Pulse Shaping Based on Integrated Waveguide Gratings...2.10 Flat-top pulse shaping using pulse...
Transcript of Pulse Shaping Based on Integrated Waveguide Gratings...2.10 Flat-top pulse shaping using pulse...
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.
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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.
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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
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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
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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
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E.2 Linearly Width-Chirped Gratings . . . . . . . . . . . . . . . . . . . . . . 137
E.3 Gaussian-Apodized Gratings . . . . . . . . . . . . . . . . . . . . . . . . . 139
E.4 Apodized and Linearly-Chirped Gratings . . . . . . . . . . . . . . . . . . 141
Bibliography 154
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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
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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
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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
C.2 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 now taken into account. . . . . . . . . . . . . . . . . . . . . . . . . 112
C.3 Reflection response of a uniform grating with a waveguide width of w �1.4 µm, a recess depth of rd � 100 nm, and a grating period of Λ �249.5 nm. The grating length is ∆z � 100µm. The effective index disper-
sion is taken into account. . . . . . . . . . . . . . . . . . . . . . . . . . . 113
C.4 Reflection response of a uniform grating with a waveguide width of w �1.4 µm, a recess depth of rd � 25 nm, and a grating period of Λ �249.5 nm. The grating length is ∆z � 200µm. The effective index disper-
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
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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
E.6 Response of a grating generated by the inverse scattering algorithm with
the target response from a width-chirped grating. . . . . . . . . . . . . . 139
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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
E.9 Response of a grating generated by the inverse scattering algorithm with
the target response from a Gaussian-apodized grating. . . . . . . . . . . 141
E.10 Complex coupling coefficient calculated for a response of a Gaussian-
apodized and linearly-chirped grating. . . . . . . . . . . . . . . . . . . . . 142
E.11 Matched waveguide width and recess depth profiles. . . . . . . . . . . . . 142
E.12 Response of a grating generated by the inverse scattering algorithm with
the target response from a Gaussian-apodized and linearly-chirped grating. 143
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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
Chapter 1. Introduction 3
demonstrated [13–15]. It is based on introducing suitable time delays between pulse
replicas generated from beam splitters and recombining them. However, the free-space
pulse shaping techniques require bulky optical elements and strict alignment.
Temporal pulse shaping implemented in optical fibers is more compatible with 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,
Chapter 1. Introduction 4
provided that the grating has the right structure.
1.4 Thesis Work
In this work, pulse shaping using integrated waveguide grating is studied. The grating is
chosen due to its merits of compactness and ability for arbitrary pulse shaping. Most of
the integrated gratings previously fabricated have a surface-corrugated structure. This
type of grating structure requires additional fabrication steps: the index perturbation is
introduced by etching the top of the waveguide after the waveguide is defined. Another
grating is a sidewall grating, where the waveguide is etched on the sides thereby causing
index perturbation [28–32]. One beneficial point of this grating structure is that the
perturbation can be simultaneously generated at the same time as the waveguide is
defined, leading to a less complicated fabrication procedure. Another advantage of the
sidewall-etching geometry is the easy control over the coupling coefficient and the Bragg
wavelength profiles [29,30] by altering the grating profiles.
The integrated sidewall Bragg grating for pulse shaping is the main focus of this 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
Chapter 1. Introduction 5
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
Chapter 1. Introduction 6
(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
Chapter 2. Literature Review 9
realize either a very weak or a very strong coupling constant. Also, the weak coupling
limit results in low reflectivity ( 20%) and shows poor energy efficiency. These facts
reveal the limitations of the first-Born approximation method.
To solve the issue of low energy efficiency of the first-Born approximation, the use
of apodized linearly-chirped fiber Bragg grating (LC-FBG) is proposed [16]. The 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.
Chapter 2. Literature Review 10
Another pulse shaping method is named direct space-to-time (DST) pulse shaping.
The setup of this method is close to that of the 4-f configuration of the Fourier pulse
shaping (which will be discussed in the next section) but with some differences. This
shaping method is suitable for applications where a direct mapping between a spatial
pattern and an output waveform is required, such as in parallel-to-serial conversion [26].
The underlying principle is the diffraction theory. The temporal output is proportional
to the convolution between the temporal input field and the time-scaled spatial mask
function, [26],
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
Chapter 2. Literature Review 11
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
Chapter 2. Literature Review 12
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
Chapter 2. Literature Review 13
and lenses, by using the reflection setup the similarity of the sets is ensured.
A diffractive optical element is used to spectrally disperse the incoming light: This 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
Chapter 2. Literature Review 14
beam. Since, different frequencies fall onto different locations, the output field experiences
a sophisticated coupled function of space and frequencies (or time). This space-time
coupling is discussed in more detail in [41]. To shape a desired temporal waveform, the
space complication can be decoupled by an appropriately designed mask. For example,
the mask function is re-rendered by the spatial distribution of the beam mode leading to a
complex mask function 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
Chapter 2. Literature Review 15
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
Chapter 2. Literature Review 16
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
Chapter 2. Literature Review 17
modulator array can perform independently from other pixels within the same beam spot.
A periodic structure of the pixel array under the beam frequency spot is proposed in [46].
It scatters light into many angular orders. Should only the zeroth order be collected after
reflection, only a fraction of energy carrying by the zeroth order is received at the output.
On the other hand, the phase of the light is affected by the average phase of the periodic
structure. Since the average phase and scattering orders are independent from each other,
the amplitude and phase modulations are independently delivered by the LCoS using the
periodic structure in the pixel groups.
Acousto-Optic Modulators
Another famous form of a programmable SLM is the acousto-optic modulator. A 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].
Chapter 2. Literature Review 18
Figure (2.3): A basic setup for an acousto-optic modulator. Reprinted from [11], © (2011) with
permission from Elsevier.
2.2.2 Direct Space-to-Time Pulse Shaping
Direct space-to-time (DST) pulse shaping techniques result in a temporal output 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
Chapter 2. Literature Review 19
Figure (2.4): A conceptual schematic diagram for DST pulse shaping [26], reprinted with permission
© 2001 IEEE.
Fourier plane of the lens is the complimentary Fourier transform of the spatial profile
of the incoming electric field. The grating in the diagram angularly disperses frequency
components so that they are located separately on the Fourier plane. Recall that if the
incoming field of a particular frequency is finite in extent, its Fourier transform spreads
across the plane as well. If a single slit is placed at the Fourier plane to collect the output
light, light of all frequencies passes through the slit but with different content on their
respective Fourier transform profiles. Therefore, the output field is described by both the
input frequency content and the Fourier transform of the spatial mask; more specifically
the temporal waveform of the output is proportional to the convolution of the temporal
profile of the input pulse and the spatial mask scaled to time domain:
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.
Chapter 2. Literature Review 20
Figure (2.5): Direct space-to-time pulse shaping setup [26], reprinted with permission © 2001 IEEE.
2.2.3 Pulse Stacking
Pulse stacking as discussed earlier refers to a technique that combines several pulses with
certain time delays. Splitting a single pulse into several subpulses and imposing time
delays among them is usually done in the interferometry setup.
In free-space optics regime, the beam splitters and mirrors constitute the main 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].
Chapter 2. Literature Review 21
Figure (2.6): Pulse stacking using free-space optics interferometry setup. The shaded area represent
the actual pulse shaping section [13], reprinted with permission © 2007 IEEE.
2.2.4 Performance and Main Drawbacks
Depending on the setup, a temporal resolution for shaping of 20-30 femtoseconds is
possible [6] as well as a spectral resolution of 0.06 nm/pixel [12]. Some of impressive
results are shown in Fig. 2.7. The data packet could be yielded from the DST method,
which in this case nine bits are generated whereas the center bit was rendered off, shown
in Fig. 2.7a. Flat-top pulse shaping was also demonstrated with a pulse duration of
2 picoseconds from a 100-fs input pulse in Fig. 2.7b. If the SML array is assigned a
linear phase function, in Fig. 2.7c the output pulse is delayed compared to the input
pulse. Pulse shaping devices can also function as a dispersion compensator as shown in
Fig. 2.7d.
Even though free-space optics has been providing excellent pulse shaping performance,
it bears some drawbacks. Bulk optics is bulky: Optical components are in a macroscopic
scale and free space propagation takes a lot of space. This characteristic goes against
the trend of miniaturization. Another issue regards the alignment problem. Precise
alignment is usually critical to obtain good results. The more optical elements in the
pulse shaping system, the more complicated the alignment will be, and it results in bad
tolerance. The last major shortcoming is the fact that high quality optical elements
Chapter 2. Literature Review 22
Figure (2.7): (a) A femtosecond data packet, (b) a 2-ps flat-top pulse, (c) a delayed pulse using linear
spectral phase, and (d) a recompressed pulse. Reprinted from [11], © (2011) with permission from
Elsevier.
are required to achieve good results, which culminates in high cost of the pulse shaping
system.
The aforementioned drawbacks drive researchers to contrive other alternatives to 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
Chapter 2. Literature Review 23
ubiquitous in many systems, other application areas gain advantages from fiber-based
pulse shaping as well.
Pulse shaping in optical fibers can utilize many properties from plain optical fibers
or from structures in optical fibers. Plain optical fibers exhibit both linear and nonlinear
dispersion. The linear dispersion is related to the group velocity dispersion, in which
different wavelengths experience different effective indices, resulting in pulse broadening
or contracting after propagating through a certain distance in a fiber. The nonlinear
dispersion arises from the power-dependent refractive index resulting in the self-phase
modulation, which represents another source of dispersion. These two dispersions are
fundamental to any optical fibers and can be used to perform pulse shaping.
The main drawback of the use of a plain optical fiber is its low dispersion value and
hence a long optical fiber might be necessary to accumulate enough phase difference
or dispersion. Fortunately, the grating structure could enhance dispersion due to its
periodic index structure. The total dispersion in the fiber grating is a combination of the
material dispersion and the structural dispersion, which is the dispersion resulted from
the grating structure. Mathematical analysis can calculate the final dispersion effect
of the fiber grating, which will reveal the relationship between the dispersion and the
physical grating structure. In this sense, the required dispersion could be achieved by
appropriately creating the grating in the fiber.
Technological advance in fiber fabrication allows implementing different kinds of 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
Chapter 2. Literature Review 24
lead to mode walk-off. The following discussion assumes that the fibers are single-mode
otherwise stated explicitly.
The grating period, which is the period of index modulation introduced to the fiber,
determines the direction of operation. If the period is short enough, the grating could
introduce interaction between the forward- and backward-propagating modes and this
is the situation in FBGs. On the other hand, if the grating period is relatively long,
the interaction will occur between the modes of the same direction, i.e. two 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)
Chapter 2. Literature Review 25
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-
Chapter 2. Literature Review 26
limited pulse cannot be achieved by solely employing the waveguide dispersion, which
only acts as a filter. In this situation, the chirp must be introduced to the unchirped
pulse in time domain by modulation, which could be implemented using electro-optic
materials or self-phase modulation (SPM) [47, Chapter 22]. The latter method is quite
convenient because a silica optical fiber also exhibits a nonlinear effect especially in the
case of short pulses. The time domain phase modulation is introduced by means of Kerr
nonlinearity
∆φptq � �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
Chapter 2. Literature Review 27
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
Chapter 2. Literature Review 28
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
Chapter 2. Literature Review 29
having opposite dispersion results in a cancelation. However, passing the pulse through
the same grating but in the opposite direction serves a better cancelation in practice due
to the difficulty of fabricating two gratings with exactly complementary responses. The
effective results is the imposition of an amplitude response without a phase response to
the input pulse [20].
2.3.3 Pulse Stacking in Fiber-Based Devices
Other than working with the Fourier transform pulse shaping, fiber gratings can also
operate under the pulse stacking paradigm. For example, in [51], N concatenating
weak uniform FBGs were proposed and demonstrated Gaussian pulse generation from
a continuous-wave source, whereas using the same technique a flat-top pulse was the
target for [52]. The schematic diagram is shown in Fig. 2.8. Physically, the incoming
pulse propagates through a series of uniform gratings and it is partially reflected by each
grating. Assuming that the grating is weak, multiple reflection between gratings is 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
Chapter 2. Literature Review 30
Figure (2.8): Pulse stacking in a fiber-based device by N uniform FBGs, [51], reprinted with permission
© 2006 IEEE. (a) The temporal profile of the input pulse, (b) Reflected pulses from a series of fiber
grating separated by time delays, and (c) The resultant pulse due to interference of all reflected pulses.
core effective index. At the second LPG, LPG2, a fraction of the cladding mode is 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
Chapter 2. Literature Review 31
Figure (2.9): Pulse stacking in a fiber-based device by LPGs [53], reprinted with permission
© 2008 IEEE.
Gaussian pulse and its first-order derivative as proposed in [21]. The relevant fact is that
the temporal differentiation could be carried out using a uniform LPG. The efficiency of
differentiation depends on the matching between the central frequency of the incoming
pulse and the designed resonance frequency of the grating, in which an ideal operation
occurs at zero detuning. If the mismatch is present, a part of energy of the incoming
pulse goes through the differentiation and the other part remains in the original pulse.
With appropriate detuning, two pulses coexist in the fiber core, namely the original pulse
and its derivative. The two pulses then interfere and give rise to a resulting output which
appears flat-top-like as shown in Fig. 2.10. Introducing a strain to a fiber by stretching
serves as a detuning mechanism as used in [22].
The main drawback in pulse stacking is that the rise and fall times of the resulting
output is determined by those of the input. As shown in Fig. 2.10, the flat-top pulse
for ∆λ � �1.3 nm still has rise/fall times of 2 picoseconds, a characteristic of the input
pulse represented in a black solid curve. Ultrafast features might be achievable by adding
many subpulses possibly with different shapes and time delays, but this will only lead to
increasing complexity of the overall system.
2.3.4 Other Fiber-Based Pulse Shaping
Another related application is pulse repetition rate multiplication, which is mainly based
on a temporal Talbot effect, discussed in detail in [54]. The Talbot effect is an effect
of dispersion, which could be represented by a phase response of a device. The pulse
Chapter 2. Literature Review 32
Figure (2.10): Flat-top pulse shaping using pulse stacking and pulse differentiation in LPGs [21],
reprinted with permission © 2006 OSA. (a)The intensity of the output pulses. The black solid curve
represents the intensity of the input pulse. The red dotted, green dashed, and blue solid lines are the
intensity profiles of the output pulses for different detuning parameters between the pulse central
frequency and the designed central frequency of the grating temporal differentiators. (b) The phase of
the pulses.
repetition rate multiplication could be included in a waveform pulse shaping to acquire
both effects simultaneously. For example, in [55,56], a linearly chirped fiber Bragg grating
was used to realize a combined response including flat-top pulse shaping and pulse rate
multiplication, which was achieved up to 80 GHz. Pulse repetition rate multiplication
can also be achieved by using superimposed FBG structures. The pulse repetition rate
as high as 170 GHz was demonstrated [57,58].
For signal processing applications, optical waveform differentiation and integration
are among basic building blocks. In fibers, FBGs and LPGs are proposed to fulfill the
operation by providing appropriate filtering functions. N -order differentiation should
be achieved from a few gratings, instead of concatenating N first-order differentiators
because of the energy loss at each of the differentiators is very high due to the ideal
Fourier pair of the temporal differentiation, Hpωq � �jpω�ω0q. A series of uniform LPGs
Chapter 2. Literature Review 33
separated by π-phase shifts could provide N -order differentiation by designing correct
grating lengths for each gratings. Differentiations up to the fifth-order were simulated
in [23] and later demonstrated experimentally in [59] showing great results and operating
bandwidth of 10 nanometers. Other than using LPGs, temporal pulse differentiation
could also be realized in LC-FBGs working in transmission [24, 25, 60, 61]. The major
difference between the uses of LPGs and FBGs for temporal pulse differentiation is their
bandwidths. In general, the LPG-based differentiators have a large operating bandwidth
which could be in a terahertz range. The FBG-based differentiators on the other hand
have a bandwidth in the order of gigahertz.
Optical waveform integration is more challenging due to the fact that the ideal Fourier
transform of the operation suggests the filtering of magnitude larger than one near the
central frequency, i.e. 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.
Chapter 2. Literature Review 34
Similar to fiber-based devices, an integrated optical device must have a waveguide
part, which is a channel for light to propagate, and a functional part, which performs
a particular function such as pulse shaping. Most of the integrated optical circuit is
fabricated in semiconductor materials, such as a silicon-on-insulator (SOI) wafer or a
III-V semiconductor wafer. For the waveguide part, the cross-sectional refractive index
profile is responsible for the modes and their properties. The overall dispersion is a
combination of the material dispersion and the waveguide dispersion. In the functional
part, additional dispersions and also responses depend on the nature of that part. For
example, if the grating is fabricated in an integrated waveguide, the dispersion property
pertaining to the grating index perturbation will contribute to the overall dispersion
characteristic and it could dominate other contributions.
In terms of material choices, there are a lot of semiconductor systems that are 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
Chapter 2. Literature Review 35
versatility, the AlGaAs system is chosen for this thesis work.
The basic ideas for pulse shaping in free-space optics could be carried out in the
integrated configurations with some modifications. Recall that Fourier transform pulse
shaping based on the 4-f configuration and the direct-space-to-time (DST) pulse shaping
in bulk optics requires diffraction gratings, whose major functionality is to spatially
disperse spectral components of light. This functionality is achieved by using an 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
Chapter 2. Literature Review 36
(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.
Chapter 2. Literature Review 37
(a) Analog AWG-based 4-f pulse shaping. (b) Digital AWG-based 4-f pulse shaping.
Figure (2.12): Implementation of 4-f pulse shaping configuration, operating in reflection, in integrated
optics using arrayed-waveguide gratings [27], reprinted with permission © 2008 IEEE.
Among a few studies, integrated gratings in an AlGaAs system were demonstrated
to generate digital bit streams [28] and to compensate chirp from a semiconductor laser
[29]. In the first work, the grating structure was designed using the first-order Born
approximation or the weak grating limit associating the impulse response in reflection to
the apodization profile of the grating. Digital bit streams composing of 0 and 1 bits are
designed with appropriate time-to-space scaling such that adjacent bits are 2 picoseconds
apart, representing the temporal resolution of the device. In the experiment, the results
show fair performance but the distinction ratio between 0 and 1 was poor. Imperfections
could result from difficulties in fabrication of III-V semiconductor as well as from the use
of the weak grating assumption.
In the second work [29], a more involved consideration to the grating design was
employed aiming for an on-chip dispersion control for a semiconductor mode-locked laser
(MLL), which exhibits pulse chirping in the range of 0.1-10 ps/nm over a few nanometer
bandwidth [70]. The grating structure was derived using an inverse scattering method,
which basically suggests the device structure from its response. The fabricated gratings
yielded measured responses close to the simulated values, which could provide a time
Chapter 2. Literature Review 38
delay span of about 10 picoseconds to the incident pulse. Some results are displayed in
Fig. 2.13.
Figure (2.13): Reflectivity and time delay responses of measured and simulated grating whose
structure is generated from the inverse scattering algorithm for a quadratic delay [29], reprinted with
permission © 2010 IEEE
Both of the two studies show unprecedented control over the coupling strength and
the chirp of the grating because the sidewall-etching geometry was used to introduce
grating perturbation. However, one of the most common perturbation types is a surface
corrugation, displayed in Fig. 2.14a, in which a perturbation is introduced to the top part
of a waveguide. The strength of coupling coefficient depends on the depth of the 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,
Chapter 2. Literature Review 39
by changing the waveguide width along the grating the chirp is easily introduced even
the perturbation period remains constant. Hence, by varying the waveguide width and
the recess depth, the sidewall-etching geometry provides great controls to the grating
design. Another main advantage of this perturbation type is that the waveguide and the
grating are defined simultaneously including the ability to tune the coupling strength
and the chirp without any further steps. Since arbitrary pulse shaping most likely 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
Chapter 2. Literature Review 40
response and to retrieve the structure of the grating are crucial and should be robust and
rigorous.
Chapter 3
Grating Responses
In this chapter, direct scattering to determine the response of the grating is considered.
The numerical modeling is based on the coupled-mode theory and the transfer matrix
method, as discussed at length in Appendix A.
3.1 Waveguide Design
Since an integrated grating is fabricated on a waveguide, it is necessary to consider the
structure of the waveguide. The electric field of the guided modes is the solution to the
following equation, [33],
∇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.
Chapter 3. Grating Responses 43
(a) AlGaAs layer structure. (b) Cross-sectional waveguide profile.
Figure (3.2): Cross-sections of a layer structure and a AlGaAs waveguide.
Lumerical MODE Solutions is used to find the guided modes and their effective 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.
Chapter 3. Grating Responses 44
(a) Cross-sectional index profile.
(b) Electric field magnitude of the TE-like mode. (c) Electric field magnitude of the TM-like mode.
Figure (3.3): The simulated index profile, a corresponding fundamental TE electric field mode, and a
corresponding fundamental TM electric field mode.
The modes and their effective indices depend on the wavelength and the refractive
index profile, which is particularly altered by changing the waveguide width and the
etch depth. Fig. 3.4 shows the dependence of the effective indices of the modes to the
waveguide width, with a fixed etch depth at 1.0 µm. The effective index increases with
increasing waveguide width because the guided modes sense more high refractive index
Chapter 3. Grating Responses 45
of AlGaAs. However, increasing the width beyond some value results in a multimode
waveguide. For this particular layer structure as shown in Fig. 3.2a, the 1.6-µm-wide
waveguide has multiple guided modes. The dispersion as a function of the waveguide
width is compared between the TE and TM modes in Fig. 3.5. It can be seen that near
w � 1.4 µm both TE and TM modes have approximately the same effective index such
that they propagate in the same manner in an unperturbed waveguide.
(a) Effective index of the TE-like mode. (b) Effective index of the TM-like mode.
Figure (3.4): Effective indices of the fundamental TE-like and TM-like modes of the waveguide as a
function of waveguide width and the light wavelength at the etch depth of 1 micron.
The effect of the etch depth on the effective index can be explained in the same way
as the effect of the waveguide width. A deeper etch depth exposes the waveguide to more
air; the modes then sense more low refractive index of air. Therefore, the effective index
of the mode decreases with increasing the etch depth. However, in an actual device the
etch depth is the same throughout the whole grating for a simple fabrication procedure,
rendered it out of the degrees of freedom. Therefore, the dependency of the effective
index on the waveguide width and the wavelength is more prominent providing a key
method to introduce chirp to the grating.
Chapter 3. Grating Responses 46
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
Chapter 3. Grating Responses 47
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
Chapter 3. Grating Responses 48
inverse scattering theory for grating design, a simple version of CMT is in use; the
waveguide is assumed to be lossless and only the first-order Fourier coefficient of the
index perturbation is considered, as laid out in [74]. In actual integrated gratings, 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)
Chapter 3. Grating Responses 49
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.
Chapter 3. Grating Responses 50
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.
Chapter 3. Grating Responses 51
Table (3.3): Cross-coupling coefficients as a function of recess depths for a constant waveguide width of
1.4 microns and λ � 1.55 µm.
Recess depth (nm) 0 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.
Chapter 3. Grating Responses 52
Table (3.4): Self-coupling coefficients as a function of recess depths for a constant waveguide width of
1.4 microns and λ � 1.55 µm.
Recess depth (nm) 0 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
Chapter 3. Grating Responses 53
large such that the frequency span is much larger than the significant bandwidth of the
grating. It is then necessary to only simulate the grating response within the grating
bandwidth and assume zero response elsewhere in the frequency space; otherwise the
simulation will consume a lot of computing time. As a result, the interested frequency
range centering around the main frequency is specified by a 1�Nν vector:
λ � 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
��� c1pzqc�1pzq
�� , (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)
m22pz, z0q � cosh�spz � z0q
� 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
Chapter 3. Grating Responses 54
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)
Chapter 3. Grating Responses 55
The rest of the frequency points in the frequency axis are set to zero.
The algorithm is verified by simulating a set of gratings with familiar responses. 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)
Chapter 4. Retrieval of the Gratings 58
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
Chapter 4. Retrieval of the Gratings 59
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)
Chapter 4. Retrieval of the Gratings 60
From its dimension, ζ is interpreted as the Fourier pair of the spatial variable z. It is also
a one-to-one function to the wavelength. The set of sampled reflection response points
represents responses of different independent variables
r � rris � 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
Chapter 4. Retrieval of the Gratings 61
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
Chapter 4. Retrieval of the Gratings 62
corresponding to the TE-like mode at that waveguide width. From the Bragg wavelength
relation, λB � 2neff,0Λ0, the grating period could be set to Λ0 � 250 nm. If the generated
waveguide width deviates from 1.4 µm considerably, the grating period could be adjusted
to reduce the deviation due to Eq. 4.26.
The unknown grating is broken down to Ng subgratings with equal length of ∆z. Since
the GLM solution, summarized in Section 4.1, is solved for each grating, its continuous
nature allows several sampling points, say Nsg, in each subgrating. The total number of
sampling points are then 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
Chapter 4. Retrieval of the Gratings 63
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;
Chapter 4. Retrieval of the Gratings 64
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
Chapter 4. Retrieval of the Gratings 65
−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.
Chapter 4. Retrieval of the Gratings 66
4.6 Verification of the Inverse Scattering Algorithm
To test the theory and the algorithm, a test grating is defined and its (test) response
is determined using the direct scattering. The inverse scattering (IS) algorithm receives
the test response, calculates the complex coupling coefficient, and yields waveguide width
and recess depth profiles. From the generated physical profiles, the grating response is
computed and compared to the test response. In Appendix E, a variety of test gratings are
used to validate the algorithm and the results are reported therein. Those gratings include
uniform gratings, linearly width-chirped gratings, and Gaussian-apodized gratings.
In this section, one type of gratings, i.e. an apodized and chirped grating, is considered
for verification. The test grating was chirped by varying the waveguide width linearly
and also Gaussian-apodized by a suitable recess depth profile. The parameters for the
inverse scattering algorithm were set as Ng � 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,
Chapter 4. Retrieval of the Gratings 67
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.
Chapter 4. Retrieval of the Gratings 68
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
Chapter 5. Pulse Shaping Simulations 71
−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
Chapter 5. Pulse Shaping Simulations 72
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
Chapter 5. Pulse Shaping Simulations 73
−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 �
Chapter 5. Pulse Shaping Simulations 74
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.
Chapter 5. Pulse Shaping Simulations 75
such that the they interfere destructively. This fact leads to a compromise between
performance and footprint of the grating. In other words, a longer grating is needed to
produce an exact flat-top pulse with minimal stray subpulses. In the frequency domain,
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
(a) Amplitude response
−5 −4 −3 −2 −1 0 1 2 3 4 5
−10
0
10
20
τ (
ps)
ν−νc (THz)
Simulated
Target
(b) Time delay response
−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
Chapter 5. Pulse Shaping Simulations 76
ps are shown in Fig. 5.6 when the parameters of the inverse scattering algorithm remain
similar to the previous case. They were generated from gratings about 200-350 micron
long. All of the waveforms have rise and fall times of approximately 0.2 ps.
−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
Chapter 5. Pulse Shaping Simulations 77
about the fabrication critical dimension, i.e. 5 nm. Adding various random deviation
profiles to the generated grating for 2-ps flat-top pulses, the responses are displayed in
Fig. 5.7. Within the flat-top duration, the maximum difference of electric field 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
(a) Amplitude response
−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
(b) Time delay response
−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
Chapter 5. Pulse Shaping Simulations 78
for the ones investigated in [5, 82]. The rise and fall times of the device is in about 200
ps, close to the pulse duration of the input pulse. This result is better than the rise/fall
times 700 ps as reported in [5].
5.3 Triangular Pulse Shaping
A triangular pulse envelope could be expressed as
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.
Chapter 5. Pulse Shaping Simulations 79
−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.
Chapter 5. Pulse Shaping Simulations 80
−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
(a) Amplitude response
−4 −2 0 2 40
5
10
15
20
τ (
ps)
ν−νc (THz)
Simulated
Target
(b) Time delay response
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.
Chapter 5. Pulse Shaping Simulations 81
5.4 One-to-Many Pulse Shaping
Previously pulse shaping is assumed to be one-to-one; however, in this section, this
assumption is relaxed.
First of all, a periodic property of the discrete Fourier transform, which delineates the
fast Fourier transform algorithm, should be recapitulated. A finite information in a time
domain is assumed fundamentally to be periodic over a period of N data points, where N
is the number of data points representing information. The corresponding discrete Fourier
transform is also periodic in N . If the spacings between points are ∆t and ∆ν in the
time and frequency domains respectively, the periodicity in time and frequency becomes
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
Chapter 5. Pulse Shaping Simulations 82
amplitude was one. Hence, it is appropriate to subvert the problem by allowing α � 0.95,
and from the definition of the target the time delay is set to τd � 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.
Chapter 5. Pulse Shaping Simulations 83
−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
Chapter 5. Pulse Shaping Simulations 84
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.
Chapter 5. Pulse Shaping Simulations 85
It should be expected that if many output pulses are desired from a single pulse, the
magnitude of the coupling coefficient of each grating section will become lower until it is
not realizable by the fabrication technology, which dictates, in this context, the matching
algorithm.
5.5 Summary
In this chapter, both the direct and inverse scattering algorithms are employed to 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
Chapter 6. Conclusions and Future Direction 88
supporting the potentials of the integrated sidewall gratings for pulse shaping purposes.
6.2 Future Directions
The theoretical framework and numerical modeling developed through the course of this
work could serve as a starting point to the design of the pulse shaping grating in the
integrated regime. The obvious next step to take is to fabricate the grating as the
algorithm suggests the physical form and measure the spectral response and shaping
performance. This step ultimately validates the algorithms and the theories behind it.
In terms of theory, a modification to the coupled-mode theory to accommodate other
leaky modes and absorption or gain could be done as well to account for strong coupling
regimes. This work also neglects the dependence of the effective index on the recess depth
of the grating structure, which is not strictly valid especially in strong perturbations and
nano-waveguides. It might also be interesting to see the effect of different perturbation
periodicities between the left and right sidewall etchings, which is beyond the scope of the
current algorithm. For the inverse scattering algorithm, the improvement could be in the
matching method. In this work, the grating period is assumed to be constant. However,
the algorithm that allows variations in the grating period will achieve one more degree
of freedom in the design. The decoupling between the width, the recess depth, and the
grating period (if included) should be more effective to eliminate error in the result. More
accurate results might be speculated if the dispersion is allowed in the algorithm as well.
A better method should be devised to address multiple output generation. Additionally,
the inverse scattering should also extend to include targeted response in transmission,
which could be useful in situations where transmission responses of a grating is at work
such as optical waveform differentiation and integration.
On the other hand, both the described direct and inverse scattering theories are
within the linear regime. Extending the theories to encompass nonlinearity will enhance
Chapter 6. Conclusions and Future Direction 89
accuracy and yield more functionalities. In doing so, the two theories must be recast in
other forms, thereby a new sets of algorithms for numerical modeling.
Appendix A
Coupled-Mode Theory (CMT)
In linear pulse shaping, a pulse shaping device is mathematically represented by its 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
Appendix A. Coupled-Mode Theory (CMT) 92
have many guided modes. However, for certain applications, a single-mode waveguide
proves to be a better choice because of no mode walk-off or mode beating.
A.2 Coupled-Mode Theory
The coupled-mode theory (CMT) has been rigorously investigated, revised, and applied
to many situations, such as in directional couplers, waveguide gratings, ring resonators,
and wireless charging. It is used to analyze a system which has small perturbation from
its original configuration.
In the case of gratings, the isolated systems are particularly the modes themselves in
the waveguide. Without any perturbation, the modes do not interact with one another.
A small perturbation in the form of gratings seeds interaction among those modes and
results in energy exchange. Despite complexity of the grating system, it is still represented
mathematically by its permittivity 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
Appendix A. Coupled-Mode Theory (CMT) 93
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)
Appendix A. Coupled-Mode Theory (CMT) 94
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)
Appendix A. Coupled-Mode Theory (CMT) 95
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
Appendix A. Coupled-Mode Theory (CMT) 96
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)
Appendix A. Coupled-Mode Theory (CMT) 97
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
Appendix A. Coupled-Mode Theory (CMT) 98
will be useful in the transfer matrix method;��� c1pzqc�1pzq
�� �
���m11pz, z0q m12pz, z0qm21pz, z0q m22pz, z0q
�� ��� c1pz0qc�1pz0q
�� � Mpz, z0q
��� c1pzqc�1pzq
�� (A.27)
where
m11pz, z0q � cosh�spz � z0q
� 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)
m22pz, z0q � cosh�spz � z0q
� 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)
Appendix A. Coupled-Mode Theory (CMT) 99
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.
Appendix A. Coupled-Mode Theory (CMT) 100
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
��� c1pz0qc�1pz0q
��
��� c1pz2qc�1pz2q
�� � M2pz2, z1q
��� c1pz1qc�1pz1q
��
...��� c1pzNqc�1pzNq
�� � MNpzN, zN�1q
��� c1pzN�1qc�1pzN�1q
��
Therefore, it could be written that��� c1pzNqc�1pzNq
�� � MNMN�1 � � �M2M1
��� c1pz0qc�1pz0q
��
��� c1pzNqc�1pzNq
�� � M
��� c1pz0qc�1pz0q
�� �
���M11 M12
M21 M22
�� ��� c1pz0qc�1pz0q
�� (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 .
Appendix A. Coupled-Mode Theory (CMT) 101
For a Bragg grating, reflection response is of particular interest. In isolation from
other optical components, the boundary condition of the problem at hands is that at the
back of the Bragg grating the backward-propagating field is zero, i.e. 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
Appendix A. Coupled-Mode Theory (CMT) 102
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
Appendix A. Coupled-Mode Theory (CMT) 103
maximum reflectivity, occurs at the Bragg wavelength. The presence of the self-coupling
constant shifts this maximum reflectivity to a nearby frequency. Another source of peak
shifting is the effective index dispersion against the wavelength.
Magnitude and Phase Responses
Gratings in general do not have a close-form reflection response as uniform gratings
do. The reflection response is calculated using the transfer matrix method as described
previously. It is usually a complex function allowing one to write
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)
Appendix B. Fourier Transforms 106
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)
Appendix B. Fourier Transforms 107
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)
Appendix B. Fourier Transforms 108
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.
Appendix C. Simulation Results for Grating Responses 111
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.
Appendix C. Simulation Results for Grating Responses 112
Another way to increase the reflectivity, apart from increasing the coupling constant,
is by increasing the grating length, ∆z. For example, let the grating has the length of
∆z � 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|
(a) Amplitude response.
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.
Figure (C.2): 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 now taken into account.
Appendix C. Simulation Results for Grating Responses 113
1.54 1.55 1.56 1.570
0.2
0.4
0.6
0.8
1
λ (µm)
|r|
(a) Amplitude response.
1.54 1.55 1.56 1.570
0.5
1
1.5
2
λ (µm)
τ (p
s)
(b) Group delay response.
Figure (C.3): Reflection response of a uniform grating with a waveguide width of w � 1.4 µm, a recess
depth of rd � 100 nm, and a grating period of Λ � 249.5 nm. The grating length is ∆z � 100µm. The
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.
Appendix C. Simulation Results for Grating Responses 114
1.54 1.55 1.56 1.570
0.2
0.4
0.6
0.8
1
λ (µm)
|r|
(a) Amplitude response.
1.54 1.55 1.56 1.572
4
6
8
λ (µm)
τ (p
s)
(b) Group delay response.
Figure (C.4): 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 � 200µm. The
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.
Appendix C. Simulation Results for Grating Responses 115
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
Appendix C. Simulation Results for Grating Responses 116
1.53 1.54 1.55 1.56 1.57 1.580
0.2
0.4
0.6
0.8
λ (µm)
|r|
(a) Amplitude response.
1.53 1.54 1.55 1.56 1.57 1.58−5
0
5
10
15
λ (µm)
τ (p
s)
(b) Group delay response.
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.
Appendix C. Simulation Results for Grating Responses 117
The group delay responses, shown in Fig. C.6b, are complimentary. The up-tapered
grating behaves similarly to the positively-chirped grating as the group delay increases
in a linear manner in a range λ P r1.54, 1.56sµm. 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|
(a) Amplitude response.
1.52 1.53 1.54 1.55 1.56 1.57−5
0
5
10
15
λ (µm)
τ (p
s)
(b) Group delay response.
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
Appendix C. Simulation Results for Grating Responses 118
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|
(a) Amplitude response.
1.53 1.54 1.55 1.56 1.57−4
−2
0
2
4
λ (µm)
τ (p
s)
(b) Group delay response.
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.
Appendix C. Simulation Results for Grating Responses 119
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
Appendix C. Simulation Results for Grating Responses 120
1.54 1.55 1.56 1.570
0.2
0.4
0.6
0.8
1
λ (µm)
|r|
(a) Amplitude response.
1.54 1.55 1.56 1.571
1.5
2
2.5
3
3.5
4
λ (µm)
τ (p
s)
(b) Group delay response.
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.
Appendix C. Simulation Results for Grating Responses 121
1.53 1.54 1.55 1.56 1.57 1.580
0.2
0.4
0.6
0.8
1
λ (µm)
|r|
(a) Amplitude response.
1.53 1.54 1.55 1.56 1.57 1.58−5
0
5
10
15
λ (µm)
τ (p
s)
(b) Group delay response.
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)
Appendix D. Inverse Scattering Theory 124
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
Appendix D. Inverse Scattering Theory 125
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)
Appendix D. Inverse Scattering Theory 126
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.
Appendix D. Inverse Scattering Theory 127
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.
Appendix D. Inverse Scattering Theory 128
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)
Appendix D. Inverse Scattering Theory 129
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)
Appendix D. Inverse Scattering Theory 130
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)
c�1pzq � c�1pzq exp
��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
dzc�1 � jκpzq� exp
��j2
» z
0
σpz1q dz1c1 � j
∆β0
2c�1. (D.37b)
Appendix D. Inverse Scattering Theory 131
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)
Appendix D. Inverse Scattering Theory 132
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
�j�1 (D.44a)
d
dzc�1 � jσc�1 � jκ�c1e
j�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)
c�1pzq � c�1pzq exp
��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
dzc�1 � jκpzq� exp
��j2
» z
0
σpz1q dz1c1 � j
∆β0
2c�1. (D.48b)
The desired reflection response is defined by
r � c�1pz0qc1pz0q . (D.49)
Appendix D. Inverse Scattering Theory 133
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
Appendix E. Simulation Results for Grating Retrieval 136
the minimum recess depth feature is 5 nm, the waveguide width and the recess depth
profiles calculated from the modified matching algorithm are calculated and shown in
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.
Appendix E. Simulation Results for Grating Retrieval 137
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
Appendix E. Simulation Results for Grating Retrieval 138
deteriorates in the recess depth profile if its value is large. However, since the region
of significant coupling can be determined from the magnitude of the complex coupling
coefficient, pieces of the calculated grating to be used could be picked manually. The
reflection response of this calculated grating when the subgrating sections from Ni � 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
Appendix E. Simulation Results for Grating Retrieval 139
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 1.535 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
(b) Time delay response
Figure (E.6): Response of a grating generated by the inverse scattering algorithm with the target
response from a width-chirped grating.
E.3 Gaussian-Apodized Gratings
In the previous examples, the recess depth is kept constant while the waveguide width
changes along the grating profile. Now consider the opposite situation with a special case
of a Gaussian apodization. Firstly from the known relationship between the coupling
coefficient and the recess depth calculated for the direct scattering problem, the recess
depth profile to yield a Gaussian coupling coefficient profile is determined. The grating
has a constant waveguide width of 1.4 µm and a grating period of 250 nm. In calculating
the response of this grating, it is divided into 200 uniform pieces with a length of 1 µm,
leading to a total length of 200 µm. This response will now be the target response for
the inverse scattering algorithm, whose initial parameters maintain the values used in
the preceding sections.
The determined complex coupling coefficient is displayed in Fig. E.7. Its amplitude
corresponds very well with the input cross-coupling coefficient. The relative phase shows
a flat Gaussian feature in the significance region reflecting the self-coupling coefficient in
its expression. The matching algorithm yields the profiles of the waveguide width and
the recess depth as plotted in Fig. E.8. The waveguide width is determined to be nearly
constant at 1.4 µm as expected and the recess depth assumes the values in excellent
Appendix E. Simulation Results for Grating Retrieval 140
agreement with that of the original grating within the significance region. With the
subgrating pieces from 48 to 248, the response of this generated grating is found to be
those shown in Fig. E.9.
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
Appendix E. Simulation Results for Grating Retrieval 141
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 1.535 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
(b) Time delay response
Figure (E.9): Response of a grating generated by the inverse scattering algorithm with the target
response from a Gaussian-apodized grating.
E.4 Apodized and Linearly-Chirped Gratings
In the previous cases so far, either the waveguide width or the recess depth is held constant
along the grating. Now consider the situation when both of them vary. Specifically,
the grating will be linearly-chirped and Gaussian-apodized in such a way that it is a
combination of the gratings in Section E.2 and Section E.3.
The complex coupling coefficient is determined and plotted in Fig. E.10. Its 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.
Appendix E. Simulation Results for Grating Retrieval 142
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.
Appendix E. Simulation Results for Grating Retrieval 143
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 (E.12): Response of a grating generated by the inverse scattering algorithm with the target
response from a Gaussian-apodized and linearly-chirped grating.
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