Pulse Shaping Based on Integrated Waveguide Gratings - TSpace

Pulse shaping based on integrated waveguide gratings

by

Pisek Kultavewuti

A thesis submitted in conformity with the requirements for the degree of Master of Applied Science Graduate Department of Electrical and Computer Engineering University of Toronto

Copyright © 2012 by Pisek Kultavewuti

Abstract Pulse shaping based on integrated waveguide gratings Pisek Kultavewuti Master of Applied Science Graduate Department of Electrical and Computer Engineering University of Toronto 2012 Temporal pulse shaping based on integrated Bragg gratings is investigated in this work to achieve arbitrary output waveforms. The grating structure is simulated based on the sidewall-etching geometry in an AlGaAs platform. The inverse scattering employin the Gel’fan-Levithan-Marchenko theorem and the layer peeling method provides a tool to determine grating structures from a desired spectral reflection response. Simulations of pulse shaping considered flat-top and triangular pulses as well as one-to-one and oneto-many pulse shaping. The suggested grating profiles revealed a compromise between performance and grating length. The integrated grating, a few hundred microns in length, could generate flat-top pulses with pulse durations as short as 500 fs with rise/fall times of 200 fs; the results are comparable to previous work in free-space optics and fiber optics. The theories and the devised algorithms could serve as a design station for advanced grating devices for, but not restricted to, optical pulse shaping.

ii

Acknowledgements To the completion of this work, I would like to acknowledge and thank Prof. Stewart Aitchison, my mentor and supervisor. He always give valuable insights and suggestions as well as support to this work. I especially thank Dr. Ksenia Dolgaleva, my colleague, for her support, mentorship, discussion about the work. I thank Dr. Sean Wagner for his scripts to determine the refractive index of AlGaAs. I thank my committee members, Prof. Joyce Poon and Prof. Nazir Kherani, for their important suggestions during the defense. I thank Arash Joushaghani for advices and thesis revisions. I also thank to my family for their unconditional support and love.

iii

Contents

1 Introduction

1

1.1

Pulse Shaping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Applications of Pulse Shaping . . . . . . . . . . . . . . . . . . . . . . . .

2

1.3

Approaches for Pulse Shaping . . . . . . . . . . . . . . . . . . . . . . . .

2

1.4

Thesis Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.5

Organization of This Thesis . . . . . . . . . . . . . . . . . . . . . . . . .

5

2 Literature Review

7

2.1

Principles of Pulse Shaping . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2

Free-Space Pulse Shaping

2.3

2.4

7

. . . . . . . . . . . . . . . . . . . . . . . . . .

11

2.2.1

Fourier Pulse Shaping . . . . . . . . . . . . . . . . . . . . . . . .

11

2.2.2

Direct Space-to-Time Pulse Shaping . . . . . . . . . . . . . . . .

18

2.2.3

Pulse Stacking

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

2.2.4

Performance and Main Drawbacks . . . . . . . . . . . . . . . . . .

21

Fiber Optics Pulse Shaping . . . . . . . . . . . . . . . . . . . . . . . . .

22

2.3.1

Pulse Compression . . . . . . . . . . . . . . . . . . . . . . . . . .

24

2.3.2

Temporal Waveform Pulse Shaping . . . . . . . . . . . . . . . . .

27

2.3.3

Pulse Stacking in Fiber-Based Devices . . . . . . . . . . . . . . .

29

2.3.4

Other Fiber-Based Pulse Shaping . . . . . . . . . . . . . . . . . .

31

Integrated Optics Pulse Shaping . . . . . . . . . . . . . . . . . . . . . . .

33

iv

2.5

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 Grating Responses

39 41

3.1

Waveguide Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

3.2

Coupling Coefficients of Sidewall Gratings . . . . . . . . . . . . . . . . .

47

3.3

Grating Responses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

3.4

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

4 Retrieval of the Gratings

56

4.1

Equations at Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

4.2

GLM Equations to the Coupled-Mode Equations . . . . . . . . . . . . . .

58

4.3

Massaging the Equations . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

4.4

Algorithm of the Inverse Scattering . . . . . . . . . . . . . . . . . . . . .

61

4.5

Matching to Physical Parameters . . . . . . . . . . . . . . . . . . . . . .

64

4.6

Verification of the Inverse Scattering Algorithm . . . . . . . . . . . . . .

66

4.7

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

5 Pulse Shaping Simulations

69

5.1

Deriving the Targeted Grating Response . . . . . . . . . . . . . . . . . .

69

5.2

Flat-top Pulse Shaping . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

5.3

Triangular Pulse Shaping . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

5.4

One-to-Many Pulse Shaping . . . . . . . . . . . . . . . . . . . . . . . . .

81

5.5

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

6 Conclusions and Future Direction

86

6.1

Aspects, Approaches, and Results of This Work . . . . . . . . . . . . . .

86

6.2

Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

A Coupled-Mode Theory (CMT)

90

A.1 Integrated Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

91

A.2 Coupled-Mode Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

A.2.1 First-Order Gratings . . . . . . . . . . . . . . . . . . . . . . . . .

95

A.2.2 Uniform Gratings . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

A.2.3 Fourier Series of Permittivity Perturbation . . . . . . . . . . . . .

98

A.2.4 Grating Responses by CMT and Transfer Matrix Method . . . . . 100 A.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 B Fourier Transforms

104

B.1 Discrete Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . 104 B.2 Implementing Fourier Transform with Discrete Fourier Transform . . . . 106 C Simulation Results for Grating Responses

109

C.1 Uniform Gratings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 C.2 Chirped and Apodized Gratings . . . . . . . . . . . . . . . . . . . . . . . 114 C.2.1 Linearly Chirped Gratings . . . . . . . . . . . . . . . . . . . . . . 114 C.2.2 Apodized gratings . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 C.3 π-phase-shift and Sampled Gratings . . . . . . . . . . . . . . . . . . . . . 119 C.3.1 π-phase-shift Gratings . . . . . . . . . . . . . . . . . . . . . . . . 119 C.3.2 Sampled Gratings . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 D Inverse Scattering Theory

122

D.1 Inverse Scattering Theory: GLM equations . . . . . . . . . . . . . . . . . 123 D.2 Layer Peeling Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 D.3 GLM with Layer Peeling Method . . . . . . . . . . . . . . . . . . . . . . 127 D.4 GLM Equations to the Coupled-Mode Equations . . . . . . . . . . . . . . 129 D.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 E Simulation Results for Grating Retrieval

134

E.1 Uniform Gratings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 vi

E.2 Linearly Width-Chirped Gratings . . . . . . . . . . . . . . . . . . . . . . 137 E.3 Gaussian-Apodized Gratings . . . . . . . . . . . . . . . . . . . . . . . . . 139 E.4 Apodized and Linearly-Chirped Gratings . . . . . . . . . . . . . . . . . . 141 Bibliography

154

vii

List of Figures 2.1

4-f Fourier pulse shaping setup. . . . . . . . . . . . . . . . . . . . . . . .

12

2.2

Schematic structures of a liquid crystal pixel. . . . . . . . . . . . . . . . .

15

2.3

A basic setup for an acousto-optic modulator. . . . . . . . . . . . . . . .

18

2.4

A conceptual schematic diagram for DST pulse shaping . . . . . . . . . .

19

2.5

Direct space-to-time pulse shaping setup . . . . . . . . . . . . . . . . . .

20

2.6

Pulse stacking using interferometry setup. . . . . . . . . . . . . . . . . .

21

2.7

Pulse shaping results using bulk optics . . . . . . . . . . . . . . . . . . .

22

2.8

Pulse stacking in a fiber-based device by N uniform FBGs. . . . . . . . .

30

2.9

Pulse stacking in a fiber-based device by LPGs . . . . . . . . . . . . . . .

31

2.10 Flat-top pulse shaping using pulse stacking and pulse differentiation. . .

32

2.11 DST pulse shaping implemented with integrated arrayed-waveguide grating 36 2.12 Implementation of 4-f pulse shaping configuration, operating in reflection, in integrated optics using arrayed-waveguide gratings. . . . . . . . . . . .

37

2.13 Results of inverse scattering algorithm for dispersion compensation. . . .

38

2.14 Integrated waveguide gratings. . . . . . . . . . . . . . . . . . . . . . . . .

39

3.1

AlGaAs refractive index as a function of aluminum concentrations. . . . .

42

3.2

Cross-sections of a layer structure and a AlGaAs waveguide. . . . . . . .

43

3.3

The simulated index profile, a corresponding fundamental TE electric field mode, and a corresponding fundamental TM electric field mode. . . . . . viii

44

3.4

Effective indices of the fundamental TE-like and TM-like modes of the waveguide as a function of waveguide width and the light wavelength at the etch depth of 1 micron. . . . . . . . . . . . . . . . . . . . . . . . . . .

3.5

45

Effective indices of the fundamental TE-like and TM-like modes as a function of waveguide width at λ 1.55 µm and the etch depth of 1 micron. .

46

3.6

Index profile with etched area shaded.

49

3.7

Cross-coupling coefficients as a function of recess depths and waveguide

. . . . . . . . . . . . . . . . . . .

widths by the surface fitting function at the wavelength of 1.55 microns. . 3.8

51

Self-coupling coefficients as a function of the recess depths and the waveguide widths by the surface fitting function at the wavelength of 1.55 microns. 52

4.1

Windowing function. f1 pxq corresponds to x1{2 f2 pxq is plotted for x1{2

4.2

xd 3.

3 and xd 1 wherease

. . . . . . . . . . . . . . . . . . . . . .

65

The complex coupling coefficient, calculated from the inverse scattering algorithm, for a response of a Gaussian-apodized and chirped grating. . .

67

4.3

Matched waveguide width and recess depth profiles. . . . . . . . . . . . .

67

4.4

Responses of a grating generated by the inverse scattering algorithm compared with the targeted responses from a Gaussian-apodized and chirped grating. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

5.2

Fourier transform of a 2-ps flat-top pulse. . . . . . . . . . . . . . . . . . .

73

5.3

Inverse scattering algorithm results for a grating to generate a 2-ps flat-top pulse from a 150-fs Gaussian pulse. . . . . . . . . . . . . . . . . . . . . . ix

74

5.4

An amplitude (a) and time delay (b) responses from a generated grating with a targeted 2-ps flat-top pulse. In (c), electric field amplitudes of the output pulses from a generated grating (blue solid) and the targeted waveform (black dash). The legend simulated and target refers to that of the generated grating and the targeted grating. The scaled input is shown in red. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.5

Electric field magnitudes of output waveforms corresponding to generated gratings with different sets of subgrating involved. . . . . . . . . . . . . .

5.6

76

Responses and performance of the generated grating when random deviations are introduced to the waveguide width and the recess depth profiles.

5.8

76

Output waveforms from generated gratings aiming to produce flat-top pulses with durations of 0.5, 1, and 2 picoseconds. . . . . . . . . . . . . .

5.7

75

77

(a) Power spectrum of the triangular pulse envelope with the FWHM duration of 2 picoseconds and (b) The magnitude of the complex coupling

5.9

coefficient calculated from the inverse scattering algorithm. . . . . . . . .

79

Matched waveguide width and the recess depth profiles. . . . . . . . . . .

79

5.10 Grating response taking upto the the point of z

600 µm of the IS-

generated grating. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

5.11 Electric field amplitudes of the output pulses from a generated grating involved upto z

600 µm.

The blue solid curve represents the output

whereas the black dashed curve is the targeted output waveform. The green dot-dash curve represents the output waveform from the grating with add random deviations. . . . . . . . . . . . . . . . . . . . . . . . . .

80

5.12 Simulated results including the waveguide width, recess depth, and electric field profiles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

82

5.13 Amplitude responses to achieve an output waveform containing two 2-ps flat-top pulses with 10-ps center-to-center separation. (a) The response from the suggested grating. (b) The ideal response. . . . . . . . . . . . .

83

5.14 Output waveforms for two 2-ps flat-top pulses with a separation of 10 picoseconds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.1 Reflection response of a uniform grating with a waveguide width of w 1.4 µm, a recess depth of rd

25 nm, and a grating period of Λ

249.5 nm. The grating length is ∆z

84

100 µm. The effective index disper-

sion is not taken into account. . . . . . . . . . . . . . . . . . . . . . . . . 111 C.2 Reflection response of a uniform grating with a waveguide width of w 1.4 µm, a recess depth of rd




sion is now taken into account. . . . . . . . . . . . . . . . . . . . . . . . . 112 C.3 Reflection response of a uniform grating with a waveguide width of w 1.4 µm, a recess depth of rd




sion is taken into account. . . . . . . . . . . . . . . . . . . . . . . . . . . 113 C.4 Reflection response of a uniform grating with a waveguide width of w 1.4 µm, a recess depth of rd




sion is taken into account. . . . . . . . . . . . . . . . . . . . . . . . . . . 114 C.5 Reflection response of a linearly chirped grating with ∆Λ

4 nm and

250 nm. The simulation is implemented with Ng 200 subgratings and m 8. (a) Amplitude response. (b) The blue line corresponds to Λ0

a postively-chirped grating and the red line corresponds to a negativelychirped grating. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 xi

C.6 Reflection response of a linearly tapered grating with the waveguide width increasing from 1.0 µm to 1.6 µm. The grating period is 250 nm and the recess depth is 50 nm, throughout the grating. The simulation is run with Ng

400 and m 4.

(a) Amplitude response. (b) The blue line

corresponds to a up-tapered grating and the red line corresponds to a down-tapered grating. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 C.7 Gaussian-apodized cross-coupling constant and its corresponding recess depth profile for a 1.4-µm-wide uniform waveguide. . . . . . . . . . . . . 118 C.8 Reflection responses of a Gaussian-apodized grating with a uniform waveguide width of 1.4 µm, corresponding to an effective index of 3.1062 for a TE-like mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 C.9 Reflection responses of a pi-phase-shift grating (blue solid line) and a complementary continuous grating (red dashed line). All grating sections are uniform: a waveguide width of 1.4 µm, a recess depth of 50 nm, and a grating period of 250 nm. Subgratings in the a pi-phase-shift grating are 100 µm long whereas a continuous uniform grating is 200 µm long. . . . . 120 C.10 Reflection responses of a sampled grating. . . . . . . . . . . . . . . . . . 121 E.1 Calculated complex coupling coefficient of a uniform grating response. . . 135 E.2 The waveguide width and the recess depth profiles matched from the corresponding complex coupling coefficient of a uniform grating response . . 136 E.3 Response of a grating generated by the inverse scattering algorithm with the target response from a uniform grating. . . . . . . . . . . . . . . . . . 137 E.4 Complex coupling coefficient calculated for a response of a width-chirped grating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 E.6 Response of a grating generated by the inverse scattering algorithm with the target response from a width-chirped grating. . . . . . . . . . . . . . 139 xii

E.7 Complex coupling coefficient calculated for a response of a Gaussianapodized grating. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 E.8 Matched waveguide width and recess depth profiles . . . . . . . . . . . . 140 E.9 Response of a grating generated by the inverse scattering algorithm with the target response from a Gaussian-apodized grating.

. . . . . . . . . . 141

E.10 Complex coupling coefficient calculated for a response of a Gaussianapodized and linearly-chirped grating. . . . . . . . . . . . . . . . . . . . . 142 E.11 Matched waveguide width and recess depth profiles. . . . . . . . . . . . . 142 E.12 Response of a grating generated by the inverse scattering algorithm with the target response from a Gaussian-apodized and linearly-chirped grating. 143

xiii

List of Tables 3.1

Propagating effective indices of TE-like and TM-like modes of a ridge waveguide with corresponding waveguide widths, w, and the etch depth of 1 µm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2

46

Third-order polynomial coefficients for the fitting functions of (a) the TElike effective index, neff,T E pw, λq

zpx, yq, and (b) the TM-like effective

index, neff,T M pw, λq z px, y q. The free variables x and y are the waveguide widths, w, and the wavelengths, λ. . . . . . . . . . . . . . . . . . . . . . 3.3

Cross-coupling coefficients as a function of recess depths for a constant waveguide width of 1.4 microns and λ 1.55 µm. . . . . . . . . . . . . .

3.4

47

51

Self-coupling coefficients as a function of recess depths for a constant waveguide width of 1.4 microns and λ 1.55 µm. . . . . . . . . . . . . .

xiv

52

Chapter 1 Introduction Lasers have shown great capabilities in both scientific study and real-world applications. In terms of temporal shapes of light generated by lasers, there are two classes of lasers: the continuous-wave (CW) lasers and the pulsed lasers. The temporal shapes of the pulses can play significant roles in the physics of light-matter interactions. Hence, it is important to manipulate the shapes of the pulses, leading to performance enhancement and new application areas.

1.1

Pulse Shaping

Temporal pulse shaping refers to attempts and techniques to control the waveform of the electromagnetic radiations in time domain. Even though pulse shaping could be carried out directly by engineering the lasing conditions of the laser system involving the resonator and the gain medium, this choice is limited by the complixity of the lasing mechanism [1–3]. Therefore, it is more practical to sculpture the waveform of the pulsed light after being emitted from a laser. Temporal pulse shaping actually applies to electromagnetic pulses at any frequencies. Practical implementations differ from one range of frequency to another, however. This results from different length and time scales of the problems at hand. For instance, pulsed radio-frequency (RF) signals have the pulse 1

Chapter 1. Introduction

2

duration in the order of microsecond. At this time scale, pulse shaping is achievable using CMOS electronic circuits. However, in the optical domain, a pulse duration could be easily be a few hundreds femtosecond. The speed of CMOS circuits cannot catch up effectively with this time scale. Therefore, if ultrafast time scale is of interest, temporal pulse shaping usually employs optical methods.

1.2

Applications of Pulse Shaping

Applications that involve the shape of the pulsed light could benefit from pulse shaping. For example, nonlinear switching, such as employed in optical-time-domain demultiplexing, could perform with better error rates if either the control pulses or the signal pulses assume flat-top shapes [4, 5]. Shaped ultrashort pulses are also used to optically control transitions of states of molecules in quantum coherent control, where an amplitude, a phase, and a bandwidth of the utilized light are crucial. An excellent review on pulse shaping for coherent control could be found in [6, 7]. Another applications of interest are spectroscopy and imaging, especially ones involving nonlinear optical processes such as multidimensional spectroscopy [8] and multiphoton imaging [9].

1.3

Approaches for Pulse Shaping

A lot of pulse shaping techniques have been proposed and developed. They could be categorized into three regimes: free-space optics, fiber optics, and integrated optics. In free space, pulse shaping is achieved by employing bulky optical elements such as lenses, gratings, and spatial light modulators (SMLs) [10–12]. The method relies on spreading light in space using a grating, spatially filtering by SMLs, and recombining the modulated light. Shaping temporal resolutions of 20-30 femtoseconds was achieved with this approach along with reprogrammability associated with arrayed spatial SMLs [6]. An interferometer-based technique, or referred to as pulse stacking, was proposed and


3

demonstrated [13–15]. It is based on introducing suitable time delays between pulse replicas generated from beam splitters and recombining them. However, the free-space pulse shaping techniques require bulky optical elements and strict alignment. Temporal pulse shaping implemented in optical fibers is more compatible with communications industry because coupling in and out of the fibers to free space can be avoided. The most prevailing design for temporal pulse shaping in fiber optics is a fiber grating [16–21]. The grating can modulate the spectral components of light due to it index perturbation, which leads to resonance and interactions between guided modes. These gratings are employed not only for pulse shaping but also for optical signal processing such as optical differentiation and integration [22–25]. However, technological limitations in creating large index perturbations in optical fibers leads to the grating dimension in centimeter scale. The third category of pulse shaping is carried out in optical integrated circuits. The analog of free-space pulse shaping, which employs gratings to spatially disperse light, is achieved in integrated circuits by using arrayed-waveguide gratings (AWG) and appropriate filters [26,27]. At the output end of the AWG, spectral components of the incident light are separated among waveguides and each component could be modulated using a spatial mask or modulator. Since another optical component that performs filtering functions is required for arbitrary shaping, the devices become complicated to fabricate even though reconfigurability could be accomplished [27]. A potential candidate is an integrated waveguide grating, which offers higher index perturbation compared to that of a fiber grating. Pulse shapers employing integrated waveguide gratings can provide arbitrary output waveforms by tailoring the appropriate grating profiles [28, 29]. Since the grating structure is introduced to a single waveguide, the size of the device is small, in the order of the waveguide. Even though post-fabrication tuning for grating pulse shapers is limited, an integrated waveguide grating can serve as a compact and robust pulse shaping device for arbitrary output waveform specification,


4

provided that the grating has the right structure.

1.4

Thesis Work

In this work, pulse shaping using integrated waveguide grating is studied. The grating is chosen due to its merits of compactness and ability for arbitrary pulse shaping. Most of the integrated gratings previously fabricated have a surface-corrugated structure. This type of grating structure requires additional fabrication steps: the index perturbation is introduced by etching the top of the waveguide after the waveguide is defined. Another grating is a sidewall grating, where the waveguide is etched on the sides thereby causing index perturbation [28–32]. One beneficial point of this grating structure is that the perturbation can be simultaneously generated at the same time as the waveguide is defined, leading to a less complicated fabrication procedure. Another advantage of the sidewall-etching geometry is the easy control over the coupling coefficient and the Bragg wavelength profiles [29, 30] by altering the grating profiles. The integrated sidewall Bragg grating for pulse shaping is the main focus of this thesis. The grating is implemented on a ridge waveguide in an AlGaAs platform. Physical parameters that dictate the behavior of the grating are the waveguide width and the recess depth profiles. In order to achieve grating designs for pulse shaping, a response of a given grating must be computed. The coupled-mode theory (CMT) [17, 33] is used in combination with a transfer matrix method (TMM) [34] to determine the grating response. This situation is referred to as direct scattering (DS). An algorithm for direct scattering is devised and is capable of handling any sidewall gratings with symmetric perturbations on the waveguide sides. For a reverse situation, i.e. the retrieval of grating structures from a desired reflection response, the inverse scattering theory (IST) [35], based on the Gel’fan-Levithan-Marchenko theory, is employed in conjunction with the layer peeling method (LPM) [36]. The algorithm for the inverse scattering theory is also


5

devised and shows ability to deal with high-reflectivity grating. Numerical implementations of both the direct scattering and inverse scattering are evaluated with a set of familiar grating profiles. Afterwards, actual grating designs for pulse shaping are investigated. An input and a targeted output are defined and a corresponding reflection response is computed. One-toone and one-to-many are demonstrated by using the inverse scattering (IS) algorithm to generate the grating and using the direct scattering algorithm to determine the response of the generated grating. Pulse shapes are considered to be either flat-top or triangular. Simulation results obtained prove that integrated sidewall gratings could perform arbitrary pulse shaping empowering by the inverse scattering to design the grating.

1.5

Organization of This Thesis

The organization of this thesis is as the followings. In Chapter 2, the work in the area of pulse shaping is reviewed starting from free space optics to fiber optics to integrated optics. The significance of a grating structure shines in fiber optics due to the success in fabrication and versatility of the structure itself. Many gratings have been proposed to carry out several functionalities. To decrease the size of the device, the question to ask is ‘Can the grating be implemented in the integrated platform?’, which is the thesis of this work. In order to analyze the grating, the coupled-mode theory (CMT) is employed and is rigorously described in Appendix A. The assumption of small perturbation to the waveguide is held and the solution to the coupled equations is derived and used to construct the response of any grating by the assistance of the transfer matrix method. Chapter 3 describes the algorithm for CMT and discusses simulation results aiming to validate the theory. For the grating design, Chapter 4 focuses on the algorithm devised based on the inverse scattering theory, whose core is consisted of the Gel’fan-Levithan-Marchenko


6

(GLM) theory and the layer peeling method (LPM). The inverse scattering itself is discussed rigorously in Appendix D. Simulation results will show that the algorithm is capable of retrieving the grating structure. Pulse shaping is discussed and demonstrated in detail in Chapter 5. The rectangular pulse and triangular pulses are the targeted waveforms. A single-pulse and multi-pulse output cases are discussed in the context of practical feasibility. It appears that the grating can perform a one-to-one pulse shaping but the performance degrades as the number of pulses increases. The last chapter, Chapter 6, draws the big picture and pulls all importance messages of the whole thesis.

Chapter 2 Literature Review In this chapter, the work in pulse shaping is reviewed. The chapter starts with the various theoretical principles underlying the manipulation of light in both time and frequency domains. Employing these principles, different pulse shaping techniques are discussed and categorized into three groups: free-space optics, fiber optics, and integrated optics.

2.1

Principles of Pulse Shaping

Pulsed light is thought to comprise many planewaves of different frequencies. These different planewaves can have different magnitudes and phases. The combination of these planewaves results in the temporal behavior of the pulsed light. Actually, this is the essence of the Fourier transform between time and frequency domains. One can change the temporal characteristics of the pulsed light by changing its spectral ingredients in amplitude and/or phase. Pulse shaping that manipulates these spectral ingredients is usually called Fourier pulse shaping. There are many ways to access the spectral components and alter them. For example, a diffraction grating could be used to angularly disperse the spectral components [37]; afterwards, these spatially dispersed spectral components could be controlled using a spatial amplitude-and/or-phase mask [10]. Another example is to use a dispersive element, such as a long optical fiber 7

8

Chapter 2. Literature Review

or a linearly-chirped grating, to temporally disperse different frequencies due to different group delays [38] and employ an optoelectronic modulator, receiving a controlling electrical waveform, to filter the temporally dispersed signal [39]. This later example is usually referred to as temporal pulse shaping. For a time-invariant linear pulse shaping, the implemented device could be represented by a response (transfer) function, H pω q, or an impulse response, hptq. If the input and output waveforms are xptq and y ptq in a time domain, with corresponding X pω q and Y pω q in a frequency domain, there exist the relations y ptq hptq xptq,

(2.1a)

Y pω q H pω qY pω q.

(2.1b)

Pulse shaping using gratings is also under rigorous investigation. Most of the work operates within the Fourier pulse shaping boundary. Many design principles are proposed in order to achieve a required output waveform especially by using fiber-based devices such as fiber Bragg gratings (FBGs) and long-period fiber gratings (LPFGs). The FBG can be designed to produce a required reflection impulse response using the first-Born approximation or the weak-grating limit. It was shown that the temporal impulse response of a uniform FBG, described by npz q nav

∆nmax Apxq cosp Λ2π0 z

φpz qq, is proportional

to the scaled apodization profile, Apz q, and the phase profile, φpz q, [16]: (

hr ptq 9 Apz qejφpzq zct{2nav ,

(2.2)

ct{2nav is the space-to-time The first-Born approximation requires that κL ! 1, where κ is the

where nav is the average refractive index of the FBG and z scaling relationship.

coupling constant and L is the grating length, for a uniform FBG. It physically means that the input light can propagate through the whole grating where each grating section contributes equally to the output waveform. A longer temporal waveform requires a longer grating length and leads to a lower coupling constant. Since the coupling constant depends on the index modulation, at some point, it is not technologically possible to

9


realize either a very weak or a very strong coupling constant. Also, the weak coupling limit results in low reflectivity ( 20%) and shows poor energy efficiency. These facts reveal the limitations of the first-Born approximation method. To solve the issue of low energy efficiency of the first-Born approximation, the use of apodized linearly-chirped fiber Bragg grating (LC-FBG) is proposed [16]. The requirement of this method is that the apodized LC-FBG must have a constant first-order

:ν dispersion coefficient, Φ

B BΦνpνq , that is large enough. Different spectral components of 2

2

the input pulse are reflected by different local sections of the LC-FBG with different reflectivity corresponding to the local apodization. Sufficient dispersion is required in order to efficiently separate different frequencies. This scheme is termed space-to-frequency-totime mapping. The mapping [16] is mathematically expressed as hr ptq 9

j π t2 e Φ: ν

tanh m A z

ct 2nav

(2.3)

where m is a constant. High reflectivity up to 60% has been numerically shown in [16] demonstrating the ability to overcome the weak coupling limit. The first-order dispersion coefficient for LC-FBG, [16], could be expressed as

:ν Φ

av L 2nc∆ν

(2.4)

where ∆ν is the chirp bandwidth of the LC-FBG whose spatial reflected frequency is written as ν pz q

ν0 pz L2 q ∆νL .

From Eq. 2.4, the dispersion coefficient is directly

proportional to the grating length, L, but inversely proportional to the chirp bandwidth, ∆ν. The grating length determines the interaction time between the input pulse and the grating. In other words, the longer the grating is, the longer output pulses will be. The chirp bandwidth represents the bandwidth of frequencies that the grating can separate. It is usually required that the chirp bandwidth covers the spectral bandwidth of the input pulse. This principle actually limits the shortest achievable output to that of the input pulse itself.

10


Another pulse shaping method is named direct space-to-time (DST) pulse shaping. The setup of this method is close to that of the 4-f configuration of the Fourier pulse shaping (which will be discussed in the next section) but with some differences. This shaping method is suitable for applications where a direct mapping between a spatial pattern and an output waveform is required, such as in parallel-to-serial conversion [26]. The underlying principle is the diffraction theory. The temporal output is proportional to the convolution between the temporal input field and the time-scaled spatial mask function, [26], Eout ptq 9 Ein ptq sptq.

(2.5)

Instead of managing spectral components of the pulsed light, pulse shaping could be done by combining many pulses with appropriate time delays. This method is similar to interferometry and is usually called interferometry-based pulse shaping or pulse stacking. In this scheme, the input pulse is split into many replica, possibly with different pulse powers. These pulses pass through different optical paths to initiate appropriate time delays among them. Afterwards, they are recombined to form the output. Coherent pulse stacking takes into account the phase information of these pulse replica and results in an temporal interference pattern. Mathematically, the output electric field is the superposition of the electric fields of the replica: Eout ptq

N ¸

Ei pt ti q.

(2.6)

i 1

The vector and complex nature of the electric field results in inference terms when the fields add together. For example, consider the pulsed planewaves expressed as E ptq

Aptqejωc t , where Aptq is the pulse envelope taken to be real without loss of generality and ωc is the central angular frequency. Pulse stacking with two replica and a time delay τ is written as Eout ptq ejωc t Aptq

Apt τ qejωc τ .

(2.7)

The term ejωc τ leads to interference. However, if the time delay is adjusted such that

11

Chapter 2. Literature Review ωc τ

2nπ, the envelope of the output could be expressed as the summation of the

envelopes of the replica. Note that Apt

τ q Aptq δpt τ q.

Therefore, Eq. 2.7

becomes Eout ptq ejωc t Aptq δ ptq

δ pt τ qejωc τ

(

.

(2.8)

The last term in the parenthesis is actually the impulse response of the interferometer. If the input pulse is incoherent, the phase information is lost and the resultant output intensity waveform is the summation of the intensity profiles of the replica. Iout ptq

N ¸

Ii pt ti q

(2.9)

i 1

This could be referred to as incoherent pulse shaping. Light sources that generate incoherent light could be an amplified spontaneous emission (ASE) source or a superluminescent LED [40].

2.2

Free-Space Pulse Shaping

Due to a long history of optics, free-space optics has been used to demonstrate predicted optical phenomena including pulse shaping. Light mostly travels in free space and its path is changed by macroscopic optical elements such as lenses, beam splitters, mirrors, and diffraction gratings. Researchers have reported a number of impressive results employing different proposed techniques, and pulse shaping instruments are designed and commercialized. Several review articles are published in the literature and provide valuable detailed background for further research. Instances of good review papers include [11, 12].

2.2.1

Fourier Pulse Shaping

Fourier pulse shaping is probably the most widely adopted pulse shaping method. The pulse shaping is carried out in the frequency domain, which makes ultrafast waveform


12

generation possible without the use of an ultrafast modulator. Albeit setup configurations are abundant, common-ground features exist among these varieties, which is shown in Fig. 2.1. This group of configurations is usually called a 4-f or a zero-dispersion configuration [10]. It consists of two diffraction gratings, two focusing lenses, and a spatial light modulator (or a mask).

Figure (2.1): A 4-f Fourier pulse shaping (or zero-dispersion) setup. Reprinted from [11], © (2011) with permission from Elsevier.

An input pulse is illuminated onto the first grating and its frequency components are angularly dispersed in space. The first lens focuses this diffracted light onto the Fourier plane. The spatial light modulator is placed at this Fourier plane to alter amplitude and phase of the light. The second lens and the second grating recombine the spatially dispersed and modulated frequency components into an output pulse. The name of the configuration comes from the total length that light passes within the shaping device, which equals to four times of the focal length. Additionally, this configuration imposes no extra dispersion to the pulsed light if the two gratings are identical so that the effects cancel each other at the output, leading to the name zero-dispersion. Note that it is possible to reduce the setup path length by half by placing a mirror just behind the mask. Actually, the reflection configuration is preferred not only because of its shorter path but also its reduced complexity in both setup and fabrication. Since the zerodispersion requires the exact similarities between the first and second sets of gratings


13

and lenses, by using the reflection setup the similarity of the sets is ensured. A diffractive optical element is used to spectrally disperse the incoming light: This element determines how different frequencies are separated. A diffractive grating is generally used for this purpose. Other possible elements include virtually imaged phased arrays (VIPA etalons), prisms, and arrayed waveguide gratings (usually in micro-structured pulse shaping). The first lens controls how light and its frequencies are focused onto the Fourier plane. Specifically, it determines the spot size of each frequency. Should the chromatic dispersion of the lens pose limitations to pulse shaping, other focusing elements such as a curved mirror can avoid this difficulty. The spatial light modulator (SLM) is the key part that performs pulse shaping. A static mask can be used as the SLM and it is usually fabricated by microlithographic patterning. To accommodate programmability in the pulse shaping, researchers utilize reconfigurable SLMs including a liquid crystal modulator (LCM) and an acousto-optic modulator (AOM). Other SLMs are holographic masks, deformable mirrors, and micromirror arrays. Of course, programmable shaping masks are more popular and widely employed in the practical operations. This pulse shaping can be cast in the linear system theory. In the 4-f configuration, the spectral content of the input pulse is spatially dispersed and subsequently focused on the Fourier plane in which the mask is placed. Let Ein pω q be the input electric field spectrum right before the mask. A mapping relation exists between the locations on the plane and the frequencies. The mask addresses pulse shaping by altering frequencies through this mapping relation. Hence, the electric field spectrum after the mask is Eout pω q

M pxpωqq Einpωq, where M pxq is the mask function and xpωq is the mapping

function. However, complications exist in realization as the beam cannot be focused into an infinitely small spot. For every frequency, the beam is focused to a finite spot. Should an abrupt change exist in the mask and in the spot, diffraction of the light beam behind the mask is present and as a result changes the spatial distribution of the


14

beam. Since, different frequencies fall onto different locations, the output field experiences a sophisticated coupled function of space and frequencies (or time). This space-time coupling is discussed in more detail in [41]. To shape a desired temporal waveform, the space complication can be decoupled by an appropriately designed mask. For example, the mask function is re-rendered by the spatial distribution of the beam mode leading to a complex mask function M pxqg pxq, where g pxq represents the mode distribution function. In this case, the original mask function is smeared by the mode function and is replaced by the new complex mask function. Additionally, the size of the mode also dictates the spectral resolution. Overall, the resolution of the pulse shaping system is the smaller of the spectral resolution and the finest feature of the SLM.

Liquid Crystal Modulators A liquid crystal is a material that exhibits properties between a liquid and a crystal. It lends itself to a numerous number of applications among which a liquid crystal display is the most abundant. For pulse shaping, it could serve as a spatial light modulator. More strictly, a liquid crystal modulator is an array of liquid crystal pixels. A simplified schematic structure of the liquid crystal pixel is shown in Fig. 2.2. The liquid crystal molecules are placed between two electrodes. Without an external electric potential, all liquid crystal molecules orient in the same direction resulting in a crystalline structure showing anisotropy or birefringence such that the x-polarized and y-polarized electric fields experience different refractive indices. In the figure, the long axis of the liquid crystal molecule aligns with the y-axis. When the electric potential is applied across the two electrodes, i.e. in the z-direction, the electric field rearranges the orientation of the liquid crystal molecules to comply more to the z-axis. This reorientation manifests in the change of the refractive index for the y-polarized electric field, altering the anisotropy. The degree of a phase change as light propagate through the pixel depends on the magnitude of the applied potential as well as the thickness of the pixel. A useful liquid crystal

15

Chapter 2. Literature Review modulator must be able to achieve 2π phase shift.

(a) Without applied voltage.

(b) With applied voltage.

(c) Phase change plot.

Figure (2.2): Schematic structures of a liquid crystal pixel. Reprinted from [11], © (2011) with permission from Elsevier.

A conventional liquid crystal modulator array is a one-dimensional array of 128 to 640 pixels. Electrodes on one side of the array are connected to ground whereas those on the other side are attached to external potential sources, which are usually computercontrolled. With appropriate potential differences, the liquid crystal modulator array could be held constantly as a mask function. The reconfiguration time depends on the dynamics of the liquid crystal and the control circuit. Since the liquid crystal pixel modulates the phase of the light, it works as a phaseonly filter. The phase-only filtering scheme has a merit of reserving the amplitude of the electric field and leads to good energy efficiency. However, it is usually needed to have more degrees of freedom. An independent amplitude and phase control is achievable by using two liquid crystal modulators attached back to back [42]. The orientation of the two liquid crystal layers need to be offset by 90 . For example, assuming the coordinate as shown in Fig. 2.2, the long axis of the liquid crystal molecules lie on the xy-plane.

45 with respect to the y-axis. Assume that the input pulse is the y-polarized light, E yˆE0 cospωtq. The electric field The two liquid crystal modulators align at

45 and

16


of the input could be decomposed onto x1 and y 1 axes forming along the long axes of the

?

two liquid crystal modulators: E pxˆ1 E0

yˆ1 E0 q cospωtq{ 2. After passing through, the

electric field experiences phase differences: E xˆ1 E0 cospωt

xˆ E20 pcospωt yˆ

?

∆φ1 q cospωt

E0 pcospωt 2

∆φ1 q

?

yˆ1 E0 cospωt

∆φ1 q{ 2

∆φ2 q{ 2

(2.10a)

∆φ2 qq

cospωt

∆φ2 qq ,

(2.10b)

Should the polarizer at the output be aligned to the y-axis, the final electric field is only the yˆ component: #

Eout

yˆ

E0 cos

∆φ

1

+

∆φ2 2

cos ωt

∆φ1 2

∆φ2 .

(2.11)

Hence, the amplitude and phase could be controlled independently through the first and second factors accordingly via adjusting the correct pair of ∆φ1 and ∆φ2 . The size of the conventional liquid crystal pixel is in the order of 100 µm. Usually the optics can focus the beam to a spot size smaller than the liquid crystal pixel. Therefore, in conventional liquid crystals, the pixel usually modulates a group of frequencies and the pixel size ultimately determines the spectral resolution. Technological advancement in liquid crystal fabrication could improve this resolution limit. One important example is the liquid crystal on silicon (LCoS) [43–45], which utilizes the advancement in CMOS microfabrication to reduce the pixel size. 2-dimensional pixels and electrode arrays are patterned to a silicon CMOS circuit and the layers of liquid crystal and transparent electrodes are deposited on top. Since the CMOS platform is opaque, a reflective layer is deposited between the substrate and the liquid crystal layer. The pixel size in the order of 10 µm is easily achievable. In this scheme, each frequency spot focused on the Fourier plane encompasses a set of liquid crystal pixels. It works complementarily to the conventional one if all the pixels in the enclosed area of the frequency spot size deliver the same response. However, a single pixelated liquid crystal


17

modulator array can perform independently from other pixels within the same beam spot. A periodic structure of the pixel array under the beam frequency spot is proposed in [46]. It scatters light into many angular orders. Should only the zeroth order be collected after reflection, only a fraction of energy carrying by the zeroth order is received at the output. On the other hand, the phase of the light is affected by the average phase of the periodic structure. Since the average phase and scattering orders are independent from each other, the amplitude and phase modulations are independently delivered by the LCoS using the periodic structure in the pixel groups.

Acousto-Optic Modulators

Another famous form of a programmable SLM is the acousto-optic modulator. A radiofrequency signal from a waveform generator is applied to a piezoelectric transducer, which actually is the spatial modulator. The piezoelectric material transduces the driving electric potential to the acoustic wave propagating through the material. This mechanical wave changes the lattice structure and hence optical properties of the material. The acoustic wave pattern in space across the material is similar to the temporal radiofrequency wave with an appropriate time-to-space scaling. The pulse shaping setup employing the acousto-optic modulator is depicted in Fig. 2.3. The pulse shaping is actually accomplished due to diffraction created by the pattern of the acoustic wave. The incident wave could scatter into many spatial orders and a specific order could be collected by choosing an appropriate angle for the output wave. The filtering function from the acousto-optic modulator is time varying because the acoustic wave propagates. The reconfigurability time depends on the speed of the acoustic wave, which is usually in the order of tens of microseconds [10].


18

Figure (2.3): A basic setup for an acousto-optic modulator. Reprinted from [11], © (2011) with permission from Elsevier.

2.2.2

Direct Space-to-Time Pulse Shaping

Direct space-to-time (DST) pulse shaping techniques result in a temporal output waveform similar to a spatial mask function. It is useful in producing pulse bursts or in parallel-to-serial conversion [26]. In free-space optics, the setup of DST pulse shaping looks similar to that of the Fourier transform pulse shaping. The schematic diagram conceptually capturing the DST principle is shown in Fig. 2.4. The input consisting of several frequency components is incident on a mask, which transfers a spatial function to the spatial distribution of the input beam. After the mask, the spatially patterned beam passes through a grating and its frequency components are angularly dispersed. The lens collects and focuses the light onto its Fourier plane. At this plane a narrow slit is placed and the output pulse is actually the part of light that can transmit through the opening. Physically, the mask transfers its spatial pattern to the frequency components of the input beam. For each frequency the spatial profile of the electric field at the back

19


Figure (2.4): A conceptual schematic diagram for DST pulse shaping [26], reprinted with permission © 2001 IEEE.

Fourier plane of the lens is the complimentary Fourier transform of the spatial profile of the incoming electric field. The grating in the diagram angularly disperses frequency components so that they are located separately on the Fourier plane. Recall that if the incoming field of a particular frequency is finite in extent, its Fourier transform spreads across the plane as well. If a single slit is placed at the Fourier plane to collect the output light, light of all frequencies passes through the slit but with different content on their respective Fourier transform profiles. Therefore, the output field is described by both the input frequency content and the Fourier transform of the spatial mask; more specifically the temporal waveform of the output is proportional to the convolution of the temporal profile of the input pulse and the spatial mask scaled to time domain: eout ptq where γ

cd cosλ θ

d

9

ein ptq s

β t ,

(2.12)

γ

is the spatial dispersion term and β

cos θ cos θ

i

d

is the astigmatism term,

in which θi and θd are the incident and diffracted angles from the grating. The actual setup could look like the one displayed in Fig. 2.5. The optics to the left of the dotted line constructs the imaging section in which the input beam is spatially patterned and the optics to the right comprises the DST pulse shaping.


20

Figure (2.5): Direct space-to-time pulse shaping setup [26], reprinted with permission © 2001 IEEE.

2.2.3

Pulse Stacking

Pulse stacking as discussed earlier refers to a technique that combines several pulses with certain time delays. Splitting a single pulse into several subpulses and imposing time delays among them is usually done in the interferometry setup.

In free-space optics regime, the beam splitters and mirrors constitute the main elements to create subpulses and time delays. One beam splitter and two mirrors comprise a single interferometer unit at which a single pulse is splitted with even energy into two subpulses, whose relative time delay depends on the positions of the two mirrors. As a result, n interferometer units will effectively create 2n subpulses. Fig. 2.6 schematically shows the interferometry-based pulse shaping system with two interferometer units in which a single incoming pulse is split into four identical subpulses. The system becomes tunable by adjusting the positions of the mirrors, for example by microactuator stages. This pulse shaping technique, especially as shown in Fig. 2.6 has been used to generate flat-top and triangular pulses with the full-width-at-half-maximum (FWHM) duration of a few picoseconds from a transform-limited input pulse of 600-800 femtoseconds in duration [13].


21

Figure (2.6): Pulse stacking using free-space optics interferometry setup. The shaded area represent the actual pulse shaping section [13], reprinted with permission © 2007 IEEE.

2.2.4

Performance and Main Drawbacks

Depending on the setup, a temporal resolution for shaping of 20-30 femtoseconds is possible [6] as well as a spectral resolution of 0.06 nm/pixel [12]. Some of impressive results are shown in Fig. 2.7. The data packet could be yielded from the DST method, which in this case nine bits are generated whereas the center bit was rendered off, shown in Fig. 2.7a. Flat-top pulse shaping was also demonstrated with a pulse duration of 2 picoseconds from a 100-fs input pulse in Fig. 2.7b. If the SML array is assigned a linear phase function, in Fig. 2.7c the output pulse is delayed compared to the input pulse. Pulse shaping devices can also function as a dispersion compensator as shown in Fig. 2.7d. Even though free-space optics has been providing excellent pulse shaping performance, it bears some drawbacks. Bulk optics is bulky: Optical components are in a macroscopic scale and free space propagation takes a lot of space. This characteristic goes against the trend of miniaturization. Another issue regards the alignment problem. Precise alignment is usually critical to obtain good results. The more optical elements in the pulse shaping system, the more complicated the alignment will be, and it results in bad tolerance. The last major shortcoming is the fact that high quality optical elements

22


Figure (2.7): (a) A femtosecond data packet, (b) a 2-ps flat-top pulse, (c) a delayed pulse using linear spectral phase, and (d) a recompressed pulse. Reprinted from [11], © (2011) with permission from Elsevier.

are required to achieve good results, which culminates in high cost of the pulse shaping system. The aforementioned drawbacks drive researchers to contrive other alternatives to maneuver light with goals in compactness, robustness, and integrability. The prominent areas being explored include fiber optics and integrated optics as platforms for pulse shaping. Albeit the physical platform is changed, the underlying principle in pulse shaping remains the same and it is appropriately transferred to the new physical platform of interest.

2.3

Fiber Optics Pulse Shaping

An optical fiber is inarguably one of the most important optical devices. It is an excellent waveguide especially at the telecommunication wavelength due to its low loss nature; it stands as the backbone of the communication network especially in a global scale. It is therefore very logical to shape pulses in optical fibers. Since optical fibers are also


23

ubiquitous in many systems, other application areas gain advantages from fiber-based pulse shaping as well. Pulse shaping in optical fibers can utilize many properties from plain optical fibers or from structures in optical fibers. Plain optical fibers exhibit both linear and nonlinear dispersion. The linear dispersion is related to the group velocity dispersion, in which different wavelengths experience different effective indices, resulting in pulse broadening or contracting after propagating through a certain distance in a fiber. The nonlinear dispersion arises from the power-dependent refractive index resulting in the self-phase modulation, which represents another source of dispersion. These two dispersions are fundamental to any optical fibers and can be used to perform pulse shaping. The main drawback of the use of a plain optical fiber is its low dispersion value and hence a long optical fiber might be necessary to accumulate enough phase difference or dispersion. Fortunately, the grating structure could enhance dispersion due to its periodic index structure. The total dispersion in the fiber grating is a combination of the material dispersion and the structural dispersion, which is the dispersion resulted from the grating structure. Mathematical analysis can calculate the final dispersion effect of the fiber grating, which will reveal the relationship between the dispersion and the physical grating structure. In this sense, the required dispersion could be achieved by appropriately creating the grating in the fiber. Technological advance in fiber fabrication allows implementing different kinds of gratings. One of the technique is the use of irreversible nonlinearity-induced refractive index change. Several kinds of fiber gratings are under research investigation as well as realworld applications. Some common types are a fiber Bragg grating (FBG), a linearlychirped fiber Bragg grating (LC-FBG), and a long-period grating (LPG). A good review about the fiber grating could be found in [17]. In general, the single-mode fiber is usually preferable because of its performance, for example the lack of mode dispersion, which occurs in a multi-mode fiber where different modes propagate with different velocities and

24


lead to mode walk-off. The following discussion assumes that the fibers are single-mode otherwise stated explicitly. The grating period, which is the period of index modulation introduced to the fiber, determines the direction of operation. If the period is short enough, the grating could introduce interaction between the forward- and backward-propagating modes and this is the situation in FBGs. On the other hand, if the grating period is relatively long, the interaction will occur between the modes of the same direction, i.e. two forwardpropagating modes, as it occurs in LPGs. In the single-mode fiber, LPGs could engage the normal fundamental mode and the cladding mode in interaction. In fabrication, the grating period is not necessarily constant along the grating; it could be varied at will. Linearly changing the grating period results in a unique behavior and the grating is termed a linearly-chirped grating for a reason that its time delay response becomes linear in the frequency domain. Several types of pulse shaping could be achieved by these gratings: pulse compression, temporal waveform shaping, real-time Fourier transform, pulse rate multiplication, optical temporal differentiation, and optical temporal integration.

2.3.1

Pulse Compression

The first simple form of pulse shaping regards pulse compression. Optical pulses propagating through a dispersive waveguide, e.g. an optical fiber, will experience chirping in their instantaneous frequencies, which can eventually broadens or compresses the temporal durations of the pulses, as a result of waveguide dispersion. The waveguide dispersion refers to the frequency dependence of the effective index of the propagating mode, which consequently leads to the definition of the group velocity and the group velocity dispersion. The group velocity dispersion (GVD) coefficient could be derived as ω d npω q d 1 . dω v c dω 2 2

Dω

g

(2.13)

25

Chapter 2. Literature Review Other notations include Dν

dpvg1q{dν

and Dλ

dpvg1q{dλ when vg

ω dn c dω

is the

group velocity. If the GVD coefficient Dω or Dν is positive, the medium is said to have normal dispersion or positive GVD. Oppositely, if the GVD coefficient is negative, the medium exhibits anomalous dispersion or negative GVD. Note that the sign of Dλ will be opposite to that of Dω or Dν . In the case of normal dispersion, light of higher frequency (shorter wavelength) possesses slower group velocity compared to light of lower frequency. The situation is reverse in the anomalous dispersion: the higher frequency component propagates with faster group velocity. The group velocity dispersion is associated with the quadratic term of the phase response. Therefore, the manifestation of GVD is actually the linear frequency chirping of the pulse in the time domain. Since the group velocity dispersion as defined above appears as the first term in distorting the shape of the pulse, it is sometimes referred to as the first order dispersion. If the initial pulse is transform-limited, the propagation of the pulse through a dispersive waveguide results in temporal pulse broadening because as the pulse propagates different frequency components traverse with different group velocities and therefore gradually separate from one another. Effectively the pulse duration is increased as measured by either the magnitude of the intensity or the complex envelope in time domain. In the case that the input pulse is initially chirped, the waveguide dispersion adds the chirp term into the complex wavefunction representing the pulse and results in a new effective chirp expression. Should the initial chirp of the pulse and the dispersion of the waveguide have opposite signs, the pulse will become momentarily unchirped at a certain point along the waveguide such that the chirp introduced by the waveguide cancels the original chirp. In this situation, the duration of the pulse is decreased; in other words, the pulse is compressed. After this point, the pulse will begin to broaden because the accumulated dispersion chirp outweighs the initial chirp. From the above discussion, it is obvious that the chirped pulses could be compressed by the appropriate dispersive waveguide. However, compressing the originally transform-

26


limited pulse cannot be achieved by solely employing the waveguide dispersion, which only acts as a filter. In this situation, the chirp must be introduced to the unchirped pulse in time domain by modulation, which could be implemented using electro-optic materials or self-phase modulation (SPM) [47, Chapter 22]. The latter method is quite convenient because a silica optical fiber also exhibits a nonlinear effect especially in the case of short pulses. The time domain phase modulation is introduced by means of Kerr nonlinearity ∆φptq n2 I ptqk0 z.

(2.14)

It can be shown, for a Gaussian pulse with parabolic approximation, that the self-phase modulation results in the following phase factor to the pulse in time domain 2 2 ej2n2 I0 k0 zt {τ

(2.15)

where n2 is the optical Kerr coefficient, I0 is the maximum intensity of the pulse, k0 is the wavenumber, and τ is the pulse duration. The result suggests that the self-phase modulation introduces chirp to the propagating pulse with the chirp sign depending on the sign of the nonlinear index n2 . In Eq. 2.15, SPM introduces a linear chirp to the pulse and could make a linearly chirped pulse from a transform-limited pulse. This phenomena opens the floor for pulse compression in a fiber if it has an appropriate dispersion behavior. Another interesting interaction between SPM and complementary dispersion of the waveguide spurs the research topic of solitons and solitary waves. A silica fiber has normal dispersion for wavelength shorter than 1.3 µm but has anomalous dispersion for longer wavelength. If pulse compression is carried out for pulses having central frequency shorter than 1.3 µ and the phase modulation is imposed by means of SPM, the use of an external anomalously dispersive delay line is mandatory to compress the pulses, as done in [38]. On the other hand, the fiber can function as an internal distributed recompressing element if the operating wavelength is well above 1.3 µm, for example at 1.5 µm. This configuration has been demonstrated to shrink 7-ps

27

Chapter 2. Literature Review optical pulses with

1/27 compression ratio.

As mentioned early, the material dispersion of the fiber could be very low. Introducing grating structures could enhance the overall dispersion. The grating structures were used to balance the self-phase modulation to maintain the pulse shape as the pulse propagates resulting in solitons [48, 49].

2.3.2

Temporal Waveform Pulse Shaping

Accomplishing arbitrary pulse shaping with simple design criteria is usually needed; directly relating the physical grating profile to the temporal waveform target represents one strategy of addressing the need. A fiber grating could be regarded as a transfer function, H pω q, that acts on an input pulse spectrum, X pω q, in a way that an output pulse spectrum, Y pω q, becomes Y pω q H pω qX pω q.

(2.16)

This is exactly the underlying principle of Fourier pulse shaping discussed earlier. The required transfer function is identified with a complete knowledge of the input and the output. Approximations simplify the strict requirement of this complete knowledge, which is sometimes not available. For instance, if the input pulse duration is short enough, such as in a femtosecond scale, compared to the desired output pulse duration, the input pulse could be represented by an impulse whose Fourier transform is unity. Another restriction on the transfer function when working a passive device is that the modulus of the transfer function must not exceed one, i.e.

|H pωq| ¤

1. In principle

arbitrary temporal waveform is achievable. The central problem after determining the transfer function becomes the retrieval of the grating structure. If the reflection amplitude is weak, i.e. |H pω q| ¤ 0.2, the first Born approximation can apply and yield a relation between the impulse response of the grating and the magnitude

28


of index perturbation, or apodization Apz q, essentially captured in Eq. 2.2 for FBGs [16]: hr ptq 9

!

Apz qe p q jφ z

)

(2.17)

{

z ct 2nav

From the relationship, if the input is regarded as an impulse, the output pulse will behave the same as the impulse response, i.e. eout ptq

hr ptq.

For a flat-top pulse target, the

apodization profile becomes a constant. In other words, a weak uniform grating could produce a flat-top pulse from an ultrashort input pulse. A more accurate method will include the shape of the input pulse as well. This relation has been used to generate 20-ps flat-top pulses from 2.5-ps soliton pulses, which are assumed to be in a hyperbolic secant form, [18]. As discussed earlier, working in the weak grating limit compromises between the coupling strength and the duration of the output pulse. Note that the previous relation appears as a mapping between space and time. The space-to-frequency-to-time mapping is proposed by J. Azana and L.R. Chen [16] to remedy the weak grating limit with a cost of an extra chirp introduced to the output pulse. The mapping relation is expressed as hr ptq 9

j π t2 e Φ: ν

tanh mA z

ct 2nav

,

(2.18)

: ν is the first-order dispersion coefficient of a linearly chirped fiber Bragg grating where Φ (LC-FBG), m is a constant, and Apz q is the apodization profile. Basically the grating reflects different frequencies at different positions along its length, specifically with a linear relationship. The reflected amplitudes of different frequencies depend on the coupling strengths at the reflection positions, i.e. the apodization profile, Apz q. Effectively, the LC-FBG imposes an amplitude response related to the apodization profile and a linear phase response due to the linear grating period chirp, which is a wavelength-to-time mapping. This technique was used to create an arbitrary temporal waveform signal [19], which could be extended to an electrical signal by employing an optical-to-electrical transducer such as a photodiode [50]. If it is needed to eliminate the extra chirp introduced by a single pass through the LC-FBG, passing the pulse through a complementary LC-FBGs


29

having opposite dispersion results in a cancelation. However, passing the pulse through the same grating but in the opposite direction serves a better cancelation in practice due to the difficulty of fabricating two gratings with exactly complementary responses. The effective results is the imposition of an amplitude response without a phase response to the input pulse [20].

2.3.3

Pulse Stacking in Fiber-Based Devices

Other than working with the Fourier transform pulse shaping, fiber gratings can also operate under the pulse stacking paradigm. For example, in [51], N concatenating weak uniform FBGs were proposed and demonstrated Gaussian pulse generation from a continuous-wave source, whereas using the same technique a flat-top pulse was the target for [52]. The schematic diagram is shown in Fig. 2.8. Physically, the incoming pulse propagates through a series of uniform gratings and it is partially reflected by each grating. Assuming that the grating is weak, multiple reflection between gratings is negligible and the overall reflected signal is composed of a series of pulses separated in time by the distance between the grating. The output waveform is effectively the interference of these pulses. The technique described in [51] will breakdown when the gratings have strong coupling leading to significant multiple reflection decreasing energy of propagating original pulse. This will demand a more accurate model and a careful design. LPGs also lend themselves to the pulse stacking technique as proposed by [53]. Recall that the physical mechanism underlying the LPG is the coupling between the core and the cladding modes. Since the core and cladding modes have different propagation constants or equivalently different effective indices, a phase difference can develop when the two modes propagate in the same distance. The proposed technique could be schematically displayed as in Fig. 2.9. The first LPG, LPG1, couples a fraction of energy of the incoming pulse to the cladding mode, where this out-coupled pulse will propagate with the cladding effective index while the remaining input pulse resumes its travel with the


30

Figure (2.8): Pulse stacking in a fiber-based device by N uniform FBGs, [51], reprinted with permission © 2006 IEEE. (a) The temporal profile of the input pulse, (b) Reflected pulses from a series of fiber grating separated by time delays, and (c) The resultant pulse due to interference of all reflected pulses.

core effective index. At the second LPG, LPG2, a fraction of the cladding mode is incoupled to the core mode at the same time as a fraction of the core mode is out-coupled. When considering the waves that remain traveling in the core region after the second LPG, those waves are the core mode that remains untouched and the in-coupled pulses from the cladding to the core. These two pulses propagate the same physical distance intervening the first and the second grating, but they develop a phase difference due to different effective indices. Essentially, in the core region after the second LPG, the resultant pulse is the interference between these two pulses. If the LPGs operate at 50% coupling strength, the two pulses will assume the same shape and magnitude and they could be called replicas of the original input pulses but each containing a quarter of energy of the original pulse. The relative time delay between the pulses depends on the distance between the two gratings. It can be shown that with an appropriate time delay two Gaussian pulses can interfere to generate a flat-top-like pulse as suggested in [53]. In actuality, pulse stacking works for any shapes of involved pulses. The two cases discussed previously simplify the discussion by considering the same shape for all subpulses. For flat-top pulse shaping, another possible technique involves the pulse stacking of a


31

Figure (2.9): Pulse stacking in a fiber-based device by LPGs [53], reprinted with permission © 2008 IEEE.

Gaussian pulse and its first-order derivative as proposed in [21]. The relevant fact is that the temporal differentiation could be carried out using a uniform LPG. The efficiency of differentiation depends on the matching between the central frequency of the incoming pulse and the designed resonance frequency of the grating, in which an ideal operation occurs at zero detuning. If the mismatch is present, a part of energy of the incoming pulse goes through the differentiation and the other part remains in the original pulse. With appropriate detuning, two pulses coexist in the fiber core, namely the original pulse and its derivative. The two pulses then interfere and give rise to a resulting output which appears flat-top-like as shown in Fig. 2.10. Introducing a strain to a fiber by stretching serves as a detuning mechanism as used in [22]. The main drawback in pulse stacking is that the rise and fall times of the resulting output is determined by those of the input. As shown in Fig. 2.10, the flat-top pulse for ∆λ 1.3 nm still has rise/fall times of 2 picoseconds, a characteristic of the input pulse represented in a black solid curve. Ultrafast features might be achievable by adding many subpulses possibly with different shapes and time delays, but this will only lead to increasing complexity of the overall system.

2.3.4

Other Fiber-Based Pulse Shaping

Another related application is pulse repetition rate multiplication, which is mainly based on a temporal Talbot effect, discussed in detail in [54]. The Talbot effect is an effect of dispersion, which could be represented by a phase response of a device. The pulse

32


Figure (2.10): Flat-top pulse shaping using pulse stacking and pulse differentiation in LPGs [21], reprinted with permission © 2006 OSA. (a)The intensity of the output pulses. The black solid curve represents the intensity of the input pulse. The red dotted, green dashed, and blue solid lines are the intensity profiles of the output pulses for different detuning parameters between the pulse central frequency and the designed central frequency of the grating temporal differentiators. (b) The phase of the pulses.

repetition rate multiplication could be included in a waveform pulse shaping to acquire both effects simultaneously. For example, in [55,56], a linearly chirped fiber Bragg grating was used to realize a combined response including flat-top pulse shaping and pulse rate multiplication, which was achieved up to 80 GHz. Pulse repetition rate multiplication can also be achieved by using superimposed FBG structures. The pulse repetition rate as high as 170 GHz was demonstrated [57, 58]. For signal processing applications, optical waveform differentiation and integration are among basic building blocks. In fibers, FBGs and LPGs are proposed to fulfill the operation by providing appropriate filtering functions. N -order differentiation should be achieved from a few gratings, instead of concatenating N first-order differentiators because of the energy loss at each of the differentiators is very high due to the ideal Fourier pair of the temporal differentiation, H pω q j pω ω0 q. A series of uniform LPGs

33


separated by π-phase shifts could provide N -order differentiation by designing correct grating lengths for each gratings. Differentiations up to the fifth-order were simulated in [23] and later demonstrated experimentally in [59] showing great results and operating bandwidth of 10 nanometers. Other than using LPGs, temporal pulse differentiation could also be realized in LC-FBGs working in transmission [24, 25, 60, 61]. The major difference between the uses of LPGs and FBGs for temporal pulse differentiation is their bandwidths. In general, the LPG-based differentiators have a large operating bandwidth which could be in a terahertz range. The FBG-based differentiators on the other hand have a bandwidth in the order of gigahertz. Optical waveform integration is more challenging due to the fact that the ideal Fourier transform of the operation suggests the filtering of magnitude larger than one near the central frequency, i.e. H pω q

1{j pω ω0q.

It means that the gain is required to im-

plement a device close to the ideal integrator, as carried out in [62], making the situation more complicated than a passive device. Fortunately the passive device method is proposed when the integration in time domain is considered [63,64], in which a weak uniform FBG working in reflection was used such that the reflected signal is the integration of the input pulse with a temporal integration windows associated with the length of the grating. Other configurations being explored to improve the performance include the use of a π-phase-shift FBG working in transmission [65] and a Er-Yb-doped FBG [66].

2.4

Integrated Optics Pulse Shaping

Integrated optics has been receiving momentum considerably due to many advantages such as more compactness and functionalities. Additionally, with the advent of semiconductor lasers and detectors, one can imagine to implement a whole optical circuit in a microscopic chip. In the optical circuit where optical pulse contains important messages or functionalities, pulse shaping is therefore necessary within the integrated circuit itself.


34

Similar to fiber-based devices, an integrated optical device must have a waveguide part, which is a channel for light to propagate, and a functional part, which performs a particular function such as pulse shaping. Most of the integrated optical circuit is fabricated in semiconductor materials, such as a silicon-on-insulator (SOI) wafer or a III-V semiconductor wafer. For the waveguide part, the cross-sectional refractive index profile is responsible for the modes and their properties. The overall dispersion is a combination of the material dispersion and the waveguide dispersion. In the functional part, additional dispersions and also responses depend on the nature of that part. For example, if the grating is fabricated in an integrated waveguide, the dispersion property pertaining to the grating index perturbation will contribute to the overall dispersion characteristic and it could dominate other contributions. In terms of material choices, there are a lot of semiconductor systems that are investigated for integrated optics. Popularized by the microelectronics, a silicon-on-insulator (SOI) system promises compactness and integrability with electronics circuit. Microfabrication for SOI is relatively easier compared to other choices of materials due to readily available knowledge and facilities. The major problem in the SOI system is that silicon has an indirect bandgap and therefore it is a inefficient material for generating light. Its nonlinearity is also smaller compared to the III-V semiconductor systems. The aluminum gallium arsenide, Alx Ga1x As, is one of many important III-V systems and it has aluminum and gallium, where x is the concentration fraction of the aluminum, as the III elements and arsenic as the V element. Physical properties, such as refractive index, of the AlGaAs system is adjustable by changing the aluminum concentration. One of the benefits in the AlGaAs system is that the lattice spacing is relatively constant with varying aluminum concentration from 0 to 1 resulting in a negligible mechanical strain when different aluminum concentrations are introduced to the materials being fabricated. Other important optical properties are its direct bandgap lending itself to efficient lasing and its high nonlinearity for advanced applications such as nonlinear switching. For its


35

versatility, the AlGaAs system is chosen for this thesis work. The basic ideas for pulse shaping in free-space optics could be carried out in the integrated configurations with some modifications. Recall that Fourier transform pulse shaping based on the 4-f configuration and the direct-space-to-time (DST) pulse shaping in bulk optics requires diffraction gratings, whose major functionality is to spatially disperse spectral components of light. This functionality is achieved by using an arrayedwaveguide grating (AWG). Fig. 2.11a shows the conventional configuration of the arrayedwaveguide grating, which includes the input waveguide, two slap waveguide regions, a waveguide array, and output waveguides. Light from the input waveguide enters the multimode slap waveguide, spreads and propagates to waveguides, and recombines after the second multimode slap. Due to interference and phase differences, the result is a wavelength separation at the output waveguides. The output waveguide gives the output signal of the DST pulse shaping since it is complimentary to the slit in the conventional DST setup shown in Fig. 2.5. The mask is then placed correspondingly for transmission (Fig. 2.11b) and reflection (Fig. 2.11c) operations [67]. Pulse bursts with the overall duration of 10 ps have been created from a single output channel with this method by employing a phase mask with a reflection AWG setup [68]. In a similar manner, the AWGs lend itself to implementation of the 4-f configuration in which two AWGs are needed to work as the two gratings. The reflection operation is preferred due to the difficulty in fabricating two identical AWGs. The schematic diagrams shown in Fig. 2.12 display two modes: the analog and digital filtering [27]. In the analog mode, the filtering device is the conventional spatial light modulation such as a liquid crystal array, and the lens collects and focuses light diffracted from the AWG. Since the diffracted light is not discretized, the mode of operation is termed analog. On the other hand if the wavelengths are divided by waveguides as shown in Fig. 2.12b and amplitude and phase modulators are fabricated for each waveguide, the digital pulse shaping is realized. A rectangular pulse with the duration of 12.5 ps was generated by this digital

36


(a) Conventional AWGs.

(b) AWG-based DST pulse

(c) AWG-based DST pulse shaping

shaping operating in transmission.

operating in reflection.

Figure (2.11): DST pulse shaping implemented with integrated arrayed-waveguide grating [67], reprinted with permission © 2004 OSA.

AWG-based DST pulse shaper fabricated on a silica platform [27]. This technique was also used to demonstrate dispersion management in InP-InGaAsP material [69]. AWG-based devices are relatively large because they are composed of many bended waveguides. Also they require amplitude and/or phase modulators being large and complex even in integrated implementation. Especially in the analog 4-f pulse shaper, light has to be coupled in and out of the waveguides leading to unnecessary loss. Hence, if generating a data packet or reconfigurability are not the main target, AWG-based devices does not provide advantages over integrated waveguide gratings discussed below. In the previous section, fiber gratings are reviewed and show great potentials to achieve various shaping functions. In integrated optics, gratings are mostly used as couplers, wavelength isolators in a WDM system, Bragg reflectors, and integrated chemical/biological sensors. A pulse shaping capability of integrated waveguide gratings is less investigated; however, progresses in this area will prove very valuable to manipulating light inclusively in the integrated environment. Since the grating can be fabricated onto a waveguide, the device size is in the levels of the waveguide itself. Providing a correct grating structure, arbitrary pulse shaping could be accomplished using a single grating.

37


(a) Analog AWG-based 4-f pulse shaping.

(b) Digital AWG-based 4-f pulse shaping.

Figure (2.12): Implementation of 4-f pulse shaping configuration, operating in reflection, in integrated optics using arrayed-waveguide gratings [27], reprinted with permission © 2008 IEEE.

Among a few studies, integrated gratings in an AlGaAs system were demonstrated to generate digital bit streams [28] and to compensate chirp from a semiconductor laser [29]. In the first work, the grating structure was designed using the first-order Born approximation or the weak grating limit associating the impulse response in reflection to the apodization profile of the grating. Digital bit streams composing of 0 and 1 bits are designed with appropriate time-to-space scaling such that adjacent bits are 2 picoseconds apart, representing the temporal resolution of the device. In the experiment, the results show fair performance but the distinction ratio between 0 and 1 was poor. Imperfections could result from difficulties in fabrication of III-V semiconductor as well as from the use of the weak grating assumption. In the second work [29], a more involved consideration to the grating design was employed aiming for an on-chip dispersion control for a semiconductor mode-locked laser (MLL), which exhibits pulse chirping in the range of 0.1-10 ps/nm over a few nanometer bandwidth [70]. The grating structure was derived using an inverse scattering method, which basically suggests the device structure from its response. The fabricated gratings yielded measured responses close to the simulated values, which could provide a time


38

delay span of about 10 picoseconds to the incident pulse. Some results are displayed in Fig. 2.13.

Figure (2.13): Reflectivity and time delay responses of measured and simulated grating whose structure is generated from the inverse scattering algorithm for a quadratic delay [29], reprinted with permission © 2010 IEEE

Both of the two studies show unprecedented control over the coupling strength and the chirp of the grating because the sidewall-etching geometry was used to introduce grating perturbation. However, one of the most common perturbation types is a surface corrugation, displayed in Fig. 2.14a, in which a perturbation is introduced to the top part of a waveguide. The strength of coupling coefficient depends on the depth of the perturbation etching. With this type of corrugation, changing the coupling coefficient along a grating becomes complicated and involves many fabrication steps. Grating couplers usually employ this type of perturbation. On the other hand, a sidewall-etching geometry delivers perturbation to the sides of the waveguide, as shown in Fig. 2.14b. The coupling coefficient not only depends on the etching depth as in the surface-corrugated grating but also on the recess depth, which is the etch depth into the sides of the waveguide as denoted dpz q in the figure, providing another degree of freedom to control the grating behavior, and the waveguide width, which by itself also determines the effective index of the waveguide mode. Since the effective index is a function of the waveguide width,

39


by changing the waveguide width along the grating the chirp is easily introduced even the perturbation period remains constant. Hence, by varying the waveguide width and the recess depth, the sidewall-etching geometry provides great controls to the grating design. Another main advantage of this perturbation type is that the waveguide and the grating are defined simultaneously including the ability to tune the coupling strength and the chirp without any further steps. Since arbitrary pulse shaping most likely requires nonuniform gratings, sidewall-etching geometry becomes promising as a simple yet efficient grating perturbation method.

(a) Surface gratings [71], reprinted with permission

(b) Sidewall gratings [29], reprinted with permission

© 2008 OSA.

© 2010 IEEE.

Figure (2.14): Schematic profiles of integrated waveguide gratings.

2.5

Summary

From the above reviews, grating structures appear promising for pulse shaping and should be implemented in the integrated platform. More investigations are needed to efficiently realize the device for arbitrary pulse shaping. The grating configuration that should be in focus is the sidewall-etching geometry for its provision of efficient control over the grating behavior. To study the sidewall-etched waveguide grating, algorithms to determine the


40

response and to retrieve the structure of the grating are crucial and should be robust and rigorous.

Chapter 3 Grating Responses In this chapter, direct scattering to determine the response of the grating is considered. The numerical modeling is based on the coupled-mode theory and the transfer matrix method, as discussed at length in Appendix A.

3.1

Waveguide Design

Since an integrated grating is fabricated on a waveguide, it is necessary to consider the structure of the waveguide. The electric field of the guided modes is the solution to the following equation, [33], ∇2K epx, y q

ω 2 µw px, y q β 2 epx, y q 0,

(3.1)

where epx, y q represents the guided mode field, w px, y q is the permittivity function of a waveguide, and β is the propagation constant. The waveguide structure is based on Alx Ga1x As, where x is the fractional aluminum concentration. The refractive index of AlGaAs is adjustable by changing this aluminum concentration. This characteristic makes it easy to epitaxially grow the AlGaAs layers with refractive index variations layer-by-layer. Fig. 3.1 shows the refractive index, n, as a function of aluminum concentration, x, based on the work of Gehrsitz et al [72]. 41

Chapter 3. Grating Responses

42

3.4 3.3

n

3.2 3.1 3 2.9 2.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x Figure (3.1): AlGaAs refractive index as a function of aluminum concentrations.

The AlGaAs layer structure in use is shown in Fig. 3.2a. It is composed of, from bottom to top, GaAs substrate, (1) 4-µm Al0.7 Ga0.3 As, (2) 0.5-µm Al0.3 Ga0.7 As, (3) 0.2µm Al0.2 Ga0.8 As, (4) 0.2-µm Al0.7 Ga0.3 As, and (5) 0.1-µm GaAs. For all layers, the aluminum concentrations are chosen to be above 0.2 to reduce the effect of two-photon absorption inferred from the bandgap energy [73]. An index contrast of about 0.2 results from choosing the aluminum concentration of 0.7 and 0.2 of the cladding and the core respectively. Most of the field resides in Layer (2) and (3), which are the core region. Layer (1) is the buffer layer that prevents the mode in the core region from leaking to the substrate. Along with Layer (4), they function as lower and upper claddings. The top layer, Layer (5), is technically deposited in order to prevent oxidation of aluminum underneath it. A two-dimensional ridge waveguide is depicted in Fig. 3.2b. The waveguide is defined by the layer structure, the waveguide width, and the etch depth. These three entities are captured in a two-dimensional permittivity profile, w px, y q. The etch depth is chosen to be 1 micron throughout the work due to the available fabrication facility for AlGaAs etching. The layer with 0.2 aluminum concentration according to the layer structure is used to attract the mode field upward to the surface resulting in a more circular mode field distribution as well as improving coupling coefficients.

43


(a) AlGaAs layer structure.

(b) Cross-sectional waveguide profile.

Figure (3.2): Cross-sections of a layer structure and a AlGaAs waveguide.

Lumerical MODE Solutions is used to find the guided modes and their effective indices. The waveguide structure is drawn in the CAD module of MODE Solutions. The structure is broken into two-dimensional nodes; each node is assigned a refractive index value corresponding to the waveguide structure. The software takes the node mesh and solves the eigenvalue-eigenfunction problem, Eq. 3.1, using the finite element method. The solver gives the guided modes and their effective indices. The values of the electric fields of the modes, epx, y q, are assigned to the nodes. The effective index is related to the propagation constant via β

2πnλ

eff

.

In the simulation, the waveguide is centered at x

0.

The simulation x-axis ranges

from -3 µm to 3 µm, with 180 nodes. The simulation y-axis ranges from 2-µm below to 0.5-µm above the layer structure surface, with 180 nodes. The x range and y range form the simulation area. The boundary condition of the simulation area is set to be a perfectly matched layer (PML) boundary condition. Fig. 3.3 gives an example of a 1.4-µm-wide waveguide and its modes, with etching depth of 1 µm. The wavelength used in this simulation is in the telecommunication regime, specifically λ=1.55 µm.


44

(a) Cross-sectional index profile.

(b) Electric field magnitude of the TE-like mode. (c) Electric field magnitude of the TM-like mode. Figure (3.3): The simulated index profile, a corresponding fundamental TE electric field mode, and a corresponding fundamental TM electric field mode.

The modes and their effective indices depend on the wavelength and the refractive index profile, which is particularly altered by changing the waveguide width and the etch depth. Fig. 3.4 shows the dependence of the effective indices of the modes to the waveguide width, with a fixed etch depth at 1.0 µm. The effective index increases with increasing waveguide width because the guided modes sense more high refractive index

45


of AlGaAs. However, increasing the width beyond some value results in a multimode waveguide. For this particular layer structure as shown in Fig. 3.2a, the 1.6-µm-wide waveguide has multiple guided modes. The dispersion as a function of the waveguide width is compared between the TE and TM modes in Fig. 3.5. It can be seen that near w

1.4 µm both TE and TM modes have approximately the same effective index such

that they propagate in the same manner in an unperturbed waveguide.

(a) Effective index of the TE-like mode.

(b) Effective index of the TM-like mode.

Figure (3.4): Effective indices of the fundamental TE-like and TM-like modes of the waveguide as a function of waveguide width and the light wavelength at the etch depth of 1 micron.

The effect of the etch depth on the effective index can be explained in the same way as the effect of the waveguide width. A deeper etch depth exposes the waveguide to more air; the modes then sense more low refractive index of air. Therefore, the effective index of the mode decreases with increasing the etch depth. However, in an actual device the etch depth is the same throughout the whole grating for a simple fabrication procedure, rendered it out of the degrees of freedom. Therefore, the dependency of the effective index on the waveguide width and the wavelength is more prominent providing a key method to introduce chirp to the grating.

46

Chapter 3. Grating Responses 3.13 TE TM

3.12

neff

3.11 3.10 3.09 3.08 3.07 3.06 1

1.1

1.2

1.3 1.4 w (µm)

1.5

1.6

Figure (3.5): Effective indices of the fundamental TE-like and TM-like modes as a function of waveguide width at λ 1.55 µm and the etch depth of 1 micron.

The simulation was repeated for several wavelengths and waveguide widths to obtain the waveguide dispersion, as shown in Fig. 3.4. The effective index data were collected by simulating over the waveguide widths of w lengths of λ

1.0, 1.2, 1.3, 1.4, 1.6 µm and the wave-

1.49, 1.51, 1.53, 1.55, 1.57, 1.59, 1.61 µm. The etch depth was 1 µm.

With this simulation settings, the waveguide supports both TM-like and TE-like fundamental modes. Their effective indices are listed in Table 3.1. Table (3.1): Propagating effective indices of TE-like and TM-like modes of a ridge waveguide with corresponding waveguide widths, w, and the etch depth of 1 µm.

w (µm) neff (TE)

1.0

1.2

1.3

1.4

1.6

3.0626 3.0896 3.0988 3.1062 3.1170

neff (TM) 3.0747 3.0936 3.1003 3.1058 3.1141

The Surface Fitting toolbox in MATLAB was then used to find polynomials that closely describes the relation between the effective index, the waveguide width, and the wavelength. Specifically, the waveguide width and the wavelength are the independent variables, named x and y, respectively, whereas the effective index is the dependent variable, z. The maximum power of both x and y is chosen to be both three. This third

47

Chapter 3. Grating Responses order polynomial is expressed as z px, y q p0,0

p1,0 x

p3,0 x3

p0,1 y

p2,1 x2 y

p2,0 x2

p1,1 xy

p1,2 xy 2

p0,3 y 3

p0,2 y 2 (3.2)

where the factors pn,m are determined by the toolbox to generate the best fit. These polynomial coefficients were found and shown in Table 3.2. Table (3.2): Third-order polynomial coefficients for the fitting functions of (a) the TE-like effective index, neff,T E pw, λq z px, y q, and (b) the TM-like effective index, neff,T M pw, λq z px, y q. The free variables x and y are the waveguide widths, w, and the wavelengths, λ. (b)

(a)

Coeff.

Values

95%-confidence bounds

Coeff.

Values

95%-confidence bounds

p0,0

3.821 (2.697, 4.964)

p0,0

3.819 (3.206, 4.433)

p1,0

0.2462 (0.1235, 0.3688)

p1,0

0.1862 (0.1204, 0.2521)

p0,1

-1.194 (-3.396, 1.008)

p0,1

-1.122 (-2.304, 0.05988)

p2,0

-0.3188 (-0.3357, -0.3019)

p2,0

-0.2068 (-0.2159, -0.1978)

p1,1

0.3851 (0.2323, 0.5379)

p1,1

0.2294 (0.1473, 0.3114)

p0,2

0.3613 (-1.056, 1.778)

p0,2

0.3972 (-0.3635, 1.158)

p3,0

0.09912 (0.0963, 0.1019)

p3,0

0.05945 (0.05794, 0.06097)

p2,1

-0.1087 (-0.117, -0.1005)

p2,1

-0.0589 (-0.06333, -0.05448)

p1,2

0.001372 (-0.04744, 0.05018)

p1,2

-0.003226 (-0.02943, 0.02298)

p0,3

-0.06304 (-0.3674, 0.2413)

p0,3

-0.06839 (-0.2318, 0.09503)

3.2

Coupling Coefficients of Sidewall Gratings

As mentioned earlier, the coupled-mode theory (CMT) and the transfer matrix method (TMM) will be used to calculate the grating response. In order to combine with the

48


inverse scattering theory for grating design, a simple version of CMT is in use; the waveguide is assumed to be lossless and only the first-order Fourier coefficient of the index perturbation is considered, as laid out in [74]. In actual integrated gratings, higherorder Fourier coefficients could significantly contribute to the behaviors of the gratings, such as leading to radiation loss and reduced coupling coefficients which are very critical in distributed feedback lasers [75]. Those effects should be included for a more accurate result in the later state of the grating design. A modified CMT that considers higher-order interactions could be found in [76, 77]. Good reviews on many coupled-mode theories could be found in [78–80]. The coupled-mode theory, which is developed in Appendix A, leads to a system of equations for a first-order uniform grating in a single-mode waveguide, [17, 74],

∆β d c˜1 pz q j σ c˜1pzq jκ˜c1pzq, dz 2

∆β d c˜1 pz q jκ c˜1 pz q j σ c˜1pzq, dz 2

(3.3a) (3.3b)

where c˜1 and c˜1 represent the forward- and the backward-propagating waves, and the detuning parameter is ∆β

Λ2π 2β. The cross- and self-coupling coefficients are defined 0

as κ κ1,1 r1s σ

σnr0s

ω 2 µ xe1 |∆px, y qr1s|e1 y , 2βn xe1 |e1 y ω 2 µ xe1 |∆px, y qr0s|e1 y . 2βn xe1 |e1 y

(3.4a) (3.4b)

The electric field of the mode, e1 , was calculated in the previous section. Assume that the recess depth function is a rectangular function with 0.5 duty cycle. Using the sidewall etching geometry, the index perturbation along the z-direction for each cross section point px, y q is px, y, z q

$ ' ' &

;0¤z

0

' ' %w x, y

p q

Λ{2,

; Λ{2 ¤ z

Λ.

(3.5)

49


Figure (3.6): Index profile with etched area shaded.

The first-order permittivity modulation is then captured in ∆px, y qr1s as ∆px, y qr1s

$ ' ' &

j

w

px,yq0 π

' ' %0

; px, y q in the etched area,

(3.6)

; otherwise,

where as the zeroth-order permittivity modulation ∆px, y qr0s is ∆px, y qr0s

$ ' ' & 0 w px,yq 2

' ' %0

; px, y q in the etched area,

(3.7)

; otherwise.

From these equations, it is clear that the contribution to the coupling constants is only from the etched areas of the waveguide, as shown in Fig. 3.6. Additionally, the number of nodes in the etched areas is increased to eight times of the normal number of nodes to achieve accurate values. Note that the actual profile of the recess depth along the grating will affect the coupling coefficient. The periodic rectangular recess depth profile is easy to design, for example in a CAD module for lithography. However, the rectangular profile has infinite orders of Fourier coefficients. Since the coupled-mode theory that is used in this work takes into account only the first-order and discards the rest, some level of errors will exist. Sinusoidal profiles could be used to suppress other Fourier orders but the drawing would be a challenge.

50


The coupling constants depend on the mode shape, the effective index, the permittivity modulation, and the wavelength of light. Since the mode shape and its corresponding effective index depends on the wavelength of light, generally the modes at each frequency have to be found before the coupling constants could be calculated. This way of calculation is inefficient and time-consuming. If the spectrum of interest is not too broad, the modes of those frequencies are similar and their effective indices are close as well. Hence, the coupling constants could be approximated to be the same for all frequencies in the spectrum. Then, the cross- and self-coupling constants are calculated using a representative frequency, which is suitably the center frequency of the spectrum, λ 1.55 µm. The waveguide width values in the simulation are the same as before. The recess depth were sampled from 25 nm to 200 nm, with 25-nm step; rd 25, 50, 75, 100, 125, 150, 175, 200 µm.

(3.8)

For instance, the cross-coupling constants for a waveguide width of 1.4 µm are listed in Table 3.3. They are imaginary with negative imaginary parts. This result corresponds to the expression for ∆px, y qr1s in Eq. 3.6. A polynomial surface fit was found using a MATLAB model poly53, however, with the recess depth and the waveguide width as x and y independent variables and the cross-coupling constant as a dependent variable. Before finding the best fit, the values of zeros are added to the calculated data for zero recess depth, rd 0 nm. The form of the fit function is z px, y q 0

¤ m ¤ 5, 0 ¤ n ¤ 3, and m

n

¤ maxt5, 3u.

° m,n

pm,n xm y n , where

The fit function of the cross-coupling

constants for TE-like and TM-like modes are displayed in Fig. 3.7. For the self-coupling constants, their values depend on the the zeroth order of permittivity perturbation, ∆px, y qr0s. Since ∆px, y qr0s is a negative real number, the self-coupling constants are also real and negative. For a 1.4-µm-wide waveguide, the self-coupling constants are shown in Table 3.4. Again, the polynomial surface fit based on the poly53 in MATLAB was calculated and is displayed in Fig. 3.8.

51


Table (3.3): Cross-coupling coefficients as a function of recess depths for a constant waveguide width of 1.4 microns and λ 1.55 µm.

Recess depth (nm) 0

|κpTEq|pcm1q |κpTMq|pcm1q

25

0 46.09

50

75

100

125

150

175

97.95 167.71 262.36 388.51 552.32 759.47 1,023.2

0 46.09 113.23 209.43 339.37 507.37 717.36 972.82 1,286.1

(a) |κpTE; rd, wq|

(b) |κpTM; rd, wq|

Figure (3.7): Cross-coupling coefficients as a function of recess depths and waveguide widths by the surface fitting function at the wavelength of 1.55 microns.

3.3

200

Grating Responses

From the previous sections, the mode and their effective indices were calculated with relations to waveguide widths and a fixed etch depth. Then, the third order polynomials were found in order to smoothly predict those relations around the sample points. The coupling constants were also sampled with variations on the waveguide widths and the recess depths, and their corresponding polynomial fits were determined. These polynomial fit functions were used as a database for calculating the grating response which is the focus of this section.

52


Table (3.4): Self-coupling coefficients as a function of recess depths for a constant waveguide width of 1.4 microns and λ 1.55 µm.

Recess depth (nm) 0

|σpTEq|pcm1q |σpTMq|pcm1q

25

50

75

100

125

150

175

0 85.87 184.13 316.18 495.39 734.36 1,044.9 1,437.9 1,933.4 0 86.05 213.11

395.4 641.88 960.81 1,359.7 1,845.2 2,435.0

(a) |σ pT E; rd, wq|

(b) |σ pT M ; rd, wq|

Figure (3.8): Self-coupling coefficients as a function of the recess depths and the waveguide widths by the surface fitting function at the wavelength of 1.55 microns.

Before discussing about the grating response, some backgrounds regarding the discrete Fourier transform and the continuous Fourier transform should be reviewed as available in Appendix B. Briefly summarized, in MATLAB, the fast Fourier transform function, fft(), receives a finite vector of values and produces another vector, of similar size, of the corresponding Fourier pair. The Fourier pair of interest is the impulse response and the reflection response. The time and frequency spaces should be defined accordingly. If the length of the vector of the impulse response or the reflection response is N , the resolutions in time and frequency spaces are related by N from

N 2∆ν

to

200

N ∆ν . 2

1{∆t∆ν.

The frequency space spans

Indeed, if both time and frequency resolutions are great, N is very

53


large such that the frequency span is much larger than the significant bandwidth of the grating. It is then necessary to only simulate the grating response within the grating bandwidth and assume zero response elsewhere in the frequency space; otherwise the simulation will consume a lot of computing time. As a result, the interested frequency range centering around the main frequency is specified by a 1 Nν vector: λ tλj u or ν where j

tνj u,

(3.9)

1, 2, . . . , Nν is the frequency index.

The apodized and chirped integrated grating is based on changing the waveguide width and/or the recess depth along the grating. In actuality, the grating period could be varying as well. However, in this work the grating period is maintained constant with a cycle duty of 0.5. The non-uniform grating will be divided into many uniform sections for computing its reflection response. The solution to the coupled-mode equations for a first-order uniform grating, Eq. 3.3, is

p q m11pz, z0q c˜1 pz q m21 pz, z0 q

where, when defining s

Mpz, z0q

m12 pz, z0 q c˜1 pz0 q

c˜1 z

b

m22 pz, z0 q

|κ|2

∆β 2

c˜1 pz0 q

σ

2

j

∆β 2

σ

κ m12 pz, z0 q j sinh spz z0 q , s κ m21 pz, z0 q j sinh spz z0 q , s

m22 pz, z0 q cosh spz z0 q

c˜1 pz q

c˜1 pz q

,

(3.10)

,


j

sinh s z

p z0q s

,

(3.11a) (3.11b)

∆β 2

σ

p z0q

sinh s z

s

(3.11c)

.

(3.11d)

The whole grating response is constructed from all the grating pieces using the transfer matrix method. If the grating is divided into Ng pieces, its grating parameters are

54

Chapter 3. Grating Responses declared and initiated in 1 Ng vectors:

twiu, rd trdi u, Λ tΛi u, ∆z t∆zi u, w

where i

1, 2, . . . , Ng .

(3.12a) (3.12b) (3.12c) (3.12d)

The waveguide width profile, w, determines the effective indices

of the grating sections. If the algorithm is configured to allow effective index dispersion versus the wavelength, the detuning parameter, ∆β, is created and initiated with a Ng Nν array: ∆βi,j

pλj q 2πneff,i λ

π . Λi

j

(3.13)

Should the wavelength dispersion be discarded, the detuning parameter is implemented as a Ng

1 array where the effective index is set to be that of the central wavelength,

neff pλc q, ∆βi,1

2πnλeffpλcq

π . Λi

j

(3.14)

The self- and cross-coupling constants are calculated using the profiles of the waveguide width and the recess depth:

tσiu, κ tκi u.

σ

(3.15a) (3.15b)

Using the transfer matrix method, the system matrix is consequently

Msys

M11 M12

MN MN 1 M2M1 g

g

.

(3.16)

M21 M22

The reflection response of a specific frequency point becomes rj

21 r rj s M . M 22

(3.17)


55

The rest of the frequency points in the frequency axis are set to zero. The algorithm is verified by simulating a set of gratings with familiar responses. Extensive results for TE-like modes are reported in Appendix C. The set of gratings include uniform gratings, linearly chirped gratings, apodized gratings, π-phase-shift gratings, and sampled gratings. The generated results appear in close agreement with the work in [17], assuring the performance and the validity of the algorithm. The algorithm can include dispersion of the effective indices against the wavelength of light. The key result is the shift in the resonance frequency of the reflection response.

3.4

Summary

In this chapter, the direct scattering is in focus especially in terms of simulations. The integrated ridge waveguide and grating are chosen to be in the AlGaAs material. Waveguide modes, effective indices, and the coupling coefficients were simulated in Lumerical MODE Solutions for different waveguide widths, recess depths, and wavelengths, and these data were processed as a database for subsequent calculations. The algorithm for finding the grating responses receives discretized physical grating parameters and evaluates effective indices and coupling constants accordingly from the prepared database. The coupled-mode theory and the transfer matrix method are then used to obtain the grating responses. The capabilities of the algorithm were demonstrated on various gratings and could produce correct responses.

Chapter 4 Retrieval of the Gratings In pulse shaping, one usually asks ‘What grating should be used to achieve the required output waveform?’. The question suggests that knowing the input and output waveforms, the grating response must be worked backward to find the grating structure. The coupled-mode theory does not suggest this direction of calculation. Fortunately, the question is addressed by the inverse scattering theory, which is discussed in detail in Appendix D, [35, 36]. Using the results from the theory, this chapter focuses on the simulation algorithm and verifies the performance of the theory for the pulse shaping purpose. Two steps must be done: firstly abstract parameters including coupling coefficients and relative phases are computed and secondly the matching step is initiated to find the physical parameters, i.e. waveguide widths and recess depths, from the abstract parameters.

4.1

Equations at Work

The aim of the inverse scattering theory is to reconstruct grating physical parameters from a known or desired grating response which could be a response from an experiment or a simulated filtering function. A combination of the layer peeling method, [81], and the Gel’fan-Levithan-Marchenko (GLM) equations, [35], was proposed to solve this problem 56

57

Chapter 4. Retrieval of the Gratings [36].

The currently unknown grating is disintegrated into Ng connected uniform subgratings. The algorithm starts with the reflection response at the front of the grating and then calculates to the last piece. For each piece, the GLM theory is applied. The GLM coupled equations are d c1 dz d c2 dz

jζc1

q pz qc2 ,

(4.1a)

qpzqc1 jζc2,

(4.1b)

where ζ is the z-independent eigenvalue and q pz q is the complex coupling coefficient. Results, derived in detail in Appendix D, show that the propagation equation of the reflection response and the complex coupling coefficient are, [36], pz, ζ qs F¯2,m1 pz, ζ q rm1 p0, ζ qr1 F¯1,m 1 rm1 pz, ζ q ej2ζz pz, ζ q , ¯ r1 F1,m1pz, ζ qs rm1p0, ζ qF¯2,m 1

pz, z q, qm pz q 2K2,m 1

where 0

¤

z

¤

(4.2) (4.3)

∆zm . rm is the reflection response of the m-th subgrating. These

two equations depend on the kernel functions Ki,m1 pz, y q, which have to be calculated iteratively from, [35], K2,m1 pz, y q hm1 pz K1,m1 pz, y q

»z

8

yq

»z

8

pz, sqh ps K1,m m1 1


y q ds,

(4.4a)

y q ds,

(4.4b)

rm1 pζ qejζy dy.

(4.5)

where hpz q is the space-scaled impulse response: hm1 pz q

1 2π

»8

8

The functions F¯1 and F¯2 are defined, [36], F¯ pz, ζ q ejζz

»z

1

F¯2 pz, ζ q ejζz

8 »z

8

K1,m1 pz, sqejζs ds,

(4.6a)

K2,m1 pz, ζ qejζs ds.

(4.6b)

58

Chapter 4. Retrieval of the Gratings

The amplitude and phase of the complex coupling constant q are processed to extract the physical grating parameters.

4.2

GLM Equations to the Coupled-Mode Equations

In order to apply the inverse scattering method to decode the grating response, the GLM equations and the coupled-mode equations must be matched. Note that, as derived in Appendix A, the coupled-mode equations for nonuniform first-order gratings are, [17], d c˜1 dz d c˜1 dz

j

dφ dz

∆β 2

jκpzqc˜1 j

σ pz q

c˜1 jκpz qc˜1 ,

dφ dz

∆β 2

(4.7a)

σpzq

c˜1 ,

(4.7b)

where c˜1 and c˜1 represent a forward- and backward-propagating waves, φ is the chirp function, κpz q is the cross-coupling coefficient, and σ pz q is the self-coupling coefficient. If the uniform subgrating is considered, i.e. dφ{dz

0, the term in the bracket could be

rewritten as ∆β 2

σpzq

2πnλeff,0

π Λ0

2πδneff pz q λ

σ pz q

πδΛ Λ20

∆β2 0 σ˜ pzq.

(4.8)

By defining »z

c¯1 pz q c˜1 pz q exp j c¯1 pz q c˜1 pz q exp

0

j

σ ˜ pz 1 q dz 1 ,

»z 0

(4.9a)

σ ˜ pz 1 q dz 1 ,

(4.9b)

the coupled-mode equations could be expressed with a z-independent eigenvalue d c¯1 dz d c¯1 dz

j ∆β2 0 c¯1 jκpzq exp j2 jκpzq exp

j2

»z 0

»z 0

σ ˜ pz 1 q dz 1 c¯1 ,

(4.10a)

∆β0 σ ˜ pz 1 q dz 1 c¯1 j c¯1 . 2

(4.10b)

By directly comparing Eq. 4.1 and Eq. 4.10, the GLM and the CMT equations could

59

Chapter 4. Retrieval of the Gratings couple to each other by allowing ζ

∆β2 0 ,

q pz q jκpz q exp j2

»z 0

σ ˜ pz 1 q dz 1 .

(4.11) (4.12)

Then, the inverse scattering formalism as summarized in Section 4.1 could now be used to solve the grating structure from a specified reflection response.

4.3

Massaging the Equations

Equations of parameters in the previous section involve integration forms that can be recast so that they look similar to the Fourier transform [36]. This trick would be applied so that the problem lends itself to the fast Fourier algorithm, fft(), in MATLAB. It is helpful to review how to implement the Fourier transform by the discrete Fourier transform, which is discussed in Appendix B. The frequency resolution ∆ν and the time resolution ∆t are related to the number of sampled points by N

1 ∆t∆ν .

(4.13)

The time, t, and frequency, ν, axes are sampled by sets of N points. These axes equally cover both the negative and positive sampled points. The spatial axis is the scaled version of the time axis by using light speed factor. Conclusively, the three axes are t tti u

(4.14a)

tνiu nc ttiu tziu, ν

z

nct

eff

(4.14b) (4.14c)

eff

where i 1, 2, . . . , N is the index of the sampling points. The spatial resolution is then ∆z

c∆t{neff.

The variable ζ, which is the eigenvalue to the coupled-mode equations,

is defined as ζ

∆β2 0 2πnλeff,0 Λπ . 0

(4.15)

60


From its dimension, ζ is interpreted as the Fourier pair of the spatial variable z. It is also a one-to-one function to the wavelength. The set of sampled reflection response points represents responses of different independent variables r rris truN i1

rpλiq rpνiq rpζiq.

(4.16)

This set of points is used to calculate the impulse response hris by hpti q h fftshift(ifft(ifftshift(h)))/dt.

(4.17)

Through a direct mapping of time and space by the light speed, one can interpret hpti q as the space-dependent function: h hris hpti q hpzi q.

(4.18)

Now, consider the integral terms in the recursive equations Eq. 4.4. The integral term appears in the form similar to »z

8

A pz, sqhps

y q ds.

(4.19)

Define a new function as, [36], AD pz, sq

$ ' ' &

A pz, sq ;

' ' %0

;

s ¤ z, z

(4.20)

s.

Eq. 4.19 can be rewritten as »z

8

A pz, sqhps

y q ds

»8

8 »8 8

AD pz, sqhps

y q ds

AD pz, y sqhpsq ds.

(4.21a)

(4.21b)

The last expression resembles the convolution definition. If the Fourier transforms of AD and h are determined, the targeted integral can be found by the inverse Fourier transform

61

Chapter 4. Retrieval of the Gratings of the products of those two Fourier transforms, i.e. »z

8

!

A pz, sqhps

)

y q ds F 1 F tAD pz, y qu F thpy qu .

(4.22)

The next step is to apply this same trick for F¯1 and F¯2 in Eq. 4.6. Their integral terms are

»z

8 $ ' ' &

One can define KD

'

Ki pz, sqejζs ds.

Ki pz, sq ;

' %0

;

(4.23)

s ¤ z, z

(4.24)

s,

which leads to »z

Ki pz, sqe

jζs

8

4.4

ds

»8

8

!

)

KD pz, sqejζs ds F KD pz, sq .

(4.25)

Algorithm of the Inverse Scattering

Now that the working equations are laid down and processed such that the fast Fourier function, fft(), is at disposal, the inverse scattering algorithm could be discussed. Assume that a desired realizable reflection response is known and both the time and frequency axes are implemented and initialized. The eigenvalue to the coupled-mode equations is defined as in Eq. 4.15 ζ

∆β2 0 2πnλeff,0 Λπ .

(4.26)

0

The values of neff,0 and Λ0 are needed and chosen by the best guesses. From the requirement of the coupled-mode theory, the perturbation should be small; therefore, neff,0 should be selected from the effective indices of an unperturbed waveguide. In the previous chapter, the unperturbed waveguide at the waveguide width of 1.4 µm has close effective indices for both TE-like and TM-like modes; hence, neff,0 is chosen to be 3.1062

62


corresponding to the TE-like mode at that waveguide width. From the Bragg wavelength relation, λB

2neff,0Λ0, the grating period could be set to Λ0 250 nm. If the generated

waveguide width deviates from 1.4 µm considerably, the grating period could be adjusted to reduce the deviation due to Eq. 4.26. The unknown grating is broken down to Ng subgratings with equal length of ∆z. Since the GLM solution, summarized in Section 4.1, is solved for each grating, its continuous nature allows several sampling points, say Nsg , in each subgrating. The total number of sampling points are then Ng Nsg . Choosing these numbers are not arbitrary as the total number of grating points must not exceed the number of available data points, which is equal to the number of frequency points of the measured grating spectrum. Without loss of generality, each subgrating piece could be short enough and contain only one spatial point, Nsg

1.

It is still necessary that the subgrating length be much longer than the

grating period. Under these conditions, it is convenient to let the subgrating length be an integral multiple of the grating period. The inverse scattering algorithm determines the complex coupling coefficient q of each subgrating and stores the value in a 1 Ng array q. Note that Ng currently represents the total number of the spatial points along the grating. For each subgrating, the reflection response at the front is calculated from Eq. 4.2. The impulse response is calculated by using the discrete Fourier transform function; h = fftshift(ifft(ifftshift(r)))/zRes; where h

hm, r rm, and m is the index of the subgrating.

(4.27)

The complex coupling

coefficients are shown to be q pz q 2K2 pz, z q where 0 ¤ z

¤ ∆z,

(4.28)

as in Eq. 4.3, [35]. The algorithm tries to find q p∆z q as a presentative of the subgrating. Hence, the value of K2 pz the term hm pz

∆z, y

∆z q must be determined from Eq. 4.5. Firstly,

y q is interpreted as a function of hm py q but with a shift of z to the

63


left (right) if z is positive (negative). Since in the discrete Fourier transform theorem hm py q is periodic, hm pz

y q could be determined by appropriately shifting and cycling

the values of hm pz q. In MATLAB, this step takes the form of h2 = [h1(shift+1:end), h1(1:shift)];, where h1 and h2 are hm py q and hm pz

(4.29)

y q and the shift is the number of the sample

point shifted. For the first iteration loop, the iteration equations assume the following computational sequential order, [36]: K2 pz, y q hpz K1 pz, y q

»z

8

y q,

(4.30a)

K2 pz, sqhps

y q ds.

(4.30b)

The later iteration loops take the original forms K2 pz, y q hpz K1 pz, y q

»z

8

yq

»z

8

K1 pz, sqhps

K2 pz, sqhps

y q ds,

y q ds.

(4.31a)

(4.31b)

In both cases, one can follow the procedure previously discussed as in Eq. 4.20 and Eq. 4.21. In MATLAB, the transformation appears as KD = conj([K(1:zri); zeros(1,Ng-zri)]);

(4.32)

where K could be either K1 or K2 , and KD is the corresponding KD . The index zri is the index that corresponds to the space point z

∆z. The following lines then calculate the

integral a = fftshift(fft(ifftshift(KD)))*zRes; b = fftshift(fft(ifftshift(h0)))*zRes; c = a.*b; d = fftshift(ifft(ifftshift(c)))/zRes;

64


in which d is the discretized vector representing the integral. Therefore, the kernel functions K1 pz, sq and K2 pz, sq can be calculated iteratively. After they are determined, the complex coupling coefficient at the point z

∆z is (4.33)

q(i1) = 2conj(K2(zri));,

where i1 is the subgrating index. Before moving to the next subgrating, the propagating reflection response is calculated from Eq. 4.2. Another step must be applied, however, in order to prevent reflection amplitude to exceed unity especially at frequencies far from the central frequency. A windowing function is multiplied to the calculated reflection. The windowing function, f pxq, is defined as

f pxq

$ ' ' ' 0.5 ' ' ' ' ' ' ' &

0.5 cos

π xd

px

x1{2 xd 2

q

1

' ' ' ' '0.5 ' ' ' ' ' %0

0.5 cos

π xd

px

x1{2 xd 2

q

;

x { 2 x x x { 2x

;

x { 2x x x { 2x

;

x1{2 xd 2

1 2

1 2

d

1 2

d

x

1 2

x1{2 xd , 2

d

d

,

, (4.34)

; otherwise,

where x1{2 is the FWHM duration and xd is the decay time, which is set to be half a period of the cosine function. Examples of the shapes of the windowing functions are shown in Fig. 4.1. Note that the function could be used in both time and frequency domain by changing x, x1{2 , and xd to appropriate variables. In this algorithm the windowing function in use is f px ν q that has x1{2

25 THz and xd 5 THz.

At the end of the algorithm, the complex coupling constant representing the subgratings is calculated and it is ready to be matched to physical parameters, i.e. the waveguide width and the recess depth.

4.5

Matching to Physical Parameters

Even though the solution to the GLM equation, i.e. q pz q, is unique to a given reflection response, the matching to physical parameters can give different results depending on

65

Chapter 4. Retrieval of the Gratings 1 f1(x)

f(x)

0.8

f2(x)

0.6 0.4 0.2 0 −4 −3 −2 −1

0 x

1

2

Figure (4.1): Windowing function. f1 pxq corresponds to x1{2 for x1{2

xd 3.

3

4

3 and xd 1 wherease f2 pxq is plotted

the matching criterion. The matching algorithm is discussed in this section. The complex coupling coefficient, q

|q|ejϕ, is shown to provide |q| |κ|, ∆ϕ 2˜ σ ∆z, 2πδneff πδΛ σ ˜σ Λ2 , λ

(4.35a) (4.35b) (4.35c)

0

where κ and σ are the cross- and self-coupling constants, respectively. It can be seen from the above equations that the matching algorithm requires the initial guesses of Λ0 and neff,0 , which depends on the waveguide width. The algorithm assumes that the first subgrating has the width of a initially specified value, w0 , which corresponds to the effective index neff,0 at the central frequency λ0 . Since the cross-coupling constant, κpw, rdq, depends on the waveguide width and the recess depth, the recess depth of the first grating can be inferred by the magnitude of the complex coupling coefficient,

|q| |κ|. For successive subgratings, the initial recess depth is calculated again from |q| at the width of w0 , and it is used to calculate the self-coupling coefficient σ. The value of δneff is determined from ∆ϕ, and dictates a new value of the waveguide width. Then, the loop starts to recalculate the recess depth and the waveguide width for a designated number of iterations to reach convergence.

66


4.6

Verification of the Inverse Scattering Algorithm

To test the theory and the algorithm, a test grating is defined and its (test) response is determined using the direct scattering. The inverse scattering (IS) algorithm receives the test response, calculates the complex coupling coefficient, and yields waveguide width and recess depth profiles. From the generated physical profiles, the grating response is computed and compared to the test response. In Appendix E, a variety of test gratings are used to validate the algorithm and the results are reported therein. Those gratings include uniform gratings, linearly width-chirped gratings, and Gaussian-apodized gratings. In this section, one type of gratings, i.e. an apodized and chirped grating, is considered for verification. The test grating was chirped by varying the waveguide width linearly and also Gaussian-apodized by a suitable recess depth profile. The parameters for the inverse scattering algorithm were set as Ng and w0

1.4 µm.

400, Λ0 250 nm, ∆z 4Λ0 1 µm,

The generated complex coupling coefficient is then determined and

plotted in Fig. E.10. Its magnitude traces the magnitude of the initial cross-coupling coefficient with great correspondence. The relative phase, as shown in Fig. E.10b, exhibits a combination of linear and Gaussian features. The physical profiles are matched from the complex coupling coefficient. The waveguide width profile corresponds well with the linear increase of the starting grating, as shown in Fig. E.11a. The recess depth, as displayed in Fig. E.11b, appears similar to that of the starting grating. Selecting the subgratings within the significance region, i.e. between z

50 µm and z 250 µm, the

response of the generated grating is calculated and plotted in Fig. E.12. The above results and ones reported in Appendix E show good agreement between the test and the generated coupling coefficients. With the current algorithm, in the region where the coupling coefficient is close to zero, fluctuations appear in the relative phase profile. This situation might be related to the fact that the phase of absolute zero is indefinite and physically has no meaning. This fluctuation manifests in the waveguide width and the recess depth profiles. However, when the grating response is calculated,

67


3

x 10

4

0.4 Simulated Target grating

0.2

2 ∆ψ

−1

|q| (m )

2.5

1.5

0

1 −0.2 0.5 0 0

−0.4 0

50 100 150 200 250 300 350 400 z (µm)

(a) |q |

50 100 150 200 250 300 350 400 z (µm)

(b) ∆ϕ

Figure (4.2): The complex coupling coefficient, calculated from the inverse scattering algorithm, for a response of a Gaussian-apodized and chirped grating.

1.8 recess depth (nm)

width (µm)

1.7 1.6 1.5 1.4 1.3 1.2 1.1 0

Simulated Target grating 50 100 150 200 250 300 350 400 z (µm)

(a) Waveguide width

90 80 70 60 50 40 30 20 10 0 0

Simulated Target grating

50 100 150 200 250 300 350 400 z (µm)

(b) Recess depth

Figure (4.3): Matched waveguide width and recess depth profiles.

the contribution from the fluctuations is insignificant or could be rendered mute by neglecting it or overriding with a constant waveguide width and a zero recess depth. In terms of grating responses, all of the generated gratings show good agreement with the test responses.

68


10

1 Simulated Target

0.9 0.8 0.7

5 τ (ps)

|r|

0.6 0.5 0.4

0

0.3 0.2

Simulated Target

0.1 0 1.53 1.535 1.54 1.545 1.55 1.555 1.56 1.565 1.57

λ (µm)

−5 1.53

(a) Amplitude response

1.54

1.55 λ (µm)

1.56

1.57

(b) Time delay response

Figure (4.4): Responses of a grating generated by the inverse scattering algorithm compared with the targeted responses from a Gaussian-apodized and chirped grating.

4.7

Summary

In this chapter, the inverse scattering formalism was adjusted so that it lends itself to numerical simulations. The retrieval of physical parameters of the grating is done by the inverse scattering algorithm and the matching algorithm. Capabilities of the implemented IS algorithm were shown a test apodized and chirped grating. The results display good agreement with the input response and the test grating target. Therefore, it could be concluded that the inverse scattering algorithm is capable of generating the waveguide width and the recess depth profiles for an integrated waveguide grating that would provide responses close to the targeted ones.

Chapter 5 Pulse Shaping Simulations In the previous chapters, the direct scattering and inverse scattering algorithms are discussed. In this chapter, pulse shaping is studied by using the aforementioned algorithms to generate structures of an integrated grating that will provide suitable reflection responses.

5.1

Deriving the Targeted Grating Response

Within a linear system, the grating provides a required filtering function that is related the input to the output and written in a mathematical equation as Eout pω q rpω qEin pω q,

(5.1)

where Ein pω q, Eout pω q, and rpω q are the input pulse, output pulse, and reflection response in frequency domain. Assume that both the input and the required output waveforms are known, both Ein and Eout are then specified consequently. From the above expression, the reflection response is calculated from r pω q However, it is required that |rpω q|

Eout pω q . Ein pω q

¤ 1 since the device is linear and passive.

(5.2) In actual

implementation, the reflection amplitude approaches infinity at the frequency where Ein 69

70

Chapter 5. Pulse Shaping Simulations

is near zero, especially far from the central frequency. Therefore, a windowing function, f pω q, is required to limit the bandwidth of the reflection response within a meaningful region. Assume that both the input and output pulses oscillate at a central frequency ωc with field envelopes Ain ptq and Aout ptq, respectively. The electric fields in a time domain are in the form Ein ptq

Ainptqejω t and Eoutptq Aoutptqejω t. Consequently, the fields in a frequency domain are Ein pω q Ain pω ωc q and Eout pω q Aout pω ωc q. It is more convenient to define a baseband frequency ω 1 ω ωc and rewrite the reflection response c

as rpω 1 q where 0

c

Aout pω 1 q 1 f pω 1 q αejω τd , Ain pω 1 q

(5.3)

¤ α ¤ 1 is the scaling factor and the last exponential term introduces a time

delay to induce causality [18]. Assume that the input is a transform-limited Gaussian pulse with a field (FWHM) pulse duration of τ

150 fs, which is shown in Fig. 5.1.

The expression for the pulse

envelope is then ein ptq e4 ln 2 τ 2 . t2

(5.4)

The windowing function, f pxq, is defined previously in Eq. 4.34. In most of the simulations that follow, the spectral windowing function is set to be f px x1{2

ν 1q that has

5 THz and xd 5 THz unless stated explicitly otherwise.

The last remark involves the normalization of the power spectra of the input and targeted pulses. Since the shapes of the pulses are concerned, the pulses are defined numerically independently in terms of amplitudes. For simplicity, most pulse definitions let the maximum electric field amplitude to be unity, so as for the Gaussian input pulse above. Therefore, it is necessary to normalize the power spectrum of each pulse shape such that the maximum value is unity, usually occurring at the central frequency. In working with Eq. 5.3 subsequently in this chapter A’s are treated as being normalized

71

Ain(t) (a.u.)

Chapter 5. Pulse Shaping Simulations 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −0.4 −0.3 −0.2 −0.1

0 0.1 0.2 0.3 0.4 t (ps)

(a) Temporal envelope −13

1.75

x 10

1.25

∠Ain(ν−νc)

|Ain(ν−νc)|

1.5 1 0.75 0.5 0.25 0 −10 −8 −6 −4 −2 0 2 4 ν−νc (THz)

6

8 10

0.4 0.3 0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −10 −8 −6 −4 −2 0 2 4 ν−νc (THz)

(b) |Ain pν 1 q|

(c)

6

8 10

=Ain pν q 1

Figure (5.1): A temporal envelope, a power spectrum, and a phase spectrum of the input pulse featuring a Gaussian shape with the duration of 150 fs.

already before calculating the reflection response.

5.2

Flat-top Pulse Shaping

In many applications, flat-top or rectangular pulses are useful such as in nonlinear switching in which the flat-top pulses could be used as a switching window [5]. The pulse

72

Chapter 5. Pulse Shaping Simulations envelope is defined as eout ptq

$ ' ' &

1 ; |t| ¤

' ' %0

TFWHM 2

(5.5)

; otherwise,

where TFWHM is the duration of the pulse and is equal to the FWHM duration for perfect rectangular pulses. The Fourier transform of a rectangular pulse is known to be a sincfunction involving infinite amount of frequencies. In defining the appropriate grating response, the windowing function must be used. Assume that a 2-ps rectangular pulse with the spectrum of Fig. 5.2 is required. Also consider α

(IS) algorithm are Λ0 rdres

5 nm.

2 ps in Eq. 5.3. The parameters for the inverse scattering 250 nm, ∆z 12Λ0 3 µm, Ng 400, w0 1.4 µm, and

1 and τd

The last parameter represents the fabrication resolution for introducing

perturbation. The IS algorithm iteration loop number is 20. The results are given in Fig. 5.3. The complex coupling coefficient sports a front increasing part and a decaying tail. The increasing part could be explained as the main reflection section where most of the light is reflected. As light propagates and reflects, the amount of energy carrying by light reduces; in order to produce a flat-top pulse with a uniform electric field amplitude, the grating must possesses larger coupling coefficients in the later sections of the front body, hence the increasing trend. The section of tailing coupling coefficient magnitudes also contributes the power of reflection, and also plays a role in canceling the electric field outside the rectangular duration of the pulse. The waveguide width and the recess depth profiles were matched from the complex coupling coefficients and suggest that the grating should start from the 16th subgrating or at z

48 µm until the piece around z 1000 µm if the recess depth resolution is 5

nm. Taking all the subgratings in this region, the reflection response of the generated grating was calculated and compared with the targeted response, as shown in Fig. 5.4a and Fig. 5.4b where the legend simululated and target refers to that of the generated

73


−12

2.5

x 10

4

∠Aout(ν−νc)

|Aout(ν−νc)|

2 1.5 1

2 0 −2

0.5 0 −10 −8 −6 −4 −2 0 2 4 ν−νc (THz)

6

8 10

−4 −10 −8 −6 −4 −2

(a) |Aout |

(b)

0 2 ν−νc

4

6

8 10

=Aout

Figure (5.2): Fourier transform of a 2-ps flat-top pulse.

grating and the targeted grating, respectively. The amplitude responses appear similar to each other except the shrink in the frequency axis. The time delay response of the generated grating have an average close to that of the targeted response. Assuming a Gaussian input pulse as described earlier, which is centered at t 0 with a maximum magnitude of unity, the output pulse in the temporal domain was calculated by taking a Fourier transform of the output pulse spectrum. The magnitude of the electric field of the temporal output waveform when all the subgratings were included was shown in Fig. 5.4c. Both the simulated and targeted outputs started at about t

1 ps.

This

feature is reasonable since the targeted response involves a time shift of 2 ps. Since the flat-top pulse shape is defined such that the front edge starts at time t TFWHM {2, by shifting with 2 ps, the 2-ps pulse should start at t 1 ps, as observed in the simulation. This time shift corresponds to the region of |q |

0, i.e.

within z from 0 to 48 µm, in

Fig. 5.3a. To see the effect of taking into account different number of subgratings, different

48 µm) piece but choosing the 341st (z 1, 023 µm), 181st (z 543 µm), 83rd (z

gratings were simulated by similarly starting from the 16th (z four different ending pieces:

74

Chapter 5. Pulse Shaping Simulations 249 µm), and 51st (z

153 µm), termed as g1, g2, g3, and g4 samples respectively, whose

output waveforms are displayed in Fig. 5.5. It is obvious that all gratings provide similar rectangular waveforms, however, with different tailing subpulses. The longer the set of the subgratings included in the simulation, the smaller the subpulse is. This is previously explained that the the later part of the complex coupling coefficient is responsible for canceling electric fields in the subpulse region. Hence, the longer set of subgratings performs better in managing the magnitude and phase of the frequency components

4

2

x 10

3 2.5 2 ∆ψ

|q| (m−1)

1.5 1

1.5 1 0.5

0.5

0

0 0

200

400

−0.5 0

600 800 1000 1200 z (µm)

200

400

(a) |q pz q|

recess depth (nm)

80

1.45 width (µm)

1000 1200

(b) ∆ϕ

1.5

1.4 1.35 1.3 0

600 800 z (µm)

200

400

600 800 1000 1200 z (µm)

60 40 20 0 0

(c) Waveguide width

200

400

600 800 1000 1200 z (µm)

(d) Recess depth

Figure (5.3): Inverse scattering algorithm results for a grating to generate a 2-ps flat-top pulse from a 150-fs Gaussian pulse.

75


such that the they interfere destructively. This fact leads to a compromise between performance and footprint of the grating. In other words, a longer grating is needed to produce an exact flat-top pulse with minimal stray subpulses. In the frequency domain,

1 Simulated 0.9 Target 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −5 −4 −3 −2 −1 0 1 2 3 4 5 ν−νc (THz)

20

0

−10 −5 −4 −3 −2 −1 0 1 2 ν−νc (THz)


3

4

5


Simulated Target Input*

0.08 field amplitude

Simulated Target

10 τ (ps)

|r|

a longer grating will provide a filtering response closer to the targeted response.

0.06 0.04 0.02 0 −2

0

2

4

6

8 10 12 14 16 18 20 22 24 26 28 30 t (ps) (c) Time domain

Figure (5.4): An amplitude (a) and time delay (b) responses from a generated grating with a targeted 2-ps flat-top pulse. In (c), electric field amplitudes of the output pulses from a generated grating (blue solid) and the targeted waveform (black dash). The legend simulated and target refers to that of the generated grating and the targeted grating. The scaled input is shown in red.

The results for targeted flat-top waveforms with pulse durations of 0.5, 1.0, and 2.0

76


ps are shown in Fig. 5.6 when the parameters of the inverse scattering algorithm remain similar to the previous case. They were generated from gratings about 200-350 micron long. All of the waveforms have rise and fall times of approximately 0.2 ps. 0.08

g

field amplitude

1

g2

0.06

g

3

g4

0.04 0.02 0 −2

0

2

4

6

8 10 12 14 16 18 20 22 24 26 28 30 t (ps)

Figure (5.5): Electric field magnitudes of output waveforms corresponding to generated gratings with different sets of subgrating involved.

field amplitude

0.3 τ=2.0 ps τ=1.0 ps τ=0.5 ps

0.2

0.1

0 −2

0

2

4

6

8 10 12 14 16 18 20 22 24 26 28 30 t (ps)

Figure (5.6): Output waveforms from generated gratings aiming to produce flat-top pulses with durations of 0.5, 1, and 2 picoseconds.

In actual fabrication, the profiles of the waveguide width and the recess depth could be different from the specified profiles; some deviations will exist. Since the deviations could occur in a random manner, it could be accommodated in the simulation by using a function rand() in MATLAB. For each subgrating, the deviations are assumed to be

77


about the fabrication critical dimension, i.e. 5 nm. Adding various random deviation profiles to the generated grating for 2-ps flat-top pulses, the responses are displayed in Fig. 5.7. Within the flat-top duration, the maximum difference of electric field magnitudes of the waveforms is in about 0.0052 (arbitrary unit used to plot the field amplitude).

1 Simulated 0.9 Target 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −5 −4 −3 −2 −1 0 1 2 3 4 5 ν−ν0 (THz)

40

Simulated Target

30 20 τ (ps)

|r|

Outside this duration, the maximum difference is about 0.009.

10 0 −10 −20 −5 −4 −3 −2 −1 0 1 2 ν−νc (THz)


3

4

5


field amplitude

0.08 0.06 0.04 0.02 0 −2

0

2

4

6

8 10 12 14 16 18 20 22 24 26 28 30 t (ps) (c) Output waveform

Figure (5.7): Responses and performance of the generated grating when random deviations are introduced to the waveguide width and the recess depth profiles.

The device that is simulated in this work has the capability of generating flat-top pulses with the lower limit of pulse durations of about 500 ps, which is a good candidate

78


for the ones investigated in [5, 82]. The rise and fall times of the device is in about 200 ps, close to the pulse duration of the input pulse. This result is better than the rise/fall times 700 ps as reported in [5].

5.3

Triangular Pulse Shaping

A triangular pulse envelope could be expressed as

eout ptq

$ ' ' ' 1 ' ' ' &

1 ' ' ' ' ' ' %0

t TFWHM

T

t

FWHM

;

TFWHM ¤ t ¤ 0

; 0 ¤ t ¤ TFWHM

(5.6)

; otherwise.

Let consider a transform-limited triangular pulse with the FWHM duration TFWHM of 2 picoseconds with α

1 and τd 2 ps.

All IS parameters were initialized as in the

previous section except that the grating period is now Λ0

249.6 nm.

The algorithm

yielded the complex coupling coefficient presented in Fig. 5.8b, which was then matched to the waveguide width and the recess depth. These results exhibit the main reflection body and the grating tail responsible for subpulses in the time domain. If the calculated grating was taken up to the point at z

600 µm, whose respective

recess depth reaches 10 nm, the response of this grating is shown in Fig. 5.10 whereas the output pulse is plotted in Fig. 5.11. The output envelope is close to that of the targeted waveform except the presence of the subpulse due to the definiteness of the implemented grating. This figure also presents the output waveform when random deviations from the suggested grating profile were added. The maximum difference in the electric field magnitude is about 0.007.

79


−12

x 10

12000 10000

1.5 |q| (m−1)

|Aout(ν−νc)| (a.u.)

2

1

8000 6000 4000

0.5

2000 0 −4 −3 −2 −1 0 1 ν−νc (THz)

2

3

0 0

4

200 400 600 800 1000 1200 z (µm)

(a) |Aout p∆ν q|

(b) |q pz q|

Figure (5.8): (a) Power spectrum of the triangular pulse envelope with the FWHM duration of 2 picoseconds and (b) The magnitude of the complex coupling coefficient calculated from the inverse scattering algorithm.

70


60 width (µm)

1.44 1.42 1.4

50 40 30 20 10

1.38 0

200

400

600 800 1000 1200 z (µm)

(a) Waveguide width

0 0

200

400

600 800 1000 1200 z (µm)

(b) Recess depth

Figure (5.9): Matched waveguide width and the recess depth profiles.

80

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −4 −3 −2 −1 0 1 ν−νc (THz)

20

Simulated Target

Simulated Target

15 τ (ps)

|r|


10 5

2

3

0 −4

4

−2


2

4


Figure (5.10): Grating response taking upto the the point of z

600 µm of the IS-generated grating.

0.08

field amplitude

0 ν−νc (THz)

Simulated Simulated* Input* Target

0.06 0.04 0.02 0 −2

0

2

4

6

8 10 t (ps)

12

14

16

18

20

Figure (5.11): Electric field amplitudes of the output pulses from a generated grating involved upto z

600 µm. The blue solid curve represents the output whereas the black dashed curve is the targeted

output waveform. The green dot-dash curve represents the output waveform from the grating with add random deviations.

81


5.4

One-to-Many Pulse Shaping

Previously pulse shaping is assumed to be one-to-one; however, in this section, this assumption is relaxed. First of all, a periodic property of the discrete Fourier transform, which delineates the fast Fourier transform algorithm, should be recapitulated. A finite information in a time domain is assumed fundamentally to be periodic over a period of N data points, where N is the number of data points representing information. The corresponding discrete Fourier transform is also periodic in N . If the spacings between points are ∆t and ∆ν in the time and frequency domains respectively, the periodicity in time and frequency becomes

1{∆t∆ν. Assuming that the ∆t and ∆ν are set and the time axis is defined from N ∆t{2 to N ∆t{2 with an interval

correspondingly N ∆t and N ∆ν, with a relationship N

∆t, the information outside this time window is not captured and loses its meaning. In particular, if ∆t

4 fs and ∆ν 2.50 GHz, N 100, 104 and T N ∆t 400.42 ps.

Any information will be conceived as the information of period 400.42 ps represented by the features that occur in the time window. The input pulses from a laser system has a pulse repetition rate of R corresponding to a time separation between two adjacent pulses of TR

1{R.

If a pulse separation of

an input pulse train is greater than the Fourier data period, then the pulse train can be regarded as a single pulse. Taking the previous value of R

2.50 GHz, a laser system

operating with a pulse repetition rate reasonably below 2.50 GHz could considered as if it provides a single pulse to the grating generated based on assuming ∆t and ∆ν. This limit can be adjusted by changing the data separations ∆t and ∆ν. Since the pulse repetition rate limit is quite high compared to real laser systems, in the following discussion pulses from a pulse train are treated individually. Consider a targeted output waveform consisting of two transform-limited 2-ps rectangular pulses separated center-to-center by 10 ps. It can be seen from actual simulations that the inverse scattering algorithm now faces some difficulty if the maximum reflection

82


amplitude was one. Hence, it is appropriate to subvert the problem by allowing α 0.95, and from the definition of the target the time delay is set to τd for the IS algorithm are Λ0 rdres

5 nm.

7 ps.

The parameters

249.6 nm, ∆z 12Λ0, Ng 1, 000, w0 1.4 µm, and

The IS algorithm iteration loop number is 10. The resulted waveguide

width and recess depth profiles in fact show severe fluctuations in some insignificant coupling regions. This problem is suppressed by manually resetting the waveguide width to w0 and the recess depth to zero. The generated grating profiles are displayed in Fig. 5.12 and its amplitude response taking into account up to the grating point z

750 µm is

shown in Fig. 5.13a compared to the ideal response in Fig. 5.13b. 10000

|q| (m−1)

8000 6000 4000 2000 0 0

1000

2000

3000

z (µm)

1.44

50

1.43

40

recess depth (nm)

width (µm)

(a) |q pz q|

1.42 1.41 1.4 1.39 1.38 0

250

500

750 1000 1250 1500 z (µm)

(b) Waveguide width

30 20 10 0 0

250

500

750 1000 1250 1500 z (µm)

(c) Recess depth

Figure (5.12): Simulated results including the waveguide width, recess depth, and electric field profiles.

83

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −2 −1.5 −1 −0.5 0 0.5 ν−ν0 (THz)

|r|

|r|


1

1.5

2

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −2 −1.5 −1 −0.5 0 0.5 ν−ν0 (THz)

(a) Amplitude response of the suggested grating

1

1.5

2

(b) Targeted amplitude response

Figure (5.13): Amplitude responses to achieve an output waveform containing two 2-ps flat-top pulses with 10-ps center-to-center separation. (a) The response from the suggested grating. (b) The ideal response.

The grating clearly consists of two main subgratings which are responsible for the reflection of the two rectangular pulses. Note that the later main subgrating has higher coupling coefficients to yield higher reflection percentage that compensates for the reduced energy after the first main subgrating. The output waveforms are shown in Fig. 5.14, including the one that include variations in the grating profiles. The separation between the two generated flat-top pulses is about 12 ps, more than the target of 10 ps. This might be the result of assuming no wavelength-dependent refractive index for the spacetime mapping relation, t also exist near t

znav{c, in the inverse scattering algorithm.

25 ps.

The subpulses

With the deviations in the grating profiles, the maximum

deviation in the electric field magnitude is about 0.007. It is interesting that the deviation in the electric field magnitude after the time t

15 ps is small compared to the

other time duration. This suggests that the electric field, including the duration of the subpulse, could originate from the abrupt index change between the grating part and the unperturbed waveguide as can be seen from Fig. 5.12b. To comply with causality, the impulse response has to be zero when the reference

84


time is negative, which occurs about half of the time axis vector in the numerical implementation. Therefore, only the impulse response within the positive time frame up to the end of the time array is used. If the actual impulse response is longer than this time window, it will be misinterpreted by the algorithm. Also, the magnitude of the complex coupling constant is an order of magnitude less than a single flat-top output in Section 5.2. This fact reflects that the energy of the input pulse has to be divided into two output pulses and this affects the strength of the grating coupling coefficients.

0.05

Simulated Target Input*

field amplitude

0.04 0.03 0.02 0.01 0 −5

0

5

10

15

20 25 t (ps)

30

35

40

45

50

(a) An output waveform from the generated grating in a solid blue curve compared to a targeted waveform shown in a black dashed curve.

0.05

field amplitude

0.04 0.03 0.02 0.01 0 −5

0

5

10

15

20 25 t (ps)

30

35

40

45

50

(b) An output waveform from the generated grating with random deviations in its profiles. Figure (5.14): Output waveforms for two 2-ps flat-top pulses with a separation of 10 picoseconds.


85

It should be expected that if many output pulses are desired from a single pulse, the magnitude of the coupling coefficient of each grating section will become lower until it is not realizable by the fabrication technology, which dictates, in this context, the matching algorithm.

5.5

Summary

In this chapter, both the direct and inverse scattering algorithms are employed to perform pulse shaping based on reflection responses for integrated waveguide gratings. The targeted pulse is defined and the appropriate reflection response is calculated. The grating to complete the task is generated and simulated to find the output waveform. In particular, flap-top and triangular waveforms were considered and one-to-one and oneto-many pulse shaping were simulated. The devised algorithm is capable of deciphering the reflection response and shows important features of the required grating that results in the main lobes of the targeted pulses, especially in the pulse train generation. The subpulses exist outside the ideal target but could be eliminated by including more generated subgratings with compromise to the total length. The previously discussed results prove that the algorithms could be used to analyze and generate the grating and will definitely provide a good starting point for further integrated waveguide grating design. In particular, flat-top pulses with the pulse duration down to 500 fs could be generated with 200-fs rise/fall times, from a grating as short as 250 microns.

Chapter 6 Conclusions and Future Direction

6.1

Aspects, Approaches, and Results of This Work

In this work, arbitrary pulse shaping in integrated optics was studied with a focus on integrated Bragg gratings featuring a sidewall-etching geometry. The waveguides and gratings were chosen to be on an AlGaAs platform due to its refractive index adjustable by changing the aluminum concentration. The sidewall-etching geometry provides simple controls over the apodization and the chirp profiles, which are functions of the waveguide widths and the recess depths. In fact, the grating period could provide another degree of freedom; however, it was kept at a fixed value for any grating design in this work. The grating design for arbitrary pulse shaping is carried out mainly by the inverse scattering (IS) based on the Gel’fan-Levithan-Marchenko theory and the layer peeling method. The reflection response derived from a targeted output waveform and a 150-fs Gaussian pulse is put to the devised IS algorithm, and the complex coupling coefficient is generated, which later will be matched to the waveguide widths and the recess depths thereby yielding the suggested grating. The next part is to calculate the spectral response of the generated grating to compare with the targeted response. The computation of the grating response is based on the coupled-mode theory and the transfer matrix method, 86

Chapter 6. Conclusions and Future Direction

87

termed direct scattering (DS). Both DS and IS algorithms were tested against known grating structures and responses. For the DS algorithm, the results showed that it is capable of handling uniform gratings, non-uniform gratings, and sampled gratings. The IS algorithm was shown to generate waveguide widths and recess depths for an integrated sidewall Bragg grating that can provide a desired reflection response. Numerical simulations for grating designs to achieve flat-top pulses, with pulse durations of 0.5, 1.0, and 2.0 picoseconds, and triangular pulses were conducted; the complex coupling coefficients, the waveguide widths, the recess depths, and the responses were reported. The complex coupling coefficients are composed of two regions: the main reflection part and the tailing part. The resulting output waveforms agreed very well with the targeted waveforms, especially in the main pulse duration. The existence of subpulses is contributed to the truncation of the tailing complex coupling coefficients, revealing a compromise between the performance and the device footprint, and also the abrupt change in the waveguide width. The truncation worsens in the generation of multiple pulses from a single input pulse in that the complex coupling coefficient profile features multiple grating sections, even more than the number of the targeted pulses itself. Additionally, the more grating sections, the lower the coupling coefficients become, and they will eventually reach the limit governed by the fabricating critical dimension. Therefore, the current work can only handle a few pulses in the output signal. When random deviations are added to the grating profiles, the maximum deviation of the electric field magnitude of the output waveform is in the order 0.01in the unit normalized by the peak of the input electric field. When compared to the devices for flat-top pulse shaping reported in [5, 82], the proposed integrated sidewall grating could also produce flat-top pulses in picosecond and subpicosecond scales. The rise and fall times of about 200 ps were achieved in this work and could be superior to the previous work. Hence, this work has provided evidences


88

supporting the potentials of the integrated sidewall gratings for pulse shaping purposes.

6.2

Future Directions

The theoretical framework and numerical modeling developed through the course of this work could serve as a starting point to the design of the pulse shaping grating in the integrated regime. The obvious next step to take is to fabricate the grating as the algorithm suggests the physical form and measure the spectral response and shaping performance. This step ultimately validates the algorithms and the theories behind it. In terms of theory, a modification to the coupled-mode theory to accommodate other leaky modes and absorption or gain could be done as well to account for strong coupling regimes. This work also neglects the dependence of the effective index on the recess depth of the grating structure, which is not strictly valid especially in strong perturbations and nano-waveguides. It might also be interesting to see the effect of different perturbation periodicities between the left and right sidewall etchings, which is beyond the scope of the current algorithm. For the inverse scattering algorithm, the improvement could be in the matching method. In this work, the grating period is assumed to be constant. However, the algorithm that allows variations in the grating period will achieve one more degree of freedom in the design. The decoupling between the width, the recess depth, and the grating period (if included) should be more effective to eliminate error in the result. More accurate results might be speculated if the dispersion is allowed in the algorithm as well. A better method should be devised to address multiple output generation. Additionally, the inverse scattering should also extend to include targeted response in transmission, which could be useful in situations where transmission responses of a grating is at work such as optical waveform differentiation and integration. On the other hand, both the described direct and inverse scattering theories are within the linear regime. Extending the theories to encompass nonlinearity will enhance


89

accuracy and yield more functionalities. In doing so, the two theories must be recast in other forms, thereby a new sets of algorithms for numerical modeling.

Appendix A

Coupled-Mode Theory (CMT)

In linear pulse shaping, a pulse shaping device is mathematically represented by its response or filtering function. In the case of interest, the device is the integrated waveguide grating. The response of the grating could be calculated by many ways between numerical and analytical. For instance, FDTD numerical technique proves to be a very powerful numerical modeling tool. However, the technique requires a lot computing power and solving time with a 3-dimensional problem. Neither does it provide explanatory insight of the phenomena. On the other hand, the coupled-mode theory (CMT) is an analytical technique commonly used to model not-too-strong gratings. One of the benefits of CMT is that it provides physics behind the observed phenomena.

This chapter discuses first an integrated waveguide and its modes. Then, the formalism of the coupled-mode theory is explained and followed by special cases including first-order gratings and uniform gratings, whose solutions are solved analytically. The theory is developed based on [33, 74] In order to analyze non-uniform gratings with the solutions of uniform gratings, the combination of CMT with the transfer matrix method (TTM) is introduced [34]. 90

91

Appendix A. Coupled-Mode Theory (CMT)

A.1

Integrated Waveguides

A grating is defined by a periodic modulation of refractive index along a waveguide. In the case of a fiber grating, this periodic modulation could be created by UV illumination to the photosensitive core of the fiber, whose refractive index changes when exposed to UV. In an integrated waveguide, the grating is commonly achieved by periodically etching along the waveguide. A surface grating is defined by etching the top part of the waveguide structure. A sidewall grating is made by etching the sides of the waveguide, which usually superimpose with the core region. The cross-sectional refractive index profile of a waveguide determines how light propagates as modes. Assuming that a waveguide is made of isotropic, non-magnetic dielectrics and neglecting possible loss or gain, the waveguide could be represented by its permittivity px, y, z q px, y q. Using Maxwell’s equations, the wave equation is yielded, ∇2 Epr, tq µw px, y q

B2 Epr, tq, Bt2

(A.1)

where Epr, tq Epx, y, z, tq and w represents the unperturbed waveguide. Assume a monochromatic wave in the phasor form Epr, tq epx, y qejωtjβz

(A.2)

and put Eq. A.2 into Eq. A.1 resulting in, [33], ∇2K epx, y q

ω 2 µw px, y q β 2 epx, y q 0

(A.3)

Eq. A.3 is in the form of eigenvalue-eigenvector problem and it determines the electric field profiles of the corresponding modes epx, y q as well as their corresponding propagation constants β. The modes could be calculated analytically in one-dimensional or slab waveguides. However, the modes in 2D waveguides could not be expressed in close forms and are found numerically by various techniques or commercial mode solvers, such as Lumerical MODE Solutions or COMSOL. For a particular waveguide, it is possible to


92

have many guided modes. However, for certain applications, a single-mode waveguide proves to be a better choice because of no mode walk-off or mode beating.

A.2

Coupled-Mode Theory

The coupled-mode theory (CMT) has been rigorously investigated, revised, and applied to many situations, such as in directional couplers, waveguide gratings, ring resonators, and wireless charging. It is used to analyze a system which has small perturbation from its original configuration. In the case of gratings, the isolated systems are particularly the modes themselves in the waveguide. Without any perturbation, the modes do not interact with one another. A small perturbation in the form of gratings seeds interaction among those modes and results in energy exchange. Despite complexity of the grating system, it is still represented mathematically by its permittivity px, y, z q. However, solving Maxwell’s equations of this system is not trivial. CMT relieves mathematical difficulties by proposing that if the perturbation is small the electric field of the perturbed system could be represented as a linear combination of the electric field modes of the unperturbed system. In this section, a conventional CMT is discussed [33, Chapter 12]. The grating is periodic in z-direction with a period of Λ, px, y, z q px, y, z

Λpz qq.

For a uniform-period grating, Λpz q

Λ is a constant, whereas a grating chirp, which is a variation in a grating period, could be introduced via a z-dependent Λpz q. With the

wave equation as shown in Eq. A.1 and assuming monochromatic waves, the equation becomes ∇2 Eprq ω 2 µpx, y, z qEprq,

(A.4)

where E represents the electric field of the system. If the perturbation of the grating is not very large, the perturbation can scatter incoming light mode to interact with other guided modes. Therefore, under CMT, the

93

Appendix A. Coupled-Mode Theory (CMT) electric field E could be expressed as a linear combination of waveguide modes: Eprq

¸

cm pz qem px, y qejβm z .

(A.5)

m 0

The summation covers all possible guided modes, indicated by the subscript m, and their propagation directions, forward-propagating for m for m

0.

That is βm

βm.

¡ 0 and backward-propagating

The factor cm pz q determines the energy carried by the

mode em and it is z-dependent due to the interaction along the grating. Substituting Eq. A.5 and using Eq. A.3 in Eq. A.4, the result is " * ¸ d jβm z 2jβm cm em ω 2 µp w qcm ejβm z em , e dz m0 m0 ¸

where a slow-varying envelope approximation such that

d2 c dz 2 m

(A.6)

! 2βm dzd cm is employed.

This equation describes the development of the total electric field along the grating expressed via a linear combination of modes on the left hand side due to the grating as a source of interaction on the right hand side of the equation. For orthogonal guided modes,

³ A

en em dA 0 if n m. The following definition is

used

xA|c|By xA|cBy Therefore, operating d cn dz

³ A

» A

A pcBq dA

(A.7)

dA en to both sides of Eq. A.6 results in

j

x | p q| y c ejpβ m 2βn xen |en y

¸ ω 2 µ en ∆ x, y, z em

m 0

m

βn qz

(A.8)

where ∆ w represents the grating perturbation to the waveguide. If the perturbation is periodic, i.e. ∆pz q ∆pz Λq, it could be expanded in Fourier series: ∆px, y, z q

¸ q

∆px, y qrq s

1 Λ

∆px, y qrq s ej

»Λ 0

2πq z Λ

∆px, y, z q ej

(A.9a) 2πq z Λ

dz

(A.9b)

94


where ∆px, y qrq s is the discrete Fourier coefficients as indicated by the use of square brackets. Also note that ∆px, y qrq s

∆ px, y qrq s because ∆px, y, z q is real. If

defining κn,m rq s σn rq s

ω 2 µ xen |∆px, y qrq s|em y 2βn xen |en y 2 ω µ xen |∆px, y qrq s|en y , 2βn xen |en y

(A.10a) (A.10b)

then it can be shown that κn,m pz q

¸

σn pz q

¸

q

κn,m rq s ej σn rq s ej

2πq z Λ

2πq z Λ

.

(A.11a) (A.11b)

q

These terms are often referred to as coupling coefficients: κn,m rq s is the cross coupling between the nth mode and the mth mode via the q th grating order; and σn rq s is the self-coupling term due to the q th grating order. Using Eq. A.10 and Eq. A.11 in Eq. A.8 leads to d cn dz

¸

jσnpzqcnpzq j

κn,m pz qcm pz qej pβm βn qz

(A.12)

m 0,n

On the other hand, if a grating is aperiodic, such as a chirped grating, the perturbation could be represented by the Fourier transform ∆px, y, z q ∆px, y, k q

»8

∆px, y, k q ejkz dk

8»

1 8 ∆px, y, z q ejkz dk. 2π 8

(A.13a) (A.13b)

In a similar manner, it is possible to define κn,m pk q σn pk q

ω 2 µ xen |∆px, y, k q|em y 2βn xen |en y 2 ω µ xen |∆px, y, k q|en y 2βn xen |en y

where n m

(A.14a) (A.14b)

and we will have κn,m pz q σn pz q

»8 »8 8

8

κn,m pk q ejkz dk

(A.15a)

σn pk q ejkz dk

(A.15b)

95

Appendix A. Coupled-Mode Theory (CMT) Therefore, applying Eq. A.14 and Eq. A.15 in the same way yields d cn dz

jσnpzqcnpzq j

¸

κn,m pz qcm pz qej pβm βn qz

(A.16)

m 0,n

The equation describes that the development of the nth mode results from the selfcoupling and the cross-coupling terms via the existence of the grating. It is clear from Eq. A.16 that without the grating perturbation both σn and κn,m are zero; therefore, cross mode interaction does not exist as

A.2.1

dcn dz

0.

First-Order Gratings

Without any approximation, full numerical modelling could be employed to solve a system of differential equations, such as Eq. A.16. However, the problem could be simplified using some approximation. The first-order grating approximation takes into account only the first Fourier component of the periodic perturbation. Nevertheless, inclusion of changes in index and perturbation periodicity could be introduced by expressing the perturbation as ∆px, y, z q ∆px, y, z qr0s

∆px, y, z qrpse

j 2πpz Λ 0

pq

jφp z

j 2πpz jφp pzq . Λ0

∆px, y, z qrpse

(A.17) Note that this expression looks like Eq. A.9 except the chirp term, φp pz q, which represents the change in perturbation periodicity. Then substitute Eq. A.17 into Eq. A.8 d cn dz

j

¸ q

σn rq scn pz qe

j 2πqz Λ 0

z p q j ¸ κ rq sc pz qej βm βn 2πq Λ0 n,m m m0,n

jφq z

p q (A.18)

jφq z

q

p, 0, p, φ0pzq 0, and φppzq φpzq φppzq. The development of cn pz q along z-axis is contributed mostly from the terms on the right hand side that have slow oscillation. For the self-coupling, it is σn r0s. For the firstorder grating approximation, set βm βn 2πp 0 as |p| 1. Physically it means that Λ

where q

0

96

Appendix A. Coupled-Mode Theory (CMT) a periodic effective index modulation matches half of the Bragg wavelength: The selected cross coupling term is then κn,m rq we have β1

β1 2πnλ

eff

neff λ.

p, ps. If the waveguide is single-mode,

. Eq. A.18 could be written as:

d c1 dz d c 1 dz where Φp

ΛB 2

jσ1r0sc1 jκ1,1rpsc1ejΦ

(A.19a)

p

jσ1r0sc1 jκ1,1rpsc1ejΦ

(A.19b)

p

pβ1 β1 2πp qz φp with corresponding definitions Λ 0

κn,m pz qrq s σn pz qrq s Defining ∆β

ω 2 µ xen |∆px, y, z qrq s|em y 2βn xen |en y 2 ω µ xen |∆px, y, z qrq s|en y . 2βn xen |en y

(A.20a) (A.20b)

, the value of p 1 such that ∆β 0. Next let β1 β1 2πp Λ 0

c˜1 pz q c1 pz qej

Φp 2

and c˜1 pz q c1 pz qej

Φp 2

,

(A.21)

which will make Eq. A.19 become

∆β dφ d c˜1 pz q j σ1pzqr0s c˜1pzq jκ1,1pzqrpsc˜1pzq dz 2 dz

d ∆β dφ c˜1 pz q jκ1,1 pz qrpsc˜1 pz q j σ1pzqr0s c˜1pzq dz 2 dz

(A.22a) (A.22b)

For a single-mode waveguide, it can be shown from the definitions that

κ1,1rps κ σ1 r0s σ1 r0s σ

κ1,1 rps

because β1

(A.23a) (A.23b)

β1 and e1px, yq e1px, yq. Hence, the coupled-mode equations for the

first-order grating in a single-mode waveguide are

d ∆β dφ c˜1 pz q j σpzq c˜1pzq jκpzqc˜1pzq dz 2 dz

d ∆β dφ c˜1 pz q jκ pz qc˜1 pz q j σpzq c˜1pzq dz 2 dz

(A.24a) (A.24b)

97


It is interesting to note that the above equations suggest that the effects of the perturbation periodicity chirp,

dφ , dz

and change in modal index, ∆pz qr0s, are similar and

indistinguishable. The problem of solving Eq. A.24 is sometimes referred to as a direct scattering problem. Analytically solving this set of equations are complicated with z-dependent functions, i.e.

dφ , dz

σ pz q, and κpz q. Numerical methods can address the problem but they

involve an iterative algorithm to achieve accurate results. Another popular method, which is used here, is the transfer matrix method. In order to reach that point, the solution of a uniform grating should be considered.

A.2.2

Uniform Gratings

Considering a grating with uniform perturbation in both magnitude and periodicity, Eq. A.24 is reduced to

∆β d c˜1 pz q j σ c˜1pzq jκ˜c1pzq dz 2

d ∆β c˜1 pz q jκ c˜1 pz q j σ c˜1pzq dz 2

(A.25a) (A.25b)

Solving Section A.25 requires two boundary conditions. Let assume that c˜1 pz0 q and c˜1 pz0 q are known. The solution will be c˜1 pz q

$ & %

cosh spz z0 q

j

∆β 2

σ

sinh s z

, z0 .

s

-

κ j s sinh spz z0q c˜1pz0q κ c˜1 pz q j sinh spz z0 q c˜1 pz0 q s$ & %

cosh spz z0 q

b

where s

|κ|2

∆β 2

σ

2

j

∆β 2

p q

c˜1 pz0 q (A.26a)

σ

sinh s z

, z0 .

s

-

p q

c˜1 pz0 q,(A.26b)

. The solution could be written in a matrix form, which

98

Appendix A. Coupled-Mode Theory (CMT) will be useful in the transfer matrix method;

p q m11pz, z0q c˜1 pz q m21 pz, z0 q

Mpz, z0q

m12 pz, z0 q c˜1 pz0 q

c˜1 z

m22 pz, z0 q

c˜1 pz0 q

where


j

κ m12 pz, z0 q j sinh spz z0 q s κ m21 pz, z0 q j sinh spz z0 q s


j

∆β 2

σ

c˜1 pz q

c˜1 pz q

sinh s z

p z0q

(A.27)

(A.28a)

s

(A.28b)

∆β 2

σ

p z0q

sinh s z

s

(A.28c) (A.28d)

The above solution to the coupled-mode equation, Section A.25, is given to the configuration of a grating with a constant period and constant coupling coefficients. It

z0 to another location z z within the grating region. It can be shown that the matrix Mpz, z0 q is unitary such that det pMpz, z0 qq 1. links the energy factors c1 and c1 from a location of z

A.2.3

Fourier Series of Permittivity Perturbation

In the previous section, the solution of the uniform first-order grating is derived. In the solution Section A.26, the values of the self-coupling and cross-coupling constants, σ and κ respectively, are needed. From their definitions, the zero and first Fourier coefficients of the permittivity perturbation are required. The unperturbed uniform waveguide is represented by w px, y q. On the other hand, the uniform sidewall grating with 0.5 duty cycle, defined by one-step etching, is written $ ' ' ' w x, y ' ' ' &

p q

px, y, z q

0 ' ' ' ' ' ' %w x, y

p q

; px, y q in the unetched area ; px, y q in the etched area, 0 ¤ z ; px, y q in the etched area,

Λ 2

Λ2

¤ z Λ.

(A.29)

99

Appendix A. Coupled-Mode Theory (CMT) Therefore, the perturbation, ∆ w , becomes ∆px, y, z q

$ ' ' &

0 w px, y q ; px, y q in the etched area, 0 ¤ z

' ' %0

Λ2

(A.30)

; otherwise.

Previously, the perturbation in Fourier series is expressed as in Eq. A.9 ∆px, y, z q

¸ q

∆px, y qrq s

1 Λ

∆px, y qrq s ej

»Λ 0

2πq z Λ

∆px, y, z q ej

(A.31a) 2πq z Λ

dz.

(A.31b)

Using this relations, we fine that for the sidewall grating, ∆px, y qr0s

and ∆px, y qrq

$ ' ' &

0s '

$ ' ' & 0 w px,yq 2

' ' %0

; px, y q in the etched area

(A.32)

; otherwise

w px,y q pejπq 1q ; px, yq in the etched area j 0 2πq

' %0

(A.33)

; otherwise.

Therefore, the first-order Fourier coefficient is ∆px, y qr1s ∆ px, y qr1s

$ ' ' &

j w px,yπ q0

' ' %0

; px, y q in the etched area

(A.34)

; otherwise.

The self-coupling and cross-coupling constants for a uniform first-order grating of a single mode waveguide could be calculated. The self-coupling constant is σ1 r0s

ω 2 µ xe1 |∆px, y qr0s|e1 y 2β1 xe1 |e1 y

(A.35)

which is real because ∆px, y qr0s is real. The cross-coupling constant, with ∆px, y qr1s, is κ1,1 r1s

ω 2 µ xe1 |∆px, y qr1s|e1 y 2β1 xe1 |e1 y

|κ|ejθ j |κ|.

It is imaginary because ∆px, y qr1s is imaginary, which means that θ

π2 .

(A.36)

100


A.2.4

Grating Responses by CMT and Transfer Matrix Method

From a previous section, the solution to the uniform grating matches c˜1 and c˜1 from one location to another location along the grating. Therefore, it is possible to break the whole grating into smaller uniform sections and then connect c˜1 and c˜1 along the grating using the relationship in Eq. A.27. This method is called the transfer matrix method (TMM), as used in [34]. It is important to note that each section should be long enough. This is because for each section, the solution to the coupled-mode theory is derived with the assumption of slowly varying functions. Therefore, it is required that the grating

" λ. If the grating is divided into N sections from z z0 to z zN , there are N transfer

length is reasonably longer than the wavelength, i.e. ∆zi

matrix equations, each with a corresponding transfer matrix Mi ,

c˜1 pz0 q p q M1pz1, z0q

c˜1 pz0 q c˜1 pz1 q

c˜1 z1

c˜1 pz1 q p q M2pz2, z1q


c˜1 z2

.

. .

p q c˜1 pzN 1 q MNpzN, zN1q

c˜1 pzN q c˜1 pzN 1 q

c˜1 zN

Therefore, it could be written that

p q c˜1 pz0 q MNMN1 M2M1

c˜1 pzN q c˜1 pz0 q

c˜1 zN

p q c˜1 pz0 q M11 M

c˜1 pzN q c˜1 pz0 q M21

c˜1 zN

M12 c˜1 pz0 q M22

c˜1 pz0 q

(A.37)

This final matrix M is the system matrix and represents the whole grating. It matches the states of c˜1 and c˜1 from the front of the grating at z at z

zN .

z0 to the end of the grating

101


For a Bragg grating, reflection response is of particular interest. In isolation from other optical components, the boundary condition of the problem at hands is that at the back of the Bragg grating the backward-propagating field is zero, i.e. c1 pzN q

0 and

as a result c˜1 pzN q 0. Therefore, from Section A.37 it leads to c˜1 pz0 q c˜1 pz0 q c˜1 pzN q c˜1 pz0 q

M21 M

(A.38a)

22

M11M22M M12M21

(A.38b)

22

The reflection response, at frequency ν, of the Bragg grating is defined as r pν q Setting z0

c1 pz0 q c1 pz0 q

(A.39)

0, the reflection response becomes rpν q


M21 M

(A.40)

22

The spectral response of the grating is achieved by calculating the reflection response r of different frequencies in a spectrum of interest. On the other hand, the transmission response is tpν q

c˜1 pzN q c˜1 pz0 q

M11M22M M12M21 .

(A.41)

22

Since the elements that construct the system matrix is unitary, the system matrix is also unitary, i.e. M11 M22 M12 M21

1. Hence, tpν q

1 M22

rMpν q

(A.42)

21

Phase-Shift and Sample Gratings If the whole grating is composed of many disconnected gratings separated by unperturbed waveguide sections, such as in the phase-shift and sample gratings, a special transfer matrix is required to represent the unperturbed sections. If the forward- and backwardpropagating waves are represented by c1 pz q and c1 pz q and they undergo propagation of

102

Appendix A. Coupled-Mode Theory (CMT) length L in an unperturbed waveguide, the transfer matrix of this propagation is

c1

c1

ejβ1 L

pzz0 Lq

0 c1 jβ1 L

0

e

c1

,

(A.43)

pzz0 q

by assuming a single-mode waveguide.

Uniform Grating Response For a uniform grating, the explicit reflection response can be written from Eq. A.40 and Eq. A.28 rpν q

m21 m22

(A.44a)

jκ sinh pspz z0qq . s cosh pspz z0 qq j ∆β σ sinh pspz z0qq 2 d

Note that s

|κ|2

∆β 2

σ

(A.44b)

2

.

(A.45)

The reflectivity is defined as the square of the magnitude of the reflection coefficient, Rpν q |rpν q|2 ,

(A.46)

where

|κ| sinhps∆zq . 2 ∆β 2 s2 cosh2 ps∆z q σ sinh p s∆z q 2 The maximum value of the reflection magnitude occurs when ∆β σ 0, 2 |rpν q| b

|r|max tanhp|κ|∆zq.

(A.47)

(A.48)

Should the self-coupling constant is not present, the resonance condition becomes ∆β

0 Ñ

λ 2neff

Λ,

(A.49)

which leads to the Bragg condition for the first-order grating. This result means that in the absence of the self-coupling constant, the maximum reflection, and consequently the


103

maximum reflectivity, occurs at the Bragg wavelength. The presence of the self-coupling constant shifts this maximum reflectivity to a nearby frequency. Another source of peak shifting is the effective index dispersion against the wavelength.

Magnitude and Phase Responses Gratings in general do not have a close-form reflection response as uniform gratings do. The reflection response is calculated using the transfer matrix method as described previously. It is usually a complex function allowing one to write rpν q |rpν q|ejφpν q .

(A.50)

Separately, |r| is termed the amplitude response whereas φpν q is the phase response. The phase response φpν q informs how each frequency of light is altered in the temporal sense by the grating. The group delay, τ , of the grating is calculated from τ p2πν q τ pω q

dφ dω

2π1 dφ . dν

(A.51)

The group delay physically represents the time delay of a pulse propagation into and reflection from the device. The group velocity is calculable from the group delay and the device length: vg

A.3

Lτ

(A.52)

Summary

In this chapter, the formalism of the coupled-mode theory is discussed, and it leads to the governing equations describing the interaction between modes in the waveguide, which is assumed to be single-mode. The explicit solution for a uniform grating is derived with the assumption of the first-order grating. Combining the coupled-mode theory with the transfer matrix method provides a way to analyze a non-uniform grating.

Appendix B Fourier Transforms B.1

Discrete Fourier Transform

The one-dimensional discrete Fourier transform relates two discrete 1 Np vectors, f rns and F rms, F rms f r ns

¸

Np 1

f rnse

j 2πmn Np

(B.1a)

n 0 Np 1

1 ¸ j 2πmn F rmse Np Np m0

where n, m 0, 1, 2, . . . , Np 1 and exp

j 2πmn N p

(B.1b)

and exp j 2πmn Np

are the basis func-

tions. These functions are periodic in m and n; φrm, ns φrm, n Np s φ rm, Np ns e

j 2πmn Np

(B.2a)

φrm, ns φrm Np , ns φ rNp m, ns e

j 2πmn Np

(B.2b)

j φ¯rm, ns φ¯rm, n Np s φ¯ rm, Np ns e

2πmn Np

j φ¯rm, ns φ¯rm Np , ns φ¯ rNp m, ns e

2πmn Np

(B.2c) (B.2d)

These relations mean that F rms F rm Np s F rNp ms

(B.3a)

f rns f rn Np s f rNp ns.

(B.3b)

104

105

Appendix B. Fourier Transforms

In MATLAB, the DFT is carried out using the fast Fourier transform algorithm with the fft function. The fft function takes a 1 N vector, say a, and returns a 1 N vector, say b, which is a discrete Fourier transform counterpart; b fft(a).

(B.4)

The inverse DFT is performed with a similar algorithm by a MATLAB function ifft; a ifft(b).

(B.5)

The functions interpret the vectors, a and b, in the order from m, n

0 to Np

1,

corresponding to the basis functions φrm, ns and φ¯rm, ns. Nevertheless, most of the times the negative frequencies, m, n 1, 2, . . ., are of interest; they could be calculated from Section B.2. For example, F r1s F rNp 1s

(B.6a)

F r2s F rNp 2s

(B.6b)

.. . F rNp

1s F r1s.

(B.6c)

Fortunately, MATLAB has a function that swaps these values. That function is fftshift and its inverse function is the ifftshift, which reverses the swap. The above discussion does not mention about what f rns is measured against. In DFT, f rns is usually a set of sampled data from a particular analog signal f ptq, with a sampling interval ∆t. The reciprocal of the sampling interval is called the sampling frequency or the sampling rate, νs

1{∆t.

The well-known Nyquist criteria to avoid

aliasing is captured in the inequality, νs

¡ 2ν,

(B.7)

meaning that the sampling rate must be larger than twice the frequency of interest. With this sampling interval, the analog signal with frequency ν is sampled to a series of data f rns f pn∆tq ej2πνn∆t

ejθn

(B.8)

106

Appendix B. Fourier Transforms where θ

2πν∆t is called the digital frequency.

This digitalized sampled data will be

periodic as its original analog signal, ej2πνt , would be only with the form θ

m 2πν∆t. 2π N

(B.9)

It implies that the digital frequency is discretized. θ

m. Ñ θm 2π N

(B.10)

Consequently, the frequency resolution, ∆ν, is found to be ∆ν

N1∆t .

(B.11)

With this discrete digital frequencies θm , the expression for the basis appears to be the same as before φrm, ns e

j 2πmn Np

j and φ¯rm, ns e

2πmn Np

.

(B.12)

Also, both f rns and F rms are periodic with periodicity of Np . For f rns, it is sampled against points of time, t : rn∆ts, where as F rms is represented versus points of frequency, ν : rm∆ν s, where m, n are integer.

B.2

Implementing Fourier Transform with Discrete Fourier Transform

In the conventional Fourier transform, the transformation relates two continuous functions: F pν q f ptq

»8

8 »8 8

f ptqej2πνt dt

(B.13a)

F pν qej2πνt dν.

(B.13b)

where t and ν are real numbers. It is said that f ptq and F pν q are the Fourier pair or f ptq ðñ F pν q.

(B.14)

107


Consider a Fourier pair between the impulse and reflection response. In the simulation, the reflection response, rrn1 s, and the impulse response, hrm1 s, are represented by 1 Np discrete finite vectors, where m1 , n1

1, 2, . . . , Np are the element indices. For the reflection response, it is sampled corresponding to a set of frequency points, ν rn1 s νn1 , with a specified frequency interval, ∆ν. Similarly, the impulse response is represented on a set of time points, trm1 s tm1 , with a time interval, ∆t. These discretized vectors are required to represent the continuous counterparts in the continuous Fourier transform. The Fourier transform could be broken down to the Reimann sum, rpν q

»8

8

hptqej2πνt dt

Np 2

¸

Np m

{

T »2

T {2

hptqej2πνt dt

(B.15a)

hpm∆tqej2πνm∆t ∆t.

(B.15b)

2

In doing so, it is necessary that hptq is negligible outside the range from

T {2 to T {2. The summation takes the 1 Np vector hrms hpm∆tq. The time domain that hrms is sampled is basically tm1 tm m∆t, where m N , . . . , N2 . 2 p

p

In order to resemble DFT so that the fft function could be used, discretize the frequency in the same way into ν Eq. B.15 can be rewritten as

n∆ν νn νn1 , where n N2

, . . . , N2p . Therefore,

Np 2

¸

rpν q Ñ rrns

p

Np m

j 2πmn Np ∆t

hrmse

(B.16a)

2

DFTthrmsu ∆t.

(B.16b)

The inverse Fourier transform could also be shown in a similar way. Therefore, in MATLAB, the Fourier transform and the inverse Fourier transform via DFT are implemented as the followings: h = fftshift(ifft(ifftshift(r)))/dt

(B.17a)

r = fftshift(fft(ifftshift(h)))*dt

(B.17b)

108


where dt is a variable representing the time interval. Actually, relations in Eq. B.17 are applicable to any pairs of vectors, f rtm s and F rνm s, with the time and frequency intervals ∆t and ∆ν. Doing so is by replacing h and r with f and F, respectively. Now the time and frequency axes are defined; they are important because hrms and rrns are sampled on them, respectively. If Np is an odd number, the time and frequency axes become t: ν:

Np 2 1 ∆t, Np 2 3 ∆t, . . . , ∆t, 0, ∆t, . . . , Np 2 3 ∆t, Np 2 1 ∆t (B.18a) Np 1 ∆ν, Np 3 ∆ν, . . . , ∆ν, 0, ∆ν, . . . , Np 3 ∆ν, Np 1 ∆ν.(B.18b) 2

2

2

2

On the other hand, if Np is an even number, the time and frequency axes are then

Np Np Np Np 1 ∆t, 2 ∆t, . . . , ∆t, 0, ∆t, . . . , 1 ∆t, ∆t (B.19a) t: 2 2 2 2

Np Np Np Np ν: 1 ∆ν, 2 ∆ν, . . . , ∆ν, 0, ∆ν, . . . , 1 ∆ν, ∆ν.(B.19b) 2 2 2 2 The time and frequency axes should be defined as discussed above in order to comply with fftshift and ifftshift functions in MATLAB.

Appendix C Simulation Results for Grating Responses This chapter contains results of grating responses computed from the direct scattering, which is discussed in Chapter 3. The grating structures of interest include uniform gratings, chirped and apodized gratings, π-phase-shift gratings, and sampled gratings.

C.1

Uniform Gratings

A uniform grating has constant coupling constants and perturbation periodicity. Considering a waveguide grating with a waveguide width of w

1.4 µm and a recess depth

of rd 25 nm, the effective index of the TE-like and TM-like modes are, respectively, neff pTEq 3.1062 and neff pTMq 3.1058.

(C.1)

The self- and cross-coupling constants at λ 1.55 µm are κ 4523j

(C.2a)

7107.

(C.2b)

σ

109

Appendix C. Simulation Results for Grating Responses Also, if the Bragg wavelength is λ

110

1.55 µm, the period of the perturbation for the

first-order Bragg condition could be calculated Λ

λB 2neff

249.5 nm.

(C.3)

The last parameter that affects the grating response is the grating length, ∆z. If the dispersion of effective index against the frequency is not taken into account, the grating response of the previously described grating with ∆z

100 µm, is shown in Fig. C.1:

(a) shows the amplitude response whereas (b)–(d) depict the unwrapped phase response, the group delay response, and the group velocity, respectively. From Fig. C.1a, the peak of the amplitude response occurs very near to the desirable Bragg wavelength at 1.55 µm. The minimum value of the group delay, Fig. C.1c, also occurs near the Bragg wavelength. This fact corresponds to the optimal detuning at this frequency and consequently a better coupling from the forward to backward waves. Hence, light at this frequency can penetrate shorter into the grating, leading to shorter group delay and faster group velocity. This is the effect of the grating structure. Away from the Bragg wavelength, light experiences decreasing coupling; its phase development is then accounted mainly from propagation: φ

β∆z, for the backward-propagating

wave. Therefore, the group delay becomes τprop

dφ dω neff,0c ∆z .

Substituting values for the current case gives τprop

(C.4)

1.035 ps, which agrees well with the

simulated result for frequencies away from the Bragg wavelength. If the dispersion is allowed in the calculation, the grating responses become as in Fig. C.2. The evident shift of the peak could be explained from a better matching in the detuning parameter at another frequency. A stronger grating is the one with a larger cross-coupling constant. For example, let set the recess depth to be rd

100 nm in which the self- and cross-coupling constants

are 43433 and 27, 650j, respectively. Its reflection responses are displayed in Fig. C.3.

111

Appendix C. Simulation Results for Grating Responses

The amplitude response is increased considerably and approaches unity in the central frequency band. This central band is also wider than that of a weak grating. This is a direct consequence from increasing the coupling coefficient. The self-coupling constant becomes larger as well and results in a prominent shift from the designated Bragg wavelength. The minimum group delay decreases compared to that of the weaker grating reflecting a short penetration into the grating of the light in the central frequency lobe.

0.5

6

0.4

4

|r|

∠r

0.3 2

0.2 0

0.1 0

1.54

1.55 λ (µm)

1.56

1.57

−2

(a) Amplitude response.

1.54

1.55 λ (µm)

1.56

1.57

(b) Phase response.

0.35

1.08 1.06

0.34 vg/c0

τ (ps)

1.04 1.02

0.33

1.00 0.32 0.98 0.96

1.54

1.55 λ (µm)

1.56

(c) Time delay response.

1.57

0.31

1.54

1.55 λ (µm)

1.56

1.57

(d) Group velocity.

1.4 µm, a recess depth of rd 25 nm, and a grating period of Λ 249.5 nm. The grating length is ∆z 100 µm. The

Figure (C.1): Reflection response of a uniform grating with a waveguide width of w

effective index dispersion is not taken into account.

112


Another way to increase the reflectivity, apart from increasing the coupling constant, is by increasing the grating length, ∆z. For example, let the grating has the length of ∆z

200 µm and the recess depth of rd 25 nm; its responses are plotted in Fig. C.4.

The maximum reflection amplitude is increased and close to one. This increase in reflectivity is a result of longer grating length; reflected power at the zero detuning condition accumulates as light propagates deeper into the grating. The central frequency lobe ap-

0.5

5 4

0.4

3 |r|

∠r

0.3 0.2

2 1 0

0.1

−1

0

1.54

1.55 λ (µm)

1.56

−2

1.57


1.54

1.55 λ (µm)

1.56

1.57

(b) Phase response.

0.35

1.08 1.06

0.34 vg/c0

τ (ps)

1.04 1.02

0.33

1 0.32 0.98 0.96

1.54

1.55 λ (µm)

1.56

(c) Time delay response.

1.57

0.31

1.54

1.55 λ (µm)

1.56

1.57

(d) Group velocity.



effective index dispersion is now taken into account.

113


1

2

0.8

1.5

|r|

τ (ps)

0.6 0.4

0.5

0.2 0

1

1.54

1.55 λ (µm)

1.56

1.57

0


1.54

1.55 λ (µm)

1.56

1.57

(b) Group delay response.



effective index dispersion is taken into account.

pears narrower as compared to the grating response in Fig. C.2. The behavior could be explained that since a longer grating reflects light with a longer duration of interaction, the spectral bandwidth of a long duration signal is effectively narrow. Physically, a long grating with a relatively low coupling coefficient has a longer distance of interaction to selectively reflect light at its resonance characteristics.

114


1 8

0.8

|r|

τ (ps)

0.6 0.4

6

4 0.2 0

1.54

1.55 λ (µm)

1.56

1.57

2


1.54

1.55 λ (µm)

1.56

1.57




effective index dispersion is taken into account.

C.2

Chirped and Apodized Gratings

The grating chirp is the change in Bragg wavelength along the grating. For a first-order grating, the Bragg condition is expressed as λB

2neffΛ.

(C.5)

Therefore, the chirp could be introduced directly to the grating period, Λpz q, which is treated in Section A.2.1. Alternatively, from the above expression, the chirp can be implemented by the effective index profile neff pz q. Since the effective index depends on the waveguide structure, this translates to the chirp by the waveguide width profile wpz q.

C.2.1

Linearly Chirped Gratings

Let consider the chirp introduced by the grating period profile whereas the waveguide width is fixed at w

1.4 µm as before.

To simulate the grating response, the transfer

matrix method is used and the number of grating sections is denoted by Ng .


115

A linearly chirped grating has a linearly changing grating period, i.e. Λpz q Λ0

∆Λ L

z

L 2

;0¤z

¤ L,

(C.6)

where Λ0 is the grating period at the center of the grating, ∆Λ is the total grating chirp, and L is the total grating length. The linearly chirped grating with ∆Λ 4 nm and Λ0

250 nm is discretized into Ng

subgratings. The sign of the chirp bandwidth

represents the positively and negatively chirped gratings. In MATLAB, this statement is translated into gtPeriod = linspace(248,252,Ng)*1e-9, for a positively-chirped grating and gtPeriod = linspace(252,248,Ng)*1e-9, for a negatively-chirped grating. The subgratings are integral times as long as their corresponding period; dz = gtPeriod*m, where m is an integer. The response when Ng

200 and m

°N g

8 is shown in Fig. C.5. The total length of this grating is L

400 µm. Each subgrating has the same self- and cross-coupling constants, 15, 467 and 9, 847j respectively. The FWHM bandwidth of the amplitude response is ∆zi

i 1

approximately 25 nm, which is in agreement with the bandwidth of the Bragg wavelength chirp calculated from ∆λB

2neff∆Λ 24.8 nm.

An approximately linear group delay

is observed within the reflection amplitude bandwidth, as shown in Fig. C.5b, and it spans about ∆τ

8 ps.

This span corresponds to the difference in time delays of the

frequencies reflected by the front and the back subgratings, and it is close to the round trip time, which is approximately τrt

2Lnc 8.28 ps. Physically, the Bragg wavelength eff

of the last subgrating has to travel to and from the back of the grating resulting in a round-trip time delay compared to the frequency reflected at the front subgrating. On the other hand, the linear chirp can be imposed through the waveguide width. However, this alternative requires an appropriate recess depth profile as well in order to

116

0.8

15

0.6

10 τ (ps)

|r|


0.4 0.2 0 1.53

5 0

1.54

1.55 1.56 λ (µm)

1.57

1.58

−5 1.53


1.54

1.55 1.56 λ (µm)

1.58


Figure (C.5): Reflection response of a linearly chirped grating with ∆Λ 4 nm and Λ0 simulation is implemented with Ng

1.57

250 nm. The

200 subgratings and m 8. (a) Amplitude response. (b) The

blue line corresponds to a postively-chirped grating and the red line corresponds to a negatively-chirped grating.

achieve pure chirped gratings without apodization. For example, consider the case that the perturbation period is set constant at Λ 250 nm and the recess depth is also fixed at rd 50 nm. The grating is a taper-like waveguide with linearly increasing waveguide width from 1.0 µm to 1.6 µm. In this case, the MATLAB variables gtWidth = linspace(1.0,1.6,Ng)*1e-9 for an up-tapered waveguide grating, and gtWidth = linspace(1.6,1.0,Ng)*1e-9 for a down-tapered waveguide grating. The responses of both the up- and down-tapered gratings are shown in Fig. C.6. The amplitude responses are similar in both gratings, Fig. C.6a. The magnitude of reflection in the central part shows a decreasing trend compared to the previous linearly-chirped grating in Fig. C.5a. This is due to a non constant cross-coupling constant profile, which in turn is a result from holding the recess depth fixed but changing the waveguide width.

117


The group delay responses, shown in Fig. C.6b, are complimentary. The up-tapered grating behaves similarly to the positively-chirped grating as the group delay increases in a linear manner in a range λ

P r1.54, 1.56s µm.

The down-tapered grating performs

like a negatively-chirped grating.

15

1 0.8

10

|r|

τ (ps)

0.6 0.4

0

0.2 0 1.52

5

1.53

1.54 1.55 λ (µm)

1.56

1.57


−5 1.52

1.53

1.54 1.55 λ (µm)

1.56

1.57


Figure (C.6): Reflection response of a linearly tapered grating with the waveguide width increasing from 1.0 µm to 1.6 µm. The grating period is 250 nm and the recess depth is 50 nm, throughout the grating. The simulation is run with Ng

400 and m 4. (a) Amplitude response. (b) The blue line

corresponds to a up-tapered grating and the red line corresponds to a down-tapered grating.

C.2.2

Apodized gratings

Apodization is the change in coupling constant along the grating. In the sidewall-etching configuration, the apodization could be realized simply by changing the recess depth. Coupling constants also depend on the waveguide width. Therefore, the coupling constant is the interplay between the waveguide width and the recess depth. An algorithm is devised to create a waveguide width and a recess depth profile with given effective index and coupling constant profiles. For example, a Gaussian-apodized cross-coupling constant for a uniformly-wide waveguide grating is plotted in Fig. C.7. Its responses are presented in Fig. C.8. Apodization helps to subside side lobes in the amplitude response

118

Appendix C. Simulation Results for Grating Responses evidencing in Fig. C.8a.

1000

200 175 Recess depth (nm)

|κ| (1/m)

800 600 400 200

150 125 100 75 50 25

0 0

0 0

25 50 75 100 125 150 175 200 z (µm)

(a) Gaussian-apodized cross-coupling constant.

25

50

75 100 125 150 175 200 z (µm)

(b) Recess depth profile.

Figure (C.7): Gaussian-apodized cross-coupling constant and its corresponding recess depth profile for a 1.4-µm-wide uniform waveguide.

1 4 0.8 2

|r|

τ (ps)

0.6 0.4

−2

0.2 0 1.53

0

1.54

1.55 λ (µm)

1.56


1.57

−4 1.53

1.54

1.55 λ (µm)

1.56

1.57


Figure (C.8): Reflection responses of a Gaussian-apodized grating with a uniform waveguide width of 1.4 µm, corresponding to an effective index of 3.1062 for a TE-like mode.


C.3

119

π-phase-shift and Sampled Gratings

In this section that the algorithm is demonstrated to solve a sampled grating. A sampled grating is a whole grating structure being composed of many disconnected gratings separated by unperturbed waveguide sections. In the algorithm, the whole grating perturbation, including the separating waveguides, is represented by a 1 Ng array of the recess depth profile, gtRD. The algorithm recognizes the unperturbed waveguide sections by the element of zero in the recess depth array.

C.3.1

π-phase-shift Gratings

For instance, consider a π-phase-shift grating which is a uniform 1.4-µm-wide waveguide with two similar uniform grating sections. Each of them has a recess depth of rd 50 nm and a grating period of Λ

250 nm.

Their lengths are 400 times of the grating period.

They are separated by the unperturbed waveguide of Λ{2 in length. In the algorithm, the whole grating is gtRD = [50, 0, 50]*1e-9 gtWidth = ones(1,3)*1.4e-6 gtPeriod = ones(1,3)*250e-9 dz = [250*400, 250/2, 250*400]*1e-9. The responses shown in Fig. C.9 compare the π-phase-shift grating with a complementary continuous grating of the same length. A deep and narrow notch is evident in the reflection amplitude, which is a characteristic of a π-phase-shift grating.

C.3.2

Sampled Gratings

Similarly, a sampled grating can be simulated by employing the same defining method. For example, consider a grating whose structure is described as

120


1

4

0.8

3.5 3

|r|

τ (ps)

0.6

2.5

0.4

2

0.2 0 1.54

1.5 1.55

λ (µm)

1.56


1.57

1 1.54

1.55

λ (µm)

1.56

1.57


Figure (C.9): Reflection responses of a pi-phase-shift grating (blue solid line) and a complementary continuous grating (red dashed line). All grating sections are uniform: a waveguide width of 1.4 µm, a recess depth of 50 nm, and a grating period of 250 nm. Subgratings in the a pi-phase-shift grating are 100 µm long whereas a continuous uniform grating is 200 µm long.

gtRD = [50, 0, 50, 0, 50]*1e-9 gtWidth = ones(1,5)*1.4e-6 gtPeriod = [248, 0, 250, 0, 252]*1e-9 dz = gtPeriod*800; dz([2,4]) = 50e-6;. That is the whole structure has three subgratings with different grating periods and, as a result, different Bragg wavelengths. Their waveguide widths and recess depths are the same at 1.4 µm and 50 nm, correspondingly. Each of them are separated by 50 µm. Fig. C.10 displays its responses. Three peaks are observed which are related to three different Bragg wavelengths from the three subgratings. Additionally, the group delay response indicates gradual increase time delay due to the locations of the subgratings.

121


1

15

0.8

10

|r|

τ (ps)

0.6 0.4

0

0.2 0 1.53

5

1.54

1.55 1.56 λ (µm)

1.57


1.58

−5 1.53

1.54

1.55 1.56 λ (µm)

1.57


Figure (C.10): Reflection responses of a sampled grating.

1.58

Appendix D

Inverse Scattering Theory

Appendix A focuses on the calculate of the response of the grating whose physical parameter profiles, such as the waveguide width and the recess depth, are known. However, the desired response of the device is often known and the physical parameters of the device that lead to that response are sought. This problem is termed inverse scattering (IS) problem. Usually it applies to a system that could be mathematically expressed in two coupled equations. The grating that couples one mode to another exactly fits into this category.

This chapter discusses the Gel’fand-Levithan-Marchenko (GLM) theory following the work in [35]. The layer peeling method is introduced and refined to combine with the GLM solution [36]. The last section discusses how to fit the coupled-mode equations and their solutions to the framework of the GLM theory in order to design a required grating. 122

123

Appendix D. Inverse Scattering Theory

D.1

Inverse Scattering Theory: GLM equations

The following derivation of the GLM equations is based on [35] and [36]. Assume that a system under consideration could be written in two coupled equations d c1 dz d c2 dz where q, q

Ñ 0 as |z| Ñ 8.

jζc1

qc2

(D.1a)

q c1

jζc2

(D.1b)

ζ is the eigenvalue of the problem and it is z-independent.

Note that Eq. D.1 resembles Eq. A.22, however, with differences in z-dependence. Therefore, some manipulations are required before the inverse scattering theory could be applied to the coupled-mode equations. Eq. D.1 could be cast in a matrix form

d c1 jζ dz c q 2

q c1

jζ

.

¯ with asymptotic behaviors at z Assume linearly independent solutions, φ and φ,

φpz

1

Ñ 8, ζ q

e

(D.2)

c2

jζz

Ñ 8 (D.3a)

0

¯ pz φ With these forms, φpz

0 jζz .

e 1

Ñ 8, ζ q

(D.3b)

Ñ 8, ζ q is the forward-propagating wave whereas φ¯ pz Ñ 8, ζ q

is the backward-propagating wave. The exact solution at any point z could be written if the kernel functions apply 1

φpz, ζ q ejζz 0

¯ pz, ζ q φ

0 jζz e

1

»z

8

Kpz, sqejζs ds

»z

8

¯ pz, sqejζs ds K

(D.4a)

(D.4b)

124


where K

K1 ¯ and K

¯ K1

K¯2

K2 general solution to Eq. D.17 is

. With these linearly independent solutions, the

c1 pz, ζ q cpz, ζ q

φpz, ζ q c2 pz, ζ q

Note that for z

Ñ 8 Eq. D.5 appears

¯ pz, ζ q. rpζ qφ

(D.5)

1 cp8, ζ q ejζz 0

0 rpζ q ejζz 1

(D.6)

which has the forward-propagating wave with a magnitude of unity and the backwardpropagating wave with a magnitude of |rpζ q|. The next step is to take a close path integral on the upper half of the complex plane of Eq. D.5 after multiplied by ejζy {2π 1 2π

¾

1 cpz, ζ qejζy dy

¾

2π

C

φpz, ζ qejζyq dy

1 2π

C

¾

¯ pz, ζ qejζy dy rpζ qφ

(D.7)

C

Using the fact that cpz, ζ q is analytical in the upper half plane, the term on the left hand side becomes zero. Also use 1 2π

¾

ejζs ds δ psq,

(D.8)

C

which lead to the important result

0 Kpz, y q

0 h z

1

p

yq

where hpy q

1 2π

»z

8 ¾

¯ pz, sqhps K

rpζ qejζy dy.

y q ds

;y

z,

(D.9)

(D.10)

C

¯ In general, this process Eq. D.9 is the main iteration equation that solves for K and K. involves four functions: K1 , K2 , K¯1 , and K¯2 . Fortunately, the complexity is reduced by

125

Appendix D. Inverse Scattering Theory utilizing the symmetry in Eq. D.17. To see this, write

d c2 jζ q c2

dz c q jζ c1 1

(D.11)

by changing the order of the row. Also take the complex conjugate of Eq. D.17 and

d c1 dz c 2

receive

jζ

q c1

.

q

jζ

(D.12)

c2

This symmetry suggests the form

φ

2 ¯ φ φ1

(D.13a)

K2 ¯ K . K1

(D.14)

As a result, the iteration equation turns to be

p q K2 pz, y q

K1 z, y

0

0 h z

p

1

yq

K pz, sq

»z

8

2

p

h s

K1 pz, sq

y q ds.

(D.15)

¯ and eventually c. ¯ are solved, they can construct φ, φ, When K and K The next step is to find the relationship between K and q. Consider the linearlyindependent solution, φ, as defined before φ1 pz, ζ q ejζz φ2 pz, ζ q

»z

8

»z

8

K1 pz, sqejζs ds

K2 pz, sqejζs ds.

(D.16a)

(D.16b)

Since φ is the solution to Eq. D.1, it is possible to write d φ1 dz d φ2 dz

jζφ1 jζφ2

qφ2 q φ2 .

(D.17a) (D.17b)

126

Appendix D. Inverse Scattering Theory By putting φ into Eq. D.17, two equations result respectively 0

»z

8

dK1 dz

dK1 ds

qpzqK2

»z

0 p2K2 pz, z q q q ejζz

8

ejζs ds dK2 dz

(D.18a) dK2 ds

q pz qK

1

ejζs ds.

(D.18b)

It is shown that it is necessary and sufficient that 0 0

dK1 dz dK2 dz

dK1 ds dK2 ds

with the boundary condition K2 pz, z q

qpzqK2

(D.19a)

qpzqK1

(D.19b)

q pz q . 2

(D.20)

In summary, consider a system whose response is described by the coupled-mode equations as expressed in Eq. D.1 and assume that the response rpζ q is desired and known. The coupling parameters q pz q could be extracted from the known response by the relationship q pz q 2K2 pz, z q.

(D.21)

where K2 pz, z q is calculated from the iteration equations K2 pz, y q hpz K1 pz, y q

»z

8

yq

»z

8

K1 pz, sqhps

K2 pz, sqhps

where hpy q

1 2π

¾

y q ds

(D.22a)

y q ds

(D.22b)

rpζ qejζy dy.

(D.23)

C

In computational implementation, the accuracy of the result depends on how many rounds of iteration are used to calculate K1 and K2 in order to achieve convergence. Therefore, for a long grating, this process might be time-consuming. The error in calculation also stems from the discrete and limited nature of computational variables, which could not be reduced by increasing the number of iteration.


D.2

127

Layer Peeling Method

To avoid large iteration in the GLM method, another method is proposed. This new method is called a layer peeling method (LPM). The layer peeling method has been investigated and employed by many researchers [81, 83–86]. The essence of the layer peeling method is to divide the grating into many small uniform grating sections and consecutively calculate their coupling constants. The front edge of the impulse response by causality is due to the closest grating section because the presence of the other later grating sections will manifest in later time. Therefore, the information of this front edge can be used to calculate the reflection and the coupling constant of the closest grating section. Then, the calculation moves to the next grating section and continues until the end of the grating. In the implementation, the required response is represented by a discrete and finite values or vectors. This discreteness and limit bandwidth are the sources of error. In the normal layer peeling method, this error accumulates and propagates along the calculation. The situation becomes worse for a strong grating and the calculated grating profile will not be accurate.

D.3

GLM with Layer Peeling Method

Rosenthal et al. proposed the integral layer-peeling (ILP) method to calculate the profiles of strong gratings [36]. The method basically combines the advantages of the pure layer peeling and the GLM method. Like the original layer peeling method, the grating is divided into several grating sections. However, a local reflection response is used to calculate the local coupling constant of the closet grating section via the GLM method. Then, the next reflection response is calculated and the next grating section is considered. The followings will illustrate this principle.

128


Recall in Section D.1 that the general solution, cpz, ζ q, to the coupled-mode equations could be written as »z

c1 pz, ζ q e

jζz

c2 pz, ζ q Note that at z

»z

8

K1 pz, sqe

jζs

8

K2 pz, sqejζs ds

ds

r pζ q

»z

8

rpζ qejζz

K2 pz, sqejζs ds

rpζ q

»z

8

K1 pz, sqejζs ds.

(D.24a)

(D.24b)

Ñ 8, c1 looks like a forward propagating wave whereas c2 appears as a

backward-propagating wave. Therefore, a local reflection is expressed as c2 pz, ζ q c1 pz, ζ q ¯1 pz, ζ q ¯ r p ζ q 1 F ej2ζz 1 F¯ pz, ζ q rpζ qFF¯2ppz,z, ζζqq 1 2

rpz, ζ q

(D.25a) (D.25b)

where F¯1 pz, ζ q ejζz F1 pz, ζ q ejζz F¯2 pz, ζ q ejζz F2 pz, ζ q

ejζz

»z

K1 pz, sqejζs ds

(D.26a)

K2 pz, sqejζs ds.

(D.26b)

8

»z

8

That is, with the known rpζ q and corresponding K1 and K2 calculated from Eq. D.22, the local reflection at a distance z away from the point with rpζ q is calculable via Section D.25. To illustrate more on the use of GLM in combination with the layer peeling method, consider a grating which is divided into N sections with a section index m 1, 2, . . . , N . For the m-th section, which is ∆zm in length, the local reflections at the front and the

rm1p∆zM q. Note that at the first section, the front local reflection will be r0 rpζ q, which is the actual required grating reflection response. rm is derived by first calculating rm1 pz, ζ q where 0 ¤ z ¤ ∆zm , back of the section are rm1 and rm

pz, ζ q F¯2 pz, ζ q r p0, ζ q 1 F¯1,m 1 j2ζz m1 rm1 pz, ζ q e ¯ 1 F1 pz, ζ q rm1 p0, ζ qF¯2 pz, ζ q

(D.27)

129

Appendix D. Inverse Scattering Theory where F¯1 pz, ζ q ejζz F¯2 pz, ζ q ejζz

»z

K1,m1 pz, sqejζs ds

(D.28a)

K2,m1 pz, ζ qejζs ds.

(D.28b)

8

»z

8

Then, rm pζ q rm1 p∆zm , ζ q. The kernel functions are iteratively calculated K2,m1 pz, y q hm1 pz K1,m1 pz, y q

»z

8

yq

»z

8


K2 pz, sqhm1 ps

where hm1 pz q

1 2π

¾

y q ds

y q ds

(D.29a)

(D.29b)

rm1 pζ qejζy dy.

(D.30)

C

The iteration starts by setting K2,m1 pz, y q hm1 pz y q. The coupling constant along the m-th section is given by

pz, z q qm pz q 2Km 1

(D.31)

The grating profile is then achieved by moving to the next section of the grating until reaching the end.

D.4

GLM Equations to the Coupled-Mode Equations

This section will apply the inverse scattering theory to the coupled-mode equations for the sidewall grating. Recall the coupled-mode equations, Eq. A.24 describing the interaction between the forward-propagating and backward-propagating waves in a single-mode sidewall Bragg grating d c˜1 dz d c˜1 dz

j

∆β 2

dφ dz

jκpzqc˜

1

j

σ pz q

∆β 2

c˜1 jκpz qc˜1

dφ dz

(D.32a)

σpzq

c˜1 .

(D.32b)

130

Appendix D. Inverse Scattering Theory and also the starting equations for the inverse scattering problem, Eq. D.1, d c1 dz d c2 dz

jζc1

qc2

(D.33a)

qc1 jζc2

(D.33b)

where ζ is z-independent. This fact restricts a direct comparison between these two sets of equation. Therefore, some mathematical manipulations are required to transform Eq. D.32 such that the form resembles Eq. D.33. In Appendix A it is shown the effect of perturbation periodicity chirp is indistinguishable from the change in effective index of the waveguide; therefore, to reduce complexity one can assume that the grating period is constant, i.e.

0.

dφ dz

on z, ∆β 2

Consider the term in the parenthesis in Eq. D.32, which depends

σpzq β1 Λπ σpzq 2πnλeff pzq Λπ σpzq.

(D.34)

It is rewritten as ∆β 2

σ pz q

where

∆β0 2

2πnλeff,0 Λπ 0 ∆β0 σ˜ pzq. 2

2πδneff pz q λ

σ pz q

πδΛ Λ20

(D.35a) (D.35b)

is the former z-independent term and σ ˜ pz q is the later z-dependent term.

Define »z

c¯1 pz q c˜1 pz q exp j c¯1 pz q c˜1 pz q exp

0

j

σ ˜ pz 1 q dz 1

»z 0

σ ˜ pz 1 q dz 1

(D.36a)

(D.36b)

such that Eq. D.32 becomes d c¯1 dz d c¯1 dz


j2

»z 0

»z 0

σ ˜ pz 1 q dz 1 c¯1 ,

(D.37a)

∆β0 σ ˜ pz 1 q dz 1 c¯1 j c¯1 . 2

(D.37b)

131

Appendix D. Inverse Scattering Theory This form of the equations can be directly compared with Eq. D.33 such that ζ

∆β2 0 2πnλneff,0 Λπ ,

q pz q jκpz q exp j2

»z 0

(D.38a)

0

σ ˜ pz 1 q dz 1 .

(D.38b)

When ILP inverse scattering algorithm is initiated, the value of ζ remains unchanged throughout the grating sections, i.e. assuming the basic grating structure with an effective index of neff,0 and a perturbation period of Λ0 . Apodization and periodicity chirp are captured in the complex coupling constant, respectively κ and σ ˜ in q. If the grating is divided into many small uniform grating sections, one can write σ ˜ pz q σ ˜

σ

2πδneff λ

πδΛ Λ2

(D.39)

0

which becomes z-independent within the length of the grating section. Considering the m-th grating section with is ∆zm long, therefore, the complex coupling constant appears q

jκej2˜σ∆z

m

.

(D.40)

Previously a cross-coupling constant of a first-order grating could be written as κ j |κ| in Eq. A.36. Therefore, the complex coupling constant becomes q

|q|ejϕ |κ|ej2˜σ∆z

m

.

(D.41)

After calculating q from the inverse scattering problem, the above result could determine the grating profiles of each grating section

|q| |κ| ϕ 2˜ σ ∆zm

(D.42a) π.

(D.42b)

However, the phase difference between the front and the back of the grating section is of particular importance. Hence, σ ˜ is calculated from ∆ϕ 2˜ σ ∆zm

(D.43)

132

Appendix D. Inverse Scattering Theory Remarks on ‘c’

In the previous discussions, one have seen many forms of the coupled-mode equations, which differ in what eigenfunctions, c, are used. Firstly, the general coupled-mode equations:

jσc1 jκc1ejΦ

d c1 dz d c 1 dz

(D.44a)

1

jκ c1 ejΦ1 .

jσc1

(D.44b)

These c1 and c1 are the complex amplitude of the forward- and backward-propagating waves. They determine the energy that is carried by those waves. One then defines c˜1 pz q c1 pz qej

and c˜1 pz q c1 pz qej

Φp 2

Φp 2

(D.45)

such that the second form looks like d c˜1 dz d c˜1 dz

dφ j σ c˜1 jκ˜c1 dz

∆β dφ jκ c˜1 j 2 dz σ c˜1. ∆β 2

(D.46a) (D.46b)

The final form that is used to comply with the inverse scattering setup. In this form, one defines again »z

c¯1 pz q c˜1 pz q exp j

c¯1 pz q c˜1 pz q exp

0

j

σ ˜ pz 1 q dz 1

»z 0

(D.47a)

σ ˜ pz 1 q dz 1 ,

(D.47b)

and this allows to write the third form d c¯1 dz d c¯1 dz


j2

»z 0

»z 0

σ ˜ pz 1 q dz 1 c¯1

(D.48a)

∆β0 σ ˜ pz 1 q dz 1 c¯1 j c¯1 . 2

(D.48b)

The desired reflection response is defined by r

cc1ppzz0qq . 1

0

(D.49)

133

Appendix D. Inverse Scattering Theory For a uniform grating setting z0 r

0, it could be shown that

cc1pp00qq c˜c˜1pp00qq c¯c¯1pp00qq . 1

1

(D.50)

1

Hence, one can still use this desired reflection response r in the starting reflection response rpζ q for the inverse scattering.

D.5

Summary

The GLM theory is discussed and found that the unique solution exists for a finite grating by solving the iteration equation. In order to reduce the number of iteration loops, the layer peeling method is introduced to the GLM theory process by breaking the grating into many small pieces. Lastly, the coupled-mode equations are rewritten and inserted into the inverse scattering framework to be ready for disposal, i.e. in the Chapter 4.

Appendix E Simulation Results for Grating Retrieval E.1

Uniform Gratings

Considering a uniform grating of 1.4-µm wide and 200-µm long with 50-nm recess depth and 250-nm grating period. The coupling constants are σ

1.546 104 m1

κ j9.847 103 m1 .

The reflection response of this grating, after multiplied with a 1-ps time delay, is the input to the inverse scattering algorithm, whose initial starting parameters are Ng 400, ∆z

1 µm, w0 1.4 µm, and Λ0 250 nm.

The calculated complex coupling

coefficient is shown in Fig. E.1. The result shows a significant cross-coupling coefficient

50 to 250 µm. The coupling magnitude agreement with |κ| 9, 847 m1 of the original

in the 200-µm-long region, i.e. between z is close to 10, 000 m1 , which is in

input grating. The phase relationship between adjacent subgratings, ∆ϕ, is relatively constant within the range of significant coupling coefficient implying a uniform width across the grating. The waveguide width and the recess depth profiles are matched from the matching algorithm and are depicted in Fig. E.2. The recess depth profile shows the 134

135

Appendix E. Simulation Results for Grating Retrieval

perturbation approximately at 50 nm within the region of 200-µm long and negligible perturbation outside this region. A constant waveguide width around 1.4 µm is apparent in the result as well. However in Fig. E.5a, it also shows severe fluctuations outside the aforementioned region. The meaningful values of the waveguide width nonetheless lies within the perturbation region from the recess depth profile.

12000

0.1

8000

0 ∆ψ

−1

|q| (m )

10000

6000 4000

−0.1

2000 0 0

50 100 150 200 250 300 350 400 z (µm)

−0.2 0

50 100 150 200 250 300 350 400 z (µm)

(a) |q |

(b) ∆ϕ

Figure (E.1): The complex coupling constant calculated from the input response from a uniform grating of 1.4-µm wide and 200-µm long with 50-nm recess depth and 250-nm grating period.

This fluctuation is mainly due to the matching algorithm. In the subgratings where their local coupling coefficient magnitudes are found to be very small, the recess depth values, which is determined solely based on the magnitude of the complex coupling coefficient q, are low and exhibit no fluctuation. On the other hand, the waveguide width is inferred from a relative phase between two consecutive complex coupling coefficients, i.e. by ∆ϕ between

2˜σ∆z.

Particularly in the region of |q |

Ñ 0, the phase can fall anywhere

π to π; hence, the relative phase can vary significantly.

Therefore, the cal-

culated waveguide width fluctuates considerably in this region. The algorithm can be improved by imposing a criteria of minimum recess depth resolution, which could be set from a fabrication capability perspective. The adjustment is made such that if the calculated recess depth is smaller than the minimum recess depth feature, it is reset to zero and the waveguide width assumes the specified bare waveguide width. Assume that


136

the minimum recess depth feature is 5 nm, the waveguide width and the recess depth profiles calculated from the modified matching algorithm are calculated and shown in

1.5

60

1.45

50

recess depth (nm)

width (µm)

Fig. E.2c and Fig. E.2d.

1.4 1.35 1.3 1.25 0

40 30 20 10 0 0

50 100 150 200 250 300 350 400 z (µm)

(a) Waveguide width

50 100 150 200 250 300 350 400 z (µm)

(b) Recess depth 60 recess depth (nm)

width (µm)

1.5 1.45 1.4

50 40 30 20 10

1.35 0

50 100 150 200 250 300 350 400 z (µm)

(c) Waveguide width

0 0

50 100 150 200 250 300 350 400 z (µm)

(d) Recess depth

Figure (E.2): The waveguide width and the recess depth profiles matched from the corresponding complex coupling coefficient of a uniform grating response.

The reflection response of the calculated grating when the subgrating points are chosen from Ni

48 to Ni 248 is then determined using the direct scattering algorithm

as previously investigated. The result is shown in Fig. E.3. The response of the simulated grating is in good agreement with the target response especially near the central frequency around 1.55 µm.

137


1

6 Simulated Target

0.9

5

0.8 0.7

4

τ (ps)

|r|

0.6 0.5 0.4

3 2

0.3 0.2

1

Simulated Target

0.1 0 1.53 1.535 1.54 1.545 1.55 1.555 1.56 1.565 1.57

0 1.53 1.535 1.54 1.545 1.55 1.555 1.56 1.565 1.57

λ (µm)

λ (µm)

(a) Reflection amplitude

(b) Time delay

Figure (E.3): Response of a grating generated by the inverse scattering algorithm with the target response from a uniform grating.

E.2

Linearly Width-Chirped Gratings

The inverse scattering algorithm is now tested with a more complicated example. Consider a grating with a linear chirp in the waveguide width profile. The waveguide width increases from 1.2 µm to 1.4 µm with a uniform subgrating increment of 1 µm in length and a grating period of 250 nm. The recess depth profile is kept constant at 50 nm across the grating. The total number of subgratings is 200 corresponding to the total length of 200 µm. The initial parameters for the inverse scattering algorithm remain similar to the previous section. The magnitude of the complex coupling coefficient is plotted in Fig. E.4, with the magnitude of the cross-coupling coefficient of the target grating. The result shows very good agreement between magnitudes of the calculated and input coupling coefficients. The relative phase exhibits a linear increase within the region of significant coupling. This result is deconvoluted to retrieve the waveguide width and the recess depth, which are shown in Fig. E.5. In the region of significant coupling value, the recess depth appears close to 50 nm on average and the waveguide width increases from 1.2 µm and 1.4 µm as it should be. Even though the modified matching algorithm is used, the fluctuations in both profiles exist outside the region of significant coupling. The effect of this fluctuation

138


deteriorates in the recess depth profile if its value is large. However, since the region of significant coupling can be determined from the magnitude of the complex coupling coefficient, pieces of the calculated grating to be used could be picked manually. The reflection response of this calculated grating when the subgrating sections from Ni to Ni

248 are chosen is shown in Fig. E.6.

48

The amplitude responses are similar. The

time delay responses show anomalies when the two are compared together. However, both of them possess similar increase in time delay against wavelength. 4

2

x 10


0.3 0.2 0.1 ∆ψ

−1

|q| (m )

1.5 1

0 −0.1 −0.2

0.5

−0.3 0 0

−0.4 0

50 100 150 200 250 300 350 400 z (µm)

(a) |q |

50 100 150 200 250 300 350 400 z (µm)

(b) ∆ϕ

Figure (E.4): Complex coupling coefficient calculated for a response of a width-chirped grating

1.8

60 recess depth (nm)

1.7 width (µm)

1.6 1.5 1.4 1.3 1.2

40 30 20 10

1.1 1 0

50

50 100 150 200 250 300 350 400 z (µm)

(a) Waveguide width

0 0

50 100 150 200 250 300 350 400 z (µm)

(b) Recess depth

Figure (E.5): Matched waveguide width and recess depth profiles


1

139

8 Simulated Target

0.9

6

0.8

Simulated Target

0.7 4

τ (ps)

|r|

0.6 0.5 0.4 0.3

2 0

0.2 0.1

−2

0 1.53 1.535 1.54 1.545 1.55 1.555 1.56 1.565 1.57

1.53 1.535 1.54 1.545 1.55 1.555 1.56 1.565 1.57

λ (µm)

λ (µm)



Figure (E.6): Response of a grating generated by the inverse scattering algorithm with the target response from a width-chirped grating.

E.3

Gaussian-Apodized Gratings

In the previous examples, the recess depth is kept constant while the waveguide width changes along the grating profile. Now consider the opposite situation with a special case of a Gaussian apodization. Firstly from the known relationship between the coupling coefficient and the recess depth calculated for the direct scattering problem, the recess depth profile to yield a Gaussian coupling coefficient profile is determined. The grating has a constant waveguide width of 1.4 µm and a grating period of 250 nm. In calculating the response of this grating, it is divided into 200 uniform pieces with a length of 1 µm, leading to a total length of 200 µm. This response will now be the target response for the inverse scattering algorithm, whose initial parameters maintain the values used in the preceding sections. The determined complex coupling coefficient is displayed in Fig. E.7. Its amplitude corresponds very well with the input cross-coupling coefficient. The relative phase shows a flat Gaussian feature in the significance region reflecting the self-coupling coefficient in its expression. The matching algorithm yields the profiles of the waveguide width and the recess depth as plotted in Fig. E.8. The waveguide width is determined to be nearly constant at 1.4 µm as expected and the recess depth assumes the values in excellent


140

agreement with that of the original grating within the significance region. With the subgrating pieces from 48 to 248, the response of this generated grating is found to be those shown in Fig. E.9. 4

2

x 10


0.2 ∆ψ

−1

|q| (m )

1.5 1

−0.2

0.5 0 0

0

−0.4 0

50 100 150 200 250 300 350 400 z (µm)

(a) |q |

50 100 150 200 250 300 350 400 z (µm)

(b) ∆ϕ

Figure (E.7): Complex coupling coefficient calculated for a response of a Gaussian-apodized grating.

recess depth (nm)

1.9 width (µm)

1.8 1.7 1.6 1.5 1.4 1.3 0

50 100 150 200 250 300 350 400 z (µm)

(a) Waveguide width

90 80 70 60 50 40 30 20 10 0 0


50 100 150 200 250 300 350 400 z (µm)

(b) Recess depth

Figure (E.8): Matched waveguide width and recess depth profiles

141


1

8 Simulated Target

0.9

6

0.8 0.7

4

τ (ps)

|r|

0.6 0.5 0.4

2 0

0.3 0.2

−2

Simulated Target

0.1 0 1.53 1.535 1.54 1.545 1.55 1.555 1.56 1.565 1.57

−4 1.53 1.535 1.54 1.545 1.55 1.555 1.56 1.565 1.57

λ (µm)

λ (µm)



Figure (E.9): Response of a grating generated by the inverse scattering algorithm with the target response from a Gaussian-apodized grating.

E.4

Apodized and Linearly-Chirped Gratings

In the previous cases so far, either the waveguide width or the recess depth is held constant along the grating. Now consider the situation when both of them vary. Specifically, the grating will be linearly-chirped and Gaussian-apodized in such a way that it is a combination of the gratings in Section E.2 and Section E.3. The complex coupling coefficient is determined and plotted in Fig. E.10. Its magnitude traces the magnitude of the initial cross-coupling coefficient with great correspondence. The relative phase, as shown in Fig. E.10b, exhibits the combination of linear and Gaussian features. The physical profiles are matched from the complex coupling coefficient. The waveguide width profile corresponds well with the linear increase of the starting grating, and so does the recess depth, as displayed in Fig. E.11. Selecting the subgratings within the significance region, the response of the generated grating is calculated and plotted in Fig. E.12.


3

x 10

4


2.5

0.2

2 ∆ψ

−1

|q| (m )

142

1.5

0

1 −0.2 0.5 0 0

−0.4 0

50 100 150 200 250 300 350 400 z (µm)

50 100 150 200 250 300 350 400 z (µm)

(a) |q |

(b) ∆ϕ

Figure (E.10): Complex coupling coefficient calculated for a response of a Gaussian-apodized and linearly-chirped grating.


width (µm)

1.7 1.6 1.5 1.4 1.3 1.2 1.1 0

Simulated Target grating 50 100 150 200 250 300 350 400 z (µm)

(a) Waveguide width

90 80 70 60 50 40 30 20 10 0 0


50 100 150 200 250 300 350 400 z (µm)

(b) Recess depth

Figure (E.11): Matched waveguide width and recess depth profiles.

143


10

1 Simulated Target

0.9 0.8 0.7

5 τ (ps)

|r|

0.6 0.5 0.4

0

0.3 0.2

Simulated Target

0.1 0 1.53 1.535 1.54 1.545 1.55 1.555 1.56 1.565 1.57

λ (µm)


−5 1.53

1.54

1.55 λ (µm)

1.56

1.57


Figure (E.12): Response of a grating generated by the inverse scattering algorithm with the target response from a Gaussian-apodized and linearly-chirped grating.

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