Pulse Shaping Based on Integrated Waveguide Gratings...2.10 Flat-top pulse shaping using pulse stacking and pulse di erentiation.. .32 2.11 DST pulse shaping implemented with integrated

Pulse shaping based on integrated waveguide gratings

by

Pisek Kultavewuti

A thesis submitted in conformity with the requirementsfor the degree of Master of Applied Science

Graduate Department of Electrical and Computer EngineeringUniversity 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 one-

to-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 . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Free-Space Pulse Shaping . . . . . . . . . . . . . . . . . . . . . . . . . . 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

2.3 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

2.4 Integrated Optics Pulse Shaping . . . . . . . . . . . . . . . . . . . . . . . 33

iv

2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3 Grating Responses 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

v

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. . . . . . 44

viii

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. . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.5 Effective indices of the fundamental TE-like and TM-like modes as a func-

tion 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. . 51

3.8 Self-coupling coefficients as a function of the recess depths and the waveg-

uide widths by the surface fitting function at the wavelength of 1.55 microns. 52

4.1 Windowing function. f1pxq corresponds to x1{2 � 3 and xd � 1 wherease

f2pxq is plotted for x1{2 � xd � 3. . . . . . . . . . . . . . . . . . . . . . . 65

4.2 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 com-

pared 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. . . . . . . . . . . . . . . . . . . . . . 74

ix

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.5 Electric field magnitudes of output waveforms corresponding to generated

gratings with different sets of subgrating involved. . . . . . . . . . . . . . 76

5.6 Output waveforms from generated gratings aiming to produce flat-top

pulses with durations of 0.5, 1, and 2 picoseconds. . . . . . . . . . . . . . 76

5.7 Responses and performance of the generated grating when random devia-

tions are introduced to the waveguide width and the recess depth profiles. 77

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. . . . . . . . . 79

5.9 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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

x

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

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 � 100µm. The effective index disper-

sion is not taken into account. . . . . . . . . . . . . . . . . . . . . . . . . 111


sion is now taken into account. . . . . . . . . . . . . . . . . . . . . . . . . 112


sion is taken into account. . . . . . . . . . . . . . . . . . . . . . . . . . . 113


sion is taken into account. . . . . . . . . . . . . . . . . . . . . . . . . . . 114

C.5 Reflection response of a linearly chirped grating with ∆Λ � 4 nm and

Λ0 � 250 nm. The simulation is implemented with Ng � 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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

xi

C.6 Reflection response of a linearly tapered grating with the waveguide width

increasing from 1.0µm to 1.6µm. 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 waveg-

uide 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 com-

plementary continuous grating (red dashed line). All grating sections are

uniform: a waveguide width of 1.4µm, 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 cor-

responding 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


the target response from a width-chirped grating. . . . . . . . . . . . . . 139

xii

E.7 Complex coupling coefficient calculated for a response of a Gaussian-

apodized grating. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

E.8 Matched waveguide width and recess depth profiles . . . . . . . . . . . . 140


the target response from a Gaussian-apodized grating. . . . . . . . . . . 141

E.10 Complex coupling coefficient calculated for a response of a Gaussian-

apodized and linearly-chirped grating. . . . . . . . . . . . . . . . . . . . . 142

E.11 Matched waveguide width and recess depth profiles. . . . . . . . . . . . . 142


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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2 Third-order polynomial coefficients for the fitting functions of (a) the TE-

like effective index, neff,TEpw, λq � zpx, yq, and (b) the TM-like effective

index, neff,TMpw, λq � zpx, yq. The free variables x and y are the waveguide

widths, w, and the wavelengths, λ. . . . . . . . . . . . . . . . . . . . . . 47

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. . . . . . . . . . . . . . 51

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. . . . . . . . . . . . . . 52

xiv

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 car-

ried 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 elec-

tromagnetic 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 demultiplex-

ing, 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


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 com-

munications 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 pro-

cessing 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 appro-

priate 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,


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 the-

sis. 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 re-

sponse. 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


devised and shows ability to deal with high-reflectivity grating. Numerical implemen-

tations 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-to-

one 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 triangu-

lar. 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 waveg-

uide 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


(GLM) theory and the layer peeling method (LPM). The inverse scattering itself is dis-

cussed 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 spec-

tral ingredients in amplitude and/or phase. Pulse shaping that manipulates these spec-

tral 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 dis-

persed 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

Chapter 2. Literature Review 8

or a linearly-chirped grating, to temporally disperse different frequencies due to differ-

ent 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, Hpωq,or an impulse response, hptq. If the input and output waveforms are xptq and yptq in a

time domain, with corresponding Xpωq and Y pωq in a frequency domain, there exist the

relations

yptq � hptq xptq, (2.1a)

Y pωq � HpωqY pωq. (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 ap-

proximation or the weak-grating limit. It was shown that the temporal impulse response

of a uniform FBG, described by npzq � nav�∆nmaxApxq cosp 2πΛ0z�φpzqq, is proportional

to the scaled apodization profile, Apzq, and the phase profile, φpzq, [16]:

hrptq 9 Apzqejφpzq(

z�ct{2nav, (2.2)

where nav is the average refractive index of the FBG and z � ct{2nav is the space-to-time

scaling relationship. The first-Born approximation requires that κL ! 1, where κ is the

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


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 re-

quirement of this method is that the apodized LC-FBG must have a constant first-order

dispersion coefficient, :Φν � B2ΦpνqBν2 , that is large enough. Different spectral components of

the input pulse are reflected by different local sections of the LC-FBG with different re-

flectivity corresponding to the local apodization. Sufficient dispersion is required in order

to efficiently separate different frequencies. This scheme is termed space-to-frequency-to-

time mapping. The mapping [16] is mathematically expressed as

hrptq 9 ej π:Φνt2

tanh

�mA

�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

:Φν � �2navL

c∆ν(2.4)

where ∆ν is the chirp bandwidth of the LC-FBG whose spatial reflected frequency is

written as νpzq � ν0 � pz � L2q∆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.


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],

Eoutptq 9 Einptq 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:

Eoutptq �N

i�1

Eipt� tiq. (2.6)

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 Eptq �Aptqejωct, 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

Eoutptq � ejωct�Aptq � Apt� τqe�jωcτ� . (2.7)

The term e�jωcτ leads to interference. However, if the time delay is adjusted such that


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

envelopes of the replica. Note that Apt � τq � Aptq δpt � τq. Therefore, Eq. 2.7

becomes

Eoutptq � ejωct Aptq �

δptq � δpt� τqe�jωcτ�( . (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.

Ioutptq �N

i�1

Iipt� tiq (2.9)

This could be referred to as incoherent pulse shaping. Light sources that generate incoher-

ent 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, mir-

rors, 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 in-

clude [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


generation possible without the use of an ultrafast modulator. Albeit setup configura-

tions 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 con-

figuration [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 zero-

dispersion requires the exact similarities between the first and second sets of gratings


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 ele-

ment 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 micro-

mirror 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 Einpωq 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

Eoutpωq � M pxpωqqEinpωq, where Mpxq 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


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 Mpxqgpxq, where gpxq 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 simpli-

fied 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 magni-

tude of the applied potential as well as the thickness of the pixel. A useful liquid crystal


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 computer-

controlled. 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 phase-

only 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.

The two liquid crystal modulators align at �45� and �45� with respect to the y-axis.

Assume that the input pulse is the y-polarized light, E � yE0 cospωtq. The electric field


of the input could be decomposed onto x1 and y1 axes forming along the long axes of the

two liquid crystal modulators: E � px1E0� y1E0q cospωtq{?2. After passing through, the

electric field experiences phase differences:

E � x1E0 cospωt�∆φ1q{?

2� y1E0 cospωt�∆φ2q{?

2 (2.10a)

� xE0

2pcospωt�∆φ1q � cospωt�∆φ2qq

�yE0

2pcospωt�∆φ1q � cospωt�∆φ2qq , (2.10b)

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

the y component:

Eout � y

#E0 cos

�∆φ1 �∆φ2

2

+cos

�ωt� ∆φ1 �∆φ2

2

. (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


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 radio-

frequency 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 radio-

frequency 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].


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 wave-

form 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


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:

eoutptq 9 einptq s

��βγt

, (2.12)

where γ � λcd cos θd

is the spatial dispersion term and β � cos θicos θd

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.


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 ele-

ments 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].


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


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 ma-

neuver 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 shap-

ing 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


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 grat-

ings. 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 real-

world applications. Some common types are a fiber Bragg grating (FBG), a linearly-

chirped 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 oc-

curs in a multi-mode fiber where different modes propagate with different velocities and


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 forward-

propagating 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 compres-

sion, 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 propa-

gating through a dispersive waveguide, e.g. an optical fiber, will experience chirping in

their instantaneous frequencies, which can eventually broadens or compresses the tempo-

ral 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 disper-

sion. The group velocity dispersion (GVD) coefficient could be derived as

Dω � d

dω

1

vg� ω

c

d2npωqdω2

. (2.13)


Other notations include Dν � dpv�1g q{dν and Dλ � dpv�1

g q{dλ when vg � ωcdndω

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 dis-

persive waveguide results in temporal pulse broadening because as the pulse propagates

different frequency components traverse with different group velocities and therefore grad-

ually 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 waveg-

uide 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-


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 � �n2Iptqk0z. (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

ej2n2I0k0zt2{τ2

(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


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, Hpωq, that acts on an input pulse spectrum, Xpωq, in a way that an output

pulse spectrum, Y pωq, becomes

Y pωq � HpωqXpωq. (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. |Hpωq| ¤ 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. |Hpωq| ¤ 0.2, the first Born approximation can

apply and yield a relation between the impulse response of the grating and the magnitude


of index perturbation, or apodization Apzq, essentially captured in Eq. 2.2 for FBGs [16]:

hrptq 9!Apzqejφpzq

)z�ct{2nav

(2.17)

From the relationship, if the input is regarded as an impulse, the output pulse will behave

the same as the impulse response, i.e. eoutptq � hrptq. 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

hrptq 9 ej π:Φνt2

tanh

�mA

�z � ct

2nav

�, (2.18)

where :Φν is the first-order dispersion coefficient of a linearly chirped fiber Bragg grating

(LC-FBG), m is a constant, and Apzq is the apodization profile. Basically the grating re-

flects 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, Apzq. 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 map-

ping. 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


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 neg-

ligible 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


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 in-

coupled 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 dis-

cussed 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


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


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, Hpωq � �jpω�ω0q. A series of uniform LPGs


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. Hpωq � �1{jpω � 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 pro-

posed 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 semicon-

ductor 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.


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 inves-

tigated for integrated optics. Popularized by the microelectronics, a silicon-on-insulator

(SOI) system promises compactness and integrability with electronics circuit. Microfab-

rication 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 alu-

minum gallium arsenide, AlxGa1�xAs, 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


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 arrayed-

waveguide grating (AWG). Fig. 2.11a shows the conventional configuration of the arrayed-

waveguide 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


(a) Conventional AWGs. (b) AWG-based DST pulse

shaping operating in transmission.

(c) AWG-based DST pulse shaping

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 com-

plex 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 chemi-

cal/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.


(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


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 per-

turbation 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 dpzq 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,


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 re-

quires nonuniform gratings, sidewall-etching geometry becomes promising as a simple yet

efficient grating perturbation method.

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

© 2008 OSA.

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

© 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


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],

∇2Kepx, yq � �

ω2µεwpx, yq � β2�epx, yq � 0, (3.1)

where epx, yq represents the guided mode field, εwpx, yq is the permittivity function of a

waveguide, and β is the propagation constant.

The waveguide structure is based on AlxGa1�xAs, 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

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12.8

2.9

3

3.1

3.2

3.3

3.4

x

n

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 Al0.7Ga0.3As, (2) 0.5-µm Al0.3Ga0.7As, (3) 0.2-

µm Al0.2Ga0.8As, (4) 0.2-µm Al0.7Ga0.3As, 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, εwpx, yq. 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.


(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 in-

dices. 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, yq, are assigned to the nodes. The effective index is related to

the propagation constant via β � 2πneff

λ.

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.


(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


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.


1 1.1 1.2 1.3 1.4 1.5 1.63.06

3.07

3.08

3.09

3.10

3.11

3.12

3.133.13

w (µm)

neff

TE

TM

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

lengths of λ � 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 funda-

mental 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) 1.0 1.2 1.3 1.4 1.6

neff(TE) 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


order polynomial is expressed as

zpx, yq � p0,0 � p1,0x� p0,1y � p2,0x2 � p1,1xy � p0,2y

2

�p3,0x3 � p2,1x

2y � p1,2xy2 � p0,3y

3 (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,TEpw, λq � zpx, yq, and (b) the TM-like effective index, neff,TM pw, λq � zpx, yq. The free

variables x and y are the waveguide widths, w, and the wavelengths, λ.

(a)

Coeff. Values 95%-confidence bounds

p0,0 3.821 (2.697, 4.964)

p1,0 0.2462 (0.1235, 0.3688)

p0,1 -1.194 (-3.396, 1.008)

p2,0 -0.3188 (-0.3357, -0.3019)

p1,1 0.3851 (0.2323, 0.5379)

p0,2 0.3613 (-1.056, 1.778)

p3,0 0.09912 (0.0963, 0.1019)

p2,1 -0.1087 (-0.117, -0.1005)

p1,2 0.001372 (-0.04744, 0.05018)

p0,3 -0.06304 (-0.3674, 0.2413)

(b)

Coeff. Values 95%-confidence bounds

p0,0 3.819 (3.206, 4.433)

p1,0 0.1862 (0.1204, 0.2521)

p0,1 -1.122 (-2.304, 0.05988)

p2,0 -0.2068 (-0.2159, -0.1978)

p1,1 0.2294 (0.1473, 0.3114)

p0,2 0.3972 (-0.3635, 1.158)

p3,0 0.05945 (0.05794, 0.06097)

p2,1 -0.0589 (-0.06333, -0.05448)

p1,2 -0.003226 (-0.02943, 0.02298)

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


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, higher-

order 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],

d

dzc1pzq � j

�∆β

2� σ

c1pzq � jκc�1pzq, (3.3a)

d

dzc�1pzq � jκ�c1pzq � j

�∆β

2� σ

c�1pzq, (3.3b)

where c1 and c�1 represent the forward- and the backward-propagating waves, and the

detuning parameter is ∆β � 2πΛ0�2β. The cross- and self-coupling coefficients are defined

as

κ � κ1,�1r�1s � ω2µ xe1|∆εpx, yqr�1s|e1y2βn xe1|e1y , (3.4a)

σ � σnr0s � ω2µ xe1|∆εpx, yqr0s|e1y2βn xe1|e1y . (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, yq is

εpx, y, zq �

$''&''%ε0 ; 0 ¤ z Λ{2,

εwpx, yq ; Λ{2 ¤ z Λ.

(3.5)


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

The first-order permittivity modulation is then captured in ∆εpx, yqr�1s as

∆εpx, yqr�1s �

$''&''%�j εwpx,yq�ε0

π; px, yq in the etched area,

0 ; otherwise,

(3.6)

where as the zeroth-order permittivity modulation ∆εpx, yqr0s is

∆εpx, yqr0s �

$''&''%

ε0�εwpx,yq2

; px, yq in the etched area,

0 ; otherwise.

(3.7)

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.


The coupling constants depend on the mode shape, the effective index, the permittiv-

ity 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 calcula-

tion 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 represen-

tative 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 ∆εpx, yqr1s 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 zpx, yq � °m,n pm,nx

myn, where

0 ¤ m ¤ 5, 0 ¤ n ¤ 3, and m � n ¤ maxt5, 3u. 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 per-

mittivity perturbation, ∆εpx, yqr0s. Since ∆εpx, yqr0s 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.


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 25 50 75 100 125 150 175 200

|κpTEq|pcm�1q 0 46.09 97.95 167.71 262.36 388.51 552.32 759.47 1,023.2

|κpTMq|pcm�1q 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 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 polyno-

mial fit functions were used as a database for calculating the grating response which is

the focus of this section.


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 25 50 75 100 125 150 175 200

|σpTEq|pcm�1q 0 85.87 184.13 316.18 495.39 734.36 1,044.9 1,437.9 1,933.4

|σpTMq|pcm�1q 0 86.05 213.11 395.4 641.88 960.81 1,359.7 1,845.2 2,435.0

(a) |σpTE; rd, wq| (b) |σpTM ; 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 � 1{∆t∆ν. The frequency space spans

from �N∆ν2

to N∆ν2

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


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 � tλju or ν � tνju, (3.9)

where j � 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

�� c1pzqc�1pzq

��

��m11pz, z0q m12pz, z0qm21pz, z0q m22pz, z0q

�� c1pz0qc�1pz0q

�� Mpz, z0q


�� , (3.10)

where, when defining s �b|κ|2 � �

∆β2� σ

�2,

m11pz, z0q � cosh�spz � z0q

� j

�∆β

2� σ

sinh�spz � z0q

s

, (3.11a)

m12pz, z0q � �j κs

sinh�spz � z0q

, (3.11b)

m21pz, z0q � jκ�

ssinh

�spz � z0q

, (3.11c)


� j

�∆β

2� σ

sinh�spz � z0q

s

. (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


declared and initiated in 1�Ng vectors:

w � twiu, (3.12a)

rd � trdiu, (3.12b)

Λ � tΛiu, (3.12c)

∆z � t∆ziu, (3.12d)

where i � 1, 2, . . . , Ng. 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 � �2πneff,ipλjqλj

� π

Λi

. (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,

neffpλcq,∆βi,1 � �2πneffpλcq

λj� π

Λi

. (3.14)

The self- and cross-coupling constants are calculated using the profiles of the waveg-

uide width and the recess depth:

σ � tσiu, (3.15a)

κ � tκiu. (3.15b)

Using the transfer matrix method, the system matrix is consequently

Msys � MNgMNg�1 � � �M2M1 �

��M11 M12

M21 M22

�� . (3.16)

The reflection response of a specific frequency point becomes

rj � rrjs � �M21

M22

. (3.17)


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. Ex-

tensive 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. Waveg-

uide 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 evalu-

ates 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 wave-

forms, 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 coeffi-

cients 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

Chapter 4. Retrieval of the Gratings 57

[36].

The currently unknown grating is disintegrated into Ng connected uniform subgrat-

ings. 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

dzc1 � jζc1 � qpzqc2, (4.1a)

d

dzc2 � q�pzqc1 � jζc2, (4.1b)

where ζ is the z-independent eigenvalue and qpzq 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],

rm�1pz, ζq � ej2ζzrm�1p0, ζqr1� F �

1,m�1pz, ζqs � F2,m�1pz, ζqr1� F1,m�1pz, ζqs � rm�1p0, ζqF �

2,m�1pz, ζq, (4.2)

qmpzq � 2K�2,m�1pz, zq, (4.3)

where 0 ¤ z ¤ ∆zm. rm is the reflection response of the m-th subgrating. These

two equations depend on the kernel functions Ki,m�1pz, yq, which have to be calculated

iteratively from, [35],

K2,m�1pz, yq � �hm�1pz � yq �z»

�8

K�1,m�1pz, sqhm�1ps� yq ds, (4.4a)

K1,m�1pz, yq � �z»

�8

K�2,m�1pz, sqhm�1ps� yq ds, (4.4b)

where hpzq is the space-scaled impulse response:

hm�1pzq � 1

2π

8»�8

rm�1pζqe�jζy dy. (4.5)

The functions F1 and F2 are defined, [36],

F1pz, ζq � e�jζzz»

�8

K1,m�1pz, sqejζs ds, (4.6a)

F2pz, ζq � ejζzz»

�8

K2,m�1pz, ζqejζs ds. (4.6b)


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

dzc1 � j

�∆β

2� dφ

dz� σpzq

c1 � jκpzqc�1, (4.7a)

d

dzc�1 � jκ�pzqc1 � j

�∆β

2� dφ

dz� σpzq

c�1, (4.7b)

where c1 and c�1 represent a forward- and backward-propagating waves, φ is the chirp

function, κpzq is the cross-coupling coefficient, and σpzq 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 �

��2πneff,0

λ� π

Λ0

��σpzq � 2πδneffpzq

λ� πδΛ

Λ20

� ∆β0

2� σpzq. (4.8)

By defining

c1pzq � c1pzq exp

�j

» z

0

σpz1q dz1, (4.9a)

c�1pzq � c�1pzq exp

��j

» z

0

σpz1q dz1, (4.9b)

the coupled-mode equations could be expressed with a z-independent eigenvalue

d

dzc1 � j

∆β0

2c1 � jκpzq exp

�j2

» z

0

σpz1q dz1c�1, (4.10a)

d

dzc�1 � jκpzq� exp

��j2

» z

0

σpz1q dz1c1 � j

∆β0

2c�1. (4.10b)

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


couple to each other by allowing

ζ � ∆β0

2, (4.11)

qpzq � �jκpzq exp

�j2

» z

0

σpz1q dz1. (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 � ttiu (4.14a)

ν � tνiu (4.14b)

z � ct

neff

� c

neff

ttiu � tziu, (4.14c)

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 � ∆β0

2� �2πneff,0

λ� π

Λ0

. (4.15)


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 � truNi�1 � rpλiq � rpνiq � rpζiq. (4.16)

This set of points is used to calculate the impulse response hris by

hptiq � h � fftshift(ifft(ifftshift(h)))/dt. (4.17)

Through a direct mapping of time and space by the light speed, one can interpret hptiqas the space-dependent function:

h � hris � hptiq � hpziq. (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� yq ds. (4.19)

Define a new function as, [36],

ADpz, sq �

$''&''%A�pz,�sq ; � s ¤ z,

0 ; z �s.(4.20)

Eq. 4.19 can be rewritten as

z»�8

A�pz, sqhps� yq ds �8»

�8

ADpz,�sqhps� yq ds (4.21a)

�8»

�8

ADpz, y � sqhpsq ds. (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


of the products of those two Fourier transforms, i.e.

z»�8

A�pz, sqhps� yq ds � F�1!FtADpz, yquFthpyqu

). (4.22)

The next step is to apply this same trick for F1 and F2 in Eq. 4.6. Their integral

terms arez»

�8

Kipz, sqejζs ds. (4.23)

One can define

KD �

$''&''%Kipz, sq ; s ¤ z,

0 ; z s,

(4.24)

which leads to

z»�8

Kipz, sqejζs ds �8»

�8

KDpz, sqejζs ds � F!KDpz, sq

). (4.25)

4.4 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 � ∆β0

2� �2πneff,0

λ� π

Λ0

. (4.26)

The values of neff,0 and Λ0 are needed and chosen by the best guesses. From the re-

quirement 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 pre-

vious 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


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 NgNsg. 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; (4.27)

where h � hm, r � rm, and m is the index of the subgrating. The complex coupling

coefficients are shown to be

qpzq � 2K�2 pz, zq where 0 ¤ z ¤ ∆z, (4.28)

as in Eq. 4.3, [35]. The algorithm tries to find qp∆zq as a presentative of the subgrating.

Hence, the value of K2pz � ∆z, y � ∆zq must be determined from Eq. 4.5. Firstly,

the term hmpz � yq is interpreted as a function of hmpyq but with a shift of z to the


left (right) if z is positive (negative). Since in the discrete Fourier transform theorem

hmpyq is periodic, hmpz � yq could be determined by appropriately shifting and cycling

the values of hmpzq. In MATLAB, this step takes the form of

h2 = [h1(shift+1:end), h1(1:shift)];, (4.29)

where h1 and h2 are hmpyq and hmpz � yq 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]:

K2pz, yq � �hpz � yq, (4.30a)

K1pz, yq � �z»

�8

K�2 pz, sqhps� yq ds. (4.30b)

The later iteration loops take the original forms

K2pz, yq � �hpz � yq �z»

�8

K�1 pz, sqhps� yq ds, (4.31a)

K1pz, yq � �z»

�8

K�2 pz, sqhps� yq ds. (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;


in which d is the discretized vector representing the integral. Therefore, the kernel

functions K1pz, sq and K2pz, sq can be calculated iteratively. After they are determined,

the complex coupling coefficient at the point z � ∆z is

q(i1) = 2conj(K2(zri));, (4.33)

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, fpxq, is defined as

fpxq �

$''''''''''&''''''''''%

0.5� 0.5 cos�πxdpx� x1{2�xd

2q

; � x1{2�xd2

x �x1{2�xd2

,

1 ; � x1{2�xd2

x x1{2�xd2

,

0.5� 0.5 cos�πxdpx� x1{2�xd

2q

;x1{2�xd

2 x x1{2�xd

2,

0 ; otherwise,

(4.34)

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 fpx � νq that has x1{2 � 25 THz and xd � 5 THz.

At the end of the algorithm, the complex coupling constant representing the subgrat-

ings 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. qpzq, is unique to a given reflection

response, the matching to physical parameters can give different results depending on


−4 −3 −2 −1 0 1 2 3 40

0.2

0.4

0.6

0.8

1

x

f(x)

f1(x)

f2(x)

Figure (4.1): Windowing function. f1pxq corresponds to x1{2 � 3 and xd � 1 wherease f2pxq is plotted

for x1{2 � xd � 3.

the matching criterion.

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

q � |q|ejϕ, is shown to provide

|q| � |κ|, (4.35a)

∆ϕ � 2σ∆z, (4.35b)

σ � σ � 2πδneff

λ� πδΛ

Λ20

, (4.35c)

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.


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 � 400, Λ0 � 250 nm, ∆z � 4Λ0 � 1 µm,

and w0 � 1.4 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,


0 50 100 150 200 250 300 350 4000

0.5

1

1.5

2

2.5

3x 10

4

z (µm)

|q| (

m−

1)

Simulated

Target grating

(a) |q|

0 50 100 150 200 250 300 350 400−0.4

−0.2

0

0.2

0.4

z (µm)

∆ψ

(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.

0 50 100 150 200 250 300 350 4001.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

z (µm)

wid

th (

µm

)

Simulated

Target grating

(a) Waveguide width

0 50 100 150 200 250 300 350 4000

10

20

30

40

50

60

70

80

90

z (µm)

rece

ss d

epth

(n

m)

Simulated

Target grating

(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.


1.53 1.535 1.54 1.545 1.55 1.555 1.56 1.565 1.571.570

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

11

λ (µm)

|r|

Simulated

Target

(a) Amplitude response

1.53 1.54 1.55 1.56 1.57−5

0

5

10

τ (p

s)

λ (µm)

Simulated

Target

(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 dis-

cussed. In this chapter, pulse shaping is studied by using the aforementioned algorithms

to generate structures of an integrated grating that will provide suitable reflection re-

sponses.

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

Eoutpωq � rpωqEinpωq, (5.1)

where Einpωq, Eoutpωq, and rpωq 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

rpωq � EoutpωqEinpωq . (5.2)

However, it is required that |rpωq| ¤ 1 since the device is linear and passive. In actual

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

69

Chapter 5. Pulse Shaping Simulations 70

is near zero, especially far from the central frequency. Therefore, a windowing function,

fpωq, 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 Ainptq and Aoutptq, respectively. The electric fields in a time domain are

in the form Einptq � Ainptqejωct and Eoutptq � Aoutptqejωct. Consequently, the fields in

a frequency domain are Einpωq � Ainpω � ωcq and Eoutpωq � Aoutpω � ωcq. It is more

convenient to define a baseband frequency ω1 � ω�ωc and rewrite the reflection response

as

rpω1q � Aoutpω1qAinpω1q fpω

1q � αe�jω1τd , (5.3)

where 0 ¤ 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

einptq � e�4 ln 2 t2

τ2 . (5.4)

The windowing function, fpxq, is defined previously in Eq. 4.34. In most of the

simulations that follow, the spectral windowing function is set to be fpx � ν 1q that has

x1{2 � 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


−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

11

t (ps)

Ain

(t)

(a.u

.)

(a) Temporal envelope

−10 −8 −6 −4 −2 0 2 4 6 8 100

0.25

0.5

0.75

1

1.25

1.5

1.751.75x 10

−13

ν−νc (THz)

|Ain

(ν−

νc)|

(b) |Ainpν1q|

−10 −8 −6 −4 −2 0 2 4 6 8 10−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

ν−νc (THz)

∠A

in(ν

−ν

c)

(c) =Ainpν1q

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 switch-

ing in which the flat-top pulses could be used as a switching window [5]. The pulse


envelope is defined as

eoutptq �

$''&''%

1 ; |t| ¤ TFWHM

2

0 ; otherwise,

(5.5)

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 sinc-

function 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 � 1 and τd � 2 ps in Eq. 5.3. The parameters for the inverse scattering

(IS) algorithm are Λ0 � 250 nm, ∆z � 12Λ0 � 3 µm, Ng � 400, w0 � 1.4 µm, and

rdres � 5 nm. 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


−10 −8 −6 −4 −2 0 2 4 6 8 100

0.5

1

1.5

2

2.5x 10

−12

ν−νc (THz)

|Ao

ut(ν

−ν

c)|

(a) |Aout|

−10 −8 −6 −4 −2 0 2 4 6 8 10−4

−2

0

2

4

ν−νc

∠A

ou

t(ν−

νc)

(b) =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

gratings were simulated by similarly starting from the 16th (z � 48 µm) piece but choosing

four different ending pieces: the 341st (z � 1, 023 µm), 181st (z � 543 µm), 83rd (z �


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

0 200 400 600 800 1000 12000

0.5

1

1.5

2x 10

4

z (µm)

|q| (

m−

1)

(a) |qpzq|

0 200 400 600 800 1000 1200−0.5

0

0.5

1

1.5

2

2.5

3

z (µm)

∆ψ

(b) ∆ϕ

0 200 400 600 800 1000 12001.3

1.35

1.4

1.45

1.5

z (µm)

wid

th (

µm

)

(c) Waveguide width

0 200 400 600 800 1000 12000

20

40

60

80

z (µm)

rece

ss d

epth

(nm

)

(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.


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,

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

−5 −4 −3 −2 −1 0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

11

ν−νc (THz)

|r|

Simulated

Target


−5 −4 −3 −2 −1 0 1 2 3 4 5

−10

0

10

20

τ (

ps)

ν−νc (THz)

Simulated

Target


−2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 300

0.02

0.04

0.06

0.08

t (ps)

fiel

d a

mp

litu

de

Simulated

Target

Input*

(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


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.

−2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 300

0.02

0.04

0.06

0.08

t (ps)

fiel

d a

mp

litu

de

g1

g2

g3

g4

Figure (5.5): Electric field magnitudes of output waveforms corresponding to generated gratings with

different sets of subgrating involved.

−2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 300

0.1

0.2

0.3

t (ps)

fiel

d a

mp

litu

de

τ=2.0 ps

τ=1.0 ps

τ=0.5 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


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 magni-

tudes of the waveforms is in about 0.0052 (arbitrary unit used to plot the field amplitude).

Outside this duration, the maximum difference is about 0.009.

−5 −4 −3 −2 −1 0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

11

ν−ν0 (THz)

|r|

Simulated

Target


−5 −4 −3 −2 −1 0 1 2 3 4 5−20

−10

0

10

20

30

40

τ (

ps)

ν−νc (THz)

Simulated

Target


−2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 300

0.02

0.04

0.06

0.08

t (ps)

fiel

d a

mpli

tude

(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


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

eoutptq �

$''''''&''''''%

1� tTFWHM

; � TFWHM ¤ t ¤ 0

1� tTFWHM

; 0 ¤ t ¤ TFWHM

0 ; otherwise.

(5.6)

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.


−4 −3 −2 −1 0 1 2 3 40

0.5

1

1.5

2x 10

−12

ν−νc (THz)

|Ao

ut(ν

−ν

c)|

(a.u

.)

(a) |Aoutp∆νq|

0 200 400 600 800 1000 12000

2000

4000

6000

8000

10000

12000

z (µm)

|q| (m

−1)

(b) |qpzq|

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.

0 200 400 600 800 1000 12001.38

1.4

1.42

1.44

1.46

z (µm)

wid

th (

µm

)

(a) Waveguide width

0 200 400 600 800 1000 12000

10

20

30

40

50

60

70

z (µm)

rece

ss d

epth

(nm

)

(b) Recess depth

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


−4 −3 −2 −1 0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

11

ν−νc (THz)

|r|

Simulated

Target


−4 −2 0 2 40

5

10

15

20

τ (

ps)

ν−νc (THz)

Simulated

Target


Figure (5.10): Grating response taking upto the the point of z � 600 µm of the IS-generated grating.

−2 0 2 4 6 8 10 12 14 16 18 200

0.02

0.04

0.06

0.08

t (ps)

fiel

d a

mpli

tude

Simulated

Simulated*

Input*

Target

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.


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

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

∆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 rectan-

gular 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


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 � 7 ps. The parameters

for the IS algorithm are Λ0 � 249.6 nm, ∆z � 12Λ0, Ng � 1, 000, w0 � 1.4 µm, and

rdres � 5 nm. The IS algorithm iteration loop number is 10. The resulted waveguide

width and recess depth profiles in fact show severe fluctuations in some insignificant cou-

pling 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.

0 1000 2000 30000

2000

4000

6000

8000

10000

z (µm)

|q| (m

−1)

(a) |qpzq|

0 250 500 750 1000 1250 15001.38

1.39

1.4

1.41

1.42

1.43

1.44

z (µm)

wid

th (

µm

)

(b) Waveguide width

0 250 500 750 1000 1250 15000

10

20

30

40

50

z (µm)

rece

ss d

epth

(nm

)

(c) Recess depth

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


−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

11

ν−ν0 (THz)

|r|

(a) Amplitude response of the suggested grating

−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

11

ν−ν0 (THz)

|r|

(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 re-

flection of the two rectangular pulses. Note that the later main subgrating has higher cou-

pling 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 space-

time mapping relation, t � znav{c, in the inverse scattering algorithm. The subpulses

also exist near t � 25 ps. With the deviations in the grating profiles, the maximum

deviation in the electric field magnitude is about 0.007. It is interesting that the devi-

ation 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


time is negative, which occurs about half of the time axis vector in the numerical im-

plementation. 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.

−5 0 5 10 15 20 25 30 35 40 45 500

0.01

0.02

0.03

0.04

0.05

t (ps)

fiel

d a

mpli

tud

e

Simulated

Target

Input*

(a) An output waveform from the generated grating in a solid blue curve

compared to a targeted waveform shown in a black dashed curve.

−5 0 5 10 15 20 25 30 35 40 45 500

0.01

0.02

0.03

0.04

0.05

t (ps)

fiel

d a

mpli

tude

(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.


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 per-

form pulse shaping based on reflection responses for integrated waveguide gratings. The

targeted pulse is defined and the appropriate reflection response is calculated. The grat-

ing 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 one-

to-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 gener-

ated 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 re-

sponses. 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 dura-

tions 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 reflec-

tion 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. Addi-

tionally, 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 de-

viations 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


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


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 re-

sponse 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 numeri-

cal 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 for-

malism 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

Appendix A. Coupled-Mode Theory (CMT) 91

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 propa-

gates 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 permit-

tivity εpx, y, zq � εpx, yq. Using Maxwell’s equations, the wave equation is yielded,

∇2Epr, tq � µεwpx, yq B2

Bt2 Epr, tq, (A.1)

where Epr, tq � Epx, y, z, tq and εw represents the unperturbed waveguide.

Assume a monochromatic wave in the phasor form

Epr, tq � epx, yqejωt�jβz (A.2)

and put Eq. A.2 into Eq. A.1 resulting in, [33],

∇2Kepx, yq � �

ω2µεwpx, yq � β2�epx, yq � 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, yq 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


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 εpx, y, zq. 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, εpx, y, zq � ε px, y, z � Λpzqq.For a uniform-period grating, Λpzq � Λ is a constant, whereas a grating chirp, which is

a variation in a grating period, could be introduced via a z-dependent Λpzq. With the

wave equation as shown in Eq. A.1 and assuming monochromatic waves, the equation

becomes

∇2Eprq � �ω2µεpx, y, zqEprq, (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


electric field E could be expressed as a linear combination of waveguide modes:

Eprq �¸m�0

cmpzqempx, yqe�jβmz. (A.5)

The summation covers all possible guided modes, indicated by the subscript m, and

their propagation directions, forward-propagating for m ¡ 0 and backward-propagating

for m 0. That is βm � �β�m. The factor cmpzq 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

¸m�0

e�jβmz"

2jβmd

dzcm

*em �

¸m�0

ω2µpε� εwqcme�jβmzem, (A.6)

where a slow-varying envelope approximation such that d2

dz2 cm ! 2βmddzcm 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,³A

e�n � em dA � 0 if n � m. The following definition is

used

xA|c|By � xA|cBy �»A

A� � pcBq dA (A.7)

Therefore, operating³A

dA e�n� to both sides of Eq. A.6 results in

d

dzcn � �j

¸m�0

ω2µ xen|∆εpx, y, zq|emy2βn xen|eny cme

�jpβm�βnqz (A.8)

where ∆ε � ε� εw represents the grating perturbation to the waveguide.

If the perturbation is periodic, i.e. ∆εpzq � ∆εpz�Λq, it could be expanded in Fourier

series:

∆εpx, y, zq �¸q

∆εpx, yqrqs ej 2πqΛz (A.9a)

∆εpx, yqrqs � 1

Λ

» Λ

0

∆εpx, y, zq e�j 2πqΛz dz (A.9b)


where ∆εpx, yqrqs is the discrete Fourier coefficients as indicated by the use of square

brackets. Also note that ∆εpx, yqr�qs � ∆ε�px, yqrqs because ∆εpx, y, zq is real. If

defining

κn,mrqs � ω2µ xen|∆εpx, yqrqs|emy2βn xen|eny (A.10a)

σnrqs � ω2µ xen|∆εpx, yqrqs|eny2βn xen|eny , (A.10b)

then it can be shown that

κn,mpzq �¸q

κn,mrqs ej2πqΛz (A.11a)

σnpzq �¸q

σnrqs ej2πqΛz. (A.11b)

These terms are often referred to as coupling coefficients: κn,mrqs is the cross coupling

between the nth mode and the mth mode via the qth grating order; and σnrqs is the

self-coupling term due to the qth grating order. Using Eq. A.10 and Eq. A.11 in Eq. A.8

leads to

d

dzcn � �jσnpzqcnpzq � j

¸m�0,n

κn,mpzqcmpzqe�jpβm�βnqz (A.12)

On the other hand, if a grating is aperiodic, such as a chirped grating, the perturbation

could be represented by the Fourier transform

∆εpx, y, zq �» 8

�8

∆εpx, y, kq ejkzdk (A.13a)

∆εpx, y, kq � 1

2π

» 8

�8

∆εpx, y, zq e�jkzdk. (A.13b)

In a similar manner, it is possible to define

κn,mpkq � ω2µ xen|∆εpx, y, kq|emy2βn xen|eny where n � m (A.14a)

σnpkq � ω2µ xen|∆εpx, y, kq|eny2βn xen|eny (A.14b)

and we will have

κn,mpzq �» 8

�8

κn,mpkq ejkz dk (A.15a)

σnpzq �» 8

�8

σnpkq ejkz dk (A.15b)


Therefore, applying Eq. A.14 and Eq. A.15 in the same way yields

d

dzcn � �jσnpzqcnpzq � j

¸m�0,n

κn,mpzqcmpzqe�jpβm�βnqz (A.16)

The equation describes that the development of the nth mode results from the self-

coupling 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 dcndz

� 0.

A.2.1 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

∆εpx, y, zq � ∆εpx, y, zqr0s �∆εpx, y, zqrpsej 2πpzΛ0

�jφppzq �∆εpx, y, zqr�pse�j 2πpzΛ0

�jφppzq.

(A.17)

Note that this expression looks like Eq. A.9 except the chirp term, φppzq, which represents

the change in perturbation periodicity.

Then substitute Eq. A.17 into Eq. A.8

d

dzcn � �j

¸q

σnrqscnpzqej2πqzΛ0

�jφqpzq� j¸

m�0,nq

κn,mrqscmpzqe�j�βm�βn�

2πqΛ0

z�jφqpzq (A.18)

where q � �p, 0, p, φ0pzq � 0, and φppzq � φpzq � �φ�ppzq.The development of cnpzq along z-axis is contributed mostly from the terms on the

right hand side that have slow oscillation. For the self-coupling, it is σnr0s. For the first-

order grating approximation, set βm � βn � 2πpΛ0

� 0 as |p| � 1. Physically it means that


a periodic effective index modulation matches half of the Bragg wavelength: ΛB2� neffλ.

The selected cross coupling term is then κn,mrq � p,�ps. If the waveguide is single-mode,

we have β1 � �β�1 � 2πneff

λ. Eq. A.18 could be written as:

d

dzc1 � �jσ1r0sc1 � jκ1,�1rpsc�1e

�jΦp (A.19a)

d

dzc�1 � �jσ�1r0sc�1 � jκ�1,1r�psc1e

jΦp (A.19b)

where Φp � pβ�1 � β1 � 2πpΛ0qz � φp with corresponding definitions

κn,mpzqrqs � ω2µ xen|∆εpx, y, zqrqs|emy2βn xen|eny (A.20a)

σnpzqrqs � ω2µ xen|∆εpx, y, zqrqs|eny2βn xen|eny . (A.20b)

Defining ∆β � β�1 � β1 � 2πpΛ0

, the value of p � �1 such that ∆β � 0. Next let

c1pzq � c1pzqejΦp2 and c�1pzq � c�1pzqe�j

Φp2 , (A.21)

which will make Eq. A.19 become

d

dzc1pzq � j

�∆β

2� dφ

dz� σ1pzqr0s

c1pzq � jκ1,�1pzqrpsc�1pzq (A.22a)

d

dzc�1pzq � �jκ�1,1pzqr�psc1pzq � j

�∆β

2� dφ

dz� σ�1pzqr0s

c�1pzq (A.22b)

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

κ1,�1rps � � κ��1,1r�ps � κ (A.23a)

σ1r0s � � σ�1r0s � σ (A.23b)

because β1 � �β�1 and e1px, yq � e�1px, yq. Hence, the coupled-mode equations for the

first-order grating in a single-mode waveguide are

d

dzc1pzq � j

�∆β

2� dφ

dz� σpzq

c1pzq � jκpzqc�1pzq (A.24a)

d

dzc�1pzq � jκ�pzqc1pzq � j

�∆β

2� dφ

dz� σpzq

c�1pzq (A.24b)


It is interesting to note that the above equations suggest that the effects of the per-

turbation periodicity chirp, dφdz

, and change in modal index, ∆εpzqr0s, are similar and

indistinguishable.

The problem of solving Eq. A.24 is sometimes referred to as a direct scattering prob-

lem. Analytically solving this set of equations are complicated with z-dependent func-

tions, i.e. dφdz

, σpzq, and κpzq. 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

d

dzc1pzq � j

�∆β

2� σ

c1pzq � jκc�1pzq (A.25a)

d

dzc�1pzq � jκ�c1pzq � j

�∆β

2� σ

c�1pzq (A.25b)

Solving Section A.25 requires two boundary conditions. Let assume that c1pz0q and

c�1pz0q are known. The solution will be

c1pzq �$&%cosh

�spz � z0q

� j

�∆β

2� σ

sinh�spz � z0q

s

,.- c1pz0q

�j κs

sinh�spz � z0q

c�1pz0q (A.26a)

c�1pzq � jκ�

ssinh

�spz � z0q

c1pz0q

�$&%cosh

�spz � z0q

� j

�∆β

2� σ

sinh�spz � z0q

s

,.- c�1pz0q,(A.26b)

where s �b|κ|2 � �

∆β2� σ

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


will be useful in the transfer matrix method;�� c1pzqc�1pzq

��

��m11pz, z0q m12pz, z0qm21pz, z0q m22pz, z0q


�� Mpz, z0q


�� (A.27)

where


� j

�∆β

2� σ

sinh�spz � z0q

s

(A.28a)

m12pz, z0q � �j κs

sinh�spz � z0q

(A.28b)

m21pz, z0q � jκ�

ssinh

�spz � z0q

(A.28c)


� j

�∆β

2� σ

sinh�spz � z0q

s

(A.28d)

The above solution to the coupled-mode equation, Section A.25, is given to the con-

figuration of a grating with a constant period and constant coupling coefficients. It

links the energy factors c1 and c�1 from a location of z � z0 to another location z � z

within the grating region. It can be shown that the matrix Mpz, z0q is unitary such that

det pMpz, z0qq � 1.

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 εwpx, yq. On the other hand,

the uniform sidewall grating with 0.5 duty cycle, defined by one-step etching, is written

εpx, y, zq �

$''''''&''''''%

εwpx, yq ; px, yq in the unetched area

ε0 ; px, yq in the etched area, 0 ¤ z Λ2

εwpx, yq ; px, yq in the etched area, Λ2¤ z Λ.

(A.29)


Therefore, the perturbation, ∆ε � ε� εw, becomes

∆εpx, y, zq �

$''&''%ε0 � εwpx, yq ; px, yq in the etched area, 0 ¤ z Λ

2

0 ; otherwise.

(A.30)

Previously, the perturbation in Fourier series is expressed as in Eq. A.9

∆εpx, y, zq �¸q

∆εpx, yqrqs ej 2πqΛz (A.31a)

∆εpx, yqrqs � 1

Λ

» Λ

0

∆εpx, y, zq e�j 2πqΛz dz. (A.31b)

Using this relations, we fine that for the sidewall grating,

∆εpx, yqr0s �

$''&''%

ε0�εwpx,yq2

; px, yq in the etched area

0 ; otherwise

(A.32)

and

∆εpx, yqrq � 0s �

$''&''%j ε0�εwpx,yq

2πqpe�jπq � 1q ; px, yq in the etched area

0 ; otherwise.

(A.33)

Therefore, the first-order Fourier coefficient is

∆εpx, yqr1s � ∆ε�px, yqr�1s �

$''&''%j εwpx,yq�ε0

π; px, yq in the etched area

0 ; otherwise.

(A.34)

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

σ1r0s � ω2µ xe1|∆εpx, yqr0s|e1y2β1 xe1|e1y (A.35)

which is real because ∆εpx, yqr0s is real. The cross-coupling constant, with ∆εpx, yqr�1s,is

κ1,�1r�1s � ω2µ xe1|∆εpx, yqr�1s|e1y2β1 xe1|e1y � |κ|ejθ � �j|κ|. (A.36)

It is imaginary because ∆εpx, yqr1s is imaginary, which means that θ � �π2.


A.2.4 Grating Responses by CMT and Transfer Matrix Method

From a previous section, the solution to the uniform grating matches c1 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 c1 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

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

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

matrix equations, each with a corresponding transfer matrix Mi,�� c1pz1qc�1pz1q

�� M1pz1, z0q


��


�� M2pz2, z1q


��

...�� c1pzNqc�1pzNq

�� MNpzN, zN�1q

�� c1pzN�1qc�1pzN�1q

��

Therefore, it could be written that�� c1pzNqc�1pzNq

�� MNMN�1 � � �M2M1


��

�� c1pzNqc�1pzNq

�� M


��

��M11 M12

M21 M22


�� (A.37)

This final matrix M is the system matrix and represents the whole grating. It matches

the states of c1 and c�1 from the front of the grating at z � z0 to the end of the grating

at z � zN .


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. c�1pzNq � 0 and

as a result c�1pzNq � 0. Therefore, from Section A.37 it leads to

c�1pz0qc1pz0q � �M21

M22

(A.38a)

c1pzNqc1pz0q �

M11M22 �M12M21

M22

(A.38b)

The reflection response, at frequency ν, of the Bragg grating is defined as

rpνq � c�1pz0qc1pz0q (A.39)

Setting z0 � 0, the reflection response becomes

rpνq � c�1pz0qc1pz0q � �M21

M22

(A.40)

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

tpνq � c1pzNqc1pz0q �

M11M22 �M12M21

M22

. (A.41)

Since the elements that construct the system matrix is unitary, the system matrix is also

unitary, i.e. M11M22 �M12M21 � 1. Hence,

tpνq � 1

M22

� �rpνqM21

(A.42)

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 backward-

propagating waves are represented by c1pzq and c�1pzq and they undergo propagation of


length L in an unperturbed waveguide, the transfer matrix of this propagation is�� c1

c�1

�� pz�z0�Lq

�

��e�jβ1L 0

0 ejβ1L

�� c1

c�1

�� pz�z0q

, (A.43)

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

rpνq � �m21

m22

(A.44a)

� �jκ� sinh pspz � z0qqs cosh pspz � z0qq � j

�∆β2� σ

�sinh pspz � z0qq

. (A.44b)

Note that

s �d|κ|2 �

�∆β

2� σ

2

. (A.45)

The reflectivity is defined as the square of the magnitude of the reflection coefficient,

Rpνq � |rpνq|2, (A.46)

where

|rpνq| � |κ| sinhps∆zqbs2 cosh2ps∆zq � �

∆β2� σ

�2sinh2ps∆zq

. (A.47)

The maximum value of the reflection magnitude occurs when ∆β2� σ � 0,

|r|max � tanhp|κ|∆zq. (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


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

rpνq � |rpνq|ejφpνq. (A.50)

Separately, |r| is termed the amplitude response whereas φpνq is the phase response. The

phase response φpνq 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

τp2πνq � τpωq � �dφdω

� � 1

2π

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 � L

τ(A.52)

A.3 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 rnsand F rms,

F rms �Np�1¸n�0

f rnse�j 2πmnNp (B.1a)

f rns � 1

Np

Np�1¸m�0

F rmsej 2πmnNp (B.1b)

where n,m � 0, 1, 2, . . . , Np � 1 and exp��j 2πmn

Np

and exp

�j 2πmn

Np

are the basis func-

tions. These functions are periodic in m and n;

φrm,ns � φrm,n�Nps � 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)

φrm,ns � φrm,n�Nps � φ�rm,Np � ns � ej 2πmnNp (B.2c)

φrm,ns � φrm�Np, ns � φ�rNp �m,ns � ej 2πmnNp (B.2d)

These relations mean that

F rms � F rm�Nps � F �rNp �ms (B.3a)

f rns � f rn�Nps � f�rNp � ns. (B.3b)

104

Appendix B. Fourier Transforms 105

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 toNp � 1,

corresponding to the basis functions φrm,ns and φrm,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 r�1s � F �rNp � 1s (B.6a)

F r�2s � F �rNp � 2s (B.6b)

...

F r�Np � 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 fptq, 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 � fpn∆tq � ej2πνn∆t � ejθn (B.8)


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 � 2πm

N� 2πν∆t. (B.9)

It implies that the digital frequency is discretized.

θ Ñ θm � 2π

Nm. (B.10)

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

∆ν � 1

N∆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 and φrm,ns � ej 2πmnNp . (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∆νs, where m,n are integer.

B.2 Implementing Fourier Transform with Discrete

Fourier Transform

In the conventional Fourier transform, the transformation relates two continuous func-

tions:

F pνq �8»

�8

fptqe�j2πνt dt (B.13a)

fptq �8»

�8

F pνqej2πνt dν. (B.13b)

where t and ν are real numbers. It is said that fptq and F pνq are the Fourier pair or

fptq ðñ F pνq. (B.14)


Consider a Fourier pair between the impulse and reflection response. In the simula-

tion, the reflection response, rrn1s, and the impulse response, hrm1s, 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, νrn1s � νn1 ,

with a specified frequency interval, ∆ν. Similarly, the impulse response is represented on

a set of time points, trm1s � 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,

rpνq �8»

�8

hptqe�j2πνt dt �T {2»

�T {2

hptqe�j2πνt dt (B.15a)

�Np2

m��Np

2

hpm∆tqe�j2πνm∆t∆t. (B.15b)

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 � �Np2, . . . , Np

2.

In order to resemble DFT so that the fft function could be used, discretize the

frequency in the same way into ν � n∆ν � νn � νn1 , where n � �Np2, . . . , Np

2. Therefore,

Eq. B.15 can be rewritten as

rpνq Ñ rrns �

��

Np2

m��Np

2

hrmse�j 2πmnNp

�� ∆t (B.16a)

� DFTthrmsu∆t. (B.16b)

The inverse Fourier transform could also be shown in a similar way. Therefore, in MAT-

LAB, 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)


where dt is a variable representing the time interval. Actually, relations in Eq. B.17 are

applicable to any pairs of vectors, f rtms and F rνms, 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 : �Np � 1

2∆t,�Np � 3

2∆t, . . . ,�∆t, 0,∆t, . . . ,

Np � 3

2∆t,

Np � 1

2∆t (B.18a)

ν : �Np � 1

2∆ν,�Np � 3

2∆ν, . . . ,�∆ν, 0,∆ν, . . . ,

Np � 3

2∆ν,

Np � 1

2∆ν.(B.18b)

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

t : ��Np

2� 1

∆t,�

�Np

2� 2

∆t, . . . ,�∆t, 0,∆t, . . . ,

�Np

2� 1

∆t,

Np

2∆t (B.19a)

ν : ��Np

2� 1

∆ν,�

�Np

2� 2

∆ν, . . . ,�∆ν, 0,∆ν, . . . ,

�Np

2� 1

∆ν,

Np

2∆ν.(B.19b)

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. Con-

sidering a waveguide grating with a waveguide width of w � 1.4µm and a recess depth

of rd � 25 nm, the effective index of the TE-like and TM-like modes are, respectively,

neffpTEq � 3.1062 and neffpTMq � 3.1058. (C.1)

The self- and cross-coupling constants at λ � 1.55µm are

κ � �4523j (C.2a)

σ � �7107. (C.2b)

109

Appendix C. Simulation Results for Grating Responses 110

Also, if the Bragg wavelength is λ � 1.55µm, the period of the perturbation for the

first-order Bragg condition could be calculated

Λ � λB2neff

� 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 � 100µm, 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,0∆z

c. (C.4)

Substituting values for the current case gives τprop � 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.


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 wave-

length. 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.

1.54 1.55 1.56 1.570

0.1

0.2

0.3

0.4

0.5

λ (µm)

|r|

(a) Amplitude response.

1.54 1.55 1.56 1.57−2

0

2

4

6

λ (µm)

∠r

(b) Phase response.

1.54 1.55 1.56 1.570.96

0.98

1.00

1.02

1.04

1.06

1.08

λ (µm)

τ (p

s)

(c) Time delay response.

1.54 1.55 1.56 1.570.31

0.32

0.33

0.34

0.35

λ (µm)

vg/c

0

(d) Group velocity.

Figure (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 � 100µm. The

effective index dispersion is not taken into account.


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 � 200µm 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 reflec-

tivity 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-

1.54 1.55 1.56 1.570

0.1

0.2

0.3

0.4

0.5

λ (µm)

|r|


1.54 1.55 1.56 1.57−2

−1

0

1

2

3

4

5

λ (µm)

∠r

(b) Phase response.

1.54 1.55 1.56 1.570.96

0.98

1

1.02

1.04

1.06

1.08

λ (µm)

τ (p

s)

(c) Time delay response.

1.54 1.55 1.56 1.570.31

0.32

0.33

0.34

0.35

λ (µm)

vg/c

0

(d) Group velocity.



effective index dispersion is now taken into account.


1.54 1.55 1.56 1.570

0.2

0.4

0.6

0.8

1

λ (µm)

|r|


1.54 1.55 1.56 1.570

0.5

1

1.5

2

λ (µm)

τ (p

s)

(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.


1.54 1.55 1.56 1.570

0.2

0.4

0.6

0.8

1

λ (µm)

|r|


1.54 1.55 1.56 1.572

4

6

8

λ (µm)

τ (p

s)




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, Λpzq, which is

treated in Section A.2.1. Alternatively, from the above expression, the chirp can be

implemented by the effective index profile neffpzq. Since the effective index depends on

the waveguide structure, this translates to the chirp by the waveguide width profile wpzq.

C.2.1 Linearly Chirped Gratings

Let consider the chirp introduced by the grating period profile whereas the waveguide

width is fixed at w � 1.4µm as before. To simulate the grating response, the transfer

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


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

Λpzq � 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 � 8 is shown in Fig. C.5. The total length of this grating is L �°Ngi�1 ∆zi � 400µm. Each subgrating has the same self- and cross-coupling constants,

�15, 467 and �9, 847j respectively. The FWHM bandwidth of the amplitude response is

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 � 2Lneff

c� 8.28 ps. Physically, the Bragg wavelength

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


1.53 1.54 1.55 1.56 1.57 1.580

0.2

0.4

0.6

0.8

λ (µm)

|r|


1.53 1.54 1.55 1.56 1.57 1.58−5

0

5

10

15

λ (µm)

τ (p

s)


Figure (C.5): Reflection response of a linearly chirped grating with ∆Λ � 4 nm and Λ0 � 250 nm. The

simulation is implemented with Ng � 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 1.0µm to 1.6µm. 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.


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, 1.56sµm. The down-tapered grating performs

like a negatively-chirped grating.

1.52 1.53 1.54 1.55 1.56 1.570

0.2

0.4

0.6

0.8

1

λ (µm)

|r|


1.52 1.53 1.54 1.55 1.56 1.57−5

0

5

10

15

λ (µm)

τ (p

s)


Figure (C.6): Reflection response of a linearly tapered grating with the waveguide width increasing

from 1.0µm to 1.6µm. 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


evidencing in Fig. C.8a.

0 25 50 75 100 125 150 175 2000

200

400

600

800

1000

z (µm)

|κ| (1

/m)

(a) Gaussian-apodized cross-coupling constant.

0 25 50 75 100 125 150 175 2000

25

50

75

100

125

150

175

200

z (µm)

Rec

ess

dep

th (

nm

)(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.

1.53 1.54 1.55 1.56 1.570

0.2

0.4

0.6

0.8

1

λ (µm)

|r|


1.53 1.54 1.55 1.56 1.57−4

−2

0

2

4

λ (µm)

τ (p

s)


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 π-phase-shift and Sampled Gratings

In this section that the algorithm is demonstrated to solve a sampled grating. A sam-

pled 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


1.54 1.55 1.56 1.570

0.2

0.4

0.6

0.8

1

λ (µm)

|r|


1.54 1.55 1.56 1.571

1.5

2

2.5

3

3.5

4

λ (µm)

τ (p

s)


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 1.4µm, 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.


1.53 1.54 1.55 1.56 1.57 1.580

0.2

0.4

0.6

0.8

1

λ (µm)

|r|


1.53 1.54 1.55 1.56 1.57 1.58−5

0

5

10

15

λ (µm)

τ (p

s)


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

Appendix D

Inverse Scattering Theory

Appendix A focuses on the calculate of the response of the grating whose physical pa-

rameter 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

Appendix D. Inverse Scattering Theory 123

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

dzc1 � jζc1 � qc2 (D.1a)

d

dzc2 � �jζc2 � q�c1 (D.1b)

where q, q� Ñ 0 as |z| Ñ 8. 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. There-

fore, some manipulations are required before the inverse scattering theory could be ap-

plied to the coupled-mode equations.

Eq. D.1 could be cast in a matrix form

d

dz

��c1

c2

��

��jζ q

q� �jζ

�� c1

c2

�� . (D.2)

Assume linearly independent solutions, φ and φ, with asymptotic behaviors at z Ñ �8

φpz Ñ �8, ζq �

��1

0

�� ejζz (D.3a)

φpz Ñ �8, ζq �

��0

1

�� e�jζz. (D.3b)

With these forms, φpz Ñ �8, ζq is the forward-propagating wave whereas φpz Ñ �8, ζqis the backward-propagating wave. The exact solution at any point z could be written if

the kernel functions apply

φpz, ζq �

��1

0

�� ejζz �

z»�8

Kpz, sqejζs ds (D.4a)

φpz, ζq �

��0

1

�� e�jζz �

z»�8

Kpz, sqe�jζs ds (D.4b)


where K �

��K1

K2

�� and K �

��K1

K2

�� . With these linearly independent solutions, the

general solution to Eq. D.17 is

cpz, ζq �

��c1pz, ζqc2pz, ζq

�� φpz, ζq � rpζqφpz, ζq. (D.5)

Note that for z Ñ �8 Eq. D.5 appears

cp�8, ζq �

��1

0

�� ejζz � rpζq

��0

1

�� e�jζz (D.6)

which has the forward-propagating wave with a magnitude of unity and the backward-

propagating wave with a magnitude of |rpζq|.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 e�jζy{2π

1

2π

¾C

cpz, ζqe�jζy dy � 1

2π

¾C

φpz, ζqe�jζyq dy � 1

2π

¾C

rpζqφpz, ζqe�jζy dy (D.7)

Using the fact that cpz, ζq is analytical in the upper half plane, the term on the left hand

side becomes zero. Also use

1

2π

¾C

ejζs ds � δpsq, (D.8)

which lead to the important result

0 � Kpz, yq �

��0

1

�� hpz � yq �

z»�8

Kpz, sqhps� yq ds ; y z, (D.9)

where

hpyq � 1

2π

¾C

rpζqe�jζy dy. (D.10)

Eq. D.9 is the main iteration equation that solves for K and K. In general, this process

involves four functions: K1, K2, K1, and K2. Fortunately, the complexity is reduced by


utilizing the symmetry in Eq. D.17. To see this, write

d

dz

��c2

c1

��

��jζ q�

q jζ

�� c2

c1

�� (D.11)

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

receive

d

dz

��c1

c2

��

�

��jζ q�

q jζ

�� c1

c2

��

. (D.12)

This symmetry suggests the form

φ �

��φ�2φ�1

�� (D.13a)

K �

��K�

2

K�1

�� . (D.14)

As a result, the iteration equation turns to be

0 �

��K1pz, yqK2pz, yq

��

��0

1

�� hpz � yq �

z»�8

��K�

2 pz, sqK�

1 pz, sq

�� hps� yq ds. (D.15)

When K and K are solved, they can construct φ, φ, and eventually c.

The next step is to find the relationship between K and q. Consider the linearly-

independent solution, φ, as defined before

φ1pz, ζq � ejζz �z»

�8

K1pz, sqejζs ds (D.16a)

φ2pz, ζq �z»

�8

K2pz, sqejζs ds. (D.16b)

Since φ is the solution to Eq. D.1, it is possible to write

d

dzφ1 � jζφ1 � qφ2 (D.17a)

d

dzφ2 � �jζφ2 � q�φ2. (D.17b)


By putting φ into Eq. D.17, two equations result respectively

0 �z»

�8

�dK1

dz� dK1

ds� qpzqK2

ejζs ds (D.18a)

0 � p2K2pz, zq � q�q ejζz �z»

�8

�dK2

dz� dK2

ds� q�pzqK1

ejζs ds. (D.18b)

It is shown that it is necessary and sufficient that

0 � dK1

dz� dK1

ds� qpzqK2 (D.19a)

0 � dK2

dz� dK2

ds� q�pzqK1 (D.19b)

with the boundary condition

K2pz, zq � q�pzq2

. (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 rpζq is desired and

known. The coupling parameters qpzq could be extracted from the known response by

the relationship

qpzq � 2K�2 pz, zq. (D.21)

where K2pz, zq is calculated from the iteration equations

K2pz, yq � �hpz � yq �z»

�8

K�1 pz, sqhps� yq ds (D.22a)

K1pz, yq � �z»

�8

K�2 pz, sqhps� yq ds (D.22b)

where

hpyq � 1

2π

¾C

rpζqe�jζy dy. (D.23)

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 calcu-

lation also stems from the discrete and limited nature of computational variables, which

could not be reduced by increasing the number of iteration.


D.2 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.


Recall in Section D.1 that the general solution, cpz, ζq, to the coupled-mode equations

could be written as

c1pz, ζq � ejζz �z»

�8

K1pz, sqejζs ds� rpζqz»

�8

K�2 pz, sqe�jζs ds (D.24a)

c2pz, ζq �z»

�8

K2pz, sqejζs ds� rpζqe�jζz � rpζqz»

�8

K�1 pz, sqe�jζs ds. (D.24b)

Note that at z Ñ �8, c1 looks like a forward propagating wave whereas c2 appears as a

backward-propagating wave. Therefore, a local reflection is expressed as

rpz, ζq � c2pz, ζqc1pz, ζq (D.25a)

� ej2ζzrpζq �1� F �

1 pz, ζq�� F2pz, ζq�

1� F1pz, ζq�� rpζqF �

2 pz, ζq(D.25b)

where

F1pz, ζq � e�jζzF1pz, ζq � e�jζzz»

�8

K1pz, sqejζs ds (D.26a)

F2pz, ζq � ejζzF2pz, ζq � ejζzz»

�8

K2pz, sqejζs ds. (D.26b)

That is, with the known rpζq and corresponding K1 and K2 calculated from Eq. D.22, the

local reflection at a distance z away from the point with rpζq 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

back of the section are rm�1 and rm � rm�1p∆zMq. Note that at the first section, the

front local reflection will be r0 � rpζq, which is the actual required grating reflection

response. rm is derived by first calculating rm�1pz, ζq where 0 ¤ z ¤ ∆zm,

rm�1pz, ζq � ej2ζzrm�1p0, ζq

�1� F �

1,m�1pz, ζq�� F2pz, ζq�

1� F1pz, ζq�� rm�1p0, ζqF �

2 pz, ζq(D.27)


where

F1pz, ζq � e�jζzz»

�8

K1,m�1pz, sqejζs ds (D.28a)

F2pz, ζq � ejζzz»

�8

K2,m�1pz, ζqejζs ds. (D.28b)

Then, rmpζq � rm�1p∆zm, ζq. The kernel functions are iteratively calculated

K2,m�1pz, yq � �hm�1pz � yq �z»

�8

K�1,m�1pz, sqhm�1ps� yq ds (D.29a)

K1,m�1pz, yq � �z»

�8

K�2 pz, sqhm�1ps� yq ds (D.29b)

where

hm�1pzq � 1

2π

¾C

rm�1pζqe�jζy dy. (D.30)

The iteration starts by setting K2,m�1pz, yq � �hm�1pz�yq. The coupling constant along

the m-th section is given by

qmpzq � 2K�m�1pz, zq (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 interac-

tion between the forward-propagating and backward-propagating waves in a single-mode

sidewall Bragg grating

d

dzc1 � j

�∆β

2� dφ

dz� σpzq

c1 � jκpzqc�1 (D.32a)

d

dzc�1 � jκ�pzqc1 � j

�∆β

2� dφ

dz� σpzq

c�1. (D.32b)


and also the starting equations for the inverse scattering problem, Eq. D.1,

d

dzc1 � jζc1 � qc2 (D.33a)

d

dzc2 � q�c1 � 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. dφdz� 0. Consider the term in the parenthesis in Eq. D.32, which depends

on z,

∆β

2� σpzq � �β1 � π

Λ� σpzq � �2πneffpzq

λ� π

Λ� σpzq. (D.34)

It is rewritten as

∆β

2� σpzq �

��2πneff,0

λ� π

Λ0

��σpzq � 2πδneffpzq

λ� πδΛ

Λ20

(D.35a)

� ∆β0

2� σpzq. (D.35b)

where ∆β0

2is the former z-independent term and σpzq is the later z-dependent term.

Define

c1pzq � c1pzq exp

�j

» z

0

σpz1q dz1

(D.36a)


��j

» z

0

σpz1q dz1

(D.36b)

such that Eq. D.32 becomes

d

dzc1 � j

∆β0

2c1 � jκpzq exp

�j2

» z

0

σpz1q dz1c�1, (D.37a)

d


��j2

» z

0

σpz1q dz1c1 � j

∆β0

2c�1. (D.37b)


This form of the equations can be directly compared with Eq. D.33 such that

ζ � ∆β0

2� �2πnneff,0

λ� π

Λ0

, (D.38a)

qpzq � �jκpzq exp

�j2

» z

0

σpz1q dz1. (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

σpzq � σ � σ � 2πδneff

λ� πδΛ

Λ20

(D.39)

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 � �jκej2σ∆zm . (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|ej2σ∆zm . (D.41)

After calculating q from the inverse scattering problem, the above result could determine

the grating profiles of each grating section

|q| � |κ| (D.42a)

ϕ � 2σ∆zm � 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 � 2σ∆zm (D.43)


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 equa-

tions:

d

dzc1 � �jσc1 � jκc1e

�jΦ�1 (D.44a)

d

dzc�1 � jσc�1 � jκ�c1e

jΦ�1 . (D.44b)

These c1 and c�1 are the complex amplitude of the forward- and backward-propagating

waves. They determine the energy that is carried by those waves. One then defines

c1pzq � c1pzqejΦp2 and c�1pzq � c�1pzqe�j

Φp2 (D.45)

such that the second form looks like

d

dzc1 � j

�∆β

2� dφ

dz� σ

c1 � jκc�1 (D.46a)

d

dzc�1 � jκ�c1 � j

�∆β

2� dφ

dz� σ

c�1. (D.46b)

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

defines again

c1pzq � c1pzq exp

�j

» z

0

σpz1q dz1

(D.47a)


��j

» z

0

σpz1q dz1, (D.47b)

and this allows to write the third form

d

dzc1 � j

∆β0

2c1 � jκpzq exp

�j2

» z

0

σpz1q dz1c�1 (D.48a)

d


��j2

» z

0

σpz1q dz1c1 � j

∆β0

2c�1. (D.48b)

The desired reflection response is defined by

r � c�1pz0qc1pz0q . (D.49)


For a uniform grating setting z0 � 0, it could be shown that

r � c�1p0qc1p0q � c�1p0q

c1p0q � c�1p0qc1p0q . (D.50)

Hence, one can still use this desired reflection response r in the starting reflection response

rpζq 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 m�1 κ � �j9.847� 103 m�1.

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

in the 200-µm-long region, i.e. between z � 50 to 250 µm. The coupling magnitude

is close to 10, 000 m�1, which is in agreement with |κ| � 9, 847 m�1 of the original

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

Appendix E. Simulation Results for Grating Retrieval 135

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.

0 50 100 150 200 250 300 350 4000

2000

4000

6000

8000

10000

12000

z (µm)

|q| (

m−

1)

(a) |q|

0 50 100 150 200 250 300 350 400−0.2

−0.1

0

0.1

z (µm)

∆ψ

(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 co-

efficient 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 � 2σ∆z. Particularly in the region of |q| Ñ 0, the phase can fall anywhere

between �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


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

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

0 50 100 150 200 250 300 350 4001.25

1.3

1.35

1.4

1.45

1.5

z (µm)

wid

th (

µm

)

(a) Waveguide width

0 50 100 150 200 250 300 350 4000

10

20

30

40

50

60

z (µm)

rece

ss d

epth

(n

m)

(b) Recess depth

0 50 100 150 200 250 300 350 400

1.35

1.4

1.45

1.5

z (µm)

wid

th (

µm

)

(c) Waveguide width

0 50 100 150 200 250 300 350 4000

10

20

30

40

50

60

z (µm)

rece

ss d

epth

(nm

)

(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 cho-

sen 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 simu-

lated grating is in good agreement with the target response especially near the central

frequency around 1.55 µm.


1.53 1.535 1.54 1.545 1.55 1.555 1.56 1.565 1.571.570

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

11

λ (µm)

|r|

Simulated

Target

(a) Reflection amplitude

1.53 1.535 1.54 1.545 1.55 1.555 1.56 1.565 1.571.570

1

2

3

4

5

6

τ (p

s)

λ (µm)

Simulated

Target

(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. Con-

sider 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


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 � 48

to Ni � 248 are chosen is shown in Fig. E.6. 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.

0 50 100 150 200 250 300 350 4000

0.5

1

1.5

2x 10

4

z (µm)

|q| (

m−

1)

Simulated

Target grating

(a) |q|

0 50 100 150 200 250 300 350 400−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

z (µm)

∆ψ

(b) ∆ϕ

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

0 50 100 150 200 250 300 350 4001

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

z (µm)

wid

th (

µm

)

(a) Waveguide width

0 50 100 150 200 250 300 350 4000

10

20

30

40

50

60

z (µm)

rece

ss d

epth

(nm

)

(b) Recess depth

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


1.53 1.535 1.54 1.545 1.55 1.555 1.56 1.565 1.571.570

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

11

λ (µm)

|r|

Simulated

Target


1.53 1.535 1.54 1.545 1.55 1.555 1.56 1.565 1.571.57

−2

0

2

4

6

8

τ (p

s)

λ (µm)

Simulated

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


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.

0 50 100 150 200 250 300 350 4000

0.5

1

1.5

2x 10

4

z (µm)

|q| (

m−

1)

Simulated

Target grating

(a) |q|

0 50 100 150 200 250 300 350 400−0.4

−0.2

0

0.2

0.4

z (µm)

∆ψ

(b) ∆ϕ

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

0 50 100 150 200 250 300 350 400

1.3

1.4

1.5

1.6

1.7

1.8

1.9

z (µm)

wid

th (

µm

)

(a) Waveguide width

0 50 100 150 200 250 300 350 4000

10

20

30

40

50

60

70

80

90

z (µm)

rece

ss d

epth

(n

m)

Simulated

Target grating

(b) Recess depth

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


1.53 1.535 1.54 1.545 1.55 1.555 1.56 1.565 1.571.570

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

11

λ (µm)

|r|

Simulated

Target


1.53 1.535 1.54 1.545 1.55 1.555 1.56 1.565 1.571.57−4

−2

0

2

4

6

8

τ (p

s)

λ (µm)

Simulated

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 magni-

tude traces the magnitude of the initial cross-coupling coefficient with great correspon-

dence. 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.


0 50 100 150 200 250 300 350 4000

0.5

1

1.5

2

2.5

3x 10

4

z (µm)

|q| (

m−

1)

Simulated

Target grating

(a) |q|

0 50 100 150 200 250 300 350 400−0.4

−0.2

0

0.2

0.4

z (µm)

∆ψ

(b) ∆ϕ

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

linearly-chirped grating.

0 50 100 150 200 250 300 350 4001.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

z (µm)

wid

th (

µm

)

Simulated

Target grating

(a) Waveguide width

0 50 100 150 200 250 300 350 4000

10

20

30

40

50

60

70

80

90

z (µm)

rece

ss d

epth

(nm

)

Simulated

Target grating

(b) Recess depth

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


1.53 1.535 1.54 1.545 1.55 1.555 1.56 1.565 1.571.570

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

11

λ (µm)

|r|

Simulated

Target


1.53 1.54 1.55 1.56 1.57−5

0

5

10

τ (p

s)

λ (µm)

Simulated

Target



response from a Gaussian-apodized and linearly-chirped grating.

Bibliography

[1] L. M. Frantz and J. S. Nodvik, “Theory of Pulse Propagation in a Laser Amplifier,”

Journal of Applied Physics, vol. 34, no. 8, pp. 2346–2349, 1963.

[2] J. Midwinte, “Theory of Q-Switching Applied to Slow Switching and Pulse Shaping

for Solid State Lasers,” British Journal of Applied Physics, vol. 16, no. 8, pp. 1125–

1133, 1965.

[3] J. Macomber, “Theory of Pulse-Shaping by Saturable Optical Materials,” Journal

of Applied Physics, vol. 38, no. 9, pp. 3525–3530, 1967.

[4] R. Slavık, L. Oxenløwe, M. Galili, H. Mulvad, Y. Park, J. Azana, and P. Jeppesen,

“Demultiplexing of 320-Gb/s OTDM data using ultrashort flat-top pulses,” IEEE

PHOTONICS TECHNOLOGY LETTERS, vol. 19, no. 22, pp. 1855–1857, 2007.

[5] L. Oxenløwe, R. Slavık, M. Galili, H. Mulvad, A. Clausen, Y. Park, J. Azana,

and P. Jeppesen, “640 Gb/s timing jitter-tolerant data processing using a long-

period fiber-grating-based flat-top pulse shaper,” IEEE Journal on Selected Topics

in Quantum Electronics, vol. 14, no. 3, pp. 566–572, 2008.

[6] D. Goswami, “Optical pulse shaping approaches to coherent control,” Physics Re-

ports, vol. 374, pp. 385–481, 2003.

[7] K. Ohmori, “Wave-packet and coherent control dynamics,” Annual review of physical

chemistry, vol. 60, pp. 487–511, 2009.

144

Bibliography 145

[8] S.-H. Shim and M. T. Zanni, “How to turn your pump–probe instrument into a

multidimensional spectrometer: 2D IR and Vis spectroscopies via pulse shaping,”

Physical Chemistry Chemical Physics, vol. 11, no. 5, pp. 748–761, 2009.

[9] Y. Silberberg, “Quantum Coherent Control for Nonlinear Spectroscopy and Mi-

croscopy,” Annual review of physical chemistry, vol. 60, pp. 277–292, 2009.

[10] A. Weiner, “Femtosecond optical pulse shaping and processing,” Progress in Quan-

tum Electronics, vol. 19, no. 3, pp. 161–237, 1995.

[11] A. M. Weiner, “Ultrafast optical pulse shaping: A tutorial review,” Optics Commu-

nications, vol. 284, pp. 3669–3692, July 2011.

[12] A. Monmayrant and S. Weber, “A newcomer’s guide to ultrashort pulse shaping and

characterization,” J. Phys. B: At. Mol. Opt. Phys., vol. 43, 2010.

[13] Y. Park and J. Azana, “Optical pulse shaping technique based on a simple interfer-

ometry setup,” Conference Proceedings - Lasers and Electro-Optics Society Annual

Meeting-LEOS, pp. 274–275, 2007.

[14] Y. Park, M. Asghari, T.-J. Ahn, and J. Azana, “Transform-limited picosecond pulse

shaping based on temporal coherence synthesization,” Optics Express, vol. 15, no. 15,

pp. 9584–9599, 2007.

[15] Y. Park, J. Azana, and R. Slavık, “Ultrafast all-optical first- and higher-order dif-

ferentiators based on interferometers,” Optics Letters, vol. 32, no. 6, pp. 710–712,

2007.

[16] J. Azana and L. Chen, “Synthesis of temporal optical waveforms by fiber Bragg

gratings: A new approach based on space-to-frequency-to-time mapping,” Journal

of the Optical Society of America B: Optical Physics, vol. 19, no. 11, pp. 2758–2769,

2002.

Bibliography 146

[17] T. Erdogan, “Fiber grating spectra,” Journal of Lightwave Technology, vol. 15, no. 8,

pp. 1277–1294, 1997.

[18] P. Petropoulos, M. Ibsen, A. Ellis, and D. Richardson, “Rectangular pulse generation

based on pulse reshaping using a superstructured fiber Bragg grating,” Journal of

Lightwave Technology, vol. 19, no. 5, pp. 746–752, 2001.

[19] C. Wang and J. Yao, “Microwave arbitrary waveform generation based on optical

spectral shaping and wavelength-to-time mapping using a chirped fiber Bragg grat-

ing,” in Microsystems and Nanoelectronics Research Conference, pp. 57–60, 2008.

[20] C. Wang and J. Yao, “Fourier transform ultrashort optical pulse shaping using a sin-

gle chirped fiber bragg grating,” IEEE PHOTONICS TECHNOLOGY LETTERS,

vol. 21, no. 19, pp. 1375–1377, 2009.

[21] Y. Park, M. Kulishov, R. Slavık, and J. Azana, “Picosecond and sub-picosecond

flat-top pulse generation using uniform long-period fiber gratings,” Optics Express,

vol. 14, no. 26, pp. 12670–12678, 2006.

[22] R. Slavık, Y. Park, and J. Azana, “Tunable dispersion-tolerant picosecond flat-top

waveform generation using an optical differentiator,” Optics Express, vol. 15, no. 11,

pp. 6717–6726, 2007.

[23] M. Kulishov, D. Krcmarık, and R. Slavık, “Design of terahertz-bandwidth arbitrary-

order temporal differentiators based on long-period fiber gratings,” Optics Letters,

vol. 32, no. 20, pp. 2978–2980, 2007.

[24] L. Rivas, K. Singh, A. Carballar, and J. Azana, “Arbitrary-Order Ultrabroadband

All-Optical Differentiators Based on Fiber Bragg Gratings,” IEEE PHOTONICS

TECHNOLOGY LETTERS, vol. 19, no. 16, pp. 1209–1211, 2007.

Bibliography 147

[25] L. Rivas, S. Boudreau, Y. Park, R. Slavık, S. LaRochelle, A. Carballar, and J. Azana,

“Experimental demonstration of ultrafast all-fiber high-order photonic temporal dif-

ferentiators,” Optics Letters, vol. 34, no. 12, pp. 1792–1794, 2009.

[26] D. Leaird and A. Weiner, “Femtosecond direct space-to-time pulse shaping,” IEEE

Journal of Quantum Electronics, vol. 37, no. 4, pp. 494–504, 2001.

[27] H. Tsuda, Y. Tanaka, T. Shioda, and T. Kurokawa, “Analog and digital optical

pulse synthesizers using arrayed-waveguide gratings for high-speed optical signal

processing,” Journal of Lightwave Technology, vol. 26, pp. 670–677, 2008.

[28] L. M. Rivas, M. J. Strain, D. Duchesne, A. Carballar, M. Sorel, R. Morandotti, and

J. Azana, “Picosecond linear optical pulse shapers based on integrated waveguide

Bragg gratings,” Optics Letters, vol. 33, pp. 2425–2427, Nov. 2008.

[29] M. Strain and M. Sorel, “Design and fabrication of integrated chirped Bragg gratings

for on-chip dispersion control,” IEEE Journal of Quantum Electronics, vol. 46, no. 5,

pp. 774–782, 2010.

[30] M. Strain and M. Sorel, “Post-growth fabrication and characterization of integrated

chirped Bragg gratings on GaAs-AlGaAs,” IEEE PHOTONICS TECHNOLOGY

LETTERS, vol. 18, no. 24, pp. 2566–2568, 2006.

[31] K. A. Rutkowska, D. Duchesne, M. J. Strain, J. Azana, R. Morandotti, and

M. Sorel, “Ultrafast all-optical temporal differentiation in integrated Silicon-on-

Insulator Bragg gratings,” in Lasers and Electro-Optics (CLEO) and Quantum Elec-

tronics and Laser Science Conference (QELS), pp. 1–2, 2010.

[32] X. Wang, W. Shi, R. Vafaei, N. Jaeger, and L. Chrostowski, “Uniform and Sam-

pled Bragg Gratings in SOI Strip Waveguides With Sidewall Corrugations,” IEEE

PHOTONICS TECHNOLOGY LETTERS, vol. 23, pp. 290–292, Mar. 2011.

Bibliography 148

[33] A. Yariv and P. Yeh, Photonics: optical electronics in modern communications. New

York: Oxford University Press, 6 ed., 2007.

[34] M. Yamada and K. Sakuda, “Analysis of almost-periodic distributed feedback slab

waveguides via a fundamental matrix approach,” Applied Optics, vol. 26, no. 16,

pp. 3474–3478, 1987.

[35] M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform. USA:

Society for Industrial and Applied Mathematics, 1981.

[36] A. Rosenthal and M. Horowitz, “Inverse scattering algorithm for reconstructing

strongly reflecting fiber Bragg gratings,” IEEE Journal of Quantum Electronics,

vol. 39, no. 8, pp. 1018–1026, 2003.

[37] E. Treacy, “Compression of picosecond light pulses* 1,” Physics Letters A, vol. 28A,

no. 1, pp. 34–35, 1968.

[38] H. Nakatsuka, D. Grischkowsky, and A. C. Balant, “Nonlinear Picosecond-Pulse

Propagation through Optical Fibers with Positive Group Velocity Dispersion,” Phys-

ical Review Letters, vol. 47, pp. 910–913, Sept. 1981.

[39] R. E. Saperstein, N. Alic, D. Panasenko, R. Rokitski, and Y. Fainman, “Time-

domain waveform processing by chromatic dispersion for temporal shaping of optical

pulses,” Journal of Optical Society of America B, vol. 22, no. 4, pp. 2427–2436, 2005.

[40] V. Torres-Company, J. Lancis, and P. Andres, “Arbitrary Waveform Generator

Based on All-Incoherent Pulse Shaping,” IEEE PHOTONICS TECHNOLOGY

LETTERS, vol. 18, pp. 2626–2628, Dec. 2006.

[41] M. Wefers and K. Nelson, “Space-time profiles of shaped ultrafast optical wave-

forms,” IEEE Journal of Quantum Electronics, vol. 32, no. 1, pp. 161–172, 1996.

Bibliography 149

[42] M. Wefers, “Analysis of programmable ultrashort waveform generation using liquid-

crystal spatial light modulators,” Journal of Optical Society of America B, vol. 27,

no. 4, pp. 742–756, 1995.

[43] Q. Mu, Z. Cao, L. Hu, D. Li, and L. Xuan, “An adaptive optics imaging system

based on a high-resolution liquid crystal on silicon device,” Optics Express, vol. 14,

no. 18, pp. 8013–8018, 2006.

[44] J. E. Wolfe and R. A. Chipman, “Polarimetric characterization of liquid-crystal-on-

silicon panels,” Applied Optics, vol. 45, no. 8, pp. 1688–1703, 2006.

[45] X. Wang, B. Wang, J. Pouch, F. Miranda, J. E. Anderson, and P. J. Bos, “Per-

formance evaluation of a liquid-crystal-on-silicon spatial light modulator,” Optical

Engineering, vol. 43, no. 11, pp. 2769–2774, 2004.

[46] J. Vaughan, T. Hornung, T. Feurer, and K. Nelson, “Diffraction-based femtosec-

ond pulse shaping with a two-dimensional spatial light modulator,” Optics Letters,

vol. 30, no. 3, pp. 323–325, 2005.

[47] B. E. A. Saleh and M. C. Teich, Fundamentals of photonics. New Jersey: John

Wiley & Sons, Inc., 2 ed., Feb. 2007.

[48] B. Eggleton, R. Slusher, C. de Sterke, and J. Sipe, “Bragg grating solitons,” Physical

Review Letters, vol. 76, no. 10, pp. 1627–1630, 1996.

[49] B. Eggleton, C. de Sterke, and R. Slusher, “Nonlinear pulse propagation in Bragg

gratings,” Journal of the Optical Society of America B: Optical Physics, vol. 14,

no. 11, pp. 2980–2993, 1997.

[50] C. Wang, “Microwave and Millimeter-Wave Arbitrary Waveform Generation and

Processing Using Fiber-Optics-Based Techniques,” Broadband Network & Multime-

dia Technology, 2009.

Bibliography 150

[51] N. K. Berger, B. Levit, and B. Fischer, “Reshaping periodic light pulses using cas-

caded uniform fiber Bragg gratings,” Journal of Lightwave Technology, vol. 24, no. 7,

pp. 2746–2751, 2006.

[52] J. Rothhardt, S. Haedrich, T. Gottschall, J. Limpert, A. Tuennermann, M. Roth-

hardt, M. Becker, S. Brueckner, and H. Bartelt, “Generation of flattop pump pulses

for OPCPA by coherent pulse stacking with fiber Bragg gratings,” Optics Express,

vol. 17, no. 18, pp. 16332–16341, 2009.

[53] R. Slavık, Y. Park, and J. Azana, “Long-period fiber-grating-based filter for gen-

eration of picosecond and subpicosecond transform-limited flat-top pulses,” IEEE

PHOTONICS TECHNOLOGY LETTERS, vol. 20, no. 10, pp. 806–808, 2008.

[54] J. Azana and M. Muriel, “Temporal self-imaging effects: Theory and application for

multiplying pulse repetition rates,” IEEE Journal on Selected Topics in Quantum

Electronics, vol. 7, no. 4, pp. 728–744, 2001.

[55] S. Longhi, M. Marano, P. Laporta, and V. Pruneri, “Multiplication and reshaping of

high-repetition-rate optical pulse trains using highly dispersive fiber Bragg gratings,”

IEEE PHOTONICS TECHNOLOGY LETTERS, vol. 12, no. 11, pp. 1498–1500,

2000.

[56] S. Longhi, M. Marano, P. Laporta, O. Svelto, and M. Belmonte, “Propagation,

manipulation, and control of picosecond optical pulses at 1.5 mu m in fiber Bragg

gratings,” Journal of the Optical Society of America B: Optical Physics, vol. 19,

no. 11, pp. 2742–2757, 2002.

[57] J. Azana, P. Kockaert, R. Slavık, L. Chen, and S. LaRochelle, “Generation of a 100-

GHz optical pulse train by pulse repetition-rate multiplication using superimposed

fiber Bragg gratings,” IEEE PHOTONICS TECHNOLOGY LETTERS, vol. 15,

no. 3, pp. 413–415, 2003.

Bibliography 151

[58] J. Azana, R. Slavık, P. Kockaert, L. Chen, and S. LaRochelle, “Generation of cus-

tomized ultrahigh repetition rate pulse sequences using superimposed fiber Bragg

gratings,” Journal of Lightwave Technology, vol. 21, no. 6, pp. 1490–1498, 2003.

[59] R. Slavık, Y. Park, M. Kulishov, and J. Azana, “Terahertz-bandwidth high-order

temporal differentiators based on phase-shifted long-period fiber gratings,” Optics

Letters, vol. 34, no. 20, pp. 3116–3118, 2009.

[60] M. A. Preciado and M. A. Muriel, “Ultrafast all-optical Nth-order differentiator

and simultaneous repetition-rate multiplier of periodic pulse train,” Optics Express,

vol. 15, no. 19, pp. 12102–12107, 2007.

[61] N. Berger, B. Levit, B. Fischer, and M. Kulishov, “Temporal differentiation of optical

signals using a phase-shifted fiber Bragg grating,” Optics Express, vol. 15, pp. 371–

375, 2007.

[62] N. Q. Ngo, “Optical integrator for optical dark-soliton detection and pulse shaping,”

Applied Optics, vol. 45, no. 26, pp. 6785–6791, 2006.

[63] J. Azana, “Proposal of a uniform fiber Bragg grating as an ultrafast all-optical

integrator,” Optics Letters, vol. 33, no. 1, pp. 4–6, 2008.

[64] Y. Park, T.-J. Ahn, Y. Dai, J. Yao, and J. Azana, “All-optical temporal integration

of ultrafast pulse waveforms,” Optics Express, vol. 16, no. 22, pp. 17817–17825, 2008.

[65] N. Quoc Ngo, “Design of an optical temporal integrator based on a phase-shifted

fiber Bragg grating in transmission,” Optics Letters, vol. 32, no. 20, pp. 3020–3022,

2007.

[66] R. Slavık, Y. Park, N. Ayotte, S. Doucet, T.-J. Ahn, S. LaRochelle, and J. Azana,

“Photonic temporal integrator for all-optical computing,” Optics Express, vol. 16,

no. 22, pp. 18202–18214, 2008.

Bibliography 152

[67] D. Leaird and A. Weiner, “Femtosecond direct space-to-time pulse shaping in an

integrated-optic configuration,” Optics Letters, vol. 29, no. 13, pp. 1551–1553, 2004.

[68] A. Krishnan, L. De Peralta, V. Kuryatkov, A. Bernussi, and H. Temkin, “Direct

space-to-time pulse shaper with reflective arrayed waveguide gratings and phase

masks,” Optics Letters, vol. 31, no. 5, pp. 640–642, 2006.

[69] M. J. R. Heck, P. Munoz, B. W. Tilma, E. A. J. M. Bente, Y. Barbarin, Y.-S.

Oei, R. Noetzel, and M. K. Smit, “Design, fabrication and characterization of an

InP-based tunable integrated optical pulse shaper,” IEEE Journal of Quantum Elec-

tronics, vol. 44, pp. 370–377, 2008.

[70] M. Hofmann, S. Bischoff, T. Franck, L. Prip, S. Brorson, J. Mørk, and K. Fro-

jdh, “Chirp of monolithic colliding pulse mode-locked diode lasers,” Applied Physics

Letters, vol. 70, no. 19, pp. 2514–2516, 1997.

[71] A. W. Fang, E. Lively, H. Kuo, D. Liang, and J. E. Bowers, “A distributed feedback

silicon evanescent laser,” Optics Express, vol. 16, no. 7, pp. 4413–4419, 2008.

[72] S. Gehrsitz, F. Reinhart, C. Gourgon, N. Herres, A. Vonlanthen, and H. Sigg, “The

refractive index of AlxGa1-xAs below the band gap: Accurate determination and

empirical modeling,” Journal of Applied Physics, vol. 87, no. 11, pp. 7825–7837,

2000.

[73] S. Giugni and T. Tansley, “Comment on the compositional dependence of bandgap

in AlGaAs and band-edge discontinuities in AlGaAs-GaAs heterostructures,” Semi-

conductor Science and Technology, vol. 7, no. 8, pp. 1113–1116, 1992.

[74] A. Yariv, “Coupled-Mode Theory for Guided-Wave Optics,” IEEE Journal of Quan-

tum Electronics, vol. QE-9, no. 9, pp. 919–933, 1973.

Bibliography 153

[75] R. Millett, K. Hinzer, A. Benhsaien, T. J. Hall, and H. Schriemer, “The impact

of laterally coupled grating microstructure on effective coupling coefficients,” Nan-

otechnology, vol. 21, no. 13, pp. –, 2010.

[76] W. Streifer, D. Scifres, and R. Burnham, “Analysis of grating-coupled radiation

in GaAs:GaAlAs lasers and waveguides,” IEEE Journal of Quantum Electronics,

vol. 12, pp. 422–428, July 1976.

[77] R. R. Millett, K. Hinzer, T. J. Hall, and H. Schriemer, “Simulation Analysis of

Higher Order Laterally-Coupled Distributed Feedback Lasers,” IEEE Journal of

Quantum Electronics, vol. 44, pp. 1145–1151, 2008.

[78] H. A. Haus and W. Huang, “Coupled-mode theory,” Proceedings of the IEEE, vol. 79,

no. 10, pp. 1505–1518, 1991.

[79] W.-P. Huang, “Coupled-mode theory for optical waveguides: an overview,” Journal

of the Optical Society of America A: Optics and Image Science, and Vision, vol. 11,

no. 3, pp. 963–983, 1994.

[80] H. Kogelnik and C. Shank, “Coupled-wave theory of distributed feedback lasers,”

Journal of Applied Physics, vol. 43, no. 5, pp. 2327–2335, 1972.

[81] R. Feced, M. Zervas, and M. Muriel, “Efficient inverse scattering algorithm for the

design of nonuniform fiber Bragg gratings,” IEEE Journal of Quantum Electronics,

vol. 35, no. 8, pp. 1105–1115, 1999.

[82] A. Weiner, J. HERITAGE, and E. KIRSCHNER, “High-Resolution Femtosecond

Pulse Shaping,” Journal of the Optical Society of America B: Optical Physics, vol. 5,

no. 8, pp. 1563–1572, 1988.

Bibliography 154

[83] P. Frangos and D. Jaggard, “Inverse Scattering - Solution of Coupled Gelfand-

Levitan-Marchenko Integral-Equations Using Successive Kernel Approximations,”

Ieee Transactions on Antennas and Propagation, vol. 43, no. 6, pp. 547–552, 1995.

[84] J. Skaar, L. Wang, and T. Erdogan, “On the synthesis of fiber Bragg gratings by

layer peeling,” IEEE Journal of Quantum Electronics, vol. 37, no. 2, pp. 165–173,

2001.

[85] J. Skaar and O. Waagaard, “Design and characterization of finite-length fiber grat-

ings,” IEEE Journal of Quantum Electronics, vol. 39, no. 10, pp. 1238–1245, 2003.

[86] L. Dong and S. Fortier, “Formulation of time-domain algorithm for fiber Bragg grat-

ing simulation and reconstruction,” IEEE Journal of Quantum Electronics, vol. 40,

no. 8, pp. 1087–1098, 2004.

Documents

Pulse Shaping Based on Integrated Waveguide Gratings...2.10 Flat-top pulse shaping using pulse stacking and pulse di erentiation.. .32 2.11 DST pulse shaping implemented with integrated