocity. µ is the substrate-dependent mobility of graphene. However, the effect of the substrate on the scattering rate is weak considering the low mobility of graphene on SiO 2 from 4 400 to 17 000 cm 2 V -1 s -1 (Figure 2.4 a). As a result, the substrate of graphene as well as the scattering rate almost have not effect on the conductivity of graphene (Figure 2.4 b, c). This situation is different across the mid-and far-infrared spectra [START_REF] Mak | Measurement of the optical conductivity of graphene[END_REF]. Moreover, for graphene plasmonic from the mid-to the far-infrared spectrum, higher carrier mobility translates to plasmon quality factors [START_REF] Woessner | Highly confined lowloss plasmons in graphene-boron nitride heterostructures[END_REF]. In the near-infrared range, due to the inplane electrical field selectivity of graphene, the main effect of substrate of graphene is degraded absorption because of uneven of some substrates such as SiO 2 and SiC. The Fermi level E f is tunable by electrical doping, which is given by [77]

E f = hv f π 0 d et |V d + V 0 | (2.8)
where t and d are respectively the thickness and permittivity of dielectric of capacitor structure which used to apply a drive voltage V d , V 0 is the voltage offset caused by natural doping, which can be set to be 0. The electrical tunable Fermi level calculated by Equation 2. By changing drive voltage and hence Fermi level, a drastic modulation of graphene conductivity from 1.27 µS to 60.85 µS can be achieved at 1.55 µm due to low density of states of graphene near the Dirac point [START_REF] Ak Geim | The rise of graphene[END_REF]. The contribution of interband and intraband terms are clearly shown in Figure 2.5 (b). As we see, the real part of conductivity, related to the light absorption by graphene, only depends on the interband term when the Fermi level is below half of the photon energy 0.4 eV at 1.55 µm. When the Fermi level is higher than 0.4 eV, interband transitions are diminished due to Pauli blocking, in this case the real part of conductivity is determined mainly by the intraband term. However, both interband and intraband terms contribute to the imaginary part of conductivity. In addition, another alternative approach of graphene modeling of anisotropic bulk permittivity model is available in Appendix A.

Optical properties of graphene 2.3.1 Graphene absorption derived from Fresnel formula

According to the Fresnel formula, for normal incidence light onto undoped monolayer graphene (Figure 2.6 a), frequency independent absorption (A) is solely determined by the universal conductivity of graphene (a constant conductivity over a wide spectral range from visible to infrared (Figure 2.6 b) [START_REF] Ab Kuzmenko | Universal optical conductance of graphite[END_REF]:

A = 1 -T -R = 1 -(1 + 0.5πα) -2 ≈ 2.3% , (2.9)
where α = σ 0 /(π 0 c) [START_REF] Rahul Raveendran Nair | Fine structure constant defines visual transparency of graphene[END_REF], σ 0 is conductivity of undoped graphene (E f = 0 eV), c is the speed of light. T = (1 + 0.5πα) -2 is transmittance. The reflectance is weak and written as

R = [σ 0 /(2 0 c)] 2 T = 1.3 × 10 -4 [120]
. So the broadband absorption from visible to infrared is about 2.3%. However, doped graphene (E f = 0 eV) has different behavior due to the different real part of conductivity and the imaginary part of conductivity is no longer 0 (Figure 2.6 b). We plotted the absorption coefficient of graphene as a function of wavelength (Figure 2.6 c) and Fermi level (Figure 2.6 d) in near-infrared using Equation 2.9, which is marked as method i. As shown in Figure 2.6 (c), the absorption coefficient of undoped graphene is constant in near-infrared range, but decreases to 0.06% at longer wavelength in doped graphene. According to Figure 2.6 d, graphene is transparent only for Fermi energy levels of graphene above the E f = hω/2 = 0.4 eV at 1.55 µm.

Universal graphene absorption formula

Equation 2.9 is only used for normal incident light onto isolated graphene.

In other cases, an universal graphene absorption formula is required. According to Ohm's law in its differential form, the surface current density J S in graphene is [START_REF] Chatzidimitriou | Rigorous calculation of nonlinear parameters in graphenecomprising waveguides[END_REF]:

J S = σE =    σ xx 0 0 0 σ yy 0 0 0 0       E x E y E z    . (2.10)
The total rate of doing work by the electric field at the surface of graphene S is [142]

Q grp = 1 2 S J S • EdS = 1 2 S σ|E | 2 dS , (2.11)
which implies that only the electric field components parallel to graphene contribute to absorption. Here, E is the in-plane electric field components within graphene layer. In order to enhance graphene absorption, special structures that can strongly confine in-plane electric field components in graphene are required by this in-plane components selectivity of graphene absorption. 32 Chapter 2. Graphene on silicon photonics as 2D active material for EOM

97.7% (a) (b) (c) E f =0eV: 99.9% E f =0.5eV: (d)
The intensity of input light is given by

I i = 0 cE i E * i /2 [69]
, where E i is the electric field of incident light. The absorption coefficient can be taken as:

A = Q grp I i = σE E * 0 cE i E * i .
(2.12)

We calculated graphene absorption in Figure 2.6 (c) and (d) using this universal graphene absorption formula (marked as method ii). The result agrees with Equation 2.9. In the next part, we calculate the light absorption of graphene on silicon waveguide configuration using the universal graphene absorption formula 2.12, which is more powerful than Equation 2.9.

EO properties of graphene on hBN

Hexagonal boron nitride (hBN) has layered structure similar to the graphene lattice constant and similar characteristics, and is sometimes named as white graphene. Its hexagonal lattice consist of B atoms and N atoms alternately in a 2D plane (Figure 2.7 a). Monolayer hBN combines each other by weak van der Waals forces. It is a direct-bandgap semiconductor with a bandgap of 5.9 eV [START_REF] Watanabe | Direct-bandgap properties and evidence for ultraviolet lasing of hexagonal boron nitride single crystal[END_REF]. It is an excellent substrate material for graphene-based devices because it is atomically flat and free of dangling bonds and surface charge traps [START_REF] Cory R Dean | Boron nitride substrates for high-quality graphene electronics[END_REF][START_REF] Xue | Scanning tunnelling microscopy and spectroscopy of ultra-flat graphene on hexagonal boron nitride[END_REF]. SiO 2 is the most common substrate of graphene, but on the SiO 2 surface there are usually impurities and has high roughness, which can cause the scattering of charge or charge traps. These impurity-induced charge traps tend to cause the graphene to electronically break up into electron-and holedoped regions at low charge density, which both limit device performance and make the Dirac point physics inaccessible [START_REF] Chen | Charged-impurity scattering in graphene[END_REF]. For example, low mobility in the range from 4 400 to 17 000 cm 2 V -1 s -1 [START_REF] Hong | Quantum scattering time and its implications on scattering sources in graphene[END_REF][START_REF] Hoang Tien | Study of graphene-based 2D-heterostructure device fabricated by all-dry transfer process[END_REF] have been reported for graphene on SiO 2 . One method to eliminate substrate interactions is to use suspended graphene due to the drastic improvement in mobility as high as 200 000 cm 2 V -1 s -1 [START_REF] Cory R Dean | Boron nitride substrates for high-quality graphene electronics[END_REF]. However, the fabrication of suspended graphene is difficult and leads to undesired strain [START_REF] Bao | Controlled ripple texturing of suspended graphene and ultrathin graphite membranes[END_REF].

Graphene devices have taken advantage from the optical and electrical properties of graphene/hBN heterostructures stacked by van der Waals forces [START_REF] Wang | Graphene, hexagonal boron nitride, and their heterostructures: properties and applications[END_REF] (Figure 2.7 b). By using hBN, the carrier mobilities in exfoliated graphene on hBN reaches 40 000 cm 2 V -1 s -1 at room temperature [START_REF] Cory R Dean | Boron nitride substrates for high-quality graphene electronics[END_REF]. Carrier mobilities of up to 37 000 cm 2 V -1 s -1 were measured with chemical vapor deposited (CVD) graphene on hBN [START_REF] Gannett | Boron nitride substrates for high mobility chemical vapor deposited graphene[END_REF]. Lower carrier mobility in CVD graphene is attributed to the transfer process but advanced transfer technique allows mobilities as high as 350 000 cm 2 V -1 s -1 [START_REF] Banszerus | Ultrahigh-mobility graphene devices from chemical vapor deposition on reusable copper[END_REF]. Relative high mobility are measured in encapsulated graphene on hBN from 100 000 cm 2 V -1 s -1 [START_REF] Hoang Tien | Study of graphene-based 2D-heterostructure device fabricated by all-dry transfer process[END_REF] to 140 000 cm 2 V -1 s -1 [START_REF] Cory R Dean | Boron nitride substrates for high-quality graphene electronics[END_REF][START_REF] Wang | One-dimensional electrical contact to a two-dimensional material[END_REF], which are near the mobility of suspended graphene at room temperature (200 000 cm 2 V -1 s -1 [START_REF] Kirill I Bolotin | Ultrahigh electron mobility in suspended graphene[END_REF][START_REF] Du | Approaching ballistic transport in suspended graphene[END_REF]). High mobility of graphene on different substrates are summarized in Table 2.1. Note that these values are maximum mobility on different substrates fabricated by optimized process, the fabrication of high mobility graphene is still a challenge.

In addition to maintain high carrier mobility in graphene, hBN is an exceptional gate insulator for graphene as it possesses low permittivity of 3.92 34 Chapter 2. Graphene on silicon photonics as 2D active material for EOM [77], high breakdown electric field strength (0.5 V per layer) [START_REF] Ks Novoselov | 2d materials and van der waals heterostructures[END_REF]. The encapsulation of graphene in hBN improves the potential for the development of high-performance interconnects compare to graphene/hBN heterostructure [START_REF] Jain | Graphene interconnects fully encapsulated in layered insulator hexagonal boron nitride[END_REF]. Indeed, the electrical properties of encapsulated graphene devices remain unchanged whereas that of graphene/hBN devices were influenced by the external environment [START_REF] Hoang Tien | Study of graphene-based 2D-heterostructure device fabricated by all-dry transfer process[END_REF]. Moreover, edge-contacts can be fabricated in encapsulated graphene on hBN, which allows improved contacts with ultralow contact resistance [START_REF] Wang | One-dimensional electrical contact to a twodimensional material[END_REF][START_REF] Shiue | High-responsivity graphene-boron nitride photodetector and autocorrelator in a silicon photonic integrated circuit[END_REF].

Owing to excellent electrical performances of encapsulated graphene, the integration of encapsulated graphene on Si photonics platform has allowed the demonstration of ultrafast, high-responsivity electro-optic devices such as photodetector [START_REF] Shiue | High-responsivity graphene-boron nitride photodetector and autocorrelator in a silicon photonic integrated circuit[END_REF] and electro-optic modulator [START_REF] Gao | High-speed electro-optic modulator integrated with graphene-boron nitride heterostructure and photonic crystal nanocavity[END_REF] (Figure 2.7 c). In addition, graphene plasmon polaritons, hBN phonon polaritons [START_REF] Li | Hyperbolic phonon-polaritons in boron nitride for near-field optical imaging and focusing[END_REF], and hybrid plasmon-phonon polaritons supported by graphene/hBN heterostructures have been investigated. It was demonstrated for instance, with the use of near-field microscopy images, that graphene plasmons in graphene encapsulated between two layers of hBN present low damping and high field confinement relative to plasmons in bare graphene [START_REF] Woessner | Highly confined lowloss plasmons in graphene-boron nitride heterostructures[END_REF]. Across the midand far-infrared spectra, plasmon polaritons in graphene and phonon polaritons in hBN couple, leading to plasmon-phonon polariton modes [START_REF] Victor W Brar | Hybrid surface-phonon-plasmon polariton modes in graphene/monolayer h-BN heterostructures[END_REF]. Tunable plasmon-phonon polaritons in graphene/hBN/graphene heterostructure were also demonstrated in the far-field with Fourier transform infrared spectroscopy [START_REF] Jia | Tunable plasmon-phonon polaritons in layered graphene-hexagonal boron nitride heterostructures[END_REF] and in the near-field via scattering-type scanning nearfield optical microscopy [START_REF] Dai | Graphene on hexagonal boron nitride as a tunable hyperbolic metamaterial[END_REF].

Thanks to the improved electrical properties of graphene/hBN and encapsulated graphene on hBN heterostructures, it has been widely discussed in different frequency spectra. In the visible spectrum, encapsulated graphene on hBN was used for incandescent light emission at high temperatures [START_REF] Seok-Kyun Son | Graphene hot-electron light bulb: incandescence from hBN-encapsulated graphene in air[END_REF]. From mid-infrared to far-infrared spectra, graphene plasmon polaritons, hBN phonon polaritons, and hybrid plasmon-phonon polaritons supported by graphene/hBN heterostructures have been investigated. Across this frequency range, the electrons transition in graphene is primarily intraband transition [START_REF] Jiang | Nonlinear plasmonic dispersion and coupling analysis in the symmetric graphene sheets waveguide[END_REF]. The intraband transition leads to metallic Drude-like conductivity, make it able to support surface plasmon. Nevertheless, the optical properties of graphene/hBN and encapsulated graphene on hBN heterostructures across the telecom spectrum has not been completely discussed yet. In this range, interband transition tends to dominate [START_REF] Steven | Waveguide-coupled graphene optoelectronics[END_REF] and hBN does not support phonon polaritons. In the next section, the interaction between light and graphene or graphene-hBN heterostructures in near-infrared range is discussed.

Integration of graphene/hBN on Si waveguides

Graphene/hBN van der Waals (vdW) heterostructures integrated on photonic waveguide configurations have been widely investigated in graphenebased devices such as optical waveguides [START_REF] Vakil | Transformation optics using graphene[END_REF], electro-optic modulators [START_REF] Liu | A graphene-based broadband optical modulator[END_REF], photodetectors [START_REF] Shiue | High-responsivity graphene-boron nitride photodetector and autocorrelator in a silicon photonic integrated circuit[END_REF], and polarization controllers [START_REF] Bao | Broadband graphene polarizer[END_REF]. By integrating graphene/ hBN vdW heterostructures on a photonic waveguide, the lightgraphene interaction can be enhanced by improved overlap of light on 2D graphene and so complete absorption can be achieved.

In the following subsection, we discuss the interaction between photonic guided modes and graphene/hBN vdW heterostructures in telecommunication wavelengths based on in-plane electric-field selectivity of graphene. 36 Chapter 2. Graphene on silicon photonics as 2D active material for EOM

Configuration and supported photonic modes

A general graphene integrated on photonic waveguide configuration is shown in Figure 2.8. Here, we consider integrated graphene/hBN van der Waals heterostructures on top of a 500 nm by 220 nm silicon photonic waveguide. hBN is used to improve the electrical and optical performances of graphene. The thickness of the hBN layer is set as 10 nm. I calculated the dispersion curve of graphene/hBN on Si waveguide configuration by MODE solution of Lumerical in Figure 2.8. Only fundamental TE and TM modes are supported at 1.55 µm (Figure 2.9 a). As a 2D material, graphene does not change the real part of effective index (Figure 2.9 a), even if the Fermi level of graphene is changed (Figure 2.9 c). But graphene changes the loss of the device. Due to the electric tunability of graphene conductivity, the loss of graphene/hBN on Si waveguide configuration is tunable by Fermi level (Figure 2.9 c). The loss also depends on the photonic guided mode because of different in-plane electric field intensity of different modes. Here, loss = 1 -|E t | 2 /|E i | 2 = 2k 0 Im(n eff ), is the power absorbed by graphene per unit length of the waveguide normalized to the waveguide mode power. By the universal graphene absorption Equation (2.12), both the real part of graphene conductivity and electric field components parallel to graphene contribute to absorption, this is confirmed again by Figure 2.9 (c).

In-plane confinement and modes absorption of graphene/ hBN on a Si waveguide

The electrical tunability of graphene allows the absorption modulation, which is basic principle of electro-optic modulators. Nevertheless, high modulation depth of EOMs, which comes from high light absorption in off state (E f < 0.4 eV according to Figure 2.9 c), depends on the in-plane electric field intensity.

To quantify the in-plane electric field confinement in graphene, we define the confinement ratio γ = ρ /ρ total , where ρ and ρ total are the in-plane and total electric field densities, respectively. In Figure 2.10 we calculated in-plane electric field density and in-plane confinement ratio under fundamental TE and TM modes incidence, respectively. For TE input, the highest in-plane electric field density occurs in the middle of Si waveguide. By contrast, when the input light is fundamental TM mode, the highest in-plane electric field density move to the upper and lower boundary of Si waveguide. Considering graphene/hBN on a Si waveguide is lossless when E f > 0.4 eV (Figure 2.9 c), the loss in Figure 2.11, which is mode absorption by graphene, correspond to the modulation depth of graphene. Hence, Figure 2.11 reveals low theoretical limit of modulation depth of graphene-based EOMs on silicon photonic waveguide of 0.22 dB/µm for TE 0 mode and 0.12 dB/µm for TM 0 mode, match experimental results (Table 1.4). Low in-plane electric field density and then low theoretical limit of modulation depth of graphene on Si photonic waveguide EOMs are reason of weak overlap of in-plane component of photonic modes into one-atom-thick graphene. The integrated graphene-hBN heterostructure on silicon photonics was fabricated by successive stacking of exfoliated hBN and CVD graphene (Figure 2.12). An hBN flake is mechanically exfoliated onto a Si substrate with 285 nm thick SiO 2 . After exfoliation, the first hBN flake is selected and transferred on top of a selected Si waveguide of a Si chip by dry transfer technique (Figure 2.12 a). Graphene was synthesized by CVD on a 100-µm-thick copper foil from Nilaco, 99.96% [START_REF] Van Luan | Seamless stitching of graphene domains on polished copper (111) foil[END_REF]. The foil was annealed at 1075 • C for 2 h under a H 2 gas flow rate of 500 sccm 1 and Ar flow rate of 1000 sccm. During CVD growth, the H 2 was reduced to 50 sccm and 3 sccm of CH 4 was injected. PMMA was spun on top of the Cu foil then an aqueous HCl solution was used to etch the Cu for 1h. The graphene on PMMA was rinsed in deionized water several times and deposited on top of the Si chip (Figure 2.12 b). The sample was soaked in acetone to dissolve the PMMA, then rinsed in IPA and dried with air. A second hBN flake is selected from the SiO 2 /Si substrate, aligned by optical microscopy and dry transferred on top of the first hBN flake (Figure 2.12 c).

(a) (b) (c) (f) (e) (d) (g) (h) (i)
FIGURE 2.13: The process of etching graphene.

To remove excess graphene and hBN from the Si chip, a pattern is defined around the graphene-hBN heterostructure by electron beam lithography (EBL). Before EBL, we spin coat a 200 nm thick positive EBL resist polymethyl methyl acrylate (PMMA) on the Si chip using a 30 g/L PMMA solution (dissolved in methyl isobutyl ketone MIBK solvent) at 4000 rpm speed and 3000 rpm/s acceleration for 30 s and followed by annealing 3 h at 160 • C (Figure 2.13 b). And then we spin coat a conductive polymer layer (espacer 300Z) over the PMMA layer using 6000 rpm speed and 4000 rpm/s acceleration for 60 s. EBL is performed using the Raith e-Line EBL system at 20 kV with a dose of 300 µC/cm 2 and a working distance of 7 mm (Figure 2.13 c). We then develop the sample by successively immersing the sample in water for 60 s to remove the conductive polymer, in 3:1 MIBK/IPA solution 60 s for development and in IPA (isopropyl alcohol) 15 s to rinse MIBK (Figure 2.13 d). Once EBL is done, 5-nm-thick Al was the evaporated on the patterned sample using PLASSYS MEB 400 evaporator with a speed of 0.15 nm/s at ∼ 10 -6 mbar (Figure 2.13 e) and lift-off as carry out with acetone 4 ∼ 12 h (Figure 2.13 f). The pattern was transferred to graphene with the use of reactive ion etching (RIE). To etch graphene only, oxygen plasma with a flow rate of 10 sccm is used for 1 min (Figure 2.13 (g), Table 2.2). In the case of etching graphene-hBN heterostructure, a mix gas of 4 sccm O 2 and 30 sccm CHF 3 is used. The etch time depends on the thickness of hBN as a rate of 11.2 nm/min. After etching process, the aluminum mask was removed by 20% HCl solution for 4 minutes without effects in the quality of graphene and hBN (Figure 2.13 h). Finally, a patterned hBN-graphene-hBN heterostructure integrated on Si waveguide was obtained (Figure 2

Raman spectroscopy

Raman spectroscopy is one of the most powerful and non-invasive techniques for characterizing graphene, hBN and other carbon based materials. This technique gives an insight into defect [START_REF] Graf | Spatially resolved raman spectroscopy of single-and few-layer graphene[END_REF][START_REF] Cançado | Quantifying defects in graphene via raman spectroscopy at different excitation energies[END_REF], the number of layers [START_REF] Ferrari | Raman spectrum of graphene and graphene layers[END_REF], doping level and doping type of graphene [START_REF] Yan | Electric field effect tuning of electron-phonon coupling in graphene[END_REF][START_REF] Pisana | Breakdown of the adiabatic born-oppenheimer approximation in graphene[END_REF][START_REF] Stampfer | Raman imaging of doping domains in graphene on si o 2[END_REF][START_REF] Das | Monitoring dopants by raman scattering in an electrochemically topgated graphene transistor[END_REF], the defect and thickness of hBN [START_REF] Roman V Gorbachev | Hunting for monolayer boron nitride: optical and raman signatures[END_REF]. To test the properties of our sample, far-field Raman spectroscopy is performed using a He:Ne laser operated at a wavelength of 632.8 nm on graphene with and without hBN layers. The spot size of the laser was about 8 µm using a 50× objective lens with numerical aperture 0.8, which collected the backscattered Raman signal. A Raman spectroscopy on the different integrated graphene/hBN vdW heterostructures on silicon photonic waveguide is shown in Figure 2.15. The Raman spectra demonstrate the quality of a monolayer graphene in different vdW heterostructures (Figure 2.15 a) [START_REF] Ferrari | Raman spectrum of graphene and graphene layers[END_REF]. Two prominent Raman peaks are measured, the G and 2D peaks at 1587 cm -1 (Figure 2.15 b) and 2648 cm -1 (Figure 2.15 c), respectively. The D peak, which indicates defect of graphene, occurs only in region ii, where graphene is on SiO 2 [START_REF] Cançado | Quantifying defects in graphene via raman spectroscopy at different excitation energies[END_REF][START_REF] Graf | Spatially resolved raman spectroscopy of single-and few-layer graphene[END_REF]. hBN is identified by the peak at 1364 cm -1 [START_REF] Roman V Gorbachev | Hunting for monolayer boron nitride: optical and raman signatures[END_REF]. The G peak position is at around 2 cm -1 lower in region iv compared to that in regions ii and iii (Figure 2.15 b), which is consistent with previous report [START_REF] Chattrakun | Gate dependent Raman spectroscopy of graphene on hexagonal boron nitride[END_REF]. The full width at half maximum (FWHM) of G peak are respectively 20 cm -1 , 19 cm -1 , and 18 cm -1 in regions ii, iii, and iv, which is a result of higher charge density in region iv [START_REF] Pisana | Breakdown of the adiabatic born-oppenheimer approximation in graphene[END_REF]. The intensity ratio I 2D /I G in regions ii, iii, and iv are respectively 2.1, 2.8, 3.8. These values indicate higher graphene quality in region iv than that in other regions due to low surface roughness and a reduction of changed impurities of hBN. Investigations of Raman properties of graphene on hBN showed that lower charge fluctuations and scattering rate are expected on hBN as compared to graphene on SiO 2 [START_REF] Wang | Negligible environmental sensitivity of graphene in a hexagonal boron nitride/graphene/h-bn sandwich structure[END_REF][START_REF] Chattrakun | Gate dependent Raman spectroscopy of graphene on hexagonal boron nitride[END_REF].

Near-field Scanning Optical Microscopy

In this part, I experimentally investigate the optical properties of different graphene/hBN heterostructures on silicon waveguide at communication frequencies on the NSOM setup at L2n/ICD UTT, which allow us to measure in-plane electric field and its attenuation, as well as modes excited. To probe the optical near field on the integrated vdW heterostructures on Si photonics, a transmission-based NSOM was used with an atomic force microscope (AFM) tip made of Si and with a nominal tip radius of about 10 nm (Figure 2.16 a). An avalanche photodetector is used to collect the transmitted optical signal through the Si waveguide, which is proportional to the local intensity at the tip [START_REF] Robinson | Imaging highly confined modes in sub-micron scale silicon waveguides using transmissionbased near-field scanning optical microscopy[END_REF][START_REF] Fevrier | Giant coupling effect between metal nanoparticle chain and optical waveguide[END_REF][START_REF] Salas-Montiel | Quantitative analysis and near-field observation of strong coupling between plasmonic nanogap and silicon waveguides[END_REF]. The sample is illuminated with the fundamental quasi-TE photonic mode of the Si waveguide at a wavelength of 1550 nm. The NSOM measurement simultaneously records the topography image.

Results and discussion

The experimental NSOM results are plotted in Figure 2.17. The fundamental photonic TE mode at λ = 1550 nm excited from the input port propagates through the different integrated vdW heterostructures. First, the mode propagates without interaction with any vdW heterostructure (region i). Then it interacts with graphene (region ii), hBN/graphene (region iii), and hBN/graphene/hBN heterostructures (region iv). The topography (Figure 2.17 b) and NSOM images (Figure 2.17 c) are obtained simultaneously. Besides hBN, graphene can be clearly seen on the topography image but the thickness cannot be measured because of the lack of precision of the system. The thickness of the top and bottom hBN layers are measured to be 30 ± 5 nm. The difference of NSOM signal in different regions come from different detection distance from NSOM tip to Si waveguide in different regions. The intensity of the NSOM signal decreases at every region along the silicon waveguide that implies different mode attenuation in each region (Figure 2.17 d). To confirm the nature of the NSOM signal decrease in different regions, I performed 3D finite-difference time-domain (FDTD) numerical simulations in order to calculate the electric field distribution of the sample. Four field monitors were placed at 20 nm above the surface of each vdW heterostructure. The input port of the Si waveguide is illuminated with its fundamental TE mode at 1500 nm by a mode source. The hBN layers were modeled with in-plane refractive index of 1.98 [START_REF] Ansell | Hybrid graphene plasmonic waveguide modulators[END_REF], graphene was modeled according to Equation 2.4, 2.5, and 2.6. Different scattering rates were used to model graphene on top of hBN and SiO 2 . The Fermi velocity was set to v f = 1.0 × 10 6 m/s and the Fermi level for graphene was set to 0.2 eV.

The numerical result for the integrated heterostructures on Si waveguide clearly shows that our measured NSOM signal in Figure 2.17 is mainly inplane electric field component due to weak out-plane component of fundamental TE mode (Figure 2.18 a, b). To further analyze the results, the integrated electric energy density with ρ and without ρ 0 graphene are calculated and plotted in Figure 2.18 (c). Comparing with the case without graphene, weak but clear decrease of in-plane density in every regions are obtain, in agreement with the NSOM image (Figure 2.17 d). According to universal graphene absorption formula (Section 2.3.2), the difference of in-plane density matches a difference of the optical absorption of graphene in different vdW heterostructure regions. This is confirmed by normalized graphene absorption Q S of different regions, which is quantized by Equation 2.11 and plotted in Figure 2.18 (c).

In addition, NSOM signal also give us modes information (Appendix B). We also present TM 0 excited NSOM signal and its numerical analyses of graphene/hBN heterostructures in Appendix B.

Conclusion

Because of weak overlap of in-plane component of photonic modes into oneatom-thick graphene, in this chapter I theoretically, numerically, and experimentally demonstrated intrinsic weak interaction between graphene and photonic modes based on derived in-plane components selectivity of graphene absorption. Numerical results revealed theoretical limit of modulation depth of the graphene-based EOMs integrated on silicon photonic waveguide, which are respectively 0.22 dB/µm with TE 0 mode illumination and 0.12 dB/µm for TM 0 mode input. To achieve 3 dB modulation strength, they require 13.6 µm or 25 µm device length, not suitable for high-density optical interconnect. Finally, with a fabricated graphene-hBN heterostructure on Si waveguide, weak graphene-photonic mode interaction was confirmed by NSOM measurement of the sample, where in-plane electric field intensity and its attenuation are measured.

To reduce the footprint of graphene-based EOM, breaking diffraction limit plasmonic structures would be a potential candidate in the enhancement of graphene-light interaction, which will be discussed in next chapter.

Chapter 3

Plasmonic Bloch modes for the enhancement of graphene properties

Introduction

As we discussed in Chapter 2, the strength of interaction between graphene and light depends on the in-plane electric field overlapped in graphene layer. Graphene on photonic waveguide configuration can enhance the contact area between graphene and light, however, such a wavelength scale confinement of photonic modes leads to weak overlap of electric field in the atomic-thick graphene. Plasmonic structures provides a way to shrink the mode size down to a few cubic nanometers for enhanced interaction with graphene nevertheless, there is a serious challenge to obtain high in-plane components in graphene layer due to the intrinsic transverse magnetic field of surface plasmon-polariton (SPP) modes supported by plasmonic waveguides.

In this chapter, we focus on in-plane enhancement of graphene-light interaction using plasmonic waveguides. We describe first plasmonic modes supported by standard plasmonic waveguide such as metal-dielectric interface, metal-dielectric-metal (MIM), nano-strip, and nano-slot plasmonic waveguides. We will then introduce the periodic metal nanoparticles chain (MNC) waveguide and compare the enhancement of the in-plane confinement. As an application of the strong in-plane field components, we present the design, fabrication, and characterization of a nano-scale TM-pass polarizer on Si waveguide based on a periodic MNC waveguide.

Chapter 3. Plasmonics for the enhancement of graphene properties

In-plane field enhancement of plasmonic modes

Some parameters have been used to assess the performances of plasmonic waveguides depending on its particular application, for example, propagation length or loss for waveguide application [START_REF] Rupert F Oulton | A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation[END_REF], confinement factor [START_REF] Zia | Geometries and materials for subwavelength surface plasmon modes[END_REF], mode area [START_REF] Rupert F Oulton | A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation[END_REF][START_REF] Rf Oulton | Confinement and propagation characteristics of subwavelength plasmonic modes[END_REF], or effective mode volume [START_REF] Maier | Plasmonic field enhancement and sers in the effective mode volume picture[END_REF] toward the application in mode confinement such as high density interconnect and surface enhanced Raman scattering (SERS). Different figures of merit have also been used for the trade-off between confinement and attenuation [START_REF] Maier | Plasmonic field enhancement and sers in the effective mode volume picture[END_REF][START_REF] Berini | Figures of merit for surface plasmon waveguides[END_REF][START_REF] Buckley | Figures of merit for 2d surface plasmon waveguides and application to metal stripes[END_REF][START_REF] Sun | Confinement and integration density of plasmonic waveguides[END_REF]. For the application on the enhancement of in-plane components, we use electric density and confinement defined in Chapter 2. The in-plane enhancement of plasmonic modes can be quantified by the in-plane electric field density ρ || across the graphene plane and by the in-plane confinement ratio γ. In this part, we calculate the in-plane electric field density and confinement ratio of various plasmonic waveguides.

Mode confinement in photonic and plasmonic waveguides

Because of the diffraction limit of light, the light confinement, represented by the effective mode volume, cannot decrease infinitely. For gradually decreasing waveguide dimensions, multi-mode photonic waveguides turn into single-mode waveguides in the scale λ 0 /2n, where λ 0 is wavelength in vacuum and n is the refractive index of the medium [START_REF] Bozhevolnyi | Plasmonic nanoguides and circuits[END_REF]. Further reduction on its size will induce modes to leak out the waveguide (Figure 3.1 a). Because of the mismatch of mode size from hundreds nanometer to several micrometers and the thickness of graphene about 0.34 nm, the overlap of photonic modes in graphene layer are intrinsic low, which restrict graphene-light interaction supported by photonic waveguides. By contrast, the diffraction limit breaks in plasmonic structures, allowing subwavelength confinement as small as a few nanometers. When the size of plasmonic waveguides are reduced gradually, the mode size also decrease monotonously even for plasmonic waveguides smaller than the wavelength (Figure 3.1 b). The mode size is only restricted by the atomic structure of matter [START_REF] Dmitri | Plasmonics beyond the diffraction limit[END_REF]. This provides an opportunity to improve the overlap of light into graphene layer and hence decreases the length of graphene-based EOMs from tens micrometer to subwavelength scale.

In-plane field confinement in 1D plasmonic waveguides

Nano-scale confinement of plasmonic waveguides come from a strong interaction between photons in light and electrons in matter generating surface plasmons-polaritons (SPPs). SPPs are electromagnetic excitations supported in a metal-dielectric interface, and decay exponentially with increasing distance into each medium from the interface.

Confinement and enhancement of SPP in a metal-dielectric interface

The most basic plasmonic waveguide is a single metal-dielectric interface. It consists of a non-absorbing dielectric with positive real permittivity ε d , Chapter 3. Plasmonics for the enhancement of graphene properties in the half-space z > 0, and a metal characterized by frequency dependent complex permittivity ε m (ω), in the half-space z < 0 (Figure 3.2 a). The SPP mode guided in this plasmonic waveguide can be excited from light in free space using suitable coupling system such as prisms, gratings, and near field probes to match the wave vectors of free space light and that of a SPP mode.

The free-space light can be both transverse electric (TE or s) or transverse magnetic (TM or p) polarized, however, SPP modes can be excited only by TM polarization light as TE polarized SPP cannot exist [START_REF] Bozhevolnyi | Plasmonic nanoguides and circuits[END_REF][START_REF] Maier | Plasmonics: fundamentals and applications[END_REF], so SPPs are intrinsic TM polarized modes but still has a weak in-plane electric field components. Using TM polarization light excitation, the electric field of SPP mode can be written as [START_REF] Maier | Plasmonics: fundamentals and applications[END_REF]:

E x (z) = iA k d ωε 0 ε d e ik SPP x e -k d z (3.1) E z (z) = -A k SPP ωε 0 ε d e ik SPP x e -k d z (3.2)
for z > 0 and

E x (z) = -iA k m ωε 0 ε m e ik SPP x e -k m z (3.3) E z (z) = -A k SPP ωε 0 ε m e ik SPP x e -k m z , (3.4)
for z < 0, where

k 2 m = k 2 SPP -k 2 0 ε m (3.5) k 2 d = k 2 SPP -k 2 0 ε d . (3.6)
And

k SPP (ω) = k 0 ε m (ω)ε d ε m (ω) + ε d , (3.7)
is the dispersion relation of the SPP mode propagating at the interface. Here, constant A can be calculated by the normalization condition and k 0 = 2π/λ 0 is the wave vector in vacuum.

Au dielectric As a example, we plotted both E x , E z components and normalized total electric field E of SPP mode in gold-glass interface at 1.55 µm (Fig. 3 To enhance the interaction between graphene and this kind of SPP mode, firstly graphene should be placed in the metal-dielectric interface due to the maximum in-plane electric field in this position (Figure 3.2 a), and then increase the in-plane electric field in the metal-dielectric interface. This can be done by increasing the refractive index of dielectric part, as shown in Figure 3.3. Actually, both in-plane confinement ratio and mode size (both in-plane and out-plane confinements) are improved at the same time (Inset of Figure 3.3). The reason is that k SPP is increased to the frequency closer its ω p when n is increased, and hence SPP mode shows stronger confinement and high loss. Here, mode size of in-plane confinement l E x is calculated according to 1/e times of maximum in-plane electric field. Another basic plasmonic waveguide consists of a thin metal film surrounded by two dielectrics known as IMI plasmonic waveguide (Figure 3.4 a). In this case, the SPP modes associated with the individual top and bottom metal-dielectric interfaces start interacting with each other under suitable thickness of metal (1 × penetration depth to 2 × penetration depth. For example, from 25 nm to 50 nm in the near-infrared region). Because the direction of SPP modes in different interface, both symmetric and anti-symmetric modes can propagate in this plasmonic waveguide. In the symmetric IMI configuration, symmetric SPP mode has symmetric out-plane component E z and anti-symmetric E z for anti-symmetric SPP mode (Figure 3.4 a). The anti-symmetric SPP mode has stronger in-plane and out-plane confinement, and hence higher attenuation, compared to symmetric SPP mode (Figure 3.4 b). Therefore, the symmetric and anti-symmetric modes are conventionally called long-range SPP mode and short-range SPP mode, respectively. As single metal-dielectric interface, the in-plane confinement of IMI configuration can be enhanced with higher refractive index of dielectric in both metal-dielectric interfaces. However, in the most of case, only one graphene layer is applied that means we need high in-plane confinement in only one interface using high refractive index in one dielectric and low refractive index in the another dielectric (Figure 3.4 a). When the dielectric in the two sides are different, the long-range SPP mode is no longer guided SPP mode due to radiation losses, which causes the light to leak out into the dielectric substrate. By contrast, short-range SPP mode is still guided SPP mode and the in-plane confinement can be enhanced by increase the refractive index of one dielectric (Figure 3.5). In addition, the thickness of metal film also has effect on the intensity of in-plane electric field (Figure 3

SPP in dielectric-metal-dielectric interface

Lateral in-plane confinement of 2D plasmonic waveguides

Because SPP modes supported by single metal-dielectric and IMI plasmonic waveguide are intrinsically TM-polarized, their in-plane components contain only the component along the propagation direction E x and no transverse component E y . Transverse in-plane component E y occurs when the width of the plasmonic structures are not infinite since they provide 2D confinement in the plane transverse to the propagation direction. For practical applications in waveguiding, many different 2D plasmonic waveguides with strong lateral confinement have been demonstrated, for example, metal nano-strip on a dielectric substrate [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 | Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of asymmetric structures[END_REF][START_REF] Joachim R Krenn | Nondiffraction-limited light transport by gold nanowires[END_REF][START_REF] Zia | Near-field characterization of guided polariton propagation and cutoff in surface plasmon waveguides[END_REF], nano-slot waveguides in the form of rectangular nanogaps in thin metal films [START_REF] Liu | Novel surface plasmon waveguide for high integration[END_REF][START_REF] Veronis | Guided subwavelength plasmonic mode supported by a slot in a thin metal film[END_REF][START_REF] David Fp Pile | Two-dimensionally localized modes of a nanoscale gap plasmon waveguide[END_REF][START_REF] David Fp Pile | On long-range plasmonic modes in metallic gaps[END_REF], sharp metal wedges [START_REF] David Fp Pile | Theoretical and experimental investigation of strongly localized plasmons on triangular metal wedges for subwavelength waveguiding[END_REF][START_REF] Boltasseva | Triangular metal wedges for subwavelength plasmon-polariton guiding at telecom wavelengths[END_REF][START_REF] Moreno | Channel plasmon-polaritons: modal shape, dispersion, and losses[END_REF][START_REF] Yan | Guided plasmon polariton at 2d metal corners[END_REF], nano-grooves in metal substrates [START_REF] Moreno | Channel plasmon-polaritons: modal shape, dispersion, and losses[END_REF][START_REF] Yan | Guided plasmon polariton at 2d metal corners[END_REF][START_REF] Iv Novikov | Channel polaritons[END_REF][START_REF] David | Channel plasmon-polariton in a triangular groove on a metal surface[END_REF][START_REF] Dmitri | Single-mode subwavelength waveguide with channel plasmon-polaritons in triangular grooves on a metal surface[END_REF][START_REF] Sergey I Bozhevolnyi | Channel plasmon-polariton guiding by subwavelength metal grooves[END_REF], metal nanoparticle chains (MNCs) [START_REF] Quinten | Electromagnetic energy transport via linear chains of silver nanoparticles[END_REF][START_REF] Maier | Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides[END_REF]. In view of difficult fabrication and integration of graphene on sharp wedges and nanogrooves, nano-strip, nano-slot, and MNC plasmonic waveguides are more suitable in the enhancement of graphene-light interaction, although groove and wedge waveguides are respectively preferred at optical and telecommunications frequencies as nano-scale waveguides [START_REF] Dmitri | Plasmonics beyond the diffraction limit[END_REF]. The dielectric nanowire on metal surface also provides strong mode confinement nevertheless, the confinement is mostly out-plane [START_REF] Rupert F Oulton | A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation[END_REF].

In-plane confinement and enhancement of nano-strip plasmonic waveguide

Metal nano-strip placed on dielectric substrate (Figure 3.7 a) is the most natural 2D plasmonic waveguide nevertheless, it is more complicated than 1D plasmonic waveguides due to more metal-dielectric interface SPP modes.

The main problem of this configuration is that the modes supported by nanostrips are very near to the light line, and prone therefore leaking out into the dielectric substrate due to large refractive index of dielectric and limited cutoff sizes, which significantly limits its application [START_REF] Bozhevolnyi | Plasmonic nanoguides and circuits[END_REF][START_REF] Berini | Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of asymmetric structures[END_REF].

For waveguide application, large width (w > λ) strip waveguides have been widely demonstrated [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 | Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of asymmetric structures[END_REF][START_REF] Zia | Near-field characterization of guided polariton propagation and cutoff in surface plasmon waveguides[END_REF][START_REF] Charbonneau | Experimental observation of plasmon-polariton waves supported by a thin metal film of finite width[END_REF][START_REF] Nikolajsen | Surface plasmon polariton based modulators and switches operating at telecom wavelengths[END_REF][START_REF] Lamprecht | Surface plasmon propagation in microscale metal stripes[END_REF] which support longrange SPP mode. However, long-range SPP mode means low propagation loss as well as weak confinement. To enhance the modes confinement supported by strip waveguides, small sizes of strips are necessary as its 1D case supporting short-range/anti-symmetric SPP mode [START_REF] Leosson | Long-range surface plasmon polariton nanowire waveguides for device applications[END_REF]. In addition, if the strip has an aspect ratio w/t < 1, then the lateral in-plane component E y dominates rather than out-plane component E z [START_REF] Berini | Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of asymmetric structures[END_REF].

In general, strip plasmonic waveguides support four main modes named ss 0 b mode, sa 0 b mode, as 0 b mode, and aa 0 b mode. Two letters s and a describing the symmetry of the main transverse electric field component, the superscript 0 expresses fundamental mode, and subscript b and l denote bound and leaky modes, respectively [START_REF] Berini | Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures[END_REF]. Nevertheless, in the asymmetric strip plasmonic waveguides, in which the refractive index of substrate and superstrate are different, long-range modes are absent due to phase mismatch Chapter 3. Plasmonics for the enhancement of graphene properties 3.2. In-plane field enhancement of plasmonic modes 59 between the SPPs at two different metal-dielectric interface [START_REF] Bozhevolnyi | Plasmonic nanoguides and circuits[END_REF][START_REF] Berini | Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of asymmetric structures[END_REF]. Especially, in the case of both w and t are shorter than the wavelength in the near-infrared region, only one fundamental plasmonic mode sa 0 b is bound mode, other three plasmonic modes ss 0 b , as 0 b , and aa 0 b are leaky modes. The electric field of dominant component of this mode is symmetric (s) with respect to the z axis and asymmetric (a) with respect to the y axis. The typical electric field of the sa 0 b mode at 1.55 µm is shown in Figure 3.7, E y component can be dominant with careful choice of its sizes, and the largest in-plane density occurs at gold-glass interface (Figure 3.7 e). For in-plane enhancement, the thickness of nano-strip should be larger than 30 nm considering the penetration depth and mode confinement. As its 1D case, the refractive index of substrate n also has effect on inplane electric field. When the refractive index of substrate is increased, the in-plane electric field is more confined in the left and right laterals of nanostrip (Figure 3.9 a), however, the in-plane component density decreases with higher n (Figure 3.9 b). That means lower n is preferable for enhancement of graphene-light interaction. Note that sa 0 b mode becomes leaky mode at n > 2.1 (Figure 3.9 c).

In-plane confinement and enhancement of slot plasmonic waveguide

Slot plasmonic waveguides (Fig. 3.10 a) are more used plasmonic waveguide than strip waveguides due to its remarkable mode confinement, reasonable propagation length and intrinsic broadband properties [START_REF] Cai | Elements for plasmonic nanocircuits with three-dimensional slot waveguides[END_REF]. It has one fundamental mode with strong lateral confinement, which rapidly enhance for smaller width [START_REF] Dmitri | Plasmonics beyond the diffraction limit[END_REF]. At the same time, the mode effective index increases and the mode is more lossy. In the limit of w ≈ 0, the slot SPP mode asymptotically approximates SPP mode supported by MIM plasmonic waveguide [START_REF] Cai | Elements for plasmonic nanocircuits with three-dimensional slot waveguides[END_REF]. For symmetric slot plasmonic waveguides, the slot SPP modes are always 3.2. In-plane field enhancement of plasmonic modes 63 bound modes, nevertheless, the application of asymmetric slot plasmonic waveguides are restricted by its cutoff gap width w and cutoff metal height t [START_REF] Veronis | Modes of subwavelength plasmonic slot waveguides[END_REF]. Specifically, cutoff gap width and cutoff metal height decrease as higher refractive index of substrate.

To enhance graphene-light interaction placed in the gold-dielectric interface (Figure 3 We note that for a the fundamental slot mode, the gap does not affect inplane electric field (Figure 3.11 b). But when gap w is wider, both the mode effective index and the loss decrease (Figure 3.11 c). Note that the cutoff range of gap is w < 40nm and the fundamental slot SPP mode is leaky for w > 85nm (Figure 3.11 c). In Figure 3.11, we used the first higher order SPP mode for w > 85nm. The thickness t of gold has same effect that t does not change the intensity of in-plane component and in-plane confinement ratio, it only changes the mode effective index and loss. By contrast, more important parameter is the refractive index n of substrate.

According to Figure 3.12, higher in-plane electric density and in-plane confinement ratio are achieved with low n substrate. The reason is that low n avoids mode leak into the substrate, at the same time weaken effects of cutoff w and t.

By comparison, the in-plane confinement of nano-slot and nano-strip plasmonic waveguides are in same level (Figure 3.7 e, Figure 3.10 e). Nevertheless, nano-slot plasmonic waveguides have smaller propagation loss (Figure 3.8 c, Figure 3.11 c, Figure 3.9 c, Figure 3.12 c). Actually, slot plasmonic waveguides are the most used configuration for the enhancement of graphenelight interaction.

In-plane enhancement of periodic plasmonic waveguide integrated on Si waveguide

Lateral in-plane component E y occurs in 2D plasmonic waveguides such as nano-strip and nano-slot configurations, nevertheless, in-plane component along the propagation direction E x are low in both nano-strip (Figure 3.7 b) and slot waveguides (Figure 3.10 b). E x and E y in-plane components can be enhanced with the use of periodic 3D plasmonic waveguides such as metal nanoparticle chain (MNC). In this part, we introduce the in-plane confinement of MNC plasmonic waveguide and its integration with standard silicon waveguide at telecommunication frequencies.

Metal nanoparticle chain (MNC) waveguide

Single metallic nanoparticle is characterized by localized surface plasmon resonance (LSPR), which can be shifted in resonance frequencies by alterations in particle shape, size, and surrounding medium. When a series of identical nanoparticles are arranged as a metal nanoparticle linear chain with a constant period a, the MNC can be used as either a plasmonic waveguide in the condition of a λ 0 , a Bragg reflection mirror when a ∼ λ 0 , or a grating if a λ 0 [START_REF] Halir | Waveguide sub-wavelength structures: a review of principles and applications[END_REF]. For closely spaced nanoparticle chain, electromagnetic 3.3. In-plane enhancement of periodic plasmonic waveguide 65 energy can be confined or squeezed into volumes smaller than the diffraction limit (λ 0 /2n) 3 [START_REF] Maier | Plasmonics: fundamentals and applications[END_REF].

Strong mode confinement is due to strong near field coupling LSPR mode of adjacent nanoparticles, this leads to the suppression of scattering field into far-field and light propagation by plasmonic modes excitation in particles along the chain axis [START_REF] Joachim R Krenn | Squeezing the optical near-field zone by plasmon coupling of metallic nanoparticles[END_REF]. Depending on the polarization direction of the exciting light, two in-plane polarized modes and one out-plane polarized mode can propagate in the MNC plasmonic waveguides (Figure 3.13). L mode (longitudinal mode) has the main electric field along the chain, and hence E x component enhancement in the gap between adjacent nanoparticles. The main electric field of T 1 and T 2 mode are respectively E y and E z components. For free-space far-field excitation, T 1 mode can be excited by s-polarized light, and p-polarized illumination with an incidence angle can excite both T 2 mode and L mode [215].

A lot of reports have proved the existence of T 1 and L modes both in theory [START_REF] Quinten | Electromagnetic energy transport via linear chains of silver nanoparticles[END_REF] and experiment [START_REF] Joachim R Krenn | Squeezing the optical near-field zone by plasmon coupling of metallic nanoparticles[END_REF]. T 2 mode has also been experimentally demonstrated [215]. In this manuscript, we focus on T 1 mode due to its strong in-plane electric field enhancement.

Re(E x )

Re(E y ) A y-oriented electric dipole source, placed near the MNC waveguide, excites the T 1 mode and an electric field monitor is used to record the electric field component across the xy-plane. As shown by the electric field distribution of T 1 mode of MNC (Figure 3.14), this kinds of MNC plasmonic waveguides have strong in-plane components, and very weak out-plane component. The nanoparticles have characteristic of dipole along y axis (Figure 3.14 b), and electromagnetic energy transports along x axis due to near-field coupling between adjacent nanoparticles (Figure 3.14 a). Periodic inversion of electric field direction reveals the nature of T 1 mode as plasmonic Bloch mode. The inversion period of electric field direction depends on medium surrounding MNC waveguide.

Re(E z ) Abs(E z ) (c) (d) (b) (a)
MNC waveguides can be analyzed using dispersion relation based on approximate point-dipole model [START_REF] Mark L Brongersma | Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit[END_REF][START_REF] Yong | Surface-plasmon dispersion relations in chains of metallic nanoparticles: An exact quasistatic calculation[END_REF]. Nevertheless, this approximate model is valid only when the ratio of nanoparticles size and distance is smaller than 1/3 [START_REF] Weber | Propagation of optical excitations by dipolar interactions in metal nanoparticle chains[END_REF]. To obtain exact dispersion relation of MNC waveguides, here the finite-difference time-domain (FDTD) method implemented by Lumerical is used to simulate the band structure of electromagnetic modes supported by the MNC waveguide. For the simulation of band structure of MNC, a few number of TE polarized point dipoles are placed randomly into a unitary cell consisting of a nanoparticle in order to excite TE-like transverse modes of the MNC. Bloch boundary conditions are used along the x axis while PML are used for the y and z axes. A set of 10 randomly placed time monitors are also distributed within the unit cell to measure the electric field (Figure 3.15 a). The electric field recorded by the time monitors are post-processed by Fourier transform to obtain the resonances in the frequency space. The entire band structure is obtained after repetition of this process for different k vectors in the Bloch boundary conditions. The simulation time depends on the size of mesh. For simulation with the mesh of 3 nm, one sweep value of k takes approximately 20 minutes in an ordinary computer. Between 10 and 30 dipole sources were used in the simulations with consistent equal results.

The simulated dispersion relation of a periodic MNC waveguide embedded in a homogeneous medium with refractive index of 1.67 is plotted in Figure 3.15 (b). Two plasmonic mode bands, fundamental T 1 mode and the first higher order T 1 mode, are obtained below the light line of the surrounding medium. Following photonic crystal nomenclature, such modes are labeled as E(o, N), where N is the index of mode. Hence the first band represents the E(o,1) mode and the second band is the E(o,2) mode. The intensity of in-plane electric field of E(o,1) and E(o,2) modes are plotted in inset. By contrast, outplane component can be neglected. Depending on working frequency of the MNC plasmonic waveguide, the position of two mode bands can be modulated by the refractive index n of the homogeneous medium (Figure 3.15 c). Simulated relationship between mode bands and the sizes of nanoparticles are also possible with same method.

Metal nanoparticle chain integrated on Si waveguide

The T 1 mode supported by MNC waveguides can be excited by different methods, such as far-field excitation with an angle [215,[START_REF] Femius Koenderink | Experimental evidence for large dynamic effects on the plasmon dispersion of subwavelength metal nanoparticle waveguides[END_REF], near-field excitation by a tip of a near-field scanning optical microscope [START_REF] Maier | Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides[END_REF], using fiber tapers placed on top of the waveguide [START_REF] Maier | Low-loss fiber accessible plasmon waveguide for planar energy guiding and sensing[END_REF][START_REF] Maier | Experimental demonstration of fiber-accessible metal nanoparticle plasmon waveguides for planar energy guiding and sensing[END_REF]. For the applications in integrated optical circuits including intra-chip optical interconnect, more convenient excitation is achieved by the near-field coupling of photonic waveguide integrated with MNC plasmonic waveguide [START_REF] Février | Giant coupling effect between metal nanoparticle chain and optical waveguide[END_REF][START_REF] Apuzzo | Observation of near-field dipolar interactions involved in a metal nanoparticle chain waveguide[END_REF].

Figure 3.16 (a) shows a 3D illustration configuration of MNC plasmonic waveguide integrated on Si photonic waveguide. By TE 0 mode illumination of Si photonic waveguide, two photonic-plasmonic hybrid modes, even and odd symmetries hybrid modes, occur in the overlap region of two waveguides due to the coupling between photonic TE 0 mode in isolated Si photonic waveguide and plasmonic T 1 mode supported by isolated MNC plasmonic waveguide (Figure 3.16 b). The interaction between even and odd symmetries hybrid modes leads to a electromagnetic field behaves as a beating pattern in the MNC and Si waveguides overlap region (Figure 3.16 c). By Figure 3.16 (c) and (d), a period across 8 MNP can be clearly seen. This period equal to two times of the wavelength of TE 0 mode in Si waveguide that means phase matching between MNC and Si waveguides, and hence high coupling efficiency. By changing the overlap length between MNC and Si waveguides, the output light can be controlled to transmit either in MNC waveguide or in Si waveguide (Figure 3.16 b, d).

The coupling between MNC and Si waveguide can be qualitatively described by the strongly coupled-mode theory as presented by Chuang [START_REF] Chuang | A coupled mode formulation by reciprocity and a variational principle[END_REF][START_REF] Chuang | Application of the strongly coupled-mode theory to integrated optical devices[END_REF] if we consider MNC as a normal waveguide made by a homogeneous material. Nevertheless, the quantitative calculation is still a big challenge when the 2D strongly coupled-mode theory is used in the 3D MNC. The strongly coupled-mode theory analyze the coupling properties between two normal waveguides. Because of the homogeneity of normal waveguides in the propagation direction, electric and magnetic field in a arbitrary cross section are used in the strongly coupled-mode theory. A problem occurs in the analysis of MNC-waveguide coupling: as a 3D plasmonic waveguide, the electric and magnetic field of MNC in every cross section are different, and hence the 2D strongly coupled-mode theory will not apply to MNCwaveguide coupling.

Some theories have been developed to analyze the coupling between unperturbed normal waveguide and periodic perturbation. There are two distinct coupling mechanisms of unperturbed waveguide and periodic perturbation: (1) One waveguide with a periodic perturbation; (2) The coupling between two isolated waveguides. The first one supposes that a normal waveguide is decorated by periodic perturbation. The coupling properties between two eigenmodes of the normal waveguide before decoration can be calculated by the coupled-mode theory [START_REF] Stoll | Coupled-mode analysis of periodic dielectric waveguides[END_REF][START_REF] Yariv | Coupled-mode theory for guided-wave optics[END_REF]. The second method analyze the coupling properties of two eigenmodes of two different waveguides. One of them is decorated by a periodic perturbation. Two coupled-mode theories are presented to the coupling between normal waveguide and periodic perturbation decorated waveguide respectively based on variational principle [START_REF] Huang | Power exchange in grating-assisted couplers[END_REF][START_REF] Huang | Nonorthogonal coupled-mode theory of grating-assisted codirectional couplers[END_REF] and reciprocity theorem [START_REF] Griffel | Frequency response and tunability of grating-assisted directional couplers[END_REF][START_REF] Griffel | Coupled mode formulation for directional couplers with longitudinal perturbation[END_REF]. Nevertheless, all of them only apply to the case with small perturbation which will not change the effective index of waveguide after perturbation decoration. A coupled local-mode theory has also been presented [START_REF] Snyder | Optical waveguide theory[END_REF][START_REF] Barthes | A coupled lossy local-mode theory description of a plasmonic tip[END_REF][START_REF] Mu | Complex coupled-mode theory for tapered optical waveguides[END_REF], but this theory hypothesizes that the cross section of waveguide in different positions must change over a distance large compared to the beat length between two modes, which is not suitable for MNC. In this part, we will only make the qualitative analyses using the strongly coupled-mode theory in [START_REF] Chuang | A coupled mode formulation by reciprocity and a variational principle[END_REF][START_REF] Chuang | Application of the strongly coupled-mode theory to integrated optical devices[END_REF], and corresponding quantitative analyses simulated by the finite-difference time-domain (FDTD) method implemented by Lumerical.

The simulated dispersion relation of MNC integrated on Si waveguide are shown in Figure 3.17. The dispersion relation of the periodic MNC waveguide embedded in a homogeneous medium is plotted in Figure 3.17 (a). The plasmonic mode band E(o,1) is obtained below the light line of the surrounding homogeneous medium. We also plotted the dispersion curve of TE 0 mode of isolated Si waveguide.

As is shown in Figure 3.17 (a), the E(o,1) mode band is crossed by the TE 0 mode at a point where frequency and momentum are conserved, which would result in an effective mode coupling. The refractive index of the homogeneous medium was chosen as 1.67 in order to achieve a mode matching condition between MNC and photonic waveguide at 1.55 µm wavelength (Figure 3.15). The coupling strength between adjacent particles is increased by high aspect ratio d 1 /d 2 [START_REF] Maier | Observation of coupled plasmon-polariton modes in Au nanoparticle chain waveguides of different lengths: Estimation of waveguide loss[END_REF]. The sizes of gold elliptical nanocylinders are d 1 = 250 nm and d 2 = 55 nm with a period a = 150 nm. The parameters of the metallic nanoparticles were chosen to obtain the lowest transmission at wavelength 1.55 µm.

The strong coupling between E(o,1) plasmonic Bloch mode and the TE 0 photonic mode is clearly demonstrated in the dispersion relation of the hybrid structure (Figure 3.17 b). Two hybrid modes within the active area of hybrid structure appear as a result of a coherent superposition of the modes supported by the isolated MNC and Si waveguides. The anti-crossing effect between the two hybrid mode bands results from a mode matching condition (crossing point in Figure 3.17 a) between the two isolated waveguides. At the mode crossing point 0.0967(a/λ), two hybrid modes lead to an efficient Chapter 3. Plasmonics for the enhancement of graphene properties energy transitions from the photonic mode to the plasmonic Bloch mode. According to Figure 3.17 (b), the propagation constants of even and odd hybrid modes are respectively β even = 7.52 ± 0.26µm -1 and β odd = 12.31 ± 0.21µm -1 , therefore, the beating length calculated by 2π/(β evenβ odd ) is 1.30 ± 0.05µm [START_REF] Salas-Montiel | Quantitative analysis and near-field observation of strong coupling between plasmonic nanogap and silicon waveguides[END_REF].

In-plane enhancement of MNC on Si waveguide

To assess the in-plane electric field confinement and enhancement of MNC integrated on Si waveguide, we take four configurations into account: standard Si waveguide as reference (Fig. 3 We consider the length of 1.05 µm for both strip and slot structures, which equal to the length of MNC consisted by 7 nanocylinders. Refer to the analyses of in-plane enhancement in isolated nano-strip and nano-slot plasmonic waveguides, the thickness of strip and slot are respectively set as 30 nm and Here, an analysis group transmission box of Lumerical was used during c,d,e) and the fundamental plasmonic mode of nano-slot (Figure 3.10), obviously the fundamental photonic TE mode of Si waveguide is coupled to the fundamental plasmonic mode of nano-slot plasmonic waveguide.

This has been demonstrated both in theory [START_REF] Salas-Montiel | Quantitative analysis and near-field observation of strong coupling between plasmonic nanogap and silicon waveguides[END_REF] and experiment [START_REF] Delacour | Efficient directional coupling between silicon and copper plasmonic nanoslot waveguides: toward metal-oxide-silicon nanophotonics[END_REF]. As shown in Figure 3.21 (b), the reflection and scattering of nano-slot on Si waveguide is larger than that of nano-strip on Si waveguide, this leads to lower in-plane density than that of the nano-strip configuration. The in-plane density of nano-slot on Si waveguide is expected to present higher values than that of the nano-strip on Si waveguide configuration. It can be achieved by a taper waveguide in the couple region of photonic waveguide and nanoslot so that the reflection can be significantly decreased [START_REF] Chen | Subwavelength confinement in an integrated metal slot waveguide on silicon[END_REF][START_REF] Ding | Efficient electro-optic modulation in low-loss grapheneplasmonic slot waveguides[END_REF].

With optimized sizes (w Au = 290 nm, w gap = 330 nm, w MNC = 250 nm), the in-plane confinement of three hybrid plasmonic waveguides and a Si reference waveguide are plotted in Figure 3.22 (a). For the hybrid plasmonic waveguides, the highest in-plane density is located in the interface between hBN and plasmonic structures. In this location, nano-strip and nano-slot hybrid plasmonic waveguides respectively have about 10 and 6 times in-plane density improvement comparing to a photonic waveguide. The MNC hybrid plasmonic waveguide has 40 times enhancement. In this part, all the electric field density were integrated in 1.2 µm length by 1.0 µm width region.

In summary, MNC hybrid plasmonic waveguide present the highest inplane electric field density, but it requires careful alignments during the fabrication process (i.e. during electron beam lithography) because of its fine structure and strong resonance property. Nano-strip hybrid plasmonic waveguide is also a resonant structure, but it does not has strict fabrication requirement. A drawback is weaker in-plane confinement deriving from intrinsic low in-plane component ratio of nano-strip plasmonic waveguides. For the trade-off between in-plane confinement and broadband operation, non-resonant nano-slot hybrid plasmonic waveguide is a preferred choice. In addition, the integral of ρ along z axis is the volume density of in-plane polarization components, which are respectively 0.12, 0.10, 0.39 and 0.05 for ρ of different configurations in Figure 3.22 (a). That means MNC hybrid plasmonic waveguide also has high in-plane polarization proportion, this is As an example, we add a graphene/hBN heterostructure between the plasmonic and photonic waveguides and we numerically analyzed the modulation depth of four different configurations (Figure 3.22 b). The MNC hybrid plasmonic waveguide shows the highest modulation depth due to 40 times in-plane enhancement, which matches graphene absorption formula (Equation 2.11). Here, we use the Fermi level of 0.53 eV for on state and 0.2 eV for off state.

Integrated periodic plasmonic waveguide on silicon photonics as TM-pass polarizer

Besides the application on the enhancement of light interaction with graphene, periodic MNC waveguide on Si waveguide configuration is also a suitable candidate for integrated TM-pass polarizer in Si photonics. As mentionned previously, MNC waveguide supports three different modes. Using MNC on Si waveguide configuration, both T 1 and T 2 modes can be respectively excited by the fundamental TE [START_REF] Février | Giant coupling effect between metal nanoparticle chain and optical waveguide[END_REF][START_REF] Apuzzo | Observation of near-field dipolar interactions involved in a metal nanoparticle chain waveguide[END_REF] and TM [START_REF] Tellez-Limon | Theoretical analysis of bloch mode propagation in an integrated chain of gold nanowires[END_REF] photonic modes under suitable conditions. By careful design, a TM-pass polarizer based on MNC on Si waveguide configuration was proposed. In the following subsections, we briefly present the mechanism and performances of such integrated TM-pass polarizer.

State-of-the-art of integrated plasmonic polarizer on silicon photonics

Benefit by high light confinement and effective suppression of parasitic capacitance and resistance, SOI (Silicon-on-insulator) platform has become the most competitive candidate in both electric and photonic integrated circuit. Unfortunately, high-index contrast makes SOI-based waveguide highly polarization dependent [START_REF] Mz Alam | Compact and siliconon-insulator-compatible hybrid plasmonic te-pass polarizer[END_REF]. Therefore, a photonic integrated circuit need polarizers to extinct the unwanted polarization state. Depending on which polarization state is need, polarizer can be divided into TE-pass and TM-pass polarizers. TE-and TM-pass polarizers can be formed respectively by vertical [START_REF] Dai | Compact broadband polarizer based on shallowly-etched silicon-on-insulator ridge optical waveguides[END_REF] and lateral [START_REF] Chen | Compact and integrated tm-pass waveguide polarizer[END_REF][START_REF] Wang | Ultracompact tm-pass silicon nanophotonic waveguide polarizer and design[END_REF][START_REF] Xiaowei Guan | Low-loss ultracompact transverse-magnetic-pass polarizer with a silicon subwavelength grating waveguide[END_REF] defects of photonic waveguide. Nevertheless, these polarizers based on mode cut-off reflect or radiate unwanted polarization, which affect neighboring polarization-sensitive devices and degrade the signal-to-noise performance of the optical chip [START_REF] Xiaowei Guan | Low-loss ultracompact transverse-magnetic-pass polarizer with a silicon subwavelength grating waveguide[END_REF][START_REF] Sun | Hybrid plasmonic waveguidebased ultra-low insertion loss transverse electric-pass polarizer[END_REF]. In addition, large size from tens micrometers [START_REF] Chen | Compact and integrated tm-pass waveguide polarizer[END_REF][START_REF] Wang | Ultracompact tm-pass silicon nanophotonic waveguide polarizer and design[END_REF][START_REF] Xiaowei Guan | Low-loss ultracompact transverse-magnetic-pass polarizer with a silicon subwavelength grating waveguide[END_REF] to millimeter scale [START_REF] Dai | Compact broadband polarizer based on shallowly-etched silicon-on-insulator ridge optical waveguides[END_REF] weaken their competitiveness in high density integrated circuits.

Hybrid plasmonic waveguides promise small device size due to the enhanced light-matter interaction. TE-pass polarizers based on hybrid plasmonic waveguide loss have been proposed both in theory [START_REF] Mz Alam | Compact and siliconon-insulator-compatible hybrid plasmonic te-pass polarizer[END_REF][START_REF] Zhu | Ridge waveguide assisted highly efficient transverse-electric-pass polarizer based on a hybrid plasmonic waveguide[END_REF] and experiment [START_REF] Sun | Hybrid plasmonic waveguidebased ultra-low insertion loss transverse electric-pass polarizer[END_REF][START_REF] Sun | Experimental demonstration of a hybrid plasmonic transverse electric pass polarizer for a silicon-on-insulator platform[END_REF] with the theoretical shortest device length of 4.2 µm [START_REF] Zhu | Ridge waveguide assisted highly efficient transverse-electric-pass polarizer based on a hybrid plasmonic waveguide[END_REF] and experimental device length of 30 µm [START_REF] Sun | Experimental demonstration of a hybrid plasmonic transverse electric pass polarizer for a silicon-on-insulator platform[END_REF][START_REF] Sun | Hybrid plasmonic waveguidebased ultra-low insertion loss transverse electric-pass polarizer[END_REF]. Nevertheless, because of plasmonic modes are intrinsic TM polarized, strong interaction between TE polarized modes and plasmonic structures remain a massive challenge. As a result, TM-pass polarizer based on plasmonic loss are still not be reported. Although some TM-pass polarizers are demonstrated theoretically using hybrid plasmonic waveguide [START_REF] Alam | Compact hybrid tm-pass polarizer for silicon-on-insulator platform[END_REF][START_REF] Ng | Analysis of cmos compatible cu-based tm-pass optical polarizer[END_REF][START_REF] Xiaowei Guan | Ultracompact broadband tm-pass polarizer using a silicon hybrid plasmonic waveguide grating[END_REF], the mechanism of operation is based on TE mode cut-off and hence, suffer from the problem of influence in neighboring polarization-sensitive devices and degeneration in signal-to-noise performance of the optical chip.

Configuration and mechanism

Here, we proposed to use the periodic MNC integrated on Si waveguide for the design of a TM-pass polarizer. The MNC of the proposed TM-pass polarizer consists of 5 identical nano-rods (i.e. cylinder with a rectangular crosssection) with length 210 nm, width 30 nm, thickness 30 nm, and a invariant period of 150 nm (Figure 3.23 b). The MNC is directly integrated on a 500 nm by 220 nm Si photonic waveguide (Figure 3. The incident TE 0 mode is efficiently coupled into the plasmonic mode, while the TM 0 mode pass without strong interaction with the plasmonic mode (Figure 3.24). The mechanism of the TM-pass polarizer can be explained with the use of the dispersion relation. The dispersion relation were calculated with the FDTD method implemented by Lumerical. A single unit cell with a single MNC on top of a Si waveguide was used. Bloch boundary condition along the x-axis and PML along y and z axes were employed. In the case of TE 0 illumination (i.e. electric dipole sources along y-axis), two hybrid mode bands occur at 1.59 µm (Figure 3.25 a). According to previous discussion, the hybrid plasmonic mode bands are the result of strong coupling between TE 0 mode of the isolated Si waveguide (red curve in Figure 3.25 a) and plasmonic mode of MNC plasmonic waveguide consisted of 5 nano-rods. By contrast, only TM 0 mode can be found at 1.59 µm under TM 0 illumination (i.e. electric dipole sources along z-axis), which means that the TM 0 mode can not couple with such a MNC waveguide at this wavelength (Figure 3.25b). The number of nano-rod is chosen according to beating length of hybrid plasmonic mode. With 5 nano-rods, TE 0 mode in Si waveguide is only coupled to MNC without coupling back from the MNC to the Si waveguide (Figure 3.24). The hybrid plasmonic mode in MNC is mainly absorbed and remaining is scattered, this allows us to increase the loss of TE 0 mode further.

Experimental results

Si photonic waveguides of 500 nm × 220 nm, fabricated at CEA/Leti within the framework of an old project, were defined by standard deep-UV lithography followed by reactive ion etching (RIE). At UTT, I fabricated the MNC on top of the Si waveguide with the use of electron beam lithography (EBL).

The EBL was conducted with the Raith eLine equipment, which allow me the proper alignment of the MNC relative to the Si waveguide. I used positive e-beam resist (PMMA) to define the MNC plasmonic waveguide on top of Si waveguide. The residual e-beam resist after development was removed with an O 2 plasma descum process. A 3-nm-thick chromium adhesion layer and 30-nm-thick gold layer were deposited respectively by electron-beam and thermal evaporation. A 630-nm-length TM-pass polarizer was finally fabricated after lift-off (Figure 3.23 b).

The transmission spectrum of fabricated TM-pass polarizer, characterized by me at UTT, is shown in Figure 3.26 (a). According to the experimental results, the extinction ratio of the TM-pass polarizer at 1.592 µm is about 23 dB/µm, the insertion loss is about 2.4 dB/µm in the 1.54 µm to 1.64 µm wavelength range, and its length is only 0.63 µm. The far-field scattering photos (top view) indicate more scattering of TE 0 mode than TM 0 mode (Inset of Figure 3.26 a). The simulated transmission spectra in Figure 3.26a were obtained with the FDTD method implemented by Lumerical. I used a mode expansion monitor to calculate the transmission for TE 0 and TM 0 input modes. The size of the calculation window was set to 3 µm × 2.5 µm by 2.5 µm. The TE 0 or TM 0 mode source is injected into the input side of Si waveguide, transmission spectra of TE 0 and TM 0 modes are recorded by a mode expansion monitor combined with a power monitor in the output side of Si waveguide. The mesh size in the MNC region is 3 nm.

The light source in the transmission measurement is produced by a tunable laser with 0.3 nm wavelength sweep step size and a modulation frequency f 0 of 10 000 Hz. The polarization state is controlled by a fiber polarization controller (FPC), and then injected into sample by a tapered lensed polarization maintaining (PM) fiber using end-fire coupling. The transmitted light is collected by another lensed PM fiber and then transformed into electrical signal by a detector. A lock-in amplifier (LIA) was used to enhance the signal-to-noise ratio by demodulation detected AC signal (Figure 3.26 b). Finally, the detected signal is normalized by the signal of reference waveguide without MNC.

Conclusion

The improvement of graphene-based EOMs performances requires enhanced overlap of in-plane component of light into one-atom-thick graphene. This is achieved by MNC hybrid plasmonic waveguide in this chapter. Comparison with 10 and 6 times in-plane electric field density enhancement of optimized nano-strip and nano-slot hybrid plasmonic waveguides, MNC hybrid plasmonic waveguide shows 40 times in-plane enhancement relative to a configuration without plasmonic structure. This is a result of strong coupling between TE 0 photonic mode of Si waveguide and TE-polarized T 1 plasmonic mode supported by MNC plasmonic waveguide. In addition, the birefringence of this kind of coupling allows us numerically and experimentally demonstrates a nano-scale TM-pass polarizer. The 40 times in-plane enhancement of MNC hybrid plasmonic waveguide is expected to improve the performances of graphene-based EOMs. Next chapter, we will present a subwavelength graphene-based EOM integrated on MNC hybrid plasmonic waveguide and its improved performances.

Chapter 4

Plasmonic-enhanced graphene based EO modulator 4.1 Introduction

Optical modulators are essential components in optical communication and photonic neuromorphic networks. Graphene on silicon waveguides have been proposed for optical modulation. In Chapter 2, based on in-plane components selectivity of graphene, we have confirmed weak graphene-light interaction of graphene on Si waveguide configuration. Following by that, we demonstrated giant in-plane enhancement of MNC hybrid plasmonic waveguide in Chapter 3, which is helpful to enhance graphene-light interaction and hence high-performance electro-optic modulator (EOM).

In this chapter, we will propose a plasmonic improved graphene-based EOM integrated on MNC hybrid plasmonic waveguide. The optical performances of proposed EOM including modulation depth and insertion loss will be numerically investigated, the optimization and influence factors of them will be analyzed. Its electrical performances such as modulation speed and energy consumption will be calculated using equivalent circuits with using of simulated resistances and capacitances. Finally, we will introduce the progress of the fabrication toward double-layer graphene-based EOM such as the fabrication of multilayer graphene/hBN heterostructures, electrode and MNC.

Description of the EO modulator

Graphene on silicon waveguides have been widely reported for optical modulation but the modulation depth is limited to 0.1 dB/µm and a modulator length of about 40 µm [START_REF] Liu | A graphene-based broadband optical modulator[END_REF][START_REF] Hu | Broadband 10 Gb/s operation of graphene electro-absorption modulator on silicon[END_REF]. Higher modulation depth and shorter modulators can be achieved with the help of plasmonic modes, however, due to the transverse magnetic nature of plasmonic modes and in-plane selectivity of graphene, traditional plasmonic modes weakly interact with graphene. In this chapter, a plasmonic Bloch mode supported by MNC plasmonic waveguide is used to overcome this problem. Its high in-plane electric field strongly interacts with graphene and hence, decreases the device area down to 0.63 λ by 0.3 λ without sacrifice of its modulation depth.

Configuration and principle of operation

The hybrid plasmonic electro-optic modulator (HPM) consists of a MNC waveguide on a 500 nm by 220 nm Si photonic waveguide (Figure 4.1 a). The plasmonic waveguide and the Si waveguide are separated by a 20-nmthick hexagonal boron nitride (hBN) layer that allows graphene to maintain its high mobility [START_REF] Kim | A role for graphene in silicon-based semiconductor devices[END_REF] and to control the modes coupling. Graphene is placed on top of the hBN layer, beneath the plasmonic MNC waveguide, where the in-plane electric field components are expected to be high. Graphene and hBN have the same sizes, a length of 3 µm and a width of 2.5 µm. Graphene is modeled as an anisotropic 2D surface conductivity with scattering rate of 6.2 meV and 5.5 meV in Fermi level of 0.53 eV and 0.2 eV, as discussed in Chapter 2, so it has no thickness. The refractive index of hBN is set to 1.98. Theoretical parameters such as modulation depth and insertion loss spectra are simulated using FDTD solution by Lumerical in a 3 µm by 2.5 µm by 2.5 µm region with the mesh size of 2 nm in the region of MNC.

The MNC waveguide is composed of 7 identical gold elliptical nanocylinders with major axis dimension d 1 = 250 nm, minor axis dimension d 2 = 55 nm, thickness 30 nm, and invariant with a period a = 150 nm. The size and number of nanocylinders were chosen to produce the highest modulation depth and low insertion loss. The optical parameters such as modulation depth and insertion loss spectra are simulated using FDTD solution of Lumerical. The p-type doping concentration of the Si waveguide and 50-nmthick Si connection layer are respectively 10 17 cm -3 and 10 19 cm -3 [START_REF] Sorianello | Design optimization of single and double layer graphene phase modulators in SOI[END_REF]. The electrodes are placed at 500 nm from the edges of the Si waveguide. The electrodes consist of 80-nm-thick Au layer and 20-nm-thick Ti layer to decrease the graphene contact resistance [START_REF] Robinson | Contacting graphene[END_REF]. DEVICE of Lumerical was used to calculate electrical performances of the EOM. The input port of the Si waveguide is illuminated with its fundamental TE mode. The plasmonic Bloch mode in the MNC waveguide is efficiently excited with the TE mode at 1550 nm as we discussed using dispersion relation (Figure 3.17). The strong in-plane electric field confinement of the plasmonic mode brings a high in-plane electric field density (Figure 3.22) and thus enhances interaction with graphene. The optical transmission is modulated by applying a voltage across the capacitor formed by the top electrode on graphene, the hBN dielectric layer, and the bottom electrode on a Si connection layer (Figure 4.1 b). The on/off light modulation is achieved by shifting the driving voltage and then Fermi level (Figure 2.5 a). Indeed, when the Fermi level is lower than the transition threshold -hω 0 /2 (equal to 0.4 eV at 1.55 µm wavelength), positive charges accumulates in the valence band and therefore, no electrons are available for interband absorption. In this case, the incident photons propagate through a transparent graphene. When the Fermi level is higher than hω 0 /2, there are no interband transition since all electron states in the conduction band are filled up. In this case also, the incident photons are not absorbed. Only when the Fermi level approaches the Dirac point (-hω 0 /2 < E f < hω 0 /2), the incident photon energy promotes electrons from the valence band to conduction band, this process is known as interband absorption, the EOM is in the off state (inset of

Optical performances 4.3.1 Modulation depth and insertion loss spectra

Modulation depth and insertion loss of EOMs are defined by its transmission (Equation 1.3). To simulate light transmission in the HPM, the finitedifference time-domain (FDTD) method implemented by Lumerical was used in a 3 µm × 2.5 µm × 2.5 µm region with the mesh size of 2 nm in the region of MNC. Transmission spectra into exactly fundamental TE mode are obtained by eigenmode expansion. Accurate gold dispersion data, measured by ellipsometry, were used in the simulations. The refractive index of hBN is set to 1.98 [START_REF] Ansell | Hybrid graphene plasmonic waveguide modulators[END_REF]. Graphene is considered using a strict 2D conductivity tensor described in Chapter 2. The anisotropic 2D surface conductivity model with scattering rate of 6.2 meV and 5.5 meV in Fermi level of 0.53 eV and 0.2 eV (Chapter 2) was used in the simulations. The refractive index of Si is about 3.48 according to the data of Palik. The doping of Si is not taken into consideration in optical simulations.

The simulated modulation depth spectrum is higher than 3 dB/µm in a wavelength range ranging from 1.491 µm to 1.685 µm, which covers the entire C, L, and U bands of the telecommunication window (Figure 4.2 a). Furthermore, the length of the HPM is 1.05 µm and the modulation depth reaches 20.9 dB/µm at resonant wavelength (λ = 1.551 µm), which are, to the best of our knowledge, the shortest graphene-based EOM with the highest modulation depth [START_REF] Liu | A graphene-based broadband optical modulator[END_REF][START_REF] Yang | Low-chirp highextinction-ratio modulator based on graphene-silicon waveguide[END_REF][START_REF] Phatak | Design of electro-optic modulators based on graphene-onsilicon slot waveguides[END_REF][START_REF] Li | Ultrafast all-optical graphene modulator[END_REF][START_REF] Liu | Double-layer graphene optical modulator[END_REF][START_REF] Christopher | Graphene electro-optic modulator with 30 GHz bandwidth[END_REF][START_REF] Qiu | Efficient modulation of 1.55 µm radiation with gated graphene on a silicon microring resonator[END_REF][START_REF] Ding | Effective electrooptical modulation with high extinction ratio by a graphene-silicon microring resonator[END_REF][START_REF] Xu | Design of an electro-optic modulator based on a siliconplasmonic hybrid phase shifter[END_REF][START_REF] Koester | High-speed waveguide-coupled graphene-on-graphene optical modulators[END_REF]. Nevertheless, the insertion loss is high near the resonant wavelength (dash-dotted line in Figure 4.2 a).

For a practical implementation, we can choose a spectral range ranging from 1.636 µm to 1.686 µm along which the modulation depth is greater than 3 dB/µm while the insertion loss is less than 3 dB/µm. This working spectrum is broader than that of resonant EOMs based on micro-ring resonator and graphene [START_REF] Christopher | Graphene electro-optic modulator with 30 GHz bandwidth[END_REF][START_REF] Qiu | Efficient modulation of 1.55 µm radiation with gated graphene on a silicon microring resonator[END_REF]. At 1.65 µm, the modulation depth is 4 dB/µm while the insertion loss is 2.6 dB. The electric field distribution in on state presents higher plasmonic enhancement than that at its off state (Figure 4.2 b, c). Note that the resonance can be shifted by changing the size of the MNC. A blueshift is obtained for shorter major axis dimension d 1 (Inset in Figure 4.2 a). In the simulation of transmission spectra, we did not take the contribution of doped silicon into account because it is only of 0.0041 dB/µm for modulation depth and 0.0042 dB/µm for insertion loss. We deal with silicon as undoped semiconductor in optical simulations (section 4.3) and doped semiconductor in electrical simulations (see section 4.4). Reflection and scattering of the HPM are 0.1 dB and 0.36 dB respectively, which are negligible across the spectrum considered here thanks to the high coupling efficiency between photonic and plasmonic guided modes.

Parametric study: influence of the sizes

The number N of gold nanoparticles is chosen based on the beating length of the hybrid plasmonic waveguide in order to produce the highest modulation depth. According to

Optical loss analysis

Loss caused by doped silicon

Because doped silicon is also an active material of EOM, insertion loss due to doped silicon is unavoidable. Nevertheless, the insertion loss of doped silicon is much smaller than the insertion loss of plasmonic structure (2.5 dB/µm at 1.65µm as shown in Figure 4.2 a). As shown in Table 1.3, the modulation depth and insertion loss of doped silicon-based EOMs are in milli dB/µm scale. By contrast, the modulation depth and insertion loss of graphene-based EOMs are about one hundred times of doped silicon-based EOMs (Table 1.4). Nevertheless, both of them are much smaller than plasmonic-enhanced EOM in this paper, which are respectively 4 dB/µm and 2.5 dB/µm at 1.65 µm (Figure 4.2 a). As a result, the effect of doped silicon in modulation depth and insertion loss in our case can be neglected.

To calculate the exact contribution of doped silicon and graphene undecorated by plasmonic structure in our plasmonic-enhanced EOM (Figure 4.1), we simulated transmission spectra of graphene on undoped silicon waveguide (Fig. 4.4 b) and doped silicon waveguide without graphene (Fig. 4.4 c) under different drive voltage V d at 1.65 µm. Firstly, we can clearly see that doped silicon waveguide is lossy, but the loss is weak. Secondly, the transmission of doped silicon waveguide will change as drive voltage is changed. Lastly, the dynamically changing of transmission of doped silicon waveguide affects both modulation depth and insertion loss of our plasmonic-enhanced EOM, but those effects are weak compared to effects of plasmonic structure. At on state voltage (V d = 8.45 V), the insertion loss is 2.5 dB/µm, by contrast, the contribution of doped silicon and graphene are respectively 0.0042 dB/µm and 0.0027 dB/µm, which are low. Modulation depth is calculated by the difference between the transmission at off state and on state, which is 4 dB/µm in our plasmonic-improved EOM (Figure 4.2 a). Nevertheless, the contribution of doped silicon and graphene in modulation depth are respectively 0.0041 dB/µm and 0.0745 dB/µm, which are also low. In the simulations of The relationship between applied voltage and carrier density can be thus simulated. The change of carrier density is then transferred to refractive index of semiconductor by Plasma-Drude model [START_REF] Henry | Spectral dependence of the change in refractive index due to carrier injection in gaas lasers[END_REF]. Especially, the change of electric doped silicon can be calculated by Equation 1.6. For graphene, electrical doping leads to a change of Fermi level, it is calculated by:

n 2D = 2(kT) 2 πh 2 v 2 F F 1 (η), (4.1)
where the first term is the 2D effective density of states (DOS), E F is the Fermi level, k is the Boltzmann constant, T is the temperature, and F 1 (η) is the Fermi-Dirac integral of order 1 with η = E F /k T (https://apps.lumerical.

com/other_application_graphene_modulator_electrical.html). The electric modulated conductivity of graphene can be obtained by Kubo formalism (Equations 2.4,2.5, 2.6). The data of carrier concentration of doped silicon corresponding to drive voltage is then loaded into optical simulation software MODE Solution by Lumerical, finally drive voltage is swept in MODE Solution and the relationship between loss of doped silicon and drive voltage is obtained. The relationship between transmission of graphene undecorated by plasmonic structure and drive voltage (blue curve in Figure 4.4 a) is simulated only using MODE Solution. Fermi level of graphene is swept in MODE Solution to obtain the dynamically changing of transmission of graphene under different Fermi levels, and then Fermi levels are translated to drive voltage by Equation 2.8.

Loss caused by graphene

The fabrication of high quality graphene is still a challenge. Defective graphene has influence in insertion loss. Actually, the defect of graphene in our simulations have been considered by suitable scattering rate Γ in Equation 2.5 and Equation 2.6 because scattering rate is strongly dependent on the graphene quality. We have measured mobility of graphene on silicon wafer in different drive voltage, which are 2 000 cm 2 /V•s at 0.2 eV and 1 000 cm 2 /V•s at 0.53 eV, respectively. And then scattering rate calculated by Equation 2.7 are respectively 5.5 meV and 6.2 meV at 0.2 eV and 0.53 eV. Here, our measured mobility is much lower than the limit of mobility of graphene on SiO 2 (17 000 cm 2 /V•s [START_REF] Hong | Quantum scattering time and its implications on scattering sources in graphene[END_REF]), graphene on hBN (37 000 cm 2 /V•s [START_REF] Gannett | Boron nitride substrates for high mobility chemical vapor deposited graphene[END_REF]), and suspended graphene (200 000 cm 2 /V•s [START_REF] Du | Approaching ballistic transport in suspended graphene[END_REF]), which means that impurities and residues of graphene have been considered. According to Equation 2.5 and 2.6, surface conductivity of graphene mainly depends on scattering rate Γ (mobility) and Fermi level E f . As shown in Figure 2.4 (b) and Figure 2.5 (b), the effect of mobility can be neglected by comparison with the effect of Fermi level (drive voltage), which matches results in Ref. [START_REF] Hu | Broadband 10 Gb/s operation of graphene electro-absorption modulator on silicon[END_REF]. In addition, as shown by the blue curve in Figure 4.4 (a), the contribution of graphene on insertion loss is only 0.0027 dB/µm, which can be neglected.

Loss caused by reflection, scattering, and absorption

For a coupling system, reflection is negligible for efficient mode coupling.

In our configuration, almost all the energy of input TE 0 mode in Si waveguide is coupled to MNC plasmonic waveguide and then coupled back to Si waveguide (Figure 4.2 c) due to the beating of hybrid plasmonic mode within coupling area and suitable number of Au elliptical nanocylinder (Figure 3.16 c). By the same method used in Figure 3.19 (b), we calculated the reflection, transmission, scattering, and absorption spectra of the EOM under on and off state (Figure 4.5). Reflection and transmission spectra are recorded by mode expansion monitor, which only recorded the power of TE 0 mode. Nevertheless, total scattering and absorption are calculated by recorded data of power monitors. According to the simulated reflection spectra of our EOM (Figure 4.5 a), the reflection of TE 0 mode is only 0.47 % at on state and 0.13 % at off state at 1.65 µm. They are much smaller than transmission of our device at 1.65 µm, which are 54.2 % at on state and 20.4 % at off state (Figure 4.5 b). We also simulated the scattering and absorption spectra of all the modes in four different directions surrounding device. As shown in Figure 4.5 (c), the scattered light is low thanks to the high coupling efficiency between photonic and plasmonic waveguides. By Figure 4.5 (d), total absorption at on state and off state at 1.65 µm wavelength are respectively 42.4 % (insertion loss) and 67.2 %. The insertion loss comes from doped silicon and graphene undecorated by plasmonic structure have been analyzed which are much smaller than optical loss of plasmonic structure Here, scattering is measured by four power monitors surrounding device, therefore, scattering and absorption spectra include the power of all the modes within the HPM.

and plasmonic enhanced graphene. Compared to the transmission spectra without graphene and with graphene at on state in Figure 4.2 (a), almost all the insertion loss comes from plasmonic structure, and graphene at off state contributes almost all the modulation depth.

Electrical performances 4.4.1 Equivalent electric circuit

In addition to the modulation depth, modulation speed and energy consumption are two important parameters. These parameters depend on the electrical properties of modulator. In our case, the equivalent electrical circuit can be modeled as a low pass filter (Figure 4.6) [START_REF] Sorianello | Design optimization of single and double layer graphene phase modulators in SOI[END_REF][START_REF] Xiao | 25 gbit/s silicon microring modulator based on misalignment-tolerant interleaved pn junctions[END_REF], where R C and R grp are respectively graphene contact and sheet resistances, C is hBN capacitance, R Si is Si resistance, C air is the capacitance between two metal pads through air, R sub is the resistance of substrate, C box-L and C box-R are capacitances of the buried oxide layer. The contact resistance of the right electrode and the Si connection layer can be neglected for large contact area [START_REF] Boberg | Contact resistance measurements of platinum-silicide and chromium contacts to highly doped n and p silicon[END_REF]. Values ranging from 100 Ω/sq to 500 Ω/sq have been reported for graphene sheet resistance1 and from 100 Ωµm to 1000 Ωµm for the graphene contact resistance depending on the metal choice. To evaluate the worse case scenario, R grp = 500 Ω/sq and R C = 1000 Ωµm were used. The capacitances can be calculated using parallel plate capacitance formula nevertheless, for accuracy, we use an electrical simulation software Device by Lumerical to simulate capacitances at an applied voltage of 10 V, both fringing parasitic capacitance and parasitic capacitance due to doped silicon are taken into account. We considered a 100-nm-thick Au metal for each electrode and 0.5 µm distance between the graphene electrode and the waveguide core edge on both sides. The device region capacitance, including C and C air , is 1.671 fF (Figure 4.7 b), and the device region resistance R Si due to p-doped silicon is 2800 Ω.

V d C air C box-L R sub C box-R R c R grp C R Si graphene hBN Au Si SiO 2
The typical thickness of buried oxide layer and Si handle are respectively ranging from 0.5 µm to 5 µm and 300 µm to 700 µm. And the resistivity of Si substrate ranges from 1 Ω•cm to 10 Ω•cm [START_REF]SOI wafer[END_REF]. If we consider 2-µm-thick buried oxide layer and 500-µm-thick Si substrate with 1×10 15 cm -3 p-type

Modulation speed and energy consumption

Usually, the modulation speed of a modulator, also known as 3 dB cutoff frequency [START_REF] Dalir | Athermal broadband graphene optical modulator with 35 GHz speed[END_REF], is calculated using Equation 1.1 if it is modeled as a first order low pass filter. Nevertheless, for a complicated low-pass filter like Figure 4.6, we can not calculate modulation speed by 1/(2πRC). By the definition of 3 dB cutoff frequency, it is the frequency for which the power applied in the filter fall to 3 dB corresponding approximately to half input power. For a modulator, the output optical power is approximately proportional to the magnitude of voltage, therefore, the 3 dB cutoff frequency corresponds to the 3 dB roll-off frequency of the magnitude of voltage V C across the device capacitance C [START_REF] Xiao | 25 gbit/s silicon microring modulator based on misalignment-tolerant interleaved pn junctions[END_REF]. Figure 4.8 (a) shows the equivalent circuit model for the calculation of modulation speed [START_REF] Hu | Broadband 10 Gb/s operation of graphene electro-absorption modulator on silicon[END_REF][START_REF] Xiao | 25 gbit/s silicon microring modulator based on misalignment-tolerant interleaved pn junctions[END_REF]. Here, both source impedance Z d and cable impedance Z 0 are set as 50 Ω, total impedance of the EOM is:

Z -1 EOM = Z -1 air + Z -1 device + Z -1 sub (4.2)
where

Z air = 1 jωC air (4.3)
Chapter 4. Plasmonic-enhanced graphene based EO modulator

Z device = 1 jωC + R C + R grp + R Si (4.4) Z sub = 1 jωC box-L + 1 jωC box-R + R sub (4.5)
And the ratio between the magnitude of V C and V d calculated by voltage divider rule:

|V C | |V d | = Z EOM Z EOM + Z 0 • 1/(j2π f C) Z device (4.6)
where V C is the voltage applied in the capacitor, V d is drive voltage, f is frequency. We plot the relationship between f and [START_REF] Ido | Ultra-high-speed multiple-quantumwell electro-absorption optical modulators with integrated waveguides[END_REF] GHz. Note that here the calculated modulation speed is electrical response speed corresponding to the magnitude of S 11 measured by network analyzer. In addition, the electrical frequency response in Figure 4.8 (b) is calculated at 10 V bias, for 0 V bias, the modulation speed increases to 50 GHz due to decrease of capacitance (Figure 4.7 b, c).

|V C | |V d | in
|V C | |V d | = 0.5, is
The energy consumption, calculated by Equation 1.2 and 2.8, is 21.2 fJ/b, reaching future needs [START_REF] Graham T Reed | Silicon optical modulators[END_REF][START_REF] Yoo | Future prospects of silicon photonics in next generation communication and computing systems[END_REF]. Here, we fixed Fermi level E on f = 0.53 eV (V d = 8.45 V) and E off f = 0.2 eV (V d = 1.20 V).

Performances evaluation: figure of merit

A "good" EOM is one that optimizes either modulation depth, or insertion loss, or footprint, or modulation speed, or energy consumption depending on the intended application. In Table 4.1 we summarized different parameters of graphene-based EOM integrated on photonic waveguide. By comparison, a ultra-compact and high modulation depth EOM is achieved, and all the parameters match the performances requirements of EOM for recent optical interconnect (Table 1.1). In addition, global optimality could been defined as maximizing a figure of merit (FoM) which contains footprint, modulation speed and energy consumption (Equation 1.4). The figure of merit of our nanomodulator is 3.5 at 1.65 µm, beyond the state-of-the-art (Table 4.1).

4.6. Progress towards the fabrication of the EOM 99

Progress towards the fabrication of the EOM

Because of its fabrication complexity (i.e. graphene/hBN heterostructures, MNC plasmonic waveguide, Si waveguide, and electrodes together), the final fabrication of the EOM sample has not yet finished. Currently, we succeed in the fabrication of the MNC waveguides on Si substrate and Si waveguide, the electrodes, and patterned graphene/hBN heterostructures. As shown by Figure 4.9 (a), MNC plasmonic waveguide with required sizes has been fabricated on silicon wafer, as well as its integration on Si waveguide (Figure 4.9 b).

Figure 4.10 shows the micrograph of a fabricated electrode on Si waveguide sample. We fabricated 500 × 400 µm electrode with a 60 µm width contact arm (Figure 4.10 a). In the contact region, the distance between electrode arm and Si waveguide is only 1 µm (Figure 4.10 b). The electrode consists of 20-nm-thick titanium and 80-nm-thick gold, which has low graphene contact resistance [START_REF] Robinson | Contacting graphene[END_REF][START_REF] Nagashio | Contact resistivity and current flow path at metal/graphene contact[END_REF].

After fabrication of the first electrode, we transferred and etched the first hBN-graphene layer by the methods introduced in Section 2.5.3. Firstly, 29 nm thickness hBN and monolayer graphene are successive transferred to the

Conclusion

In this chapter we proposed and numerically demonstrated a graphene-based EOM integrated on MNC hybrid plasmonic waveguide. Due to the remarkable enhancement of in-plane electric field components of hybrid plasmonic mode within one-atom-thick graphene, sub-λ scale footprint of proposed EOM was achieved without sacrifice of the modulation depth. Numerical analyses show that the EO nanomodulator presents a modulation depth as high as 4 dB/µm at non-resonant wavelength of 1.65 µm, with a transmission loss of 2.6 dB. Due to the sub-λ scale footprint of the proposed EO nanomodulator, the computed modulation speed is up to 39 GHz and the computed energy consumption is as low as 21.2 fJ/b.

General conclusion and future work

Electro-optic modulators (EOMs) link electrical and optical binary signals.

They are key short-reach interconnect components in current and future optical communication networks. EOMs based on lithium niobate, germanium, and silicon have been widely discussed, however, they suffer from low modulation depth and bulky footprint due to weak light-matter interaction (Table 1.3). The integration of graphene on silicon photonics for optical modulation is a viable solution for enhancement of light-matter interaction however, the modulation depth is still limited to 0.16 dB/µm (Table 1.4). Several configurations that combine graphene and plasmonic waveguides have been considered in nanophotonic and plasmonic scientific communities, nevertheless, hybrid modulators still present limited modulation depth of 0.5 dB/µm (Table 1.5). In this context, the aim of this work is to enhance the interaction between light and graphene integrated on photonic waveguide with the help of plasmonic structures and to design a plasmonic-enhanced graphene-based EOM integrated on photonic waveguide that reaches future performances requirements of EOM (Table 1.1).

In view of intrinsic weak interaction between graphene and photonic modes, further enhancement of graphene-light interaction was achieved by MNC hybrid plasmonic waveguide. After introduction of optimization of standard 1D and 2D plasmonic waveguides toward enhanced in-plane electric field, we numerically analyzed and compared in-plane confinement of different hybrid plasmonic modes supported by nano-strip, nano-slot, and MNC hybrid plasmonic waveguides. Comparison with 10 and 6 times in-plane electric field density enhancement of nano-strip and nano-slot hybrid plasmonic waveguides, MNC hybrid plasmonic waveguide show 40 times in-plane enhancement relative to a configuration with graphene on a Si waveguide. This is a result of strong coupling between TE 0 photonic mode of Si waveguide and TE-polarized T 1 plasmonic mode supported by MNC plasmonic waveguide.

Chapter 4. Plasmonic-enhanced graphene based EO modulator Benefit from in-plane enhancement of MNC hybrid plasmonic waveguide, we finally proposed and numerically demonstrated a graphene-based EOM integrated on MNC hybrid plasmonic waveguide. The remarkable enhancement of in-plane electric field components and then strongly overlap with one-atom-thick graphene allow sub-λ scale footprint without sacrifice of the modulation depth. Numerical analyses show that the EO nanomodulator presents a modulation depth as high as 4 dB/µm at non-resonant wavelength of 1.65 µm, with a transmission loss of 2.6 dB. Due to the sub-λ scale footprint of the proposed EO nanomodulator, the modulation speed is up to 39 GHz and the energy consumption is as low as 21.2 fJ/b.

In addition, the coupling between TE 0 photonic mode and TE-polarized T 1 plasmonic mode has strong birefringence for TE-and TM-polarized photonic modes, this offers a opportunity for some polarization sensitive operation. On basis of this birefringence we numerically and experimentally demonstrates a nano-scale TM-pass polarizer. According to the experimental results, the extinction ratio of 23 dB/µm and insertion loss of 2.4 dB/µm were achieved, and its length is only 0.63 µm. We are currently working on the fabrication of several EOM samples with double graphene and hBN layers (Figure 4.12 a). This configuration replaces doped silicon by another graphene layer to connect with the right electrode, and hence hopeful to further improve electro-optic performances of graphenebased EOMs. Its working bandwidth increases to 70 nm if we consider the After final fabrication, we will measure the transmission spectra under different drive voltage. NSOM measurements will be also conducted to investigate different near-field characteristics and field attenuation in different drive voltage.

Future work

The electro-optical modulation speed of the fabricated EOM will be measured using a network analyzer. The dynamic energy consumption will be estimated according to used drive voltages in on and off states.

Although the optical performances of the designed EOM have been well optimized, further optimization in the electrical performances are possible. For example, the modulation speed is limited by capacitance and resistance. The resistance can be improved by graphene quality, electrodes contact with graphene, using double-layer graphene to avoid high Si doping resistance. And the capacitance expected to decrease by thicker hBN, however, it leads to the change of coupling and therefore effects on optical performances. In addition, there is another trade-off that thicker hBN leads to higher drive voltage and then higher energy consumption. As strong thermal radiation of high density integrated circuits, the investigation of thermal stability of the proposed EOM is required, nevertheless, both numerically and experimentally investigation of thermal stability would be a challenge.

As a complementary device, high-performance graphene-based photodetector is also limited by weak graphene-light interaction. Some plasmonicenhanced PDs have been experimentally demonstrated. For example, nanostrip, nano-slot, and metal bowtie chain on photonic waveguides. Due to the in-plane enhancement of plasmonic modes in MNC plasmonic waveguide, high photo-sensitivity is expected.

Finally, integrated nanophotonics with 2D materials is foreseen. La vitesse de modulation ou 3dB bande passante est le paramètre le plus important du système d'interconnexion optique. Pour un EOM, la vitesse de Chapter 5. Résumé en Français modulation peut être calculée comme suit [START_REF] Peng | Highly efficient graphene-on-gap modulator by employing the hybrid plasmonic effect[END_REF]:

f 3dB = 1 2πRC
(5.1)

Ici, R et C sont respectivement la résistance totale et la capacité totale de la EOM. D'autre part, la consommation d'énergie d'un EOM peut être calculée comme suit [START_REF] Christopher | Graphene electro-optic modulator with 30 GHz bandwidth[END_REF][START_REF] Qiu | Efficient modulation of 1.55 µm radiation with gated graphene on a silicon microring resonator[END_REF]:

E bit = 1 4 C∆V 2 (5.2)
où ∆V est la différence entre les tensions V d à l'état marche et à l'état arrêt. Étant donné que la taille de l'équivalent électrique de l'EOM, le transistor à effet de champ (FET),est court de plusieurs dizaines de nanomètres, l'objectif des EOMs est de fabriquer des dispositifs à longueur d'onde sans sacrifier les performances [START_REF] Fratalocchi | Nano-optics gets practical[END_REF]. Le faible encombrement au sol apporte également une faible résistance et capacité et fournit non seulement une vitesse de modulation élevée mais aussi une faible consommation d'énergie [START_REF] Sun | Optical modulators with 2D layered materials[END_REF]. La intensité de modulation est la performance la plus importante des EOM, définie comme la différence de transmission d'un EOM à l'état marche et à l'état arrêt. Afin de prendre en compte à la fois la performance (force de modulation) et la taille, la profondeur de modulation (MD) est redéfinie par dB/µm comme suit [START_REF] Graham T Reed | Silicon optical modulators[END_REF]: (5.4) où S est la superficie de la région active. Pour l'application récente des EOMs dans l'interconnexion optique, une profondeur de modulation supérieure à 3 dB/µm, une perte d'insertion inférieure à 3 dB/µm, avec un niveau de consommation d'énergie de 10 fJ/bit, une vitesse de modulation supérieure à 10 GHz et une taille de l'échelle de sub-longueur d'onde sont requises en 2020 [START_REF] Miller | Device requirements for optical interconnects to CMOS silicon chips[END_REF].

MD = 1 L 10
FoM = f 3dB S • E bit

Etat de l'art de l'EOM photonique

Les premières démonstrations des EOMs sont principalement basées sur l'effet électro-optique tel que l'effet Pockel (LiNbO 3 , polymère EO), l'effet d'électroabsorption (semi-conducteurs composés Ge et III-V) et l'effet de dispersion plasma (Si, ITO). La configuration de base des EOMs comprend une structure de diodes à broches (type p/couches intrinsèques/ type n), une jonction pn ou un condensateur MOS (diode métal-oxyde-semicon ductor). Néanmoins, en raison de la faible interaction entre la lumière et les matériaux électrooptiques, l'interféromètre Mach-Zehnder (MZI) et le résonateur à micro-anneau (MRR) sont largement utilisés pour améliorer l'interaction entre la lumière et les matériaux électro-optiques. De nombreux matériaux électro-optiques ont été utilisés dans les EOM, tels que LiNbO 3 , Ge, GeSi, Ge/SiGe puits quantiques (QWs), Si, polymère EO, ITO, etc.

Le niobate de lithium (LiNbO 3 ) est le premier matériau électro-optique appliqué dans les EOM. Néanmoins, la longueur de ce type de EOM se situe dans une plage de l'ordre du millimètre en raison de changement électrique d'indice est faible, par conséquent, de grands tensions ou de longues longueurs d'électrodes sont nécessaires pour obtenir une modulation suffisante. En fait, les EOMs à base de LiNbO 3 sont principalement utilisées pour la communication par fibre optique. Germanium (Ge) et des semiconducteurs composés III-V, tels que les composés de Ga, As, P, In, est une meilleure approche pour obtenir une faible consommation d'énergie de 10-100 fJ/bit et une modulation à grande vitesse grâce à son empreinte micrométrique (puis à sa faible capacité) et son temps de fonctionnement souspicosecondes [START_REF] Schmitt-Rink | How fast is excitonic electroabsorption?[END_REF]. Cependant, ces modulateurs d'électro-absorption sont intrinsèquement à bande étroite, et afin d'atteindre une grande profondeur de modulation, tous les modulateurs d'électro-absorption Ge-on-Si ont jusqu'à présent des niveaux de tension requis qui ne sont pas compatibles avec les niveaux de commande CMOS [START_REF] Timurdogan | An ultralow power athermal silicon modulator[END_REF]. Le silicium est le matériau le plus étudié pour les EOMs, car la prédominance du silicium en tant que semi-conducteur de choix pour la microélectronique permet d'attirer de manière rentable l'intégration des EOMs à l'électronique. Cependant, le Si a un effet d'électroabsorption intrinsèquement faible et ne présente pas d'effet électro-optique linéaire (effet Pockels) [START_REF] Soref | Electrooptical effects in silicon[END_REF]. En fait, la plupart des EOMs à base de Si sont conçues par un effet de dispersion plasmatique relativement faible. Pour améliorer l'interaction silicium-lumière, la plupart des EOMs à base de Si utilisent soit un MZI, soit un MRR. Le modulateur MZI au silicium mesure habituellement quelques millimètres de long et consomme donc des dizaines ou des centaines de milliwatts, une perte d'insertion relativement élevée (∼3-5 dB) et une tension appliquée (∼2-5 V). Le modulateur à micro-anneau en silicium a une faible consommation d'énergie, mais la longueur d'onde de travail est très sensible aux imperfections de fabrication et à la température [START_REF] Barwicz | Silicon photonics for compact, energy-efficient interconnects[END_REF].

EOMs photoniques à base de graphène

Le graphène est un matériau électro-optique émergent en raison de sa grande accordabilité électrique et de sa forte interaction lumière-matière. Le principe de fonctionnement des EOMs à base de graphène repose sur l'effet de blocage de Pauli qui, seulement lorsque le niveau de Fermi E f approche du point Dirac (-hω 0 /2 < E f < hω 0 /2), les photons incidents sont absorbés [START_REF] Wang | Gate-variable optical transitions in graphene[END_REF][START_REF] Li | Dirac charge dynamics in graphene by infrared spectroscopy[END_REF]. Le niveau de Fermi est modulé par la tension appliquée V d à travers un condensateur.

Le premier EOM à base de graphène intégré sur guide d'ondes Si a été présenté en 2011 par le groupe de Zhang [START_REF] Liu | A graphene-based broadband optical modulator[END_REF]. (Figure 5.2). En intégrant le graphène sur le guide d'onde photonique, le graphène se couple fortement avec le champ évanescent du mode optique guidé du guide d'onde photonique. Cette configuration a grandement amélioré la intensité d'interaction entre le graphène et la lumière, et il n'est déterminé que par la longueur de l'appareil. Ainsi, une absorption complète de la lumière peut être obtenue par une longueur d'appareil appropriée (Figure 5.2 a-b). Bien que l'absorption optique du graphène ait considérablement changé par rapport à celle du silicium et que la longueur de la EOM ait été optimisée à 40 µm, la profondeur de modulation d'une telle EOM est encore limitée à 0,1 dB/µm en raison du faible chevauchement entre le mode de propagation et le graphène, qui est au même niveau que les EOM à semiconducteur composé Ge et III-V. Le chevauchement lumière-graphène peut être légèrement augmenté par une configuration graphène double couche, mais la profondeur de modulation est toujours de 0,16 dB/µm [START_REF] Liu | Double-layer graphene optical modulator[END_REF]. L'amélioration de l'interaction graphènelumière est démontrée par l'intégration du graphène sur un résonateur microanneau en 2014 par Qiu et ses collègues [START_REF] Qiu | Efficient modulation of 1.55 µm radiation with gated graphene on a silicon microring resonator[END_REF]. La configuration a un encombrement au sol plus faible, et une modulation vitesse plus rapide. Néanmoins, ces dispositifs souffert d'une bande passante de travail étroite (< 0.1 nm [START_REF] Dalir | Athermal broadband graphene optical modulator with 35 GHz speed[END_REF]), d'une sensibilité élevée à la tolérance de fabrication et d'une faible stabilité thermique due à l'application du résonateur. Comme LiNbO 3 et les EOM à base de Si, le MZI a été utilisé dans certains modèles théoriques de EOM à base de graphène [START_REF] Hao | Ultra-compact optical modulator by graphene induced electro-refraction effect[END_REF][START_REF] Sorianello | Design optimization of single and double layer graphene phase modulators in SOI[END_REF]. Mais plusieurs centaines de micromètres de long ne conviennent pas aux EOMs à base de graphène. Jusqu'à présent, les EOMs à base de graphène sont plus prometteuses que les EOMs à base de Si. En raison de l'interaction plus forte des matériaux photosensibles, la longueur des EOMs à base de graphène a été réduite à l'échelle des dizaines de microns, qui est beaucoup plus petite que celle des EOMs à base de silicium. De plus, les EOMs à base de graphène ont une plus grande profondeur de modulation, le même niveau de vitesse de modulation, mais elles souffrent d'un petit inconvénient en termes de perte d'insertion et surtout de consommation d'énergie. A l'exception d'une consommation d'énergie plus élevée, les autres performances des EOMs à base de graphène sont au même niveau que celles des EOMs à base de semiconducteurs composés Ge et III-V.

EOMs photoniques à base de graphène améliorées par plasmonique

Comme l'interaction faible entre le graphène et la lumière, l'amélioration de cette interaction est le principal défi des EOMs à haute performance. Les polaritons plasmoniques de surface (SPP) confinent fortement la lumière bien en dessous de la limite de diffraction [START_REF] Maier | Plasmonics: fundamentals and applications[END_REF] et sont donc des candidats appropriés pour l'amélioration de l'interaction graphène-lumière en augmentant le chevauchement des modes plasmoniques dans le graphène. En effet, le simple fait de placer un matériau 2D sur la surface d'un guide d'ondes plasmonique n'entraînera pas un chevauchement accru malgré la forte concentration du champ près de cette interface métal-diélectrique. Li et ses collègues ont démontré que le graphène a une forte sélectivité des composantes du champ électrique, ce qui signifie que seules les composantes dans le plan, parallèles au graphène, interagissent fortement avec le graphène [START_REF] Li | Optical absorption in graphene integrated on silicon waveguides[END_REF]. Une telle sélectivité nécessite une structure plasmonique spéciale, la Au bande plasmonic sur guide d'onde en silicium n'est plus adaptée aux 5.2. Graphène sur silicium photonique comme matériau actif pour EOM113 EOMs hautes performances. La conception théorique suivante a montré que la profondeur de modulation d'une telle configuration de 0.31 dB/µm à 1.23 dB/µm [START_REF] Huang | Waveguide-coupled hybrid plasmonic modulator based on graphene[END_REF][START_REF] Peng | Highly efficient graphene-on-gap modulator by employing the hybrid plasmonic effect[END_REF][START_REF] Sun | Highlyefficient graphene-based silicon hybrid plasmonic waveguide modulator[END_REF][START_REF] Zhu | Modeling of 2D graphene material for plasmonic hybrid waveguide with enhanced near-infrared modulation[END_REF]. Et il a aussi été démontré expérimentalement qui est environ 0.25 dB/µm [START_REF] Khan | Sub-voltage graphene-plasmon based electro-absorption modulator[END_REF]. En revanche, les EOMs améliorées plasmoniques à fente Au sont plus prometteuses dans les EOM à hautes performances (Figure 5.3 b). Néanmoins, théoriquement, la profondeur de modulation de prédiction de la modulation du plasmonique à Au fente sur une configuration de guide d'onde hybride silicium est de 0.524 dB/µm [START_REF] Luan | Design and optimization of a graphene modulator based on hybrid plasmonic waveguide with double lowindex slots[END_REF] et la valeur mesurée est de 0.13 dB/µm [START_REF] Ding | Efficient electro-optic modulation in low-loss grapheneplasmonic slot waveguides[END_REF] ce qui ne est pas encore suffisants.

Objectif de la thèse

Dans ce contexte, ce manuscrit de doctorat se concentre sur l'amélioration de la conception d'un EOM plasmonique à base de graphène qui répond à toutes les exigences de performance des récentes applications d'interconnexion optique. Un mode de Bloch plasmonique spécial supporté par une chaîne de nanoparticules métalliques (MNC) sera présenté. En raison du confinement bien en dessous de la limite de diffraction et du fort champ électrique dans le plan du MNC en configuration guide d'ondes Si, il a un potentiel énorme en EOM haute performance à base de graphène, par exemple, l'empreinte sublongueurs d'onde entraîne une vitesse de modulation élevée pouvant atteindre 42 GHz et une consommation énergétique de 12 fJ/bit. Et une profondeur de modulation élevée de 4 dB/µm et une perte d'insertion de 2.5 dB/µm lui permettent de répondre à toutes les exigences de performances d'une interconnexion optique récente.

Graphène sur silicium comme matériau actif 2D pour EOM

Propriétés électroniques du graphène

Le graphène, un semi-métal à bande interdite nulle, est une monocouche plate d'atomes de carbone compactée dans un réseau hexagonal bidimensionnel (2D) [START_REF] Ak Geim | The rise of graphene[END_REF]. En raison de ses propriétés électriques et optiques uniques, le graphène a attiré une attention particulière dans les domaines de l'électricité (transistors, électrodes transparentes en raison de sa très grande mobilité), de la photonique (modulateurs électro-optiques, photodétecteurs, polariseurs en raison de la bande énergétique du graphène accordable électrique) et de la plasmonique (graphène plasmonique induit par une transition intrabandes des électrons).

Ces merveilleuses propriétés électriques et optiques du graphène proviennent de sa structure de bande d'énergie spéciale sans bande interdite. La structure de bande d'énergie du graphène est accordable par dopage. Par exemple, le dopage induit par adsorption (ou dopage par transfert de surface) [START_REF] To Wehling | Molecular doping of graphene[END_REF][START_REF] Wang | N-doping of graphene through electrothermal reactions with ammonia[END_REF], le dopage de substitution chimique [START_REF] Liu | Chemical doping of graphene[END_REF], le dopage électrique [START_REF] Wang | Gate-variable optical transitions in graphene[END_REF]. Le dopage électrique est obtenu en changeant la tension appliquée au graphène [START_REF] Kostya S Novoselov | Electric field effect in atomically thin carbon films[END_REF][START_REF] Das | Monitoring dopants by raman scattering in an electrochemically topgated graphene transistor[END_REF], qui est donnée par [START_REF] Ansell | Hybrid graphene plasmonic waveguide modulators[END_REF]

E f = hv f π 0 d et |V d + V 0 | (5.5)
où t et d sont respectivement l'épaisseur et la permittivité du diélectrique de la structure du condensateur qui permettait d'appliquer une tension de commande V d , V 0 est le décalage de voltage causé par le dopage naturel, qui peut être fixé à 0.

Modèle de conductivité de surface anisotrope

Le graphène 2D est caractérisé par une conductivité de surface, qui consiste en un terme interbande σ inter et un terme intrabande σ intra , peut être exprimé comme un tenseur [START_REF] Chatzidimitriou | Rigorous calculation of nonlinear parameters in graphenecomprising waveguides[END_REF]:

σ s =    σ xx σ xy 0 σ yx σ yy 0 0 0 0    , (5.6)
où σ xy = σ yx ≈ 0 en l'absence d'un magnétique externe, et σ xx = σ yy = σ. La conductivité dans le plan σ peut être déduite du formalisme de Kubo par [START_REF] Hanson | Dyadic green's functions and guided surface waves for a surface conductivity model of graphene[END_REF][START_REF] Falkovsky | Optical properties of graphene[END_REF]: Le taux de diffusion Γ peut être calculé par [START_REF] Thongrattanasiri | Complete optical absorption in periodically patterned graphene[END_REF] Γ = ehv 2 f /(2µE f ) (5.10) où v f est la vitesse de Fermi modifiée par le substrat. La mobilité du graphène dépend du substrat. Mais en fait, l'effet du substrat sur le taux de diffusion est très faible compte tenu de la faible mobilité du graphène sur SiO 2 de 4 400 ∼ 17 000 cm 2 V -1 s -1 . Par conséquent, le substrat du graphène n'a presque pas d'effet sur la conductivité du graphène (Figure 5.4 a). En changeant la tension appliquée et donc le niveau de Fermi, une modulation drastique de la conductivité du graphène de 1.27 µS à 60.85 µS peut être obtenue à 1.55 µm. La contribution des termes interbande et intrabande est clairement indiquée à la Figure 5.4 (b). Comme nous le voyons, la partie réelle de la conductivité, liée à l'absorption de la lumière par le graphène, ne dépend que du terme interbande lorsque le niveau de Fermi est inférieur à la moitié de l'énergie photonique 0.4 eV à 1.55 µm. Lorsque le niveau de Fermi est supérieur à 0.4 eV, les transitions interbande sont diminuées.

σ(ω, E f , Γ, T) = σ inter (ω, E f , Γ, T) + σ intra (ω, E f , Γ, T) (5.7) σ inter (ω, E f , Γ, T) = ie 2 (ω + i2Γ) πh 2 ∞ 0 f d (-ξ) -f d (ξ) (ω + i2Γ) 2 -2ξ h 2 dξ (5.8) σ intra (ω, E f , Γ, T) = - ie 2 πh 2 (ω + i2Γ) ∞ 0 ξ ∂ f d (ξ) ∂ξ - ∂ f d (-ξ) ∂ξ dξ (5.9)

Formule universelle d'absorption du graphène

Selon la loi d'Ohm sous sa forme différentielle, la densité de courant de surface J S dans le graphène est [START_REF] Chatzidimitriou | Rigorous calculation of nonlinear parameters in graphenecomprising waveguides[END_REF]:

J S = σE =    σ xx 0 0 0 σ yy 0 0 0 0       E x E y E z    . ( 5
Q grp = 1 2 S J S • EdS = 1 2 S σ|E | 2 dS , ( 5

Graphène sur silicium photonique comme matériau actif pour EOM117

En intégrant les hétérostructures graphène/hBN van der Waals sur un guide d'ondes photonique, l'interaction lumière-graphène peut être améliorée par un meilleur chevauchement de la lumière sur la couche de graphène 2D et ainsi une absorption complète peut être obtenue. La Par la formule universelle d'absorption du graphène 5.12, la profondeur de modulation élevée des EOMs, qui provient d'une absorption lumineuse élevée à l'état off, dépend de l'intensité du champ électrique en plan. Pour quantifier le confinement du champ électrique dans le plan du graphène, nous définissons le rapport de confinement γ = ρ /ρ total , où ρ total et ρ sont respectivement la densité de champ électrique total et dans le plan. Dans la

Fabrication et test de graphène/hBN sur guide d'ondes

Si

L'hétérostructure graphène-hBN intégrée sur silicium photonique a été fabriquée par empilage successif de hBN exfolié et de graphène CVD. Un flocon de hBN est exfolié mécaniquement sur un substrat de Si. Après l'exfoliation, le premier flocon d'hBN est sélectionné et transféré sur un guide d'ondes Si sélectionné d'une puce Si par transfert à sec. Le graphène a été synthétisé par CVD sur une feuille de cuivre de 100-µm d'épaisseur de Nilaco, 99.96% [START_REF] Van Luan | Seamless stitching of graphene domains on polished copper (111) foil[END_REF]. Le PMMA a été enduit sur la feuille de Cu puis une solution aqueuse de HCl a été utilisée pour graver le Cu pendant 1h. L'échantillon a été trempé dans de l'acétone pour dissoudre le PMMA, puis rincé à l'IPA et séché à l'air. Un deuxième flocon hBN est choisi parmi le substrat SiO 2 /Si, aligné par microscopie optique et transféré à sec sur le premier flocon hBN (Figure 5.7 a).

Pour éliminer l'excès de graphène et de hBN de la puce Si, un motif est défini autour de l'hétérostructure graphène-hBN par lithographie par faisceau d'électrons (EBL). Après EBL, nous développons l'échantillon en l'im- 5.8 a) [START_REF] Ferrari | Raman spectrum of graphene and graphene layers[END_REF]. Deux pics Raman importants sont mesurés, les pics G et 2D à 1587 cm -1 (Figure 5.8 b) et 2648 cm -1 (Figure 5.8 c), respectivement. Le pic D, qui indique un défaut de graphène, ne se produit que dans la région ii, où le graphène est sur SiO 2 [START_REF] Cançado | Quantifying defects in graphene via raman spectroscopy at different excitation energies[END_REF][START_REF] Graf | Spatially resolved raman spectroscopy of single-and few-layer graphene[END_REF]. hBN est identifié par le pic à 1364 cm -1 [START_REF] Roman V Gorbachev | Hunting for monolayer boron nitride: optical and raman signatures[END_REF]. La position de pointe G est inférieure d'environ 2 cm -1 dans la région iv par rapport à celle des régions ii et iii (Figure 5.8 b), ce qui est conforme au rapport précédent [START_REF] Chattrakun | Gate dependent Raman spectroscopy of graphene on hexagonal boron nitride[END_REF]. Le rapport d'intensité I 2D /I G dans les régions ii, iii et iv est respectivement 2,1, 2,8, 3,8. Ces valeurs indiquent une qualité de graphène plus élevée dans la région iv que dans les autres régions en raison de la faible rugosité de la surface et d'une réduction des impuretés modifiées du hBN.

Modes Plasmoniques pour exalter les propriétés du graphène

L'amélioration de l'interaction du graphène-lumière requiert des composantes de champ électrique à plan élevé qui se chevauchent dans une couche de graphène d'un atome d'épaisseur. La capacité d'amélioration dans le plan des guides d'ondes est quantifiée par sa densité de champ électrique dans le plan ρ || intégral dans le plan graphène et le rapport de confinement dans le plan, γ en complément. En raison de la limite de diffraction de la lumière, la capacité de confinement de la lumière dans les guides d'ondes photoniques ne peut pas diminuer à l'infini (avec la limite d'échelle de longueur d'onde λ 0 /4n, où λ 0 est la longueur d'onde dans le vide, n est l'index réfractif entourant le guide d'ondes photonique [START_REF] Bozhevolnyi | Plasmonic nanoguides and circuits[END_REF]). En revanche, les structures plasmoniques rupture de la limite de diffraction permettent un confinement de la lumière à l'échelle nanométrique aussi petit que quelques nanomètres [START_REF] Dmitri | Plasmonics beyond the diffraction limit[END_REF]. Cela permet d'améliorer le chevauchement de la lumière dans la couche de graphène et donc de réduire la longueur des EOM à base de graphène de dizaines de micromètre à une échelle inférieure à la longueur d'onde. La composante latérale dans le plan E y se produit lorsque la largeur des structures plasmoniques n'est pas infinie puisqu'elles fournissent un confinement 2D dans le plan transversal à la direction de propagation. Pour des applications pratiques dans le domaine du guidage d'ondes, de nombreux guides d'ondes plasmoniques 2D différents à fort confinement latéral ont été démontrés, par exemple, une nano-bande métallique sur un substrat diélectrique. [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 | Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of asymmetric structures[END_REF][START_REF] Joachim R Krenn | Nondiffraction-limited light transport by gold nanowires[END_REF][START_REF] Zia | Near-field characterization of guided polariton propagation and cutoff in surface plasmon waveguides[END_REF], guides d'ondes nano-fente sous forme de nanoécarts rectangulaires dans des films métalliques minces [START_REF] Liu | Novel surface plasmon waveguide for high integration[END_REF][START_REF] Veronis | Guided subwavelength plasmonic mode supported by a slot in a thin metal film[END_REF][START_REF] David Fp Pile | Two-dimensionally localized modes of a nanoscale gap plasmon waveguide[END_REF][START_REF] David Fp Pile | On long-range plasmonic modes in metallic gaps[END_REF], cales métalliques tranchantes [START_REF] David Fp Pile | Theoretical and experimental investigation of strongly localized plasmons on triangular metal wedges for subwavelength waveguiding[END_REF][START_REF] Boltasseva | Triangular metal wedges for subwavelength plasmon-polariton guiding at telecom wavelengths[END_REF][START_REF] Moreno | Channel plasmon-polaritons: modal shape, dispersion, and losses[END_REF][START_REF] Yan | Guided plasmon polariton at 2d metal corners[END_REF], nano-rainures dans des substrats métalliques [START_REF] Moreno | Channel plasmon-polaritons: modal shape, dispersion, and losses[END_REF][START_REF] Yan | Guided plasmon polariton at 2d metal corners[END_REF][START_REF] Iv Novikov | Channel polaritons[END_REF][START_REF] David | Channel plasmon-polariton in a triangular groove on a metal surface[END_REF][START_REF] Dmitri | Single-mode subwavelength waveguide with channel plasmon-polaritons in triangular grooves on a metal surface[END_REF][START_REF] Sergey I Bozhevolnyi | Channel plasmon-polariton guiding by subwavelength metal grooves[END_REF], chaînes de nanoparticules métalliques (MNCs) [START_REF] Quinten | Electromagnetic energy transport via linear chains of silver nanoparticles[END_REF][START_REF] Maier | Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides[END_REF], et ainsi de suite. En raison de la facilité de fabrication et de la difficulté d'intégration du graphène sur les coins et rainures tranchants, les guides d'ondes plasmoniques nano-bande, nano-fente et MNC conviennent mieux à l'amélioration de l'interaction graphène-lumière.

Amélioration des guides d'ondes plasmoniques

La nano-bande métallique placée sur un substrat diélectrique (Encarts dans la Figure 5.9 a) est le guide d'onde plasmonique 2D le plus naturel, néanmoins, le problème principal de cette configuration est que les modes supportés par les nano-bandes sont très proches de la ligne lumineuse, et donc susceptibles de s'échapper dans le substrat diélectrique en raison de grand indice de réfraction du diélectrique et de la taille limitée des pastilles qui limite significativement son application [START_REF] Bozhevolnyi | Plasmonic nanoguides and circuits[END_REF][START_REF] Berini | Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of asymmetric structures[END_REF]. Dans le cas où w et t sont petits par rapport à la longueur d'onde dans la région proche infrarouge, seul le mode as 0 b peut être trouvé, les autres modes sont fuyants. La densité de champ électrique optimisée dans le plan et rapport de confinement dans le plan de as 0 b mode de 30 nm × 50 nm nanobande à 1.55 µm est représenté Rainurer les guides d'ondes plasmoniques comme indiqué dans l'encadré de la Figure 5.9 (b) sont plus utilisés que les guides d'ondes plasmoniques 2D en raison de leur mode de confinement remarquable, de leur longueur de propagation raisonnable et de leurs propriétés intrinsèques à large bande [START_REF] Cai | Elements for plasmonic nanocircuits with three-dimensional slot waveguides[END_REF]. Il n'a qu'un seul mode fondamental avec un fort confinement latéral, 5.3. Modes Plasmoniques pour exalter les propriétés du graphène 123 qui s'améliore rapidement pour une largeur de fente plus petite [START_REF] Dmitri | Plasmonics beyond the diffraction limit[END_REF]. Bénéficier d'un confinement en mode fort dans la composante latérale, dans le plan est fortement confiné dans l'espace avec une intensité plus élevée que la composante hors plan. Selon la densité de champ électrique dans le plan optimisée et le rapport de confinement de 80 nm × 80 nm nano-fente à 1.55 µm (Figure 5.9 b), la densité de champ électrique dans le plan la plus élevée est de 0.015 dans l'interface verre or. Pour améliorer l'interaction graphènelumière placée dans l'interface or-diélectrique, nous avons simulé la densité de champ électrique dans le plan en fonction de la largeur de fente w. Nous notons que pour un intervalle fondamental donné, la largeur de l'intervalle du mode SPP w n'affecte pas le champ électrique dans le plan, mais lorsque l'intervalle w est plus large, l'indice effectif du mode et la perte diminuent. Notez que la plage de coupure de la largeur de fente est w < 40nm, et que le mode SPP de la fente fondamentale fuit lorsque w > 85nm. L'épaisseur t de l'or a le même effet que t ne change pas l'intensité de la composante dans le plan et le ratio de confinement dans le plan, il change seulement l'indice et la perte effectifs du mode. En revanche, le paramètre le plus important est l'indice de réfraction n du substrat. Une densité électrique dans le plan et un rapport de confinement dans le plan plus élevés sont obtenus par un substrat à faible n. La raison en est que le substrat à faible n évite les fuites de mode dans le substrat, tout en affaiblissant les effets de la coupure w et t.

Par comparaison, la capacité de confinement dans le plan des guides d'ondes plasmoniques à nano-fentes et des guides d'ondes plasmoniques à nano-bandes est au même niveau (Figure 5.9 a, b). Néanmoins, les guides d'ondes plasmoniques nano-fente ont une perte de propagation beaucoup plus faible. En fait, les guides d'ondes plasmoniques à fente sont les configurations les plus utilisées dans l'amélioration de l'interaction graphènelumière. En résumé, le guide d'onde plasmonique hybride MNC a la densité de champ électrique dans le plan la plus élevée, mais il nécessite une petite erreur de fabrication en raison de sa structure fine et de sa forte propriété de résonance. Le guide d'onde plasmonique hybride nano-bande est également une structure résonnante, mais il n'a pas d'exigence de fabrication stricte. Un inconvénient est le confinement plus faible dans le plan. Pour les applications de confinement dans le plan et d'exploitation à large bande, le guide d'ondes plasmonique hybride nano-fente non résonnant est un bon choix. De plus, l'intégrale de ρ || le long de l'axe z est la densité volumique des composantes de polarisation dans le plan, qui sont respectivement 0.05, 0.12, 0.10 et 0.39 pour ρ || des différentes configurations de la Figure 5.11. Cela signifie que le guide d'onde plasmonique hybride MNC a également la plus grande fraction de polarisation dans le plan, ce qui est utile dans certaines applications de manipulation de polarisation telles que le polariseur passe-TM. TE 0 est plus diffusant que le mode TM 0 et que le mode TM 0 est donc confiné dans le guide d'ondes Si (Encarts de la Figure 5.13 a). La source lumineuse de la mesure de transmission est produite par un laser à super-continuum accordable avec un pas de balayage de 0.3 nm. L'état de polarisation est contrôlé par un contrôleur de polarisation de fibre (FPC), puis injecté dans l'échantillon par une fibre optique de maintien de polarisation (PM) à l'aide d'un couplage d'extrémité. La lumière à la sortie de l'échantillon est captée par une autre fibre PM optique et ensuite transformée en signal électrique par un détecteur (Figure 5.13 b). Enfin, le signal détecté est normalisé par le signal du guide d'ondes de référence sans MNC.

Chaine de nanoparticules métalliques intégrée au guide d'ondes Si

Modulateur électro-optique à base de graphène et plasmons

Les modulateurs optiques sont des composants essentiels de la communication optique et des réseaux neuromorphes photoniques. Le graphène sur guides d'ondes en silicium a été largement rapporté pour la modulation optique mais la profondeur de modulation est limitée à 0.1 dB/µm et une longueur de modulateur d'environ 40 µm [START_REF] Liu | A graphene-based broadband optical modulator[END_REF][START_REF] Hu | Broadband 10 Gb/s operation of graphene electro-absorption modulator on silicon[END_REF]. Des modulateurs plus courts et une profondeur de modulation supérieure peuvent être obtenus avec des modes plasmonic, mais, grâce à la nature magnétique transversale des modes plasmonic et la sélectivité en plan de graphène, des modes traditionnels plasmonic interagissent légèrement avec graphène. En nous basant sur la sélectivité du graphène dans le plan, nous avons démontré une amélioration géante du guide d'ondes plasmonique hybride MNC dans le plan, ce qui est utile pour améliorer l'interaction graphène-lumière et donc le modulateur électro-optique (EOM) haute performance. Dans cette partie, nous proposons un EOM plasmonique amélioré à base de graphène sur silicium combinant les résultats précédents, diminue la surface du dispositif à 0.63 λ de 0.3 λ sans sacrifice de sa profondeur de modulation. Le port d'entrée du guide d'ondes Si est éclairé avec son mode fondamental TE. Le mode de Bloch plasmonique dans le guide d'ondes MNC est efficacement excité avec le mode TE à 1550 nm comme nous l'avons vu en utilisant la relation de dispersion (Figure 5.10). Le fort confinement du champ électrique dans le plan du mode plasmonique apporte une forte densité de champ électrique dans le plan (Figure 5.11) et améliore ainsi l'interaction avec le graphène. La transmission optique est modulée en appliquant une voltage à travers le condensateur formé par l'électrode supérieure sur le graphène, la couche diélectrique hBN et l'électrode inférieure sur une couche de connexion Si (Figure 5.14 b). La modulation de la lumière on/off est obtenue en décalant la voltage de commande puis le niveau de Fermi (Équation 5.5, encarts de la Figure 5.14 a).

Description du modulateur électro-optique

Performances optiques

La profondeur de modulation et la perte d'insertion des EOM sont définies par leur transmission (Équation 5.3). Pour simuler la transmission de la lumière dans l'HPM, on a utilisé la méthode du domaine temporel par différence finie mise en oeuvre par Lumerical. Les spectres de transmission dans le mode TE fondamental sont obtenus par expansion en mode propre. Des données précises sur la dispersion de l'or, mesurées par ellipsométrie, ont été utilisées dans les simulations. L'indice de réfraction du hBN est réglé à 1,98 [START_REF] Ansell | Hybrid graphene plasmonic waveguide modulators[END_REF]. Le graphène est pris en compte à l'aide d'un tenseur de conductivité 2D strict. (Équation 5.6). Le spectre de profondeur de modulation simulé est supérieur à 3 dB/µm dans une gamme de longueurs d'onde allant de 1.491 µm à 1.685 µm, qui couvre la totalité des bandes C, L et U de la fenêtre de télécommunications (Figure 5.15 a). De plus, la longueur de la HPM est de 1.05 µm et la profondeur de modulation atteint 20.9 dB/µm à la longueur d'onde de résonance (λ = 1.551 µm), qui sont, à notre connaissance, les plus courtes EOM à base de graphène avec la plus grande profondeur de modulation [START_REF] Christopher | Graphene electro-optic modulator with 30 GHz bandwidth[END_REF][START_REF] Qiu | Efficient modulation of 1.55 µm radiation with gated graphene on a silicon microring resonator[END_REF][START_REF] Liu | A graphene-based broadband optical modulator[END_REF][START_REF] Yang | Low-chirp highextinction-ratio modulator based on graphene-silicon waveguide[END_REF][START_REF] Phatak | Design of electro-optic modulators based on graphene-onsilicon slot waveguides[END_REF][START_REF] Li | Ultrafast all-optical graphene modulator[END_REF][START_REF] Liu | Double-layer graphene optical modulator[END_REF][START_REF] Ding | Effective electrooptical modulation with high extinction ratio by a graphene-silicon microring resonator[END_REF][START_REF] Xu | Design of an electro-optic modulator based on a siliconplasmonic hybrid phase shifter[END_REF][START_REF] Koester | High-speed waveguide-coupled graphene-on-graphene optical modulators[END_REF]. Néanmoins, la perte d'insertion est élevée près de la longueur d'onde de résonance (ligne pointillée sur la Figure 5.15 a).

Pour une mise en oeuvre pratique, nous pouvons choisir une plage spectrale allant de 1.636 µm à 1.686 µm sur laquelle la profondeur de modulation est supérieure à 3 dB/µm tandis que la perte d'insertion est inférieure à 3 dB/µm. Ce spectre de travail est plus large que celui des EOM résonnantes basées sur le résonateur en micro-ring et le graphène [START_REF] Christopher | Graphene electro-optic modulator with 30 GHz bandwidth[END_REF][START_REF] Qiu | Efficient modulation of 1.55 µm radiation with gated graphene on a silicon microring resonator[END_REF]. A 1.65 µm, la profondeur de modulation est de 4 dB/µm tandis que la perte d'insertion est de 2.6 dB. La distribution du champ électrique à l'état On présente une amélioration plasmonique supérieure à celle à l'état Off (Figure 5.15 b, c). Notez que la résonance peut être décalée en changeant la taille du MNP. Un décalage bleu est obtenu pour la dimension plus courte de l'axe principal d 1 (Encart dans la Figure 5.15 a). Dans la simulation des spectres de transmission, nous n'avons pas pris en compte la contribution du silicium dopé car elle n'est que de 0.0041 dB/µm pour la profondeur de modulation et 0.0042 dB/µm pour la perte d'insertion. Nous traitons le silicium comme semi-conducteur non dopé dans les simulations optiques et le semiconducteur dopé dans les simulations électriques. La réflexion et la diffusion de la HPM sont respectivement de 0.1 dB et 0.36 dB, ce qui est négligeable dans l'ensemble du spectre considéré ici grâce à l'efficacité de couplage élevée entre les modes guidés photonique et plasmonique.

Performances électriques

Outre la profondeur de modulation, la vitesse de modulation et la consommation d'énergie sont deux paramètres importants. Ces paramètres dépendent des propriétés électriques du modulateur. Dans notre cas, le circuit électrique équivalent peut être modélisé comme un filtre passe-bas (Figure 5.16) [START_REF] Sorianello | Design optimization of single and double layer graphene phase modulators in SOI[END_REF][START_REF] Xiao | 25 gbit/s silicon microring modulator based on misalignment-tolerant interleaved pn junctions[END_REF], où R C et R grp sont respectivement la résistance des contacts en graphène et celle des plaques, C est la capacité hBN, R Si est la résistance Si, C air est la capacité entre deux plaques métalliques à travers l'air, R sub est la résistance du substrat, C box-L et C box-R sont les capacités de la couche d'oxyde enterrée. La résistance de contact de l'électrode droite et de la couche de raccordement Si peut être négligée pour une grande surface de contact [START_REF] Boberg | Contact resistance measurements of platinum-silicide and chromium contacts to highly doped n and p silicon[END_REF]. Des valeurs allant de 100 Ω/sq à 500 Ω/sq ont été rapportées pour la résistance de la feuille de graphène et de 100 Ωµm à 1000 Ωµm pour la résistance de contact du graphène selon le métal choisi. Pour évaluer le pire des scénarios, on a utilisé R grp = 500 Ω/sq et R C = 1000 Ωµm.

Les capacités peuvent être calculées à l'aide d'une formule de capacité à plaques parallèles. Néanmoins, pour plus de précision, nous utilisons un logiciel de simulation électrique Device by Lumerical pour simuler les capacités à une voltage appliquée de 10 V, en tenant compte à la fois de la capacité Habituellement, la vitesse de modulation d'un modulateur, aussi connu sous le nom de fréquence de coupure de 3 dB [START_REF] Dalir | Athermal broadband graphene optical modulator with 35 GHz speed[END_REF], est calculée en utilisant l'équation 5.1 si elle est modélisée comme un filtre passe-bas du premier ordre. Néanmoins, pour un filtre passe-bas compliqué comme Figure 5.16, nous ne pouvons pas calculer la vitesse de modulation par 1/(2πRC). Par la définition de la fréquence de coupure de 3 dB, c'est la fréquence pour laquelle la puissance appliquée dans le filtre tombe à 3 dB correspondant approximativement à la moitié de la puissance d'entrée. Pour un modulateur, la puissance optique de sortie est approximativement proportionnelle à l'amplitude de la voltage, par conséquent, la fréquence de coupure de 3 dB correspond à la fréquence de coupure de 3 dB de l'amplitude de la voltage V C sur la capacité du dispositif C [START_REF] Xiao | 25 gbit/s silicon microring modulator based on misalignment-tolerant interleaved pn junctions[END_REF]. Le rapport entre la magnitude de V C et V d calculée par la règle du diviseur de voltage est le suivant:

|V C | |V d | = Z EOM Z EOM + Z 0 • 1/(j2π f

L'évaluation des performances : figure de mérite

1 . 1

 11 Optical communications and interconnects from deep-space to on-chip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 A typical configuration of optical communication/interconnect system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 A typical EOM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Typical configuration of MZI and MRR based EOM . . . . . . 1.5 Typical configuration of cross section of EOM . . . . . . . . . . 1.6 Working principle of graphene-based EOM . . . . . . . . . . . 1.7 The first graphene-based waveguide-integrated EOM . . . . . 1.8 Other configurations of graphene-based waveguide-integrated EOM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Configurations of plasmonic enhanced graphene-based EOMs 2.1 The lattice structure of graphene . . . . . . . . . . . . . . . . . 2.2 The energy band structure of graphene . . . . . . . . . . . . . 2.3 Doping of graphene . . . . . . . . . . . . . . . . . . . . . . . . . 2.

3 . 5 138 B. 1 142 xiiiof Tables 1 . 1

 35138114211 In-plane electric field of short-range SPP mode under different refractive index of dielectric part . . . . . . . . . . . . . . . . . 3.6 In-plane electric field of short-range SPP mode under different thickness of gold . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 SPP mode supported by nano-strip plasmonic waveguide . . 3.8 The effect of nano-strip width w on in-plane confinement . . . 3.9 The effect of n on in-plane confinement . . . . . . . . . . . . . 3.10 SPP mode supported by nano-slot plasmonic waveguide . . . 3.11 The effect of gap width on in-plane confinement . . . . . . . .3.12 The effect of refractive index of substrate on in-plane confinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.13 Schematic diagram of periodic plasmonic waveguide and its modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.14 Typical electric field distribution of T 1 mode of MNC . . . . . 3.15 Dispersion curve of MNC embedded in a homogeneous medium 3.16 Hybrid mode of MNC on Si waveguide configuration . . . . . 3.17 Dispersion curve of MNC plasmonic waveguide integrated on Si photonic waveguide . . . . . . . . . . . . . . . . . . . . . . . 3.18 Schematic diagrams of in-plane enhancement structures in this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.19 In-plane confinement of nano-strip on Si waveguide as a function of strip width . . . . . . . . . . . . . . . . . . . . . . . . . .xi 3.20 Electric field comprison of photonic mode in Si waveguide, plasmonic mode in isolated nano-strip waveguide, and hybrid plasmonic modes in the nano-strip hybrid plasmonic waveguide 3.21 In-plane confinement of nano-slot on Si waveguide as a function of gap width . . . . . . . . . . . . . . . . . . . . . . . . . . 3.22 In-plane confinement of different plasmonic waveguides . . . 3.23 Configuration of proposed TM-pass polarizer . . . . . . . . . . 3.24 Schematic diagram and electric field profile of proposed TMpass polarizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.25 Dispersion curve of proposed TM-pass polarizer . . . . . . . . 3.26 Transmission spectrum of proposed TM-pass polarizer . . . . 4.1 3D artwork of the plasmonic-enhanced modulator on Si photonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Modulation depth and transmission spectra of the EOM . . . 4.3 Influence of parameters into the modulation depth . . . . . . . 4.4 Insertion loss of doped-Si . . . . . . . . . . . . . . . . . . . . . 4.5 Reflection, scattering, and absorption spectra of proposed EOM 4.6 Equivalent electric circuit of the device . . . . . . . . . . . . . . 4.7 Simulated capacitances and resistance . . . . . . . . . . . . . . 4.8 Frequency response of proposed EOM . . . . . . . . . . . . . . 4.9 Scanning electron micrograph of fabricated MNC plasmonic waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10 Fabricated electrode . . . . . . . . . . . . . . . . . . . . . . . . . 4.11 Etching of graphene on hBN on electrode and Si waveguide . 4.12 Double-graphene layer plasmonic-enhanced sub-λ EOM . . . 4.13 Future work on the fabrication of a double-graphene plasmonicenhanced EOM . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 EOM typique . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 La première EOM intégrée à guide d'ondes à base de graphène 5.3 Configurations d'EOM à base de graphène amélioré par plasmoniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Conductivité électrique accordable du graphène à 1.55 µm . . 5.5 Schéma de la configuration graphique générale/hBN sur guide d'ondes Si . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Densité de champ électrique et rapport de confinement dans le plan en entrée des modes fondamentaux TE et TM . . . . . 5.7 Une hétérostructure hBN-graphène-HBN sur guide d'ondes Si xii 5.8 Spectroscopie Raman de différentes hétérostructures graphiques/hBN vdW sur guide d'ondes Si . . . . . . . . . . . . . . . . . . . . . 120 5.9 Confinement dans le plan de guides d'ondes plasmoniques à nano-bande et à nano-fente . . . . . . . . . . . . . . . . . . . . 122 5.10 Mode hybride de configuration MNC sur guide d'ondes Si . . 124 5.11 Confinement dans le plan de différents guides d'ondes plasmoniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 5.12 Schéma schématique du polariseur passe-TM proposé . . . . . 127 5.13 Spectre de transmission de polariseur TM proposé . . . . . . . 128 5.14 3D illustration du modulateur plasmonique amélioré sur la photonique Si . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 5.15 Profondeur de modulation et spectres de transmission de l'EOM131 5.16 Circuit électrique équivalent de l'appareil . . . . . . . . . . . . 133 A.1 The in-plane component of permittivity and refractive index of graphene at 1.55 µm . . . . . . . . . . . . . . . . . . . . . . . Modes analyses of graphene-hBN hetrostructures on Si waveguide using NSOM signal . . . . . . . . . . . . . . . . . . . . . . 140 B.2 NSOM measurement results under TM 0 mode incidence . . . 141 B.3 Calculated electric near-field distributions of the TM 0 mode propagating through different vdW heterostructures on Si waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 B.4 Simulated electric field distribution of TM 0 mode propagating through different integrated vdW heterostructures on Si photonics in xz plane . . . . . . . . . . . . . . . . . . . . . . . . . . List Performances requirements of EOM . . . . . . . . . . . . . . . 1.2 Classification of traditional EOM . . . . . . . . . . . . . . . . . 1.3 Performances of traditional EOMs . . . . . . . . . . . . . . . . 1.4 Experimental performances of graphene-based EOMs . . . . . 1.5 Experimental and theoretical performances of plasmonic enhanced graphene-based EOMs . . . . . . . . . . . . . . . . . . 2.1 Mobility of graphene on different substrates and configurations 2.2 RIE process parameters to etch graphene and graphene/hBN 4.1 Performances of graphene-based EOM integrated on photonic waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Performances de l'EOM à base de graphène intégré sur guide d'onde photonique . . . . . . . . . . . . . . . . . . . . . . . . .

FIGURE 1 . 2 :

 12 FIGURE 1.2: A typical configuration of optical communication/interconnect system. (a) The basic components of optical transmitter part. (b) Processing wavelengths of optical binary signals on the receiver part. [12]

FIGURE 1 . 3 :

 13 FIGURE 1.3: A typical EOM. (a) Schematic diagram of IBM's EOM. (b) Basic principle of an EOM. The input light is a continuous wave (CW) and the output light is optical binary signal.

FIGURE 1 . 4 :

 14 FIGURE 1.4: Typical configuration of MZI (a) and MRR (b)based EOM[START_REF] Rao | Heterogeneous microring and Mach-Zehnder modulators based on lithium niobate and chalcogenide glasses on silicon[END_REF].

FIGURE 1 . 5 :

 15 FIGURE 1.5: Typical configuration of MOS capacitor (a) [45], pn junction (b) and pin diode structure (c). Both MOS capacitor and pin diode structure form injection of electrons and holes by applied voltage around the waveguide. But the pn junction forms a large carrier depletion area with increasing reverse bias voltage [23].

FIGURE 1 . 6 :

 16 FIGURE 1.6: Energy band schematic diagrams illustration of graphene-based EOM. (a) In intrinsic graphene, incidence photons are absorbed in terms of symmetric interband transition (Both hω 1 and hω 2 ). (b) In n-doped graphene, an optical photon (hω 2 ) with energy less than 2E F cannot be absorbed. (c) In pdoped graphene, an optical photon (hω 2 ) with energy less than 2E F cannot be absorbed. (d) Interband plasmon absorption (hω 2 ) and intraband plasmon (hω 1 ) absorption. Here, Fermi level E F (n-doped and p-doped) is modulated by electric field (applied drive voltage). The optical photon (hω 1 ) always can be absorbed due to high photon energy [75].

FIGURE 1 . 7 :

 17 FIGURE 1.7: The first graphene-based EOM integrated on Si waveguide. (a) Under normal incidence, graphene has a universal absorption coefficient of 2.3%. (b) By integrating graphene on a photonic waveguide, 100% light absorption can be achieved by increasing the length of the device [69]. (c) 3D schematic illustration of the EOM. (d) Cross section of the EOM with electric field distribution. The right inset is the crosssection through the center of Si waveguide and its electric field profile [73].

FIGURE 1 . 8 :

 18 FIGURE 1.8: Other configurations of graphene-based waveguide-integrated EOM. (a) Double-layer graphene based EOM [80]. (b) Multilayer graphene in horizontal slot waveguide which is proved is not appropriate for graphenebased EOMs for TM modes incidence. (c) Graphene-based EOM on microring resonator [85]. (d) MZI-based graphene EOM integrating in Si waveguide [86].

FIGURE 1 . 9 :

 19 FIGURE 1.9: Configurations of plasmonic enhanced graphenebased EOMs. (a) Enhanced EOM based on Au plasmonic strip on silicon integrated waveguide [102]. (b) Au plasmonic slot on silicon integrated waveguide configuration [107].

FIGURE 2 . 1 :

 21 FIGURE 2.1: The lattice structure of graphene and other carbon materials. Graphene can be wrapped up into 0D bucky balls (fullerenes), rolled into 1D nanotubes (carbon nanotubes)or stacked into 3D graphite[START_REF] Ak Geim | The rise of graphene[END_REF].

Figure 2 .

 2 2 (b). Here the valence and conduction bands of graphene are cone-shaped and meet at the K point of the Brillouin zone, so undoped pristine graphene is a zero bandgap semimetal.

FIGURE 2 . 2 :

 22 FIGURE 2.2: The energy band structure of graphene. (a) Brillouin zone of hexagonal lattice structure of graphene. The Dirac cones are located at the K and K points. (b) Electronic dispersion in the hexagonal lattice of (a). Left is full band structure of graphene. Right inset is a zoom close to the Dirac points (at the K or K point in the Brillouin zone) [123].

FIGURE 2 . 3 :

 23 FIGURE 2.3: (a) Adsorbate-induced doping by SiC substrate. The left and right panels show n-type and p-type doping, respectively, while the center panels show the pure graphene layers [127]. (b) Electrical doping. The insets show the position of Fermi energy E f and Dirac point of graphene as a function of drive voltage V g [110].

FIGURE 2 . 4 :

 24 FIGURE 2.4: The effect of substrate on conductivity of graphene at 1.55 µm. (a) The effect of mobility on scattering rate. The real part and imaginary part of conductivity of graphene as function of mobility (b) and scattering rate (c) at 1.55 µm. Here, the Fermi level of 0.2 eV is used.

  8 and simulated by optoelectronics simulation software DEVICE of Lumerical are shown in Figure 2.5 (a).

FIGURE 2 . 5 :

 25 FIGURE 2.5: Electrical tunable conductivity of graphene at 1.55 µm. (a) Electrical tunability of Fermi level of graphene. The dielectric is 7 nm thick Al 2 O 3 , Al 2 O 3 substrate modified Fermi velocity is 1.17 ×10 6 m/s [140]. (b) The real part and imaginary part of conductivity of graphene as function of Fermi level E f .The interband and intraband terms are also plotted.

FIGURE 2 . 6 :

 26 FIGURE 2.6: Graphene absorption spectra under normal incident light at Fermi levels. (a) Schematic of normal incident light onto graphene. (b) Conductivity of undoped (E f = 0 eV) and doped graphene (E f = 0.5 eV). (c) Light absorption of undoped and doped graphene calculated by method i (Equation 2.9) and method ii (Equation 2.12). (d) Light absorption of graphene as a function of Fermi level at 1.55 µm.

FIGURE 2 . 7 :

 27 FIGURE 2.7: hBN and its application as substrate of graphene. (a) The lattice structure of hBN. N and B are nitrogen and boron atoms [131]. (b) High-resolution cross section ADF-STEM image of hBN/graphene/hBN heterostructures [71]. (c) Schematic of an EOM based on hBN/graphene/hBN heterostructures [146]. (d) Tunable hybrid plasmon-phonon polaritons supported by graphene/hBN heterostructures. Zoomin shows cartoon of graphene plasmon coupling to hBN phonon [122].

FIGURE 2 . 8 :

 28 FIGURE 2.8: Schematic of general graphene/hBN on Si waveguide configuration. (a) Three-dimensional schematic illustration of the graphene/hBN van der Waals heterostructures integrated on Si waveguide. (b) Longitudinal cross section of the integrated structure.

FIGURE 2 . 9 :

 29 FIGURE 2.9: Dispersion curve of graphene/hBN on Si waveguide configuration in Figure 2.8 (b). The real part of effective index (a) and loss (b) of the configuration as a function of wavelength of different photonic guide modes with different Fermi level. (c) Effective index and loss of different guide modes under different Fermi level.

FIGURE 2 . 10 :

 210 FIGURE 2.10: Electric field density and in-plane confinement ratio under fundamental TE (a) and TM modes (b) input. Insets show in-plane electric field distribution in yz plane of fundamental TE and TM modes.

FIGURE 2 . 11 :FIGURE 2 . 12 :

 211212 FIGURE 2.11: Loss under fundamental TE and TM modes input as a function of graphene position z graphene .

FIGURE 2 .

 2 FIGURE 2.14: A patterned hBN-graphene-hBN heterostructure on Si waveguide. Micrograph of the heterostructure on Si waveguide before (a) and after (b) pattern. The bottom and top hBN are indicated by blue and graphene dashed line, respectively. The mask of pattern is shown by red dashed line.

FIGURE 2 . 15 :

 215 FIGURE 2.15: Raman spectroscopy of different graphene/hBN vdW heterostructures on Si waveguide. (a) Raman spectroscopy of SiO 2 (region i), graphene (region ii), hBN-graphene vdW heterostructure (region iii) and hBN-graphene-hBN vdW heterostructures (region iv). Inset shows micrograph of the measured sample. (b) and (c) are respectively zoom in of G peak and 2D peak in different regions.

FIGURE 2 . 16 :

 216 FIGURE 2.16: Optical near-field setup and structure. (a) Setup for the measurement of the optical near field with a transmission NSOM. (b) Longitudinal cross section of the fabricated integrated structure. Regions i, ii, iii and iv indicate: bare Si waveguide, graphene, hBN/graphene, and hBN/graphene/hBN on Si waveguide, respectively. (c) Transverse cross section across region iv. The vdW heterostructures are separated by 100-nm-thick SiO 2 from the Si waveguide.

FIGURE 2 .

 2 FIGURE 2.17: NSOM imaging of the integrated photonic device. (a) Micrograph of the integrated vdW heterostructures on a Si waveguide. AFM topography (b) and NSOM images (c) at λ = 1550 nm. Graphene covered region is indicated by white dashed line. (d) AFM and NSOM signal profiles along the propagation direction of Si waveguide.

FIGURE 2 . 18 :

 218 FIGURE 2.18: Simulated electric near-field distribution of the TE mode propagating through different integrated vdW heterostructures on Si photonics at λ = 1550 nm. In-plane (a) and out-plane (b) electric field taken 20 nm above the surface of the structure. (c) Integrated electric field density ρ across the y-axis and and normalized absorption Q S of graphene calculated using Equation 2.11. Both case with and without (Indicated by subscript 0) graphene are plotted.

  photonic waveguide

FIGURE 3 . 1 :

 31 FIGURE 3.1: Light confinement in photonic and plasmonic waveguides. (a) Mode size of the fundamental TM mode in a typical photonic waveguide under different waveguide size d. (b) Mode size of SPP mode in cylindrical metal nanowire plasmonic waveguide for different waveguide size. λ 0 and λ 0sp are respectively the wavelength of photonic and plasmonic waveguide. The mode size is indicated by dashed lines and is calculated at e -1 level of the electric field [188].

2 FIGURE 3 . 2 :

 232 FIGURE 3.2: In-plane electric field of SPP mode supported by a metal-dielectric interface. (a) Configuration of metal-dielectric interface. The only mode supported by interface is shown in red dashed line. (b) Normalized electric field components and in-plane confinement ratio of Au-SiO 2 interface at the wavelength of 1.55 µm. Note that for 1D plasmonic waveguides, the in-plane electric field density ρ || = |E || | 2 = |E x | 2 , and total electric field density ρ total = |E| 2 .

FIGURE 3 . 3 :

 33 FIGURE 3.3: In-plane electric field of SPP under different refractive index n of dielectric part at the wavelength of 1.55 µm. Inset shows the relationship between n and mode size, inplane confinement ratio at metal-dielectric interface. Note that |E x | 2 = ρ || .

  .2 b). Note that in the 1D case, the in-plane electric field density ρ || = |E || | 2 = |E x | 2 , and total electric field density ρ total = |E| 2 . As this kind of SPP mode is TM polarized, almost all the electric field is out-plane component and the intensity of in-plane component |E x | 2 is only 0.02 at the gold-glass interface (Figure 3.2 b), and the in-plane confinement ratio γ is 0.08.

FIGURE 3 . 4 :

 34 FIGURE 3.4: SPP modes supported by dielectric-metaldielectric (IMI) interface. (a) Configuration of IMI plasmonic waveguide and SPP modes (blue curves) supported by the waveguide. (b) Normalized electric field of short-range and long-range SPP modes in SiO 2 -Au-SiO 2 interface at the wavelength of 1.55 µm. The thickness of gold is 30 nm.

FIGURE 3 . 5 :

 35 FIGURE 3.5: In-plane electric field of short-range SPP mode supported by a dielectric-gold-air interface under different refractive index at the wavelength of 1.55 µm. Inset shows the relationship between refractive index of dielectric and mode size, in-plane confinement ratio at metal-dielectric interface. The thickness of gold is set as 30 nm. Note that |E x | 2 = ρ || .

2 mFIGURE 3 . 6 :

 236 FIGURE 3.6: In-plane electric field of short-range SPP mode under different thickness of gold at the wavelength of 1.55 µm. Inset shows the relationship between the thickness of gold and maximum in-plane electric field at a SiO 2 -Au-air interface. Here, |E x | 2 = ρ || .

FIGURE 3 . 7 :

 37 FIGURE 3.7: SPP mode supported by nano-strip plasmonic waveguide. (a) Configuration of nano-strip plasmonic waveguide considered. Normalized real part of electric field E x (b), E y (c), and E z (d) of sa 0 b mode of nano-strip plasmonic waveguide at the wavelength of 1.55 µm. The thickness and width of gold strip are respectively 30 nm and 50 nm, substrate is SiO 2 . (e) In-plane electric field density and in-plane confinement ratio along z axis. Note that for 2D waveguide, ρ || ≈ ρ Ey is integral along y axis in a 1 µm range.

FIGURE 3 . 8 :

 38 FIGURE 3.8: The effect of nano-strip width w on in-plane confinement. (a) In-plane component E y under different w at 1.55 µm. Here, |E y | 2 ≈ |E || | 2 . Electric field density at gold-dielectric interface (b) and loss (c) of sa 0 b mode as function of w at 1.55 µm. The thickness of nano-strip is set as 30 nm, and substrate is SiO 2 .

FIGURE 3 . 9 :

 39 FIGURE 3.9: The effect of refractive index of substrate n on inplane confinement. (a) In-plane component E y with varying n at 1.55 µm. Inset shows the intensity of E y at gold-dielectric interface. Here, |E y | 2 ≈ |E || | 2 . In-plane confinement ratio (b) and loss (c) of sa 0 b mode under different n at 1.55 µm. The thickness of nano-strip is 30 nm.

FIGURE 3 . 10 :

 310 FIGURE 3.10: SPP mode supported by nano-slot plasmonic waveguide. (a) Configuration of nano-slot plasmonic waveguide. Normalized real part of E x (b), E y (c), and E z (d) of slot SPP mode at the wavelength of 1.55 µm. Both width and height of air gap are 80 nm, substrate is SiO 2 . (e) In-plane electric field density and confinement ratio along z axis. Note that ρ || ≈ ρ Ey .

  Figures 3.10(b), (c) and Figure 3.10 (d) respectively show the in-plane and out-plane electric field components of an asymmetric slot plasmonic waveguide with different refractive indices of substrate and superstrate. Benefit from strong lateral mode confinement, the in-plane component is confined to the gap with higher intensity than out-plane component.

FIGURE 3 . 11 :

 311 FIGURE 3.11: The effect of gap width w on in-plane confinement. (a) In-plane component E y as a function of gap width at 1.55 µm. In the 2D case, |E y | 2 ≈ |E || | 2 . Electric field density at gold-dielectric interface (b) and loss (c) of slot SPP mode under different w at 1.55 µm. The thickness of gold is set as 80 nm, and substrate is SiO 2 .

FIGURE 3 . 12 :

 312 FIGURE 3.12: The effect of refractive index of substrate n on in-plane confinement. (a) In-plane component E y with varying n at 1.55 µm. Here, |E y | 2 ≈ |E || | 2 . Inset shows the intensity of E y at gold-dielectric interface. Electric field density (b) and loss (c) of slot SPP mode under different n at 1.55 µm. Both the thickness of gold and width of gap are set as 80 nm.

FIGURE 3 . 13 :

 313 FIGURE 3.13: Schematic diagram of periodic plasmonic waveguide (a) and modes supported by this kind of plasmonic waveguide (b-d), which are respectively named as L mode (longitudinal mode along propagation direction), T 1 mode (transverse mode parallel to MNC-substrate interface), and T 2 mode (transverse mode perpendicular to MNC-substrate interface) dependent on the direction of dipole moment (blue arrows). MNC plasmonic waveguide consists of metallic elliptical nanocylinders with major axis dimension d 1 and minor axis dimension d 2 .

FIGURE 3 . 14 :

 314 FIGURE 3.14: Typical electric field distribution of T 1 mode of MNC at 1.55 µm. Real part of in-plane components E x (a), E y (b) and out-plane component E z (c) in the MNC-substrate interface. The electric field is showed in (d). Here, T 1 mode is excited by a dipole source with polarization along y axis. Red dashed line is the outline of nanoparticle. Electric field direction extracted form (d) is redrawn by blue dotted lines in (a) and (b).

EFIGURE 3 . 15 :

 315 FIGURE 3.15: Dispersion curve of MNC embedded in a homogeneous medium. (a) Simulation of the structure's band diagram of MNC. (b) Dispersion curve of MNC where the refractive index of the homogeneous medium is 1.67. The sizes of nanocylinders are set as d 1 =250 nm and d 2 =55 nm. Inset are |E || | of the E(o,1) and E(o,2) modes. (c) The relationship between the refractive index of the homogeneous medium and position of E(o,1) mode of MNC, the wavevector is chosen as 0.23. Red dotted line demonstrates the position of 1.55 µm.

FIGURE 3 . 16 :

 316 FIGURE 3.16: Hybrid mode of MNC on Si waveguide configuration. (a) 3D illustration of MNC plasmonic waveguide integrated on Si photonic waveguide. (b) Schematic diagram of coupling mechanism between MNC plasmonic waveguide and Si photonic waveguide. Cyan arrow highlights the energy transfer between two waveguides. (c) Electric field distribution across xz plane with different number of nanoparticles. (d) Power evolution in the middle of MNC and Si waveguide along the propagation direction (N = 25).

3. 3 . 71 EFIGURE 3 . 17 :

 371317 FIGURE 3.17: Dispersion curve of MNC plasmonic waveguide integrated on Si photonic waveguide. (a) Plasmonic and photonic modes in isolated MNC and Si waveguides. The black and white dashed lines are respectively light line of homogeneous medium and dispersion of TE 0 mode of the isolated Si waveguide. (b) Dispersion of the hybrid plasmonic waveguide showing anticrossing mode coupling points. Light lines of Si (white dashed line), SiO 2 (black dotted line), and air (white dotted line) are plotted. Modes of isolated Si waveguide simulated by mode solver are indicated by coloured lines. The dotted yellow curves are even and odd hybrid mode bands.

  .18 a), to compare the in-plane confinement enhancement of nano-strip (Figure 3.18 b), nano-slot (Figure 3.18 c), and MNC plasmonic waveguides (Figure 3.18 d) integrated on Si waveguide.

FIGURE 3 . 18 :

 318 FIGURE 3.18: Configurations of standard Si waveguide (a), nano-strip (b), nano-slot (c), and MNC plasmonic waveguides (d) integrated on Si photonic waveguide. The thickness of hBN is set as 20 nm. All the configurations are illuminated by TE 0 mode of Si waveguide at 1.55 µm.

3. 3 .FIGURE 3 .

 33 FIGURE 3.19: (a) In-plane confinement of nano-strip on Si waveguide as a function of strip width w Au . (b) Transmission, reflection, scattering, and absorption spectra of nano-strip on Si waveguide with the strip width of 290 nm. The spectrum for total and TE are plotted. The real part of electric field components E x , E y , and E z at w Au = 290 nm are plotted in (c), (d), and (e).

FIGURE 3 . 20 :

 320 FIGURE 3.20: Normalized real part of electric field components E x , E y , and E z of photonic mode TE 0 in Si waveguide (a)-(c), plasmonic mode ss 0 l in isolated nano-strip waveguide (d)-(e), hybrid plasmonic modes H e (g)-(i) and H o (j)-(l) in the nano-strip hybrid plasmonic waveguide excited by TE 0 and ss 0 l mode coupling. The outline of Si waveguide, isolated nanostrip plasmonic waveguide, and nano-strip hybrid plasmonic waveguide are plotted by dotted lines.

3. 3 .FIGURE 3 . 76 Chapter 3 .

 33763 FIGURE 3.21: (a) In-plane confinement of nano-slot on Si waveguide as a function of gap width w gap . (b) is the transmission, reflection, scattering and absorption spectrum at the gap width of 330 nm. The real part of electric field components E x , E y , and E z at w gap = 330 nm are plotted in (c), (d), and (e), respectively.

3. 4 .FIGURE 3 .

 43 FIGURE 3.22: (a) Calculated in-plane electric field density of nano-strip, nano-slot, MNC hybrid plasmonic waveguides, and Si photonic waveguide along z axis. (b) Calculated modulation depth of graphene-based EOMs designed using nano-strip, nano-slot, MNC hybrid plasmonic waveguides, and Si waveguide. Graphene is located on hBN layer.

  23 a).

FIGURE 3 .

 3 FIGURE 3.23: (a) 3D artwork of proposed TM-pass polarizer. Inset shows the top view and sizes of MNC. (b) Scanning electron micrograph of the TM-pass polarizer in Si waveguide.

FIGURE 3 .

 3 FIGURE 3.24: (a) Schematic representation of mechanism of the TM-pass polarizer. The fundamental TE mode incidence strongly coupled to MNC plasmonic waveguide, fundamental TM mode transmit only in the Si waveguide. (b) and (c) show the electric field profile under TE 0 and TM 0 illumination at 1.59 µm wavelength.

FIGURE 3 . 25 :

 325 FIGURE 3.25: Dispersion curve of proposed TM-pass polarizer. Single unit cell with a MNC on top of a Si waveguide. Bloch boundary condition along the x-axis and PML along y and z axes. Light lines of Si (white dashed line), SiO 2 (white dotted line), and air (white dash-dotted line) are plotted. Fundamental TE and TM modes of isolated Si waveguide calculated by mode solver are indicated by red solid curve and red dashed curve, respectively. The red dotted line shows the position of 1.59 µm wavelength. Insets show electric field distributions of two hybrid plasmonic modes.

FIGURE 3 .

 3 FIGURE 3.26: (a) Transmission spectrum of proposed TM-pass polarizer. Transmission under TM 0 and TE 0 illumination are respectively plotted by red and blue curve. Upper and lower inset show far-field scattering at 1.592 µm under TM 0 and TE 0 illumination, respectively. (b) Setup of transmission spectrum measurement. FPC is a fiber polarization controller, connected with two polarization maintaining (PM) fibers. LIA is lock-in amplifier.

FIGURE 4 . 1 :

 41 FIGURE 4.1: Plasmonic-enhanced graphene-based modulator on Si photonics. (a) 3D artwork of the device. The plasmonic waveguide consist of 7 Au nanocylinders with major axis d 1 = 250 nm, minor axis d 2 = 55 nm, thickness 30 nm, and period a = 150 nm. Inset: Graphene dispersion in three Fermi level (red line), green line is the position of Dirac point under incident light energy hω 0 . (b) Cross section of the HPM.

  Figure 4.1 (a), Figure 2.6 d).

FIGURE 4 . 2 :

 42 FIGURE 4.2: (a) Modulation depth and transmission spectra of the EOM with d 1 = 250 nm. Red area highlights the wavelength range for transmission loss less that 3 dB/µm with a modulation depth higher than 3 dB/µm. Inset: modulation depth and transmission spectra for a MNC with d 1 = 235 nm. Electric field distributions across a xy-and xz-planes at (b) On and (c) Off states at 1.65 µm.

Figure 4 .

 4 3 (a), the highest modulation depth occurs for N = 7. The reason is that there is a mode coupling when the field is transmitting in the HPM. As shown in Figure3.16 (c) and (d), the beating length is 1.2 µm, which contains 8 nanoparticles. The period a and minor axis d 2 are two parameters that controls the energy transition into the MNC while the major axis d 1 controls the dipole resonance. In other words, the spectral position

FIGURE 4 . 3 :

 43 FIGURE 4.3: Influence of parameters into the modulation depth at 1.55 µm. (a) Modulation depth with different number of Au elliptical nanocylinder, (b) major axis dimension, and (c) thickness of hBN for N = 7 MNC, at 1.55 µm.

FIGURE 4 . 4 :

 44 FIGURE 4.4: Transmission as a function of drive voltage (a) of graphene on undoped silicon waveguide (b) and doped silicon waveguide without graphene (c) under different drive voltage V d . The off state and on state voltages of our plasmonicenhanced EOM are respectively indicated by red and green arrows, where transmissions of our EOM are -6.5 dB/µm and -2.5 dB/µm at 1.65 µm, respectively (Figure 4.2 a).

Figure 4 .

 4 2 (a), we did not take the optical loss of doped silicon into account, we deal with silicon as undoped semiconductor in optical simulations and doped semiconductor in electrical simulations. The dynamically changing of transmission of doped silicon (red curve in Figure 4.4 a) is simulated as following step: The change of carrier concentration of doped silicon due to different drive voltage is simulated by electrical simulation software DEVICE multiphysics by Lumerical. The solver CHARGE within DEVICE, provides the solution of some problems of semiconductor devices relate to the electrostatic potential by solving the Poisson's equation and density of free carriers by the resolution of the drift-diffusion equations.

Chapter 4 .

 4 Plasmonic-enhanced graphene based EO modulator

FIGURE 4 . 5 :

 45 FIGURE 4.5: (a) Reflection and (b) transmission spectra of TE 0 mode of our EOM under TE 0 mode illumination. (c) Scattering and (d) absorption spectra under TE 0 mode illumination.Here, scattering is measured by four power monitors surrounding device, therefore, scattering and absorption spectra include the power of all the modes within the HPM.

FIGURE 4 . 6 :

 46 FIGURE 4.6: The equivalent electric circuit, where R C , R grp , C, and R Si are graphene contact and sheet resistance, hBN capacitance, and Si resistance. C air denotes the capacitance between the electrodes through air, R sub is the resistance of Si substrate, C box-L and C box-R represents capacitances of graphene and doped-Si to Si substrate through the oxide layer.

Chapter 4 .

 4 Plasmonic-enhanced graphene based EO modulator

FIGURE 4 . 7 :

 47 FIGURE 4.7: Simulated capacitances and resistance. (a). Schematic diagram of simulation of Si substrate resistance R sub . (b) Sum of parasitic capacitance between the electrodes through top air C air and device capacitance C. The capacitance is not constant due to the effect of semiconductor. Inset is the schematic diagram of the simulation. (c) Simulated parasitic capacitance C box-L and C box-R through box. The simulation schematic diagrams are shown in (e) and (d), respectively. In the simulation of C box-L and C box-R , the thickness of Si substrate was set as 5 µm, which is enough to take the parasitic capacitance by p-doped Si substrate into account.

FIGURE 4 . 8 :

 48 FIGURE 4.8: (a) Equivalent circuit of modulation speed calculation. V d and Z d are respectively voltage and impedance of source, Z 0 is the impedance of cable and probe. (b) Frequency response of proposed EOM. Modulation speed of EOM is the frequency when |V C | |V d | = 0.5.

Figure 4 .

 4 8 (b), hence, the modulation speed of the modulator, read from Figure 4.8 (b) when

Chapter 4 .FIGURE 4 . 9 :

 449 FIGURE 4.9: Scanning electron micrograph of fabricated MNC plasmonic waveguide on silicon wafer (a) and silicon photonic waveguide (b).

FIGURE 4 . 10 :

 410 FIGURE 4.10: Fabricated electrode (a) and its zoom in (b). The distance between electrode and Si waveguide is only 1 µm.

FIGURE 4 . 11 :

 411 FIGURE 4.11: Before (a) and after (b) etching of the first graphene layer on hBN on electrode and Si waveguide. Graphene is indicated by white dotted line. The length of graphene-hBN on Si waveguide part is 2 µm. (c) Second layer of graphene on 4nm-thick hBN.

FIGURE 4 .

 4 FIGURE 4.12: (a) Double-graphene layer plasmonic-enhanced sub-λ EOM. (b) Simulated modulation depth and transmission spectra. Red area highlights the wavelength range for transmission loss lower that 3 dB/µm with a modulation depth higher than 3 dB/µm.

4. 7 .

 7 Conclusion 105 spectral range which the modulation depth is higher than 3 dB/µm with an insertion loss lower than 3 dB/µm (Figure 4.12 b), comparing to 50 nm of monolayer graphene-based EOM (Figure 4.2 a). In the wavelength of insertion loss of 2.5 dB/µm, the modulation depth is 4.3 dB/µm. The transfer of second hBN and graphene layers are still ongoing (Figure 4.13 a). Based on our previous experience, the second graphene/hBN heterostructure can be patterned in a straightforward way (Figure 4.13 b). Finally the second electrode and MNC will be fabricated (Figure 4.13 b).

FIGURE 4 . 13 :

 413 FIGURE 4.13: Future work on the fabrication of a doublegraphene plasmonic-enhanced EOM. (a) Transfer of second hBN and graphene layers and (b) etching and fabrication of MNC and second electrode.

1

 1 Modulateurs électro-optiques et leurs performancesLe modulateur électro-optique (EOM) est un composant de base du système de communication et d'interconnexion optique. Il nous permet de contrôler l'amplitude, la phase ou l'état de polarisation d'un faisceau optique électriquement. Par exemple, dans les systèmes d'interconnexion optique, ces modulateurs traduisent un signal binaire électrique en un signal binaire optique (Figure 5.1 a). Le signal binaire électrique (V d ) induit un changement de la concentration de porteurs libres de la matière active et conduit ensuite à un décalage dans la partie réelle (n) et la partie imaginaire (α) de la constante de propagation du EOM. Et finalement ∆n conduit à un déphasage et ∆α provoque une absorption différente sous différents V d (Figure 5.1 b).

FIGURE 5 . 1 :

 51 FIGURE 5.1: Une EOM typique. (a) Le diagramme schématique du EOM d'IBM. (b) Le principe d'un EOM. Ici, la lumière d'entrée est une onde continue (CW), la lumière de sortie est un signal optique binaire.

FIGURE 5 . 2 :

 52 FIGURE 5.2: La première EOM intégrée à guide d'ondes à base de graphène sur guide d'ondes Si. (a) En cas d'incident normal, le graphène a un coefficient d'absorption universel de 2,3 %. (b) En intégrant du graphène sur un guide d'ondes photonique, on peut obtenir une absorption lumineuse de 100 % en augmentant la longueur du dispositif [69]. (c) Représentation schématique tridimensionnelle de la EOM. (d) Coupe transversale de la EOM avec distribution du champ électrique en raison de l'incidence de la mode TM fondamentale. L'encart de droite est la section transversale à travers le centre du guide d'ondes Si et son profil de champ électrique. [73].

  Plusieurs configurations combinant des guides d'ondes hybrides graphène et plasmonique ont été démontrées expérimentalement, par exemple, le guide d'ondes plasmonique hybride en Au bande et en Au fente. Les conceptions théoriques d'EOMs basées sur la bande d'Au plasmonique sur guide d'onde hybride silicium ont été présentées avec une profondeur de modulation ultra-haute d'environ 10 dB/µm [96, 97, 98] (Figure 5.3 a). Néanmoins, la profondeur de modulation de ces configurations s'est avérée surestimée en considérant le graphène comme un matériau isotrope 3D avec une épaisseur et une permittivité efficaces.

FIGURE 5 . 3 :

 53 FIGURE 5.3: Configurations d'EOM à base de graphène amélioré par plasmoniques. (a) Configurations d'EOM sur guide d'onde en silicium avec Au bande plasmonic améliorée [102]. (b) Configuration de guide d'onde plasmonique à Au fente sur guide d'onde hybride au silicium [107].

5. 2 .FIGURE 5 . 4 :

 254 FIGURE 5.4: Conductivité électrique accordable du graphène à 1.55 µm. (a) L'effet de la mobilité sur la partie réelle et la partie imaginaire de la conductivité du graphène. Ici, on utilise le niveau de Fermi de 0.2 eV. (b) La partie réelle et la partie imaginaire de la conductivité du graphène en fonction du niveau de Fermi E f . Les termes interbande et intrabande sont également tracés.

5 . 2 . 4 FIGURE 5 . 5 :

 52455 FIGURE 5.5: Schéma de la configuration graphique générale/hBN sur guide d'ondes Si. (a) Illustration schématique tridimensionnelle des hétérostructures graphiques/hBN van der Waals intégrées sur guide d'ondes Si. (b) Coupe transversale longitudinale de la structure intégrée.

Figure 5 .

 5 5 présente un graphène général intégré à la configuration du guide d'ondes photonique. Ici, nous considérons une hétérostructure intégrée graphène/hBN van der Waals sur un guide d'onde photonique en silicium de 500 nm par 220 nm, dont seuls les modes fondamentaux TE et TM peuvent être supportés dans le dispositif à 1.55 µm. Le hBN est utilisé pour améliorer les performances électriques et optiques du graphène. L'épaisseur de la couche hBN est fixée à 10 nm. En tant que matériau 2D, le graphène ne modifie pas la partie réelle de l'indice effectif des modes photoniques et du profil des modes.

FIGURE 5 . 6 :

 56 FIGURE 5.6: Densité pour les champ électrique et rapport de confinement dans le plan en modes fondamentaux TE (a) et TM (b). Les encarts montrent la distribution du champ électrique dans le plan dans le plan yz. (c) Pertes sous les modes fondamentaux TE et TM en fonction de la position du graphène z graphene .

Figure 5 . 6 ,

 56 nous avons calculé la densité de champ électrique dans le plan et le rapport de confinement dans le plan par les modes fondamentaux TE et TM, respectivement. Pour l'entrée TE, la densité de champ électrique dans le plan la plus élevée se produit au milieu du guide d'ondes Si. En revanche, lorsque la lumière d'entrée est en mode TM fondamental, la densité de champ électrique la plus élevée dans le plan se déplace vers les limites supérieure et inférieure du guide d'ondes Si. Pour obtenir l'absorption la plus élevée, nous avons changé la position du graphène et calculé la perte du dispositif en mode fondamental TE et TM en fonction de la position du graphène z graphene (Figure 5.6 c). La perte la plus élevée se produit au milieu du guide d'ondes Si pour le mode fondamental TE et dans les limites supérieure et inférieure du guide d'ondes Si pour le mode fondamental TM, qui correspondent au changement de la densité de champ électrique dans le plan sur la Figure 5.6 (a) et (b).

5. 2 .FIGURE 5 . 7 :

 257 FIGURE 5.7: Une hétérostructure hBN-graphène-hBN sur guide d'ondes Si. Micrographie de l'hétérostructure sur guide d'ondes Si avant (a) et après (b) motif. Le hBN inférieur et le hBN supérieur sont indiqués par une ligne pointillée bleue et une ligne pointillée en graphène, respectivement. Le masque de motif est représenté par une ligne pointillée rouge.

FIGURE 5 . 8 :

 58 FIGURE 5.8: Spectroscopie Raman de différentes hétérostructures graphiques/hBN vdW sur guide d'ondes Si. (a) Spectroscopie Raman des hétérostructures SiO 2 (région i), graphène (région ii), hBN-graphène vdW (région iii) et hBN-graphène-hBN vdW. L'encart montre la micrographie de l'échantillon mesuré. (b) et (c) sont respectivement le zoom avant du pic G et du pic 2D dans différentes régions.

Chapter 5 . 2 FIGURE 5 . 9 :

 5259 FIGURE 5.9: Confinement dans le plan de guides d'ondes plasmoniques à nano-bande et à nano-fente. Densité de champ électrique dans le plan et rapport de confinement dans le plan des guides d'ondes plasmoniques nano-bande (a) et nano-fente (b) selon l'axe z. Ici, l'épaisseur et la largeur de la bande d'or sont respectivement de 30 nm et 50 nm, la largeur et la hauteur de l'entrefer sont de 80 nm, le substrat est SiO 2 . Notez que pour les guides d'ondes 2D, ρ || ≈ ρ Ey est intégral le long de l'axe des y dans une fourchette de 1 µm. Les configurations des guides d'ondes plasmoniques nano-bande et nano-fente sont illustrées en encarts.

FIGURE 5 . 10 :

 510 FIGURE 5.10: Mode hybride de configuration MNC sur guide d'ondes Si. (a) Illustration en 3D d'un guide d'ondes plasmonique MNC intégré sur un guide d'ondes photonique Si. (b) Schéma du mécanisme de couplage entre le guide d'ondes plasmonique MNC et le guide d'ondes photonique Si. (c) Modes plasmonique et photonique dans des guides d'ondes MNC et Si isolés. Les lignes pointillées en noir et blanc sont respectivement la ligne de lumière du milieu homogène et la dispersion du mode TE 0 du guide d'ondes Si isolé. (d) Dispersion du guide d'ondes plasmonique hybride montrant les points de couplage en mode anticroisement. Les lignes claires de Si (ligne pointillée blanche), SiO 2 (ligne pointillée noire) et air (ligne pointillée blanche) sont tracées. Les courbes jaunes pointillées sont les modes hybrides paire et impaire.

FIGURE 5 . 11 :

 511 FIGURE 5.11: Densité de champ électrique calculée et rapport de confinement dans le plan du guide d'ondes photonique Si (a), du guide d'ondes plasmonique en bande (b), du guide d'ondes plasmonique en fente (c) et du guide d'ondes plasmonique MNC (d) intégrés au guide d'ondes photonique Si. Les encarts montrent les configurations des guides d'ondes considérés.

5. 3 . 3

 33 Chaîne de nanoparticules métalliques pour la conception d'un Polariseur TM En favorisant l'interaction graphène-lumière, la configuration MNC sur guide d'onde Si est également une solution appropriée pour concevoir un polariseur TM intégré. Pour cela, la chaine MNC doit interagir fortement avec le mode TE 0 plutôt que le mode TM 0 du guide d'ondes Si. Ici, nous utilisons le MNC optimisé intégré sur guide d'onde Si (Figure 5.12 a, b), la lumière d'incidence TE 0 est couplée au maximum dans MNC et absorbée par MNC, alors que le mode TM 0 transmet uniquement dans guide Si car il ne peut être couplé dans MNC (Figure 5.12 c).

FIGURE 5 .

 5 FIGURE 5.12: (a) Dessin 3D du polariseur passe-TM proposé. (b) Micrographie électronique à balayage de MNC bien alignés sur guide d'ondes Si. (c) Représentation schématique du mécanisme du polariseur passe-TM.

FIGURE 5 .

 5 FIGURE 5.13: (a) Spectre de transmission du polariseur TM proposé. Les encarts supérieur et inférieur montrent une diffusion en champ lointain à 1.592 µm sous une illumination TM 0 et TE 0 , respectivement. (b) Mise en place de la mesure du spectre de transmission.

Chapter 5 .

 5 Résumé en Françaiscouche Au de 80 nm d'épaisseur et d'une couche Ti de 20 nm d'épaisseur pour diminuer la résistance de contact du graphène[START_REF] Robinson | Contacting graphene[END_REF].

FIGURE 5 . 14 :

 514 FIGURE 5.14: Modulateur à base de graphène et chaine de nanoparticules plasmoniques . (a) illustration en 3D du dispositif. Encarts: Dispersion du graphène en trois niveaux de Fermi (ligne rouge), la ligne verte est la position du point Dirac sous l'énergie lumineuse incidente hω 0 . (b) Coupe transversale de la HPM.

FIGURE 5 .

 5 FIGURE 5.15: (a) Profondeur de modulation et spectres de transmission de l'EOM avec d 1 = 250 nm. La zone rouge met en évidence la plage de longueurs d'onde pour une perte de transmission inférieure à 3 dB/µm avec une profondeur de modulation supérieure à 3 dB/µm. Encart: profondeur de modulation et spectres de transmission pour une MNC avec d 1 = 235 nm. Répartition du champ électrique sur les plans xy-et xz-à (b) états On et (c) Off à 1.65 µm.

2 FIGURE 5 . 16 :

 2516 FIGURE 5.16: Circuit électrique équivalent, où R C , R grp , C, et R Si sont la résistance des contacts et des feuilles de graphène, la capacité hBN et la résistance Si. C air représente la capacité entre les électrodes par l'air, R sub est la résistance du substrat Si, C box-L et C box-R représente la capacité du graphène et dopé-Si au substrat Si par la couche d'oxyde.

FIGURE B. 1 :

 1 FIGURE B.1: Modes analyses of graphene-hBN hetrostructures on Si waveguide using NSOM signal. (a) The modes exist in four different regions of device in Figure 2.16. (b) Calculated beating length of different modes. (c) NSOM measurement result in which both TE 0 mode and TM 0 mode are excited. (d) FFT spectra in region iii and region iv of amplitude curve in (c).A clearly FFT peak can be found which is the result of beating between fundamental TE and TM mode.

FIGURE B. 2 :

 2 FIGURE B.2: NSOM measurement results under TM 0 mode incidence. (a) AFM topography and NSOM image under TM 0 mode incidence. (b) AFM and NSOM signal profile along the silicon waveguide. (c) FFT spectra of the amplitude curve in region iii (blue line) and region iv (red dotted line). In both region iii and region iv, a FFT peak at 2.4 µm -1 occurs due to the beating of forward and backward TM 0 modes, which indicates TM 0 mode is excited.

  µm 2.5 Electrical tunable conductivity of graphene at 1.55 µm . . . . . 2.6 Graphene absorption spectra under normal incident light at different Fermi levels . . . . . . . . . . . . . . . . . . . . . . . . 2.7 hBN and its application as substrate of graphene . . . . . . . . 2.8 Schematic of a graphene/hBN on Si waveguide configuration 2.9 Dispersion curve of graphene/hBN on Si waveguide configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 Electric field density and in-plane confinement ratio under fundamental TE and TM modes input . . . . . . . . . . . . . . . . 2.11 Loss under fundamental TE and TM modes input as a function of graphene position . . . . . . . . . . . . . . . . . . . . . . . . .13 The process of etching graphene . . . . . . . . . . . . . . . . . x 2.14 A patterned hBN-graphene-hBN heterostructure on Si waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.15 Raman spectroscopy of different graphene/hBN vdW heterostructures on Si waveguide . . . . . . . . . . . . . . . . . . . . . . . 2.16 Optical near-field setup and structure . . . . . . . . . . . . . . 2.17 NSOM imaging of the integrated photonic device . . . . . . . 2.18 Simulated electric near-field distribution of the TE mode propagating through different integrated vdW heterostructures on Si photonics at λ = 1550 nm . . . . . . . . . . . . . . . . . . . . 3.1 Light confinement in photonic and plasmonic waveguide . . . 3.2 In-plane electric field of SPP mode supported by metal-dielectric interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 In-plane electric field of SPP under different refractive index of dielectric part . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 SPP modes supported by dielectric-metal-dielectric interface .

2.12 Transfer of a hBN-graphene-hBN heterostructure on Si waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

TABLE 1 .

 1 1: Performances requirements of EOM for recent optical interconnect. * For micro-scale and millimeter-scale EOMs, more suitable unit of MD and IL are dB,

	Footprint	MD *	IL **	f 3dB	E bit
	subwavelength > 3 dB/µm < 3 dB/µm	> 10 GHz	∼ 10 fJ/bit

* MD: modulation depth, IL: insertion loss, f 3dB : modulation speed, E bit : energy consumption. *

TABLE 1 .

 1

	Material	Principle	Electrically control	Configuration
	LiNbO 3	Pockel's effect	∆n = n(V d )	MZI/MRR
	bulk GeX *	Franz-Keldysh effect	∆α = α(V d )	pin	MZI/MRR
	GeX * QWs	Quantum Confirmed Stark effect	∆α = α(V d )	pin	MZI/MRR
	Si	Plasma dispersion effect	∆α = α(V d ) ∆n = n(V d )	pn/pin /MOS	MZI/MRR /Micro-disk
	Others	EO polymer, ITO, Si-LiNbO 3 , Si-polymer, plasmonic-polymer

2: Classification of traditional EOM.

  8 × 10 -22 ∆N e + 8.5 × 10 -18 (∆N h ) 0.8 ] ∆α = ∆α e + ∆α h = 8.5 × 10 -18 ∆N e + 6.0 × 10 -18 ∆N

h ,

(1.6)

TABLE 1 . 3 :

 13 Performances of traditional EOMs at telecom frequencies.

TABLE 1 .

 1 4: Experimental performances of graphene-based EOMs at telecommunication wavelength.

	Material	Footprint µm	MD dB/µm	IL dB/µm	f 3dB GHz	E bit fJ/b
	graphene [73]	40	0.1	-	1	1000
	graphene [79]	50	0.05	0.08	2.6-5.9 350
	graphene (2-layers) [80]	40	0.16	0.1	1	2000
	graphene (2-layers) [82]	300	0.053	0.011	0.67	-
	graphene (2-layers) [83]	90	0.05	-	2.5	-
	graphene (2-layers) [84]	30	0.067	0.03	35	1400
	graphene (MRR) [85]	10 × 10	0.07	0.2	80	-
	graphene (MRR) [92]	100 × 100	0.07	0.03	-	-
	graphene (MRR) [93]	40 × 40	0.12	0.10	30	800
	several order of magnitude shorter than Si-based EOMs. Moreover, graphene-
	based EOMs have higher modulation depth, same level in modulation speed,
	but it suffers from a little drawback in insertion loss and especially in energy
	consumption. Except higher energy consumption, the other performances of
	graphene-based EOMs are in same level with Ge and III-V compound semi-
	conductors based EOMs (Table 1.3 and Table 1.4). According to Table 1.1,
	higher performances of graphene-based EOMs are still needed for recent op-
	tical interconnect.

TABLE 1 .

 1

	5: Experimental and theoretical performances of plas-
	monic enhanced graphene-based EOMs at telecommunication
		wavelength.
	Plasmonic structure	Footprint µm	MD dB/µm	IL dB/µm	f 3dB GHz	E bit fJ/bit
	Experimental performances:
	Gold strip [106]	8	0.25	-	-	18
	Gold slot [108]	-	0.13	0.68	-	-
	Gold Wedge [100]	12	0.03	-	-	-
	Theoretical performances:
	Strip [104]	8.5	0.37	0.12	400	17
	Strip [20]	3.6	0.84, 1.23	0.09	480	140
	Strip [105]	10	0.95	0.5	25	-
	Slot [107]	-	0.524	0.05	150	607
	Patch structure [109]	0.4	10	2.5	400	0.5

TABLE 2 .

 2 1: Mobility of graphene on different substrates and configurations (Unit of mobility: cm 2 V -1 s -1 ).

	grp on SiO 2	grp on hBN	hBN-graphene-hBN suspended grp
	4 400 ∼ 17 000	35 000 ∼ 40 000	100 000 ∼ 140 000	200 000
	[148, 149]	[144, 149, 151, 155]	[144, 149, 71, 156]	[153, 154]

TABLE 2 .

 2

	2: RIE process parameters to etch graphene and
		graphene/hBN
	Parameters	Etch graphene	Etch graphene-hBN
	Process name Root\process_3 Root\process_graphene-hBN
	RIE power	50 W	60 W
	ICP power	0 W	0 W
	Gas mode	RIBE	RIBE
	Pressure	20 mTorr	20 mTorr
	Gas	O 2 (10 sccm)	O 2 (4 sccm) & CHF 3 (30 sccm)
	Etch time	1 min	11.2 nm/min

TABLE 4 .

 4 1: Performances of graphene-based EOM integrated on photonic waveguide.

	Configuration	Footprint (µm) 2	MD dB/µm	IL dB	f 3dB GHz	E bit fJ/bit	FoM
	1 × grp [exp] * [73]	40 × 0.60	0.1	n/a 1.2	1000	10 -5
	1 × grp [exp] [79]	50 × 0.75	0.05	3.8	2.6	350	10 -4
	2 × grp [exp] [80]	40 × 0.60	0.16	4.0	1	2000	10 -5
	2 × grp [exp] [82]	300 × 0.52	0.05	3.3 0.67 517500 10 -9
	2 × grp [exp] [83]	90 × 2.10	0.05	n/a 2.5	n/a	n/a
	2 × grp [exp] [84]	30 × 0.60	0.07	0.9	35	1400	10 -3
	1 × grp, MRR [exp] * * [85]	10 × 10	2.22	0.1	80	n/a	n/a
	1 × grp, MRR [exp] [92]	100 × 100	8	4	n/a	n/a	n/a
	2 × grp, MRR [exp] [93]	40 × 40	15	13	30	800	10 -5
	1 × grp, Strip SPP [exp] [106]	8 × 0.63	0.25	n/a n/a	18	n/a
	2 × grp, Slot SPP [exp] [108]	20 × 0.60	0.13	13.6 n/a	n/a	n/a
	2 × grp, Strip SPP [sim] [104]	8.5 × 0.19	0.35	1	400	17	14.6
	4 × grp, Strip SPP [sim] [20]	3.6 × 0.67	0.84	0.32 480	140	1.4
	2 × grp, Slot SPP [sim] [264]	1 × 0.75	0.52	0.05 150	607	0.3
	1 × grp, MNC [sim] [this work] 1.05 × 0.5	4	2.6	39	21.2	3.5

* grp: graphene; MRR: micro-ring resonator; SPP: surface plasmon polaritons. ** For MRR based-EOMs, only total MD and insertion loss with unit dB are listed.

  log(T max /T min )(5.3) où L est la longueur d'efficacité de la région active de la EOM qui est généralement la taille de la région active le long du guide d'ondes, T max et T min sont respectivement la transmission en état marche et arrêt exprimée en dB. Bien qu'une profondeur de modulation élevée de > 7 dB soit toujours préférée pour l'interconnexion optique, 4 ∼ 5 dB est souvent suffisant si le système global doit faire un compromis entre les performances dans un domaine et celles d'un autre[START_REF] Graham T Reed | Silicon optical modulators[END_REF]. De plus, un autre paramètre important de la EOM est la perte d'insection (IL), qui est la perte intrinsèque d'une EOM en état de marche. Afin d'établir une comparaison des performances entre les différentes EOM, un paramètre de référence unifié du mérite (FoM) est défini comme[START_REF] Amin | Attojoule-efficient graphene optical modulators[END_REF]:

  mergeant successivement dans de l'eau 60 s pour éliminer le polymère conducteur, dans une solution 3:1 MIBK/IPA 60 s pour le développement et dans IPA (alcool isopropylique) 15 s pour rincer MIBK. Une fois l'EBL terminée, une couche mince d'aluminium de 3 nm d'épaisseur a été évaporée sur l'échantillon à l'aide du nouvel évaporateur PLASSYS MEB 4000 et décollée avec de l'acétone 4 ∼ 12 h. Le modèle a été transféré au graphène avec l'utilisation de la gravure par ions réactifs (RIE). Après le processus de gravure, le masque en aluminium a été retiré avec une solution de HCl à 20 % pendant 4 minutes.Pour tester les propriétés de notre échantillon, une spectroscopie Raman en champ lointain est réalisée, une spectroscopie Raman sur les différentes hétérostructures intégrées graphène/hBN vdW sur guide d'onde photonique silicium est représentée sur la Figure5.8. Les spectres Raman démontrent la qualité d'un graphène monocouche dans différentes hétérostructures vdW (Figure

  Le confinement à l'échelle nanométrique des guides d'ondes plasmoniques résulte d'une forte interaction entre la lumière et les électrons en mode SPP. Les SPP sont des excitations électromagnétiques qui se propagent le long d'une interface métallique et diélectrique et s'affaiblir de façon exponentielle à mesure que la distance entre chaque milieu et l'interface augmente. Les modes SPP sont intrinsèquement perpendiculaires à sa surface, d'où des composantes dans le plan très faibles en général. Par exemple, le mode SPP supporté par des interfaces métal-diélectrique simples et le mode SPP à courte portée de diélectrique-métal-diélectrique ne contiennent que la composante dans le plan dans le sens de la transmission E x et aucune composante latérale E y .

  Le modulateur électro-optique plasmonique hybride (HPM) est constitué d'un guide d'ondes MNC sur un guide d'ondes Si de 500 nm par 220 nm (Figure5.14 a). Le guide d'onde plasmonique et le guide d'onde Si sont séparés par une couche de nitrure de bore hexagonal (hBN) de 20 nm d'épaisseur qui permet au graphène de maintenir sa grande mobilité[START_REF] Kim | A role for graphene in silicon-based semiconductor devices[END_REF] et de contrôler le couplage des modes. Le graphène est placé sur la couche hBN, sous le guide d'ondes plasmonique MNC, où les composantes du champ électrique dans le plan devraient être élevées. Le guide d'ondes MNC est composé de 7 nanocylindres elliptiques en or identiques avec la dimension de l'axe majeur d 1 = 250 nm, la dimension de l'axe mineur d 2 = 55 nm, épaisseur 30 nm, et invariant avec une période a = 150 nm. La taille et le nombre de nanocylindres ont été choisis pour produire la profondeur de modulation la plus élevée et une faible perte d'insertion. La concentration de dopage de type p du guide d'ondes Si et de la couche de connexion Si de 50 nm d'épaisseur est respectivement de 10 17 cm -3 and 10 19 cm -3[START_REF] Sorianello | Design optimization of single and double layer graphene phase modulators in SOI[END_REF]. Les électrodes sont placées à 500 nm des bords du guide d'ondes Si. Les électrodes sont constituées d'une

  C) Z device (5.13) où V C est la voltage appliquée dans le condensateur, V d est la voltage de commande, f est la fréquence. Ici, l'impédance de source Z d et l'impédance du câble Z 0 sont réglées à 50 Ω, l'impédance totale de la EOM est Z -1 EOM = Z -1 air + Z -1 device + Z -1 sub , où Z air = 1/jωC air , Z device = 1/jωC + R C + R grp + R Si , et Z sub = 1/jωC box-L + 1/jωC box-R + R sub . Par conséquent, la vitesse de modulation du modulateur, lue à |V C | |V d | = 0.5, est 39 GHz. La consommation d'énergie, calculée par l'équation 5.2 et 5.5, est de 21.2 fJ/b, répondant aux besoins futurs [23, 263]. Ici, nous avons fixé le niveau de Fermi E on f = 0.53 eV (V d = 8.45 V) et E off f = 0.2 eV (V d = 1.20 V).

TABLE 5 .

 5 1: Performances de l'EOM à base de graphène intégré sur guide d'onde photonique.

	Configuration	Taille (µm) 2	MD dB/µm	IL dB	f 3dB GHz	E bit fJ/bit	FoM
	1 × grp [exp] * [73]	40 × 0.60	0.1	n/a 1.2	1000	10 -5
	1 × grp [exp] [79]	50 × 0.75	0.05	3.8	2.6	350	10 -4
	2 × grp [exp] [80]	40 × 0.60	0.16	4.0	1	2000	10 -5
	2 × grp [exp] [82]	300 × 0.52	0.05	3.3 0.67 517500 10 -9
	2 × grp [exp] [83]	90 × 2.10	0.05	n/a 2.5	n/a	n/a
	2 × grp [exp] [84]	30 × 0.60	0.07	0.9	35	1400	10 -3
	1 × grp, MRR [exp] * * [85]	10 × 10	2.22	0.1	80	n/a	n/a
	1 × grp, MRR [exp] [92]	100 × 100	8	4	n/a	n/a	n/a
	2 × grp, MRR [exp] [93]	40 × 40	15	13	30	800	10 -5
	1 × grp, bande SPP [exp] [106]	8 × 0.63	0.25	n/a n/a	18	n/a
	2 × grp, fente SPP [exp] [108]	20 × 0.60	0.13	13.6 n/a	n/a	n/a
	2 × grp, bande SPP [sim] [104] 8.5 × 0.19	0.35	1	400	17	14.6
	4 × grp, bande SPP [sim] [20]	3.6 × 0.67	0.84	0.32 480	140	1.4
	2 × grp, fente SPP [sim] [264]	1 × 0.75	0.52	0.05 150	607	0.3
	1 × grp, MNC [sim] [this work] 1.05 × 0.5	4	2.6	39	21.2	3.5

(a) (b)

Standard Cubic Centimeters per Minute, a flow measurement term indicating cm 3 /min in standard conditions for temperature and pressure of the fluid.

Sheet resistance is a special case of resistivity for a uniform sheet thickness. The unit are in ohms per square, which is dimensionally equal to an ohm, but is exclusively used for sheet resistance.

Acknowledgements

Acknowledgements

iii

I would like to give my acknowledges to the China Scholarship Council

Appendix A

Anisotropic bulk permittivity modeling of graphene

Another alternative approach of graphene modeling is the the equivalent bulk medium approach, where graphene is modeled as a bulky material by using a uniaxial anisotropic permittivity if we assume that the graphene layer has a finite sub-nanometer thickness t grp . The anisotropic permittivity tensor stem from surface conductivity tensor is

where zz = ⊥ is the out-plane component which is equal to the background permittivity of graphene, xx = yy = are in-plane components of permittivity, which is given by

where ω is angular frequency, 0 is vacuum permittivity, t grp ≈ 0.34 nm.

The refractive index n is simply taking the square-root of the permittivity expression. The in-plane permittivity and refractive index of graphene are shown in When the Fermi level is lower than 0.52 eV, both the real and imaginary parts of the permittivity are positive so that the graphene layer behaves like a dielectric material. When the Fermi level is higher than 0.52 eV, the real part of permittivity becomes negative and the imaginary part is approaching zero which means graphene has behave of metal. As a result, the imaginary part of refractive index is lowest at 0.49 eV where the imaginary part of in-plane conductivity is 0 (Figure

where first term cos[0.5(k 2k 1 )x] is irrelevant with time which lead to a beating pattern. The beating length can be expressed as L beating = λ/((n eff2n eff1 )), where n eff1 and n eff2 are respectively the effective index of mode 1 and mode 2.

The beating length between different modes in four different regions in our device is plotted in