Dynamic Optical Arbitrary Waveform Generation (O-AWG) in a...
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Dynamic Optical Arbitrary Waveform Generation (O-AWG) in a Continuous Fiber by Yu Yeung (Kenny) Ho A thesis submitted in conformity with the requirements for the degree of Master of Applied Science Graduate Department of Electrical and Computer Engineering University of Toronto Copyright c 2008 by Yu Yeung (Kenny) Ho
Transcript of Dynamic Optical Arbitrary Waveform Generation (O-AWG) in a...
Dynamic Optical Arbitrary Waveform Generation (O-AWG)
in a Continuous Fiber
by
Yu Yeung (Kenny) Ho
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 c© 2008 by Yu Yeung (Kenny) Ho
Abstract
Dynamic Optical Arbitrary Waveform Generation (O-AWG)
in a Continuous Fiber
Yu Yeung (Kenny) Ho
Master of Applied Science
Graduate Department of Electrical and Computer Engineering
University of Toronto
2008
We present a low-loss dynamic waveform shaping technique for high-repetition-rate sig-
nals by independent phase and amplitude control of spectral lines in a continuous fiber.
Our system employs uniform FBGs to separate the spectral lines of a modulated CW
signal, and provides independent amplitude and phase control for each line via in-line
polarization controller and in-line fiber stretcher respectively. Our system offers the ad-
vantage of negligible insertion loss, and thus can be scaled up to control many lines to
achieve high temporal resolution and better shape control.
We modelled and simulated our O-AWG system and investigated the effects of finite
linewidth and finite number of line. Then a testbed was assembled to test a 3-line and
a 5-line system. Several waveforms were experimentally demonstrated with a spectral
resolution of 0.12nm and a temporal resolution of 17ps for the 5-line system; 0.16nm
spectral resolution and 25ps temporal resolution for the 3-line system.
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Acknowledgements
First, I would like to thank my supervisor Professor Li Qian for all her advice and patient.
I appreciate her rigour and persistent in consideration of my work.
I would like to acknowledge and thank Professor Xijia Gu for allowing me to use his
FBG fabrication equipments and to his lab technician Jiang Li for assisting me in the
making of the FBGs. I would also like to thank Dr. Waleed Mohammad, Chris Sapiano
and Dr. Aaron Zilkie for their knowledge and many useful discussion that motivated me
and sparked new ideas.
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Contents
1 Motivation for Arbitrary Waveform Generation 1
1.1 Applications of AWG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 AWG Applications in the Electrical Domain . . . . . . . . . . . . 2
1.1.2 AWG Applications in the Optical Domain . . . . . . . . . . . . . 4
1.2 Arbitrary Waveform Generation Methods . . . . . . . . . . . . . . . . . . 4
1.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Existing O-AWG Methods 6
2.1 Direct Temporal Shaping . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Direct Frequency Shaping . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.1 O-AWG via Static Frequency Manipulation . . . . . . . . . . . . 10
2.2.2 O-AWG via Dynamic Frequency Manipulation . . . . . . . . . . . 12
Spatial Light Modulator . . . . . . . . . . . . . . . . . . . . . . . 12
Shaping via Electro-Optic Effect . . . . . . . . . . . . . . . . . . . 13
Shaping via Acousto-Optic Effect . . . . . . . . . . . . . . . . . . 16
2.3 Thesis Contribution — Dynamic O-AWG in a Continuous Fiber . . . . . 17
3 Modeling and Simulation 19
3.1 Modelling of Our O-AWG System . . . . . . . . . . . . . . . . . . . . . . 19
3.1.1 Practical Aspects of Modelling Optical Signals . . . . . . . . . . . 21
3.2 Effects of Finite Spectral Sampling . . . . . . . . . . . . . . . . . . . . . 27
3.2.1 Bandwidth limit due to finite number of spectral lines . . . . . . . 28
3.2.2 Effect of coarse spectral sampling due to finite spectral resolution 30
3.2.3 Arbitrary Waveform Generation with 101 Spectral Lines . . . . . 31
3.3 Waveform Control and Error Minimization . . . . . . . . . . . . . . . . . 31
3.4 Effects of Non-Zero Linewidth . . . . . . . . . . . . . . . . . . . . . . . . 38
3.4.1 Phase Recovery for the Spectral Line Through Gerchberg-Saxton
Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
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3.4.2 Spectral Line Modelling Through Coherence Time Simulation in
Time Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4 Experimental O-AWG System 48
4.1 The Testbed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2 Principle Operations of the O-AWG System . . . . . . . . . . . . . . . . 51
4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.3.1 3-line O-AWG System . . . . . . . . . . . . . . . . . . . . . . . . 53
4.3.2 5-line O-AWG System . . . . . . . . . . . . . . . . . . . . . . . . 55
4.3.3 Waveform Stability . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.4 Comparisons Between Different Dynamic O-AWG Methods . . . . . . . . 62
5 Conclusion 64
5.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Appendices 67
A MATLAB Code for Characterizing Spectrum Coarse Sampling 67
Appendices 70
B MATLAB Code for Generating the 101-line Example 71
Appendices 72
C MATLAB Code for Simulating Finite Linewidth with Coherence Time
Model 73
Appendices 76
D MATLAB Code for Characterizing Limited Bandwidth 77
Appendices 78
E MATLAB Code for Recovering the Phase of Measured Data 79
Bibliography 80
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List of Tables
3.1 Phase relationships (as appeared in (3.7)) of the three spectral lines for
the 5 example cases in Figure 3.3. . . . . . . . . . . . . . . . . . . . . . . 23
3.2 Phase increments from one example to another for each phase relationship. 27
3.3 Simulation conditions and related physical constraints. . . . . . . . . . . 27
4.1 3-FBG Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.2 5-FBG Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.3 Comparison between various existing methods O-AWG methods with our
proof of concept system. . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
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List of Figures
1.1 Electromagnetic spectrum [1]. . . . . . . . . . . . . . . . . . . . . . . . . 2
2.1 Transfer function diagram of an arbitrary waveform generator. . . . . . . 6
2.2 Illustration of a finite impulse response (FIR). . . . . . . . . . . . . . . . 8
2.3 A basic delay-line photonic signal processor that can be modelled with
FIR [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Static pulse shaping using amplitude and/or phase mask [3]. . . . . . . . 11
2.5 Pulse shaping using Spatial Light Modulator (SLM) [4]. . . . . . . . . . . 13
2.6 Pulse shaping via Electro-Optic effect [5]. . . . . . . . . . . . . . . . . . . 14
2.7 Pulse shaping via Electro-Optic amplitude modulation with FBGs as dis-
persive elements [6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.8 Pulse shaping using Acousto-Optic effect via a birefringent crystal [7]. . . 16
2.9 All-fiber spectral line-by-line O-AWG system incorporating FBGs, polar-
ization controllers, and fiber stretchers, demonstrated in this thesis. Plot
A, B and C depict the signal spectra at points A, B and C of the system,
respectively. The spectral lines of the signal (λ1, λ2, . . . , λn) match the
central wavelengths of the FBGs (FBG1, FBG2, . . . , FBGn). For a 5-line
O-AWG, n=5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.1 Illustration of the transfer function for a line-by-line shaping system verses
a block shaping system. . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2 Illustration of several system parameters: Temporal Resolution or Band-
width, Spectral Resolution or Period. . . . . . . . . . . . . . . . . . . . . 21
3.3 Simulated 3-line shaping with L1 = 0.5, L2 = 1, L3 = 1.5. . . . . . . . . 24
3.4 Simulated 3-line shaping with L1 = 1, L2 = 1, L3 = 1. . . . . . . . . . . 25
3.5 Simulated 3-line shaping with L1 = 1, L2 = 0.5, L3 = 1.5. . . . . . . . . 26
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3.6 Error introduced by representing a transform limited Gaussian signal with
5 discrete spectral lines, plotted as a function of the bandwidth of the
Gaussian signal. Line separation is 0.12 nm resulting in total 0.48 nm
bandwidth with 5 lines. Solid lines represent the target time function
and target spectrum, dotted lines represent the shaped function and the
“sampled” spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.7 Time plots of a Gaussian spectrum varied by polynomial with 4, 5, 7 or
9 degree. Solid lines represent the target signal in the time domain and
the target spectrum, dotted lines represent the generated waveform and
spectrum with a 5-line system. In the 5-degree case, the target waveform
is duplicated twice and shifted to show the aliasing of the waveform in
the time domain (shaded lines). The shaded rectangle indicates the time
aperture as defined by the repetition rate (15 GHz or 66.66 ps). The target
spectrum is shown below the time domain plot. . . . . . . . . . . . . . . 32
3.8 Waveform example 1. Top Left: Target (dotted) and generated (solid)
waveform in time domain. Top Right: Spectral amplitude and phase of
the target waveform. Bottom Right: Spectral amplitude and phase of the
waveform generated by the 101-line O-AWG. Bottom Left: Deviation error
from the target waveform. The average error is 0.88%. . . . . . . . . . . 33
3.9 Waveform example 2. Top Left: Target (dotted) and generated (solid)
waveform in time domain. Top Right: Spectral amplitude and phase of
the target waveform. Bottom Right: Spectral amplitude and phase of the
waveform generated by the 5-line O-AWG. Bottom Left: Deviation error
from the target waveform. The average error is 25.85%. . . . . . . . . . . 34
3.10 Flow diagram of GS Algorithm for pulse shaping applications [8]. . . . . 35
3.11 a) Inverse Fourier transform of the target (dotted) and 5-line spectrum
(solid), b) amplitude and phase of the continuous (target), c) “sampled”
amplitude and phase of the target spectrum, d) “sampled” amplitude and
phase of the target spectrum with error introduced to the amplitude, e)
Error between the target and generated signal over 100 iteration of GS
algorithm showing minimal error already achieved through direct sam-
pling, f) Error between the target and generated signal after the spectral
amplitude was altered showing reduction in error through phase adjustment. 37
x
3.12 Simulation of a 5-line rectangular spectrum with zero linewidth. a) time
domain signal over a short period (250 ps), b) time domain signal over a
long period (8×105 ps) showing the overall envelope, c) spectral amplitude,
d) spectral phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.13 Simulation of a 5-line rectangular spectrum with 100 kHz linewidth with
zero phase (non-physical). a) time domain signal over a short period (250
ps), b) time domain signal over a long period (8 × 105 ps) showing the
overall envelope, c) spectral amplitude, d) spectral phase. . . . . . . . . . 39
3.14 Simulation of a 5-line rectangular spectrum with 100 kHz linewidth with
random phase (non-physical). a) time domain signal over a short period
(250 ps), b) time domain signal over a long period (8 × 105 ps) showing
the overall envelope, c) spectral amplitude, d) spectral phase. . . . . . . . 40
3.15 Simulation of a single Gaussian spectrum with 100 kHz FWHM with a
phase function estimated by GS algorithm. a) time domain signal over a
short period (250 ps), b) time domain signal over a long period (8 × 105
ps) showing the overall average, c) spectral amplitude, d) spectral phase. 42
3.16 Simulation of a 5-line spectrum with 100 kHz Gaussian linewidth and a
phase function estimated by GS algorithm. a) time domain signal over a
short period (250 ps), b) time domain signal over a long period (8 × 105
ps) showing the overall envelope, c) spectral amplitude, d) spectral phase. 43
3.17 Simulation of a single spectral line with 100 kHz FWHM by modelling
coherence time in the time domain. a) time domain signal over a short
period (250 ps), b) time domain signal over a long period (8 × 105 ps)
showing the overall envelope, c) spectral amplitude, d) spectral phase. . . 45
3.18 Simulation of a 5-line spectrum with 100 kHz Gaussian linewidth by mod-
elling coherence time. a) time domain signal over a short period (250 ps),
b) time domain signal over a long period (8× 105 ps) showing the overall
envelope, c) spectral amplitude, d) spectral phase. . . . . . . . . . . . . . 46
4.1 Pulse shaper testbed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2 Pulse shaping via an array of FBGs using polarization control (PC1, 2
. . . n) together with a polarizer to achieve amplitude control, and using
fiber stretching for phase control. Point A, B and C depict the spectral
information of the signal at various stages of the system. . . . . . . . . . 51
4.3 Spectral response (solid line) and signal input (dotted line) of a conceptual
three-FBG array. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
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4.4 Spectral response of a three-FBG O-AWG. . . . . . . . . . . . . . . . . . 54
4.5 Waveforms generated by the 3-line O-AWG. . . . . . . . . . . . . . . . . 56
4.6 Spectral response of a five-FBG O-AWG (solid line) and the input of a
phase modulated CW laser (dotted line). . . . . . . . . . . . . . . . . . . 57
4.7 Experimental results showing various temporal waveforms from a five-line
arbitrary waveform generator. The insets show the measured spectral
amplitudes of the lines. a) and b) illustrate phase control resulting in
different pulse shape for the same spectral amplitude. c) has a shape close
to a saw-tooth form, and d) has a near “flat-top” shape. . . . . . . . . . 58
4.8 Persistence plot of a CW (left) and a signal from the 3-line O-AWG system
(right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.9 Persistence plots of the 3-line O-AWG for 10 seconds (left), for 50 seconds
(right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.10 Persistence plots of the 3-line O-AWG after it is submerged in gel. Persis-
tence for a) 6 seconds, b) 10 minutes, c) 20 minutes. . . . . . . . . . . . . 60
4.11 Picture of the 5-line O-AWG system. a) Circulator, b) Polarizer (in a box
to reduce disturbance), c) fiber optic embedded in a tub of sodium poly-
acrylate gel, d) in-line fiber stretcher/ phase shifter, e) in-line polarization
controller. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
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List of Acronyms
AOM Acousto-optic Modulation
AWG Arbitrary Waveform Generation
CW Continuous-wave
DAC Digital-to-Analog Converter
DCA Digital Communication Analyzer
DDS Direct Digital Synthesizer
EDFA Erbium-Doped Fiber Amplifier
EOM Electro-optic Modulation
FBG Fiber Bragg Grating
FIR Finite Impulse Response
IIR Infinite Impulse Response
LCM Liquid Chrystal Modulator
LPF Low Pass Filter
MMW Millimetre Wave
O-AWG Optical Arbitrary Waveform Generation
OSA Optical Spectrum Analyzer
PC Polarization Controller
PLL Phase-Locked Loop
SLM Spatial Light Modulator
UWB UltraWide Band
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Chapter 1
Motivation for Arbitrary Waveform
Generation
Electromagnetic waves are an essential component of modern society as they play signif-
icant roles in areas such as communication, manufacturing and public safety. We utilize
electromagnetic waves to fulfil those roles by using it to transmit information and inter-
act with matters. These applications require us to manipulate the energy content of the
wave over time, which is achieved by altering the temporal shape of the waveform. We
can accomplish this requirement through Arbitrary Waveform Generation (AWG), a tool
that allows its user to manipulate the amplitude and phase of electric field.
There are many ways to generate an waveform with user-defined characteristics, and
they generally falls into one of two categories: static or dynamic. With static AWG,
the response of the system is finalized during the fabrication of the system. Therefore,
users cannot change the shape of the generated waveform to adapt to the changing
conditions and requirements in later applications. In contrast, with dynamic AWG,
users can customize the waveform in-the-field, giving them greater degrees of freedom
for optimization and control. For this reason, dynamic AWG will be the main focus of
this thesis. An example of such needs is in the radar and wireless communication areas
where electromagnetic waves are transmitted through the atmosphere. The atmosphere
is a medium in constant fluctuation due to change in factors such as pressure, humidity
and temperature. Therefore, the transmitted signal needs to be adaptive to optimize
the performance of the communication systems. In this chapter, we will describe some
of the motivating applications for AWG and how their needs are met by existing AWG
techniques.
1
2 Chapter 1. Motivation for Arbitrary Waveform Generation
Op
tic
al D
om
ain
Ele
ctr
ica
l D
om
ain
Figure 1.1: Electromagnetic spectrum [1].
1.1 Applications of AWG
There are many applications for AWG at different parts of the electromagnetic spectrum
(Figure 1.1). According to their application areas and generation methods, they can be
classified into the electrical domain or the optical domain. In this thesis, we use the
wavelength of 100 µm (or 3 THz in frequency) as the threshold between “electrical” and
“optical” domain. Details about the applications in each area are discussed below.
1.1.1 AWG Applications in the Electrical Domain
The main fields of application for AWG in the electrical domain are wireless communica-
tion and radar, particularly those involving Ultra-wideband (UWB) signals. The Federal
Communications Commission (FCC) of the United States defines a radio frequency signal
as UWB when its bandwidth exceeds 500 MHz or 20% of its center frequency. The UWB
signal is important because they have several useful properties. In the communication
area, UWB signals are strongly immune to multipath interference, and they allow the
transmission of data at high rates and throughputs [9]. They also have a low probability
of being intercepted or detected, making them ideal for secure communication [9]. In
addition, UWB signals have strong material penetration characteristics (of ground, hills,
and buildings) which enable better non-line-of-sight communication. For radar appli-
1.1. Applications of AWG 3
cations, this penetration property allows radar to have a better range, resolution and
accuracy [9, 10].
The arbitrary generation of UWB signals is fundamental for the transmission of infor-
mation over the wireless communication channel [11]. The ability to arbitrarily generate
UWB signals allows the user to fully exploit the wireless channel despite its changing
characteristics such as variations in the environment, noise condition and the movement
of wireless transmitters and receivers. In addition, AWG allows the user to compen-
sate for non-ideal conditions in the communication system such as antenna dispersion
[12]. In the case of antenna dispersion compensation, the signal is first generated by the
AWG and then fed to the antenna. Then, the compensation is achieved by adjusting the
generated waveform according to the feedback measurement of the antenna emission.
For radar applications, UWB AWG enables the implementation of novel radar sys-
tems such as Impulsed Radars [13] and Ground Penetrating Radars [14] because UWB
signals are strongly immune to multipath interference, have strong material penetration
capability and other beneficial characteristics mentioned above. For example, a wideband
radar system can be used for collision avoidance to detect objects in the range of 1 foot
to 350 feet without the large false alarm rates of Doppler-based system [15]. In addition,
AWG in electrical and optical domain can be used together to implement a lidar-radar
hybrid system for detection and ranging of under water objects [16].
As long as the signal’s bandwidth is sufficiently small, the above benefits can be ob-
tained by electrical AWG. For UWB signals beyond 1 GHz, however, generating and
transmitting UWB over optical fiber becomes necessary [2]. Furthermore, transmitting
wireless signals over fiber allows the range of the wireless signals to extend without vi-
olating power emission regulations because the signals are not distributed over the air
[17]. In addition, distributing UWB signals over fiber can simplify the implementations
of certain applications. For example, a remote millimetre-wave (MMW) antenna base
station often requires local oscillators for up- and down-conversion when a signal is trans-
mitted to the base station at a lower (intermediate) frequency. If the MMW signal can
be transmitted to the base station directly over an optical fiber (since the electrical UWB
signal has comparatively narrowband in the optical domain), the local oscillators and the
conversion requirements at the remote base station can be eliminated and thus simplify-
ing its design [18]. That is the reason why generation and transmission of UWB signals
over optical fibers has received much attention [10, 11, 12, 17, 18, 19].
4 Chapter 1. Motivation for Arbitrary Waveform Generation
1.1.2 AWG Applications in the Optical Domain
Other than the applications in the electrical domain, there are many applications in the
optical domain as well, which are generally categorized using the term Optical AWG (O-
AWG). First, O-AWG can be used to modify the temporal characteristics of a laser source
such as converting a continuous-wave (CW) laser into a pulsed laser. The advantage of
this technique is that it can generate a more stable pulsed laser than mode-locked laser
[20, 21, 22]. Second, an O-AWG can also increase the repetition rate of a pulsed source
via repetition rate multiplication [23, 24]. Third, an O-AWG can generate customized
waveforms to provide coherent control of various quantum dynamics of a chemical pro-
cess [25, 26], which is important in the investigation of light-matter interaction [27] as
well as physicochemical processes [28]. Forth, O-AWG can be used in the telecommuni-
cation sector to compensate for the signal dispersion induced over long-haul fiber-optic
communication. O-AWG can also serve to increase the fractional bandwidth utilization
of optical signals. According to [29], the fractional bandwidth utilization at the optical
communication band is approximately 0.1%, which is much lower than the 50% utiliza-
tion of RF signal at 2 GHz carrier frequency. Fifth, an O-AWG can be used to optimize
the intensity profile of a laser pulse in micromachining [30, 31], as a means to optimize
energy transfer and control thermal effects.
1.2 Arbitrary Waveform Generation Methods
AWG can be achieved through a variety of methods depending on the application do-
main. In the electrical domain, Phase-Locked Loop (PLL) based frequency synthesizer is
a popular electronic waveform generator design, but its bandwidth is limited to hundreds
of kHz to tens of MHz [32, 33]. Direct Digital Synthesizer (DDS) is another electronic
synthesizer capable of generating signals with wider bandwidth than PLL, but it is still
limited to a bandwidth of 400 MHz [34, 35]. One of the challenges for increasing the band-
width for DDS is the design and fabrication of its Digital-to-Analog Converter (DAC)
component [36]. In a DDS, the arbitrary signal is first represented digitally and then
converted to the analog signal via the DAC. Nonlinearities or spur can be introduced
by the skew of the signal-representing DAC bits, which causes distortion as bandwidth
increases. Other challenges include fabrication process limitations, which affect design-
ers’ ability to control the device’s parasitic capacitance. That limits the performance
of the frequency synthesizer because the transfer function (poles and zeros) that model
the synthesizer is based on the impedance of the circuit. Due to the speed limitation of
1.3. Thesis Outline 5
electronics and challenges mentioned above, direct generation of electrical signals with
bandwidth exceeding 1 GHz is a challenge [2].
A solution to this problem of broadband electrical signal processing is to conduct it
in the optical domain [2, 37], then convert the optical signal into electrical signal using
wideband photo detector subsequently[38]. Temporal pulse shaping with picoseconds or
subpicoseconds resolution is more convenient in optical domain because a variety of meth-
ods can be exploited to manipulate the frequency characteristics of optical signals, thus
avoiding the requirements of fast temporal response of the pulse shaping/generating sys-
tem. In addition, optical signal processing is also immune to electromagnetic interference
and related noise.
As we will show in Chapter 2, there are several types of O-AWG techniques. These
techniques are not without fault however, as they typically require users to couple the
optical signal in and out of optical fiber, thus causing significant loss. While there are
existing fiber-based techniques [39, 40], they do not provide dynamic O-AWG. In this
thesis, we will demonstrate an all-fiber dynamic spectral line-by-line O-AWG technique
to solve these issues. Our technique is competitive in performance compared with many
of the existing methods while solving many of the short comings mentioned above.
1.3 Thesis Outline
In Chapter 2, we will review the fundamentals of pulse shaping and present a survey of
prominent techniques used to date for O-AWG. Then, in Chapter 3, we will discuss the
modelling and simulation of our spectral line-by-line O-AWG system to investigate the
effects of having a limited number of spectral line. In Chapter 4, we will describe our
experimental setup and demonstrate a number of waveforms using our system. Finally,
we will conclude this thesis in Chapter 5 and highlight the important aspects of the work.
We will also discuss the limitations of our system and its potential for improvement in
the future.
Chapter 2
Existing O-AWG Methods
While there are many ways to shape a signal, all pulse shaping system can be described
by fundamental signal processing theory as indicated by the block diagram in Figure 2.1.
Given an input signal s(t) with carrier frequency ωc,
s(t) = Re{e(t)ejωct} (2.1)
The shaped signal y(t) is the result of the convolution between the input signal in time
and the impulse response h(t) of the system. Since a convolution made by a Linear Time
Invariant (LTI) system in time domain is equivalent to a multiplication in frequency
domain, this impulse response has a corresponding transfer function H(ω). This function
is called a transfer function because it relates the input S(ω) and the output Y (ω) in the
frequency domain.
y(t) = s(t)⊗ h(t) (2.2)
Y (ω) = S(ω) ·H(ω) (2.3)
In all pulse shaping applications, the ultimate goal is to achieve arbitrary control of the
filter response h(t) or H(ω). All waveform generation methods can generally be divided
into two categories: direct temporal shaping and frequency shaping. The direct temporal
Pulse Shaping / AWG
optical input filter response
transfer functionin frequency
F
Figure 2.1: Transfer function diagram of an arbitrary waveform generator.
6
2.1. Direct Temporal Shaping 7
shaping method is better established since its modelling techniques and methodology are
already in use for lower frequency signal with narrower bandwidths. Because with direct
temporal shaping, the signal is sampled and altered in time domain, the time resolution of
the arbitrary waveform depends on the sampling rate [41, 42]. With frequency shaping
case, the desired waveform is generated by modifying the signal’s spectrum. In some
implementations, the input signal is dispersed spatially as a function of frequency before
the transfer function H(ω) is applied [4, 5, 43]. In other instances, the transfer function is
applied directly to the signal [7, 39, 40]. Descriptions of these AWG methods are detailed
next.
2.1 Direct Temporal Shaping
In the first category of direct shaping in time, the solution to AWG is to use the Finite
Impulse Response (FIR) model from discrete time signal processing theory. The FIR
model essentially describes the summing of successive samples of an input signal. Each
of these samples is delayed with respect to an absolute time reference for a different
amount. Their amplitude is also weighted according to the desired output. This model
can be described by the following equation:
y(t) =N∑
n=0
Wns(t− nT ) (2.4)
where y(t) is the output or the shaped waveform, N is the total number of samples or
the number of “taps”, s(t) is the optical input, n is the sample count, T is the sampling
period, and Wn is the amplitude weighting on the nth sample of the input. Figure
2.2 illustrates this model by having a single impulse input to the FIR model. Thus, the
output waveform is the result of a series of impulses with various amplitudes. If the signal
is then processed by a Low-Pass Filter (LPF), then the waveform will be smoothed out
as shown by the trace in Figure 2.2.
Alternatively, one can implement recursive filters such that the output of the filter is
delayed and fed back into the filter as an input. In that case, the system is termed to be
an Infinite Impulse Response (IIR) system as oppose to FIR.
This model has been widely adopted in electrical signal processing and it has also been
used to model photonic signal processors of microwave signal in both FIR ([22, 41, 44])
and IIR [45] form. Figure 2.3 shows a basic delay-line photonic signal processor that is
modelled using FIR. The delay elements are simply the spatial separation between the
8 Chapter 2. Existing O-AWG Methods
tt t
* =
Figure 2.2: Illustration of a finite impulse response (FIR).
weighting elements or taps, which is a piece of optical fiber of a specific length. Since
the delay is the distance divided by the speed of light in the fiber, the precision of the
delay is very much dependent on the precision of these physical separations. As for the
weighting elements, they can be couplers with specific coupling coefficient [46] or FBGs
with specific reflectivity [41, 44].
The shaping function of the FIR and IIR system is implemented through the manipu-
lation of the optical intensity of the input signal. In addition, the minimum delay time is
designed to be greater than the coherence time of the optical source [2]. This is done to
avoid intensity fluctuation due to phase variation cause by environmental perturbations.
While this model is useful in the electrical domain, there are complications when
it is used in the optical domain. For electrical signals, including those that are in the
microwave regime, the fractional bandwidth of the signal remains negligible when it is
put in the context of optical domain. Thus, the sampling rate requirement as defined
by the Nyquist criterion is not a concern. When the desired signal is in the optical
domain (such as the case of O-AWG) however, the sampling rate must be very high for
the O-AWG system to have acceptable temporal resolution. Therefore, the sampling rate
becomes the limiting factor of the system.
The incoherent operation through intensity manipulation also creates another com-
plication. Because the implementations operates on the intensity, the weight (Wn) of the
taps are all positive, limiting this approach to a subset of functions. In other words, this
approach cannot implement a completely arbitrary filter response h(t) [2, 46]. To imple-
ment both positive and negative weighting, a differential detection scheme must be used
to turn two positive filters into a two-signed filters [42]. While this method is useful for
shaping in microwave system, it is not applicable in all-optical systems as the differential
detection would require the conversion of optical signals into electrical domain.
2.1. Direct Temporal Shaping 9
Figure 2.3: A basic delay-line photonic signal processor that can be modelled with FIR[2].
10 Chapter 2. Existing O-AWG Methods
2.2 Direct Frequency Shaping
For frequency shaping methods, the transfer function H(ω) of the filter response h(t) is
implemented in the frequency domain such that the spectrum of the optical input signal
can be directly manipulated. These methods are more desirable for AWG in the optical
domain because there are many passive structures, such as Fiber Bragg Gratings (FBGs),
Arrayed Waveguide Grating and Diffraction Gratings, that can be used for separating
an optical signal according to its spectral distribution. Within this class of system, the
AWG methods can be differentiated into Static or Dynamic. In the static case, the
transfer function for a desired waveform with a given input is fixed. In other words, the
implemented transfer function cannot be altered to adapt to either changes in the input
properties or changes in the desired output waveform. In contrast, the dynamic methods
allow in-the-field customizations so that the O-AWG system can be adjusted to adapt
to the conditions of the applications. The extra degrees of freedom also allow the users
to optimize the generated waveform base on the outcome of their applications through
feedback. In this section, we will describe several methods for direct frequency shaping
in both static and dynamic cases.
2.2.1 O-AWG via Static Frequency Manipulation
There are many ways one can implement the shaping function H(ω) statically. Two of
the well established methods are shaping via FBGs [39, 40] and shaping via time-to-
frequency-to-space mapping using diffraction grating.
The FBG method takes advantage of the fact that an optical fiber with a particularly
modulated refractive index would result in a corresponding frequency response. In the
weak FBG regime where the change in refractive index is small, the frequency response
of the FBG is the Fourier transform of the refractive index modulation profile along the
length of the fiber, A(z) [47]:
H(k) =1
2π
∫ +∞
−∞A(z)ejkzdz (2.5)
where k is the wave vector, which relates to ω by the dispersion of the fiber. In the
strong reflection FBG regime, in which the refractive index change is great enough that
light may not penetrate the full length of the FBG, its frequency response H(ω) can
be obtained using other algorithms such as the inverse scattering method [48]. The
resulting FBG is called a complex super-structure FBG. In addition to constructing the
static filter with a complex super-structure FBG described above, the spectral filter can
2.2. Direct Frequency Shaping 11
Figure 2.4: Static pulse shaping using amplitude and/or phase mask [3].
also be implemented using a cascade of uniform FBG [40]. Although the FBGs in the
cascade scheme is simpler to model, the phase relationships between the cascading FBGs
must be tuned properly, which requires active monitoring and control during fabrication
[40]. The active monitoring is achieved by sending a probe laser during fabrication and
analyzing the reflection from both the FBGs and the fiber end with an optical spectrum
analyzer. With the phase relationship measured, [40] corrected the phase relationships
by adjusting the refractive index of the fiber in between the six cascaded FBGs through
UV irradiation.
The second static O-AWG method involves diffraction gratings and the principle of
the 4-f imaging system (Figure 2.4) [49]. First, the optical signal is dispersed angularly
by a diffraction grating:
θq(λ) = θi + qλ
Λ(2.6)
where q is the integer diffraction order, θi is the incident angle, and Λ is the periodicity
of the diffraction grating. With the frequency components angularly separated, these
components are transmitted through a lens with focal length f . If the diffraction grating
is located one focal length away from the lens (which is the case for a 4-f system), then
the intensity distribution of the signal, g(x), is wavelength dependent (assuming first
order, q = 1):
g(x) = θ(λ)f, (2.7)
= θ( c
ν
)f
12 Chapter 2. Existing O-AWG Methods
Then, an amplitude and/or a phase mask can be placed at the focal plane to implement
H(ω) according to the distribution described by g(x). After the signal is manipulated,
it passes through the above lens-grating setup for recombination to become the shaped
time signal.
2.2.2 O-AWG via Dynamic Frequency Manipulation
Even though static manipulation of a signal’s spectral content may be sufficient for
certain applications, it has limited usage because it is not capable of adapting to changes
in the conditions of the input signal, or the changes in the requirements of the output
waveform. To eliminate this restriction, one can perform arbitrary waveform generation
in the time domain (as has been demonstrated in the sub-nanosecond range [41]). It is,
however, difficult to improve such system base on this principle as the temporal features
of a desired waveform are reduced to picoseconds or femtoseconds scale. That is because,
as mentioned in Section §2.1, the temporal resolution of a direct temporal shaping system
is limited by the sampling rate of the system.
One way to remove the above constraints from O-AWG is to perform the function
via dynamic frequency manipulation. Similar to static frequency manipulation, dynamic
frequency manipulation also utilizes the strength of many passive optical devices, that is,
their ability to change their property as a function of frequency. Spatial gratings [4], Fibre
Bragg Gratings, and array waveguide gratings [43] are some of the passive optical devices
being used for dynamic O-AWG. Additional techniques, such as the use of liquid-crystal
modulator (LCM) [4], have been developed to be used in conjunction with these devices to
achieve the dynamic aspects of the O-AWG system. In addition, the frequency response
can be generated in electrical domain and then applied to the optical signal via electro-
optic [5] or acousto-optic effects [7]. While this method seems to be simply shifting the
shaping problem away from one domain into another, the acousto-optics case allows the
filter response to be generated with frequency in MHz range, which can be easily handled
by electronic devices. In the following subsections, we will provide more details about
several existing O-AWG systems using dynamic frequency manipulation. In addition, we
will discuss some of the advantages and disadvantages of each system.
Spatial Light Modulator
The Spatial Light Modulator (SLM) [4], depicted in Figure 2.5, uses the same principle
as the second static shaping method mentioned in §2.2.1. That is, they both use the
diffraction grating to disperse and recombine optical signal spatially as a function of
2.2. Direct Frequency Shaping 13
Figure 2.5: Pulse shaping using Spatial Light Modulator (SLM) [4].
frequency. Instead of using a static mask, a liquid-crystal modulator is placed at the
Fourier plane to provide the dynamic adjustments of the O-AWG system. The liquid-
crystal modulator (LCM) would modify both the amplitude and phase of the frequency
components. Instead of recombining the spatially distributed signal with a second set
of diffraction grating, the manipulated signal is reflected by a mirror and recombined
through the same grating used to disperse the light.
This method is very flexible in the sense that it provides fully independent amplitude
and phase control of the optical signal. The spectral resolution of this method is depen-
dent on the configuration of the diffraction grating, the related optics, and the resolution
of the liquid crystal modulator. A state-of-the-art system can provide a spectral resolu-
tion of 5 GHz and independent control of more than 100 spectral lines [50]. This system
has the disadvantage of large insertion loss in the order of 13 dB however, as optical
signal must be coupled into free space. In addition to the lack of integration with the
fiber optic system, the SLM methods also have the complexity associated with free-space
optic systems.
Shaping via Electro-Optic Effect
For pulse shaping using the electro-optic effect [5], it uses two pieces of Dispersion Com-
pensated Fiber (DCF) of opposite sign and an electro-optic phase modulator to imple-
ment the shaping response H(ω). The input is initially dispersed by a factor of β1L1 such
14 Chapter 2. Existing O-AWG Methods
1
2
3t
1
2
3t
1 2 3
t
1 2 3
t
Ultrashort Input Dispersed Pulse Phase Modulated Shaped Output
Figure 2.6: Pulse shaping via Electro-Optic effect [5].
that different frequency components of the pulse is located at different time slots. The
shaping function is then applied via the phase modulator and it is modelled as follows:
u(t)e−jAm(t) (2.8)
where u(t) is the dispersed input pulse, m(t) is the electrical modulation function and
A is the modulation amplitude. After the shaping function is applied, the manipulated
signal is compressed back together by another DCF with an equal but opposite dispersion
factor β2L2 = −β1L1. If the e−jAm(t) is used to implement a Fourier series:
e−jAm(t) ∝∞∑
k=−∞ake
jkωmt (2.9)
such that,
ak = ωm
∫
Tm
e−jAm(t)e−jkωmtdt (2.10)
2.2. Direct Frequency Shaping 15
Figure 2.7: Pulse shaping via Electro-Optic amplitude modulation with FBGs as disper-sive elements [6].
where ωm is the repetition rate of m(t) and Tm is the equivalent period (Tm = 2πωm
). Then
the filter response of the pulse shaping system is found to be the following [5]:
h(t) ∝ akδ(t− kTR) (2.11)
where ak is the Fourier series and TR = β1L1ωm. Alternatively, one can use chirped
FBGs as the dispersive elements in place of the DCF and use amplitude modulation as
the shaping mechanism as oppose to phase modulation [6, 51].
There are several limitations for this system. First, this system requires a good match
between the conjugate dispersion elements and low level of higher order dispersions [52].
As shown in the simulations performed by [52], mismatch between the dispersion elements
and higher order dispersions create distortion in the generated waveform. In addition,
this system requires strict synchronization between the electrical modulation signal and
the input optical signal. Synchronization becomes a greater problem at lower repetition
rates because it requires longer tunable delay to correct the mismatch. For example, a
functioning experimental system needed to operate with a repetition rate greater than
40 GHz [5]. Lastly, both amplitude and phase modulation is required to generate asym-
metrical waveform [52], which further restrict the synchronization requirement.
16 Chapter 2. Existing O-AWG Methods
tCrystal Slow Axis
Fast Axis
acoustic grating: z(w)
1
23
Figure 2.8: Pulse shaping using Acousto-Optic effect via a birefringent crystal [7].
Shaping via Acousto-Optic Effect
In the case of pulse shaping via acousto-optic effect [7], the system uses an acousto-optic
birefringent crystal, such as the tellurium dioxide crystal, to transfer the desired shaping
response from the electrical domain to the optical domain. An electrical RF signal is used
to drive an acoustic wave in the crystal such that optical modes of the birefringent axes
(fast and slow axis) would couple with each other. The input optical signal is adjusted
such that its polarization is aligned with the fast axis while the output polarization is
aligned with the slow axis as indicated by Figure 2.8. The acoustic wave act as an acoustic
grating to provide the phase matching condition required for the coupling. The coupling
efficiency can be viewed as the amplitude control of H(ω). By introducing acoustic
grating of different frequencies at different part of the crystal via the RF signal, different
frequency components would be coupled at different locations and hence introduces phase
relationships between the frequency components.
The limiting factor for this system is the refreshing rate of the acoustic wave. During
the shaping process, the acoustic wave (which implements the transfer function) appears
to be stationary because of its low speed relative to that of the optical wave. The transfer
function changes away from the ideal response over time, however, as the acoustic wave
drifts across the crystal. For the system to generate the target waveform again, the
acoustic wave must be refreshed. Since the refresh rate for the acoustic wave is slow, this
O-AWG method operates with limited repetition rates, in the kHz range [7].
2.3. Thesis Contribution — Dynamic O-AWG in a Continuous Fiber 17
2.3 Thesis Contribution — Dynamic O-AWG in a
Continuous Fiber
In this thesis, we will demonstrate dynamic optical arbitrary waveform generation (O-
AWG) through spectral line-by-line shaping using a continuous piece of optical fiber.
As shown in Figure 2.9, this system employs uniform Fiber Bragg Gratings (FBGs) to
separate the spectral lines, and provides independent amplitude and phase control for
each line via in-line polarization controllers and in-line fiber stretchers respectively. Since
the pulse shaping is carried out in a continuous fiber, the system has negligible loss, and
thus can be used to control many lines to achieve higher temporal resolution and better
shape control. An all-fiber approach also makes the system compatible with existing
fiber communication networks, making the system robust and low cost. In addition, the
fiber-based waveform shaping system can be encapsulated in other materials to protect
it from environmental perturbations. By conducting AWG in the optical domain, we are
able to generate arbitrary waveforms with a bandwidth of 60GHz, well beyond the 2GHz
limit of electronic AWG.
18 Chapter 2. Existing O-AWG Methods
1550nm
phasemodulator
PC1PC2PCn
EDFA
FSS: Fiber Stretching Stages
FSS2FSSn
FBGn
FBG2
FBG1
......n 2 1
1
32
Polarizer
CW
A
CW A
B
PhaseModulated
B
C
Shaped C
Figure 2.9: All-fiber spectral line-by-line O-AWG system incorporating FBGs, polar-ization controllers, and fiber stretchers, demonstrated in this thesis. Plot A, B andC depict the signal spectra at points A, B and C of the system, respectively. Thespectral lines of the signal (λ1, λ2, . . . , λn) match the central wavelengths of the FBGs(FBG1, FBG2, . . . , FBGn). For a 5-line O-AWG, n=5.
Chapter 3
Modeling and Simulation
While there are many intrinsic advantages offered by the O-AWG approach presented
in this thesis, we must also quantify its capabilities and limits as well as the effects of
practical constraints. To do so, we have modelled the system and conducted several
simulations. First, we modelled our system using Fourier analysis and using parameters
related to system components we use in the experiment. Then, we investigated the
effects of several constraints base on practical considerations. One of the constraints is
the practical limit of having a finite number of spectral line controls. As we will show
later, having finite number of control will limit the flexibility of the O-AWG system.
Then, we explored the effects of non-zero line width with the aid of Gerchberg-Saxton
(GS) algorithm. Through that discussion, we will show that finite spectral linewidth in
a typical system does not have a significant effect on the temporal shape of the O-AWG
output, and therefore simulations can be conducted using discrete, ideal, spectral lines.
3.1 Modelling of Our O-AWG System
We begin with Fourier theorem, which states that an arbitrary periodic signal, modulated
at carrier frequency ω0, can be represented by a Fourier series, or discrete spectral lines,
with suitable amplitudes |am| and phases φm:
s(t) = Σm=∞m=−∞|am|ejm2πft+jφmejω0t (3.1)
where f is the repetition rate of the periodic signal, which corresponds to the frequency
separation of the discrete spectral lines. One can also define a complex phasor am asso-
19
20 Chapter 3. Modeling and Simulation
f ff
* <=>
Figure 3.1: Illustration of the transfer function for a line-by-line shaping system verses ablock shaping system.
ciated with each spectral line as:
am = |am|ejφm =1
T
∫
T
s(t)e−jm2πft−jω0tdt (3.2)
(3.1) also implies that if one can independently control |am| and φm, one can generate
arbitrarily-shaped periodic signals, with a fundamental repetition rate corresponding to
the frequency separation of the discrete spectral lines. This is the operation principle of
a line-by-line O-AWG system. The Fourier series representation (3.1) exists as long as
(3.2) converge. Any signal that has finite energy over a single period (∫
T|s(t)|2dt < ∞,
i.e. any physical signal) has absolutely converging Fourier series [54].
In a spectrum-shaping scheme of finite frequency resolution, the transfer function of
the shaping system can be viewed as a train of impulses convoluted with a rectangular-
like function as shown in Figure 3.1. The rectangular-like function corresponds to the
finite frequency resolution of the shaping system such as the pixel of a liquid crystal
modulator in the SLM system. The convolution in the frequency domain translates
into a multiplication of a wide envelope on top of the arbitrary waveform in the time
domain. With increasing spectral resolution, the corresponding width of the rectangular-
like function would decrease, which results in a wider envelope in the time domain.
Infinite spectral resolution is not practical, however, as it requires infinite number of
controls. Therefore, the system to be demonstrated in this thesis performs spectral
line-by-line manipulation as oppose to block-spectrum shaping so that we can precisely
control the shape of the waveform without the adverse effect of the envelope.
While an infinite Fourier series can represent any physical periodic signal, it is im-
possible to implement a practical system with infinite controls and infinitely fine spectral
resolution. On the other hand, because we are usually concerned with signals of a finite
bandwidth, a compromise can be made to approximate a signal with a finite Fourier
series. For a Fourier series with N values, we can define an error function eN(t) and a
3.1. Modelling of Our O-AWG System 21
1 31 2 4 N
Sp
ec
tra
l A
mp
litu
de
Te
mp
ora
l A
mp
litu
de Total # of line: N = 11
t
(100us) Period, T Frequency Res., (10GHz)
(10ps) Time Res., ∆t Bandwidth (N =100GHz)
......
Figure 3.2: Illustration of several system parameters: Temporal Resolution or Bandwidth,Spectral Resolution or Period.
total integrated error parameter EN as follows:
sN(t) =N−1∑m=0
|am|ejmft+jφmejω0t (3.3)
eN(t) = s(t)− sN(t) (3.4)
EN =1
T
∫
T
|eN(t)|2dt (3.5)
With (3.3) representing the signal, several parameters of the O-AWG system can be
defined for the purpose of analysis and discussion. First, by definition, f is the frequency
resolution of our shaping system because it is the frequency separation between adjacent
spectral lines. It is also the repetition rate of the system and it defines a window in
the time domain (the time aperture) in which an arbitrary waveform can be generated.
Second, the bandwidth of the system is defined to be (N − 1)f with N being the total
number of spectral lines generated by the system. The inverse of the bandwidth is the
temporal resolution achievable by the system. For example, if an O-AWG system with
f = 10GHz with N = 11, it will have a bandwidth of 100GHz achieving a temporal
resolution of 10 ps. An illustration of these system parameters are presented in Figure
3.2.
3.1.1 Practical Aspects of Modelling Optical Signals
To simulate optical signals oscillating at very high frequencies, a large amount of points
is required to achieve sufficient temporal resolution and, at the same time, to cover long
22 Chapter 3. Modeling and Simulation
enough duration to achieve high spectral resolution. One way to avoid this problem is
to simulate the signal without the carrier, and calculate the intensity outcome instead.
This approach is acceptable because most applications in the optical domain utilize the
intensity profile of a shaped waveform without concerning the phase of the carrier. To
illustrate how the carrier can be disregarded for intensity-only calculations, consider a
signal with three spectral lines with arbitrary amplitudes (Am = |am|) and phase (φm).
Using complex representation:
s(t) = ejω0t[A1e
−j(ft−φ1) + A2ejφ2 + A3e
j(ft+φ3)]
(3.6)
where ω0 is the carrier frequency and f is the repetition rate or the frequency separation
between the lines, one can calculate the intensity of the signal by:
I(t) = s(t)s∗(t)
= ejω0t[A1e
−j(ft−φ1) + A2ejφ2 + A3e
j(ft+φ3)]×
e−jω0t[A1e
j(ft−φ1) + A2e−jφ2 + A3e
−j(ft+φ3)]
= A21 + A2
2 + A23+
A1A2e−j(ft−φ1+φ2) + A1A2e
j(ft−φ1+φ2)+
A1A3e−j(2ft−φ1+φ3) + A1A3e
j(2ft−φ1+φ3)+
A2A3e−j(ft−φ2+φ3) + A2A3e
j(ft−φ2+φ3)
= A21 + A2
2 + A23 + 2A1A2 cos(ft− φ1 + φ2)+ (3.7)
2A1A3 cos(2ft− φ1 + φ3) + 2A2A3 cos(ft− φ2 + φ3)
As shown in the above calculations, the carrier component is cancelled out. Therefore
the intensity profile can be simulated with the spectral lines centered at the zero frequency
position instead of the carrier frequency position in the frequency domain. At the same
time, it is also clear that both the phase and the amplitude of each frequency components
contribute to the final intensity outcome. We can further generalize the above derivation
to obtain the intensity formula for a N -line spectrum with N > 2:
N∑n=1
A2n + 2
{N−1∑n=1
N∑m=n+1
AnAm cos [(m− n)ft + φm − φn]
}(3.8)
giving us N(N−1)2
number of cosines, each having a pairwise phase difference (φm−φn) in
the argument.
3.1. Modelling of Our O-AWG System 23
Table 3.1: Phase relationships (as appeared in (3.7)) of the three spectral lines for the 5example cases in Figure 3.3.
A B C D Eφ2 − φ1 −3
7π −2
7π −1
7π 0 −1
7π
φ3 − φ1 −37π −1
7π 1
7π 3
7π 0
φ3 − φ2 0 17π 2
7π 3
7π 1
7π
To illustrate the principle of spectral line-by-line shaping embodied in (3.8), we simu-
lated an O-AWG system with 3 spectral lines, using a MATLAB program. The spectral
lines used in the simulation are ideal, i.e. zero linewidth, with varying amplitude and
phase relationships. In Figure 3.3, 3.4 and 3.5, we present the simulated temporal wave-
forms at the output of the 3-line O-AWG system. The spectral amplitudes are given in
the inset of each figure. The pairwise phase relationships are swept from 0 to π, with π7
increments.
As expected, a variety of waveform can be generated by modifying either the ampli-
tude or the phase relationships between spectral lines, or both. Also, when the spectral
lines are of equal amplitude (Figure 3.4) we get a train of sinc2 function as expected.
In addition, we would like to note that, for each of the Figures 3.3, 3.4, 3.5, the shapes
of the waveform along the diagonal are the same. This is the case because the intensity
profile, as revealed in (3.7), is a function of pairwise phase relationships between any pair
of spectral lines in the spectrum. Sweeping of the phase in the simulation coincidentally
produced equal phase increments on each of those phase relationships. To illustrate this
fact, we have selected 5 waveforms from the one of the sweep (Figure 3.3) and framed
them with thick dash lines and labelled them A to E as shown.
Table 3.1 shows the relevant phase relations as appeared in (3.7) for waveforms A, B,
C, D, E. Although the phase relationships are different in value, they are in fact shifted
by the same amount in phase (as shown in Table 3.2). While the value of the increment
for the φ3 − φ1 relation is different from other relations, it is in fact the same amount in
phase because the cosine for that particular relation is twice the frequency of the other
cosine (the argument is 2ft−φ1 +φ3 as oppose to ft−φ1 +φ2 or ft−φ2 +φ3.) A counter
example is E, which has different phase increments on those cosine and therefore the
shape differ. Therefore, if the O-AWG application is indifferent about the phase of the
shaped waveform (which is just a time delay), we can fix one pairwise phase relationship
and anchor the shaped waveform to a fixed time axis. Doing so will allow us to reduce
the phase control of an N-line O-AWG system from N − 1 to N − 2.
The simulation parameters used in the MATLAB program such as temporal and
24 Chapter 3. Modeling and Simulation
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ps)
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e (
ps)
01
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005
10
Tim
e (
ps)
φ1 -
φ2 =
L1
L2
L3
0π
1/7
π2
/7 π
3/7
π4
/7 π
5/7
π6
/7 π
φ3 - φ
2 =
0 π1/7 π 2/7 π 3/7 π 4/7 π 5/7 π 6/7 π
A
B
C
D
E
Figure 3.3: Simulated 3-line shaping with L1 = 0.5, L2 = 1, L3 = 1.5.
3.1. Modelling of Our O-AWG System 25
15
49
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49
.51
55
01
55
0.5
15
51
0
0.1
0.2
0.3
0.4
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0.8
0.91
Wa
ve
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gth
(n
m)
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01
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ps)
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005
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Tim
e (
ps)
φ1 -
φ2 =
L1
L2
L3
0π
1/7
π2
/7 π
3/7
π4
/7 π
5/7
π6
/7 π
φ3 - φ
2 =
0 π1/7 π 2/7 π 3/7 π 4/7 π 5/7 π 6/7 π
Figure 3.4: Simulated 3-line shaping with L1 = 1, L2 = 1, L3 = 1.
26 Chapter 3. Modeling and Simulation
01
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005
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15
49
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55
01
55
0.5
15
51
0
0.51
1.5
Wa
ve
len
gth
(n
m)
Amplitude (a.u.)
φ1 -
φ2 =
L1
L2
L3
0π
1/7
π2
/7 π
3/7
π4
/7 π
5/7
π6
/7 π
φ3 - φ
2 =
0 π1/7 π 2/7 π 3/7 π 4/7 π 5/7 π 6/7 π
Figure 3.5: Simulated 3-line shaping with L1 = 1, L2 = 0.5, L3 = 1.5.
3.2. Effects of Finite Spectral Sampling 27
Table 3.2: Phase increments from one example to another for each phase relationship.A → B B → C C → D D → E
φ2 − φ117π 1
7π 1
7π −1
7π
φ3 − φ127π 2
7π 2
7π −3
7π
φ3 − φ217π 1
7π 1
7π −2
7π
Table 3.3: Simulation conditions and related physical constraints.Physical Constraints Simulation Parameters
OSA Bandwidth * Bandwidth 10 nm @ 1550 nmDCA Bandwidth 65 GHz ≈ 1.3 THzDCA Resolution 15.38 ps Temporal Resolution 764 fsOSA Resolution 0.01 nm Spectral Resolution 0.01 pm
HP8168 CW minimum line width:≈ 50 MHz ≈ 0.4 pm @ 1550 nm
FBG FWHM ≈ 90 pm
spectral resolutions are listed in Table 3.3. They are chosen based on the physical char-
acteristics of our measurement system. Details will be discussed in Chapter 4.
3.2 Effects of Finite Spectral Sampling
As described in the introduction of this chapter, representing a signal with a finite number
of spectral lines can induce error quantified by EN as defined in (3.5). Several simulations
were conducted on a five-line O-AWG system to observe the effects of using a finite
number of spectral lines to represent a target signal. The spectral resolution of the
simulated O-AWG system was set to 15 GHz or 0.120 nm at 1550 nm giving a total
bandwidth of 60 GHz or 0.48 nm at 1550 nm between the first and the fifth (last)
spectral line. The spectral resolution of the O-AWG system was kept constant for this
set of simulations because the goal of these simulations is to investigate the effects of
having finite number of spectral lines keeping all other variables constant. The simulation
parameters were as described in Table 3.3.
There are two main contributing factors to the error parameter EN : the bandwidth
and coarse spectral sampling of the signal. The bandwidth is a concern because a finite
number of spectral lines cannot adequately represent a signal having a bandwidth wider
than the cumulative bandwidth of the lines (N − 1)f . If the target signal’s bandwidth is
under represented, then the sharp feature of the target signal in the time domain would
be smoothed out.
The coarseness of the spectrum translates into a window in time domain where an
28 Chapter 3. Modeling and Simulation
arbitrary waveform is defined. Alternatively, one can view this window as the time
domain dual of the bandwidth quantity in frequency domain. One can view the effect
of coarse spectral sampling as the dual of the discrete temporal sampling described in
Nyquist sampling theorem. To appropriately sample a time domain signal, we must
sample the signal above the Nyquist rate (twice the bandwidth), otherwise the spectrum
will overlap in the frequency domain and cause aliasing. Instead of sampling in the time
domain, here we are sampling the spectrum of the target waveform in the frequency
domain. As we will show later, under sampling a spectrum with dense spectral features
leads to the overlapping of the time windows and causes error.
To fully characterize these two issues, we generally divided any arbitrary spectrum
profile into two types: 1) smooth and, 2) rough. We first characterize a smooth target
spectrum to investigate the error contribution due to the finite bandwidth limitation.
That is because a smooth spectrum would not induce error of the second type and thus
allow us to keep the characterization separate. Then, several rough spectrum profiles
with limited bandwidths were examined. The bandwidth limits were determined from
the results of the first characterization. Again, this was done to separate the effects of
the two issues for the characterizations.
3.2.1 Bandwidth limit due to finite number of spectral lines
First, to illustrate the effects of representing a smooth spectrum with a finite number
of discrete spectral lines, we use a transform limited Gaussian function as the target
spectrum. We then vary the bandwidth (Full Width at Half Maximum/FWHM) of the
Gaussian spectrum from 0.01 nm to 1 nm using the script in Appendix D. From the results
of the simulation shown in Figure 3.6, it is clear that there is an optimal bandwidth range
where the O-AWG can generate a signal with minimum error. When the target signal
has a FWHM bandwidth beyond 0.3 nm, the error begins to increase. That is because
the extra bandwidth is not represented by any additional spectral lines. The effect of
this unaccounted spectrum is the extra ripple in time domain. These ripples in time
can be viewed as a result of a sinc function. Imagine the limiting case of representing
a spectrum of uniform amplitude and infinite bandwidth with five spectral lines. The
sampled spectrum would have a rectangular shape, transforming into a periodic sinc
function in time domain. The error between the target and the generated signal begins
to increase again as the bandwidth is reduced below 0.2 nm FWHM. That is because the
narrow bandwidth is close to the frequency resolution of the O-AWG.
Like many practical systems, the acceptable error level is a trade off against other
3.2. Effects of Finite Spectral Sampling 29
Error Threshold 1
Region of insufficient bandwidth coverageRegion of
insufficient
spectral
resolution
Error Threshold 2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
10
20
30
40
50
60
70
80
90
X: 0.48Y: 6.015
Pe
rce
nta
ge
err
or
(EN
) d
ivid
ed
by
Ta
rge
t P
uls
e E
ne
rgy
) (%
)
X: 0.25Y: 0.004664
X: 0.15Y: 2.228
X: 0.1Y: 16.48
Bandwidth (nm)
0 20 40 600
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (ps)
Am
plit
ud
e (a
.u.)
1.549 1.5495 1.55 1.5505 1.551
x 10−6
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Wavelength (m)
Bandwidth = 0.1 nm
70
) d
ivid
ed
by
Ta
rge
t P
uls
e E
ne
rgy
) (%
)
X: 0.1Y: 16.48
0 20 40 600
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (ps)
Am
plit
ud
e (a
.u.)
1.549 1.5495 1.55 1.5505 1.551
x 10−6
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Wavelength (m)
Bandwidth = 0.15 nm
Region of insufficient bandwidth coverage
X: 0.15Y: 2.228
0 20 40 600
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (ps)
Am
plit
ud
e (a
.u.)
1.549 1.5495 1.55 1.5505 1.551
x 10−6
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Wavelength (m)
Bandwidth = 0.25 nm
1 0.2 0.3
Bandwidth (nm)
0 20 40 600
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (ps)
Am
plit
ud
e (a
.u.)
1.549 1.5495 1.55 1.5505 1.551
x 10−6
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Wavelength (m)
Bandwidth = 0.48 nm
0.8
Bandwidth (nm)
Figure 3.6: Error introduced by representing a transform limited Gaussian signal with5 discrete spectral lines, plotted as a function of the bandwidth of the Gaussian signal.Line separation is 0.12 nm resulting in total 0.48 nm bandwidth with 5 lines. Solid linesrepresent the target time function and target spectrum, dotted lines represent the shapedfunction and the “sampled” spectrum.
30 Chapter 3. Modeling and Simulation
factors that are also important to a particular application. In the case of our O-AWG
system, the error level is the trade off against the operating bandwidth range as well
as the shape of the target spectrum profile. The shaded area in Figure 3.6 depicts the
trade off between the bandwidth range and error level. If the applications can tolerate
greater level of error (Error Threshold 1), then the O-AWG system can be operated over
a larger range of bandwidth as indicated by the white and light grey areas. If the end
user require lower error level such as the one indicated by Error Threshold 2, then the
O-AWG can only generate a waveform with bandwidth that falls inside the white area.
The shape of the spectrum profile also affects the bandwidth capability of the O-AWG
system because it describes the spectral energy distribution of the signal. In our example,
we simulated the system using a Gaussian function. Given this spectrum profile, the error
will never be zero because the Gaussian function only approaches zero at infinity. On
the other hand, if the target spectrum is a rectangular function that does not exceed
the cumulative span of the discrete spectral lines, then it is possible to represent the
spectrum without error. We define the O-AWG bandwidth capacity by the base width
of all the spectral lines. In this simulation, the 5-line O-AWG bandwidth capacity is 60
GHz or 0.48 nm @ 1550 nm.
3.2.2 Effect of coarse spectral sampling due to finite spectral
resolution
A second set of simulations is conducted for signals with greater variation in the frequency
domain. i.e. signals with a “rough” spectrum. To represent this category of signals, a
Gaussian function with FWHM of 0.3 nm is multiplied with polynomials of various
degrees, and it is used as the target spectrum (Appendix A). This is done to ensure the
bandwidth of the target spectrum is within the bandwidth capacity of the O-AWG system
while introducing different variations to the spectrum. As shown in Figure 3.7, the effect
of under sampling by coarse spectral lines in the frequency domain can be described
as aliasing in the time domain. The spectral lines “sample” the target spectrum in
the frequency domain with a “sampling rate” directly related to the repetition rate of
the system. As the variation in the frequency domain increases, the target waveform
in the time domain begins to broaden. When the variation increases beyond a point
for which the spectral resolution of the O-AWG system can accommodate, the adjacent
waveforms in the time domain begin to overlap with each other and cause aliasing (clearly
shown in the 5-degree polynomial case in Figure 3.7.) Alternatively, one can think of the
frequency dual of “Nyquist” threshold in the time domain as the point where the temporal
3.3. Waveform Control and Error Minimization 31
waveform has broadened beyond the period or time aperture defined by the repetition
rate. In effect, the finite spectral resolution puts a lower bound on the repetition rate of
the target spectrum.
3.2.3 Arbitrary Waveform Generation with 101 Spectral Lines
As mentioned in Chapter 2, in our OAWG system, the arbitrary waveform generation
is carried out in a continuous fiber, the system therefore has negligible loss and can be
scaled up to control many lines for higher temporal resolution and better shape control.
Here we show an example using 101 spectral lines which we believe can be achieved
in a practical system. This simulated 101-line O-AWG system (see Appendix B)has a
spectral resolution of 12.5 GHz giving a total bandwidth of 1.25 THz. The system was
simulated to generate a 66.66 ps long rectangular waveform defined in time domain with
consideration of temporal and spectral resolution as mentioned in previous sections. As
depicted in Figure 3.8, the waveform generated by the 101-line system closely follow the
target shape giving an average deviation error of 0.88%. In contrast, for the same target
waveform generated by a 5-line system as shown in Figure 3.9, the error is much greater.
The average deviation error for the 5-line generated waveform is 25.85%. As discussed
in previous section, these errors are due to insufficient representation of the bandwidth
in frequency domain. The error can also be attributed to the Gibbs phenomenon, which
states that truncated Fourier series representation of a discontinuous signal will in general
exhibit high-frequecy ripples and overshoot near discontinuities.
3.3 Waveform Control and Error Minimization
Since our O-AWG system allows the independent control of amplitude and phase, it
implements the Fourier series and can generate a waveform with minimum error as long
as the waveform satisfy the frequency and time domain requirements mentioned in the
previous section. That is, the time domain target waveform is within the time aperture
or period defined by the repetition rate, and the bandwidth of the target spectrum is
within the bandwidth capacity of the O-AWG system. On the other hand, if there
is a malfunction in the intensity control of the O-AWG system (a broken polarization
controller for example) an arbitrary waveform can still be generated with an error that
can be minimized through phase correction. One useful tool for minimizing error is
the Gerchberg-Saxton (GS) Algorithm or the error-reduction algorithm, which will be
discussed next.
32 Chapter 3. Modeling and Simulation
1549.5 1550 1550.50
0.2
0.4
0.6
0.8
1
1.2
1.4
Wavelength (nm)
Sp
ect
ral A
mp
litu
de
(a
.u.)
1549.5 1550 1550.50
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Wavelength (nm)
Sp
ect
ral A
mp
litu
de
(a
.u.)
1549.5 1550 1550.50
0.2
0.4
0.6
0.8
1
1.2
1.4
Wavelength (nm)
Sp
ect
ral A
mp
litu
de
(a
.u.)
1549.5 1550 1550.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Wavelength (nm)
Sp
ect
ral A
mp
litu
de
(a
.u.)
600 650 700 750 800 850 900 950 10000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (ps)
Am
pli
tud
e (
a.u
.)
0 100 200 300 400 500 600 7000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (ps)
Am
pli
tud
e (
a.u
.)
0 100 200 300 400 500 600 7000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (ps)
Am
pli
tud
e (
a.u
.)
0 100 200 300 400 500 600 7000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (ps)
Am
pli
tud
e (
a.u
.)
Gaussian Spectrum Varied by a 4-degree polynomial Gaussian Spectrum Varied by a 5-degree polynomial
Gaussian Spectrum Varied by a 7-degree polynomial Gaussian Spectrum Varied by a 9-degree polynomial
Figure 3.7: Time plots of a Gaussian spectrum varied by polynomial with 4, 5, 7 or 9degree. Solid lines represent the target signal in the time domain and the target spectrum,dotted lines represent the generated waveform and spectrum with a 5-line system. In the5-degree case, the target waveform is duplicated twice and shifted to show the aliasing ofthe waveform in the time domain (shaded lines). The shaded rectangle indicates the timeaperture as defined by the repetition rate (15 GHz or 66.66 ps). The target spectrum isshown below the time domain plot.
3.3. Waveform Control and Error Minimization 33
0 10 20 30 40 50 60 70 800
0.02
0.04
0.06
0.08
0.1
0.12
Time (ps)
Nor
mal
ized
Am
plitu
de (
a.u.
)
1545 1550 15550
5
10
Wavelength (nm)
Am
plitu
de (
a.u.
)
1545 1550 1555−4000
−2000
0
Pha
se (
Deg
)
0 10 20 30 40 50 60 70 80−20
−15
−10
−5
0
5
10
15
20
Time (ps)
Err
or fr
om T
arge
t Wav
efor
m (
%)
1545 1550 15550
5
10
Wavelength (nm)
Am
plitu
de (
a.u.
)
1545 1550 1555−200
0
200
Pha
se (
Deg
)
Figure 3.8: Waveform example 1. Top Left: Target (dotted) and generated (solid) wave-form in time domain. Top Right: Spectral amplitude and phase of the target waveform.Bottom Right: Spectral amplitude and phase of the waveform generated by the 101-lineO-AWG. Bottom Left: Deviation error from the target waveform. The average error is0.88%.
34 Chapter 3. Modeling and Simulation
0 10 20 30 40 50 60 70 800
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Time (ps)
Nor
mal
ized
Am
plitu
de (
a.u.
)
1545 1550 15550
5
10
Wavelength (nm)
Am
plitu
de (
a.u.
)
1545 1550 1555−4000
−2000
0
Wavelength (nm)
Pha
se (
Deg
)
0 10 20 30 40 50 60 70 80−80
−60
−40
−20
0
20
40
60
80
Time (ps)
Err
or o
f Tar
get W
avef
orm
(%
)
1545 1550 15550
5
10
Wavelength (nm)
Am
plitu
de (
a.u.
)
1545 1550 1555−200
0
200
Pha
se (
Deg
)
Figure 3.9: Waveform example 2. Top Left: Target (dotted) and generated (solid) wave-form in time domain. Top Right: Spectral amplitude and phase of the target waveform.Bottom Right: Spectral amplitude and phase of the waveform generated by the 5-lineO-AWG. Bottom Left: Deviation error from the target waveform. The average error is25.85%.
3.3. Waveform Control and Error Minimization 35
Start
End
FFT of
sqrt(I(ω))
Replace |Ek(t)|
with |Etar(t)|
Replace |E’k(ω)|
with |Emeas(ω)|
FFT to “ω”
domain
Ek(t)
E’k(t)E’k(ω)
E’k+1(ω) Ek+1(t)FFT to “t”
FFT to “ω”
Figure 3.10: Flow diagram of GS Algorithm for pulse shaping applications [8].
As described in various publications [8, 55, 56, 57], the GS Algorithm is an iterative
algorithm that calculates the phase of a signal when the modulus (the absolute value)
of the signal is known in both frequency and time domain. This algorithm is used in
a variety of areas such as astronomy, x-ray, electron microscopy [58] and pulse shaping
[8] for phase retrieval when phase measurement is difficult. This algorithm is useful in
the pulse shaping or O-AWG applications because those applications often focus on the
intensity profile with no knowledge of the phase of the underlying carrier of the signal.
Through this algorithm, the user can optimize the intensity profile by adjusting the phase
of the carrier. The GS Algorithm is said to excel in error reduction because of its relation
to convex optimization theory, but a complete explanation has not yet been developed
[57].
Ek(t) = |Ek(t)|eiφk(t) = FFT−1{E ′k(ω)} (3.9)
E ′k(t) = |Etar(t)|eiφk(t) (3.10)
E ′k(ω) = |E ′
k(ω)|eiΦk+1(ω) = FFT{E ′k(t)} (3.11)
E ′k+1(ω) = |Emeas(ω)|eiΦk+1(ω) (3.12)
The GS Algorithm involves four steps as described by (3.9) (3.10) (3.11) (3.12) where
|Emeas(ω)| and |Etar(t)| are the two fixed modulus constraint in frequency and time
domain respectively. The algorithm starts with the inverse FFT of the spectral modulus.
The phase component of the transform is retained but the amplitude is replaced with the
time modulus constraint (|Etar(t)|) as indicated in (3.10). Then the newly constructed
waveform is transformed back into frequency domain. Its phase component is again
36 Chapter 3. Modeling and Simulation
retained while |Emeas(ω)| is put in place of the transformed amplitude. These steps are
reiterated until the phase function converges. That is, when the error (as defined in (3.5))
no longer decrease or has been reduced below an acceptable level. With this solution, the
error between the two constraints will be minimized. Figure 3.10 describe this process
using a flow chart.
Using this tool, we can investigate the possibility of minimizing error in waveform
generation. First, a target waveform is defined in time domain with consideration of
the O-AWG system constrains and its end application. As shown in Figure 3.11 a), the
waveform is defined for a time aperture of 66.66 ps, which corresponds to a repetition
rate of 15 GHz. The corresponding spectral property of the signal is shown in Figure 3.11
b). Clearly, this continuous spectrum cannot be completely represented by a 5-line O-
AWG system since the spectrum extends to infinity. As shown in 3.11 c) the continuous
spectrum is “sampled” and the corresponding periodic signal in time is shown in 3.11 a)
(solid line).
To apply the GS Algorithm, the time domain target is repeated to serve as the
modulus constraint in time domain, while the 5-line “sampled” spectrum is used as the
amplitude constraint in frequency domain. Since the GS-Algorithm tend to converge
quickly [8], the GS Algorithm only needs to be computed for 100 iterations.
Since the mapping between time and frequency domain by Fourier transform is unique
(one-to-one) and the time domain profile is dependent on both spectral phase and am-
plitude as described by (3.8), the direct sampling of the target spectrum should results
in minimal error. As shown in Figure 3.11 e), the error remains constant indicating that
the error is already minimized as the 5-line O-AWG implements the Fourier transform.
Two minor errors were introduced to the amplitude of the spectrum as simulated mal-
functions in the intensity control to demonstrate the GS Algorithm. First, as indicated
in Figure 3.11 d), the amplitude of the second spectral line was increased by 10% while
the forth spectral line was decreased by 10%. Then the GS Algorithm is again computed
for 100 iterations in an attempt to reduce the error by adjusting the phase. As shown in
Figure 3.11 e), the error reduces rapidly and converges to a minimum value. Note that
the error is still greater than the direct sampling of the target spectrum and the error
is only reduced for a small amount. That is because the intensity profile is determined
by (3.8) and altering either the amplitude or the phase of the spectrum invariably leads
to a change in the output temporal waveform. Given the spectral phase as a degree of
freedom, however, the GS algorithm can attempt to reduce the error caused by the am-
plitude change and bring the output waveform closer to the target waveform by changing
the phase.
3.3. Waveform Control and Error Minimization 37
1.549 1.5495 1.55 1.5505 1.551
x 10−6
0
100
200
300
400
Am
pli
tud
e (
a.u
.)
1.549 1.5495 1.55 1.5505 1.551
x 10−6
−2
−1.8
−1.6
−1.4
−1.2x 10
6
Ph
ase
Wavelength (m)
10 20 30 40 50 60 70 80 90 1001.1078
1.1079
1.108
1.1081
1.1082
1.1083
1.1084x 10
4
Iteration #
Err
or
(a.u
.)
10 20 30 40 50 60 70 80 90 1001.1067
1.1068
1.1068
1.1069
1.107
1.107x 10
4
Iteration #
Err
or
0 10 20 30 40 50 600
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (ps)
Inte
nsi
ty (
a.u
.)
Target
5−line
1.549 1.5495 1.55 1.5505 1.551
x 10−6
0
100
200
300
400
Am
pli
tud
e (
a.u
.)
1.549 1.5495 1.55 1.5505 1.551
x 10−6
−4
−2
0
2
4
Ph
ase
Wavelength (m)
a) b)
e)
c)
1.549 1.5495 1.55 1.5505 1.551
x 10−6
0
100
200
300
400
Am
pli
tud
e (
a.u
.)
1.549 1.5495 1.55 1.5505 1.551
x 10−6
−4
−2
0
2
4P
ha
se
Wavelength (m)
d)
f)
+10% -10%
Err
or
(a.u
.)
Figure 3.11: a) Inverse Fourier transform of the target (dotted) and 5-line spectrum(solid), b) amplitude and phase of the continuous (target), c) “sampled” amplitude andphase of the target spectrum, d) “sampled” amplitude and phase of the target spectrumwith error introduced to the amplitude, e) Error between the target and generated signalover 100 iteration of GS algorithm showing minimal error already achieved through directsampling, f) Error between the target and generated signal after the spectral amplitudewas altered showing reduction in error through phase adjustment.
38 Chapter 3. Modeling and Simulation
1.549 1.5495 1.55 1.5505 1.551
x 10−6
0
0.2
0.4
0.6
0.8
1
Am
pli
tud
e (
a.u
.)
Wavelength (m)
1.549 1.5495 1.55 1.5505 1.551
x 10−6
−1
−0.5
0
0.5
1
Ph
ase
Wavelength (m)
0 50 100 150 2000
0.2
0.4
0.6
0.8
1
Time (ps)
Inte
nsi
ty (
a.u
.)
Short time scale
0 2 4 6 8
x 105
0
0.2
0.4
0.6
0.8
1
Time (ps)
Inte
nsi
ty (
a.u
.)
Long time scale
Frequency DomainTime Domain
a)
b)
c)
d)
Figure 3.12: Simulation of a 5-line rectangular spectrum with zero linewidth. a) timedomain signal over a short period (250 ps), b) time domain signal over a long period(8× 105 ps) showing the overall envelope, c) spectral amplitude, d) spectral phase.
3.4 Effects of Non-Zero Linewidth
Although all the simulation conducted thus far were made under the assumption that the
spectral lines have zero width, it does not entirely reflect the reality. That is because all
lasers have a finite linewidth due to various physical effects such as spontaneous emission.
For example, the HP8168 CW laser has a linewidth ranging from 100 kHz to 500 MHz
depending on the mode of operation [59]. We will show that the finite linewidth only
contribute to a long term envelope dependant on the amplitude and phase characteristic
of the linewidth. Therefore, the shape of the arbitrary waveform can be appropriately
model with ideal zero-width lines.
To simulate the effect of non-zero linewidth, a linewidth of 100 kHz was assumed with
a Gaussian shape and it is then convoluted with five zero-width lines of equal amplitude.
3.4. Effects of Non-Zero Linewidth 39
1.5495 1.55 1.5505
x 10−6
0
0.2
0.4
0.6
0.8
1
Am
pli
tud
e (
a.u
.)
Wavelength (m)
1.5495 1.55 1.5505
x 10−6
−1
−0.5
0
0.5
1
Ph
ase
Wavelength (m)
0 50 100 1500
0.2
0.4
0.6
0.8
1
Time (ps)
Inte
nsi
ty (
a.u
.)
Short time scale
0 0.5 1 1.5 2
x 104
0
0.2
0.4
0.6
0.8
1
Time (ps)
Inte
nsi
ty (
a.u
.)
Long time scale
Frequency DomainTime Domain
a)
b)
c)
d)
Figure 3.13: Simulation of a 5-line rectangular spectrum with 100 kHz linewidth withzero phase (non-physical). a) time domain signal over a short period (250 ps), b) timedomain signal over a long period (8 × 105 ps) showing the overall envelope, c) spectralamplitude, d) spectral phase.
40 Chapter 3. Modeling and Simulation
1.5495 1.55 1.5505
x 10−6
0
0.2
0.4
0.6
0.8
1
Am
pli
tud
e (
a.u
.)
Wavelength (m)
1.5495 1.55 1.5505
x 10−6
−4
−2
0
2
4
Ph
ase
Wavelength (m)
0 50 100 1500
0.05
0.1
0.15
0.2
Time (ps)
Inte
nsi
ty (
a.u
.)
Short time scale
0 2 4 6 8
x 105
0
0.2
0.4
0.6
0.8
1
Time (ps)
Inte
nsi
ty (
a.u
.)
Long time scale
Frequency DomainTime Domain
a)
b)
c)
d)
Figure 3.14: Simulation of a 5-line rectangular spectrum with 100 kHz linewidth withrandom phase (non-physical). a) time domain signal over a short period (250 ps), b) timedomain signal over a long period (8 × 105 ps) showing the overall envelope, c) spectralamplitude, d) spectral phase.
3.4. Effects of Non-Zero Linewidth 41
In the ideal case of zero linewidth (Figure 3.12), five spectral lines of equal amplitude
(Figure 3.12 c) result in a periodic sinc2 function. When the spectral lines are convolved
with the 100 kHz Gaussian linewidth with zero phase (Figure 3.13), there is an overall
envelope on top of the repeating sinc2 function. This is because in Fourier theory, a
convolution in one domain (time or frequency) transform into a multiplication in the
other domain (frequency or time). The envelope in the time domain is directly related
to both amplitude and phase of the linewidth profile in the frequency domain. As shown
in Figure 3.14, we would have a different envelope on top of the repeating sinc2 simply
by randomizing the phase of the linewidth profile without altering its amplitude.
As the reader will see in Chapter 4, both the zero phase 3.13 and the random phase
3.14 case do not correspond to reality because the overall envelope is not observed in
experiment. But how would we estimate the phase of the linewidth profile such that its
effect can be properly simulated? Although the linewidth is known to be the result of
phase and amplitude noise of the laser, the phase function of the linewidth is difficult
to measure experimentally. Therefore, we use two methods to account for the unknown
phase of the spectral line. For the first method, we attempt to recover the phase of the
line using the GS algorithm mentioned in previous section. For the second method, we
will model the spectral line base on our understanding of coherence time by generating
the laser signal in time domain. In the end, both methods lead us to conclude that the
phase of the spectral line only affects the long term intensity noise and not the shape of
the arbitrary waveform. More importantly, we will show that as long as the initial CW
laser has a stable power level, the overall envelop for the shaped waveform will be stable
as well.
3.4.1 Phase Recovery for the Spectral Line Through Gerchberg-
Saxton Algorithm
Since we know a CW laser generates constant power, we can use that as one of the
constraints for the GS algorithm and use the algorithm to find a possible phase profile
that agrees with physical observation. With a Gaussian linewidth, the GS algorithm was
computed for 10000 iterations and the result is shown in Figure 3.15. With the calculated
phase, the linewidth was again convolved with the five equal-amplitude spectral lines to
simulate the effect.
As shown in Figure 3.16, the periodic sinc2 function is unaffected except having the
additional envelope on top of the intensity waveform over a long time period. Given this
result, it is clear that for the purpose of simulating the shape of the arbitrary waveform,
42 Chapter 3. Modeling and Simulation
50
100
150
200
250
300
350
Am
pli
tud
e (
a.u
.)
−220
−200
−180
−160
−140
−120
−100
Ph
ase
0 50 100 1500
0.2
0.4
0.6
0.8
1
Time (ps)
Inte
nsi
ty (
a.u
.)
Short time scale
0 2 4 6 8
x 105
0
0.2
0.4
0.6
0.8
1
Time (ps)
Inte
nsi
ty (
a.u
.)
Long time scale
1549.99 1549.995 1550 1550.005 1550.01
Wavelength (nm)
1549.99 1549.995 1550 1550.005 1550.01
Wavelength (nm)
Frequency DomainTime Domain
a)
b)
c)
d)
Figure 3.15: Simulation of a single Gaussian spectrum with 100 kHz FWHM with a phasefunction estimated by GS algorithm. a) time domain signal over a short period (250 ps),b) time domain signal over a long period (8 × 105 ps) showing the overall average, c)spectral amplitude, d) spectral phase.
we can assume ideal zero linewidth. That is because the amplitude profile and phase
of the non-zero linewidth has negligible contribution to generated waveform in practice.
As long as the CW laser that is driving the O-AWG system has a stable intensity, the
spectral lines generated through electro-optic phase modulation as well as the subsequent
shaped arbitrary waveform will be stable over the long term.
3.4. Effects of Non-Zero Linewidth 43
1.5496 1.5498 1.55 1.5502 1.5504
x 10−6
0
50
100
150
200
250
300
350
Am
pli
tud
e (
a.u
.)
Wavelength (m)
1.5496 1.5498 1.55 1.5502 1.5504
x 10−6
−250
−200
−150
−100
−50
0
50
Ph
ase
Wavelength (m)
0 50 100 1500
0.2
0.4
0.6
0.8
1
Time (ps)
Inte
nsi
ty (
a.u
.)
Short time scale
0 2 4 6 8
x 105
0
0.2
0.4
0.6
0.8
1
Time (ps)
Inte
nsi
ty (
a.u
.)
Long time scale
Frequency DomainTime Domain
a)
b)
c)
d)
Figure 3.16: Simulation of a 5-line spectrum with 100 kHz Gaussian linewidth and aphase function estimated by GS algorithm. a) time domain signal over a short period(250 ps), b) time domain signal over a long period (8 × 105 ps) showing the overallenvelope, c) spectral amplitude, d) spectral phase.
44 Chapter 3. Modeling and Simulation
3.4.2 Spectral Line Modelling Through Coherence Time Simu-
lation in Time Domain
The spectral width of a line is related to the coherence time by the following equation
[49]:
∆νc =1
τc
(3.13)
A linewidth of 100 kHz results in a coherence time of 10 µs. Given this, we can generate
a time domain waveform in our simulation with constant amplitude but with randomized
phase at random intervals. Because the degree of correlation between different parts of
the waveform in time defines the coherence time, the distribution of the random intervals
is also set by it. The distribution is assumed to have a Gaussian distribution with 1 ns
variance. As shown in Figure 3.17, a 100 kHz linewidth is resulted in frequency domain
with a particular phase function. We would like to note that even though the line does
not have a smooth profile, it does not affect our conclusion. That is because we can only
simulate a signal with limited time interval and one can expects a smoother line profile
as the time interval increase.
After replicating the line in the frequency domain, we have Figure 3.18, showing
again the amplitude and phase of the line only affects the long term envelope (which is
constant in this case). In the short time scale, the waveform is once again a periodic
sinc2 function.
3.5 Summary
In this chapter, we modelled the O-AWG system and conducted several simulations to
investigate various aspects of the system. We began by modelling the optical signal
without the carrier to reduce computation complexity. Through (3.8), we have shown
that the shape of the waveform depends on the amplitude of the spectral lines as well as
the pairwise phase relationships between the lines. Then, we showed that for an N-line
O-AWG system, the number of phase control can be reduced if the overall delay of shaped
waveform is not important for the application.
We then investigated the effects of having finite sampling of the spectrum. As ex-
pected, with limited number of spectral lines, the O-AWG system can only generate an
arbitrary waveform of finite bandwidth within the period defined by the repetition rate.
The spectral resolution of the O-AWG system puts a lower bound on the repetition rate
3.5. Summary 45
1549.99 1549.995 1550 1550.005 1550.010
0.5
1
1.5
2
2.5x 10
9
Am
pli
tud
e (
a.u
.)
Wavelength (nm)
1549.99 1549.995 1550 1550.005 1550.01−9.28
−9.26
−9.24
−9.22
−9.2
−9.18
−9.16x 10
5
Ph
ase
(d
eg
)
Wavelength (nm)
0 50 100 150 2000
0.2
0.4
0.6
0.8
1
Time (ps)
Inte
nsi
ty (
a.u
.)
Short time scale
0 2 4 6 8
x 105
0
0.2
0.4
0.6
0.8
1
Time (ps)
Inte
nsi
ty (
a.u
.)
Long time scale
Frequency DomainTime Domain
a)
b)
c)
d)
Figure 3.17: Simulation of a single spectral line with 100 kHz FWHM by modellingcoherence time in the time domain. a) time domain signal over a short period (250 ps),b) time domain signal over a long period (8 × 105 ps) showing the overall envelope, c)spectral amplitude, d) spectral phase.
46 Chapter 3. Modeling and Simulation
1549 1549.5 1550 1550.5 15510
2
4
6
8
10
12x 10
4
Am
pli
tud
e (
a.u
.)
Wavelength (nm)
1549 1549.5 1550 1550.5 1551−2.3
−2.2
−2.1
−2
−1.9
−1.8
−1.7
−1.6x 10
4
Ph
ase
Wavelength (nm)
0 50 100 150 2000
0.2
0.4
0.6
0.8
1
Time (ps)
Inte
nsi
ty (
a.u
.)
Short time scale
0 2 4 6 8
x 105
0
0.2
0.4
0.6
0.8
1
Time (ps)
Inte
nsi
ty (
a.u
.)
Long time scale
Frequency DomainTime Domain
a)
b)
c)
d)
Figure 3.18: Simulation of a 5-line spectrum with 100 kHz Gaussian linewidth by mod-elling coherence time. a) time domain signal over a short period (250 ps), b) time domainsignal over a long period (8×105 ps) showing the overall envelope, c) spectral amplitude,d) spectral phase.
3.5. Summary 47
of the target waveform because waveform will begin to “alias” in time domain if the
target repetition rate is below the bound.
Next, we discussed the effects of having finite linewidth. We have shown that the
linewidth only affects the long term intensity noise of the shaped waveform. The shape
of the envelope depends on the amplitude and phase of the spectral line. Through two
independent methods, namely phase recovery with the Gerchberg-Saxton algorithm and
coherence time modelling, we showed that there exists a phase function for a non-zero-
width spectral line to have a stable, long term intensity noise. While the phase function
of the spectral line is determined by the driving CW laser, the phase relationship between
the lines are fixed since they are generated through modulation of the CW laser line. It
is through the uniform FBGs and in-line fiber stretchers that we are able to modify these
fixed phase relationship to generate arbitrary waveforms.
Chapter 4
Experimental O-AWG System
As shown in Chapter 3, arbitrary waveforms can be generated by manipulating the
amplitude and phase of individual spectral lines of a signal. Despite various practical
constraints such as limited number of lines and finite linewidth, we have shown that a
line-by-line O-AWG can generate a target waveform if it falls within the operating range
of an O-AWG system. In this chapter, we will demonstrate an O-AWG using a continuous
piece of optical fiber, in which we implemented the amplitude and phase control via in-
line polarization controller, fiber stretcher, and polarizer. It is the combination of these
devices that allows us to manipulate the input signal in the form of a Fourier series.
By implementing the O-AWG system in a continuous fiber, the system has negligible
loss, and thus can be used to control many lines for high temporal resolution and better
shape control. An all-fiber approach also makes the system compatible with existing fiber
communication networks, making the system robust and low cost. Two test systems, a
3-line, and a 5-line O-AWG system were built to demonstrate the feasibility of such
a system. We will discuss the implementations and results from these systems in this
chapter.
4.1 The Testbed
In Figure 4.1, the experiment testbed is shown and it consists of three stages: source,
device-under-test (DUT) and measurement/observation. In the source stage, it is a mod-
ulated CW laser. With an amplitude-modulated CW, three spectral lines (at frequency
48
4.1. The Testbed 49
CW @ ~ 1550nm
RF @ ~ 15 GHz
EOM
Variable DC Supply
DCA
90%
10%
1%
99%
DUT
OSA
Input
RF Input
AM PM
DC Bias
Output
Trigger
RF Amplifier
EDFA
Electrical Signal
Optical Signal
Source
Figure 4.1: Pulse shaper testbed.
ωc, ωc − ωm, and ωc + ωm) are generated:
s(t) = [A + M cos(ωmt)] sin(ωct) (4.1)
= A sin(ωct) + M cos(ωmt) sin(ωct)
= A sin(ωct) +M
2{sin[(ωc + ωm)t] + sin[(ωc − ωm)t]}
where ωc is the CW wavelength in angular frequency, ωm is the modulation frequency,
A is the CW output amplitude and M is the modulation amplitude. To generate more
than three spectral lines, however, a phase modulation is required:
s(t) = Aejωct+jM sin(ωmt) (4.2)
= Aejωct {cos[M sin(ωmt)] + j sin[M sin(ωmt)]}
From [60] p.361 equation (9.1.42) and (9.1.43), we know that,
cos(z sin θ) = J0(z) + 2∞∑
k=1
J2k(z) cos(2kθ) (4.3)
sin(z sin θ) = 2∞∑
k=0
J2k+1(z) sin[(2k + 1)θ] (4.4)
50 Chapter 4. Experimental O-AWG System
Substitute (4.3) and (4.4) into (4.2) we have,
s(t) = Aejωct
{J0(M) + 2
∞∑
k=1
J2k(M) cos(2kωmt) + 2j∞∑
k=0
J2k+1(M) sin[(2k + 1)ωmt]
}
(4.5)
= Aejωct
{J0(M) +
∞∑
k=1
J2k(M)(ej2kωmt + e−j2kωmt) +∞∑
k=0
J2k+1(M)[ej(2k+1)ωmt − e−j(2k+1)ωmt
]}
= Aejωct
(J0(M) +
∞∑
k=1
Jk(M)ejkωmt +∞∑
k=1
(−1)kJk(M)e−jkωmt
)
where Jk(M) is the Bessel functions of the first kind. The amplitude modulated CW
source was used for the 3-line O-AWG while a phase modulated source was used for the
5-line O-AWG. In addition to the modulated CW source, it can be replaced with a high-
repetition-rate pulse source if necessary. For example, if the application for the O-AWG
requires large bandwidth in the range of terahertz and a smooth spectral profile, then the
user may need use an Optical Frequency Comb Generator (OFCG) as the source [61].
The Device-Under-Test (DUT) is where the 3-line and 5-line O-AWG system are
implemented. As mentioned previously, the O-AWG system consists of uniform FBGs,
in-line fiber stretchers, in-line polarization controller and polarizer. A circulator, also
part of the DUT stage, is there to couple the optical signal into and out of the O-AWG
system.
The observation stage consists of a Digital Communication Analyzer (DCA) and an
Optical Spectrum Analyzer (OSA) to observe both temporal and spectral information
concurrently. For higher bandwidth signal, an autocorrelator can be used in place of the
DCA to observe the temporal signal.
The following components were used for the testbed:
• Tunable Continuous-Wave (CW) laser around 1550 nm
• 10 GHz Electro-optic Modulator - Amplitude (EOM-A) - for modulating the CW
laser to generate the sidebands
• 40 GHz Electro-optic Modulator - Phase (EOM-P) - for modulating the CW laser
to generate the sidebands
• 20 GHz RF Generator - to drive the EOM
• Optical Spectrum Analyzer - for observing the signal in frequency domain
4.2. Principle Operations of the O-AWG System 51
Scope
OSA
ShapedPC1PC2PCn
Trigger
EDFA
Observation
......
......
n 2 1
1
32
Polarizer
PhaseModulated
B
B
C
C
FSS: Fiber Stretching Stages
FSS2FSSn
Figure 4.2: Pulse shaping via an array of FBGs using polarization control (PC1, 2 . . . n)together with a polarizer to achieve amplitude control, and using fiber stretching forphase control. Point A, B and C depict the spectral information of the signal at variousstages of the system.
• Digital Communication Analyzer (DCA) - for observing the signal in time domain
• RF amplifier - to increase the RF voltage as required by the EOM
• RF coupler - to split the RF signal between the EOM and the DCA
• Optical coupler (1x2) - for monitoring the signal in time and frequency domain
simultaneously
• 3-port circulator - for routing signal between input, gratings and output
• Polarization controllers and polarizer - work in conjunction to amplitude modulate
different part of the spectrum
• Erbium-doped Fiber Amplifier (EDFA) - for increasing the signal strength of the
shaped signal
• Piezo fiber stretcher - for adjusting the phase relation between the spectral lines
• Piezo driver - for driving the fiber stretcher
4.2 Principle Operations of the O-AWG System
As mentioned previously, we use a modulated continuous-wave (CW) laser as the source of
the system. The sinusoidal RF signal used to modulate the CW laser generates a number
of sidebands, or spectral lines (as shown in inset B of Figure 4.2). The spatial separation
of these spectral lines is achieved by sending the modulated CW signal through an array
52 Chapter 4. Experimental O-AWG System
|Reflectivity| |Input|
123 123
Figure 4.3: Spectral response (solid line) and signal input (dotted line) of a conceptualthree-FBG array.
of uniform FBGs with high reflectivity (above 90%). The central wavelengths of the
FBGs are spaced equally, and the separation corresponds to the modulation frequency of
the RF signal. Either the FBGs or the RF signal can be tuned to satisfy this condition.
In our case, we tuned the RF frequency to match the spectral separation of the FBGs,
and we tuned the wavelength of the CW laser to ensure each spectral line correspond
to the peak reflection wavelength of an FBG. Hence, each FBG would only reflect one
spectral line and thus spatially separating the lines for further manipulation.
As shown in Figure 4.3, this approach allows partial overlap of the FBG spectrum and
thus provides better fabrication tolerance. In addition, unlike direct temporal shaping,
fabrication tolerance increases as repetition rate increases because the requirement for
spectrum spacing between FBGs widen with repetition rate.
Amplitude manipulation of individual spectral lines is achieved through the combi-
nation of polarization controllers and a polarizer rather than the features of the FBGs.
The polarization controllers are inserted between the FBGs while the polarizer is placed
at the output port (port 3) of the circulator as shown in Figure 4.2. As the spectral
lines are separated by the FBGs, each of the lines passes through a different number of
polarization controllers, allowing independent control of polarization for each line as long
as the controllers are adjusted in the right order (PC1, PC2, . . . then PCn). Thus, as
the signal (containing frequency-dependent polarization variations) passes through the
polarizer at the output, the amplitude of each spectral line is changed as the polarizer
filters out light in the orthogonal polarization.
Phase relationships between the spectral lines are modified by varying the optical
4.3. Results 53
path length between the FBGs through fiber stretching. Pulse shaping is achievable as
long as the phase relationships between the frequency lines are controllable over 2π. The
operating wavelength for the phase shifter is between 970 nm and 1650 nm and it is
capable of producing a phase shift of 15 to 8 π. The half-wave (π) phase shifting voltage
is between 10 and 20 volts, which means the accuracy of the phase shifter is dependent
on the voltage source that drives it. Since FBGs are fabricated into the fiber, and
polarization controllers and fiber stretchers can be inserted between the FBGs without
breaking the fiber, this technique provides a way to shape a waveform in a continuous,
splice-free fiber, and hence, ensures the lowest possible insertion loss.
4.3 Results
To demonstrate the O-AWG concept, two arrays of FBGs were fabricated at Ryerson
University with the help of Prof. Gu. The FBGs are apodized uniform FBGs made from
a single phase mask. The centre wavelength shift of the different FBGs were introduced
by applying tension to the fiber during fabrication using weights. The first array consists
of three FBGs. It was made for a proof of principle demonstration. The second array
consists of five FBGs. It is an improved system based on the experience learnt from the
first system and it is made to generate better controlled waveform with higher bandwidth
and flexibility. All gratings were fabricated using an existing phase mask of 25 mm and
apodization masks.
To measure the reflectivity of the FBGs, the branch of fiber is attached to a circulator.
A broadband source is used as the input to the circulator while the reflected output from
the FBGs is routed to an OSA for observation. Since the bandwidths of the FBGs are
wider than their frequency separation, some FBGs were mechanically stretched while
others were being measured. For example, in the 3-line O-AWG case, when the spectral
response of FBG1 is being measured, FBG2 and FBG3 are stretched such that their
reflections are far away from that of FBG1. Doing so allow us to observe the edge of the
FBG reflection. The edges of the FBGs are important because each FBG should only
reflect one spectral line. The edges identify the spectral position where the spectral lines
should be tuned to.
4.3.1 3-line O-AWG System
Figure 4.4 shows the measured reflection spectrum of the three FBGs used for the O-
AWG and their properties is described in Table 4.1. Based on these properties, the CW
54 Chapter 4. Experimental O-AWG System
-1.20E+01
-8.00E+00
-4.00E+00
0.00E+00
1550.5 1551 1551.5 1552
Wavelength (nm)
Re
fle
cti
vit
y (
a.u
.)
FBG1
FBG2
FBG3
Figure 4.4: Spectral response of a three-FBG O-AWG.
Table 4.1: 3-FBG PropertiesFBG 1 FBG 2 FBG 3
FWHM (pm) 173 209 259Centre Wavelength (nm) 1551.021 1551.203 1551.336
4.3. Results 55
Table 4.2: 5-FBG PropertiesFBG 1 FBG 2 FBG 3 FBG 4 FBG 5
FWHM (pm) 88 89 90 88 72Centre Wavelength (nm) 1546.740 1546.895 1547.003 1547.136 1547.264
laser was tuned to 1551.22 nm and amplitude modulated at 20 GHz (0.16 nm @ 1551.22
nm) such that each FBG would reflect only one spectral line.
Several waveforms were generated to demonstrate the ability of the 3-line O-AWG.
As shown in the first plot of Figure 4.5, we were able to demonstrate the periodic sinc2
simulated in Figure 3.4.
4.3.2 5-line O-AWG System
With the success of the 3-line O-AWG, an array of five FBGs was fabricated. Its reflection
spectrum is shown in Figure 4.6 with their spectral information detailed in Table 4.2. The
3dB bandwidth of each FBG is approximately 90 pm, and the centre-to-centre frequency
separation of the FBGs is 0.12 nm, which corresponds to 15GHz of modulation frequency
on the CW signal, and 15GHz fundamental repetition rate of the shaped signal. The total
spectral bandwidth consist of 5 spectral lines is 0.48nm, which corresponds to 60GHz (at
1550 nm) of bandwidth for the shaped signal. This bandwidth translates into a temporal
resolution of approximately 17ps (assuming Gaussian time-bandwidth product.)
While our system can be implemented in a continuous, splice-free, fiber, the com-
ponents of the systems (FBGs, polarization controller, fiber stretcher) were fabricated
separately and splice together into one system for the convenience of the experiment.
This resulted in progressively higher loss as the splice loss accumulate when the number
of stages increases. This is reflected in the roll off of reflectivity in Figure 4.6. Because
the long wavelength stages were cascaded after th short wavelength stage, the accumu-
lation of splice loss and other connection losses manifested as a reduction in reflectivity.
Fortunately, since the mechanism for amplitude control does not depend on the exact
reflective profile of the FBGs, the arbitrary waveform generation capability of the system
is not compromised.
In Figure 4.7, we show a variety of shaped pulse trains generated by our system. Fig-
ure 4.7 a) and b) demonstrate the independent control of phase by showing two different
waveform resulting from the same spectral amplitude. Waveform c) has a temporal shape
similar to a saw-tooth shape, and waveform d) shows a near “flat-top” temporal shape.
We would like to remark that the bandwidth of the shaped signal (60GHz) is similar to
56 Chapter 4. Experimental O-AWG System
Figure 4.5: Waveforms generated by the 3-line O-AWG.
4.3. Results 57
-12
-10
-8
-6
-4
-2
0
1546.5 1546.7 1546.9 1547.1 1547.3 1547.5Wavelength (nm)
No
rma
lize
d A
mp
litu
de
(d
B)
FBG Array Reflection Spectrum
Spectral lines of a modulated CW source
Figure 4.6: Spectral response of a five-FBG O-AWG (solid line) and the input of a phasemodulated CW laser (dotted line).
the bandwidth of the digital sampling scope (65GHz) used to record these waveforms,
and therefore some of the sharp temporal features in the shaped waveform may not be
accurately reproduced by the scope.
Plot along the experimentally measured results are the corresponding simulated re-
sults used to recover the spectral phases. The measured temporal and spectral amplitude
were used as the modulus constraints of the G-S algorithm to recover the phase. Since
the OSA has insufficient spectral resolution to resolve the line shape, an ideal line profile
is assumed.
As shown in Figure 4.7, the simulated waveforms from the generated with the recov-
ered phase are very similar with small amount of deviation from the measured waveforms.
There are several reasons. First, the DCA used to measure the temporal waveform has
limited memory depth and therefore can only measure a waveform for a limited duration.
Consequently, the limited period in time measurement reduces the spectral resolution of
the simulation leading to the coarse, triangular-shaped line in the frequency domain.
This slight deviation from the actual measurement may cause small amount of error.
In addition, there is a small delay between the time measurement by the DCA and the
spectrum captured by the OSA. During this small delay, the waveform may have drifted,
contributing to the discrepancies.
4.3.3 Waveform Stability
We studied both the long term and short term stability of the O-AWG system. Figure
4.8 shows two scope plots with persistence for 10 second for the short term stability
characterization. The left plot is the measurement of a CW laser while the trace on
58 Chapter 4. Experimental O-AWG System
0 50 100 150 2000
0.5
1
Time (ps)
Inte
nsi
ty (
a.u
.)
0
0.2
0.4
Sp
ect
ral A
mp
litu
de
1546 1546.5 1547 1547.5 15480
2000
4000
6000
Wavelength (nm)
Ph
ase
(D
eg
)
Measured
Simulated
0 50 100 150 2000
0.5
1
Time (ps)
Inte
nsi
ty (
a.u
.)
1546 1546.5 1547 1547.5 15480
0.2
0.4
Wavelength (nm)
Sp
ect
ral A
mp
litu
de
−1000
0
1000
2000
Ph
ase
(D
eg
ree
)
Measured
Simulated
0 100 200 3000
0.5
1
Time (ps)
Inte
nsi
ty (
a.u
.)
1546 1546.5 1547 1547.5 15480
0.1
0.2
0.3
0.4
0.5
Wavelength (nm)
Sp
ect
ral A
mp
litu
de
1546 1546.5 1547 1547.5 1548−500
0
500
1000
1500
2000
Ph
ase
(D
eg
ree
)
Measured
Simulated
0 100 200 3000
0.5
1
Time (ps)
Inte
nsi
ty (
a.u
.)
1546 1546.5 1547 1547.5 15480
0.5
1
Wavelength (nm)
Sp
ect
ral A
mp
litu
de
1546 1546.5 1547 1547.5 1548−4000
−2000
0
Ph
ase
(D
eg
ree
)
Measured
Simulated
b)
c) d)
a)
Figure 4.7: Experimental results showing various temporal waveforms from a five-linearbitrary waveform generator. The insets show the measured spectral amplitudes of thelines. a) and b) illustrate phase control resulting in different pulse shape for the samespectral amplitude. c) has a shape close to a saw-tooth form, and d) has a near “flat-top”shape.
time (ps)
In
ten
sit
y (
a.u
.)
0 100 200
time (ps)
In
ten
sit
y (
a.u
.)
0 100 200
Figure 4.8: Persistence plot of a CW (left) and a signal from the 3-line O-AWG system(right).
4.3. Results 59
time (ps)
In
ten
sit
y (
a.u
.)
5000 100 200 300 400
time (ps)
In
ten
sit
y (
a.u
.)
5000 100 200 300 400
Figure 4.9: Persistence plots of the 3-line O-AWG for 10 seconds (left), for 50 seconds(right).
the right is the measurement of a modulated CW laser after it has passed through the
O-AWG system. Both plots exhibits similar range of fluctuation, which can be caused
by the spontaneous emission of the EDFA or the noise at the detector. Since this small
fluctuation is independent of the O-AWG system (as it occurs with or without the O-AWG
system), the characterization of the shaping performance were done using a 16-sample
average to remove this small variation.
For long term stability, a script was written in Python to periodically capture data
from the DCA and OSA over GPIB. As shown in Figure 4.9, the generated waveform has
large fluctuation in its shape over a very short period (50 seconds) even though the fibers
for the O-AWG system is enclosed to prevent disturbance from airflow. These fluctuations
came from both the instability of phase and polarization. First, the amplitude of the
spectral lines varies with time, which indicate the polarization between the signal and
the polarizer varies with time. Second, even when the observation from the OSA remain
stable momentarily, the temporal waveform continues to shift. That indicates the change
in the phase relationship between the spectral lines.
To mitigate the instability caused by vibrational and thermal fluctuations, the entire
branch of fiber, except where the polarization controllers and phase shifters were located,
was submerged in several containers of gel to insulate it from the environment. This
gel is essentially water captured by sodium polyacrylate, which is a super absorbent.
Since water is the dominant component, the gel can insulate the fiber from temperature
fluctuation though the high specific heat capacity of water (4.181 Jcm3K
). In addition,
the mechanical properties of the gel help insulate the fiber from physical disturbance. As
shown in Figure 4.10, this insulation method achieved at least 10 minutes of stability.
The stability of the 5-line O-AWG system was also characterized. It was initially
stable only for 10 seconds but the stability was increased to 1 minute after several revisions
were made on the design of the gel containers. While the stability of the system has
60 Chapter 4. Experimental O-AWG System
a)
b) c)
Figure 4.10: Persistence plots of the 3-line O-AWG after it is submerged in gel. Persis-tence for a) 6 seconds, b) 10 minutes, c) 20 minutes.
4.3. Results 61
a)
b)
c)
d)
e)
Figure 4.11: Picture of the 5-line O-AWG system. a) Circulator, b) Polarizer (in a boxto reduce disturbance), c) fiber optic embedded in a tub of sodium polyacrylate gel, d)in-line fiber stretcher/ phase shifter, e) in-line polarization controller.
62 Chapter 4. Experimental O-AWG System
improved, it remains to be much less stable than the 3-line O-AWG system (10 minutes.)
There are several reasons for this disparity. First, the 5-line O-AWG has greater degree of
control but those controls (the polarization controllers and phase shifters) are additional
sources of physical disturbance. That is because both of those controls function through
mechanical actions (transverse compressing and twisting for the polarization controllers
and axial stretching for the phase shifters.) In addition, the branch of fiber for the 5-line
O-AWG is considerably longer making it more susceptible to disturbance in spite of the
gel insulation.
4.4 Comparisons Between Different Dynamic O-AWG
Methods
Among the surveyed dynamic O-AWG methods and our system, there are several com-
mon criteria that we can use to compare them. First, we can compare these systems
according to their bandwidth capability. This criterion determines the temporal resolu-
tion achievable by the system because the bandwidth of the signal relates to the fastest
feature of the generated waveform in time. Second, we can compare the spectral resolu-
tion of the O-AWG systems. In line-by-line shaping, spectral resolution is the repetition
rate because it determines the separation of the spectral lines. Third, O-AWG system can
have different types of optical signal source, which has stability implications. Arbitrary
waveforms have been shown to be more stable when generated from a CW rather than
from a mode-locked pulse laser [20]. Last but not least, we can compare these systems
according to their insertion loss. Many existing O-AWG methods require the optical
signal to be coupled in and out of the optical fiber causing an insertion loss between 3
and 13 dB. Table 4.3 summarizes the presented dynamic O-AWG methods with their
features.
As we have shown, we were able to demonstrate a 5-line O-AWG system (shown
here in Figure 2.9) with temporal resolution of 17 ps and spectral resolution of 0.12 nm.
Although the temporal resolution of our system is only better than the direct temporal
shaping system, our system can be scaled up by increasing the number of stages to control
greater number of spectral lines and bandwidth. That is possible for our system because
our continuous-fiber implementation has low insertion loss. In addition, the fiber-based
nature of the system allows it to be integrated to communication system more readily. In
terms of the spectral resolution, we are only exceeded by the SLM implementations. Our
system can be improved further in principle by implementing FBG tuning [53]. Since our
4.4. Comparisons Between Different Dynamic O-AWG Methods 63
Table 4.3: Comparison between various existing methods O-AWG methods with ourproof of concept system.Features Direct
TemporalShaping[2]
EOM [5] AOM [7] SLM [50] This Thesis
Source Pulse Pulse Pulse CW/Pulse CW/High reprate pulse
TemporalResolution
ns n/a 0.1 ps 10 fs 17 ps (5-line), 25ps (3-line)
SpectralResolution
n/a n/a 0.6 nm 5 GHz 0.12 nm (5-line),0.16 nm (3-line)
InsertionLoss
est. > 1.5NdB
3 dB 3 dB 13 dB 0.02N
AdditionalComments
signalsconvertedto electricaldomain
demonstratedfor rep ratemultipli-cation,O-AWG insimulationonly
kHz repeti-tion rate
free-spacecouplingloss, bulkoptic com-plexity andlack ofintegration
All-fiber (lowloss), GHz rep-etition rate,signal remainin the opticaldomain
system does not require stringent synchronization like the EOM system, or the refreshing
of the acoustic wave as in the AOM case, our system can be made to operate at a wide
range of repetition rates above 12.5 GHz.
In terms of insertion loss, our system perform better than all of the reviewed systems.
Given a scattering loss 0.01 dB scattering loss per FBG per pass, the total insertion loss
(the loss of the system excluding any effect of amplitude shaping) is 0.02N, where N is
the number of lines for the system. This is much better than the 1.5N dB loss for the
direct temporal shaping or the 13 dB loss for the SLM system.
Chapter 5
Conclusion
For the first time, we have presented a novel method to perform arbitrary waveform
generation in a continuous fiber. Our O-AWG system performs arbitrary waveform gen-
eration via spectral line-by-line manipulations. Our system uses uniform Fiber Bragg
Gratings (FBGs) to separate the spectral lines, and provides independent amplitude and
phase control for each line via in-line polarization controller and in-line fiber stretcher
respectively. Since the AWG is carried out in a continuous fiber, the system has negligible
loss, and thus can be scaled up to control many lines for high temporal resolution and
better shape control.
Several state-of-the-art pulse shaping methods were reviewed and compared with
our system. Most of the reviewed systems require the optical signal to be coupled out
of the fiber causing significant insertion loss. There are in-fiber O-AWG systems but
they are not dynamic, therefore they cannot adapt to the changing requirements of the
applications. The spectral resolution of our system is limited by the slope of the FBG
spectrum, which can be controlled to be within 100 pm/20 dB [62], giving a spectral
resolution of 12.5GHz. While the spectral resolution of our system is only comparable to
that of the state-of-the-art SLM system [50], the advantage of low insertion loss remains.
In Chapter 3, the novel O-AWG system was modelled and simulated. The effects
and constraints due to finite number of spectral lines and finite linewidth were also
characterized. The error between the generated waveform and the target waveform is
minimized when the following two conditions are met: 1) the duration of the target
waveform is shorter than the periodicity/spectral resolution of the O-AWG system, and
2) the target bandwidth is well represented by the O-AWG system. If the bandwidth
of the target waveform is within the capacity of the system, the error can be well below
5%. For effect of finite linewidth, we used GS phase recovery method and coherence time
modelling method to verify that finite linewidth is only of consequence on the long term
64
5.1. Future Work 65
envelope. In addition, this envelope is flat and stable as long as the driving CW laser
have stable power level. Furthermore, we demonstrated the possibility of expanding the
system to 101 lines.
To test this novel system, a testbed was established. A 3-line and a 5-line O-AWG
system was fabricated and characterized. Our 3-line system can achieve a bandwidth
of 40 GHz or 25 ps temporal resolution with 0.16 nm spectral resolution. With our
5-line system, we can generate waveforms with 60 GHz bandwidth or 17 ps temporal
resolution with 0.12 nm spectral resolution. Several techniques were explored and realized
to improve the stability of the system. For the 3-line O-AWG, we were able to achieve
a stability of greater than 10 minutes while the 5-line O-AWG were able to achieve a
stability of approximately 60 seconds.
As a result of the work completed in this thesis, a journal paper was published in
the Optics Letters [63], and a refereed conference paper was published at the Coherent
Optical Technologies and Applications (COTA) Topical Meeting [64].
5.1 Future Work
While we have demonstrated the feasibility of the O-AWG system, it is desirable to scale
the system beyond 5 spectral lines to at least 10 lines and above. Doing so will give the
user greater flexibility and control to generate more meaningful waveforms.
Consider scattering loss at the FBG and small loses at the PC, an estimated worst
case double-pass transmission loss of 0.1 dB per stage can be achieved. A 100-line
system will therefore have a worse-case spectral line loss of 10 dB. This does not reflect
the system total insertion loss, however, as this worse-case scenario only applies to the
last reflected line. For example, for a 5 nm FWHM Gaussian input, a 100-line system
with 0.1 nm spectral resolution would only produce a loss of 4.64 dB. That is because
the total insertion loss of the system is dependent on the energy distribution of the input
spectrum. Alternatively, one can use FBG tuning techniques to optimize the system such
that bulk of the energy is reflected first and exploit the transmission loss as part of the
amplitude controls.
Two major obstacles must be overcome in order to scale this O-AWG system. First,
the manufacturing techniques of the system needs to be automated. Instead of making
FBGs one by one and splice them together with a phase shifter, an automated process
would allow us to fabricate the FBGs with greater accuracy and allow us to place the
piezo component directly onto the fiber. Second, a computerized controlling scheme is
needed to manage the increasing number of variables as the system scales up. For each
66 Chapter 5. Conclusion
additional spectral line there will be an additional variable for phase control and an
additional variable for amplitude control. If the polarizer is not made to filter out linear
polarization, the polarization control would have two degrees of freedom giving three
variables per spectral line addition.
Other than the scaling up of the O-AWG system, improving the stability performance
of this system is also important. In addition to building better insulation methods for the
fiber, overcoming the two obstacles mentioned above will also improve stability. With
an automated manufacturing process, the system can be made much more compact and
reduce the chances of the fiber getting disturbed. Currently, the total length of the 5-line
O-AWG is approximately 2.5 meters but only 0.6 meter of the fiber is the functioning part
of the system. The rest of the fiber length came from the slack left behind for splicing
error. With a computerized control system, active feedback control can be employ to
further increase the stability of the system.
Appendix A
MATLAB Code for Characterizing
Spectrum Coarse Sampling
1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
2 % codeHighVariation.m
3 % By Kenny Ho, Aug 31, 2007
4 %
5 % For characterizing coarse samplign of bandwidth with high variation
6 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
7
8 clear all;
9 close all;
10
11 c = 299792458; % Speed of light [m]
12
13 %% Simulation Parameters
14 % Grating resolution 0.1 nm
15 % OSA resolution 0.01 nm
16 % HP8168F line width 50 - 500 MHz
17 % using df = c/lambda^2*dlam with lambda = 1550nm, 50MHz is ~ 0.4 pm
18 centreWL = 1550e-9; % Centre wavelength [m]
19 centreFreq = c/centreWL;
20
21 % set simulating spectral resoution to 0.01 pm
22 specRes = 0.01e-12;
23
24 % use bandwidth of 10 nm for high temporal resolution
25 bandwidthWL = 10e-9; %simulation bandwidth 10 nm [m] -->
26 %simulation temporal resolution of 764.5fs
27 bandwidthFreq = c/centreWL^2*bandwidthWL;
28 numPoint = bandwidthWL/specRes;
29 numPoint = 2^length(dec2bin(numPoint)); %round up to the cloest 2^n for FFT
30 actualSimWLBW = numPoint*specRes;%Actual simulation bandwidth after roundup
31 timeRes = centreWL^2/c/actualSimWLBW; %Simulation temporal resolution
32 tAx = linspace(0,timeRes*1e12*(numPoint-1),numPoint); % time axis in ps
33
34 % set frequency/wavelength axis --> at this point, start with lowest
67
68Chapter A. MATLAB Code for Characterizing Spectrum Coarse Sampling
35 % wavelength without ifftshift so this is -pi to pi
36 wl = linspace(0,specRes*(numPoint-1),numPoint) + centreWL - actualSimWLBW/2;
37 f = c./wl;
38 spectrum = zeros(1,numPoint);
39 specCont = zeros(1,numPoint);% For a continuous spectrum instead of 5 lines
40 tTarget = zeros(1,numPoint);
41
42 repRate = 15e9; % repetition rate at approximately 15 GHz [Hz]
43 repRateWL = repRate/c*centreWL^2;
44 spacingCount = round(repRateWL/specRes);
45 repRateWL =spacingCount*specRes;%Calculate backward to account for rounding
46 repRate = repRateWL/centreWL^2*c;
47
48 % view one period
49 timeWin = round(1/repRate/timeRes); %round down from 87.2287 1/0.2287
50 %mean every 4.3729 period there’s a
51 %1 time error
52 tStart = numPoint/2 - round(timeWin/2);
53 tEnd = numPoint/2 + round(timeWin/2);
54 wl = wl.*1e9;
55 %% Set bandwidth and then set variation
56 % Set maximum bandwidth to reduce error
57 FWHM = 0.3e-9;
58 stdDev = FWHM/2/sqrt(2*log(2));
59
60 % Set variation by multiplying a polynomial
61 % First, get the full width at 1% from FWHM and use that as the width of
62 % the poly variation
63 % identify 1% location
64 % pct1point1 = sqrt(2*log(100))*stdDev+centreWL;
65 % pct1point2 = -sqrt(2*log(100))*stdDev+centreWL;
66
67 polyWidth = FWHM/sqrt(log(2))*sqrt(log(100)); %Full Width at 1%
68 %Set min and max width of the gaussian and then sum them to make
69 %a varying spectrum
70 polyWidthCount = round(polyWidth/specRes);
71 polyStart = floor(numPoint/2 - polyWidthCount/2);
72 polyEnd = ceil(numPoint/2 + polyWidthCount/2);
73 polyWidthCount = polyEnd-polyStart+1;
74 minWidth = round(polyWidthCount/10); %round because it’s a index
75 maxWidth = round(polyWidth/8);
76
77 peak1 = rand(1,1)*(wl(polyStart+minWidth*2)-wl(polyStart+minWidth))+...
78 wl(polyStart+minWidth);
79 peak2 = rand(1,1)*(wl(polyStart+minWidth*3)-wl(polyStart+minWidth*2))+...
80 wl(polyStart+minWidth*2);
81 peak3 = rand(1,1)*(wl(polyStart+minWidth*4)-wl(polyStart+minWidth*3))+...
82 wl(polyStart+minWidth*3);
83 peak4 = rand(1,1)*(wl(polyStart+minWidth*5)-wl(polyStart+minWidth*4))+...
84 wl(polyStart+minWidth*4);
85 peak5 = rand(1,1)*(wl(polyStart+minWidth*6)-wl(polyStart+minWidth*5))+...
86 wl(polyStart+minWidth*5);
87 peak6 = rand(1,1)*(wl(polyStart+minWidth*7)-wl(polyStart+minWidth*6))+...
88 wl(polyStart+minWidth*6);
69
89 peak7 = rand(1,1)*(wl(polyStart+minWidth*8)-wl(polyStart+minWidth*7))+...
90 wl(polyStart+minWidth*7);
91 peak8 = rand(1,1)*(wl(polyStart+minWidth*9)-wl(polyStart+minWidth*8))+...
92 wl(polyStart+minWidth*8);
93 peak9 = rand(1,1)*(wl(polyStart+minWidth*10)-wl(polyStart+minWidth*9))+...
94 wl(polyStart+minWidth*9);
95
96 %These are half width and consider them to be at 1 %
97 sig1 = peak1-wl(polyStart);
98 sig2 = peak2-wl(polyStart+minWidth);
99 sig3 = peak3-wl(polyStart+minWidth*2);
100 sig4 = peak4-wl(polyStart+minWidth*3);
101 sig5 = peak5-wl(polyStart+minWidth*4);
102 sig6 = peak6-wl(polyStart+minWidth*5);
103 sig7 = peak7-wl(polyStart+minWidth*6);
104 sig8 = peak8-wl(polyStart+minWidth*7);
105 sig9 = peak9-wl(polyStart+minWidth*8);
106 sig1 = sig1/sqrt(2*log(100));
107 sig2 = sig2/sqrt(2*log(100));
108 sig3 = sig3/sqrt(2*log(100));
109 sig4 = sig4/sqrt(2*log(100));
110 sig5 = sig5/sqrt(2*log(100));
111 sig6 = sig6/sqrt(2*log(100));
112 sig7 = sig7/sqrt(2*log(100));
113 sig8 = sig8/sqrt(2*log(100));
114 sig9 = sig9/sqrt(2*log(100));
115
116 a1 = rand(1,1)*0.7+0.3;
117 a2 = rand(1,1)*0.7+0.3;
118 a3 = rand(1,1)*0.7+0.3;
119 a4 = rand(1,1)*0.7+0.3;
120 a5 = rand(1,1)*0.7+0.3;
121 a6 = rand(1,1)*0.7+0.3;
122 a7 = rand(1,1)*0.7+0.3;
123 a8 = rand(1,1)*0.7+0.3;
124 a9 = rand(1,1)*0.7+0.3;
125
126 specCont = a1*exp(-(wl-peak1).^2/(2*sig1^2)) + ...
127 a2*exp(-(wl-peak2).^2/(2*sig2^2)) + ...
128 a3*exp(-(wl-peak3).^2/(2*sig3^2)) + ...
129 a4*exp(-(wl-peak4).^2/(2*sig4^2)) + ...
130 a5*exp(-(wl-peak5).^2/(2*sig5^2)) + ...
131 a6*exp(-(wl-peak6).^2/(2*sig6^2)) + ...
132 a7*exp(-(wl-peak7).^2/(2*sig7^2)) + ...
133 a8*exp(-(wl-peak8).^2/(2*sig8^2)) + ...
134 a9*exp(-(wl-peak9).^2/(2*sig9^2));
135
136 plot(wl,specCont);
137 xlim([1549.5 1550.5]);
138 xlabel(’Wavelength (nm)’);
139 ylabel(’Spectral Amplitude (a.u.)’);
140 %%
141 specCont = specCont./max(specCont); %normalize
142 % plot(specCont);
70Chapter A. MATLAB Code for Characterizing Spectrum Coarse Sampling
143 specCont = ifftshift(specCont);
144
145 % set 5 spectral lines --> setting according to FFT convention 0 to 2pi
146 spectrum(1) = specCont(1);
147 spectrum(1+spacingCount) = specCont(1+spacingCount);
148 spectrum(1+spacingCount+spacingCount) = ...
149 specCont(1+spacingCount+spacingCount);
150 spectrum(end-spacingCount+1) = specCont(end-spacingCount+1);
151 spectrum(end-spacingCount-spacingCount+1) = ...
152 specCont(end-spacingCount-spacingCount+1);
153
154 yLim = fftshift(fft(spectrum));%fftshift so we can see the -ve time as well
155 yCont = fftshift(fft(specCont));
156 YLim = abs(yLim).^2; %Find intensity
157 YCont = abs(yCont).^2; %Find intensity
158 YLim = YLim/sum(YLim); %Normalize
159 YCont = YCont/sum(YCont); %Normalize
160 error = sum((YLim(tStart:tEnd)-YCont(tStart:tEnd)).^2);
Appendix B
MATLAB Code for Generating the
101-line Example
1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
2 % code101linesEX.m
3 % By Kenny Ho, Aug 31, 2007
4 %
5 % For generating 101 lines example
6 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
7
8 clear all;
9 close all;
10
11 c = 299792458; % Speed of light [m]
12
13 %% Simulation Parameters
14 % Grating resolution 0.1 nm
15 % OSA resolution 0.01 nm
16 % HP8168F line width 50 - 500 MHz
17 % using df = c/lambda^2*dlam with lambda = 1550nm, 50MHz is ~ 0.4 pm
18 centreWL = 1550e-9; % Centre wavelength [m]
19 centreFreq = c/centreWL;
20
21 % set simulating spectral resoution to 0.01 pm
22 specRes = 0.01e-12;
23
24 % Minimum resolution
25 bandwidthWL = 10e-9; %simulation bandwidth 10 nm [m]
26 %--> simulation temporal resolution of 764.5fs
27 bandwidthFreq = c/centreWL^2*bandwidthWL;
28 numPoint = bandwidthWL/specRes;
29 numPoint = 2^length(dec2bin(numPoint));%round up to the cloest 2^N for FFT
30 actualSimWLBW = numPoint*specRes;%Actual simulation bandwidth after roundup
31 timeRes = centreWL^2/c/actualSimWLBW; %Simulation temporal resolution
32 tAx = linspace(0,timeRes*1e12*(numPoint-1),numPoint); % time axis in ps
33
34 % set frequency/wavelength axis --> at this point, start with lowest
71
72 Chapter B. MATLAB Code for Generating the 101-line Example
35 % wavelength without ifftshift so this is -pi to pi
36 wl = linspace(0,specRes*(numPoint-1),numPoint) + centreWL...
37 - actualSimWLBW/2;
38 f = c./wl;
39 spectrum = zeros(1,numPoint);
40 specCont = zeros(1,numPoint); % For a continuous spectrum
41
42 % For 100 line, use minimum grating resolution of 0.1 nm / 12.5GHz
43 repRate = 12.5e9; % repetition rate at approximately 12.5 GHz [Hz]
44 repRateWL = repRate/c*centreWL^2;
45 spacingCount = round(repRateWL/specRes);
46
47 % view one period
48 timeWin = round(1/repRate/timeRes);
49 tStart = numPoint/2 - round(timeWin/2);
50 tEnd = numPoint/2 + round(timeWin/2);
51
52 %% Define Target -- 12.5 GHz --> 80ps period --> 100 point with sim res
53 % 101 lines --> bandwidth 1.25 THz, res~ 800 fs
54 % let’s go for 66.66 ps
55 tTarget = zeros(1,numPoint);
56 numCount = round(66.66e-12/timeRes);
57
58 tStart = round((timeWin-numCount)/2);
59 tTarget(tStart:tStart+numCount)=5;
60
61 %%
62 specCont=fft(tTarget);
63 spectrum(1) = specCont(1);
64 for m = 1:2 % 1:50 for 101 lines, 1:2 for 5 lines
65 spectrum(1+spacingCount*m) = specCont(1+spacingCount*m);
66 spectrum(end-spacingCount*m+1) = specCont(end-spacingCount*m+1);
67 end
68
69 tTry = ifft(spectrum).*length(spectrum)./100; % scale because of sampling
70 TTry = abs(tTry).^2;
71 TTry = TTry./sum(TTry(1:timeWin))*10;
72 tTarget = tTarget./sum(tTarget)*10;
73 figure;plot(tAx(1:timeWin),tTarget(1:timeWin),tAx(1:timeWin),...
74 TTry(1:timeWin));
75 error = TTry(1:timeWin)-tTarget(1:timeWin);
76 perror = error/max(tTarget);
77 figure;plot(tAx(1:timeWin),perror*100);
78 ylim([-80 80]);
79
80 %%
81 wl=wl.*1e9;
82 figure;[AX,H1,H2]=plotyy(wl,abs(fftshift(specCont)),wl,...
83 unwrap(angle(specCont))./pi.*180,’plot’);
84 figure;[AX,H1,H2]=plotyy(wl,abs(fftshift(spectrum)),wl,...
85 unwrap(angle(spectrum))./pi.*180,’plot’);
Appendix C
MATLAB Code for Simulating
Finite Linewidth with Coherence
Time Model
1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
2 % codeNZLineWidthCoherenceTimeModel.m
3 % By Kenny Ho, Aug 31, 2007
4 %
5 % For simulating finite line width with coherence time model
6 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
7
8 clear all;
9 close all;
10
11 c = 299792458; % Speed of light [m]
12
13 %% Simulation Parameters
14 % Grating resolution 0.1 nm
15 % OSA resolution 0.01 nm
16 % HP8168F line width 50 - 500 MHz
17 % using df = c/lambda^2*dlam with lambda = 1550nm, 50MHz is ~ 0.4 pm
18 centreWL = 1550e-9; % Centre wavelength [m]
19 centreFreq = c/centreWL;
20
21 % set simulating spectral resoution to 0.01 pm
22 specRes = 0.01e-12;
23
24 % use bandwidth of 10 nm for high temporal resolution
25 bandwidthWL = 10e-9; %simulation bandwidth 10 nm [m] -->
26 %simulation temporal resolution of 764.5fs
27 bandwidthFreq = c/centreWL^2*bandwidthWL;
28 numPoint = bandwidthWL/specRes;
29 numPoint = 2^length(dec2bin(numPoint)); %round up to the cloest 2^n for FFT
30 actualSimWLBW = numPoint*specRes;%Actual simulation bandwidth after roundup
31 timeRes = centreWL^2/c/actualSimWLBW; %Simulation temporal resolution
73
74Chapter C. MATLAB Code for Simulating Finite Linewidth with Coherence Time Model
32 tAx = linspace(0,timeRes*1e12*(numPoint-1),numPoint); % time axis in ps
33
34 % set frequency/wavelength axis --> at this point, start with lowest
35 % wavelength without ifftshift so this is -pi to pi
36 wl = linspace(0,specRes*(numPoint-1),numPoint) + centreWL - actualSimWLBW/2;
37 f = c./wl;
38 spectrum = zeros(1,numPoint);
39 specCont = zeros(1,numPoint);% For a continuous spectrum instead of 5 lines
40 tTarget = zeros(1,numPoint);
41
42 repRate = 15e9; % repetition rate at approximately 15 GHz [Hz]
43 repRateWL = repRate/c*centreWL^2;
44 spacingCount = round(repRateWL/specRes);
45 repRateWL =spacingCount*specRes;%Calculate backward to account for rounding
46 repRate = repRateWL/centreWL^2*c;
47
48 % view one period
49 timeWin = round(1/repRate/timeRes); %round down from 87.2287 1/0.2287
50 %mean every 4.3729 period there’s a
51 %1 time error
52 tStart = numPoint/2 - round(timeWin/2);
53 tEnd = numPoint/2 + round(timeWin/2);
54
55 %% Prepare for non-zero line width
56 % Suppose 500 MHz linewidth ->
57 linewidthFreq = 500e6; % coherence time 2ns?
58 linewidthTau = 1/linewidthFreq;
59 % get randomize segments to model coherence with variance being 1ns
60 % (coherence time/2) also check to see if the total cumulate to the
61 % simulation windows
62 simTime = (numPoint-1)*timeRes;
63 randSize = 100;
64 randSeg = randn(1,randSize)*linewidthTau/2;
65 randSeg = round(abs(randSeg)./timeRes);
66
67 while (sum(randSeg) < numPoint)
68 randSize = randSize+1;
69 randSeg = randn(1,randSize)*linewidthTau/2;
70 randSeg = round(abs(randSeg)./timeRes);
71 end
72
73 % generate line
74 currentIdx = 1;
75 tTemp = ones(1,randSeg(1)).*1.*exp(i*(rand*2*pi-pi));
76 tTarget = tTemp;
77 for m = 2:length(randSeg)
78 tTemp = ones(1,randSeg(m)).*1.*exp(i*(rand*2*pi-pi));
79 tTarget = [tTarget tTemp];
80 end
81 tTarget = tTarget(1:numPoint);
82 fTry = fft(tTarget);
83 FTry = abs(fTry).^2;
84
85 %%
75
86 YLim = abs(tTarget).^2;
87 wl = wl.*1e9; % convert to nm
88 figure(’Position’,[80 80 800 600]);
89 subplot(2,2,2),plot(wl,abs(fftshift(fTry)).^2);
90 ylabel(’Amplitude (a.u.)’);
91 xlabel(’Wavelength (nm)’);
92 xlim([1549.99 1550.01]);
93 subplot(2,2,4),plot(wl,unwrap(angle(fftshift(fTry)))*180/pi);
94 ylabel(’Phase (deg)’);
95 xlabel(’Wavelength (nm)’);
96 xlim([1549.99 1550.01]);
97 subplot(2,2,1),plot(tAx(tStart-100:tEnd+100)-tAx(tStart-100),...
98 YLim(tStart-100:tEnd+100));
99 xlabel(’Time (ps)’);
100 xlim([0 tAx(tEnd+100)-tAx(tStart-100)]);
101 ylabel(’Intensity (a.u.)’);
102 ylim([0 1.1]);
103 title(’Short time scale’);
104 subplot(2,2,3),plot(tAx,YLim);
105 xlim([tAx(1) tAx(end)]);
106 ylim([0 1.1]);
107 xlabel(’Time (ps)’);
108 ylabel(’Intensity (a.u.)’);
109 title(’Long time scale’);
110
111 %% Zero Line Width
112
113 % set 5 spectral lines --> setting according to FFT convention 0 to 2pi
114 spectrum = fftshift(fTry);
115 spectrum = spectrum + circshift(fftshift(fTry),[1,spacingCount]);
116 spectrum = spectrum + circshift(fftshift(fTry),[1,-spacingCount]);
117 spectrum = spectrum + circshift(fftshift(fTry),[1,spacingCount*2]);
118 spectrum = spectrum + circshift(fftshift(fTry),[1,-spacingCount*2]);
119
120 spectrum = ifftshift(spectrum);
121 %%
122 yLim = ifft(spectrum);
123 YLim = abs(yLim).^2; %Find intensity
124 YLim = YLim/max(YLim);
125
126 % tTarget = tTarget/max(tTarget);
127 figure(’Position’,[80 80 800 600]);
128 subplot(2,2,2),plot(wl,abs(fftshift(spectrum)));
129 ylabel(’Amplitude (a.u.)’);
130 xlabel(’Wavelength (nm)’);
131 xlim([1549 1551]);
132 subplot(2,2,4),plot(wl,unwrap(angle(fftshift(spectrum))));
133 ylabel(’Phase’);
134 xlabel(’Wavelength (nm)’);
135 xlim([1549 1551]);
136 subplot(2,2,1),plot(tAx(tStart-100:tEnd+100)-tAx(tStart-100),...
137 YLim(tStart-100:tEnd+100));
138 xlabel(’Time (ps)’);
139 xlim([0 tAx(tEnd+100)-tAx(tStart-100)]);
76Chapter C. MATLAB Code for Simulating Finite Linewidth with Coherence Time Model
140 ylabel(’Intensity (a.u.)’);
141 title(’Short time scale’);
142 subplot(2,2,3),plot(tAx,YLim);
143 xlim([tAx(1) tAx(end)]);
144 xlabel(’Time (ps)’);
145 ylabel(’Intensity (a.u.)’);
146 title(’Long time scale’);
147
148 %% Convolve with the zero width spectrum
149 spectrum = fftshift(spectrum);
150 out = conv(line, spectrum);
151
152 %Make sure the convoluted line still matches the zero width line
153 spectrum = out(floor(linePoint/2):end-ceil(linePoint/2));
154 spectrum = fftshift(spectrum);
155
156 %% With some line width
157 yLim = ifft(spectrum);
158 YLim = abs(yLim).^2; %Find intensity
159 YLim = YLim/max(YLim);
160
161 tTarget = tTarget/max(tTarget);
162
163 figure(’Position’,[80 80 800 600]);
164 subplot(2,2,2),plot(wl,abs(fftshift(spectrum)));
165 ylabel(’Amplitude (a.u.)’);
166 xlabel(’Wavelength (m)’);
167 xlim([1549.5 1550.5]);
168 subplot(2,2,4),plot(wl,angle(fftshift(spectrum)));
169 ylabel(’Phase’);
170 xlabel(’Wavelength (m)’);
171 xlim([1549.5 1550.5]);
172 subplot(2,2,1),plot(tAx(1:200),YLim(1:200));
173 xlabel(’Time (ps)’);
174 xlim([0 tAx(200)]);
175 ylabel(’Intensity (a.u.)’);
176 title(’Short time scale’);
177 subplot(2,2,3),plot(tAx,YLim);
178 xlim([tAx(1) tAx(end)]);
179 xlabel(’Time (ps)’);
180 ylabel(’Intensity (a.u.)’);
181 title(’Long time scale’);
Appendix D
MATLAB Code for Characterizing
Limited Bandwidth
1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
2 % codeThreeLineSweepPhase.m
3 % By Kenny Ho, Aug 31, 2007
4 %
5 % For characterizing limited bandwidth
6 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
7
8 clear all;
9 close all;
10
11 c = 299792458; % Speed of light [m]
12
13 %% Simulation Parameters
14 % Grating resolution 0.1 nm
15 % OSA resolution 0.01 nm
16 % HP8168F line width 50 - 500 MHz
17 % using df = c/lambda^2*dlam with lambda = 1550nm, 50MHz is ~ 0.4 pm
18 centreWL = 1550e-9; % Centre wavelength [m]
19 centreFreq = c/centreWL;
20
21 % set simulating spectral resoution to 0.01 pm
22 specRes = 0.01e-12;
23
24 % use bandwidth of 10 nm for high temporal resolution
25 bandwidthWL = 10e-9; %simulation bandwidth 10 nm [m]
26 %--> simulation temporal resolution of 764.5fs
27 bandwidthFreq = c/centreWL^2*bandwidthWL;
28 numPoint = bandwidthWL/specRes;
29 numPoint = 2^length(dec2bin(numPoint));%round to the cloest 2^N for FFT
30 actualSimWLBW = numPoint*specRes;%Actual simulation bandwidth after roundup
31 tempRes = centreWL^2/c/actualSimWLBW; %Simulation temporal resolution
32 tAx = linspace(0,tempRes*1e12*(numPoint-1),numPoint); % time axis in ps
33
34 % set frequency/wavelength axis --> at this point, start with lowest
35 % wavelength without fft shift so this is -pi to pi
77
78 Chapter D. MATLAB Code for Characterizing Limited Bandwidth
36 wl = linspace(0,specRes*(numPoint-1),numPoint) +centreWL - actualSimWLBW/2;
37 wl = wl.*1e9; %in nm
38 f = c./wl;
39 spectrum = zeros(1,numPoint);
40
41 repRate = 20e9; % repetition rate at 20 GHz [Hz]
42 repRateWL = repRate/c*centreWL^2;
43 spacingCount = round(repRateWL/specRes);
44
45 % set spectral lines --> setting according to FFT convention 0 to 2pi
46 % instead of -pi to pi
47 spectrum(1) = 1;
48 spectrum(1+spacingCount) = 1;
49 spectrum(end-spacingCount+1) = 1;
50
51 %% sweep phase of the three-line rectangular spectrum
52 figure(’Position’,[80 0 1540 1190]);
53 phaseCount = 7;
54 for m = 0:phaseCount
55 for n = 0:phaseCount
56 spectrum(1) = 0.5; %reference phase
57 spectrum(1+spacingCount) = 1.5*exp(i*m/phaseCount*2*pi);
58 spectrum(end-spacingCount+1) = 1*exp(i*n/phaseCount*2*pi);
59 y = abs(fft(spectrum)).^2;
60 subplot(phaseCount+1,phaseCount+1,m*(phaseCount+1)+n+1),...
61 plot(tAx(1:300),y(1:300));
62 xlim([tAx(1) tAx(300)]);
63 xlabel(’Time (ps)’);
64 %ylabel(’Amplitude (a.u.)’);
65 end
66 end
67
68 % plot with -pi to pi convention
69 figure;
70 plot(wl,abs(fftshift(spectrum)));
71 xlabel(’Wavelength (nm)’);
72 xlim([1549 1551]);
73 ylabel(’Amplitude (a.u.)’);
Appendix E
MATLAB Code for Recovering the
Phase of Measured Data
1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
2 % codeFiveLinePhaseERRealData.m
3 % By Kenny Ho, Jan 31, 2008
4 %
5 % For recovering the phase of measured data
6 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
7
8 clear all;
9 % close all;
10
11 c = 299792458; % Speed of light [m]
12
13 %% Simulation Parameters
14 % Grating resolution 0.1 nm
15 % OSA resolution 0.01 nm
16 % HP8168F line width 50 - 500 MHz
17 % using df = c/lambda^2*dlam with lambda = 1550nm, 50MHz is ~ 0.4 pm
18 centreWL = 1547.014e-9; % Centre wavelength [m]
19 centreFreq = c/centreWL;
20
21 % set simulating spectral resoution to 2 pm
22 specRes = 2e-12;
23
24 % real data time resolution 61.278 fs
25 timeRes = 61.278e-15;
26 bandwidthFreq = 1/timeRes;
27 bandwidthWL = bandwidthFreq/c*centreWL^2;
28 numPoint = 4050;
29 specRes = bandwidthWL/numPoint;
30 fa = [(centreWL-specRes*(numPoint/2)):specRes:(centreWL-specRes)];
31 fb = [centreWL:specRes:(centreWL+((numPoint/2)-1)*specRes)];
32 fi = [fa fb].*1e9; % convert to nm unit
33 %%
34 Import the csv files
35 %%
79
80Chapter E. MATLAB Code for Recovering the Phase of Measured Data
36 fTarget = interp1(specSquare5(1,:),specSquare5(2,:),fi,’nearest’,0);
37 tTarget = timeSquare5(1,:);
38
39 % normalize energy
40 fTarget = fTarget./sum(fTarget);
41 tTarget = tTarget./sum(tTarget);
42
43 %% Gerchberg-Saxton Algorithm
44
45 tTry = ifft(fTarget);
46 errorGS = zeros(1,1000);
47 %% GS loop
48 for m = 1:1000
49 tTry = abs(tTarget).*exp(i.*angle(tTry));
50 fTry = fft(tTry);
51 fTry = abs(fTarget).*exp(i.*angle(fTry));
52 tTry = ifft(fTry);
53 TTry = abs(tTry).^2;
54 TTry = TTry./sum(TTry);
55 errorGS(m) = sum(abs(TTry-tTarget).^2);
56 end
57
58 %use find(abs(fTry) > 0.035) to find peaks and then alter phase to try to
59 %kick it out of the local minimum
60 %result : 2018 2022 2026 2030 2034
61
62 %%
63 figure(2);
64 subplot(2,1,1),plot([0:timeRes:(numPoint-1)*timeRes]*1e12,...
65 (tTarget/sum(tTarget)*1.4-min(tTarget))/max(tTarget/sum(tTarget)...
66 *1.4-min(tTarget)),...
67 [0:timeRes:(numPoint-1)*timeRes]*1e12,TTry/max(TTry));
68 subplot(2,1,2);
69 [AX,H1,H2]=plotyy(fi,abs(fTry),fi,unwrap(angle(fTry))./pi.*180,’plot’);
70 set(AX(1),’XLim’,[1546 1548]);
71 set(AX(2),’XLim’,[1546 1548]);
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