10 2.1.2 Nonlinear optical response

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Contents

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 refractive 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

2.3.3 Linear optical properties of graphene . . . . 25 vi

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Contents vii

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

4.2.6 Challenges . . . . 73

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Contents viii

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

5.5 Measurements: nonlinear regime . . . 117

5.5.1 Simple waveguides . . . 117

5.6 Discussion . . . 123

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Contents ix

6 Conclusions and Outlook 124

A Microscope images from the graphene-covered chip 128

Bibliography 131