Detailed characterization
for application in
integrated photonic circuits
Evdokia Dremetsika
PhD Thesis
Detailed characterization for application
in integrated photonic circuits
Evdokia Dremetsika
Thesis presented in fulfillment of the requirements for the
degree of Doctor of Engineering Sciences
Supervisor: Pascal Kockaert
Co-supervisor: Philippe Emplit
OP´ERA-Photonique Universit´e libre de Bruxelles ´
Detailed characterization for application
in integrated photonic circuits
Evdokia Dremetsika
Thesis presented in fulfillment of the requirements for the
degree of Doctor of Engineering Sciences
Supervisor: Pascal Kockaert (ULB)
Co-supervisor: Philippe Emplit (ULB)
Members of the Jury
Simon-Pierre Gorza (ULB)
Luc Henrard (UNamur)
Bart Kuyken (UGent)
Pierre Seneor (UParis Sud)
OP´ERA-PhotoniqueUniversit´e libre de Bruxelles ´
Ecole polytechnique de Bruxelles - Brussels School of engineering ——————————————————————————-Funding sources:
FRIA grant (Fonds pour la Formation ´a la Recherche dans l’Industrie et dans l’Agriculture, FNRS)
In the quest for ultrathin materials compatible with CMOS technology for all-optical signal pro-cessing applications in integrated photonics, graphene appears to be a promising candidate, with broadband1optical properties and a high and broadband optical nonlinearity. However, researchers do not agree on the value of its nonlinear refractive index, and commonly used characterization methods do not provide a clear picture of the optical nonlinearity, in terms of its tensor nature or relaxation time. In the first part of this thesis, apart from the previously used Z-scan method, we have also used the ultrafast Optical Kerr Effect method coupled to Optical Heterodyne De-tection (OHD-OKE) for the characterization of the third order optical nonlinearity of monolayer CVD graphene at telecom wavelengths. This method allows to separately measure the real and the imaginary part of the third-order nonlinearity, as well as their dynamics. With respect to the Z-scan method, OHD-OKE presents the major advantage of being robust against inhomogeneities of the sample. As such, we have demonstrated that graphene has a negative nonlinear refractive index, contrary to previously reported results. In addition, we have studied the real and imagi-nary part of graphene’s nonlinearity, when electrostatic gating is applied to change the chemical potential of graphene. Furthermore, we have proposed an enhanced version of the OHD-OKE method, together with the appropriate theoretical framework, in order to extract the tensor ele-ments of the nonlinearity including the out-of-plane tensor eleele-ments. In particular, we have mea-sured separately the time response of the two main tensor elements of the nonlinear susceptibility and we have experimentally verified that the out-of-plane tensor components are negligible. In the second part of this thesis, we have investigated, from an experimental point of view, the use of the nonlinear optical response of graphene for all-optical switching applications in integrated photonics. Namely, we have designed simple silicon nitride waveguide structures that constitute basic building blocks of switching devices, which were then fabricated and covered by graphene patches. Finally, we have experimentally tested the graphene-covered structures at low and high power levels and discussed the results.
Acknowledgements
The five-years of my doctoral studies was a long journey full of new knowledge, exceptional mo-ments, both nice and difficult, and of course, endless hours in the lab. During this journey, I was thankfully not alone.
Foremost, even if words could not be enough, I would like to express my gratitude for my super-visor Prof. Pascal Kockaert for his continuous support during these five years of my Phd Thesis, his endless patience to explain again and again my questions, his encouragement at the difficult moments of my research, the knowledge and the valuable advice that he offered to me and the fruitful discussions we had during all these years.
Besides, I would like to thank my co-advisor Prof. Philippe Emplit, who trusted me and gave me the opportunity to start this Thesis in OPERA, an opportunity to follow my dream to work as a re-searcher in physics-engineering and discover the joy of performing experimental work in physics. I thank him also for his support until the end of my Thesis and for the experience to help the stu-dents as an assistant in a Physics lab the past five years.
I am sincerely grateful to Prof. Simon-Pierre Gorza for his help and advice in different aspects of my experimental work. Also, I thank Prof. Marc Haelterman for his interest in my progress and his advice. In addition, for the last years of my Thesis, I thank Charles Ciret for his help with my experiments in the lab, especially to adjust the laser for my experiments and learn to perform measurements on a photonic chip. For the first years years of my work, I thank Jassem Safioui who introduced me to the lab and the Z-scan experiment, as well as Maïté Swaelens.
Moreover, I would like to acknowledge the contribution of the group from Unité mixte de CNRS-Thales who provided high-quality graphene samples and performed the transfer on quartz sub-strates: Bruno Dlubak, Prof. Pierre Seneor and Prof. Daniel Dolfi and everyone else that worked for these samples. I also thank Prof. Serge Massar from our group for his contribution regarding this collaboration.
In addition, I gratefully acknowledge our collaborators from Photonics Research Group (PRG) from Gent university, Alexander Koen, Bart Kuyken and Dries van Thourhout, as well as Stéphane Clem-men for the discussions, the opportunity and the realization of our SiN graphene-covered waveg-uide structures for my project. I also acknowledge the contribution of Alexander Koen and Bart Kuyken for the realization of the gated graphene sample.
Furthermore, I am grateful to Gilles Rosolen and Prof. Bjorn Maes from the university of Mons for our discussions and their help with the simulations of the MMI waveguide structure.
Also, I thank Tiriana Segato and Gilles Wallaert from the group 4MAT (ULB) for the annealing of our samples, and Prof. Martin Mittendorf from university of Maryland, whom I met at a conference,
OKE.
I thank the technicians of our lab, Lory Marchal and Michaël Fita Codina for the tools they made for me, especially Lory for the aligning tool that helped me in the endless hours of aligning the laser beam in the lab, and Michaël for the special sample holder that was very crucial for our gating measurements.
I am really grateful for the secretaries of our lab Ibtissame Malouli and Alexandra Peereboom who helped me a lot, not only with their administration work, but also with their support during my doctoral studies, and David Houssart for his support at the end of my Thesis.
I would sincerely like to thank all members of OPERA-photonique, previous and present, for their comments regarding my project, the nice company, the fruitful discussions, the funny moments and generally for the nice working environment during all these years. Especially I thank the present members of OPERA who supported me (and tolerated my stress) during the writing of this Thesis.
Abstract ii
Acknowledgements iv
List of Figures x
List of Tables xiii
1 Introduction 1
Publication list. . . 3
2 Theory 5 2.1 Basic concepts in optics and nonlinear optics . . . 5
2.1.1 Material response to an electromagnetic radiation . . . 5
2.1.1.1 Propagation equation . . . 8
2.1.1.2 Birefringence and dichroism . . . 9
2.1.1.3 Continuity of electromagnetic fields at an interface . . . 10
2.1.2 Nonlinear optical response . . . 12
2.1.2.1 Third-order nonlinear response . . . 12
2.1.2.2 Third-order susceptibility. . . 13
2.1.3 Nonlinear propagation. . . 14
2.1.3.1 Nonlinear phase shift . . . 14
2.1.3.2 Nonlinear refractive index . . . 15
2.1.3.3 Nonlinear absorption . . . 16
2.1.3.4 Full relation between third-order susceptibility and nonlinear refrac-tive index in materials with losses . . . 17
2.1.4 Nonlinear phase shift in problems involving continuity at interface . . . 17
2.1.5 Ultrafast lasers . . . 19
2.1.6 Pump-probe measurements . . . 19
2.2 Characterization of optical nonlinearities. . . 20
2.3 Linear and nonlinear optical properties of graphene . . . 22
2.3.1 Graphene: the wonder material . . . 22
2.3.1.1 Production of graphene . . . 22
2.3.1.2 Graphene structure . . . 23
2.3.2 Electronic properties . . . 24
2.3.4 Nonlinear optical properties of graphene . . . 27
2.3.4.1 Saturable absorption . . . 27
2.3.4.2 Third-order nonlinearity / nonlinear refractive index of graphene . . 28
2.4 Modeling graphene for linear and nonlinear optics: 2D or not 2D? . . . 31
2.4.1 Sheet conductivity model . . . 32
2.4.2 Effective bulk model . . . 33
2.4.3 Surface susceptibility model with non-zero out-of-plane components . . . . 37
3 Characterization of the nonlinear optical properties of graphene with Z-scan 40 3.1 Z-scan: the method. . . 40
3.1.1 The Z-scan trace . . . 41
3.1.2 Z-scan measurement in absorbing media. . . 41
3.1.3 Retrieval of parameters . . . 42
3.2 Advantages and disadvantages of the Z-scan technique . . . 43
3.2.1 Simplicity . . . 43 3.2.2 Quality of samples . . . 43 3.2.3 Beam quality . . . 44 3.2.4 Thermal effects . . . 44 3.2.5 Multiple reflections. . . 44 3.2.6 Relaxation dynamics . . . 45 3.3 Experimental setup . . . 45 3.4 Graphene samples . . . 46
3.5 Open aperture measurements . . . 46
3.6 Alternative I-scan measurement . . . 48
3.7 Closed aperture measurements. . . 50
3.8 Z-scan experiment with Ti:Sapphire laser at 780 nm . . . 52
3.9 Simulations . . . 53
3.9.1 Alternative Z-scan experiment with image processing . . . 53
3.10 Discussion . . . 54
4 Characterization of the third-order optical nonlinearity of graphene with the OHD-OKE method 56 4.1 OHD-OKE: the method . . . 56
4.1.1 Simple OKE . . . 57
4.1.2 Optical Heterodyne detection. . . 61
4.1.2.1 OHD: the principle. . . 61
4.1.2.2 OHD: example . . . 62
4.2 Experimental procedure . . . 65
4.2.1 Description of the experimental setup . . . 65
4.2.2 Building the setup . . . 66
4.2.3 SNR study . . . 66
4.2.4 Lock-in amplifier . . . 69
4.2.4.1 Lock-in detection process. . . 69
4.2.4.2 Lock-in detection in optics . . . 70
4.2.5 Preparing the OHD-OKE experiment . . . 71
4.3 Experimental Results . . . 76
4.3.1 Real part: nonlinear refraction . . . 76
4.3.2 Imaginary part: nonlinear absorption . . . 78
4.3.3 Relaxation dynamics . . . 80
4.3.4 Temperature controlled measurements . . . 81
4.4 OHD-OKE with controlled Fermi energy of graphene . . . 82
4.4.1 Gating method . . . 82
4.4.1.1 The sample . . . 83
4.4.1.2 The experimental setup . . . 83
4.4.2 Electrostatic gating measurements . . . 84
4.4.3 OHD-OKE with applied electrostatic gating . . . 85
4.5 OHD-OKE and nonlinear susceptibility tensor . . . 87
4.5.1 Tensor susceptibility of graphene . . . 87
4.5.2 Vectorial model of the nonlinear response . . . 87
4.5.3 Enhanced OHD-OKE method: 2D-OHD-OKE . . . 89
4.5.4 In-plane component measurements. . . 92
4.5.5 Out-of-plane component measurements . . . 93
4.6 Discussion . . . 95
5 Nonlinear integrated photonics with graphene 97 5.1 Integrated photonics and graphene . . . 97
5.2 Silicon nitride waveguide structures covered with graphene . . . 98
5.2.1 The platform. . . 98
5.2.2 Parameters of the waveguides . . . 99
5.2.2.1 Mode profile . . . 100
5.2.2.2 Dispersion. . . 102
5.2.2.3 Simulation of graphene-covered waveguide . . . 102
5.2.2.4 Nonlinearity. . . 103
5.2.2.5 Summary of parameters. . . 103
5.3 Silicon nitride waveguide structures: Simulations and design . . . 104
5.3.1 Directional Couplers . . . 104
5.3.2 Waveguide arrays . . . 106
5.3.3 Rectangular structure . . . 107
5.3.4 Multimode interference (MMI) coupler . . . 109
5.4 Measurements: linear regime . . . 111
5.4.1 Simple waveguides . . . 112
5.4.2 Directional Couplers . . . 113
5.4.3 Waveguide arrays . . . 114
5.4.4 Rectangular structure . . . 114
5.4.5 Multimode interference (MMI) coupler . . . 115
5.5 Measurements: nonlinear regime . . . 117
5.5.1 Simple waveguides . . . 117
5.5.2 Directional Couplers . . . 118
5.5.3 Waveguide arrays . . . 120
5.5.4 Rectangular structure . . . 120
5.5.5 Multimode interference (MMI) coupler . . . 121
6 Conclusions and Outlook 124
A Microscope images from the graphene-covered chip 128
2.1 Simplified geometry of the experiments studying the transmission of a beam through
a sample of limited thickness. . . 10
2.2 Illustration of a frequency mixing process due to the third-order nonlinear effect. . . 13
2.3 Qualitative diagram of absorption saturation. . . 16
2.4 Schematic of pump-probe experiment. . . 20
2.5 Illustration of CVD graphene growth on copper foil. . . 23
2.6 Illustration of transfer process of graphene onto a substrate. . . 23
2.7 Graphene structure and º orbitals.. . . 24
2.8 Graphene lattice, unit cell and reciprocal lattice. . . 24
2.9 Energy band structure of graphene. Dirac cone and Fermi energy level with respect to doping. . . 25
2.10 Normalized conductivity of graphene for different doping levels. . . 26
2.11 Geometry of the electromagnetic problem. . . 32
2.12 Geometry of the electromagnetic problem. . . 34
2.13 Dispersion of effective refractive index of graphene. . . 36
3.1 Basic setup for Z-scan measurement. . . 41
3.2 Typical Z-scan trace for positive and negative nonlinearity. . . 41
3.3 Principle of Z-scan technique for positive self-focusing nonlinearity. . . 42
3.4 Illustration of the illuminated area when the sample is at the focus point of the beam or far from it. Due to the inhomogneities of the sample, the transmission is different in the two cases. . . 44
3.5 Raman spectrum from monolayer graphene on quartz. . . 46
3.6 Open aperture Z-scan trace of single-layer graphene sample for different input in-tensities. . . 47
3.7 Open aperture Z-scan trace of graphene samples with various numbers of layers. . . 47
3.8 Open aperture Z-scan trace of single-layer graphene sample at different input powers. 48 3.9 Experimental data from I-scan measurement. . . 49
3.10 Experimental data from I-scan measurement. . . 49
3.11 Normalized absorption of multi-layer graphene and fitting for the calculation of sat-uration intensity . . . 50
3.12 Open aperture and closed aperture Z-scan data from monolayer graphene. . . 51
3.13 Open aperture and closed aperture Z-scan data from monolayer graphene. . . 51
3.14 Z-scan trace from monolayer CVD graphene. . . 52
3.15 Simulated Z-scan trace for a thick and a thin sample. . . 53
3.16 Spatial intensity distribution of the beam from image processing (far field). . . 54
4.1 Basic optical Kerr effect method setup. . . 58
4.2 Polarization vectors at the simple optical Kerr effect method. . . 59
4.3 Illustration of the principle of the optical heterodyne detection. . . 61
4.4 Polarization vectors in the optical Kerr effect method with optical heterodyne detection 63 4.5 Experimental setup of the OHD-OKE method. . . 65
4.6 Normalized signal-to-noise ratio.. . . 68
4.7 Schematic of lock-in amplifier process. . . 70
4.8 Lock-in detection scheme with modulation of the pump beam.. . . 71
4.9 Lock-in detection scheme with modulation of both pump and probe beams.. . . 71
4.10 Geometry for the right position of the lens used to image the focus point on the cam-era and observe the interference fringes when pump and probe pulses are synchro-nized. . . 72
4.11 Experimental OHD-OKE data from graphene with spurious interferences. . . 73
4.12 Comparison between the measured OHD-OKE signal from graphene, with and with-out the vibrations of the mirror . . . 74
4.13 Experimental results from OHD-OKE . . . 77
4.14 Dependence of OHD-OKE signal at zero delay to probe power and heterodyne angle. 77 4.15 Dependence of OHD-OKE signal at zero delay to the pump power.. . . 78
4.16 Experimental OHD-OKE results for the imaginary part of the nonlinearity . . . 79
4.17 Measured nonlinear absorption coefficient and nonlinear refractive index with re-spect to intensity. . . 79
4.18 Normalized difference and sum of OHD-OKE signals at ±µ. . . . 80
4.19 Normalized sum of OHD-OKE signals at ±µ from parallel polarization configuration. 81 4.20 Experimental results from OHD-OKE with controlled sample temperature.. . . 81
4.21 Illustration of the gated sample. . . 83
4.22 Photo of the home-made sample holder for the OHD-OKE measurement with con-trol of the Fermi level with electrostatic gating. . . 84
4.23 Transconductance measurement. . . 84
4.24 Gating measurement. . . 85
4.25 OHD-OKE results at different values of gating voltage. . . 85
4.26 Geometry of the tilted sample and schematic with the optical elements in the probe path modeled with Jones matrices.. . . 89
4.27 Experimental data and fitting of Sµ di f with respect to the pump polarization angle. . 93
5.1 Schematic of the cross-section of the chip. . . 99
5.2 Cross section of the waveguide and spatial mode profile.. . . 101
5.3 Normalized spatial mode profile of the three components of the electric field. . . 101
5.4 Group velocity dispersion.. . . 102
5.5 Schematic of the waveguide covered with an effective bulk graphene film. . . 103
5.6 Layout and propagation simulations of the directional couples. . . 105
5.7 Layout and propagation simulations of the waveguide array WA1. . . 106
5.8 Layout and propagation simulations of the waveguide array WA2. . . 106
5.9 Layout and propagation simulations of the waveguide array WA3. . . 107
5.10 Layout and propagation simulations of the waveguide array WA4. . . 107
5.11 Layout and propagation simulations of the rectangular structure RS1.. . . 108
5.12 Layout and propagation simulations of the rectangular structure RS2.. . . 109
5.14 Layout and propagation simulations of the multimode interference coupler MMI2.. 110
5.15 Layout and propagation simulations of the multimode interference coupler MMI1 at longer scale . . . 110
5.16 Photos of injection with lensed fibers . . . 111
5.17 Transmission losses in dB of three waveguides with different widths. . . 112
5.18 Fitting of graphene and waveguide losses with respect to the width of the waveguide. 113 5.19 Experimental results of directional couplers in the linear regime. . . 114
5.20 Experimental results of waveguide arrays in the linear regime. . . 115
5.21 Experimental results of rectangular structures in the linear regime. . . 115
5.22 Experimental results of MMI1 in the linear regime. . . 116
5.23 Experimental results of MMI2 in the linear regime. . . 116
5.24 Experimental setup for transmission losses measurements in the nonlinear regime. 117 5.25 Saturation in absorption for different waveguides covered by different graphene length.118 5.26 Experimental measurements of the directional coupler in the nonlinear regime.. . . 119
5.27 Experimental measurements of the directional coupler in the nonlinear regime.. . . 120
5.28 Experimental measurements of the rectangular structure in the nonlinear regime. . 120
5.29 Experimental measurements of the MMI coupler in the nonlinear regime. . . 121
5.30 Experimental measurements of the MMI coupler in the nonlinear regime. . . 122
A.1 Silicon nitride directional couplers covered with graphene. . . 128
A.2 Silicon nitride waveguide array covered with graphene. . . 129
A.3 Silicon nitride multimode interference couplers (MMI) covered with graphene. . . . 129
2.1 Generalized complex nonlinear phase shift relations for two limit cases of samples.. 18
2.2 Generalized complex phase shift relations for different graphene models.. . . 37
3.1 Table summarizing some of the values of saturation intensity of graphene found in the literature. . . 50
3.2 Comparison between published values of the nonlinear refractive index of graphene measured with the Z-scan technique. . . 55
4.1 OHD-OKE zero delay values at different gate voltage values.. . . 86
4.2 Relative values of ¬(3)tensor elements of graphene at pump intensities around 5 £ 1012W/m2. . . 92
5.1 Parameters of the silicon nitride waveguides.. . . 104
5.2 Graphene losses and waveguide losses for different waveguide widths. . . 112
5.3 Comparison between the measured losses from the reference waveguides. . . 113
5.4 Estimated peak and mean power needed to observe a power change in the nonlinear directional coupler due to the nonlinearity.. . . 119
CHAPTER
1
Introduction
In the world of rapidly evolving technology, physical limitations of electronics lead to the end of Moore’s law and researchers seek new pathways for signal-processing in photonics [1]. Taking ad-vantage of the standard semiconductor integration technology, photonic circuits provide new op-portunities for data generation, processing, transmission and storage in information technology and communications. High-speed communication rates, large bandwidth, low energy consump-tion and ecological footprint, as well as low cost and compact design are essential needs addressed by photonics. To this end, all-optical signal processing plays a leading role in recent technologi-cal advances, mainly by lifting the necessity of the electro-optitechnologi-cal conversion process in optitechnologi-cal communications. Optical nonlinearity is the key mechanism for that, therefore there is a strong demand for new materials with high and broadband nonlinearity for all-optical switching, wave-length conversion, pulse generation and modulation, among others.
The advent of graphene in 2004, the third offspring of the carbon family after the fullerenes (0D) and the carbon-nanotubes (1D), opened a new era in nanoscience: that of two dimensional (2D) materials. Graphene attracted the interest of researchers quite early with a plethora of extraordi-nary properties. It has been studied extensively for its exotic physical properties and proposed for applications in a wide range of fields; photonics could not be an exception. Graphene is a promis-ing candidate material for integrated photonics, as not only it is compatible with the standard CMOS-technology for the production of integrated circuits, but also, due to its atomic thickness, it is ultra-compact and features broadband1and tunable optical properties. As such, graphene has been proposed in various photonic and electro-optic applications, including modulators, pho-todetectors or sensors. Also, graphene has been used as a saturable absorber in mode-locked lasers.
In 2007, it has been predicted that graphene has a strong and broadband nonlinear optical re-sponse, something that was later confirmed from the first experimental demonstrations. It is therefore reasonable to consider that graphene has a great potential to be used for all-optical signal processing in photonics, as this relies on the optical nonlinearity. The first step to investigate this
1The term broadband in this Thesis refers to large spectral range.
potential, is a detailed characterization of the nonlinear optical parameters of graphene. Namely, the knowledge of the sign and the magnitude of the nonlinear refractive index is quite important, as this is the most crucial nonlinear parameter for all-optical switching. Nevertheless, there are large discrepancies between already reported values. In addition, the tensor nature and the dy-namics of the nonlinearity, as well as the separate estimation of its real and imaginary part related to nonlinear refraction and absorption respectively, are still not fully investigated by researchers, and commonly used experimental characterization methods are not sufficient to address these issues.
In view of that, the first goal of this thesis is related to the experimental characterization of the nonlinear optical properties of graphene at telecom wavelengths. To this end, apart from using the common Z-scan technique, we have implemented, for the first time in graphene, the ultrafast optical Kerr effect method, coupled to optical heterodyne detection (OHD-OKE). This method pro-vides a separate measurement of both the real and the imaginary part of the nonlinearity together with their dynamics. We have therefore demonstrated that graphene has a negative nonlinear re-fractive index in contradiction with previously reported results. In addition, we have studied the tensor nature of the nonlinearity by providing an enhanced version of the OHD-OKE method to-gether with the appropriate theoretical framework that allows to extract the tensor elements of the nonlinearity, including the - considered as negligible - out-of-plane tensor elements.
The second goal of this thesis is to demonstrate that the nonlinear optical response of graphene can be used for applications in photonics, specifically targeting all-optical switching. As such, we have experimentally explored the use of the optical nonlinearity of graphene in photonic in-tegrated structures for all-optical signal processing from an experimental point of view. Namely, we have designed a photonic chip with simple silicon nitride waveguide structures, basic build-ing blocks of switchbuild-ing devices, covered by graphene patches. The optical nonlinearity of silicon nitride is negligible with respect to that of graphene, therefore we have expected to observe non-linear behavior of the waveguide structures, due to the added graphene. Following the fabrication of the chip, we have experimentally tested the graphene-covered structures at low and high power levels.
This thesis is organized as follows:
In Chapter 3, we present the Z-scan technique giving emphasis to the advantages and the incon-veniences that led to the quest for a different characterization method. Also, our experimental measurements and results are provided, regarding the saturable absorption and the nonlinear re-fractive index of graphene.
Chapter 4 includes our work regarding the OHD-OKE method. Starting from a simple mathemat-ical description of the method, we describe in detail the experimental setup and the most impor-tant steps needed to build it. Then, we present the experimental results, starting from the measure-ment of the nonlinear refractive index of graphene, the imaginary part of the nonlinearity related to the saturation of absorption and the temporal dynamics of the different components of the linearity. In addition, we provide our measurements regarding the modification of graphene’s non-linearity with applied electrostatic gating and finally, we propose an enhanced OHD-OKE method for the measurement of the different tensor elements of the nonlinearity including the out-of-plane ones.
Chapter 5 is related to our studies on the graphene-covered integrated silicon nitride waveguide structures. To begin, the parameters of the waveguides and the simulations that leaded to the design of the chip are provided. Then, the experimental results at the linear and the nonlinear regime (that correspond to low and high input power respectively) are provided and discussed. In Chapter 6, we summarize the major conclusions drawn from our work, from both the de-tailed characterization of graphene’s optical nonlinearity and the realization of graphene nonlin-ear waveguide structures, and finally we give the perspectives for future work related to this thesis. Publication list
• E. Dremetsika et al., ’Measuring the nonlinear refractive index of graphene using the optical Kerr effect method,’ Optics Letters 41, 3281-3284, July 2016. [2]
• E. Dremetsika and P. Kockaert, ’Enhanced optical Kerr effect method for a detailed character-ization of the third order nonlinearity of 2D materials applied to graphene,’ Physical Review B 96, 235422, December 2017. [3]
• Dremetsika et al., ’Characterization of nonlinear optical properties of graphene at telecom-munication wavelength’, at ’Spatio -Temporal Complexity in Optical Fibers’ international workshop IEEE, Italy, Sept. 2013
• E. Dremetsika et al., ’Characterization of the 3rd order optical nonlinearity of graphene with the ultrafast Optical Kerr Effect method,’ ’Nice Optics 2016: 1st international conference on optics, photonics and materials’, France, Oct. 2016
CHAPTER
2
Theory
In this chapter, the basic theoretical concepts related to our work are discussed. Firstly, the theo-retical framework describing the response of a material to an optical field is provided. Then, the most common characterization methods for the optical nonlinearity are presented. After that, graphene is introduced, starting from its general properties, to discuss in detail its linear optical properties. Also, some of the existing works on the nonlinear optical properties are discussed with an emphasis on the third-order nonlinearity and finally, the different models of graphene for linear and nonlinear optics are described and discussed.
2.1 Basic concepts in optics and nonlinear optics
In this first Section, an analysis is given of the material response to electromagnetic radiation, in order to introduce the basic notions of the material’s nonlinear optical response and therefore link the experimentally measured values with the intrinsic properties of the materials found in theory. Indeed, from experimental measurements usually phase or absorption changes are extracted that are usually linked to a refractive index or absorption coefficient. However, the intrinsic properties of the materials are expressed by theoreticians with the parameters of the constitutive relations, namely the permittivity, the susceptibility or the conductivity of the material. These notions will be discussed in this section.
2.1.1 Material response to an electromagnetic radiation
To describe the optical properties of materials [5], we start from the well-known macroscopic Maxwell’s equations in a non-magnetic medium with a notation of the fields varying in space and
time as ~F ¥ ~F(~r,t), r · ~D = Ωf, (2.1) r · ~B = 0, (2.2) r £ ~E = °@~B @t , (2.3) r £ ~H = ~Jf+@~D @t , (2.4)
where Ωf and ~Jf denote the free charge and current densities, ~E and ~B the electric and magnetic
fields, and ~D and ~H the displacement and the magnetizing field respectively. The last two are
defined as ~ D = "0~E + ~P, (2.5) ~ H = 1 µ0 ~ B. (2.6)
Here, "0is the permittivity and µ0the permeability of vacuum. ~P is the electric-dipole polarization
field, a macroscopic field, linked to the microscopic polarizability.
From the conservation law of charges, for the current ~J and the charge distribution Ω we get
r ·~J+@Ω@t = 0, (2.7)
where the current is the sum of free and bound currents ~J = ~Jf+~Jband the bound current is related
to the polarization field as
~Jb=@~P
@t . (2.8)
The material properties are found in the constitutive relations, which basically imply the relation between ~D and ~E, and ~H and ~B due to the response of the material to the electromagnetic field.
As such, the polarization (linear) is related to the electric field as ~
P(~r, t) =
Z+1
°1 ¬(~r °~r
0, t ° t0) · ~E(~r0, t0)dr d t0= ¬ ≠ ~E, (2.9)
where ¬(~r, t) is the electric susceptibility of the medium (dimensionless), which describes the po-larization induced in a medium when an electric field is applied. In the following, the spatial de-pendence of the susceptibility is neglected, as we study only local phenomena. Therefore, in what follows, we shorten the notation and write the fields as F (t) = F (~r,t). Usually, the spectral ex-pression is used, obtained by Fourier transforming the temporal one, for a monochromatic plane wave:
~P(!) = "0ˆ¬(!)~E(!). (2.10)
media1. The relation of the susceptibility with the frequency defines the dispersion of the medium. For simplicity, in the following we provide the constitutive relations, omitting the dispersion.
~P = "0ˆ¬~E, (2.11)
~D = "~E, (2.12)
(2.13) where " = "0(1+ ˆ¬) is the electric permittivity of the medium, and "r= "/"0= (1+ ˆ¬) is the dielectric
constant or relative permittivity of the medium (dimensionless). The permittivity is a measure of how the material opposes to an external electric field. Another intrinsic property of a material related to the electric field is the electric conductivity æ, which describes the ability of the material to conduct an electric current, as
~J = æ~E. (2.14)
In isotropic media, in the linear regime, æ is scalar, similarly to ˆ¬ and ", while in anisotropic media they are expressed as tensors. In lossless perfect dielectrics usually æ is considered as zero and " as real, and in perfect conductors æ is infinite. To describe real materials that are neither perfect conductors, nor perfect dielectrics, we can use a complex permittivity/susceptibility or a complex conductivity. In this thesis, we consider both as complex and therefore, the following relation between the susceptibility and the conductivity is used:
ˆ˜¬ = i!"æ˜
0. (2.15)
The flow of electromagnetic energy is expressed with the Poynting vector which is defined as [6, A.13]2:
~
S = ~E £ ~H (2.16)
The time-averaged magnitude of the Poynting vector represents the light intensity, while its direc-tion is the propagadirec-tion direcdirec-tion of energy. For a plane monochromatic wave, the intensity is given by
I = |hSi| = 2"0cpRe{"r}|E0|2. (2.17)
where a phasor notation is used, so that E = E cos(~k~r ° !t + ') = E0ei (~k~r°!t)+ c.c.
In this thesis, we will mainly address two basic electromagnetic problems: the propagation of an electromagnetic wave in a medium and the continuity of electromagnetic radiation at an interface.
2.1.1.1 Propagation equation
To describe the propagation of an electromagnetic wave in a medium, we combine Maxwell’s equations to find the wave equation [7,8]:
r2~E ° 1 c2 @2~E @2t = µ0 @~P @t . (2.18)
In the spectral domain, with Fourier transformation we get the dispersion relation (with k the wavenumber) in a material without losses:
k2°!
2
c2(1 + ˆ¬(!)) = 0. (2.19)
The refractive index of the medium is defined here as
n2¥ "r= 1 + ˆ¬(!), (2.20)
so that, when the dispersion relation is fulfilled, solutions propagate in the medium with a phase-velocity c/n3. The plane wave traveling along the z-axis is expressed with the phasor notation as
E = E0ei (kz°!t), (2.21)
where k = 2º∏0n. Obviously, the phase change of the wave due to propagation is related to the
refractive index of the medium, as
¢©=2º
∏0n¢z, (2.22)
for a propagation distance equal to ¢z. From a phenomenological point of view, in a lossy medium, we can use a complex index ˜n = n + i∑ with the real part being equal to the refractive index, and
the imaginary part to the extinction coefficient. So, the amplitude of the wave is related to the ex-tinction coefficient as E(z + ¢z) = E(z)ei2º∏0n¢ze°
2º
∏0∑¢z. This is linked to the absorption coefficient
as e°∏02º∑¢z= e°Æ2¢ztherefore the absorption coefficient is given by
Æ=4º
∏∑. (2.23)
In this case, we introduce a generalized ‘phase’ shift which is complex, which is proportional to the complex index ˜n, to describe both phase shifts and amplitude changes in one equation:
¢©c=2º
∏0n¢z.˜ (2.24)
The propagation equation is used when we study the propagation of electromagnetic waves inside
a specific medium, for example an optical waveguide. In this case, we have provided the well-known relation between the phase shift and the refractive index. However, if along the propagation path, the wave is transmitted through an interface, then we have to take into account both the transmission and the reflection of the electromagnetic wave at the interface. We have therefore to consider the problem of electromagnetic field continuity, in order to calculate the phase shift or amplitude change of the output wave, which is not due to propagation only.
2.1.1.2 Birefringence and dichroism
Before dealing with the problem of electromagnetic field continuity at an interface, we consider the tensor nature of the susceptibility of a material, to introduce the concepts of birefringence and dichroism, because they will be crucial for the description of our experiments later.
Generally, the susceptibility tensor is expressed as:
¬= 2 6 6 4 ¬xx ¬x y ¬xz ¬yx ¬y y ¬yz ¬zx ¬z y ¬zz 3 7 7 5 (2.25)
In isotropic media only the diagonal elements of the linear susceptibility tensor are non-zero, and equal with each other, so the susceptibility can be expressed by a scalar. However, this is not the case for a large variety of materials, for which, as implied by the tensor nature of the susceptibil-ity, the refractive index depends on the polarization and the propagation direction of the incident wave. This anisotropy related to the refractive index is called birefringence. The same property in terms of absorption, concerning the imaginary part of a complex susceptibility, is called dichro-ism.
In this thesis, we will be interested in the most simple kind of birefringence/dichroism, the uniaxial one. In uniaxial materials, as for the isotropic, there exist a reference frame in which only the diagonal elements of the susceptibility are non-zero, but the tensor is given by
¬= 2 6 6 4 ¬xx 0 0 0 ¬xx 0 0 0 ¬zz 3 7 7 5. (2.26)
Tetragonal, trigonal and hexagonal crystal lattices are characterized by this kind of linear suscep-tibility [6, Ch. 1.5.9]. In uniaxial birefringent materials the dielectric tensor is usually written as
where no the index of refraction seen by light propagating with a polarization perpendicular to
the optical axis of the crystal (here z-axis), the so called ‘ordinary’ wave and ne the refractive
in-dex of an ‘extraordinary’ wave propagating with a polarization parallel to the optical axis. The optical axis is unique in a uniaxial medium and as implied by the tensor, the ‘ordinary’ wave ex-periences the same optical properties for any polarization rotation around the optical axis. When light propagates through the medium with mixed polarization (with an ordinary and an extraor-dinary component), its polarization is altered due to the different phase velocity and the induced phase difference. For example, a linear polarization state could become elliptical. This is the basic principle of the waveplates.
2.1.1.3 Continuity of electromagnetic fields at an interface
In our experiments, as in the majority of experiments studying the optical properties of materials, we measure the transmission of a beam through a sample of limited thickness. Trying to answer the question, ‘what exactly we measure in this kind of experiments, and how the phase and am-plitude changes are related to the optical index of the material?’, we find that in some cases, the output field differs from what results from the simple propagation in a medium without interface involved, because here we should take into account both reflection and transmission. So, we need to consider the boundary conditions of the Maxwell’s equations at the interface, and calculate the Fresnel coefficients [7,8]. Moreover, for a sample with limited thickness, and depending on the pulse duration of the laser that we use, sometimes we have to take into account multiple reflec-tions in the sample. The full reasoning, is found in many textbooks [7,9] and later in2.4.2for the case of the bulk graphene model.
FIGURE2.1: Simplified geometry of the experiments studying the transmission of a beam through a sample of limited thickness. In some cases, the consideration of the continuity of
electromag-netic fields at the interfaces is important.
In Fig.2.1the basic geometry that is usually considered in this kind of experiments is depicted. A sample of thickness d surrounded by air is studied. At normal incidence, for both TE and TM polarizations, the transmission coefficient t4of the electric field for the case where we can neglect multiple reflections (nmr), is given by the Fresnel coefficient multiplied by the propagation phase
term, while for the case that multiple reflections cannot be neglected (mr),t is given by the Fabry-Perot equations. Thus, the t coefficients for the two cases are given respectively as:
tnmr = 4 ˜n (1 + ˜n)2ei k0nd˜ , (2.28) tmr = 4 ˜n (1+ ˜n)2ei k0nd˜ 1 ° (1° ˜n1+ ˜n)2ei 2k0nd˜ . (2.29)
In the above expressions, the refractive index is complex for materials with losses. It is worth mentioning some different limit cases of this kind of problems, regarding the phase change of the output field, in order to understand the physical difference between these two transmission coefficients:
1. For an experiment in which multiple reflections in the material are negligible5we get the output field from eq. (2.28) so for a pure dielectric material without losses the output phase is k0nd, which is the same as in eq. (2.22), because it results only from the propagation in
the sample. On the contrary, if the material is lossy ( ˜n complex), there is a phase component
to be added to k0nd 6. This implies that the phase in the output, is not simply proportional
to the refractive index of a material with losses.
2. For a very thin material, verifying k0| ˜n|d ø 1, so that ei k0nd˜ º 1+ik0nd, we take into account˜
the multiple reflections to get from eq. (2.29) the output field
E(z0+ d) = E(z0)ei k0(1+ ˜n2)2 d. (2.30)
It is obvious that the phase in the output is not proportional to the refractive index in this case. In fact, it is proportional to Re{1 + ˜n2} = 1 + n2° ∑2(= Re{2 + ˜¬}).
3. For materials for which we have to take into account multiple reflections, but they are not thin enough to make the previous approximation, the relation is more complicated and it is more difficult to resolve the relation between the phase shift and the refractive index. In some cases, the common phase term ¢© = ik0nd is dominant. Comparing the phase terms
from eq. (2.29) we find the criterion to use such approximation for materials without losses:
k0nd ¿ arctan √ (1°n1+n)2sin(2k0nd) (1°n1+n)2cos(2k0nd) ° 1 ! . (2.31)
All in all, one has to be very careful, when extracting the phase or the absorption of an electromag-netic field to give the correct relation with the complex material index. Up to this point, we have studied the linear phase shift, valid when the input intensities are low. However, in this thesis, we
5This depends on the pulse duration ø. For thickness d > 2cø multiple reflections are negligible. 6Equal to ¢©
are interested in the relation between the nonlinear phase shift and absorption with the nonlinear complex index change. In the following, we introduce some basic concepts of the theory of the nonlinear optical response of a material, and then we study the propagation and continuity-at-interface problems.
2.1.2 Nonlinear optical response
Generally, when the electromagnetic field is strong, the linear constitutive relations are not suffi-cient to describe the material response. This implies that the material properties ",¬,æ are field-dependent. In this case, the response of the medium to an optical field is said to be nonlinear [5,6]. Thus, the polarization field can be written as a power series of the field, and then separated into a linear and a nonlinear term:
~P = "0ˆ¬(1)~E + "
0ˆ¬(2)~E~E + "0ˆ¬(3)~E~E~E + ... (2.32)
~P = ~P(1)+~P(2)+~P(3)+ ... (2.33)
~P = ~PL+~PN L (2.34)
The above equations are simplified. There is a big variety of nonlinear optical effects governed by this kind of equations. Sum and difference frequency generation, multi-photon absorption, sat-urable absorption, Kerr effect, self-focusing are only some of them [6]. In centrosymmetric mate-rials, second-order nonlinearity is forbidden for symmetry reasons [6], so the next most important nonlinearity term is the third-order nonlinearity, which will be the main subject of our work. 2.1.2.1 Third-order nonlinear response
Keeping only the third order term, that involves the interaction of three fields, in the nonlinear polarization we get the simplified relation
~PNL= "
0ˆ¬(3)~E~E~E,7 (2.35)
which in the temporal domain would be described as ~
PNL= "0
—
¬(3)(t ° t1, t ° t2, t ° t3)...~E(t1)~E(t2)~E(t3)d t1d t2d t3, (2.36)
while in the spectral domain as ~PNL(!) = "
0 X
(lnm)
ˆ¬(3)(! = !l+ !n+ !m)~E(!l)~E(!n)~E(!m).8 (2.37)
7The ¬(3)is generally a fourth-rank tensor. Its tensor nature is discussed later, and here it is considered as a scalar. 8In the following, in order to simplify the notations we use ¬(3)and not ˆ¬(3)even when it is expressed in the spectral
The third-order susceptibility, governs different phenomena, such as third-harmonic generation (THG), four-wave mixing (FWM), Kerr effect, two-photon absorption (TPA) and others [6]. In this thesis, we study experimentally the Kerr effect and TPA, in which ! = ! + ! ° ! and ~k =~k +~k °~k, as illustrated in Fig.2.2.
FIGURE 2.2: Illustration of a frequency mixing process due to the third-order nonlinear effect
(equal frequencies: Kerr effect), reproduced from [6].
As for the response time of the third order nonlinearity, in many cases, in order to simplify the cal-culations, we assume that the nonlinear response is ultrafast, or at least faster than the pulse dura-tion (in case we use pulsed laser), although this is not always true. We then add a phenomenolog-ical relaxation time to account for the delayed response in the experimental measurements. The response time of the third-order nonlinearity depends on its origin. Electronic nonlinearities can generally be considered as ultrafast, as their characteristic relaxation time is in the order of a few femtoseconds, which is generally smaller than the pulse duration in the range of 100 fs to 10 ps. Parametric processes are also considered as ultrafast, because they involve virtual levels, how-ever non-parametric processes are in general slower. Third-order nonlinearities from molecular re-orientation processes are slow. It is possible to confuse thermo-optic effects with third-order optical nonlinearities, because of the linear relation of the phase change with the intensity of light. Thermo-optical nonlinearities are also slow (ns).
2.1.2.2 Third-order susceptibility
The third order susceptibility ¬(3) is a fourth-rank tensor with 81 elements. Depending on the symmetry of the medium, the number of non-zero and independent elements is reduced. This is studied in detail in [6,10]. For example, in isotropic media there are 21 non-zero elements, from which 3 are independent: ¬xx y y,¬x y y xand ¬x yx y[6].
It is worth mentioning, that it is possible to induce birefringence to an isotropic and therefore non-birefringent medium, by applying a strong beam (the so-called pump) [5, Ch.16]. Therefore, the resultant total susceptibility tensor (linear and nonlinear) is non-diagonal, because of the third-order susceptibility tensor. This will be discussed in more detail in Chapter 4.
mechanical models using usually the density matrix formalism. In this thesis, we do not study the nonlinearity from the perspective of a microscopic model. We rather work with a ‘black-box’ model: basically, we experimentally measure phase and amplitude changes to extract and study the nonlinear susceptibility of graphene from a phenomenological point of view. For this reason, in the next sections, we explicate the relation between the nonlinear phase shift and the third-order susceptibility, as well as the nonlinear refractive index, similarly to their linear counterparts in Sections2.1.1.1and2.1.1.3.
2.1.3 Nonlinear propagation
To describe the propagation of light in a nonlinear medium, we have to add a nonlinear term in the propagation equation to account for the nonlinear response of the material. This is the nonlinear polarization term [5,11], shown in simplified form in (2.35). We consider a monochromatic wave propagating in a medium characterized by an ultrafast isotropic third-order nonlinearity. In the slow varying envelope approximation (SVEA), the field is written as E(~r, t) = A(~r)ei kze°i!t, with
Fourier transformation only in the time domain. We use the paraxial approximation, with@2A
@z2
neg-ligible. The resulting equation, as demonstrated in many textbooks, has the form of a Nonlinear Schr¨odinger equation (NLSE). With k = n0k0, we get
2i k@A @z + µ@2A @x2 + @2A @y2 ∂ + k02¬(3)eff|A|2A = 0, (2.38)
in which the first term accounts for the propagation, the second for diffraction and the third for the nonlinearity. The NLSE implies a lens-like behavior of the medium at high intensities, due to the nonlinear phase variations. For a gaussian beam propagating in the medium, this leads to self-focusing or defocusing of the beam depending on the sign of the nonlinearity. When the self-focusing effect is of the same order of magnitude as the diffraction effect, the one effect can counteract the other.9 In the same way, a pulse propagating in a nonlinear medium experiences similar effects from the interplay between the dispersion and the nonlinearity [11, Section 1.3.1]. This is not further discussed here, because these effects are not present in our experiments, due to the very small thickness of our samples.
2.1.3.1 Nonlinear phase shift
From the NLSE equation2.38, we will calculate the nonlinear phase shift, for a short propagation distance, for example propagation in a thin sample [12, Ch.14]. Thus the diffractive effect is negli-gible and we assume that the nonlinearity is only a perturbation. For the field on-axis, we find the solution
E(z + ¢z) = E(z)ei k0n¢zei k0
¬(3)eff
n0|E|2¢z. (2.39)
9When the beam profile varies only in one transverse dimension, for example in propagation in a planar waveguide,
It should be mentioned, that ¬(3)and n0can be complex in the presence of linear and nonlinear
absorption. In this case, k is also complex in (2.38). So, the nonlinear phase shift is given as:
¢©NL= k0Re ∑ ˜¬(3) eff ˜ n0 ∏ |E|2¢z. (2.40)
We have provided in the equation (2.40) above, the relation between the nonlinear phase shift and the ¬(3)for a wave propagating in a medium for short distance, so that diffraction is negligible (or for a plane wave). Now, we will express the nonlinear phase shift in terms of a refractive index change. This is easy, if we consider n = n0+ ±n. For a propagation distance ¢z = d, comparing
with the linear phase shift in (2.22), we get
¢©N L= k0±n d. (2.41)
2.1.3.2 Nonlinear refractive index
In Kerr media (media with third-order nonlinear response) we define the nonlinear refractive index as
n2¥±nI (2.42)
where I the incident intensity from (2.17). However, in some books it is defined as n2¥|E|±n2.
There-fore, values of the n2in the literature are expressed either in m2/W, corresponding to (2.42) or in
m2/V2. In most cases, going from one constant to the other is easily performed by multiplying or dividing by the factor appearing in (2.17), which should be written explicitly with Re(n0). It is
im-portant to note that this expression makes use of the real part of the refractive index that should be measured independently from n2. We will see later that this introduces experimental difficulties,
and explains probably why in the literature, authors refrains to compare values in m2/V2(or esu) to values in m2/W. The same applies to the comparison between n
2in m2/W and ¬(3)which is
expressed in m2/V2. We express then the nonlinear phase shift as
¢©N L= k0n2I d. (2.43)
When ¢©NLdoes not scale linearly with the intensity, meaning that higher order effects are present,
the n2is not a constant coefficient, but a function of the intensity10.In this case, a more general
definition of n2can be:
n2=@n
@I. (2.44)
10As far as the assumption of a fast nonlinearity is valid. Otherwise, the nonlinear response should be expressed as
As such, in materials without losses we can easily find the relation between the nonlinear refractive index and the nonlinear susceptibility, by comparing (2.43) with (2.40):
n2=4"3 0Re ∑ ˜¬(3) eff ˜ n0 ∏ (2.45) 2.1.3.3 Nonlinear absorption
For a material that presents nonlinear absorption, in analogy with the nonlinear refractive index we define the nonlinear extinction coefficient ∑2as
∑2=@∑
@I . (2.46)
As for its linear counterpart, the nonlinear absorption change is given by ±Æ =4º∏±∑, where ±∑ =
∑2I in case of third-order nonlinearity ∑ = ∑0+ ∑2I . As such, two-photon absorption, which is
a third-order nonlinear process, implies that the third-order susceptibility is complex. The TPA coefficient Ø in a material with negligible linear losses, is given by
∑2=4º∏ Ø=4"3 0Im ∑ ˜¬(3) eff ˜ n0 ∏ . (2.47)
In this thesis we will also study another type of nonlinear absorption, the saturable absorption [6, Section 1.2.11], which is a non-parametric process. This is not described by a single third-order coefficient, because higher order terms are also needed, as the absorption is given by
Æ(I ) = Æ0
1 +IIS
11 (2.48)
where Is the saturation intensity. The absorption saturation appears when a material is
illumi-0 5 10 15 20 25 30 0 0.2 0.4 0.6 0.8 1
Normalized intensity I/Is
Normalized absorption
FIGURE2.3: Qualitative diagram of absorption saturation.
11This relation is sometimes expressed with the fluence instead of the intensity, when the pulse duration is very short
nated by a strong field and, due to Pauli blocking, the absorption decreases with increasing inten-sity and is saturated at high inteninten-sity values (band-filling effect). A qualitative diagram of saturable absorption is given in Fig.2.3.
2.1.3.4 Full relation between third-order susceptibility and nonlinear refractive index in ma-terials with losses
In a material with linear and nonlinear losses we have to be careful to use the full relations between the complex nonlinear index ˜n2= n2+ i∑2and the complex third-order susceptibility, found by
comparing (2.43) with (2.40): ˜ n2 = 4"3 0n0 ˜¬(3)eff ˜ n0 (2.49) Re[ ˜¬(3)eff] = 4"03n0(n0n2° ∑0∑2) (2.50) Im[ ˜¬(3)eff] = 4"03n0(n0∑2° ∑0n2) (2.51)
These expressions are found in the literature [13], however, as it is discussed in the next section, one has to be very careful when using them, if transition through an interface is involved during the propagation of the wave. Comparing with (2.24), the generalized complex nonlinear ‘phase’ shift is
¢©c NL=2º
∏0± ˜n¢z = k0
˜
n2I ¢z. (2.52)
2.1.4 Nonlinear phase shift in problems involving continuity at interface
Looking at the electromagnetic problem when the interfaces of a sample are involved, for a non-linear change of the complex refractive index ˜n = ˜n0+ ±˜n, we find that the nonlinear phase shift
is not always described by2.43. Considering the same limit cases as in Section2.1.1.3, we use the same expressions and we substitute ˜n with ˜n0+ ±˜n, with |±˜n| ø | ˜n0| and we get:
1. For an experiment in which multiple reflections are negligible,12in a medium without losses, considering (2.28), we find that the common propagation nonlinear phase shift of equation (2.43) can be used, if k0d ¿ n02+1, which is usually satisfied. For example, at 1550nm for a
material with n0= 2, we should have d ¿164nm.
2. For a very thin material such that k0| ˜n0|d ø 1, so that ei k0n˜0dº 1 + ik0n˜0d, taking equation
(2.30), the generalized complex nonlinear phase shift is
¢©c NL= k0n˜0±˜n d = k0n˜0n˜2I d = k0˜¬(3)|E|2d. (2.53)
12This depends on the pulse duration ø with respect to the thickness of the sample d. For 2d > cø multiple reflections
This relation differs from the equation (2.52) of the complex nonlinear phase shift from propagation because in this case, the nonlinear phase shift (and attenuation) from the in-terfaces of the sample is important. The nonlinear phase shift is the real part of (2.53):
¢©c NL= k0Re[ ˜n0±˜n]d = k0Re[ ˜n0n˜2]I d = k0Re[ ˜¬(3)]|E|2d, (2.54)
that obviously differs from (2.41), (2.43) and (2.40). For example, in the third expression of (2.54) with the ¬(3), the complex linear index ˜n0does not appear in this case, contrary to the
expression (2.40).
3. For materials in which multiple reflections have to be taken into account, but they are not thin enough to take the previous approximation, the calculation of the nonlinear phase shift is cumbersome and out of the scope of this thesis. Therefore, the nonlinear phase shift given by the standard relations for the nonlinear phase shift from propagation (2.41), (2.43), (2.40) or (2.52) cannot be used.
To conclude and show the importance of taking into account this kind of problems in our work, we consider as an example two different samples studied with ultrashort pulses of ø = 100 fs duration and wavelength 1550 nm. In this case, if the thickness of the sample is larger than 15 µm, mul-tiple reflections are negligible and the nonlinear phase shift (according to case 1) is given by the standard relations (2.41), (2.43), (2.40) or (2.52), which corresponds to the nonlinear phase shift from the propagation in a medium, without taking into account the interfaces. On the contrary, for a sample thinner than 120 nm that corresponds to case 2 (k0n0d ø 1), we take into account
multiple reflections and the nonlinear phase shift is given by (2.54), (2.53). These conclusions are summarized in the Table2.1that follows.
TABLE 2.1: Generalized complex nonlinear phase shift relations for two limit cases of samples
studied with pulse duration ø = 100 fs ∏0=1550 nm. (The relations in the same column are
equiv-alent). ¢©c NLfor d > 15 µm ¢©c NLfor d < 123 nm k0±˜n d k0n˜0±˜n d k0n˜2I d k0n˜0n˜2I d k0˜¬ (3) eff ˜ n0 |E| 2d k 0˜¬(3)eff|E|2d
When we want to compare the nonlinear refractive index of two materials with samples that cor-respond to the two different categories presented in the Table2.1, it is sometimes useful to use an effective nonlinear refractive index for the thin sample, so that ˜neff2 = ˜n0n˜2and we can therefore
use the same relation for the nonlinear phase shift ¢©c NL = k0n˜eff2 I d and directly compare the
nonlinear index, without the need to know the linear index ˜n0. However, in this case we should be
relation between ˜n2effis not the same as the one given in Section2.1.3.4, but we get instead ˜ n2eff = 4"3 0n0˜¬ (3) eff, (2.55)
Re[ ˜¬(3)eff] = 4"03n0neff2 , (2.56)
Im[ ˜¬(3)eff] = 4"03n0∑eff2 , (2.57)
where ˜neff2 = n2eff+ i∑eff2 . This conclusion is quite important for our work, as we will see in the next chapters.
2.1.5 Ultrafast lasers
Despite the fact that in most of the expressions found in this work we use a monochromatic wave for the sake of simplicity and readability, in our experiments we used optical sources that deliver ultrashort pulses. In particular, we mainly used two lasers in our experiments: one fiber mode-locked laser amplified by an EDFA delivering picosecond pulses, and one Ti:Sapphire laser deliv-ering femtosecond pulses.
The electric field of an optical pulse is described with an envelope function, related to the pulse shape, and a carrier function related to the field oscillation, as
E(t) = Re≥E0a(t)ei (kz°!0t)
¥
. (2.58)
Usually the pulses are Gaussian or sech-shaped, so that the envelope is given respectively by
a(t) = e°(øpt ) 2 , (2.59) a(t) = sech≥ t øp ¥ , (2.60)
where øpis related to the pulse duration. The power spectrum of a Gaussian pulse is given by
P(!) / e°ø22p(!°!0)2. (2.61)
2.1.6 Pump-probe measurements
measurements are also very common. Finally, birefringence and dichroism measurements are linked to pump-induced anisotropies of the refractive index and the absorption respectively. A simple illustration of a pump-probe measurement is depicted in Fig.2.4. The simplest and
FIGURE2.4: Schematic of pump-probe experiment.
most common pump-probe experiment is the measurement of differential transmission. In a few words, this consists of measuring the transmission when the pump is unblocked (ON) and blocked (OFF). To give an idea of the pump-probe signal, the differential transmission, using equation (2.39) without taking into consideration the temporal evolution of the signal, is given by
¢T T = TON° TOFF TOFF = e °2k0Im ©¬(3)eff n0 ™ |E|2¢z º 1 ° 2k0Im n ¬(3) eff n0 o |E|2¢z = 1 ° ±Æ¢z.13 (2.62) In this thesis, we will be interested in degenerate pump-probe measurements, in which both pump and probe are at the same wavelength.
2.2 Characterization of optical nonlinearities
Several methods exist for the characterization of the third-order optical nonlinearities. The most common among them is the Z-scan technique [15], which is widely used due to its simplicity, in terms of experimental setup and interpretation of the results. Z-scan is based on the self-focusing or defocusing of a beam due to the nonlinear refraction. It is a sensitive single-beam technique that allows the determination of the sign and the magnitude of the nonlinear refractive index, as well as the nonlinear absorption. However, as will be discussed in detail in the next chapter, the experimental data should be carefully interpreted, because Z-scan is sensitive to all kinds of nonlinearities, including thermal ones. Also, the recorded data are strongly influenced by inho-mogeneities of the samples. Z-scan is one of the methods that we implemented to characterize the nonlinearity of graphene. We will therefore present it in detail in the next Chapter.
Another very common method is the four-wave mixing (FMW) technique [16], which is based on the detection of a nonlinear signal beam generated by four-wave mixing between pump and probe 13Of course, in practice the source is pulsed and a delay is introduced between the pump and probe pulses. Therefore,
fields. This method is very sensitive, but it requires more than one beam and also the appropri-ate angles between the beams, so that phase-matching is achieved (except if it is performed in photonic integrated waveguides [17]). By performing the FWM, only the magnitude of the fast electronic component of the third order susceptibility is accessible. Thus, we cannot distinguish phase from magnitude nonlinear effects. Similar to FWM, is the third-harmonic generation (THG) experiment.
More accurate, but still more complicated methods for measuring the nonlinear refractive index include interferometry. For example the spectral shearing interferometry (SPIDER) [18] or the Mach-Zehnder interferometry [19], which have the advantage to allow single shot characteriza-tion, but require a reference beam and quite high precision. Moreover, the cross-polarized auto-correlation method has been used to measure different tensor elements of second and third order susceptibilities ¬(2)and ¬(3)from second harmonic generation (SHG) and two-photon absorption (TPA) respectively [20]. The main difficulty of these methods is that they require very high stability of the experimental setup and the source.
Moreover, optical Kerr effect (OKE) methods, based on polarization changes due to nonlinearly-induced birefringence are less complicated than interferometric methods and contrary to Z-scan, they are not sensitive to thermal effects. In addition, they provide the relaxation dynamics of the nonlinearities. The second method that we used in this thesis falls into this category. Thus, it will be discussed in detail in Chapter 4.
2.3 Linear and nonlinear optical properties of graphene
In this section, we introduce the protagonist of our work: graphene, the third offspring of the carbon family after the fullerenes and the carbon-nanotubes. We discuss its unique electronic and optical properties and finally we present its nonlinear optical properties, giving emphasis to the state of the art regarding their theoretical investigation and experimental characterization.
2.3.1 Graphene: the wonder material
It all started with a scotch tape! Graphene, a ‘flat monolayer of carbon atoms tightly packed into a two-dimensional (2D) honeycomb lattice’ [24], was firstly isolated in the lab by A. Geim and K. Novoselov in 2004 with the help of adhesive tape [25]. This was surprising for the scientific community, because although graphene and other 2D materials had been theoretically studied before 2004, it was claimed that they would not be stable enough to exist in the free state [24]. The discovery of graphene, which was awarded with a Nobel prize in 2010, opened the gate for new fascinating physics [25–27], and of course, the research for a wide range of applications in many different technological domains. It was soon discovered that graphene has a number of remarkable properties that render it a promising material that could replace or enhance other materials in existing applications or even be used for new applications.
Due to its remarkable properties and its promising potential for applications, graphene has been usually referred as a wonder material. As a two-dimensional material, it is characterized by an atomic thickness around 0.33 nm. Graphene is one of the strongest materials, 200 times stronger than steel. Also, it has been reported, that it has a very high thermal conductivity, as well as an exceptional electron mobility. Exotic phenomena have been observed in graphene, such as the anomalous quantum Hall effect [26] and the Klein paradox. As will be discussed later in this thesis, graphene has many unique electronic [27] and optical, as well as plasmonic properties. Finally, graphene is easily synthesized and is compatible with the CMOS technology, so that it can be in-tegrated on a chip, or combined with other materials in 2D heterostructures.
2.3.1.1 Production of graphene
