Controlling Semiconductor Optical Amplifiers for Robust Integrated Photonic Signal Processing
by
Scott B. Kuntze
A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Electrical and Computer Engineering University of Toronto
c 2009 by Scott B. Kuntze Copyright
Abstract Controlling Semiconductor Optical Amplifiers for Robust Integrated Photonic Signal Processing Scott B. Kuntze Doctor of Philosophy Graduate Department of Electrical and Computer Engineering University of Toronto 2009 How can we evaluate and design integrated photonic circuit performance systematically? Can active photonic circuits be controlled for optimized performance? This work uses control theory to analyze, design, and optimize photonic integrated circuits based on versatile semiconductor optical amplifiers (SOAs). Control theory provides a mathematically robust set of tools for system analysis, design, and control. Although control theory is a rich and well-developed field, its application to the analysis and design of photonic circuits is not widespread. Following control theoretic methods already used for fibreline systems we derive three interrelated state-space models: a core photonic model, a photonic model with gain compression, and a equivalent circuit optoelectronic model. We validate each model and calibrate the gain compression model by pump–probe experiments. We then linearize the state-space models to design and analyze SOA controllers. We apply each linearized model to proof-of-concept SOA control applications such as suppressing interchannel crosstalk and regulating output power. We demonstrate the power of linearized state-space models in controller design and stability analysis. To illustrate the importance of using the complete equivalent circuit model in controller design, we demonstrate an intuitive bias-current controller that fails due to the dynamics of the intervening parasitic circuitry of the SOA. We use the linearized statespace models to map a relationship between feedback delay and controller strength for stable operation, and demonstrate that SOAs pose unusual control difficulties due to their ultrafast dynamics. Finally, we leverage the linearized models to design a novel and successful hybrid controller that uses one SOA to control another via feedback (for reliability) and feedforward (for speed) control. The feedback controller takes full advantage of the equivalent iii
iv
Abstract
circuit modelling by sampling the voltage of the controlled SOA and using the error to drive the bias current of the controller SOA. Filtering in the feedback path is specified by transfer function analysis. The feedforward design uses a novel application of the linearized models to set the controller bias points correctly. The modelling and design framework we develop is entirely general and opens the way to the robust optoelectronic control of integrated photonic circuits.
Preface: Light, the Internet, and everyone I suppose the first time I really appreciated the combination of light and control was as an adolescent watching Genesis concert videos of the 1980s and 90s with their impressive light shows—hundreds of robotic Varilites sweeping and changing colour in perfect synchronization with the musical dynamics. I had never seen anything like it and I was enthralled by the visual impact of the shows. My fascination with the Internet began a little later in 1994 when I realized I could find information and discussion on the obscure “progressive rock” music I loved as a teenager. Just waiting for me out there were heated debates, recommendations, reviews, and catalogues. Suddenly, the Internet was personal and useful for me. During the last three years I have been enthralled and fascinated by the control of another kind of light—the invisible kind that pulses through modern telecommunication circuits and that conveys high-definition concert video around the world via the Internet in the blink of an eye. My work would not have been possible with a number of key people. My Doctorate supervisors Stewart Aitchison and Lacra Pavel have been so very supportive throughout the life of this project. They allowed me to initiate this joint photonics–control project and encouraged me at every turn. Together, they complemented each other ideally as thesis advisors and I am fortunate to have found their perfect mix for my background and interests. My Masters supervisor Ted Sargent taught me to be a researcher and was so supportive over changes of direction. I owe much of my research process and work flow training to my time working with him. Along the way we had many discussions that were always helpful and encouraging. v
vi
Preface My colleagues Aaron Zilkie and Baosen Zhang helped me to push my work beyond a
couple of obstacles with their contributions. Prof. John Cartledge provided an excellent external appraisal of this dissertation and Prof. T.J. Lim offered an excellent external viewpoint on my work; both made valuable contributions to the refinement of my thesis through their questions and comments. My parents have always been completely supportive in everything I have done, this lengthy project included. And of course Julia, who offered her patience, support, more patience, understanding, and even more patience as I scrambled to hit deadlines, worked long hours into the night, and occasionally became consumed by simulations, papers, and reports. To these people and to the many other people who factored into my life in some way or another during this time, thank you—you are a part of the pages that follow.
Contents Abstract
ii
Preface
v
Contents
vii
List of Figures
xi
List of Tables
xiv
1 The promise of integrated photonics
1
1.1
The Internet is still too slow . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Current trends in photonic signal processing . . . . . . . . . . . . . . . .
3
1.2.1
Integration and the versatility of semiconductor optical amplifiers
3
1.2.2
SOA-based integrable functions . . . . . . . . . . . . . . . . . . .
5
1.2.3
True large-scale photonic integration . . . . . . . . . . . . . . . .
7
This work: a theoretical approach to photonic integrated circuit control .
7
1.3.1
The need for photonic regulation: optical amplifier control . . . .
7
1.3.2
Control theory as a tool for robust photonic design . . . . . . . .
9
1.3.3
Bringing control theory to SOA regulator design: an overview . .
13
Conclusion: robust control methods for integrated photonics . . . . . . .
17
1.3
1.4
2 Technical background
19
2.1
Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
2.2
Essential semiconductor optical amplifier physics . . . . . . . . . . . . . .
20
2.2.1
Principles of architecture and operation . . . . . . . . . . . . . . .
21
2.2.2
Carrier rate equation . . . . . . . . . . . . . . . . . . . . . . . . .
25
vii
viii
Contents 2.2.3 2.3
Optical propagation equation . . . . . . . . . . . . . . . . . . . .
26
Control theory methods . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
2.3.1
State-space realization . . . . . . . . . . . . . . . . . . . . . . . .
28
2.3.2
State-space model linearization . . . . . . . . . . . . . . . . . . .
29
2.3.3
State-space model solution . . . . . . . . . . . . . . . . . . . . . .
31
2.3.4
Linear model transfer function . . . . . . . . . . . . . . . . . . . .
32
2.3.5
Controller canonical form . . . . . . . . . . . . . . . . . . . . . .
33
2.3.6
Useful system properties . . . . . . . . . . . . . . . . . . . . . . .
34
2.3.7
State feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
2.3.8
Optimal least-squares state feedback . . . . . . . . . . . . . . . .
37
2.3.9
Output feedback . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
2.3.10 Dynamic single-input/single-output controllers . . . . . . . . . . .
39
2.3.11 Useful calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
2.3.12 Example: EDFA control formulation . . . . . . . . . . . . . . . .
40
3 Core photonic state-space model 3.1
43
Nonlinear state-space model . . . . . . . . . . . . . . . . . . . . . . . . .
44
3.1.1
Governing equations . . . . . . . . . . . . . . . . . . . . . . . . .
45
3.1.2
Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
3.1.3
Output relations . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
3.1.4
State update equation . . . . . . . . . . . . . . . . . . . . . . . .
49
3.1.5
Nonlinear control form summary . . . . . . . . . . . . . . . . . .
49
3.2
Linearized state-space model . . . . . . . . . . . . . . . . . . . . . . . . .
51
3.3
Feedback control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
3.3.1
State feedback: suppressing cross-talk electronically . . . . . . . .
55
3.3.2
State observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
3.3.3
Output Feedback: Suppressing Cross-talk Optically . . . . . . . .
59
3.4
Controlling phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
3.5
Conclusion: a new SOA state-space design framework . . . . . . . . . . .
65
4 Explicit photonic gain compression state-space model
67
4.1
Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
4.2
General state-space form . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
4.3
Solving the propagation equation with gain compression . . . . . . . . .
71
Contents
ix
4.4
State-space model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
4.5
Model verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
4.5.1
Experiment and simulation . . . . . . . . . . . . . . . . . . . . . .
76
4.5.2
Device and model parameters . . . . . . . . . . . . . . . . . . . .
78
4.5.3
Experiment–model comparison
. . . . . . . . . . . . . . . . . . .
79
4.6
Optical feedback control for constant output power . . . . . . . . . . . .
81
4.7
Effect of gain compression on required controller strength . . . . . . . . .
84
4.8
Conclusion: the first explicit input–output model with gain compression .
86
5 Equivalent circuit dynamic model 5.1
87
Nonlinear state-space equivalent circuit model . . . . . . . . . . . . . . .
88
5.1.1
89
5.1.2
State-space realization of the equivalent circuit . . . . . . . . . . . ¯ . . . . . . . . . . . . . . . . . . . . SOA active region current I(t)
5.1.3
SOA current–voltage relationship . . . . . . . . . . . . . . . . . .
92
5.1.4
Nonlinear equivalent-circuit space-space model . . . . . . . . . . .
92
5.2
Linearized state-space equivalent circuit model . . . . . . . . . . . . . . .
95
5.3
Conclusion: complete optoelectronic SOA state-space description . . . . .
99
6 Impact of feedback delay on closed-loop stability
91
101
6.1
State feedback into the drive current and system stability . . . . . . . . . 102
6.2
The delay margin for feedback stability . . . . . . . . . . . . . . . . . . . 104 6.2.1
Least-squares optimal control . . . . . . . . . . . . . . . . . . . . 105
6.2.2
Delay margin of the feedback controller . . . . . . . . . . . . . . . 107
6.3
Hybrid feedforward–feedback controller . . . . . . . . . . . . . . . . . . . 113
6.4
Conclusion: feedback delay constraints are severe for SOAs . . . . . . . . 118
7 Incoherent optoelectronic control of a semiconductor optical amplifier121 7.1
7.2
7.3
Controller architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 7.1.1
Principle of operation . . . . . . . . . . . . . . . . . . . . . . . . . 122
7.1.2
Device models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
Robust design using state-space modelling . . . . . . . . . . . . . . . . . 125 7.2.1
Preliminary system considerations . . . . . . . . . . . . . . . . . . 125
7.2.2
Feedforward design . . . . . . . . . . . . . . . . . . . . . . . . . . 126
7.2.3
Feedback design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
Design verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
x
Contents 7.4
Conclusion: a novel SOA-based SOA controller and design techniques . . 133
8 Contributions and conclusions
135
8.1
Original contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
8.2
Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
8.3
Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
A Single-input–single-output output feedback
141
A.1 Output relation and controlled input . . . . . . . . . . . . . . . . . . . . 141 A.2 State update equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 A.3 Closed-loop transfer function . . . . . . . . . . . . . . . . . . . . . . . . . 143 B State observer (estimator) design
145
B.1 Observer system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 B.2 Observer-based feedback control . . . . . . . . . . . . . . . . . . . . . . . 147 C Methods for deriving general state-space models
149
C.1 Two core photonic states . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 C.1.1 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 C.1.2 Equilibrium points: lasing threshold condition . . . . . . . . . . . 151 C.1.3 Linear gain conclusion . . . . . . . . . . . . . . . . . . . . . . . . 152 C.2 Gain generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 C.2.1 gi N(z, t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
C.3 Separable N, N(z, t) = Nz (z)Nt (t) . . . . . . . . . . . . . . . . . . . . . 154 C.4 Finite difference decomposition in space
. . . . . . . . . . . . . . . . . . 155
C.4.1 Propagation equation . . . . . . . . . . . . . . . . . . . . . . . . . 156 C.4.2 Carrier rate equation . . . . . . . . . . . . . . . . . . . . . . . . . 157 C.4.3 Optical power rate equation . . . . . . . . . . . . . . . . . . . . . 159 C.4.4 Finite difference conclusion . . . . . . . . . . . . . . . . . . . . . . 159 D Equivalent circuit model derivatives
161
Bibliography
163
List of Figures 1.1
Microscopic views of a ridge waveguide SOA . . . . . . . . . . . . . . . .
4
1.2
Number of papers published per year on SOAs and EDFAs . . . . . . . .
4
1.3
Example integrated photonic circuit featuring SOAs . . . . . . . . . . . .
5
1.4
Simple controller: measure the output power and adjust the bias current
8
1.5
Control theory process . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
2.1
SOA physical layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
2.2
Stimulated emission inside a semiconductor optical amplifier . . . . . . .
22
2.3
SOA active region architecture . . . . . . . . . . . . . . . . . . . . . . . .
23
2.4
Incremental model of optical gain . . . . . . . . . . . . . . . . . . . . . .
26
2.5
Linearization D of a scalar nonlinear state function . . . . . . . . . . . .
29
2.6
Block diagram of the linear system (A, B, C, D) in the Laplace domain .
32
3.1
Input–output control model of an SOA . . . . . . . . . . . . . . . . . . .
44
3.2
Using piecewise-linear gain segments to approximate nonlinear gain . . .
47
3.3
Linear model (A, b, c, D) of a semiconductor optical amplifier . . . . . .
51
3.4
Verification of the linear model . . . . . . . . . . . . . . . . . . . . . . .
54
3.5
Constant negative state feedback applied to the electrical input . . . . .
57
3.6
State observer relative error . . . . . . . . . . . . . . . . . . . . . . . . .
59
3.7
State observer relative error with estimated state feedback . . . . . . . .
60
3.8
Optical constant output feedback . . . . . . . . . . . . . . . . . . . . . .
62
3.9
Phase estimator circuit to control output phase without direct phase measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
4.1
Comparison of gain compression forms . . . . . . . . . . . . . . . . . . .
71
4.2
Pump–probe experiments
77
. . . . . . . . . . . . . . . . . . . . . . . . . . xi
xii
List of Figures 4.3
Experimental and simulated gain curves of a multi-quantum-well SOA . .
4.4
Experimental and simulated pump–probe responses of a multi-quantum-
79
well SOA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
4.5
Comparison of compressed and noncompressed state-space models . . . .
81
4.6
Feedback controller using total input–output optical powers . . . . . . .
82
4.7
Regulation of the total output power . . . . . . . . . . . . . . . . . . . .
83
4.8
Effect of compression on required controller strength
. . . . . . . . . . .
85
5.1
Electronic model signal flow . . . . . . . . . . . . . . . . . . . . . . . . .
88
5.2
SOA model with parasitic network . . . . . . . . . . . . . . . . . . . . .
88
5.3
Comparison of open-loop system responses . . . . . . . . . . . . . . . . .
94
6.1
Phase portraits for state feedback of N(t) into Is (t) . . . . . . . . . . . . 103
6.2
Optimal control schematic . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.3
Flow diagram for the Constant Matrices Test
6.4
Algorithm for generating many sets of τsup and K . . . . . . . . . . . . . 110
6.5
(q, r) ∈ (Q, R) map for kKkF calculation . . . . . . . . . . . . . . . . . . 111
6.6
Time delay margin analysis . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.7
Hybrid feedforward/feedback control design . . . . . . . . . . . . . . . . 113
6.8
Example implementation of a feedback integrator . . . . . . . . . . . . . 114
6.9
Comparison of feedforward, feedback, and hybrid controllers . . . . . . . 116
. . . . . . . . . . . . . . . 109
6.10 Eye diagrams of aggressor and victim channels under various control schemes117 7.1
All-optical SOA control architecture using incoherent feedback . . . . . . 123
7.2
Contour map showing values of design parameter µ against bias . . . . . 128
7.3
Contour maps of design parameter µ for varying input powers . . . . . . 129
7.4
Envelope response of the SOA-controlled SOA . . . . . . . . . . . . . . . 131
7.5
Step response of controller SOA for delay . . . . . . . . . . . . . . . . . . 132
7.6
Eye diagrams for 28 − 1 PRBS sequences at 10 Gb/s for SOA-controlled SOA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
A.1 SISO PID implementation for constant power. . . . . . . . . . . . . . . . 141 B.1 Simple full state observer . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 B.2 Full state observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 B.3 Constant estimated-state feedback with full state observer . . . . . . . . 148
List of Figures
xiii
C.1 Finite difference space mesh. . . . . . . . . . . . . . . . . . . . . . . . . . 156
List of Tables 2.1
Physical constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
3.1
Parameters for core photonic model simulations . . . . . . . . . . . . . .
52
4.1
Parameter definitions and values from qualifying compression model . . .
69
5.1
Parameter definitions and values for electronic model simulations . . . .
90
xiv
Chapter 1 The promise of integrated photonics 1.1
The Internet is still too slow
Despite the economic downturn of the telecommunications industry nearly a decade ago, bandwidth demand has continued to grow at ever increasing rates [1–3]. Media of all types are produced by corporations and users at astonishing rates, and are devoured in downloading frenzies. With the recent conversion of audio–visual media to high-definition formats like Blu-Ray and HDTV, bandwidth consumption can only continue to explode. Even with modern efficient filesharing schemes like BitTorrent [4], it can take several days to download a DVD’s worth of data with premiere consumer services costing $600 CDN per year. Beyond home consumers, businesses are now tied to the Internet and prosper through their electronic transactions whether or not their business models are actually Internet-centric. And with gigantic science experiments like the Large Hadron Collider coming online that produce petabytes of data [5–7], the current Internet is simply too slow. Meanwhile, miniature laptops, iPhones and Blackberries are driving a wireless revolution. These modern personal data agents (known commonly as PDAs) and cellular phones are increasingly thin clients, compact computing devices in size and memory that store only the essential data and software onboard, relying on fast transfers to remote repositories to access bulk data. Personal information—work, electronic books, digital media, etc.—is distributed and accessible anywhere, anytime without lugging around the storage. But the wireless link is just a last-few-kilometers appendage that needs a sturdy backbone in a photonic Internet. 1
2
Section 1.2 The Internet is still too slow While foundational optical fibres and ultrafast lasers enable THz bandwidth, at its
core the Internet is limited by its lack of transparency and agility. In order for data to get from one subnetwork to the next, or even just for the data to be directed towards its correct destination at an intervening node, optical–electronic–optical (OEO) conversion takes place because fully optical switching has not matured. Thus, the modern Internet lacks transparency, and the intervening opaque OEO nodes impose delays, curtail bandwidth, and consume extraordinary power. Recently there have been remarkable advances towards electronic signal processing that fully inverts the optical impairments of a fibre link [8–11], both backward [11] and forward [8] correcting. Still, these advances do not offer improvements towards agility in the current Internet. If more bandwidth is needed along a particular link, often it must be scheduled long (days, weeks) in advance and a human technician must go to a field location and physically reconfigure fibres. If a fibre is cut by a backhoe, the entire network could be vulnerable and humans must again visit multiple sites to reroute data flow. It is as if there are still human operators at the ready to plug one caller into the next. Hence, the modern Internet lacks true agility in its ability to reconfigure itself at timescales 103 or even 106 times the typical data rate of GHz (25 ms is often viewed as the threshold for switching “agility,” some 109 /(1/[25 × 10−3 ]) = 25 × 106 times slower than modest 1 GHz data). With the quantity and importance of transactions occurring across global networks, transparency and agility are no longer secondary or tertiary considerations in network design. Electronic advances towards real-time impairment correction and progress towards keeping up with Moore’s Law aside, the greatest bandwidth potential is still firmly in the optical domain. The world’s fastest electronic signal processors work at gigabits per second—what happens as data rates climb through thresholds of tens and hundreds of gigabits per second? At some point in the near future, pure electronic signal processing will become plainly inadequate and fully photonic signal processing [12] will be needed. If we are going to tap the optical domain’s full potential, our photonic signal processing capabilities must improve greatly. Photonic signal processing also offers considerable performance advantages for ultrashort intrachip connections [13] within electronic-domain circuits. Current trends in photonic signal processing are pushing towards active photonic devices that can provide the wide array of agile and transparent functionality needed in next-generation wavelength division multiplexed (WDM) systems at 100 Gb/s and well beyond.
Chapter 1 The promise of integrated photonics
1.2
3
Current trends in photonic signal processing
This present work began as the Massachussetts Institute for Technology (MIT) finished its first Communications Technology Roadmap (CTR) with the aid of leading research institutions, components and systems manufacturers [14]. To quote two of the key summary points from the report, “This [electronic–photonic convergence] direction is triggering a major shift in the leadership of the component industry from information transmission (telecom) to information processing (computing, imaging),” and “Planar integration will drive cost reduction.” Throughout, MIT’s roadmap points towards integrated photonic signal processing as the solution to overcoming today’s telecommunication and computing limitations. A particular roadmap within the CTR for 20 years in the future suggests that all-optical computing is a feasible and necessary goal [14]. The CTR is just as relevant today and is currently forming the basis for a second roadmap in progress at MIT. Below we sketch some of the trends and architectures over the last decade that form a foundation for future progress towards fully integrated photonic signal processing, and ultimately complete transparency and agility.
1.2.1
Integration and the versatility of semiconductor optical amplifiers
Erbium-doped fibre amplifiers (EDFAs), while commercially popular for photonic signal processing since the mid 1990s [15], suffer from several drawbacks for functional photonic signal processing. They have large size (roughly 10 cm diameter) due to a minimum fibre bending radius and are therefore non-integrable. EDFAs have slow gain recovery and low nonlinearity [15] and are therefore unsuitable for nonlinear switching (although very good for linear amplification). Other fibre amplifiers such as Raman [16] and parametric [17] amplifiers also suffer from large footprints due to fibre bending limits. In contrast, semiconductor optical amplifiers (SOAs) are active photonic devices that can be monolithically integrated to provide complex photonic functions such as signal amplification [19], regeneration [20], switching [21], and frequency conversion [22]. Although not a perfect photonic analogue to the electronic transistor, the SOA is a leading candidate for integrated photonic circuitry because it is so versatile. An SOA is a semiconductor diode laser coated with antireflection facets so that signals enter and exit
4
Section 1.2 Current trends in photonic signal processing
Figure 1.1: Microscopic views of a ridge waveguide diode laser or semiconductor optical amplifier [18]. Left: top-down view with the ridge waveguide running up–down. Right: edge-on view with the ridge waveguide indicated by the arrow. 300 SLA or SOA EDFA
number of papers
250
200
150
100
50
0 1980
1985
1990
1995
2000
2005
year
Figure 1.2: Number of papers published per year on SOAs (search term: (semiconductor laser amplifier) OR (semiconductor optical amplifier)) and EDFAs (search term: erbium doped (fibre OR fiber) amplifier) as indexed by Thompson ISI Web of Science, http://www.isiwebofknowledge.com/. the gain region relatively unimpeded; Fig. 1.1 shows a ridge waveguide diode laser that could be AR-coated to form a SOA.1 Depending on the electronic bias and power of the optical signal, linear amplification (through stimulated emission) [19, 25] and phase change (through gain saturation) [26, 27] are possible, two key properties for larger-scale functionality. Fig. 1.2 shows there is a continuing significant research effort driving SOA development and application. SOAs are small and may be fully integrated into monolithic waveguide-based photonic circuits, and form the basis of contemporary active photonic circuitry [28–30]. Figure 1.3 1
SOAs could just as easily be made from other diode laser architectures such as buried heterostructure [23] or lateral current injection [24] structures.
Chapter 1 The promise of integrated photonics
5 cw
linear
FWM
data
ctrl
XGM data
TOAD
data Figure 1.3: Example integrated photonic circuit featuring SOAs. Incoming data may be amplified, wavelength-converted through four-wave-mixing (FWM) or cross-gain modulation (XGM), then switched by a terahertz optical asymmetric demultiplexer (TOAD), all realized by SOAs. is a schematic illustrating several SOA-based function and suggests how several applications could be integrated together. Working clockwise from the top left corner, data attenuated from propagation enters an SOA biased in a linear amplification regime and is rejuvenated. In a second SOA, the nonlinear process of four-wave-mixing (FWM) converts the centre wavelength from one channel to another in a larger WDM context. The channel could be converted again through cross-gain modulation (XGM) where an SOA is saturated so that the data imprints its inverse on a continuous wave (cw) channel of another colour. The data—or rather its inverse (but the original data is recoverable just the same)—is finally guided via some waveguides into an interferometric terahertz optical asymmetric demultiplexing (TOAD) switch, which switches the data according to the timing of counterpropagating control pulses (ctrl). Although we might not ever design this particular circuit, photonic circuits containing these functions, other functions, or combinations thereof have been demonstrated in the literature [19–22, 25–63].
1.2.2
SOA-based integrable functions
Architectures have been presented for digital functions using interferometric SOA switches, including shift registers employing SOA feedback [31], all-optical self-starting clocks [32], XOR gates [33], and star topologies of terahertz optical asymmetric demultiplexers (TOADs) for fast timeslot demultiplexing [34]. Detailed theory and performance analysis for SOA–MZI (Mach-Zehnder interferometer) XOR gates [35], AND/OR/XOR
6
Section 1.2 Current trends in photonic signal processing
gates [36], time-domain rate equations [37] and SOA gain and phase dynamic effects [27] for SLALOM switches (including switch transfer functions), and the response of an SOA TOAD [26, 38] have all been considered. A nonlinear optical loop mirror has been proposed for simultaneous λ-conversion and demultiplexing at 40 Gb/s [39]. Detailed theory of noise and SNR in TOADs has also been published [40]. General detailed switching analysis including detailed physical SOA theory and cascaded SNR / switching time analysis, along with architectures of crossbar switches have been reported [41]. The impact of SOA switch noise and cross-talk has been evaluated in terms of system BER [42] and optical burst switching (OBS) node throughput [43]. Experimental results have been reported for comparison of TOAD, and forward and reverse-propagating MZI SOA/fibre-based switches [44], as well MZI switches with femtosecond switching [21, 45–47]. All-optical regeneration consists of 2R (re-amplification, reshaping) or 3R (when retiming is added) functionality where the signal remains entirely in the optical domain. High speed all-optical regeneration has been described in theory and architecture with comparison to OEO [48]. Recently, 40 Gb/s 3R regeneration using cascaded SOAs integrated into MZIs has been reported [49]. A regenerator architecture has been deployed in which optical clock and pump wavelength are triggered off delayed degraded data in SOA [50] regenerated data is imprinted on the pump and filtered. A complex two-section DFB laser architecture with SOA-MZI has also given rise to 3R all-optical regeneration [51]; placement within a WDM link is also discussed. SOA-based threshold detectors based on cross-connected SOAs have been built with some theoretical consideration [52]. Further 2R/3R regenerators have been proposed using SOA-based MZIs [20] (with λ-conversion too) and featuring low-polarization-sensitive 2R regeneration [53]. Performance between 2R and 3R schemes for cascaded amplifier all-optical systems have been evaluated [54]. For wavelength conversion, all-optical interferometer architectures using EDFAs and SOAs have been compared for performance experimentally [55]. There exists theory and experimental confirmation of wavelength conversion including power penalty analysis [56]. Detailed theory on chirp and extinction ratio for wavelength-converted signals has been developed [57]. Heuristic optimization analysis for conversion has been used [58]. λswitching SOA architectures have been reviewed comprehensively [59]. In order to select the resulting channels from wavelength conversion, tunable filter realization is possible using integrable microring resonators [60]. Chromatic dispersion among wavelengths has also been compensated with SOAs [61, 62].
Chapter 1 The promise of integrated photonics
1.2.3
7
True large-scale photonic integration
Photonics must become truly integrated—passive with active at a large scale—if it is ever going to mature to large-scale signal processing and computing. The functional blocks needed for photonics signal processing have been demonstrated in the lab as noted above, but large-scale photonic integration has yet to be realized. As of present, there are no commercially-available computer chips that run entirely on light, or that process information digitally and transparently from end-to-end entirely in the optical domain. Researchers have built planar lightwave circuits for wavelength selection using densely packed SOAs with AWGs [63]. Integration can also occur vertically through vertical stack integration of SOAs and ring resonators in InGaAsP where control of resonator quality factor and coupling is possible via tunable loss in the SOA [30]. Additionally, Integrated MZI TOADs [29] and demultiplexers (80 channels) [28] have been demonstrated. Yet, as photonic components become integrated and linked together in ever increasingly sophisticated ways, we must address the problem of signal integrity. How do we guarantee that functional Block A feeds Block B with properly formatted inputs? What if the operating parameters of Block A change? Can Block B’s output still be guaranteed? Will all downstream performance suffer? Can we prevent problems anywhere in the chain from propagating down the line? For SOA-based photonic circuits, there is no systematic control model in the literature, no robust control scheme, no systematic optimization of noise, device size, and device power. We will see that none of the works on SOA control documents the control theory employed nor evaluates the higher-level impacts on overall system performance. Thus there is a gap between SOA-based circuit design and robust design, and overall performance presently comes without a theoretical guarantee.
1.3
This work: a theoretical approach to photonic integrated circuit control
1.3.1
The need for photonic regulation: optical amplifier control
As active photonic circuits become more complex, it becomes increasingly important to regulate their behaviour for the circuits and receivers that follow downstream. In response, there is a growing collection of SOA control designs in the literature particularly
8
Section 1.3 This work: a theoretical approach to photonic integrated circuit control SOA data in
tap
data out
bias electronic controller Figure 1.4: Simple controller: measure the output power and adjust the bias current. in the last year or so [64–67]. The central problem, however, is that these designs rely on physical intuition and heuristics. While the underlying physics may seem sound, there are no robust evaluations of system parameters such as controllability (i.e. does the control really do as desired under all operating conditions) or stability (i.e. does the controlled system remain in a stable operating regime). The following example demonstrates that intuition is not always sufficient beyond the simplest control cases. Consider a simple control scheme to maintain a constant carrier density of an SOA: we sample the output power, compare that value to the desired value, and then adjust the SOA’s bias current accordingly as illustrated in Fig. 1.4. Initially, this seems to be a feasible control scheme and has even been demonstrated experimentally [68]. However, we can make simple intuitive counter-arguments as to why this particular controller might not be a good choice. If a channel is added to the input, the carrier density will decrease in response to the increased stimulated emission, but the net effect at the output is not clear—the overall output could go up due to the added power of the channel if there are few channels, or it could go down if there are many channels experiencing the reduction in carrier density. How should the controller respond? Moreover, consider the case where the net output power goes up, and suppose the controller is designed to supply more bias current under the assumption that increased output power means that a channel has been added and so more carriers are required. The addition of more carriers through injection increases the rates of stimulated emission across all the channels, and thus the output power climbs as a result of this controller action. The controller sees the climb in output power, and continues to deliver more current. It should be clear by now that this is a runaway process. When EDFAs were chained together into longer amplifier sequences during the late 1990s, there were a variety of regulation problems that drew the field control theory
Chapter 1 The promise of integrated photonics
9
into the design process [69]—heuristic methods could not solve all the issues. Therefore, when SOAs are assembled into larger systems, we should expect that robust control will be needed. While the engineering literature is rich in the control theory of fibre amplifiers [69–74]—it is clear that optical amplifier control is an important issue—there is no comprehensive and robust approach towards control of integrated systems.
1.3.2
Control theory as a tool for robust photonic design
We respond to the absence of control theory for integrated photonic circuits by providing the necessary models and control framework in this dissertation. We evaluate active integrated photonic circuits needed for photonic signal processing from a rigorous systems perspective. To realize robust design for photonic signal processing, we apply linear and nonlinear control methods to active photonic integrated circuits. Specifically, this research addresses state-space modelling, analysis and design of versatile semiconductor optical amplifiers. The state-space methods employed are entirely general and can be applied to design more sophisticated controllers and functions using active photonic circuitry. There are already many models available that describe SOA dynamics and applications. Detailed general physical models for quantum well (QW) SOAs have been documented [75] applied to device simulation [76] and MZI simulation [77]. There has been experimental investigation and comparison of SOAs having different parameters for linear amplification at 40 Gb/s [25]. Gain dynamics of QW SOAs have been modelled analytically. Impulse response simulations have been reported [78, 79] with experimental comparison [80] and with concatenations of SOAs for dynamic analysis [81]. The impact of fast gain dynamics have been studied in TOAD [26] and SLALOM [27] switch architectures with switch transfer functions; also noted are effects on input dynamic range [82]. Specific time-domain physical models have been developed for the modulation of an SOA [83], for general all-optical switching [21, 84], for an all-optical XOR gate (MZI) switch [85], and for a SLALOM architecture switch [37]. A transfer matrix theory has been developed for distributed feedback lasers cascaded with a MZI [86]. Noise has been considered in terms of dynamic noise [87, 88], noise figures [87], practical noise of an optical amplifier [89], and ASE noise performance has been simulated [90]. Models specific to wavelength conversion have also been developed for QW SOAs. In particular, a FWM model has been created for continuous-wave and ultrafast pulses for switching,
10
Section 1.3 This work: a theoretical approach to photonic integrated circuit control
converting, sampling, with noise analysis from ASE [22]. Detailed theory on chirp and extinction ratio for wavelength-converted signals has been presented [57]. Very highly detailed general physical models of quantum dot (QD) SOAs exist as well [91,92], including those for gain dynamics [93] and cross-gain wavelength conversion [94]. For control design and analysis, none of models listed above would be straightforward to use. Contemporary control theory is largely built around state-space models having a particular form that we will define in the very next chapter. A number of important system quantities may be derived directly from this state-space form or from its linearized counterpart without time-domain simulation—equilibria, controllability, stability and stability margin, transfer function, impulse response, etc. Such quantities are more difficult to coax out of multi-celled or implicit models, or even so-called “control models” that are described by intractable SPICE modelling [64]. More importantly, most controller design techniques depend on having the state-space form readily available. Once we have a qualified SOA state-space model, we can look to the many studies of EDFA control as guides. For example, the cross-talk penalty near saturation has been evaluated in theory and in experiment [95] . EDFA control and transient suppression has been modelled dynamically for metro rings [96] and for bursty traffic [97], the latter implemented via optical–electronic envelope detection and feedback [97]. Architectures of all-optical feedback control strategies for EDFA cascades, including dynamic all-optical automatic gain control in multichannel EDFAs, have been documented [98–103]. Theory has been developed for all-optical feedback EDFA control [104, 105]. There has been experimental comparison of pump control with co- and counter-propagating optical signals [106]. Some hybrid optical feedback structures have been employed, including acousto-optical modulator optical feedback [107], and electronic feedforward with optical feedback control [108]. Multichannel cascaded control for constant gain in saturated EDFAs has been modelled and simulated [109]. Multichannel equalization has been achieved with integrated MZ variable gain slope controllers for spectral equalization (even during power transitions) [110], using thermo-optic heaters to achieve phase shift in the MZ. Another approach achieves gain stabilization in the EDFA using a gain-clamped SOA architecture [111]. Gain stabilization of EDFAs in isolation and in generalized cascaded networks has been established [72, 112]. SNR has been optimized rigorously using noncooperative game theory [113] and central cost [114] approaches. So it is clear that there is a lot of existing control schemes on which to base future SOA control. None of the above is to suggest that controlling SOAs is a completely untapped area
Chapter 1 The promise of integrated photonics
11
of research. For electronic control, there are several control schemes that monitor changes in the DC bias current at RF frequencies and compensate via electronic bias accordingly [115–117]; monolithic integration of such a controller has been demonstrated [117]. One reported optoelectronic scheme uses control electronics with an SOA to equalize packetto-packet power variations in optical 3R regeneration: incoming power is tapped and sent to a controller which drives a pump laser electronically back into the SOA optically [116]. Other schemes use more intuitive post-plant optoelectronic monitoring and electronic feedback [68, 118]. It is also possible to use an external out-of-band laser source to clamp SOA gain and therefore reduce in-band cross-talk via optoelectronic feedback [119]. Control architectures specific to nonlinear applications have been investigated: constant output power in an SOA-based crosspoint switch [120] architecture of control structure, as well as constant power packet-to-packet using SOA in saturation (for switching or regeneration) [121]. Architectures have been examined for SOAs employed as linear preamplifiers using subcarrier multiplexing [122] and compared experimentally at 40 Gb/s [25]. Use of SOAs in cable television links as preamplifiers has been documented with some theory [19]. An all-optical feedback loop for constant SOA output power across 50 wavelengths has been demonstrated [123, 124]. Yet how can we evaluate active photonic circuit performance systematically? Can active photonic circuits be controlled for optimized performance? Indeed, control theory provides a mathematically robust set of tools for system design, analysis, and control. Although control theory is a rich and well-developed field, its application to the analysis, design, and management of integrated photonics is very sparse, including in the architectures and models noted above. It can be applied at nearly any point in the system hierarchy, whether at the device, subsystem, link, or overall network level. The essential component in robust control is feedback: somewhere in the system parameters are sampled, filtered in some useful way, and then input elsewhere in the system such that a control objective (such as operating stably with maximum SNR) is met. Control theory is ideally suited to solving regulation problems because it determines the inputs into the SOA (bias current and optical power) needed to produce the desired outputs downstream. As illustrated in Fig. 1.5, governing physical rate equations of photonic circuit components could be inputs into control theory framework; this framework can then generate robust solutions for control and optimization of the components and combinations thereof. As the literature review above demonstrates, such systematic analysis has not been reported to date. Rather, heuristic and device-level analysis
12
Section 1.3 This work: a theoretical approach to photonic integrated circuit control Physical domain
Mathematical domain
physical equations
control formulation control methods
physical interpretation
abstract results
physical realization Figure 1.5: Control theory process used to model, control and optimize physical systems. and optimization have been performed; a systematic approach—one offered by analytical control theory—is lacking. Reducing a (z, t) model to a t-dependent model is described in the literature when modelling input–output behaviour [125]. Ideally, every SOA model would include full (x, y, z) information to model effects like current spreading in the transverse direction [126]. However, input–output state-space models are particularly useful for SOA control design [127] as well as rapid system analysis [128] even if some approximations must be made for the sake of obtaining analytical closed-form expressions. Sometimes, these simplified input–output models are essential. Consider the use of ideal and input/output models of operational amplifiers: in a large circuit, simplifying op-amps is common for design and analysis; certainly some fine detail is lost, but ease-of-use and fast computation are gained. For our purposes, ease-of-use is key, because ultimately we seek a linear timeinvariant model for control design and analysis. State-space models are easily linearized to produce linear time-invariant systems for which there is a rich field of analysis and design tools (see Ref. [129] for example). The resulting controllers can be used to regulate and groom the output of single SOAs or complex photonic circuits. Furthermore, state-space models consist of sets of first-order ordinary differential equations, so system simulation is fast compared to finite-element models [128]. Moreover, once the state-space model is linearized, its eigenvalues are easily computed and indicate system performance and stability before any simulations
Chapter 1 The promise of integrated photonics
13
or experiments are run, further accelerating the design of closed-loop regulators.
1.3.3
Bringing control theory to SOA regulator design: an overview
While control theory could provide a powerful set of analysis and design tools for photonic devices, its application to integrated SOAs is not yet widespread despite having been employed extensively for fibreline amplifier systems [73, 100, 130]. And although SOAs have been used in experimental systems with empirical optical [124, 131] and electronic [68, 115–119] feedback control schemes, no analytical framework has been presented to date. Having a state-space control model for SOAs would greatly facilitate design and analysis of integrated photonic control systems. Following robust control theoretic methods already used for fibreline systems (optical fibre and EDFAs), [69, 72, 112, 114], the present work has the following seven goals: 1. Derive a analytical, closed-form, explicit, state-space dynamic model of a semiconductor optical amplifier that is as physically accurate as possible. The model should include both optical and electronic dynamics to the fullest extent. The model should be verified by experiment or by qualified reports already in the literature, or both. 2. Linearize the state-space model to form a linear time-invariant control model. This linearized model will be the essential component to robust SOA control design and analysis. 3. Using the SOA control model, examine the important system properties with regards to control design (e.g. stability, controllability). 4. Demonstrate and evaluate state feedback control of an SOA under small and large signal disturbances (e.g. channel add/drop), including crosstalk suppression. 5. Demonstrate and evaluate output feedback control of an SOA under small and large signal disturbances, including crosstalk suppression and output power regulation. 6. Determine how feedback delays affect stability with regards to the SOA’s ultrashort (sub-nanosecond) dynamics. 7. Design and demonstrate controllers that leverage the SOA’s ultrafast response.
14
Section 1.3 This work: a theoretical approach to photonic integrated circuit control With these goals in mind, in the next chapter (Chapter 2), we lay the theoretical
foundations for the largely analytical work that follows. We introduce the governing equations for SOAs, and describe their dynamic behaviour. We also introduce much of the control theory that is used repeatedly. Chapters 3–5 contain the central model development and evaluation of Goals 1–5 above. We break Goal 1 down into three primary parts: development of a core photonic model, addition of gain compression to the core model, and finally addition of electronic dynamics, each addressed by a separate chapter. Goals 2–5 are interspersed throughout these chapters as proof-of-concept applications. In Chapter 3, we derive a state-space model for a multi-quantum-well semiconductor optical amplifier from the governing equations presented in Chapter 2. The governing equations for a two-level SOA consist of an optical power propagation equation and an electronic carrier rate equation. From the propagation equation we obtain an input/output relation for the optical channels, while the carrier rate equation is manipulated algebraically into a system state equation where the state is the average population inversion along the length of the device. In our attempts to produce this state-space model we find that the form of the optical gain matters a great deal for whether or not a suitable closed-form, explicit model can be derived; we evaluate the various gain formulations and explore the limitations. In particular, we temporarily leave optical gain compression aside so that we can begin control analysis and design with a simpler, more tractable model (there is precedent in the literature for ignoring gain compression [132]). Linearizing the output and state equations about preset input conditions yields a linear model of the SOA. This linear model is verified against the nonlinear equations under various simulated input conditions. Stability and controllability characteristics are analyzed analytically and numerically, and the SOA is shown to be both internally stable and controllable. We then use the linear model to design cross-talk suppressing systems in multichannel SOAs using electronic state and optical output feedback loops. Finally, we will show that optical phase is easily incorporated into the model for both control and indirect measurement. The work in this chapter opens the way for further state-space development. Gain compression is a strong nonlinear effect that suppresses the amplification dynamics and limits the maximum gain available [133, 134]. Nonlinear gain compression is an important factor in applications such as asymmetric demultiplexers [26], loop mirrors [135], cross-gain-modulation wavelength converters [136], logic gates [137], and bit
Chapter 1 The promise of integrated photonics
15
shapers [138]. In Chapter 4, we extend the SOA state-space model [127,128] of Chapter 3 to include nonlinear gain compression to account for spectral hole burning and carrier heating. We show that a multichannel closed-form state-space model is possible with polynomial gain compression and position-independent carrier density. The compressed model is significantly more accurate than existing noncompressed models up to 100 GHz as demonstrated by comparison with pump–probe experiments on a ridge waveguide QW SOA: response magnitude and recovery time are more accurately modelled with explicit gain compression. We then apply the model to design an optical feedback controller that regulates the total optical power at the output of a gain-compressed SOA and satisfies Goal 5 above. Finally, we compare the requirements of a controller between compressed and noncompressed SOAs using analytical and numerical means, and find that compression affects electrical and optical controllers differently. Typical feedback controllers need a comparator for reference. Phase is variable out of the SOA due to chirp, so despite optical integrators, differentiators, and gain elements (such as other SOAs), it is likely that some electronics will be needed. For example, electronic feedback has been used to control an SOA in 3R regeneration schemes [116]. There are further possible control advantages to using SOA electronic dynamics. A. Wonfor and colleagues have found [115] that an RF current modulation on the bias is a nonlinear, monotonic function of average optical output power; the RF modulation is independent of DC bias over typical operating ranges. As they suggest, this RF modulation signal can then be used as a straightforward controller input. M. L. Majewski and colleagues suggest [117] that the SOA can be modelled as a photodiode due to its light absorption. With this viewpoint, the effect of state depletion due to spontaneous emission is not of concern, but rather the effect of state replenishment due to spontaneous absorption. Absorption is a function of input (or output) optical power and bias current, so the linear system interconnections from our model above could be rewritten to provide optical power signals to an electronic model. The surrounding electronic circuit can introduce significant dynamics [139] that could affect control schemes that modulate the bias current directly. If these circuit dynamics are not considered adequately, the SOA control design could fail. Furthermore, in purelyphotonic models the current drawn by the SOA is fixed and unaffected by changes in carrier depletion due to variations in input optical power; and such changes could be used in a feedback circuit [115] or to monitor amplifier performance. Thus, in Chapter 5 we extend the SOA state-space model of Chapter 4 to include equivalent circuit parasitics for
16
Section 1.3 This work: a theoretical approach to photonic integrated circuit control
bias current control and nonlinear gain compression. The first task is to capture a statespace model in terms of the circuit’s linear components by relating the SOA’s current to the optical powers incident on the SOA. We obtain the complete nonlinear state-space model by rewriting the resulting equations as coupled ordinary differential equations. Our model features dynamic impedance changes in the SOA caused by stimulated emission of the optical inputs, so that optically-induced changes in the measurable SOA current may be used as inputs to a controller. To illustrate an application of this model, we leverage the SOA’s front-end dynamic impedance to design a feedback controller that measures the source voltage and drives an optical control channel. The controller is applied to the full nonlinear SOA model to regulate optical output power. Our examples use equivalent circuits commonly cited in the literature for packaging [139–141] and for laser diode photodetection [117, 132, 142]. While it is impossible to generalize every possible equivalent circuit from every possible physical realization, our examples do illustrate plausible design issues and elucidate possible design flaws. More importantly, our interconnection framework and design methods are entirely general and can be used with any linear or linearized equivalent circuit model. The remaining Goals 6 and 7 are addressed in Chapters 6 and 7, and the models and concepts developed in the earlier work are applied to deeper analysis and design. For applications driving the bias current as a control input, the current seen by the SOA’s active region may not achieve the bias current instantaneously, or at all. Systematically analyzing the electronic control reveals interesting interplay between the electronics and optics that affects closed-loop stability. In fact, in Chapter 6 we will show that the delay through a front-end parasitic network can destabilize an SOA under the controllers we design in the previous chapters. It turns out that controlling a SOA poses a special challenge: because the SOA responds so quickly at a nanosecond timescale, a feedback controller must respond at a much faster timescale. Delays in the feedback path due to signal propagation, detection, processing and modulation can jeopardize the tight timing required for reliable control because the SOA may respond to its inputs long before the corresponding control signal arrives. If the closed-loop delay is too large the controller can actually destabilize the SOA, causing unpredictable output and possibly even damaging the SOA or other parts of the optoelectronic circuitry. In Chapter 6 we use the full compression–electronic linear model to determine the closed-loop stability conditions in the presence of delay, in line with Goal 6 above. Intuitively we might suppose that an order of magnitude increase in speed over the SOA’s response time
Chapter 1 The promise of integrated photonics
17
would be sufficient, but the conclusions of Chapter 6 are somewhat surprising in their severity for stable operation. We find that the resulting delay–feedback trade-off can be alleviated by introducing feedforward control, and by applying weak feedback only to correct longer-timescale errors. Finally, in light of the feedback restrictions found in Chapter 6, we design a new hybrid feedforward–feedback controller in Chapter 7 that uses a fast second SOA to control the first, thereby satisfying the last Goal 7. We use cross-gain modulation in the controller SOA to realize feedforward control, and we employ electronic feedback control to correct feedforward errors. Our design is informed by all the previous work and further develops novel techniques for setting the operating point of the controller SOA. We list the novel contributions of this work in Chapter 8 and look towards further applications of our findings.
1.4
Conclusion: robust control methods for integrated photonics
In this opening chapter we have set the context for the rest of this dissertation. We have seen that photonic signal processing is a key enabling technology for faster computing and a faster Internet. The current trend in photonic signal processing is clearly toward integrated solutions, and the semiconductor optical amplifier is a highly versatile, highly integrable device that has been demonstrated in a variety of important functions. As SOAs are integrated together on a large scale, it becomes more difficult to guarantee that their interplay will produce the desired outputs. Although there are many empirical and heuristic controllers proposed in the literature, there is little evidence of robust design. With control theory we can analyze and design SOA control systems with a robust theoretical approach. In the remainder of this dissertation, we bring control theory and integrated photonics together, deriving the necessary device models and demonstrating robust integrated circuit design and analysis. This work opens the way for further robust analysis, design and control of integrated active photonic circuits.
18
Section 1.4 Conclusion: robust control methods for integrated photonics
Chapter 2 Technical background This background is divided into two principal parts after dealing with some conventions immediately below in Section 2.1. The first part describes the physical operation of the semiconductor optical amplifier in Section 2.2, while the second part outlines the fundamental control theory and techniques in Section 2.3. Only the principles that reoccur throughout the dissertation are outlined here for brevity, and some deeper technical detail is left to the relevant later chapters for specific cases.
2.1
Conventions
It will be essential to keep track of matrices, vectors and scalars that are interrelated. We use bold capital letters for matrices, bold small letters for vectors, and either case unbolded for scalars. For example, the scalar a might be an element of the vector a, in turn partitioned from the matrix A. We also make use of some set notation, particularly the set of real numbers R where the dimension is noted as a superscript, and the set of non-negative real numbers R+ 0. Similarly, the set of complex numbers with dimension n is denoted Cn . There are a few places where notation may seem to collide, but context and careful differentiation should keep separate the various quantities. For example, N is the carrier density while N k is linear system matrix for a controller; q is the charge density while q 1 is the state of the linear system for “delay unit number one.” Because the core of this work is model derivation and testing, no high-level tools are used such as MatLab’s Simulink environment or Octave’s control interconnection package. 19
20
Section 2.2 Essential semiconductor optical amplifier physics
Table 2.1: Physical constant values used in numerical calculations [143]. Constant Symbol Value Unit −19 Elemental charge q 1.602 176 53 × 10 C −34 Planck’s constant h 6.626 069 3 × 10 J·s Boltzmann’s constant kB 1.380 650 5 × 10−23 J·K−1 Electric constant ǫ0 8.854 187 817 × 10−12 F·m−1 Electron mass m0 9.109 382 6 × 10−31 kg All the algebraic deductions such as system interconnections are carried out by hand. Many of the larger system interconnection formulae are derived using a text editor line-byline so that typographic errors are eliminated, particularly with long feedback expressions. The commercial software package Maple1 is used to solve larger integrals, derivatives, and differential equations, and the results are always checked either by manual computation, reverse computation (i.e. differentiating an integral solution), or both. MKS units are used throughout. The values of physical constants are given in Table 2.1. Energy is denoted by curly E. “Room temperature” is taken to be 300 K. Unless otherwise noted, room temperature is assumed. Finally, in the engineering literature it is common to find the symbol ≡ stand for “is defined by”; in this dissertation we use ≡ for its more traditional “is equivalent or identical to,” for example in x ≡ 0 if x is exactly, identically 0. For definition assignment we use the more traditional ,.
2.2
Essential semiconductor optical amplifier physics
For our modelling purposes we begin with a set of governing equations that describe the SOA phenomenologically. We need to know what happens to a photon injected into a facet and to an electron–hole pair injected into opposite terminals, on the average. While we need to know the mathematical functions relating current and input optical fields to the output optical fields, we do not have to delve down into the specifics of quantum transition theory. We begin with an overview of how SOAs are designed—their architecture and operation. We then examine the two basic governing equations, the carrier rate equation and 1
Maple version 7, Maplesoft, Waterloo, 2001.
Chapter 2 Technical background
21 current ridge
+ terminal p-type
active region substrate
n-type - terminal
(a) SOA cross-section showing the flow of current from the positive terminal on the ridge, through the active region, and down through the negative terminal of the substrate.
AR coating ingress
active waveguide
egress
(b) Top-down view showing optical antireflection coatings and angled waveguide.
Figure 2.1: SOA physical layout: (a) cross-section; (b) top-down view. the optical propagation equation.
2.2.1
Principles of architecture and operation
Early on in their development, semiconductor optical amplifiers were called semiconductor laser amplifiers because SOAs were developed from diode lasers, with Brewster-angled facets or antireflective coatings applied to the facets to suppress Fabry–P´erot feedback.2 As in lasers, electric current injects into a p-type region, passes through an active region, and exits through an n-type region as shown in Fig. 2.1(a). Inside the active region, much of the current’s energy converts from the electrical domain into the optical domain. Unlike in lasers, however, SOAs are injected with coherent light from an external source, which travels down a waveguide through the active region as illustrated in Fig. 2.1(b). Hence, SOAs can be used to process light as it travels from source to detector. 2
III-V semiconductors typically have n ≈ 3 and so there is roughly 25% reflection at the semiconductor–air interface; reflection can be reduced to less than 1% using antireflection coatings [144]. By setting the angle of the facets to the Brewster angle, polarized light in the transverse-magnetic or p polarization state experiences effectively zero reflection [144].
22
Section 2.2 Essential semiconductor optical amplifier physics ¯ I(t) e−
E2
Sin (t)
Sout (t) h+
E1
E z
Figure 2.2: Stimulated emission inside a semiconductor optical amplifier, with carriers supplied by an external current. Internally, electrons are injected from the n-type region and holes are injected from the p-type region into the active region [145]. Electron–hole pairs tend to recombine in the active region and, in turn, this recombination tends to emit photons with energy equal to the energy separation of the original electron–hole pair. Some photons result from spontaneous recombination of electron–hole pairs and propagate with random phase and direction. However, externally injected photons can trigger stimulated recombination in which a second photon is produced that is an exact copy in phase, wavelength, and direction, as demonstrated by Fig. 2.2. Coherent gain is achieved for externally injected signals by way of this stimulated emission: both the original and stimulated photons can repeat the process, leading to potentially exponential net gain as more and more photons are generated. Unlike with lasers though, there is no concept of threshold, but when the gain is equal to the cavity loss, amplifier transparency is said to occur [146]. Above transparency in the gain regime the optical beam is coherent, focused, and intense. Below transparency in the loss regime the signals are coherent but the SOA behaves as an optical attenuator. The first semiconductor diode lasers [147–149] (1962) were broad-area p–n junctions made of a direct bandgap semiconductor such as GaAs. In such lasers or laser amplifiers, recombination occurs over a large region so photon populations needed for lasing are slow to build and require high input currents. Confinement of carriers and the optical mode is poor, and nonradiative recombination is high. Radiative recombination can be enhanced by introduction of a double heterostructure between the hole and electron injectors [150– 154], which lowers transparency current and permits continuous-wave operation at room temperature [150]. With the precision of molecular beam epitaxy and chemical vapour deposition [155],
Chapter 2 Technical background
23
e− E2
wells or dots
active region E1
E x
h+
S
n x
Figure 2.3: SOA active region architecture. Above: energy profile of the quantum confinement structures (wells or dots) where electrons and holes are injected from opposite ends, leading to recombination. Below: resulting approximated refractive index that supports the guided optical mode. one-dimensional superlattices can be grown in the growth direction [156] to form multiquantum well structures. By reducing the thickness of the well layers to the order of the de Broglie of the carriers and using a material of smaller bandgap, quantum wells are formed that trap and quantize carriers in the longitudinal direction [156]. Inside the wells, carriers exist in a two-dimensional carrier gas confined in the third direction. In one-dimensional confinement, the density of states assumes a staircase series of stepfunctions constrained by the free electron dispersion relation [146, 156].3 Active region media must be lattice-matched4 to the cladding layers; this is true for AlGaAs/GaAs and true for a range of InGaAsP compositions in InGaAsP/InP heterostructures. Advantages of using multiple quantum wells include lower transparency current, reduced temperature dependence on bias current, emission wavelength tuning (via state quantization), and improved dynamic behaviour [156–159]. Full three-dimensional quantum confinement is achieved by growing quantum dots with processes such as Stranksy–Krostinow growth [160–162]. Just as there are performance improvements by using quantum wells over broad-area junctions, there are further improvements by using quantum dots. The faster dynamic response of dots compared to 3
Broadening mechanisms smooth out the staircase so that the transitions from one step to the next are continuous [156]. 4 Slight strain is permissible for very thin layers [146].
24
Section 2.2 Essential semiconductor optical amplifier physics
wells [163] is of particular interest as data rates continue to climb. The definition of the active region has an additional advantage in that the refractive indices n of the smaller-bandgap active region materials tend to be higher than those of the cladding materials for commonly used III-V material systems such as InGaAsP/InP and AlGaAs/GaAs, as shown approximately in Fig. 2.3 (the confinement materials and their barriers may differ slightly in n). This index difference creates a waveguide [144,164] that contains the optical field and guides the propagating modes down the SOA. Further reduction of the amplifier transparency condition is achieved by confining current to a roughly optical-mode-shaped region because more photons from recombination are produced for stimulated emission in a small volume. In one method of current confinement a ridge structure is etched longitudinally along one carrier injector (illustrated schematically in Fig. 2.1 and imaged in Fig. 1.1), thus providing a closed circuit only through a small lateral region [165]. The ridge is achieved by the introduction of a thin layer through one cladding region that serves as an etch-stop during the etching process; a simple mask is used to define the lateral ridge structure. Often it is the p-type contact that is shaped into a ridge, although some broad-area p-type material is retained to keep the etch-stop layer away from the active region. The ridge additionally provides moderate refractive index contrast to the optical mode and guiding occurs along the length of the ridge [166] as suggested by Fig. 2.1(b). A feedback mechanism often employed in buried-heterostructure diode lasers5 is the distributed Bragg reflector, a periodic corrugation of the refractive index along the length of the laser cavity [167–169]. One can reinforce the operational (freespace) wavelength λ0 by setting the corrugation period to be Λ = λ0 /2¯ n for average refractive index n ¯ [15]. Bragg reflectors are sometimes used in travelling-wave SOAs to reinforce the desired wavelengths and modes, but this is not the global feedback we employ as control signals later on in the technical chapters of this dissertation. In summary, we need to keep track of both the carriers (electrons and holes) supplied by the current, as well as the propagation of photons down the waveguide. We address each of these quantitatively in the next two subsections.
5
Buried-heterostructure diode lasers have distinct internal architecture from ridge-waveguide diode lasers but share the same basic operating principles.
Chapter 2 Technical background
2.2.2
25
Carrier rate equation
The basic model we employ is that of a noninteracting two-level system where the upper level has a characteristic interband decay time and the intraband decay time is negligibly quick [170] (see Fig. 2.2 for a depiction). Let the number of promoted carriers per unit volume be N2 and the number of ground state carriers per unit volume be N1 ; for example, if we count electrons, N2 is their density in the conduction band while N1 is their density in the valence band. Thus we define the population inversion density as N(z, t) , N2 (z, t) − N1 (z, t)
(2.1)
at position z along the amplifier at time t. Although strictly N(z, t) counts discrete carriers, the number of carriers is so vast (typically in excess of 1018 cm−3 ) that we let N(z, t) be smoothly-varying and continuous (and thus differentiable) for all (z, t) ∈ + 2 R+ 0 × R0 . Carrier diffusion ∇x,y N along (x, y) can be ignored if the transverse profile is
uniform and the device length is significantly longer than the diffusion length [170], an assumption that is pervasive in the literature and one we make. At position z, energetic carriers are supplied by the active region current I(z, t). As stated above and depicted in Fig. 2.2, stimulated emission occurs when injected photons stimulate electron–hole recombination that produce near-replica photons in terms of phase, direction, and wavelength. Thus, the stimulated emission rate depends on the product of a gain function g and the photon density S. It turns out that the form of the gain function g has a profound effect on deriving a state-space set of equations, so we defer discussing it until the appropriate places in the rest of this dissertation. Generally, however, gain is a function of photon wavelength λ, photon density S, inversion carrier density N, and even inherent material properties within the SOA, all of which can vary with (x, y, z) position and time.6 There are several other interactions not shown in Fig. 2.2. Carriers can recombine nonradiatively through various means. Recombination due to carriers falling into defect states (such as those between strained active region layers) is proportional to N under the high injection conditions typically found in SOAs [146]. Bimolecular recombination requires an electron–hole pair for a direct transition and is proportional to N 2 . Finally, Auger recombination occurs when the energy of a decaying electron (hole) goes into 6
Although the material is largely time-invariant, dopants can drift down their concentration gradients over time and change the finely-tuned architecture.
26
Section 2.2 Essential semiconductor optical amplifier physics ∝ g(z, t)
∝ α(z, t)
S(z, t)
S(z + ∂z, t) ∂z
Figure 2.4: Incremental model of optical gain. promoting an electron (hole) already in the conduction (valence) band to an even higher energy state, and is proportional to N 3 [146]. While it is common in the literature to approximate all nonradiative recombination by a single time constant τc as in Rnr (N, z, t) =
N(z, t) , τc
(2.2)
the full polynomial is generally more accurate, Rnr (N, z, t) = RA N(z, t) + RB N 2 (z, t) + RC N 3 (z, t),
(2.3)
where RA,B,C are the corresponding rate coefficients. Assembling the entire electronic carrier rate equation we have [83, 171] ∂N(z, t) I(z, t) vg = − Rnr (N, z, t) − g(N, S, z, t)S(z, t). ∂t qV Γ
(2.4)
In this equation, I(z, t) is the injection current, q the electronic unit charge, V the active region interaction volume, Rnr (N, z, t) the nonradiative recombination rate, vg the group velocity, Γ the modal confinement factor (essentially a fractional measure of how well the optical mode overlaps the active region), g(N, S, z, t) the stimulated optical gain, S(z, t) the instantaneous photon density, and N(z, t) the instantaneous population inversion density. This equation governs the supply and consumption of energetic carriers within the system, and forms half the basis of our core model to follow in the next chapter.
2.2.3
Optical propagation equation
Figure 2.4 illustrates how the SOA achieves optical gain across a small cell of length ∂z. Input photons are subject to additions through modal gain g(z, t) and subtractions through modal loss α(z, t), the latter due to re-absorption and scattering losses that do
Chapter 2 Technical background
27
not directly repopulate N(z, t). Hence, we can write the propagation equation ∂S(z, t) = g(N, z, t) − α(z, t) S(z, t) ∂z
(2.5)
that forms the second half of the basis for our models that follow. Again, we say that amplifier transparency occurs when the total gain and loss are equal, so that the net gain across the amplifier is unity. Thus the gain regime occurs when g(N, z, t) > α(z, t), and the loss regime when g(N, z, t) < α(z, t). Carriers can interact with vacuum photons, seemingly recombining spontaneously and producing incoherent real photons; because there is no cavity feedback in an SOA, this spontaneous emission does not contribute significantly to signal modes and so we account for it only through amplified spontaneous emission (ASE). The spontaneous emission rate Rsp can be model-specific [128,172,173], and we defer its discussion for now. Thus, ASE propagates as Γ ∂SASE (z, t) Rsp (N, z, t), = g(N, z, t) − α(z, t) SASE (z, t) + ∂z Avg
(2.6)
where A is the transverse interaction area of the active region. Due to the instantaneous refractive index of the waveguide, there is a phase shift associated with each cell of Fig. 2.4, given by [171] ∂φ 2πn0 =Γ ∆n λ, N, z, t , ∂z λ
(2.7)
where n0 is the base refractive index of the medium and ∆n the free-carrier-induced change. At the system level it is actually more practical to work with optical power P rather than photon density S. We can convert this formulation above [83] to use instantaneous optical power instead of photon density through S(z, t) =
Γ P (z, t), ~ωvg A
where ω is the optical carrier frequency,and P (z, t) the optical power.
(2.8)
28
Section 2.3 Control theory methods
2.3 2.3.1
Control theory methods
State-space realization
A plant is the device or system we wish to model and ultimately control; it may be as simple as an inverted pendulum or as complex as a nuclear reactor. For much of the following work we take an SOA as the plant, and in specific cases we may fold controllers, filters, and delay units into the plant. A plant’s inputs are the signals that drive some aspect of operation; we refer to external inputs when dealing with inputs to a system made up of several plants. Conversely, a plant’s outputs are signals generated by some aspect of operation and that can be measured in some physical way. A plant with one input and one output is conventionally single-input/single-output or SISO; with multiple inputs and outputs the plant is multi-input/multi-output or MIMO. A controller is a system that attempts to set the inputs of the plant such that the plant’s outputs and state behave in some designed manner. Open-loop refers to a plant that is uncontrolled, while closed-loop implies that feedback control is used. A plant’s state contains all the operational details of how the plant is acting at a particular time or time step. Mathematically, the state is generally a vector x ∈ Rn indexed by time that is updated by some governing rule of the plant. It is perhaps most intuitive to start in discretized time with time-step index k; in general the state-update equation has the form xk+1 = fˆ(xk , uk ),
(2.9)
where fˆ is a vector function that takes the current-time state xk and input uk ∈ Rm , and produces the state at the next time step. The outputs that are available from the plant y ∈ Rp are given by an output relation, a function of the form ˆ k , uk ) y k = h(x
(2.10)
ˆ that relates the current state and input to the current output via a vector function h. We say that these functions are affine in a variable (e.g. in x) if that variable can be factored out linearly. While discrete-time models are most commonly used when there is some kind of digital control, we focus on continuous-time models throughout this dissertation because
Chapter 2 Technical background
29 δ x˙
f (x) = x˙ Dx 0
x0
x
0
δx slope = A
Figure 2.5: Linearization D of a scalar nonlinear state function f with respect to a scalar state x at equilibrium x0 . The result is a linear function δ x(t) ˙ = Aδx(t) with slope A = Dx f (x)|x0 . the associated SOA systems are analog at sub-bit-length timescales. In continuous time the state-update equation becomes ˙ x(t) = f x(t), u(t) ,
(2.11)
y(t) = h x(t), u(t) .
(2.12)
while the output relation is
One of the central issues in this dissertation is finding f , h, and x(t) that describe SOA physical operation. It is noteworthy that continuous-time models are easily discretized [174]. There is a large branch of control theory devoted to analyzing nonlinear state equations when f is nonlinear in x(t) or u(t) or both, but some tools such as Lyapunov stability analysis can be difficult in general depending on the actual form that f takes, particularly if f is not affine. Thus we turn our attention to linearizing these state-space models to leverage linear control theory and its many robust tools and facilities.
2.3.2
State-space model linearization
A plant has an equilibrium point (x0 , u0 ) ∈ Rn × Rm if f (x0 , u0 ) ≡ 0.
(2.13)
Of course, when this condition is achieved the system is at steady-state by (2.11). The output at equilibrium is simply y 0 = h(x0 , u0 ).
(2.14)
30
Section 2.3 Control theory methods To linearize the state-space system Eqs. (2.11),(2.12) at an equilibrium point (x0 , u0 )
we apply the Jacobian operator D defined by [175] ∂f1 ··· ∂x1 . . Dxf (x) = . ∂fn ··· ∂x1
∂f1 ∂xn .. . . ∂fn ∂xn
(2.15)
Thus, we define the linear time-invariant system by the set of constant matrices (A ∈ Rn×n , B ∈ Rn×m , C ∈ Rp×n , D ∈ Rp×p ) for some positive integers (m, n, p), obtained by A =Dxf x(t), u(t) |(x0 ,u0 ) B =Duf x(t), u(t) |(x0 ,u0 ) C =Dxh x(t), u(t) |(x0 ,u0 ) D =Duh x(t), u(t) |(x0 ,u0 )
(2.16a) (2.16b) (2.16c) (2.16d)
In the above, D takes precedence over the evaluation at (x0 , u0 ). The resulting linearized state-update equation is ˙ δ x(t) = Aδx(t) + Bδu(t)
(2.17)
while the output relation becomes δy(t) = Cδx(t) + Dδu(t),
(2.18)
where the δ-quantities are the small-signal variables and xL (t) =δx(t) + x0
(2.19a)
uL (t) =δu(t) + u0
(2.19b)
y L (t) =δy(t) + y 0
(2.19c)
are the recovered linear approximations. Figure 2.5 illustrates how D linearizes function f with respect to state x for one dimension. The validity of this linearization depends entirely on the goals of the designer. If f and h are highly nonlinear at a given equilibrium point, then the errors |x(t) − xL (t)| may diverge too quickly with even small deviations in δx(t) for a particular application.
Chapter 2 Technical background
31
Such errors can also be quantified by comparing second-order coefficients generated by a second-order Jacobian. However, it is often the goal in a controlled system to keep x(t) near x0 , and so some error is usually acceptable. For the rest of this dissertation we drop the δ and subscript-L notation and let context bear out whether x(t) denotes the nonlinear state x(t), small-signal deviation δx(t), or recovered linear approximation xL (t), and likewise for u(t), y(t), and other signals we encounter. Henceforth we write ˙ x(t) = Ax(t) + Bu(t)
(2.20)
y(t) = Cx(t) + Du(t)
(2.21)
and
and keep in mind the picture of Figure 2.5.
2.3.3
State-space model solution
Unlike the general nonlinear state function (2.11) that may not have an analytical solution, the linear state function (2.20) can be integrated analytically provided the input u(t) is known over its entire duration. The solution to (2.20) is [174] A(t−t0 )
x(t) = e
x(t0 ) +
Z
t
eA(t−τ ) Bu(τ )dτ.
(2.22)
t0
With this solution we can derive the output directly as well, A(t−t0 )
y(t) = Ce
x(t0 ) + C
Z
t
eA(t−τ ) Bu(τ )dτ + Du(t).
(2.23)
t0
The types of inputs that interest us—pseudo random bit sequences hundreds of bits long, for example—are not easily represented analytically, and so we employ direct numerical integration to solve Eq. (2.20) as well as its nonlinear counterpart Eq. (2.11). All the time-evolution simulations of the state update equations are performed by direct numerical integration with the Livermore Solver for Ordinary Differential Equations (LSODE) [176] driven by the fsolve command in GNU/Octave.7 LSODE is a highly sophisticated numerical solver for first-order ordinary differential equations that is par7
Octave is an open-source numerical mathematics program that aims to replicate MatLab’s functionality and syntax, http://www.octave.org.
32
Section 2.3 Control theory methods
PSfrag D U (s)
X(s) s−1
B
Y (s) C
A Figure 2.6: Block diagram of the linear system (A, B, C, D) in the Laplace domain. ticularly adept at solving stiff8 problems by way of its backward differentiation formula method [176]. Both the step size and method order are varied automatically according to the problem, and step size is varied dynamically during operation. Often we plot the state solutions and outputs versus time. Occasionally we use phase portraits that plot the state components against each other parametrically with time. Phase plots show the trajectory of the state through the state space and can elucidate some system behaviours more clearly than simple plots against time.
2.3.4
Linear model transfer function
Although we are primarily interested in time-domain analysis, it is useful occasionally to transform into the Laplace domain via the Laplace transform X(s) = L {x(t)} ,
Z
∞
x(t)e−st dx,
(2.24)
0−
and back again via its inverse x(t) = L
−1
1 {X(s)} , 2πj
Z
j∞
X(s)est ds,
(2.25)
−j∞
where s is the Laplace frequency. The real frequency ω can be recovered via s = jω. Deriving the linearized plant’s transfer function T(s) from Eqs. (2.20),(2.21) is straightforward. First we transform (2.20), sX(s) = AX(s) + BU (s) 8
(2.26)
Stiffness is difficult to define in absolutely generic terms, but conventionally implies that there are large differences in time scale and time constant, as there is between, say, the leading edge and duration of an optical pulse.
Chapter 2 Technical background
33
(note that X 0 ≡ 0 is assumed without loss of generality), then solve for the state, X(s) = (s − A)−1 BU (s)
(2.27)
and substitute the state into the output, Y (s) = [C(s − A)−1 B + D]U (s).
(2.28)
The multivariable transfer function is then T(s) = C(s − A)−1 B + D.
(2.29)
This form is easily depicted in a block diagram as in Fig. 2.6, and allows straightforward block multiplication with other subsystems and plants. We say T(s) is a proper rational function if the degree of the denominator is greater than or equal to that of the numerator. Similarly, we say T(s) is a strictly proper rational function if the degree of the denominator is strictly greater than that of the numerator. Causal physical systems must be at least proper [177]. With a transfer function we can quickly evaluate the system’s long-term steady-state behaviour. The final value theorem states that lim y(t) = lim+ sY (s) = lim+ sT(s)U (s),
t→∞
s→0
s→0
(2.30)
provided the poles of sY (s) are in the lefthand complex plane (i.e. sT(s)U (s) is stable). So once we have (A, B, C, D) we can quickly find the final value lim y(t) = lim+ s CA−1 B + D U (s).
t→∞
s→0
(2.31)
This method is particularly useful for determining if a constant controller has residual steady-state error, in which case a dynamic controller is needed.
2.3.5
Controller canonical form
There are several methods to convert a transfer function back into a state-space realization, although not all methods yield minimal realizations.9 We are particularly interested 9
A minimal realization is one having the lowest state dimension possible.
34
Section 2.3 Control theory methods
in converting SISO filter transfer functions, and the method we employ is basically a template: for a transfer function written in the form T(s) =
c0 + c1 s + c2 s2 + · · · + cn−1 sn−1 +d a0 + a1 s + a2 s2 + · · · + an−1 sn−1 + sn
(2.32)
the controller canonical form is given by [174]
A =
0
1
0
···
0
0 .. .
0 .. .
1 .. .
··· .. .
0 .. .
0
0
0
···
1
−a0 −a1 h b = 0 ··· 0 h c = c0 c1 · · ·
−a2 · · · −an−1 iT 1 i cn−1 ,
(2.33a)
(2.33b) (2.33c)
with d read straight from the pass-through term of the transfer function, if present (i.e. if T(s) is proper, not strictly proper). Not that it matters particularly for our applications, this canonical form gets its name from the fact that it is automatically controllable (see below) and minimal.
2.3.6
Useful system properties
Equipped with the linear model and its transfer function we can assess several system properties given (A, B, C, D). That we can analyze the plant before numerical simulation is a key advantage of linear state-space models. A plant is controllable if its state can be brought from any value x1 ∈ Rn to any other value x2 ∈ Rn in a finite amount of time by a suitable choice of input u(t). Controllability can be tested in several ways, but we form the controllability matrix defined by h i 2 n−1 C(A, B) , B AB A B · · · A B ,
(2.34)
and check its rank: if C(A, B) has full rank, (A, B) is controllable. Note that we can also test the controllability of a single input or combination of inputs by choosing a subset of columns of B corresponding to those inputs. A plant is observable if we can reconstruct x(t) over a specified time interval by
Chapter 2 Technical background
35
knowing y(t) over the same interval. Essentially, observability qualifies whether we can determine the state given the input–output behaviour of the plant. The test for observability is the dual of the controllability test. We form the observability matrix defined by
C
CA O(A, C) , CA2 , . . . CAn−1
(2.35)
and again require full rank for observability. We can test observability for particular outputs by choosing a subset of rows of C corresponding to those outputs. There are several notions of system stability available, but we employ boundedinput/bounded-output stability. Thus a system is stable if any bounded input u(t) yields only a bounded output y(t). Equivalently, the state x(t) too must be bounded at all times, and we can test the stability local to (x0 , u0 ) by examining the state’s dynamics. To do so, consider ˙ x(t) = Ax(t)
(2.36)
with solution x(t) = eAt x0 ,
x0 6= 0.
(2.37)
If A is diagonal then its eigenvalues λA,i are the diagonal entries and the matrix exponential is
eAt =
eλA,1 t
0 elsewhere ..
.
0 elsewhere
eλA,n t
.
(2.38)
If A is not already diagonal, a similarity operation transforms the matrix exponential to its Jordan form [174], and again the solution contains terms with eλA,i t . Hence, in order to be stable, all the eigenvalues of A must satisfy Re{λA,i } < 0.
(2.39)
It is possible for the system to be stable if Re{λA,i } ≡ 0 depending on the imaginary part, but we will not encounter this borderline case. Thus we can evaluate several important properties of the system rigorously. These
36
Section 2.3 Control theory methods
properties factor into many of the following SOA controller designs.
2.3.7
State feedback
Under constant state feedback, the state x(t) is subjected to an affine transformation10 and fed back into the plant’s input u(t). Closing the loop from state to input yields u(t) = Kx(t) + r(t),
(2.40)
where K ∈ Rm×n is the feedback gain and r(t) ∈ Rm a reference. Substituting this control input into the linear state-update equation (2.20) yields ˙ x(t) = Ax(t) + B Kx(t) + r(t) ,
(2.41)
˙ x(t) = A + BK x(t) + Br(t).
(2.42)
y(t) = Cx(t) + D Kx(t) + r(t) ,
(2.43)
which we rewrite as
The output relation becomes
which is conventionally rewritten y(t) = C + DK x(t) + Dr(t).
(2.44)
In the rest of the dissertation, we commonly rename r(t) back to u(t) so that u(t) represents the collection of external system inputs. Feedback has the effect of shifting the system’s governing eigenvalues. If we focus just on the state dynamics,
the solution now contains
˙ x(t) = A + BK x(t), x(t) = e[A+BK]t x(0),
x(0) 6= 0,
(2.45)
(2.46)
so the eigenvalues of the combined A + BK term govern the time-evolution of the 10
An affine transformation is of the form A(x) = ax + b for some scalars (a, b) ∈ R × R.
Chapter 2 Technical background
37
system. For example, if we choose K such that controlled eigenvalues have real parts more negative than their open-loop counterparts, the system converges to x(t) → x0 more quickly; we will see just how advantageous such a design can be in SOA systems in the next chapter. State feedback is the most effective feedback because the system states contain all the current operating information. Unfortunately, the states are not always easily measured. For example, the weight of an airplane in flight might be a state of the airplane, but the weight changes during flight due to fuel consumption and there are obvious technical difficulties in weighing an aircraft at an altitude of 10 km. In these cases, an observer may be run to estimate the states: the observer measures the system inputs u(t) and outputs y(t), and generates an estimate of the state by running the system model in parallel. In order to estimate the full state, the system must be observable. We relegate further discussion on observers to Appendix B.
2.3.8
Optimal least-squares state feedback
The constant feedback controller is optimal in the least-squares sense when a cost η ∈ R+ 0 associated with control effort is minimized; conventionally the cost is given by η=
Z
0
∞
xT (t)Qx(t) + uT (t)Ru(t) dt.
(2.47)
Here, Q ∈ Rn×n is a constant positive-semidefinite matrix11 that penalizes excursions in the state from equilibrium; similarly, R ∈ Rm×m is a constant positive-definite matrix12 that penalizes effort of the controlled inputs. These penalty matrices are commonly chosen to be diagonal so that an excursion in a particular variable penalizes only that variable. Once the penalties are designed, the controller K is found by K = −R−1 B T P,
(2.48)
where P ∈ Rn×n is a symmetric matrix found numerically via the algebraic Riccati equation [174] AT P + PA − PBR−1 B T P + Q = 0.
(2.49)
When the observer gain L is designed using this optimal formulation, the observer is 11 12
A positive-semidefinite matrix Q implies xT Qx ≥ 0 for every x ∈ Cn . A positive-definite matrix R implies uT Ru > 0 for every u ∈ Cn .
38
Section 2.3 Control theory methods
called a Kalman filter. Kalman filters are particularly effective at filtering out white Gaussian noise disturbances from state measurements [174].
2.3.9
Output feedback
When the states are not easily measured and when an observer is infeasible, output feedback may provide suitable control, particularly when the system is observable. A specific but useful case of output feedback—single-input/single-output output feedback— is relegated to Appendix A. Under constant output feedback the output y(t) is fed through an affine transformation and the result drives the plant’s input u(t). Closing the loop from output to input gives u(t) = Ky(t) + r(t),
(2.50)
where K ∈ Rm×p is the feedback gain and r(t) ∈ Rm an external reference. Substituting this controlled input into the output relation (2.21) yields y(t) = Cx(t) + D Ky(t) + r(t) ,
(2.51)
which contains y(t) on both sides. Thus, unlike in the state feedback case, we must pause to solve for y(t), y(t) − DKy(t) = Cx(t) + Dr(t),
(2.52)
−1 −1 y(t) = I − DK Cx(t) + I − DK Dr(t).
(2.53)
so that we obtain
Substitution the state equation (2.20) then gives −1 −1 ˙ x(t) = A + BK I − DK C x(t) + B I + K I − DK D r(t)
(2.54)
The system’s eigenvalues now come from A + BK[I − DK]−1 C, so stability and performance are functions of (A, B, C, D) as well as K. As a result, output feedback is not as simple or effective as state feedback in general, although careful design can produce good closed-loop behaviour.
Chapter 2 Technical background
2.3.10
39
Dynamic single-input/single-output controllers
We will find that in some cases, constant feedback control (whether state or output feedback) leaves residual steady-state error. For instance, if we desire that the output achieves some designed output y(t) → y d over some prescribed timescale, it may be that a constant proportional SISO controller kp maintains some finite error, |y(t) − y d | > 0. An integrator can suppress steady-state error when added to a feedback circuit under bounded step or ramp input conditions, and has dynamics x˙ I (t) = u(t)
(2.55)
yI (t) = xI (t).
(2.56)
with output
Intuitively, an integrator accumulates error even if u(t) reaches a steady—but possibly erroneous—value, and so a integrating controller continues to drive the plant to the desired state and output. Integrators tend to act as low-pass filters, so we consider adding a differentiator that passes through higher-frequency signals and improve transient response. All together, a proportional–integral–differential or PID controller allows us to tune the feedback and ultimately the transient response of the entire closed-loop system. A SISO PID controller k(t) can be implemented with P–I–D elements in parallel, Z d 1 t k(t) = kp 1 + τd + dτ , dt τi 0
(2.57)
where kp is the proportional gain, and where τd and τi are the filter time constants. In the Laplace domain we have K(s) = kp (1 + sτd + s−1 τi−1 ),
(2.58)
which is unrealizable in state-space form due to the derivative term that yields a nonproper transfer function. We can filter the derivative term with (1+ǫs)−1 that attenuates high frequencies and yields a proper transfer function, so let K(s) = kp
sτd 1 1+ + 1 + ǫτd s τi s
(2.59)
40
Section 2.3 Control theory methods
be the PID controller transfer function in parallel implementation. By comparison, a cascade configuration has the form K(s) = kp
1 + sτd 1 + ǫτd s
1 + sτi . sτi
(2.60)
Either implementation can be realized via the controller canonical form from above.
2.3.11
Useful calculus
We need two theorems from calculus to convert our particular governing equations into nonlinear state-space equations. The Fundamental Theorem of Calculus states that if f is a continuous function on the closed interval [a, b] and F ′ = f on [a,b], then [175] Z
b
f (x)dx = F (b) − F (a).
(2.61)
a
Leibnitz’s Rule states that if f (z, t) and ∂f /∂t are continuous on z ∈ [a, b], t ∈ [t1 , t2 ], then for t ∈ [t1 , t2 ], d dt
Z
a
b
f (z, t)∂z =
Z
a
b
∂f (z, t) ∂z. ∂t
(2.62)
The proof can be found on pg. 266 of Ref. [178]. This theorem really just tells us when we can swap the order of differentiation and integration.
2.3.12
Example: EDFA control formulation
To close our discussion on state-space methods we review how the governing equations of EDFAs have been realized in state-space form [69]. This type of derivation serves as a basis for the core photonic SOA model in the next chapter. A spatially uniform two-energy-level system is assumed over the length L of the EDFA. We let N be the total number of carriers in the system and assume it is constant in time. The number carriers in the upper energy state is N2 and in the lower energy state N1 , so that N = N1 + N2 (note that this is distinctly different from how we defined N for SOAs in Section 2.2.2). Except for L we normalize all the other physical parameters by setting them identically to one.
Chapter 2 Technical background
41
The rate equation is [69] m
X ∂N2 (z, t) = −N2 (z, t) − N2 (z, t) − N1 (z, t) Pi (z, t) ∂t i=1
(2.63)
and the propagation equation is [69] ∂Pi (z, t) = N2 (z, t) − N1 (z, t) Pi (z, t), ∂z
(2.64)
where z is the spatial coordinate along the amplifier length and Pi (z, t) is the ith data channel of m channels. Again, we have normalized out all the other physical parameters for clarity.
Substituting the righthand side of (2.64) into the last term of (2.63) we get m
X ∂Pi (z, t) ∂N2 (z, t) = −N2 (z, t) − . ∂t ∂z i=1
(2.65)
Integrating this with respect to z and normalizing by the amplifier length L we have m 1 X ∂ N¯2 (t) ¯ Pi (L, t) − Pi (0, t) , = −N2 (t) − ∂t L i=1
(2.66)
where Leibnitz’s Rule was used to swap the order of differentiation and integration on the lefthand side, and where the overbar indicates the length-average. This result is a state-update equation that can be rewritten in the state-space notation from above as m
x(t) ˙ = −x(t) −
1 X yi (t) − ui (t) , L i=1
(2.67)
¯2 (t) is the state, yi(t) = Pi (L, t) the ith output, and ui (t) = Pi (0, t) the where x(t) = N ith input.
As for the propagation equation (2.64), we can integrate it with respect to z using
42
Section 2.3 Control theory methods
N1 = N − N2 and employ the Fundamental Theorem of Calculus to get
⇒ ⇒
∂Pi (z, t) = N2 (z, t) − N − N2 (z, t) Pi (z, t) ∂z ∂Pi (z, t) = 2N2 (z, t) − N ∂z Pi (z, t) ¯2 (t)L − NL ln Pi (L, t) − ln Pi (0, t) = 2N ¯2 (t)−N ]L [2N
⇒ Pi (L, t) = Pi (0, t)e
,
(2.68a) (2.68b) (2.68c) (2.68d)
which, using the state-space notations from above, can be written as yi (t) = ui (t)e[2x(t)−N ]L .
(2.69)
In turn, this output (2.69) can be substituted into the state equation (2.67) to yield m
x(t) ˙ =−
1X x(t) ui(t) e[2x(t)−N ]L − 1 . − τ0 L i=1
(2.70)
In summary, this is an affine nonlinear system having general form x(t) ˙ = f x(t) + g x(t) u(t) y(t) = k x(t) u(t),
(2.71)
which is now ready to be analyzed using all the techniques described above. In the next chapter we follow this procedure as a rough guide for producing a SOA state-space model.
Chapter 3 Core photonic state-space model In this chapter we derive a state-space model that captures the optoelectronic dynamics of an SOA at the expense of neglecting nonlinear optical effects. The goal of our derivation is to produce an input–output state-space model suitable for control applications that act on the inputs and outputs of the SOA. This model serves as a core photonic model on which we will build more complete models in the following two chapters. The dynamics responsible for the wide array of interesting amplification applications are still present in our model, even if some of the nonlinear optical effects are neglected. So while the required length-averaging does indeed remove spatial information of optical powers and carrier densities within the SOA, the linear optical input–output dynamics remain in our “lumped” model in much the same way that input–output dynamics are modelled for a bipolar junction transistor without accounting explicitly for local minority and majority carrier densities and transport, for example. In Section 3.1 we start with the SOA governing equations and cast them into a state-space form. We will find that the form of the gain function plays an important part in deriving suitable state-space equations. From the SOA propagation equation we obtain an input–output relation for the optical channels, while the carrier rate equation is manipulated into a system state equation with the state being the average population inversion along the length of the device. Linearizing the state equations about preset input conditions yields a linear model of the SOA in Section 3.2. This linear model is verified against the original partial differential equations under various simulated input conditions. In Section 3.3 we use the linear model to design and analyze two types of controllers. A state feedback controller is implemented and used to eliminate interchannel cross43
44
Section 3.1 Nonlinear state-space model ¯ I(t)
outputs ¯ I(t)
inputs
P 1,...,m (0, t)
state ¯ (t) N
P0 (0, t)
P 1,...,m (L, t) P0 (L, t)
L Figure 3.1: Input–output control model of an SOA. Inputs: m lightwave data channels P 1,...,m (0, t), auxiliary lightwave for SOA optical control P0 (0, t), bias and modulation ¯ current I(t). Outputs: lightwave channels P 1,...,m (L, t), auxiliary lightwave for SOA ¯ optical control P0 (L, t), bias and modulation current I(t). State: length-averaged carrier ¯ concentration N(t).
talk by negative feedback into the electrical drive current; we also use transfer function analysis to determine the physical characteristics of the feedback. For the second proofof-concept control example we employ a feedback control application in which the total optical power at the SOA output is sampled and used to drive an optical control channel that suppresses interchannel crosstalk among the data channels. This optical output feedback technique is commonly used in fibre amplifier control (see for example Ref. [73]) and relies only on optical field power, not phase. These derivations and demonstrations show that control theory provides systematic methods for designing and regulating integrable photonic amplifiers.
3.1
Nonlinear state-space model
In this section we start from the SOA governing equations presented in Chapter 2 and derive a dynamic state-space model (a set of first-order ordinary differential equations) that is suitable for linearization and control. From an input–output perspective, a SOA is a multi-input/multi-output system, as shown in Fig. 3.1. The inputs are the electronic ¯ and optical data and control channels P (0, t). Measurable outputs are drive current I(t) ¯ the electronic drive current and the optical channels after passing through the SOA, I(t) and P (L, t) respectively.
Chapter 3 Core photonic state-space model
3.1.1
45
Governing equations
The governing equations for a two-level multichannel SOA consist of an optical power propagation equation ∂Pi (z, t) = gi (N, P , z, t)Pi (z, t) − αi Pi (z, t) ∂z
(3.1)
and an electronic carrier rate equation m
∂N(z, t) I(z, t) 1 X gi (N, P , z, t) = − Rnr (N, z, t) − Pi (z, t), ∂t qV A i=0 ~ωi
(3.2)
converted from Eqs. (2.4) and (2.5) via Eq. (2.8). In these equations for channel i, Pi (z, t) is the instantaneous optical power with P (z, t) = P0 (z, t), . . . , Pm (z, t) , N(z, t) the instantaneous population inversion carrier density, gi (N, P , z, t) the stimulated optical gain, αi the optical loss from scattering and free carrier absorption, I(z, t) the injection current, q the electronic unit charge, V and A the interaction volume and transverse area, Rnr (N, z, t) the nonradiative recombination rate, and ωi is the optical carrier frequency. The SOA is assumed time-invariant and homogeneous along its length. This SOA model handles multiple optical channels (for wavelength-division multiplexing, for example) by indexing parameters on a channel-by-channel basis. We adopt the convention that there are m optical data channels indexed 1, . . . , m, and one auxiliary optical control channel numbered 0, as shown in Fig. 3.1. Typically with this convention, each indexed parameter in Eqs. (3.1) and (3.2) is constant over the given channel’s bandwidth, but the parameters can be unique for every channel; because each channel is indexed to a single central frequency ωi, interchannel spectral overlap is ignored, but there is no restriction on maximum channel bandwidth and interchannel crosstalk is still possible due to the coupling through the common inversion carrier density Eq. (3.2). If, however, spectral overlap and intrachannel variations must be incorporated, sets of indices can be assigned to each channel and the parameters varied piecewise over the channel bandwidths.
3.1.2
Gain
The form of the gain function has a profound impact on the algebra needed to create a state-space model. For any unknown time-varying quantity, we must have an analytical
46
Section 3.1 Nonlinear state-space model
rate equation so that the unknown quantity can be made a state of the system; however, not all the required rate equations are convenient or available. The gi N(z, t), P (z, t) Pi (z, t) products in the last term of (3.2) are nonlinear and problematic. Rearranging the propagation equation (3.1)
∂Pi (z, t) gi N, P , z, t Pi (z, t) = + αi Pi (z, t) ∂z
(3.3)
and substituting into the last term of the rate equation (3.2) yields m ∂ I(z, t) 1X 1 dN(z, t) = − Rnr (N, z, t) − + αi Pi (z, t). dt qV A i=0 ~ωi ∂z
(3.4)
Integrating with respect to z and dividing by L gives Z L m X ¯ (t) ¯ dN I(t) 1 ∂ 1 ¯ nr (N, t) − = −R + αi Pi (z, t)∂z dt qV A i=0 ~ωi L 0 ∂z m ¯ 1 X 1 Pi (L, t) − Pi (0, t) I(t) ¯ ¯ − Rnr (N, t) − + αi Pi (t) , = qV A i=0 ~ωi L
(3.5a) (3.5b)
which is the general carrier-update equation. We made several initial attempts at deriving a state-space form that was as general as possible, and these attempts are relegated to Appendix C; here we summarize briefly our methods and conclusions. The carrier rate equation (3.5) contains two unknown length¯ (t) and P¯ (t) that could be used as system states; Section C.1 averaged quantities, N documents why P¯ (t) cannot be made a state specifically in SOAs. An attempt was made to generalize gain entirely by assigning g¯(t) to a system state in Section C.2, but this method fails because deriving an input–output relation is not possible without knowing the form of the gi terms explicitly. Following from the results of Section C.2 we restrict N(z, t) so that it is separable in z and t, but this method fails too because the spatial dependence of N is needed as shown in C.3. Finally, decomposition into finite-difference space is sometimes applied to stubborn problems; Section C.4 shows that prohibitively cumbersome recursive expressions result in trying to solve the propagation equation (3.1). Drawing on the analysis in Appendix C we conclude several assumptions are required. We assume that local gain gi depends on the local charge carrier density N, but not the local optical powers P . While slightly restrictive, this assumption is particularly useful because it allows us to obtain a closed-form state equation and is still general enough
Chapter 3 Core photonic state-space model
47 an
g(N)
a1 ... Ntr N1
...
Nn
N
Figure 3.2: Using piecewise-linear gain segments to approximate nonlinear gain. to model nonlinear gain. Compressive gains that involve optical powers lead to output differential equations (3.1) without suitable closed-form solutions if N depends on z; if N is taken to be z-independent a priori, it is possible to find an output solution as shown in the following chapter. Furthermore, because our goal is to produce an input–output model, we will integrate the gain over the length of the SOA [125]. Therefore, we take the charge carrier density N to be constant over the length of the device, N(z, t) = N(t), because integrating optical output relation in the following section reduces the SOA to a lumped element and averages the internal spatial information to a single value N(t). With these assumptions, gain for each channel i can be modelled linearly [146], gi (N, t) = Γi ai N(t) − Ntr,i ,
(3.6)
or logarithmically, gi (N, t) = Γi ai ln
N(t) , Ntr,i
(3.7)
where Γ is the modal confinement factor, a the differential gain and Ntr the transparency carrier density. If desired nonlinear gain functions are not analytically tractable, they can be approximated by a sequence of linear gain functions, as shown in Fig. 3.2. For a piecewise-linear decomposition having n segments, the differential gain is given by a(N) = a1 U(N − Ntr ) − a1 U(N − N1 ) + · · · + an U(N − Nn−1 ) − an U(N − Nn ), (3.8) where ai is the ith differential gain and U(·) is the unit step function. The linear gain case is solved below, so the model and eventual controller could switch to adjacent linear
48
Section 3.1 Nonlinear state-space model
differential gains to follow the carrier concentration through a series of linear approximations. Amplifiers are often switched between constant-gain and constant-power modes, so it is noteworthy that there exists general research into switched linear system controllers (switching between different control subsystems and considering the entire switchable controller) [179].
3.1.3
Output relations
Electrical domain The drive current is not necessarily the instantaneous current seen by the SOA because the power source and SOA may have parasitic resistance, capacitance and inductance. Although augmenting the model with linear circuit equations is straightforward, for simplicity at the moment we ignore the dynamics of any electronic drive circuitry and assume ¯ directly. Hence, the electrical we can measure and adjust the active region current I(t) input and output are equal.
Optical domain To relate the optical outputs at z = L to the inputs at z = 0 we separate variables of the propagation equation (3.1), integrate and normalize on [0, z], 1 z
Z
z 0
∂Pi (ζ, t) 1 = Pi (ζ, t) z
Z
0
z
gi (N, t) − αi ∂ζ
(3.9)
and solve for the optical power at location z in terms of the input power, Pi (z, t) = Pi (0, t)e[gi(N,t)−αi ]z .
(3.10)
At the end of the device the optical output is Pi (L, t) = Pi (0, t)e[gi(N,t)−αi ]L .
(3.11)
Chapter 3 Core photonic state-space model
3.1.4
49
State update equation
Integrating the multichannel rate equation (3.2) on z ∈ [0, L] and normalizing by 1/L we have
m ¯ dN(t) 1 X gi (N, t)P¯i (t) I(t) , = − R(N, t) − dt qV A i=0 ~ωi
(3.12)
where Leibnitz’s Rule has been employed to interchange the time derivative and the spatial definite integral in (3.12). In Eq. (3.12) we have defined
and
¯ , 1 I(t) L
Z
L
1 P¯i (t) , L
Z
L
I(z, t)∂z
(3.13)
Pi (z, t)∂z.
(3.14)
0
0
Substituting Eq. (3.10) into (3.14) gives Pi (0, t)[e[gi (N,t)−αi ]L − 1] . P¯i (t) = [gi (N, t) − αi ]L
(3.15)
Because N is assumed to be spatially invariant, ¯ (t) , 1 N L
Z
L
N(t)∂z
(3.16)
0
¯ (t) = N(t). Subis simply given by N(t) everywhere and we omit the overscript bar, N stituting (3.15) into the rate equation (3.12) we get m ¯ dN(t) 1 X gi (N, t)Pi (0, t)[e[gi(N,t)−αi ]L − 1] I(t) = − R(N, t) − , dt qV V i=0 ~ωi [gi (N, t) − αi ]
(3.17)
which represents the general nonlinear state equation for any recombination R(N, t) and any set of gains g(N, t). This state update equation is nonlinear in the carrier concentration but affine in both electrical and optical inputs (i.e. the inputs can be factored out algebraically).
3.1.5
Nonlinear control form summary
We define the state variable as ¯ (t). x(t) , N
(3.18)
50
Section 3.2 Linearized state-space model
and the electrical and optical inputs as ¯ ue (t) , I(t)
(3.19)
uo (t) , P (0, t).
(3.20)
and
Finally, we define the electrical and optical outputs as ¯ ye (t) , I(t)
(3.21)
y o (t) , P (L, t).
(3.22)
and
Concatenating the electronic and optical domains yields total input and output vectors defined by u(t) ,
"
y(t) ,
"
and
#
(3.23)
#
(3.24)
ue (t)
uo (t)
ye (t)
y o (t)
.
The nonlinear system state update equation (3.17) can then be written as m ue (t) 1 X gi (x, t)[e[gi (x,t)−αi ]L − 1] d x(t) = − R(x, t) − uo,i(t), dt qV V i=0 ~ωi[gi (x, t) − αi ]
(3.25)
with nonlinear output relation (from Eq. (3.11)) h i y(t) = 1 e[g0 (x,t)−α0 ]L · · · e[gm (x,t)−αm ]L u(t).
(3.26)
We have now converted the partial differential SOA equations (3.1) and (3.2) into a state-space form suitable for linearizing.
Chapter 3 Core photonic state-space model
51
SOA D U (s)
X(s) b
(s − A)−1
Y (s) c
K(s) Figure 3.3: Linear model (A, b, c, D) of a semiconductor optical amplifier used in the control designs. Either the state X(s) or output Y (s) may be fed back to the input U (s) through controller K(s). The control simulations are applied to the nonlinear SOA model.
3.2
Linearized state-space model
The general state-space dynamic model Eqs. (3.25) and (3.26) is nonlinear. Here we seek the linearized model (A, b, c, D) by taking the Jacobian of Eqs. (3.25) and (3.26) about equilibria x0 , u0 , as in Eq. (2.16). The equilibrium point x0 is found numerically by setting x(t) ˙ = 0 for a given u0 in Eq. (3.25). The output equilibrium points are simply y 0 = y(x0 , u0 ) using Eq. (3.26). This model is depicted in Fig. 3.3 where the variables have been transformed to the Laplace domain to allow straightforward block multiplication (U (s) = L {u(t)}, X(s) = L {x(t)}, Y (s) = L {y(t)}).
Linearizing Eqs. (3.25) and (3.26) about an arbitrary equilibrium point (x0 , u0 ) [180]
52
Section 3.2 Linearized state-space model
Table 3.1: Parameters for simulations (all channels). Parameter Symbol Value Unit Length L 500 µm Active width W 3 µm Active height H 80 nm Confinement Γ 0.15 — Waveguide loss α 40 cm−1 Differential gain a 2.78 × 1020 m2 18 Transparency carrier density Ntr 1.4 × 10 cm−3 Linear recombination RA 1.43 × 108 s−1 Bimolecular recombination RB 10−16 m3 s−1 −41 Auger recombination RC 3 × 10 m6 s−1
we compute for general gain g(N, t) m X ∂R(x, t) uo,i,0 A=− − ∂x V ~ωi [gi (x0 ) − αi ] x0 i=0 [gi (x0 )−αi ]L gi (x0 )(e[gi (x0 )−αi ]L − 1) ∂gi (x) × 1 + gi (x0 )L e −1− gi (x0 ) − αi ∂x x0 h i [g0 (x0 )−α0 ]L ) gm (x0 )(1−e[gm (x0 )−αm ]L ) b = 1/qV g0 (x0V)(1−e · · · ~ω[g(x0 )−α0 ] V ~ω[g(x0 )−αm ] 0 u Le[g0 (x0 )−α0 ]L ∂g0 (x) o,0,0 ∂x x0 c = . . . [gm (x0 )−αm ]L ∂gm (x) uo,m,0 Le ∂x x0 1 0 0 e[g0 (x0 )−α0 ]L 0 elsewhere D = . .. . [gm (x0 )−αm ]L 0 elsewhere e
(3.27a)
(3.27b)
(3.27c)
(3.27d)
Hence, we have obtained a time-invariant SOA model that is linear in the state and inputs. To verify the linear model (A, b, c, D), we compare its response to the response of the nonlinear system (Eqs. (3.25) and (3.26)) numerically integrated in time. For example we consider two identical optical channels with linear gain Eq. (3.6) and recombination in the form of R(N, t) = RA N +RB N 2 +RC N 3 [146]. We use the parameters of Ref. [181] listed in Table 4.1 with λ0 = 1550 nm for both channels throughout. Taking equilibrium
Chapter 3 Core photonic state-space model
53
inputs as h i uT0 = 150 mA 0.6 mW 1 mW ,
(3.28)
x0 = 3.891875 × 1024 m−3
(3.29)
A = −9.90 × 109 s−1 , h i b = 5.20 A−1 −248 W−1 −248 W−1 × 1034 m−3 s−1 , 0A × 10−26 m3 s, c= 3.06 W 5.09 W 1 0 0 . D= 0 24.4 0 0 0 24.4
(3.30a)
we calculate the equilibrium state
and linear coefficients
(3.30b)
(3.30c)
(3.30d)
Because the pole A resides in the negative real half plane, the SOA is already internally stable by the definition of stability, Eq. (2.39). Therefore a controller is not needed to stabilize the SOA. Figure 3.4 shows simulation results comparing the nonlinear model with the linearized model for 20% input step modulations: (a) shows the optical outputs and (b) shows the state due to modulation of the optical inputs (c) and electrical input (d). For each modulation the linear model follows the nonlinear one very closely in terms of transient and steady-state responses. Cross-talk is evident between the optical channels during optical modulation because the channels are coupled together through the state (i.e. each optical channel draws from the common population inversion carrier density). Some deviation is noticeable for larger modulation (typically in excess of 20% modulation from equilibrium) but overall performance is qualitatively good. Quantitative agreement is difficult to generalize because it depends on the relative nonlinearity of the dynamics in Eqs. (3.25) and (3.26), although specific deviations can be calculated either by subtracting the linear and nonlinear models, or by calculating the second-order differential terms and weighing them against the linear terms Eq. (3.27). It is up to the
54
Section 3.2 Linearized state-space model
35
(a)
linear nonlinear
output power [mW]
30 chnl 1
25 20
crosstalk chnl 2
15
current [mA]
input power [mW]
carrier density 18 -3 [x10 cm ]
10 4.1 4
(b)
linear nonlinear
3.9 3.8 3.7 1.2 1
(c) chnl 1
0.8 chnl 2
0.6 0.4 190 170 150 130 110
(d)
0
1
2
3
4
5
6
7
time [ns]
Figure 3.4: Verification of the linear model Eq. (3.27) with nonlinear model Eqs. (3.25) and (3.26) by direct numerical integration. (a) Optical output and (b) state modulation due to (c) optical input modulation and (d) electrical input modulation. All step modulations are 20% of nominal values. Channel cross-talk is evident during optical modulation.
Chapter 3 Core photonic state-space model
55
designer how much error is permissible in a given application. In fact, the output feedback controller in Section 3.3 works well for 100% modulation because of the action of the feedback. For applications where there are several operating points, a controller may schedule (switch) its gain based on the linear model evaluated at several different sets of pre-calculated equilibria: if the SOA drifts too far from one set of equilibria, the controller can schedule a design from the linear model using another set of equilibria [179].
3.3
Feedback control
In this section we employ our linear model (A, b, c, D) to design two SOA control schemes to suppress interchannel cross-talk: electronic state feedback and optical output feedback. For both cases we use constant feedback K(s) = k for simplicity in demonstrating the concepts, although the controller could certainly be generalized to include dynamic terms (differentiators and integrators) to improve transient characteristics and eliminate steadystate errors. The linear model greatly simplifies the controller design and the controller is then applied to the nonlinear SOA model Eqs. (3.25) and (3.26) to verify the performance. Because the state space is one-dimensional (i.e. A ∈ R1×1 ), the system is trivially controllable 1 provided b 6= 0. The term 1/qV ∈ b where q is the electronic unity charge and V the active region volume, so clearly b 6= 0 and our system is controllable.
3.3.1
State feedback: suppressing cross-talk electronically
The general state feedback equation (2.42) takes the specific form x(t) ˙ = A + bk x(t),
(3.31)
while the output equation (3.32) becomes y(t) = c + Dk x(t),
(3.32)
where k is the feedback column vector that bridges the state to the inputs. Consider the SOA system verified in Section 3.2, again with linear gain Eq. (3.6), recombination R(N, t) = RA N + RB N 2 + RC N 3 , and two identical optical channels. 1
Controllability implies that the system may be steered from any one state to any other state given the appropriate input.
56
Section 3.3 Feedback control
State dynamics in the frequency domain are given by sX(s) − x(0) = AX(s) ⇒ X(s) = (s − A)−1 x(0),
(3.33)
so the SOA has a single pole at s = A. Pushing the pole further negative causes the state to converge more quickly to x0 , thereby reducing excursions in the state due to the inputs. Letting the desired pole location be s′ we calculate the control gains by solving for k in s′ −A+bk = 0. There is some flexibility in choosing a pole location that achieves the control objective (suppressing cross-talk) with reasonable controller gain. Because we employ feedback from the state x into the electrical drive current ue , k2 = k3 = 0. For illustration we choose s′ = 5A and calculate k1 =
A − s′ = 7.62 × 10−25 m3 A. b1
(3.34)
Figure 3.5 shows that state feedback into the electrical drive current (d) suppresses cross-talk between optical channels (a) due to optical input modulation (c). The negative feedback reduces fluctuations in the state (Fig. 3.5 (b)): as optical channel 1 steps up, the state wants to step down as carriers are depleted by stimulated emission, but this step down in state is converted into a step up in drive current, thereby refilling the population inversion. Because the population inversion is held relatively constant, the channels decouple and interchannel cross-talk collapses. With the decoupling, modulated channels receive better steady-state gain and better transient response. Greater controller gain k1 yields better output performance at the expense of greater and faster swings in the drive current. To illustrate, let the drive current seen by the SOA be µ1 (t), defined by µ1 (t) = u1 (t) − k1 x(t)
(3.35)
from Fig. 3.3. Using the Laplace transform of Eq. (2.42) to find the state x(t), setting u1 (t) to a constant reference, and assuming that the total change in optical input uopt (t) is a step function with amplitude ∆uopt , the magnitude of drive current change called for by the controller is given by 1 − e−(|A|+|b1 k1 |)t ∆µ1 (t) = |b1 | + |A/k1 |
m X i=2
bi
!
∆uopt .
(3.36)
As the controller gain k1 increases, the denominator of ∆µ1 (t) decreases, thus ∆µ1 (t)
Chapter 3 Core photonic state-space model
current [mA]
input power [mW]
carrier density 18 -3 [x10 cm ]
output power [mW]
30
57
(a) state feedback open-loop chnl 1
25 20
chnl 2
15 10 4
(b) state feedback open-loop
3.9 3.8 1.2 1
(c) chnl 1
0.8
chnl 2
0.6 0.4 160 155 150 145 140
(d)
0
1
2
3
4
5
6
7
time [ns]
Figure 3.5: Constant negative state feedback applied to the electrical input (k = [7.62 × 10−25 m3 A 0 0]) of the nonlinear model Eqs. (3.25) and (3.26). Cross-talk in the outputs (a) is essentially eliminated because the state (b) is forced to be constant by feedback into the drive current (d). (c) shows the optical inputs modulated with 20% steps from equilibrium.
58
Section 3.3 Feedback control
increases with k1 and so greater drive current swings are required as the transient response fades. The time constant of the exponential transient τ = (|A| + |b1 k1 |)−1 decreases as k1 increases, and so faster drive current swings are required.
3.3.2
State observer
In order to use state feedback when the state is difficult to measure, an observer is needed that estimates the state based on the SOA’s input–output behaviour and the linearized model. Appendix B discusses the method for designing an observer. Because A is scalar and c 6= 0, the system is trivially observable i.e. given the inputs and outputs, the state can be reconstructed fully. ˆ c ˆ vary ˆ b, ˆ, D) To test a state observer by simulation, we let the estimated model (A, from the actual system by up to 20%. For example, we take Aˆ = −1.148 × 1010 i h ˆ b = 5.76 −167 −167 −167 × 1034 0 1.28 ˆ= c × 10−26 2.56 3.84 0.879 0 0 0 0 12.5 0 0 ˆ D= , 0 0 12.5 0 0 0 0 12.5
(3.37a) (3.37b)
(3.37c)
(3.37d)
in standard units. We design the pole to be at 5Aˆ = −5.74 × 1010 , and calculate h i ℓ = 5.99 5.99 5.99 5.99 × 1035 .
(3.38)
The percent state error is shown in Fig. 3.6 for 20% optical input modulation. Despite the errors in the model, the relative state error fluctuates less than 0.2%. Designing the state feedback to be from optical channels into the electronic drive
Chapter 3 Core photonic state-space model
59
0.15
relative error (x - xhat)/x [%]
0.1 0.05 0 -0.05 -0.1 -0.15 -0.2 0
1
2
3
4
5
6
7
time [ns]
Figure 3.6: State observer relative error with 20% model uncertainty, 20% optical input ˆ modulation; pole set at 5A. current with dynamics 5Aˆ we calculate h i k = 7.98 × 10−25 0 0 0 .
(3.39)
Simulating under the same conditions as above, we see in Fig. 3.7 that the state estimate is still very good, although it suffers slightly because the observer dynamics are no faster than the feedback dynamics.
3.3.3
Output Feedback: Suppressing Cross-talk Optically
Measuring the state—the average population inversion along the length of the SOA—is difficult in real time. Although the observer circuit above could be constructed in parallel to estimate the state, the outputs are much easier to measure physically and so we now consider direct output feedback. Tapping the outputs and feeding them back into the plant yields constant output feedback, illustrated in the system block diagram Fig. 3.3 taking the branch that feeds the output Y (s) back to the input through constant controller K(s) = K. Optical outputs are readily available and easily measured, so this control scheme lends itself to
60
Section 3.3 Feedback control
1.5
relative error (x - xhat)/x [%]
1
0.5
0
-0.5
-1
-1.5 0
1
2
3
4
5
6
7
time [ns]
Figure 3.7: State observer relative error with estimated state feedback, 20% model unˆ certainty, 20% optical input modulation, and poles (for ℓ and k) set at 5A.
a straightforward implementation. The general output feedback governing equation (2.54) has the specific form −1 x(t) ˙ = A + bK I − DK c x(t)
(3.40)
for our particular model, and the output relation becomes −1 y(t) = I − DK cx(t).
(3.41)
Note that unlike the state feedback design above, the control K here is a matrix because it bridges the vector outputs to the vector inputs. Consider the same SOA system from above, and add a third data channel and a dedicated optical control channel with equilibrium inputs now set to uT0
h i = 150 mA 1.0 mW 1.2 mW 0.8 mW 0.0 mW .
(3.42)
The terms in uT0 are the drive current, optical power in the control channel, and optical powers in data channels 1–3 with channel 3 initially dark. The linear coefficients
Chapter 3 Core photonic state-space model
61
Eq. (3.27) are re-evaluated given these new input equilibria. To sum the optical outputs and loop their total back into the control optical input with moderate negative scaling, we set 0 0 K = −2 0 0 0
0 0 0 0
1 1 1 1 0 0 0 0 . 0 0 0 0 0 0 0 0
(3.43)
Figure 3.8 shows the results of constant output feedback into the optical control channel. When channel 2 turns off at t = 1 ns (c), crosstalk into channel 1 is suppressed (a) because the control channel responds (d) to keep the state constant (b) by keeping the total input optical power constant at the preset equilibrium value. When channel 2 returns at t = 2 ns, the transient power spike in the open loop is suppressed by the feedback action. The control channel adjusts again when channel 3 comes online at t = 4 ns to suppress crosstalk into the other channels and improve the gain of all three data channels. The control channel must be set at sufficiently high power to accommodate the total power of all added or increased data channels.
3.4
Controlling phase
So far we have only considered optical powers in our modelling and control design, but phase also plays an important role in photonic circuitry. It turns out that phase is easily incorporated into our state-space modelling framework. The phase at the SOA output for each channel i is related to the gain via [170] ∂ 1 φi (z, t) = − γigi (z, t), ∂z 2
(3.44)
where γ is the linewidth enhancement factor. Under the previous assumption that ¯ (t), the gain in Eq. (3.44) loses its z-dependence and we get N(z, t) = N 1 φi (L, t) = φi (0, t) − γiL¯ gi (t) 2
(3.45)
¯ (t) and generally by length integration. Thus the output phase φi (L, t) is a function of N
62
Section 3.4 Controlling phase
(a)
output feedback open-loop
output power [mW]
25 20 chnl 1 15 chnl 2 10 chnl 3
control input power power [mW] [mW]
carrier density 18 -3 [x10 cm ]
5 0 3.8
(b)
output feedback open-loop
3.7 3.6 3.5 1.5
(c) chnl 1
1
chnl 2
0.5 0 2 1.5 1 0.5 0
chnl 3 (d)
0
1
2
3
4
5
6
7
time [ns]
Figure 3.8: Optical constant output feedback applied from the sum of optical outputs into the optical control channel of the nonlinear model Eqs. (3.25) and (3.26). Interchannel crosstalk in the optical outputs (a) from the optical inputs (c) is suppressed. The controlled optical input (d) holds the state (b) relatively constant by negative feedback action. Optical modulations are 100% steps from equilibrium.
Chapter 3 Core photonic state-space model
63
P (0, t) through the gain, as well as the input phase: ¯ φi (L, t) = fi N(t), P (0, t), φi(0, t) .
(3.46)
Hence, we can linearize f and use the linearized output relation ¯ (t) + D φ,p P (0, t) + dφ,φ φ(0, t) φ(L, t) = cφ N
(3.47)
in the same manner we used output relations in the previous output feedback control schemes.2 Furthermore, output power and phase can be grouped together,
and
h iT u(t) , P1 (0, t) · · · Pm (0, t) φ1 (0, t) · · · φm (0, t)
(3.48)
h iT y(t) , P1 (L, t) · · · Pm (L, t) φ(L, t) · · · φm (L, t) ,
(3.49)
so that, in principle, both quantities can be fully controlled via the design criterion y(t) → y desired . However, controllability and observability should be checked for each particular control system because it is likely the case that output power and phase are ¯ alone, just as the powers not independently controllable through control of the state N(t) of two signal channels may not be independently controllable. Measuring the phase of the SOA’s output is more difficult than simply measuring the power because a coherent receiver is necessary (for example, an interferometer with one signal of known phase). Coherent measurement is further complicated by the general situation that each output signal can have its own input phase and experience its own phase change (and possibly chirp) due to the SOA’s action through Eq. (3.45). In such cases a state observer could be configured to calculate each signal’s output phase change ¯ (t) based on the inputs to (3.50)—namely P (0, t) (already needed by an observer) and N (produced by an observer by default design); phase change is given simply by ¯ + Dφ,p P (0, t). ∆φ(L, t) = cφ N(t)
(3.50)
A subsequent control circuit could then use the estimated phase as a control input, as illustrated in Fig. 3.9 (see Appendix B for details on observer design). 2
cφ , Dφ,p and dφ,φ are the appropriate linear coefficients found via the Jacobian as before.
64
Section 3.5 Controlling phase
U (s)
Y (s) SOA
controller
ˆ ∆Φ(L, s) Dφ,p ˆℓ cφ Yˆ (s) ˆ b
s−1 Aˆ
SOA model
ˆ c ˆ X(s)
ˆ D
phase-change estimator Figure 3.9: Phase estimator circuit based on a full-state observer to control output phase ˆ without direct phase measurement. ∆Φ(L, s) is the estimated output phase change (in ˆ ˆ ˆ with estimated Laplace space). The approximate linear model of the SOA is (A, b, cˆ, D) ˆ state X(s) and outputs Yˆ (s). The external inputs are U (s) and the external outputs ˆ are Y (s). Finally, the observer feedback gain is ℓ.
Chapter 3 Core photonic state-space model
3.5
65
Conclusion: a new SOA state-space design framework
We have cast the SOA governing equations in a state-space form and derived a linear multichannel control model for optical power and phase. The model can accommodate any gain function of the form g N(t) . Constant state feedback and output controllers
were employed to suppress interchannel cross-talk electronically and optically during multichannel amplification. The model can be extended to include models for the electronic drive circuitry and more advanced robust design and control (including dynamic feedback and feedforward control as will see in subsequent chapters.
While the state feedback we employed offers generally superior performance over the output feedback, a direct comparison is not fair at this point: we will see In Chapter 6 that parasitics present in the electronic feedback loop temper the type of electronic statefeedback control. The optical output feedback is more straightforward to implement and the controlled response of Fig. 3.8 should be judged on its own by a designer. Regardless, these preliminary results point us in the right direction of inquiry for the work that follows in subsequent chapters, both in terms of overall performance and practical implementation.
66
Section 3.5 Conclusion: a new SOA state-space design framework
Chapter 4 Explicit photonic gain compression state-space model In this chapter, we develop a multichannel SOA state-space model that includes gain compression for the first time; the phenomenological compression constant ǫ that we employ accounts for carrier heating and spectral hole burning [182,183]. Furthermore, we demonstrate that our compressed model is significantly more accurate than the existing uncompressed state-space models of the previous chapter and of Ref. [128]. Our model can be adapted to complex control design and system analysis up to 100 GHz. We start by rewriting the governing equations to include ASE and photonic effects on gain in Section 4.1, and by deriving a generalized SOA state-space model in Section 4.2. We then evaluate the forms of gain that permit closed-form solutions to the SOA photon propagation equation, and show that only polynomial compression yields a usable solution. In Section 4.5 we verify the resulting compressed model against pump–probe experiments that elucidate carrier recovery and output dynamics: agreement between the model and experiments is very good, while prior noncompressed models overestimate both magnitude and time constant. Finally, in Section 4.6 we use the compressed model to design and demonstrate a constant feedback controller that regulates the total output power of a single SOA. 67
68
Section 4.1 Governing equations
4.1
Governing equations
Unlike in the previous chapter, gain g is now a function of optical signal powers P and amplified spontaneous emission (ASE) powers Q. The SOA governing equations now consist of an optical power propagation equation for each optical channel i = 0, 1, . . . , m, ∂Pi (z, t) = gi (N, P , Q, z, t)Pi (z, t) − αi Pi (z, t), ∂z
(4.1)
a new set of optical power propagation equations for discretized amplified spontaneous emission (ASE) j = 1, . . . , µ, ∂Q± j (z, t) = ±(gj (N, P , Q, z, t) − αj )Q± j (z, t) + ~ωj Rsp,j (N, z, t), ∂z
(4.2)
and a population inversion density rate equation ∂N(z, t) I(z, t) = − Rnr (N, z, t) ∂t qV µ m 2 X gj (N, P , Q, z, t)Qj (z, t) 1 X gi (N, P , Q, z, t)Pi (z, t) − − A i=0 ~ωi A j=1 ~ωj
(4.3)
that includes ASE explicitly, with all parameters defined in Table 4.1. The factor of 2 in the ASE recombination term accounts for power in both transverse modes [128], while the spontaneous emission Rsp is determined by the cavity modes [172, 173], Rsp,j (N, z, t) = nsp,j (N, z, t)∆fj gj (N, P , Q, z, t) N(z, t) c = gj (N, P , Q, z, t). N(z, t) − Ntr,j 2neff,j L
(4.4a) (4.4b)
The discretization of the ASE spectrum need not necessarily imply the presence of resonant cavity modes—there can be as many discrete ASE channels as is necessary to represent the broadband noise spectrum of the device, as well as the broadband effect of ASE on the carrier population outside the data channel bandwidths [128, 172]. Indeed, our derivation is predicated on the assumption that the SOA is properly AR-coated and that any internal reflection is negligibly weak compared to the single-pass channel gain. For example, take two data channels—red and blue. To model ASE we use three discrete ASE channels: one ASE channel at the red wavelength, one ASE channel at the blue
Chapter 4 Explicit photonic gain compression state-space model
69
Table 4.1: Parameter definitions and values for the models and simulations (all optical channels are taken to be identical). Parameter Symbol Value Unit Inversion carrier density N — cm−3 Active region current I — mA ¯ Total active region current I — mA Nonradiative recombination rate R — cm−3 s−1 Linear recombination RA 2.9 × 108 s−1 −11 Bimolecular recombination RB 4.5 × 10 cm3 s−1 Auger recombination RC 1.5 × 10−29 cm6 s−1 Spontaneous radiation rate Rsp — s−1 Length L 1 mm Ridge width W 2 µm Active region height H 150 nm Active region area A W ×H m2 Active region volume V A×L m3 Effective refractive index neff 3.2 — Data channel power P — mW Set of data channels P — mW ASE channel power Q± — mW Set of ASE channels Q — mW Optical carrier number m — — ASE carrier number µ — — Optical carrier frequency ω 2πc/1560e-9 rad/s Single-pass gain g — cm−1 Confinement & gain factor Γa 75 cm−1 Transparency carrier density Ntr 5 × 1017 cm−3 17 Logarithmic gain parameter [146] Ns 2.4 × 10 cm−3 Waveguide loss α 17 cm−1 wavelength, and one ASE channel outside red and blue that represents the remaining broadband effect of ASE on the carrier population. To produce an input–output state-space model, we must integrate each of these equations over the length L of the SOA.
4.2
General state-space form
Recall that the state-space model we seek consists of a system of first-order ordinary ˙ differential equations x(t) = f x(t), u(t) that keep track of how the states x(t) change due to inputs u(t) through functions f , along with output relations y(t) = h x(t), u(t)
70
Section 4.3 General state-space form
that determine the outputs y(t) due to x(t) and u(t) through functions h. We proceed by casting the revised governing SOA equations (4.1)–(4.3) into this state-space form.
Rearranging the propagation equations (4.1) and (4.2) we have ∂ gi (N, P , Q, z, t)Pi (z, t) = αi + Pi (z, t) ∂z and gj (N, P , Q, z, t)Q± j
=
∂ ± + αj Q± j ∓ ~j ωj Rsp,j (N, z, t). ∂z
(4.5)
(4.6)
Substituting (4.5) and (4.6) into the stimulated emission terms of the rate equation (4.3) we get m 1X 1 ∂ ∂N(z, t) I(z, t) = − Rnr (N, z, t) − + αi Pi (z, t) ∂t qV A i=0 ~ωi ∂z µ 2X 1 ∂ ∂ + − − + αj Qj (z, t) + − + αj Qj (z, t) . A j=1 ~ωj ∂z ∂z
(4.7)
Integrating on z ∈ [0, L] and normalizing by L gives m X ¯ (t) ¯ ¯i (t) dN I(t) P (L, t) − P (0, t) 1 α P i i i ¯ nr (N, t) − = −R + dt qV A i=0 ~ωi L ~ωi µ + ¯ + (t) Q− (0, t) − Q− (L, t) αj Q ¯ − (t) αj Q Qj (L, t) − Q+ 2X j j j j j (0, t) − + , + + A j=1 ~ωj L ~ωj ~ωj L ~ωj
(4.8)
where the over-bars indicate length-averaged quantities, for example ¯ + (t) , 1 Q j L
Z
0
L
Q+ j (z, t)∂z.
(4.9)
For (4.8) to be useful in producing a state-space model it must have an analytical closed form, so there must exist analytical closed-form solutions for Pi (z, t) and Q± j (z, t). Hence, the solutions to (4.1) and (4.2) (the output relations) and (4.8) (the state update equation) constitute the general state-space form for any set of gains g(N, P , Q, z, t), with ¯ (t), inputs P (0, t) and Q± (0, t), and outputs P (L, t) and Q± (L, t). state N
Chapter 4 Explicit photonic gain compression state-space model
71
10
optical power function, a.u.
P P / (1 + 0.1 P) P (1 - 0.1 P) 8
6
4
2
0 0
2
4
6
8
10
optical power P, a.u.
Figure 4.1: Comparison of gain compression forms with uncompressed optical power, ǫ = 0.1.
4.3
Solving the propagation equation with gain compression
Before proceeding to the solution, it is worthwhile comparing the form g(N, P, z, t) = gN (N, t) 1 − ǫP (z, t)
(4.10)
to the more commonly used form g(N, P, z, t) =
gN (N, t) . 1 + ǫP (z, t)
(4.11)
Figure 4.1 compares P (z, t) 1 − ǫP (z, t) to P (z, t)/ 1 + ǫP (z, t) and P (z, t) for ǫ = 0.1:
when P (z, t) > 5, the polynomial form diverges considerably from the rational form and forces the optical power back down to 0; beyond P (z, t) = 10, the optical power actually becomes negative and unphysical. Granted, the interplay between N(z, t) and P (z, t) may moderate this undesired model behaviour, but there remains a power limitation in the accuracy of the polynomial compression form: above a certain optical power, the model fails.
72
Section 4.3 Solving the propagation equation with gain compression A common method of “solving” the propagation equation with compressive gain is to
make waveguide losses susceptible to gain compression [83], P (z, t) ∂P (z, t) = g(z, t) − α , ∂z 1 + ǫP (z, t)
(4.12)
and to gather like terms on each side, integrate on [0, L], and solve for the input–output relation, Z ⇒ ⇒
L
∂P (z, t) +ǫ P (z, t)
Z
L
Z
L
g(z, t) − α ∂z 0 0 0 Z L ln P (L, t) − ln P (0, t) + ǫ P (L, t) − P (0, t) = g(z, t)∂z − αL 0 Z L ln P (L, t) + ǫP (L, t) = g(z, t)∂z − αL + ln P (0, t) + ǫP (0, t) ∂P (z, t) =
⇒ P (L, t)eǫP (L,t) = P (0, t)e[
0 RL 0
g(z,t)∂z−αL+ǫP (0,t)]
(4.13a) (4.13b) (4.13c) (4.13d)
There are two problems with this approach however: first, while waveguide loss is proportional to the photon density it is not strongly affected by gain compressive effects [15, 133, 134, 146] and so the form of (4.12) is just an approximation to allow the variable separation of (4.13a) (P to one side, z to the other); second, the inevitable result (4.13d) is implicit in the output P (L, t) and thus is not a suitable input–output relation. In the multichannel case and for rational gain compression [133] we have for each channel i
gi(N, z, t)Pi (z, t) ∂Pi (z, t) − αi Pi (z, t), = ∂z 1 + ǫi Pi (z, t) + Σi (z, t)
(4.14)
where ǫi is the gain compression factor that accounts for spectral hole burning and carrier heating [182, 183], and where Σi (z, t) ,
m X k=0 k6=i
Pk (z, t) +
µ X j=1
− Q+ j (z, t) + Qj (L − z, t)
(4.15)
is used to emphasize that the sum of remaining channels k 6= i and ASE is an unknown function in z. Eq. (4.14) is an ordinary differential equation in z, specifically an Abel equation of the second type, class A. Although certain forms of Abel equations are analytically solvable, to our knowledge this particular one (4.14) has no analytical closed-form solution due to the presence of the Σi (z, t) term, even with clever algebraic rearrangement
Chapter 4 Explicit photonic gain compression state-space model
73
as in the appendix of Ref. [184]. Even if we restrict the model to only one single channel or total optical power, Eq. (4.14) reduces to ∂P (z, t) g(N, z, t)P (z, t) = − αP (z, t), ∂z 1 + ǫP (z, t)
(4.16)
which again is an Abel equation (second type, class A) without a closed-form solution. Because rational gain compression is analytically intractable, we turn to polynomial gain compression [133] in the multichannel case with form ∂Pi (z, t) = gi (N, z, t)Pi (z, t) 1 − ǫi Pi (z, t) + Σi (z, t) − αi Pi (z, t), ∂z
(4.17)
where again Eq. (4.15) is used for the remaining channels k 6= i and ASE. This ordinary differential equation in z (4.17) is a Bernoulli equation that has the analytical solution R
Pi (z, t) = R L 0
e
gi (z,t)[1−ǫi Σi (z,t)]−αi ∂z R
ǫi gi(z, t)e
gi (z,t)[1−ǫi Σi (z,t)]−αi ∂z ∂z
+C
,
(4.18)
where C is an integration constant. Unfortunately, this solution contains indefinite integrals without explicitly known integrands gi (N, z, t) and Σi (z, t), and therefore cannot be used to find a definite Pi (z, t). These indefinite integrals persist in the solution even for the single-channel case with Σi (0, t) ≡ 0, R
P (z, t) = R L 0
e
g(z,t)−α∂z R
ǫg(z, t)e
g(z,t)−α∂z ∂z
+C
.
(4.19)
From the analysis above we conclude that two assumptions are necessary to include nonlinear gain compression in a state-space model. The first assumption is that N(z, t) = N(t) so that g(N, z, t) = g(N, t). Because the final state-space form collapses the SOA into a lumped input-output device, this first assumption is not restrictive to our model. ¯ (t) The second assumption is that gain compression acts via the inversion level density N and that direct coupling via Σi (z, t) can be ignored. This second assumption is potentially more restrictive, but the compression factors ǫi can be exaggerated to compensate for the additional compression contributed by the other signal channels at the expense of shifting the SOA’s gain-recovery time constant. Furthermore, a single channel can be used to represent the total power of several channels if the channels have roughly equal operating parameters and the gain dispersion over their frequency span is relatively flat.
74
Section 4.4 Solving the propagation equation with gain compression With these two assumptions the solution to the propagation equation (4.17) is gi(N, t) − αi Pi (0, t) , Pi (z, t) = gi (N, t) − αi e−[gi (N,t)−αi ]z + ǫi 1 − e−[gi(N,t)−αi ]z gi (N, t)Pi (0, t)
(4.20)
which is definite and therefore usable with the carrier rate equation (4.8). Note that when ǫi = 0, we recover the output relation Pi (z, t) = Pi (0, t)e[gi(N,t)−αi ]z ,
(4.21)
which is the uncompressed model of Chapter 3. Using (4.20), the average power levels in the SOA are [gi (N,t)−αi ]L 1 ǫ g (N, t)(e − 1)P (0, t) i i i P¯i (t) = . ln 1 + ǫi gi (N, t)L gi (N, t) − αi
(4.22)
Because length-integration collapses the SOA into a length-symmetric lumped device, forward and backward propagating ASE powers are indistinguishable inside the cavity. − Hence we take Qj (z, t) , Q+ j (z, t) = Qj (z, t) for z ∈ (0, L). The solution for Qj (z, t) is
obtained from (4.2) with the same assumptions as for Pi (z, t), Qj (z, t) =
Gj (N, t) + ζj (N, t)ϕj (N, z, t) , 2ǫj gj (N, t)
(4.23)
where Gj (N, t) , gj (N, t) − αj q ζj (N, t) , 4ǫj gj (N, t)Rsp,j (N, t) + G2j (N, t) ξj (N, t) zζj (N, t) + arctanh ϕj (N, z, t) , tanh 2 ζj (N, t) ξj (N, t) , 2ǫj gj (N, t)Qj (0, t) − Gj (N, t).
(4.24a) (4.24b) (4.24c) (4.24d)
Averaging over the device length we obtain ξj (N,t) ξj (N,t) + 1 − 1 1 1 ζj (N,t) ζj (N,t) ¯ Qj (t) = + Gj (N, t) . ln 2ǫj gj (N, t) L [ϕj (N, L, t) + 1][ϕj (N, L, t) − 1]
(4.25)
Chapter 4 Explicit photonic gain compression state-space model
4.4
75
State-space model
The final state-space model consists of the m + 1 output relations gi (N, t) − αi Pi (0, t) Pi (L, t) = gi (N, t) − αi e−[gi(N,t)−αi ]L + ǫi 1 − e−[gi(N,t)−αi ]L gi (N, t)Pi (0, t)
(4.26)
and µ ASE equations
Qj (L, t) =
Gj (N, t) + ζj (N, t)ϕj (N, L, t) , 2ǫj gj (N, t)
(4.27)
along with the state update equation ¯ (t) I(t) ¯ dN ¯ nr (N, t) = −R dt qV µ m ¯ j (t) 4 X Qj (L, t) − Qj (0, t) Q 1 X Pi (L, t) − Pi (0, t) αi P¯i (t) − , + + − A i=0 ~ωi L ~ωi A j=1 ~ωj L ~ωj (4.28)
which contains Eqs. (4.22), (4.25)–(4.27). Single-pass gains gi (N, t) are often taken to be linear in the literature, ¯ t) = Γi ai N ¯ (t) − Ntr,i , gi (N,
(4.29)
although logarithmic forms are more accurate over the full range of carrier densities [146], ¯ t) = Γi ai ln gi (N,
¯ N + Ns,i . Ntr,i + Ns,i
(4.30)
In these gains Γ is the modal confinement fraction, a the incremental gain parameter, Ntr the transparent carrier density, and Ns a fitting parameter [146]; each parameter may be varied over the channels i to model gain dispersion. For nonradiative recombination, we employ the polynomial form 2.3. We are now ready to verify and apply this new state-space model Eqs. (4.26)–(4.28).
76
Section 4.5 Model verification
4.5
Model verification
Here we employ pump–probe experiments and simulations to verify the accuracy of the state-space model; the experiments were performed by colleague Aaron Zilkie [163, 185]. The data from pump–probe experiments are richer than digital pulse experiments because they reveal system dynamics over a wider range of time scales and are not limited by modulator rise times or detector bandwidths.
4.5.1
Experiment and simulation
The essential aspects of the pump–probe experiment are conveyed in the schematic Fig. 4.2; the complete experiment configuration is described in detail in Ref. [185]. A Ti:sapphire-pumped optical parametric oscillator produced a train of nearly transformlimited laser pulses (150 fs full width half maximum, 13 ns period) at 1560 nm. The laser pulses were split into high-energy pump and low-energy probe and reference pulses. Acousto–optic modulators shifted the reference by 52 MHz and the probe by 53.5 MHz. For each period the pulses were recombined such that the reference preceded the pump– probe pair; the time delay between the pump and probe δt was set by the variable delay and varied continuously during the experiment. For each pump–probe pulse pair, the pump pulse entered the SOA first and induced an impulse response; the following probe pulse was then variably delayed and its relative transmission used to measure the output and carrier dynamics of the SOA due to the pump pulse. Accounting for the insertion loss of the SOA, the pump pulse delivered approximately 1 pJ of energy at the input while the probe pulse delivered 0.1 pJ [163]. In the experiment, beating between the probe and reference pulses is detected by a lock-in amplifier and the pump–probe response can be calculated from the output [185]. We verified our state-space model by replicating these essential experimental elements in simulation. Using a 2-ps window about the centre of each pump pulse, the rate equation (4.28) was integrated with a resolution of 10 fs over a duration of 1.5 ns to produce a complete time-response of the SOA induced by injected pump pulses at a given bias. Then, at 10-ps increments across the 1.5 ns of the pump response, we carried out delayed ¯ = δt − 1 ps) as the probe integrations using the pump response’s inversion density N(t initial state for each probe integration (2-ps windows were employed about the centres of the probe pulses). The outputs were calculated with the output relation (4.26). ASE was modelled with a single ASE channel. As for reference–probe beating into the detector, it
Chapter 4 Explicit photonic gain compression state-space model
77
δt pump probe λ/2
reference
t
interferometer
SOA BS
objectives λ/2
BS
NPBS
pump reference VOA PBS λ/2
+53.5 MHz
AOM1
PBS λ/2
laser
probe
+52 MHz
variable delay stage
probe / reference beating 1.5 MHz AOM2 detector
BS
lock-in
1560 nm
(a) Complete laboratory pump–probe experiment, including polarizing beam splitters (PBS), half-wave plates (λ/2), acousto–optic modulators (AOM), beam splitters (BS), a variable optical attenuator (VOA), and a non-polarizing beam splitter (NPBS). Adapted from Ref. [185].
δt
variable delay stage
SOA
t
ω0 + δω probe
pump +δω
laser
ω0
(b) Essential components of the pump–probe experiment necessary for state-space simulation.
Figure 4.2: Pump–probe experiments: (a) actual implementation, (b) simulation.
78
Section 4.5 Model verification
is noted on Fig. 4.2(b) that the detector measures ω0 + δω, so in the simulation we do not actually implement a reference signal (i.e. we can separate wavelengths directly through the model), nor do we need to simulate the interferometer. This simulation method is fast and efficient by leveraging the state-space model and its capability of setting initial states.
4.5.2
Device and model parameters
The SOA under test was an antireflection-coated multi-quantum-well ridge-waveguide architecture with five In0.805 Ga0.195 As0.8 P0.2 active layers and In0.805 Ga0.195 As0.405 P0.595 barriers grown by chemical beam epitaxy on an InP substrate [163]. Its electroluminescence peak was near 1560 nm as determined by measurements of its ASE spectrum [163]. The remaining physical dimensions and parameters are given in Table 4.1. To match the model parameters to the experimental device, we first matched the net gain curves as a function of bias current as shown in Fig. 4.3. In the laboratory, net gain was measured by comparing input and output pulse magnitudes at very low input powers within the linear regime of the SOA. For the model, we set the input power to ¯0 , and then calculated zero, found the equilibrium carrier density N ¯ = g(N, t) − α, gnet (N)
(4.31)
where g(N, t) is the single-pass gain of the sole signal channel using the logarithmic gain form Eq. (4.30). We then used the coefficients of the polynomial nonradiative recombination Eq. (2.3) and the gain compression factor ǫ as fitting parameters in the pump–probe simulation to set the magnitude and time constant of the response: increasing ǫ dampens the magnitude and slows the time constant, while increasing RA (and to a lesser extent RB and RC ) has the opposite effect. If the gain curve fit was upset, we iterated until suitable gain curve and pump–probe fits were obtained; the final values are listed in Table 4.1. There is some disagreement between the observed and simulated gain curves at low bias: the observed gain curve exhibits an inflection point around 10 mA. We attribute this low-bias disagreement to diode parasitics and current leakage [126] effects that are ¯ we set is the pronounced below threshold and that are not modelled—the quantity I(t) direct active region current and is not subject to parasitic electronic effects. However,
Chapter 4 Explicit photonic gain compression state-space model
80
79
experiment model
60
-1
net gain [cm ]
40 20 0 -20 -40 -60 -80 -100 0
10
20
30
40
50
60
70
80
90
100 110
bias current [mA]
Figure 4.3: Experimental and simulated gain curves of a multi-quantum-well SOA. The gain parameters (Γa), Ntr and Ns , are adjusted so that the calculated gain matches the experimental gain. there is excellent fit over the positive gain regime where the SOA would normally operate.
4.5.3
Experiment–model comparison
Figure 4.4 shows general experiment–model agreement over a range of bias points. In the gain regime the response magnitudes and recovery time constants agree well between experiment and simulation; fitted values of ǫ are shown in the inset and they appear to settle asymptotically for larger currents. The initial spikes in the experimental data that last ≤ 1 ps are due to instantaneous two-photon absorption and ultrafast carrier heating, phenomena not captured in the governing SOA equations nor the resulting state-space model. However, up to 100 GHz the state-space model performs extremely well. By contrast, Fig. 4.5 demonstrates the inadequate modelling by the noncompressed models of the previous chapter (with ASE added) and Ref. [128] at 70 mA: with the gain curve fit of Fig. 4.3 the noncompressed model greatly overestimates the response magnitude and recovery time constant compared to the compressed model. Similar overestimations are present at the other bias points.
80
Section 4.6 Model verification
1 0.95 0.9
70 mA
0.85 0.8 experiment model 0.75
0.95 0.9
50 mA -1
ε [W ]
normalized transmission
1
0.85
0.34 0.3 0.26 0.22 0.18
0.8
30
50
70
bias [mA] 0.75 1 0.95 0.9
30 mA
0.85 0.8 0.75 0
200
400
600
800
1000
1200
1400
pump-probe delay [ps]
Figure 4.4: Experimental and simulated pump–probe responses of a multi-quantumwell SOA at bias currents of 30, 50, and 70 mA. Inset: bias-dependent values of gain compression ǫ used to fit the model.
Chapter 4 Explicit photonic gain compression state-space model
compression
1 normalized transmission
81
0.8
0.6 no compression 0.4
0.2
experiment model 0
200
400
600
800
1000
1200
1400
pump-probe delay [ps]
Figure 4.5: Comparison of compressed and noncompressed state-space models of the previous chapter (with ASE added) and Ref. [128] at 70 mA. The compressed model achieves a significantly better fit with experiment when the gain curves of Fig. 4.3 are properly matched over the gain regime.
4.6
Optical feedback control for constant output power
With the compressed model qualified, we apply it to design a controller that regulates total output power to protect devices and detectors downstream from power transients. We demonstrated in the previous chapter that maintaining constant output power keeps the inversion carrier density constant, which, in turn, decouples the data channels and suppresses interchannel crosstalk; a controller operating on total input and output powers in a multichannel system has the same effect even if the individual channels are not resolved by the controller. Figure 4.6(a) illustrates a general total-power control scheme. The total power is measured at the output, sent to the control circuit, and the resulting control signal is added to the data channels on a separate control channel P0 (0, t) so that the total input power into the SOA is P (0, t) = Σ0 (0, t) + P0 (0, t).
(4.32)
Figure 4.7 shows a numerical integration of the compressed model’s response to a series of 20% and 40% optical modulations (representing one and two-channel adds/drops for a
82
Section 4.6 Optical feedback control for constant output power
(a) data channels
P (0, t)
P (L, t) SOA
controller
(b) P (0, t) Σ0 (0, t)
P (L, t) SOA
P0 (0, t) P0 (0, 0)
k
P (L, 0)
Figure 4.6: Feedback controller using total input–output optical powers: (a) general control scheme, (b) constant output feedback controller simulated in Fig. 4.7.
group of 5 channels) as shown in Fig 4.7(c) when using the constant feedback controller of Figure 4.6(b), P0 (0, t) = P0 (0, 0) + k P (L, t) − P (L, 0) .
(4.33)
With this constant feedback, the output power is given by P (L, t) =
(fb − fc ) +
p (fb − fc )2 + 4fa fd , 2fd
(4.34)
where fa =G(N, t) Σ0 (0, t) + P0 (0, 0) − kP (L, 0) fb =kG(N, t)
fc =G(N, t)e−G(N,t)L + ǫg(N, t)fa 1 − e−G(N,t)L /G(N, t) fd =kǫg(N, t) 1 − e−G(N,t)L .
(4.35a) (4.35b) (4.35c) (4.35d)
We design k = −10 for Σ0 (0, 0) = P0 (0, 0) = 1 mW; the bias current is set to a constant 70 mA. The total output power in Fig. 4.7(a) is kept relatively constant because the population inversion density in Fig. 4.7(b) is held constant by the optical feedback channel, despite data channel adds/drops in Fig. 4.7(c). This controller could be realized by tapping off a small percentage of the output light, collecting it with a broadband
Chapter 4 Explicit photonic gain compression state-space model
output power P(L,t) [mW]
20
(a)
open-loop closed-loop
(b)
open-loop closed-loop
83
18
16
14
12
population inversion density 18 -3 N(t) [x10 cm ]
1.02 1.01 1 0.99 0.98 0.97
input power Σ0(0,t) [mW]
0.96 1.4 1.2 1 0.8 0.6
(c)
0
1
2
3
4
5
6
7
time [ns]
Figure 4.7: Regulation of the total output power, using the control scheme of Fig. 4.6(b) with constant controller k = −10: (a) total optical output, (b) inversion carrier density, (c) data channel optical input.
84
Section 4.7 Effect of gain compression on required controller strength
photodetector, and driving a laser of frequency ωi=0 through a high-speed differential amplifier [186] with a reference calibrated to a desired output P (L, 0).
4.7
Effect of gain compression on required controller strength
To probe the effect of gain compression on required controller strength, we examine statefeedback control where we can define controller magnitude easily in terms of the system parameters. Recall From Chapter 3 that for a single control input (either optical or electronic), we have ki =
A − s′ , bi
(4.36)
where i now indexes the controlled input channel. With compression, the coefficient A is given by m ∂ 1 X 1 ∂ αi L ∂ ¯ A=− Pi (N, t) , Rnr (N, t) − Pi (L, N, t) + ∂N V i=0 ~ωi ∂N ~ωi ∂N
(4.37)
while boptical
1 =− V
1 ∂ α0 L ∂ P¯0 (N, t) P0 (L, N, t) − 1 + ~ω0 ∂P0 (z = 0) ~ω0 ∂P0 (z = 0)
and belectrical =
1 , qV
(4.38)
(4.39)
and where the output equation (4.26) is used. We neglect ASE temporarily to make the analysis more straightforward. In order to isolate the effect of compression from the design of the pole s′ , let
so that
s′ = ηA
(4.40)
A ki = (1 − η) . bi
(4.41)
Chapter 4 Explicit photonic gain compression state-space model
85
1.002 optical control electrical control
normalized ki / (1-η) [no unit]
1.0015 1.001 1.0005 1 0.9995 0.999 0.9985 0.998 0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
-1
gain compression factor ε [W ]
Figure 4.8: Effect of compression on required controller strength for electronic and optical control. Thus, we can separate the design from the compression-dependent terms, ki A = , 1−η bi
(4.42)
so that the lefthand side is essentially a measure of required control strength, while the righthand side reflects the magnitude of gain compression. We fixed the bias of an SOA in the middle of the compression regime validated by the pump–probe experiments (I¯0 = 50 mA with P0 (z = 0) = 1 mW), and then varied the compression factor ǫ about the compression value we found at that bias, ǫ = 0.3 W−1 . We then calculated Eq. (4.42) numerically at each value of ǫ (i.e. finding A and bi after solving for each equilibrium point), and the normalized results are plotted in Fig. 4.8 (each dataset is normalized about the value at ǫ = 0.3 W−1 , the nominal value of ǫ at this SOA bias). Required control strength increases with gain compression for optical control (Fig. 4.8 solid line) because gain compression suppresses the effects of adding more photons to the SOA, and thus the same control becomes less effective as compression increases. Conversely, required control strength decreases for electronic control (Fig. 4.8 dashed line) because under gain compression the range of possible optical output powers is compressed, and so smaller changes in supplying the carrier density yield more effective
86
Section 4.8 Conclusion: the first explicit input–output model with gain compression
control.
4.8
Conclusion: the first explicit input–output model with gain compression
In summary, a new SOA state-space model with polynomial gain compression was derived from a generalized state-space model. Furthermore, we offered mathematical background on the forms of compression for which solutions exist. This model extends SOA state space models [127, 128] in significant, nontrivial ways. The model behaved as expected in comparison with the previously-verified uncompressed model—gain was suppressed and recovery time increased. The addition of compression to the model improves the correlation of model and experimental dynamics; this correlation is best elucidated by the pump–probe experiments and simulations we performed. The compressed model achieved significantly better agreement to experimental pump–probe data than previous noncompressed models [127, 128]. We employed the model to design an output power regulator that maintained a relatively constant output power despite disturbances at the input. This model could be used to design and analyze a variety of control schemes or evaluate communication system performance. We also introduced a novel state-space modelling technique for pump–probe experiments that has the potential to simplify and expedite their simulation.
Chapter 5 Equivalent circuit dynamic model We derived linear state-space control models for multichannel SOAs in the previous two chapters to address elementary optoelectronic and photonic control. However, these previous models only consider the SOA’s photonic operation and neglect any parasitic electronic effects of the SOA or surrounding chip [139, 141, 187–196]. Typically, the surrounding electronic circuit introduces significant dynamics that could affect control schemes that modulate the bias current directly; if these circuit dynamics are not considered adequately, the SOA control design may fail. Furthermore, in the photonic models we developed this far the current drawn by the SOA was fixed and unaffected by changes in carrier depletion due to variations in input optical power, and such changes could be used in a feedback circuit [115] or to monitor amplifier performance [132, 197–210]. There is a distinction to be made here: we are not interested in modelling the photonic operation of the SOA with circuit elements as is often done when equivalent circuits are employed [141, 188, 190, 195, 211–229]. Nor do we want cumbersome partial-differential equation models or intractable SPICE-based models [64]. Rather, we seek to incorporate the electronic front-end behaviour of the SOA via an equivalent circuit in a broader state-space model. We proceed by first deriving in Section 5.1 a SOA state-space model based on the photonic model of the previous chapter with gain compression. As shown in Fig. 5.1, our model adds an equivalent circuit to the front end of the SOA to model parasitics that can further delay the arrival of electronic control signals to the active region current. Furthermore, the resulting source voltage is a useful measure of the SOA’s photonic response [115] that we leverage in the subsequent controller design of Chapter 6. 87
88
Section 5.1 Nonlinear state-space equivalent circuit model electronic model drive circuit
ISOA (t)
SOA electronics ¯ I(t)
Is (t)
P in (t)
dynamics
existing statespace model
P out (t)
Figure 5.1: Electronic model signal flow. Is (t) is the bias current (a controllable input), ¯ is the current Isoa (t) is the current delivered into the SOA (a measurable output), and I(t) through the SOA’s active region. The state and input power of the existing SOA model may influence the dynamic behaviour of the equivalent circuit. Isoa vp Is
Rin
ip Lp
Cp
Rp v s
Rs
Cs
vd Csc
I¯
Figure 5.2: SOA model with parasitic network, adapted from Ref. [139]. The active ¯ may differ significantly from the drive current Is (t), and is a function region current I(t) of the optical inputs and population inversion density. All other components are defined in Table 5.1.
5.1
Nonlinear state-space equivalent circuit model
The SOA state-space models we developed in the previous two chapters only account for photonic operation, but for full electronic control design a complete description of SOA dynamics is needed. In this section we derive state-space model that includes SOA electronic dynamics. The equivalent circuit between source and SOA is shown in Fig. 5.2, adapted from Refs. [139,187,188] with components defined in Table 5.1. This particular circuit captures the essential shunt capacitances, series and source resistances, and series inductance of a typical diode parasitic network; further parasitic elements [189], leakage pathways [139],
Chapter 5 Equivalent circuit dynamic model
89
and further active region complexity [229, 230] are added using the same approach. In Fig. 5.2, the drive current Is (t) is set by the user or a control system, while the active ¯ depends on the diode current (through the diode voltage vd (t)) and region current I(t) stimulated current (which is drawn as carriers are used up in stimulated emission). The ¯ current between the source and the SOA, Isoa (t), is dependent on both Is (t) and I(t), and is either measured directly with a series ammeter or calculated from Is (t) − vp (t)/Rin if vp (t) is measured across the source’s terminals with a voltmeter. The first task is deriving a state-space model in terms of the circuit’s linear compo¯ to the optical powers nents (Sec. 5.1.1). We then relate the active region current I(t) incident on the SOA, P in (t) (Sec. 5.1.2). To find the SOA’s impedance, we relate the ¯ and P in (t), and recast vd (t) in terms of the SOA inversion diode voltage vd (t) to I(t) carrier density N(t) (Sec. 5.1.3). Finally, we obtain the complete nonlinear state-space model by rewriting the resulting equations as coupled ordinary differential equations (Sec. 5.1.4).
5.1.1
State-space realization of the equivalent circuit
For the equivalent circuit shown in Fig. 5.2, convenient choices for states are the capacitor voltages vp (t), vs (t), and vd (t), and the inductor current, ip (t). By relating the currents through the capacitors to the time-derivatives of the corresponding voltages, and the voltage across the inductor to the time-derivative of its current, we write all the time derivatives as a set of coupled first-order ordinary differential equations. The current through the first capacitor is Cp so rearranging we get
dvp vp =− − ip + Is , dt Rin
dvp 1 1 1 =− vp − ip + Is . dt Rin Cp Cp Cp
(5.1)
(5.2)
The voltage across the inductor is given by Lp so rearranging we have
dip = vp − Rp ip − vs , dt
dip Rp 1 1 vp − ip − vs . = dt Lp Lp Lp
(5.3)
(5.4)
90
Section 5.1 Nonlinear state-space equivalent circuit model
Table 5.1: Parameter definitions and values for the models and simulations unless otherwise noted (all optical channels are taken to be identical). Values not stated here are either stated in the text or are presented graphically. Parameter Symbol Value Unit Source current Is — mA Source resistance Rin 50 Ω Package capacitance Cp — pF Package inductance Lp — nH Package resistance Rp 1 Ω Chip capacitance Cs — pF Chip resistance Rs 1 Ω Space-charge capacitance Csc 1 pF Inversion carrier density N — cm−3 Active region current I — mA Total active region current I¯ — mA Nonradiative recombination rate R — cm−3 s−1 8 Linear recombination RA 2.9 × 10 s−1 Bimolecular recombination RB 4.5 × 10−11 cm3 s−1 Auger recombination RC 1.5 × 10−29 cm6 s−1 Spontaneous radiation rate Rsp — s−1 Length L 1 mm Ridge width W 2 µm Active region height H 150 nm Active region area A W ×H m2 Active region volume V A×L m3 Effective refractive index neff 3.2 — Data channel power P — mW Set of data channels P — mW ± ASE channel power Q — mW Set of ASE channels Q — mW Optical carrier number m — — ASE carrier number µ — — Optical carrier frequency ω 2πc/1560e-9 rad/s Single-pass gain g — cm−1 Gain compression factor ǫ 0.19 W−1 Confinement & gain factor Γa 75 cm−1 Transparency carrier density Ntr 5 × 1017 cm−3 Log gain parameter [146] Ns 2.4 × 1017 cm−3 Waveguide loss α 17 cm−1 Zero-bias inversion density Ne 5 × 1010 cm−3 Diode ideality factor n 2 — Temperature T 300 K
Chapter 5 Equivalent circuit dynamic model
91
The current through the second capacitor is Cs
dvs vs = ip − , dt Rs
(5.5)
so a final rearrangement gives dvs 1 1 ip − vs . = dt Cs Rs Cs
(5.6)
In matrix form we get −1/(Rin Cp ) −1/Cp 0 0 v˙ v 1/Cp 0 p p " # i˙ 1/L I −R /L −1/L 0 i 0 0 p p p p p p s . = + v˙ s vs 0 I¯ 0 1/C −1/(R C ) 1/(R C ) 0 s s s s s v˙ d 0 0 1/(Rs Csc ) −1/(Rs Csc ) vd 0 −1/Csc (5.7) ¯ is treated as an input in Eq. (5.7), but it is actually a The active region current I(t) ¯ nonlinear function of optical inputs P in (t) and junction voltage vd (t); we obtain I(t) from the photonic state-space model of the SOA.
5.1.2
¯ SOA active region current I(t)
¯ as a function of the In order to obtain the current drawn by the SOA’s active region I(t) optical inputs P in (t), we rearrange the length-averaged inversion carrier density equation (4.28), m
Pout,i (t) − Pin,i(t) αi P¯i (t) + ~ωi L ~ωi µ X ¯ j (t) Qout,j (t) − Qin,j (t) αj Q + 4qL . + ~ω L ~ω j j j=1
X ¯ nr (t) + qL ¯ = qV dN(t) + qV R I(t) dt i=0
(5.8)
The terms in order are the diode currents due to diffusion capacitance, recombination, ¯ is nonlinear in P in (t) (through P ¯ (t)) and N(t), and linear and stimulated emission. I(t) ˙ in N(t).
92
Section 5.1 Nonlinear state-space equivalent circuit model
5.1.3
SOA current–voltage relationship
¯ We now have the current through the SOA’s active region I(t), but need the SOA’s junction voltage vd (t) to complete the circuit model. Hence, we relate the junction voltage vd (t) to the inversion carrier density N(t), vd (t) = θ N(t) .
(5.9)
N(t) = θ−1 vd (t) .
(5.10)
In order to obtain a closed-form solution we must impose the condition that θ−1 exists with
Additionally, we need the time derivative
v˙ d (t) = N˙ (t)
∂θ(t) . ∂N(t)
(5.11)
The function θ can be the Boltzmann approximation [231,232] or another Fermi function approximation [233] provided (5.10) yields an explicit closed-form relation between N(t) and vd (t). Using (5.9) and (5.11) in the last line of the circuit equations (5.7) with the expression ¯ given in (5.8) results in for I(t) m X ¯i P − P ∂θ v θ α P out,i in,i s i ¯ nr + qL Csc N˙ = − − qV N˙ + qV R + ∂N Rs Rs ~ωi L ~ωi i=0 µ X Qout,j − Qin,j αj Q ¯ j + 4qL . + ~ω L ~ω j j j=1
(5.12)
Hence, we have replaced the circuit state vd (t) in (5.7) with the inversion carrier density N(t) of the SOA.
5.1.4
Nonlinear equivalent-circuit space-space model
With the impedance of the SOA, we now have all the pieces we need to complete the ˙ nonlinear model for the circuit of Fig. 5.2. Solving for N(t) in terms of vs (t), N(t), and P in (t) in (5.12), making the substitution of vd (t) (5.9) into v˙ s (t) in (5.7), and assembling
Chapter 5 Equivalent circuit dynamic model
93
the entire system gives # vp " # " " # h i −1/(Rin Cp ) −1/Cp 0 v˙ p 1/C p ip + = Is i˙p 1/Lp −Rp /Lp −1/Lp 0 vs
(5.13a)
ip vs θ − + (5.13b) Cs Rs Cs Rs Cs −1 m X θ Pout,i − Pin,i αi P¯i vs ∂θ ¯ ˙ − − qV Rnr − qL + qV + N = Csc ∂N Rs Rs ~ωi L ~ωi i=0 (5.13c) µ X ¯j Qout,j − Qin,j αj Q − 4qL . + ~ωj L ~ωj j=1 v˙ s =
The first matrix equation is a linear, time-invariant subsystem for the linear equivalent circuit, coupled to the nonlinear photonic rate equation of N˙ (t) through the middle equation in v˙ s (t). The photonic output relations remain unchanged by the variable substitutions and are given by Eqs. (4.26)–(4.27). Figure 5.3 compares the nonlinear (dashed line) equivalent circuit model with the previously-verified nonlinear SOA-only model (dot–dashed line) from Chapter 4 by direct numerical integration of state equations (5.13) using the Livermore solver for ordinary differential equations [176]; 20% step modulations are used to probe the responses because the linearized model (to follow) begins to deviate at this modulation depth. Gain is logarithmic Eq. (4.30) while recombination is polynomial Eq. (2.3). The Boltzmann relation is employed for the junction voltage, N(t) = Ne (eqvd (t)/(nkB T ) − 1),
(5.14)
where Ne is the average inversion level at thermal equilibrium over the whole device (usually on the order of 1010 cm−3 in InP/InGaAsP SOAs [232]). Values for the model parameters are given in Table 5.1 and are representative of previous studies in the literature [188, 232]; the remaining circuit values are Lp = 1 nH, Cp = 10 pF and Cs = 1 pF and are chosen here to be conservative to emphasize the effects of the equivalent circuit. The nominal drive current Is0 is set to 70 mA, Fig. 5.3(d), and the nominal optical power of the single channel is set to 1 mW, Fig. 5.3(c). Due to current division in the circuit, the current seen by the active region of the SOA in the equivalent circuit model is less than the full 70 mA seen by the SOA-only model, and this is reflected in the lower
94
Section 5.1 Nonlinear state-space equivalent circuit model
14 (a)
SOA alone, nonlinear SOA+circuit, nonlinear SOA+circuit, linear
(b)
SOA alone, nonlinear SOA+circuit, nonlinear SOA+circuit, linear
output [mW]
12 10 8 6 4
input [mW]
carrier density 18 -3 [x10 cm ]
2 1.1 1 0.9 0.8 (c)
1.2 1 0.8
Is [mA]
90
(d)
80 70 60 50 0
2
4
6
8
10
time [ns]
Figure 5.3: Comparison of open-loop system responses to 20% step inputs: (a) optical output, (b) inversion carrier density, (c) optical input, (d) bias current. The nonlinear SOA model without the circuit is Eqs. (4.26)–(4.28), the nonlinear SOA model with the circuit in Fig. 5.2 is Eqs. (5.13) and (4.26)–(4.27), and the linear SOA model with the circuit is Eq. (5.18).
Chapter 5 Equivalent circuit dynamic model
95
inversion carrier density, Fig. 5.3(b), and output power, Fig. 5.3(a). The other clear difference is the ringing in the inversion level and outputs due to ringing in the front end of the equivalent circuit under bias current modulation. Other than these expected deviations, there is remarkably good qualitative agreement between nonlinear equivalent circuit model and the SOA-only model. It is worthwhile noting that under optical modulation, the dynamic responses of the full model (4.26)–(4.27) and (5.13) and photonic-only model (4.26)–(4.28) are qualitatively similar. Hence, for all-optical control design the simpler model (4.26)–(4.28) from Chapter 4 can be used without significant penalty. We employ this nonlinear state-space model as the basis for linearization and controller design in the sections that follow. This derivation method above can be used to produce a nonlinear state-space model for alternative linear or linearized equivalent circuits.
5.2
Linearized state-space equivalent circuit model
For notational convenience, let the system state of Eqs. (4.26)–(4.27) and (5.13) be denoted by
h iT z(t) = vp (t) ip (t) vs (t) N(t) ,
(5.15)
iT h u(t) = Is (t) P Tin (t) QTin (t) ,
(5.16)
h iT v(t) = Isoa (t) or vp (t) P Tout (t) QTout (t) .
(5.17)
let the input be denoted by
and let the output be
The form of the linearized state-space model corresponding to (4.26)–(4.27) and (5.13) is then ˙ z(t) =F z(t) + Gu(t)
(5.18a)
v(t) =Hz(t) + J u(t),
(5.18b)
which we denote by the ordered set of the constant coefficient matrices (F , G, H, J ).
96
Section 5.2 Linearized state-space equivalent circuit model These linear system coefficients (F , G, H, J) are obtained by first finding the equi-
˙ librium points for z(t) = 0 with a given set of inputs u(t) = u0 and all the subsequent linearizations are evaluated at z 0 and u0 . To find these, we first set the operating point ˙ u(t) = u0 and then solve z(t) = 0 evaluated at z(t) = z 0 for z 0 . In practice, it is easier to first solve v˙ p (t), i˙p (t), v˙ s (t) = 0 in (5.13) for (vp0 , ip0 , vs0 ) algebraically, then solve N˙ for N0 from (5.13c) numerically. We find
vp0 =Rin (Is0 − ip0 ) Rin 1 ip0 = = Is0 + vs0 Rin + Rp Rin + Rp Rin + Rp Rs Rin Is0 + θ(N0 ). vs0 = Rin + Rp + Rs Rin + Rp + Rs
(5.19a) (5.19b) (5.19c)
Using vs0 in (5.13c) allows computation of N0 , but only numerically. Plotting N˙ against N shows there is a single equilibrium point z 0 for a given operating point u0 . With the component and physical values used throughout and taking Ne = 5 × 1010 cm−3 , we find
vp0
i p0 z0 = = vs0 N
1.19 V 0.13 A 1.06 V 3.58 × 1018 cm−3
,
(5.20)
which is physically consistent (i.e. voltages drop across the circuit and ip,0 < Is,0 = 0.15 A).
The constant coefficient matrix F relates the system state z(t) to its time-derivative ˙ z(t); the eigenvalues of F correspond to the poles of the system’s transfer function and therefore predict the nature of the system’s evolution over time. Computing the Jacobian [180] of (5.13) with respect to the states we get
F =
−1/(Rin Cp )
−1/Cp
1/Lp
−Rp /Lp
0
1/Cs
0
0
0
0
0 , −1/(Rs Cs ) F3,4 F4,3 F4,4 −1/Lp
(5.21)
Chapter 5 Equivalent circuit dynamic model
97
where F34 =
1 ∂θ Rs Cs ∂N
(5.22a)
1 (Csc ∂θ/∂N + qV )Rs Csc vs θ =− − − qV Rnr (Csc ∂θ/∂N + qV )2 Rs Rs m X 1 Pout,i − Pin,i ¯ − qL + α Pi ~ ω L i i i=0 2 µ X 1 Qout,j − Qin,j ∂ θ ¯ − 4qL + αQj ~ω L ∂N 2 j=1 j j 1 ∂θ ∂Rnr 1 − − qV + Csc ∂θ/∂N + qV Rs ∂N ∂N m X ¯ 1 ∂Pout,i ∂ Pi −q + αL ~ω ∂N ∂N i=0 i i µ X ¯ j ∂Qout,j 1 ∂Q − 4q . + αL ~ω ∂N ∂N j=1 j j
F43 = F44
(5.22b)
(5.22c)
All derivatives in (5.22) and subsequent coefficients are lengthy and are listed in Appendix D.
G relates the inputs (bias current, m data channels plus one optional optical control ˙ channel, and µ ASE channels) to the state derivative z(t), and has four rows and m+µ+2 columns as a result. The Jacobian of (5.13) with respect to the inputs yields a sparse matrix G with the nonzero elements given by G1,1 = 1/Cp
(5.23a)
q ∂Pout,i ∂ P¯i G4,i+2 = 1− − αi L ~i ωi [Csc ∂θ/∂N + qV ] ∂Pin,i ∂Pin,i ¯j ∂Qout,j ∂Q 4q 1− . − αj L G4,j+m+2 = ~j ωj [Csc ∂θ/∂N + qV ] ∂Qin,j ∂Qin,j
(5.23b) (5.23c)
The coefficients H and J relate the state z(t) and input u(t) to the output v(t). H has m + µ + 2 rows and four columns; J is square with dimension m + µ + 2. Both are
98
Section 5.3 Linearized state-space equivalent circuit model
sparse, with J diagonal. If the electronic output is taken to be Isoa (t) in Fig. 5.2, we set H1,1 = − 1/Rin
(5.24a)
J1,1 =1;
(5.24b)
if the electronic output is instead vp (t), we set H1,1 =1
(5.25a)
J1,1 =0.
(5.25b)
Regardless, the remaining nonzero entries are given by ∂Pout,i ∂N ∂Qout,j = ∂N
Hi+2,4 = Hj+m+2,4
(5.26a) (5.26b)
and ∂Pout,i ∂Pin,i ∂Qout,j = , ∂Qin,j
Ji+2,i+2 = Jj+m+2,j+m+2
(5.27a) (5.27b)
where again all the derivatives are found in Appendix D and are evaluated at the equilibria (z 0 , u0 ).
Referring back to Fig. 5.3, there is good qualitative agreement between the nonlinear (dashed line) and linear (solid line) equivalent circuit models. For input modulations typically greater than 20%, there is some qualitative separation between the linear and nonlinear models; for a given application it is the designer’s choice how much error is acceptable, although the controller can be designed to switch between several precomputed operating points to lessen discrepancies. We show in Sec. 6.3 that a controller designed at a single operating point works well even with 100% optical modulations.
Chapter 5 Equivalent circuit dynamic model
5.3
99
Conclusion: complete optoelectronic SOA state-space description
In this chapter we first derived a complete optoelectronic state-space control model of a SOA by folding in an equivalent circuit containing parasitics and dynamic impedance. We then linearized the model and validated both the linear and nonlinear models. Now that we have this complete state-space model we apply it to the analysis and design of practical SOA-based controllers. In the next chapter we demonstrate that time delay in the feedback path can destabilize the SOA through phase accumulation. We then apply linear system theory to calculate the best-case stable delay margin for a given controller norm, and find a potentially severe inverse relationship between delay margin and controller norm.
100
Section 5.3 Conclusion: complete optoelectronic SOA state-space description
Chapter 6 Impact of feedback delay on closed-loop stability in semiconductor optical amplifier control circuits In Chapter 3, interchannel crosstalk was suppressed by keeping the inversion carrier density—and therefore the gain—constant. This gain control is realized by feeding back ¯ the deviation of the SOA’s state N(t) − N0 into the active region current I(t). For the ¯ is no longer more general case of a parasitic network before the SOA as in Fig. 5.2, I(t) directly accessible and only Is (t) may be set by the user or controller. In this chapter we illustrate that delay in the feedback path—whether caused by phase delay through a parasitic network or lumped delay—causes instability in an otherwise robust controller design; the eigenvalues of the linearized model direct us to the cause of the delay in Section 6.1. In Section 6.2 we quantify the maximum delay margin for stability using an optimal state feedback controller that serves as the best case for a given device. As intuition suggests, we calculate that stronger feedback requires smaller delay margin for stable behaviour. In fact, the constraint on feedback norm may be so severe as to render the controller ineffective, a surprising result considering the nanosecond response times of SOAs. Hence, guided by the delay–controller relationship we design a hybrid feedforward–feedback controller in Section 6.3 to illustrate that good transient and steady-state regulation is obtained by carefully balancing the feedforward and feedback components.
This control configuration has fast feedforward transient response and
just a small amount of feedback to correct any feedforward modelling errors despite a 101
102
Section 6.1 State feedback into the drive current and system stability
relatively large feedback delay.
6.1
State feedback into the drive current and system stability
Using the same state feedback scheme as in Chapter 3 with the parasitic equivalent circuit present leads to Fig. 6.1(a). Analytically, this feedback is achieved by modifying the first equation in the nonlinear model (5.13) to include state feedback through a constant controller k, v˙ p (t) = −
vp (t) ip (t) kN(t) Is0 − kN0 − + + . Rin Cp Cp Cp Cp
(6.1)
Using the exact photonic model of Chapter 3 augmented with the equivalent circuit, the same operating point (150 mA bias, 1.6 mW total optical input power), and the feedback gain calculated for the SOA-only model (k = −7.62 × 10−25 m3 A), now leads to an unstable closed-loop system, as illustrated by the outward-spreading spirals in the parametric phase diagrams of Fig. 6.1(b). Under a 13% upward step modulation in optical power the components of the equivalent circuit state z(t) begin to oscillate wildly, and the magnitudes of these oscillations increase indefinitely until some part of the system fails or is damaged. Already within 2 ns ∆ip (t) < 150 mA in Fig. 6.1(b) (left panel) and thus the net current ip (t) is flowing back to the source, which implies the controller calls for Is (t) < 0 backwards through the diode. Controllability is not the issue here: the augmented system is fully controllable through the input Is (t) because its controllability matrix (formed via the formalism of Appendix A) is nonsingular [180]. Rather, to avoid these instabilities we must look deeper into the SOA’s performance. The phase diagrams of Fig. 6.1(b) indicate that the front end of the equivalent circuit is oscillating more heavily than the back end near the actual SOA—the swings in vp (t) are roughly 10 times those of vs (t) over the same duration—and so it appears that the path through the circuit is responsible for the instability. However, more insight is obtained from the linear model. First, we construct the closed-loop feedback system ˙ z(t) =(F + Gk K)z(t) + Gu u(t)
(6.2a)
v(t) =(H + J k K)z(t) + J u u(t),
(6.2b)
Chapter 6 Impact of feedback delay on closed-loop stability
103
ip vp Is
Rin
Lp
Rp v s
Cp
Rs
vd
Cs
I¯ N
Csc
k (a) SOA equivalent circuit and state feedback.
-3
cm ]
1.2
18
400 200 0 t
-200
∆ N(t) [x10
∆ ip(t) [mA]
600
0.8 0.4 0 -0.4
t
-0.8 -1.2
-4
-2
0 2 4 ∆ vp(t) [V]
6
-0.2
0 0.2 0.4 ∆ vs(t) [V]
0.6
(b) Phase diagrams; time elapsed is 2 ns.
Figure 6.1: (a) SOA current model with parasitic drive network, adapted from Ref. [139]. State feedback of the state N(t) into the drive current Is (t) is designed to suppress crosstalk between optical channels P in (t). (b) Relative phase portraits of the SOA under state feedback into the bias current from N(t) only; the flow of time is indicated by the arrows. The system—excited by a 13% step modulation in optical power for 2 ns—is unstable, spiraling ever outward until device failure. Analyzing the eigenvalues of the linear model’s closed-loop system matrix F + GK shows the instability is caused by the oscillatory dynamics of the equivalent circuit.
104
Section 6.2 The delay margin for feedback stability
where K is a sparse (m + 2) × 4 matrix with K1,4 = k and where G = [Gk |Gu ] and J = [J k |J u ] have been partitioned over the controlled inputs (subscript k) and external inputs (subscript u); note also that input vector u(t) now excludes any controlled input and thus contains only external inputs. Examining the eigenvalues (poles) of the new system matrix (F + Gk K),
11
−1
−3.33 × 10 s (+3.43 + 15.6j) × 109 s−1 λ(F + KGk ) = , 9 −1 (+3.43 − 15.6j) × 10 s −1.88 × 1010 s−1
(6.3)
reveals that the real parts of two circuit eigenvalues are positive (each +3.43 × 109 s−1 ), and are therefore the causes of the instability. The last eigenvalue belongs to the SOA and has indeed been shifted further negative by the feedback as desired (open-loop value is −7.90 × 109 s−1 ). However, the unstable circuit eigenvalues dominate and destabilize the entire system. Because the feedback gain k remains identical between these two models (with and without the front-end equivalent circuit), the closed-loop phase is responsible for the sudden instability with the circuit. In particular, the delay caused by the parasitic frontend has exceeded the stable delay margin with respect to the SOA’s state N(t). In the next section, we generalize and quantify the effect of delay margin on the closed-loop stability for SOA systems.
6.2
The delay margin for feedback stability
In the previous section our analysis led to the conclusion that the phase lag through the parasitic circuit caused closed-loop instability; equivalently, the time delay for the controller action exceeded the delay margin. Delay in the feedback loop is caused by other sources in addition to the phase lag of the SOA’s parasitics: propagation time through the SOA (∼ 10 ps for a 1-mm device); carrier diffusion times in the detector, gain elements, and drivers (up to 1 ns or more depending on controller complexity); and passive propagation delay in the feedback path, both optical and electronic (at least ∼ 10 ps for the return trip). To characterize the delay–control relationship, we employ optimal state feedback
Chapter 6 Impact of feedback delay on closed-loop stability
P in(t)
SOA + circuit
Pin,0 (t − τ )
Is (t − τ )
105
P out (t)
z(t)
Is (t) delay
K Pin,0 (t)
Figure 6.2: Optimal control schematic. SOA gain is regulated by full state feedback through the minimum-cost K into an optical control channel and drive current.
(Sec. 6.2.1) and determine the maximum delay margin for stability as a function of feedback gain (Sec. 6.2.2). Although state feedback requires measurement or estimation of all the states—a difficult requirement to meet in practice in real time—optimal control minimizes feedback gain and thus we resolve the best case. More practical closed-loop control schemes such as output feedback have stricter delay margins.
6.2.1
Least-squares optimal control
The optimal controller is depicted in Fig. 6.2: the full state z(t) is fed through the constant controller K, and used to drive both the input current Is (t) and an auxiliary optical channel Pin,0 (t). The governing equations Eq. (6.2) are modified to include the lumped delay τ that accounts for signal propagation, controller lag, etc., ˙ z(t) =F z(t) + Gk Kz(t − τ ) + Gu u(t)
(6.4a)
v(t) =Hz(t) + J k Kz(t − τ ) + J u u(t);
(6.4b)
note again that G and J are partitioned over controlled and external inputs and that u(t) contains only external inputs to the system. (We note that for the time-domain simulations the nonlinear version of this model is used, Eqs. (4.26)–(4.27) and (5.13) with the appropriate delayed feedback terms.) The least-squares optimal control framework Eqs. (2.47)–(2.49) is used for the design where we design the penalty matrices (Q, R) to
106
Section 6.2 The delay margin for feedback stability
be diagonal. For example, if we design 31 0 0 0 0 310 0 0 Q= 0 0 310 0 0 0 0 310
R=
" 3.2 0
0 3.2
#
× 104,
(6.5)
the optimal state feedback controller is then K=
"
9.5
−2.5 0.88
2.8 × 10−25
−2.5 −13 −13 −9.0 × 10
−24
#
× 10−3 .
(6.6)
This constant controller minimizes excursions in both the state and controlled input. Because the SOA states ip , vs , and N (vd ) are more model-specific compared to the source voltage vp , we penalize these states more heavily by setting the form of Q to be
q
0
0
0 10q 0 Q= 0 0 10q 0
0
0
0
0 , 0 10q
q > 0.
(6.7)
We choose the form of R to have identical diagonal elements r since neither control input should be favoured a priori. Numerically, (F , G) is poorly conditioned for solving the Riccati equation (2.49) because the matrix elements span so many orders of magnitude. For the optimal controller design we actually scale the model by first defining the scaled state ζ(t) , T z(t),
(6.8)
where T is a constant square matrix with the same dimension as F . Hence, the original state is recovered by z(t) = T −1 ζ(t),
(6.9)
which we then substitute into the general state-feedback state update equation to get ˙ = F T −1 ζ(t) + G u(t) + KT −1 ζ(t) . T −1 ζ(t)
(6.10)
Chapter 6 Impact of feedback delay on closed-loop stability
107
˙ Solving for ζ(t) gives ˙ ζ(t) = T F T −1 ζ(t) + T G u(t) + KT −1 ζ(t) ,
(6.11)
˙ ˜ u(t) + Kζ(t) ˜ ζ(t) = F˜ ζ(t) + G
(6.12)
F˜ , T F T −1
(6.13a)
˜ ,TG G
(6.13b)
˜ , KT −1 . K
(6.13c)
which we rewrite as
with definitions
˜ K, ˜ ζ(t) and then back again to avoid numerical problems In sum, we convert to F˜ , G, in the following calculations. We found that a good choice for T is 1 0 T = 0 0
0 0
0
0 , 0 1 0 0 0 rc (F ) 1 0
(6.14)
where rc (F ) is the reciprocal condition number of F .
6.2.2
Delay margin of the feedback controller
We model the delay using an (n − 1, n)-order Pad´e approximation given by [129] n−1 τ 1− s 2n n , δn (s) = τ 1+ s 2n
(6.15)
where τ is the delay and s the Laplace frequency. This particular approximation is strictly proper and thus suppresses transients at the start of the delay period [234]. We found that there was negligible qualitative improvement beyond a (4, 5)-order delay in the feedback signals, and so that is the delay function we use.
108
Section 6.2 The delay margin for feedback stability
To measure the strength of the controller K we use the Frobenius norm, kKkF =
q
trace(K T K),
(6.16)
because it is related to the magnitudes of the elements K and is therefore representative of the combined efforts of the feedback gains. Note that when K = k is scalar, kKkF = |k|. For a given optimal controller design (Q, R) we calculate the controller K and controller magnitude kKkF . To find the delay margin associated with the controller magnitude, we employ the Constant Matrices Test (Thm. 2.13 of Ref. [235]), summarized in Fig. 6.3. The first step requires composing two new matrices U= and V =
"
" I
0
0 Gk K ⊗ I
0
#
I
−I ⊗ (Gk K)T −F ⊗ I − I ⊗ F T
(6.17)
#
,
(6.18)
where ⊗ is the Kronecker tensor product.1 The closed-loop system is unconditionally stable if and only if at least one of two conditions is met: either the generalized eigenvectors γi of (U , V ) do not intersect the unit circle of the complex plane, or else all the eigenvalues of F + Gk Kγi are identically zero for any generalized eigenvector γi that does intersect the unit circle. Any controller K that violates both these conditions has conditional stability and an upper bound on the stable delay margin [235]. For conditionally stable feedback controllers, the delay margin’s upper bound can be calculated [235]. For each γi = e−jφi —that is, for each generalized eigenvector of (U , V ) that lies on the complex unit circle—we calculate ωi > 0 such that jωi is an eigenvalue of F + Gk Ke−jφi that lies on the positive imaginary axis. The stable delay margin supremum τsup is then given by [235] τsup = min i
φi , ωi
(6.20)
The Kronecker tensor product of two matrices A ∈ Rm×n and B ∈ Rp×q is defined by the block matrix a11 B . . . a1n B .. . .. A ⊗ B , ... (6.19) . . 1
am1 B
. . . amn B
Chapter 6 Impact of feedback delay on closed-loop stability
109
Form A0 = F A1 = Gk K B 1 = A0 ⊕ I B 0 = I ⊗ A1 B 2 = A1 ⊗ I
Form U (B 2 ), V (B 0 , B 1 ) Find σ(V , U)
no
yes σ(V , U ) on ∂D?
Find angles θ
unconditionally stable
Find σ(A0 + A1 e−jθ )
no
Eigenvals have form jω, ω>0
yes
τ¯ = mini (θi /ωi ) Figure 6.3: Flow diagram for the Constant Matrices Test.
110
Section 6.2 The delay margin for feedback stability repeated for many (Q, R) NL model (z 0 , u0 )
(Q, R)
linearize
(F , G) optimal control design
constant matrices test
K
τsup
Frobenius norm
kKkF
Figure 6.4: Algorithm for generating many sets of τsup and K. so that the system is stable for delays τ ∈ [0, τsup ). Thus a typical evaluation of delay margin starts with the design of the optimal parameters Q and R, proceeds with the calculation of the optimal feedback K, and ends by finding τsup , which is either infinite or given by Eq. (6.20) according to the Constant Matrices Test. To map out the complete relationship between τsup and K given system equilibria (z 0 , u0 ), we need to generate many different optimal feedback designs and then run the Constant Matrices Test on each design, as shown in Fig. 6.4. It turns out that varying the parameters (q, r) of (Q, R) exponentially produces a table of values for kKkF with the structure shown in Fig. 6.5. There are two remarkable properties about the values in Fig. 6.5: first, there is symmetry so that only the highlighted staircase needs to be calculated, thereby converting an operation order n2 steps to one of order n steps; second, the values of kKkF along the highlighted staircase increase at the same exponential rate as (q, r) (assuming both q and r are changed at the same rate). Thus, we can quickly produce a wide range of kKkF equally spaced along a logarithmic axis, and then use the Constant Matrices Test on every controller. Figure 6.6 shows a detailed analysis of the constraints on feedback magnitude imposed by finite delay margins in a two-channel system. Panel (a) illustrates the relationship between delay margin and controller magnitude kKkF calculated with Eq. (6.20) using the optimal design Eq. (6.7) of the previous section for a sequence of values of q and diagonal matrices R. The resulting solid line in Fig. 6.6(a) separates unstable closedloop system response (above) from stable response (below). At very low kKkF the delay margin is essentially unbounded, while the delay margin quickly drops for stronger controllers. The dashed lines are obtained by scaling each Lp , Cp and Cs by a factor of 10 to either reduce or increase the circuit delay from the solid line where Lp = 0.1 nH,
Chapter 6 Impact of feedback delay on closed-loop stability
q
111
r g
f
e
d c
h g
f
i
h g
j
i
k j l
b
a
e
d c
b
f
e
d c
h g
f
e
d
h g
f
e
h g
f
i
k j
m l
i
k j
i
h g
Figure 6.5: (q, r) ∈ (Q, R) map for kKkF calculation.
Cp = Cs = 10 pF, for example; hence the delay margin shifts depending on the reactive response of the circuit. For further analysis we mark locations 1, 2 and 3 about the solid line on Fig. 6.6(a). Panel (b) of Fig. 6.6 shows the stability characteristics of locations 1 and 2 of panel (a) with 20% modulation of the modulated “aggressor” channel (the “victim” channel remains idle at the input). Above the solid delay margin line at location 1 the phase portraits of the states spiral outward until the system fails (note that the location 1 spirals are scaled by a factor of 10−3 along both axes); at location 2 the system is stable and the states simply settle asymptotically at new equilibria. The performance at location 2 is further illustrated in panel (c) where the controller magnitude is high: the aggressor channel sees an improved gain profile over time while crosstalk into the idle victim channel is eliminated. By contrast, the performance at the weaker controller at location 3 in panel (a) is shown in panel (c), and it is clear this controller is significantly less effective despite the increase in delay margin. The implications of Fig. 6.6 are potentially severe: even for this slower multi-quantumwell SOA, feedback control could be unfeasible in electronic domain due to the delay margin constrains the controller—at location 2, the delay margin is on the order of mere picoseconds. Again, this particular result is for optimal state feedback and represents the best case, whereas the delay margin line shifts downwards for partial state or output feedback schemes. Hence, to complete our investigation into the effects of time delay we examine the performance improvement of feedforward control.
112
Section 6.3 The delay margin for feedback stability
1e-08
lumped delay τ [s]
1e-09
unstable region
•
1e-10 location 3 1e-11
location 1 • •
1e-12 location 2 1e-13
stable region
1e-14 1e-15 0.001
0.01
0.1 1 10 controller strength ||K||F
100
1000
(a) Delay margin as a function of controller norm: above the line is unstable while below the line is stable. The dashed lines are obtained by scaling each Lp , Cp and Cs by a factor of 10 to either reduce or increase the circuit delay from the solid line where Lp = 0.1 nH, Cp = Cs = 10 pF. -3
unstable location 1 (x 10 ) stable location 2 0.3 t
-3
∆N [cm ]
∆ip [µA]
10 0.2 0.1 0
t
0 t
-10
t
-20 -0.1 -1.2 -0.8 -0.4 0 ∆vp [µV]
0.4
-1.2 -0.8 -0.4 0 ∆vs [µV]
0.4
∆ output power [mW]
(b) Relative phase portrait of the controllers at locations 1 and 2 of (a): at location 1 the states spiral outwards until system failure, while at location 2 the states settle at new equilibria asymptotically. The flow of time is indicated by the arrows.
open-loop strong location 2 weak location 3
aggressor
0.6 0.4 0.2
victim
0 -0.2 0
1
2
3
4
5
time [ns] (c) Time-response of the controller at location 2 of (a): because the controller norm is relative high, the aggressor channel’s gain does not saturate and there is no crosstalk into the victim channel, as desired. Time-response of the controller at location 3 of (a): although the delay margin is significantly higher than at location 2, the constrained controller norm leads to poor output performance for both channels.
Figure 6.6: Time delay margin analysis for the optimal controller depicted in Fig. 6.2 using the parameters in Table I.
Chapter 6 Impact of feedback delay on closed-loop stability
113
P in(t)
feedforward
P out (t) SOA + circuit
Kf Pin,0 (t)
vp (t)
τ delay
Kb (t)
feedback Figure 6.7: Hybrid feedforward/feedback control design. K f is the constant feedforward controller, while K b (s) is the dynamic feedback controller with intrinsic delay τ (including propagation delay of the SOA and optical circuitry).
6.3
Hybrid feedforward–feedback controller
As we have just seen in Sec. 6.2, delay margin in the feedback loop has the potential to restrict the norm and performance of a feedback controller, perhaps to the point where a sufficiently strong and fast controller cannot be realized. A common solution to overcoming the speed limitations of feedback in erbium-doped fibre amplifier control is to employ feedforward control [236]. However, feedforward control has a significant limitation itself in that the model and parameters of the SOA must be well characterized and accurate because the controller cannot self-adjust based on the actual outputs of the system. Hence, we add a weak feedback controller to compensate for any errors induced by imperfect feedforward control. Moreover, the feedback is small enough to afford a delay margin large enough for optoelectronic implementation. Figure 6.7 illustrates the hybrid feedforward–feedback control design. The total power of the incoming data signals is sampled and fed through a constant controller Kf ; if the implementation of Kf has significant delay, the optical channels P in (t) can be relayed through an optical delay line such that feedforward control appears instantaneously at the input of the SOA. Kf actually needs to invert the input signal, and for this function a second SOA with a continuous-wave input at frequency ω0 can be used to generate Pin,0 (t) by cross-gain modulation [138] with negligible delay. For the feedback circuit the source voltage vp (t) is a convenient measure of the pho-
114
Section 6.3 Hybrid feedforward–feedback controller vp (t)
driver
Rf
Pin,0 (t)
vp0
Cf
Figure 6.8: Example implementation of a feedback integrator. The circuit detects the source voltage vp (t), buffers it, compares it to the set point vp0 , filters the result, and then uses the result to drive the optical control channel Pin,0 (t). tonic state of the SOA2 and is readily accessible. When P in (t) steps up or down there is some ripple in vp (t) that we filter out using a first-order low-pass R–C network with a cutoff frequency of 10 MHz, illustrated in Fig. 6.8. Although the low-pass filter slows the leading edge of the feedback signal, a high-pass filter in parallel has little effect due to the relatively low-frequency spectrum of vp (t) (vp is inherently low-pass filtered through the parasitic equivalent circuit from changes in the SOA active region). We must introduce a new state x(t) for the controller to keep track of the integrated result of the filter, x(t) ˙ = − ωc x(t) + vp (t) − vp0 Pin,0 (t) =kωc x(t),
(6.21a) (6.21b)
where ωc = 1/Rf Cf is the cutoff frequency of the filter and k is the feedback gain supplied by the combination of the buffer and driver. To the nonlinear model (5.13) we add the integrator state x(t) and adjust (4.28) specifically to accommodate the optical feedback. For the linear model, we use superposition to add the filtered feedback, # " ˙ z(t) x(t) ˙
=
# #" " F Gµkωc z(t) δ
−ωc
h v(t) = H J µkωc
x(t) " # i z(t) x(t)
+
" # G 0
u(t)
+ J u(t),
(6.22a)
(6.22b)
where µ = [0, 1, 0, . . . , 0]T is a vector that multiplexes the scalar feedback output kωc x(t) back into the inputs, and where δ = [1, 0, . . . , 0] is a vector that demultiplexes vp (t) out from z(t). This linear model is used to design the feedback control parameters k and ωc 2
As carriers are consumed in the SOA active region, the SOA current Isoa (t) increases and the result is reflected in the source voltage vp (t).
Chapter 6 Impact of feedback delay on closed-loop stability
115
based on standard tuning techniques [237]. The eigenvalues of the first matrix coefficient in (6.22a) indicate the performance and stability of the system as a whole. We have specifically designed this controller to avoid dealing with phase relations between coherent signals. The feedback controller Kb (t) measures the terminal voltage of the SOA vp (t) so that the subtraction is an incoherent operation (the subtraction could be achieved by any differential amplifier in the electrical domain, for example). The feedforward controller K f simply needs to invert the total optical power level at its input and scale the result. Hence, the controller provides an optical control channel governed by m X Pin,0 (t) = Pin,0 (0) + Kb (t − τ ) vp (t − τ ) − vp (0) + Kf Pin,i (t) − Pin,i (0) .
(6.23)
i=1
To illustrate the operation of the hybrid feedforward–feedback controller, we purposely introduce feedforward modelling error by miscalculating Kf by +40%, and introduce 1 ns of delay in the feedback path in addition to the inherent delay of the filter. Real errors in K f could result from poor device characterization, device parameter drift with age or temperature, or changing ASE at the input. Figure 6.9 demonstrates that the best combined transient and steady-state response for both channels is obtained with the hybrid controller when the aggressor channel is modulated +100% from 1 to 2 mW (over 0–10 ns) and -100% from 1 to 0 mW (over 20–30 ns). Gain is enhanced in the aggressor channel (a) while crosstalk into the idle victim channel (b) is suppressed most effectively with the hybrid control. Feedforward-only control introduces steady-state error, whereas feedback-only control suffers from very poor transient response due to the closed-loop delay. Examining the components of the hybrid control signal in (c) shows that the calculation error in feedforward component is suppressed as the feedback correction arrives. The system response illustrated in Fig. 6.9 over the relatively long time scales can be viewed as the envelope or average-power response due to the various control schemes. Figure 6.10 shows the eye diagrams for the same system as in Fig. 6.9, now modulated by a 28 − 1 pseudo-random bit sequence (PRBS) at 10 Gb/s and a depth of 20 dB (-20 dBm to 0 dBm), with no jitter and rise/fall times that are essentially zero. As with Fig. 6.9, there are two data channels and a control channel. The simulation takes place over three PRBS sequences: only the victim channel is modulated for the first two PRBS sequences, and the aggressor channel turns on for the final sequence. Furthermore, because the
Section 6.3 Hybrid feedforward–feedback controller
aggressor channel [mW]
116
6
(a)
no control feedforward + feedback feedforward only feedback only
4
2
0 4
victim channel [mW]
(b)
3
no control feedforward + feedback feedforward only feedback only
2
control [mW]
2
(c)
feedforward + feedback case
1 0 -1
feedforward + feedback feedback contribution feedforward contribution
-2 0
5
10
15
20
25
30
time [ns]
Figure 6.9: Comparison of feedforward, feedback, and hybrid controllers with +40% feedforward modelling error. On [0,10] ns the aggressor channel doubles in power, while on [20,30] ns it drops out completely. The best combined transient and steady-state performance in the outputs (a) and (b) is obtained with a hybrid controller: the feedforward component provides fast transient response while the feedback component ensures steady-state accuracy. (c) shows the components of the hybrid control including the 1-ns-delayed feedback.
Chapter 6 Impact of feedback delay on closed-loop stability
117
no control
feedback
feedback + imperfect feedforward
perfect feedforward
100 ps
100 ps
100 ps
100 ps
aggressor [mW]
5 4 3 2 1 0
victim [mW]
5 4 3 2 1 0
Figure 6.10: Eye diagrams of aggressor and victim channels under various control schemes, modulated by 20 dB 28 − 1 pseudo-random bit sequences at 10 Gb/s; each vertical division is 1 mW and each cell is 100 ps long.
118
Section 6.4 Conclusion: feedback delay constraints are severe for SOAs
controller takes time to settle after starting at t = 0, we do not plot the eye patterns for the first PRBS sequence—this gives the appearance that the controller has been running for a long time prior to the aggressor channel turning on. These eye diagrams lack the typically sloped transitions of “conventional” eye diagrams for two reasons: first, ideal step functions (within the temporal resolution of the simulation) are used so the rise and fall transients are negligible compared to the rest of the bit symbol; second, we neglect timing jitter in the bit transitions. Thus, we need only plot a single period of the eye because the transitions occur only at the exact edges of each cell shown. In Fig. 6.10, the open-loop eyes (first column) close compared to perfect feedforward control (fourth column) due to downward fluctuations in the population inversion density, whether caused by interchannel crosstalk (i.e. an aggressor channel’s input increases) or the victim channel’s own carrier depletion. With significant closed-loop delay the feedback-only controller (second column) responds too slowly to resupply the population inversion; in many cases the controller actually responds with the wrong signal at the wrong delayed time and further closes the eyes. However, with the addition of fast feedforward control (third column)—even with a feedforward gain miscalculated by +40% as before—the eyes are reopened and the average power level approaches that of perfect (feedforward) control. Thus the feedback controller works on smaller errors in the signal vp (t) − vp (0) and serves essentially to correct the average power or envelope over many bit periods.
6.4
Conclusion: feedback delay constraints are severe for SOAs
SOA feedback control is challenging because the controller must respond sufficiently quickly to the sub-nanosecond dynamics of the SOA. Signal detection, processing and routing cause time delay in the feedback path. In turn, time delay imposes an upper bound on the norm of the feedback controller: the greater the norm, the smaller the delay margin. Exceeding the delay margin causes system instability that can damage the SOA or surrounding optoelectronic circuitry. However, reducing the norm also reduces the controller performance. We calculated this delay–control trade-off by deriving a state-space SOA model that contains electronic dynamics, designing a set of best-case optimal state feedback con-
Chapter 6 Impact of feedback delay on closed-loop stability
119
trollers, and employing system stability theory. We have seen that feedback delay places an upper bound on the maximum controller strength for stable closed-loop operation. Equivalently, the stronger the feedback gain or the more parasitic the SOA equivalent circuit, the smaller the lumped time delay margin. Finally, we employed the delay–control relationship of Fig. 6.6 to guide the design of a hybrid feedforward–feedback controller that used relatively weak feedback only to correct steady-state errors due to feedforward–SOA mismatch. With slower feedback loops, we could further employ more sophistical Kalman filter feedback controllers to suppress noisy disturbances as demonstrated in Ref. [238]. These state-space methods of model derivation, performance analysis, and controller design are entirely general, and can be applied to design more sophisticated controllers and functions for active photonic circuitry.
120
Section 6.4 Conclusion: feedback delay constraints are severe for SOAs
Chapter 7 Incoherent optoelectronic control of a semiconductor optical amplifier In this final technical chapter we design a SOA controller that uses a second SOA as the controller itself. As we have just seen in Chapter 6, for any feedback loop there is a strength–delay trade-off that can be quite severe due to the SOA’s rapid response. Hence, we employ an SOA-based controller that is as fast as the SOA we wish to control. Our control objective is again to hold the population inversion density constant to regulate the SOA’s output power and suppress interchannel crosstalk. Ideally, we would like any control signal to be incoherent with the data signals, and furthermore, it should be incoherent with itself around any feedback loop. There are examples of coherent EDFA control where a spectral slice of ASE is sampled and fed back, thereby causing lasing and suppressing the amplifier gain under certain input conditions [104,107,239], but these schemes tend to have upper power limits [104] or abrupt switching characteristics [239] due to the closed-loop lasing action that is required. An incoherent controller in the closed-loop sense should avoid these limitations and further avoids phase effects like chirp. In fact, to the best of our knowledge there is no incoherent optical function that performs a linear difference, y(t) = u(t) − u0 , where u0 is a set reference point. With coherent interference we can achieve y(t) = |u(t) − u0 | when u(t) ≥ u0 by a suitable interferometer, but not y(t) = −|u(t) − u0 | when u(t) < u0 . While it may be possible to bias an interferometer halfway between its on and off output states, such a biasing scheme is limited to half the dynamic range of the interferometer because a quantity of 121
122
Section 7.1 Controller architecture
light cannot attain negative values in the way of voltage (for example). Thus, we use electronic feedback between the controlled SOA and the controller SOA to enable an effective, incoherent difference function. We saw in Chapter 5 that the terminal voltage of the controlled SOA can be used as a useful feedback signal, and we use this signal to drive the bias current of the controller SOA. We apply the conclusion of Chapter 5 that feedforward control yields the fastest control possible. Furthermore, we take full advantage of the linear state-space model developed through Chapters 3–5 to propose a novel design method for biasing the controller SOA. We also use the linear model to dictate the type of feedback that achieves our desired closed-loop objective.
7.1 7.1.1
Controller architecture
Principle of operation
The feedforward component of the controller needs to counteract changes in the inputs to SOA 1: if the average power of a data channel increases, the feedforward signal must decrease proportionally. Hence, the feedforward control must measure the data inputs and send a correction that is equal in magnitude but opposite in sign. Cross-gain modulation in a second SOA (SOA 2) can be used to obtain the needed equal-butopposite signal by sampling the data inputs and inducing interchannel cross-talk onto a dedicated control channel that is input into SOA 2. The design challenge then becomes setting SOA 2 so that it scales the cross-gain modulation in proportion to the data channels arriving at SOA 1. The practice of using cross-gain modulation in a SOA for signal inversion is not new within a signal processing context [138, 240, 241], but here we produce an innovative analytical method for biasing a SOA to set the modulation depth of the inverted signal. Feedforward control does not sample the outputs of SOA 1, and so any variations in the operating points of either SOA may lead to control errors. Operating points may shift by ASE fluctuations, parameter drift (due to ageing or temperature), or other disturbances such as power source brown-outs. By introducing a feedback signal, any variations can be corrected and the output control can be made robust. The simplest method to correct the feedforward scaling is to adjust bias current of SOA 2, and the simplest measure of the output of SOA 1 is the terminal voltage of SOA 1. Thus the
Chapter 7 Incoherent optoelectronic control of a semiconductor optical amplifier data in
κ1
SOA 1
κ2
SOA 2
F1
123
F2 data out
vp1 (t) Pout,c (t)
Pin,c Is2 (t)
out
in electronic controller
(a) Physical realization.
SOA 1 Pin,a (t) Pin,v (t)
Pout,a (t) Pout,v (t)
1−κ1 vp1 (t) κ1
SOA 2
Pin,c
κ2
vp0
Pout,c (t)
k
δ(τ )
Fk (t)
Is2 (t)
(b) Logical signal flow. The electronic feedback controller is surrounded by the dashed box.
Figure 7.1: All-optical SOA control architecture using incoherent feedback: (a) physical realization; (b) system schematic.
feedback circuit samples the SOA 1 terminal voltage, processes it appropriately, and drives the bias current of SOA 2. A physical realization of the proposed controller is shown in Fig. 7.1(a). A small fraction of the data input power is tapped by a coupler κ1 and directed into SOA 2. SOA 2 is pumped by a constant control channel Pin,c , and the action of the coupled data power induces cross-gain crosstalk onto this channel. Optical filter F1 passes only the control channel, which is in turn coupled into SOA 1 via coupler κ2 . Optical filter F2 rejects the control channel at the output of SOA 1 so that only the data channels continue downstream. The feedback controller measures the SOA 1 terminal voltage vp1 (t), and drives the bias current Is2 (t) to correct any feedforward errors over a longer timescale.
124
Section 7.1 Controller architecture
Figure 7.1(b) illustrates the decomposition of this architecture into its logical signals. The splitting and coupling operations result in the scalar multiplications by κ1 , (1 − κ1 ), and κ2 . The feedback controller is contained inside the dashed box: vp1 (t) is compared to the reference vp0 that corresponds to the desired set point of SOA 1, and the difference is then scaled by k and filtered by Fk (t). A lumped delay element δ(τ ) is also included to account for processing and propagation delays in the feedback path.
7.1.2
Device models
Each SOA is modelled with the full equivalent circuit state-space representation with gain compression, as in Eqs. (5.13), of Chapter 5 and output relation Eqs. (4.26),(4.30) of Chapter 4. There are three representative channels: an aggressor channel denoted by subscript a, a victim channel denoted by subscript v, and a control channel denoted by subscript c. The nominal powers in the aggressor and victim channels are set to be equal to represent a worst-case scenario where changes in the aggressor cause a large disturbance to the victim. As for the linearized SOA models, let the linearization of SOA 1 (as in Section 5.2) be represented by the constant coefficients (A, B, C, D) with state x(t), (1 − κ )P (t) 1 in,a i h ˙ x(t) = Ax(t) + B a B v B c (1 − κ1 )Pin,v (t) , κ2 Pout,c (t)
(7.1)
and outputs Ca Pout,a (t) Pout,v (t) = C v vp (t) 1 01×3
Da 0 0
(1 − κ1 )Pin,a (t)
x(t) + 0 Dv 0 (1 − κ1 )Pin,v (t) κ2 Pout,c (t) 0 0 0
(7.2)
and similarly let the linearization of SOA 2 be represented by the constant coefficients (F , G, H, J ) with state z(t),
h i κ1 Pin,a (t) ˙ z(t) = F z(t) + Ga Gv GI κ1 Pin,v (t) , Is2 (t)
(7.3)
Chapter 7 Incoherent optoelectronic control of a semiconductor optical amplifier
125
with sole output Pout,c (t) = H c z(t).
(7.4)
The inputs Pin,c and Is1 are held constant and so they do not affect the linearized responses, thereby simplifying the analysis. The feedback controller consists of a constant gain k, a filter Fk (t) with scalar linear model (Lk , Mk , Nk , Rk ) and state uk (t), and parasitic delay of τ provided by a 5th -order (n,n-1) Pad´e approximation, Eq. (6.15). ASE is modelled with four channels, one at each of the three signal wavelengths (λa , λv , λc ) and one auxiliary to represent the remaining ASE spectrum. ASE power levels are always taken to be 20 dB down from the corresponding signal level, and the auxiliary channel is taken to be the sum of the other three ASE channels.
7.2 7.2.1
Robust design using state-space modelling
Preliminary system considerations
For the purposes of obtaining a tangible design, we assume the coupling coefficients are fixed at k1 = 0.1, k2 = 0.9, so that a small fraction of the input power is tapped off into SOA 2, and that most of the control power is directed into SOA 1. We assume also that Is1 is fixed for a given gain through SOA 1 (note that this is not strictly true since adjustments to SOA 2 will impact the gain of SOA 1 and a designer would likely readjust Is1 to compensate). Hence the feedforward design entails setting b and Is2 such that (1 − κ1 )Pin,a (t) + κ2 Pout,c (t) → 0
(7.5)
despite unforeseen changes in Pin,a (t). Before launching into the design, it is wise to check that the two SOAs are in fact controllable with the signals we intend to use. For SOA 1, the controllability matrix through the control input CSOA1
h i 2 3 = B c AB c A B c A B c
(7.6)
126
Section 7.2 Robust design using state-space modelling
and the observability matrix through the terminal voltage
OSOA1
[1 01×3 ] [1 0 ]A 1×3 = [1 01×3 ]A2
(7.7)
[1 01×3 ]A3
each have full rank. Thus, SOA 1 can be driven by SOA 2 to any state and the terminal voltage of SOA 1 is a suitable error signal. The other quantity to check is the controllability of SOA 2 via its drive current, and the corresponding matrix h i CSOA2 = GI F GI F 2 GI F 3 GI
(7.8)
is also full rank.
7.2.2
Feedforward design
There are two SOA 2 bias points to be set in the feedforward design: the control channel power Pin,c , and the nominal drive current Is2,0 . These two parameters are constrained by the input objective Eq. (7.5), or equivalently by −
κ2 Pout,c (t) = 1, (1 − κ1 )Pin,a (t)
(7.9)
a form that we will find more useful. Let the lefthand side of 7.9 be κ2 Pout,c (t) (1 − κ1 )Pin,a (t) κ2 H c z(t) =− (1 − κ1 )Pin,a (t)
µ ,−
(7.10a) (7.10b)
Thus, in order to satisfy (7.9) we need µ = 1 by setting (Is2,0 , Pout,c ) appropriately. The object now becomes finding an expression for how SOA 2’s state z depends on these inputs to SOA 2. We can simplify the analysis at DC by ignoring the delay and transient feedback. Assuming that Is1 (t) and the average power of Pin,v (t) are constant and at their equilibrium values, we can discard these quantities. Setting the time derivatives for each SOA to zero
Chapter 7 Incoherent optoelectronic control of a semiconductor optical amplifier
127
yields 0 =Ax(t) + B c κ2 H c z(t) + B a (1 − κ1 )Pin,a (t)
(7.11a)
0 =F z(t) + Ga κ1 Pin,a (t) + Rk kx(t),
(7.11b)
where k = [k 0 0 0] to pick out the first state term vp1 (t) from x(t), and where Rk is a 4 × 4 matrix entirely zero save the value Rk in the (1, 1) position to direct the feedback into Is2 (t). Solving for the state x(t) from (7.11a) x(t) = −A−1 B c κ2 H c z(t) − A−1 B a (1 − k1 )Pin,a (t)
(7.12)
substituting x(t) into (7.11b) and solving for z(t) gives z(t) = − F − Rk kA−1 B c κ2 H c
−1
Ga κ1 − Rk kA−1 B a [1 − κ1 ] Pin,a (t).
(7.13)
Eq. (7.13) now describes how z(t) varies with Pin,a (t) and so we can now update Eq. (7.10), µ=
−1 κ2 H c F − Rk kA−1 B c κ2 H c Ga κ1 − Rk kA−1 B a [1 − κ1 ] . 1 − κ1
(7.14)
One further simplification is appropriate that restricts the form of the controller: set Rk ≡ 0 ⇒ Rk ≡ 0, which means no direct passthrough from v p (t) to Is2 (t) that would imply a proportional parallel feedback. In other words, the feedback’s transfer function must be strictly proper and any direct parasitic leakage must be relatively insignificant. With this feedback design restriction we have µ=
κ1 κ2 H c F −1 Ga . 1 − κ1
(7.15)
Hence, we have produced a design equation for the feedforward component of the controller that depends solely on the bias points of SOA 2; any feedback will act only to correct for model errors, operating point shifts, etc. from the feedforward design. The solid line in Fig. 7.2 is a contour map of µ = 1 as a function of (Pin,c , Is2,0 ) (i.e. through H c , F −1 and Ga ) with Is1,0 = 100 mA, Pin,a = 0.01 ≈ 0 mW, and Pin,v = 0.5 mW i.e. the aggressor channel starts near zero, and the average power in the victim channel is 0.5 mW. Hence, given the bias points of SOA 1 the feedforward design entails picking a point somewhere along the solid line µ = 1. At first, intuition might suggest
128
Section 7.2 Robust design using state-space modelling
120 115
SOA 2 bias [mA]
110
SOA 1 gain = 2 dB
105 100 95 µ=1
90 85 8 dB
4 dB
6 dB
80 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
control channel power [mW]
Figure 7.2: Contour map showing values of design parameter µ over ranges of SOA 2 operational points (solid line). Also shown are the gain contours for SOA 1 (dashed lines). picking the minimum near (0.3 mW, 85 mA). However, plotting contours of the input– output gain of SOA 1 also as a function of (Pin,c , Is2,0) shows that we should pick a point for as small a value of Pin,c because κ2 Pout,c tends to consume the useful gain of SOA 1. The noise floor is the ultimate limit at the lower end of Pin,c , but there is also a law of diminishing returns in effect because the bias current increases substantially for only slightly better gain. Because the data channels also enter SOA 2 (scaled by the coupling coefficient κ1 ), their average power affects the saturation of SOA 2 and thus for a given (Pin,c , Is2,0 ), µ changes with respect to Pin,a and Pin,v in our analysis. Figure 7.3 shows the variation of the µ contour curves over (Pin,c , Is2,0 ) for increasing values of Pin,a + Pin,v . As the total signal power increases, the µ curves shift upward and to the right as SOA 2 must be biased higher as the signals consume the available gain.
7.2.3
Feedback design
The feedback controller measures the terminal voltage vp1 (t), compares vp1 (t) to its equilibrium value, and drives the SOA 2 bias current Is2 (t) from the error. As we’ve seen
Chapter 7 Incoherent optoelectronic control of a semiconductor optical amplifier
129
120 115
SOA 2 bias [mA]
110 total data signal power, [0 : 0.5 : 3] mW 105 each curve µ = 1
100 95 90 85 80 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
control channel power [mW]
Figure 7.3: Contour map showing values of design parameter µ over ranges of SOA 2 operational points, for increasing values of Pin,a + Pin,v .
in Chapters 5 and 6, changes in the optical bias of SOA 1 affect the terminal voltage through carrier–current supply, but that a filter may be needed to smooth out some of the ripple caused by parasitics. Because it is not clear what the combined interaction will be between vp1 (t) and Is2 (t)—each is affected by the parasitics of the respective SOAs—we again look to the linear models to dictate the design. Specifically, we use the system transfer function to ensure their is no error remaining at steady-state over the long run. It will be helpful to specify the structures of the various linear coefficients. From the linearization of the equivalent circuit model state update equation Eq. (5.21), the structure of A is
A A12 0 0 11 A 21 A22 −A21 0 A= ; 0 A32 A33 A34 0 0 A43 A44
(7.16)
F has the same structure. The input vector for the aggressor channel Pin,a (t) is h B a = 0 0 0 Ba,4
0 0 0 0
0
iT
,
(7.17)
130
Section 7.3 Design verification
and the output vector corresponding to the population inversion density of SOA 1 is h CN = 0 0 0 1
0 0 0 0
i 0 .
(7.18)
For the full system with the optoelectronic connections between two SOAs described ˙ ˙ above, the system matrix that relates [x(t) z(t) uk (t)]T to [x(t) z(t) u˙ k (t)]T is
A11 A12 0 0 0 0 0 0 0 A21 A22 −A21 0 0 0 0 0 0 0 A32 A33 A34 0 0 0 0 0 0 A43 A44 0 0 0 AF 0 0 , Rk 0 0 0 F F 0 0 N 11 12 k 0 0 0 0 F F −F 0 0 21 22 21 0 0 0 0 0 F F F 0 32 33 34 0 0 0 0 0 0 F F 0 43 44 Mk 0 0 0 0 0 0 0 Lk
(7.19)
where AF is a coefficient that accounts for the coupling of z4 (t) into x4 (t) through Pout,c (t) (i.e. the population inversion densities are coupled through the optical control channel). Using the Final Value Theorem Eq. (2.31), the resulting steady-state transfer function T∞ from Pin,a (t) to x4 (t) (N(t) in SOA 1) has the rational form T∞ =
Ba,4 Lk f1 (A, F )Pin,a , Lk f2 (A, F , AF , Rk ) − Mk Nk f3 (A, F , AF )
(7.20)
where f1,2,3 are scalar functions and where Pin,a (t) is assumed to be a step function with magnitude Pin,a .
From the transfer function (7.20), in order to get T∞ = 0 we must have Lk = 0, Mk 6= 0 and Nk 6= 0; the value of Rk has no effect. Under these restrictions, the controller is an integrator where the value of Rk determines whether the proportional scaling is in series (Rk = 0) or in parallel (Rk 6= 0), although we are restricted to the former under the feedforward assumption we made above for µ.
Chapter 7 Incoherent optoelectronic control of a semiconductor optical amplifier
uncontrolled controlled
5 aggressor channel [mW]
131
4 3 2 1 0
victim channel [mW]
2.6 2.5 2.4 2.3 uncontrolled controlled
2.2
optical control [mW]
1.5 1
feedforward + feedback case
0.5 0 -0.5 -1 0
5
10
15
20
25
30
time [ns]
Figure 7.4: Envelope response of the SOA-controlled SOA with full feedforward–feedback. Top: aggressor response showing improved gain under control. Middle: victim response with crosstalk suppressed by control. Bottom: optical control channel.
132
Section 7.3 Design verification
0.6 N(t) dN(t)/dt critical point
normalized carrier density
0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0
0.2
0.4
0.6
0.8
1
time [ns]
Figure 7.5: Step response of SOA 2, used to determine the delay to the response peak.
7.3
Design verification
Figure 7.4 shows the controller’s envelope or average power response. Both channels are set to 1 mW, and the aggressor channel is modulated to first 2 mW, then 0 mW. Crosstalk is quickly addressed by the feedforward as predicted. After the approximate 1 ns in the feedback loop the feedback kicks in (a subtle bump in the controlled response is evident) and steers the long-term error back to zero. The initial peaks in Fig. 7.4 at 0 ns, 10 ns, and 20 ns in the victim panel are due to the finite response of SOA 2—unlike our ideal case in Chapter 5, this SOA-based feedforward takes some time to produce the control signal as the cross-gain modulation affects the SOA 2 carrier density. Of course, there will also be some lumped propagation delay through SOA 2 that would further broaden the initial peaks. To compensate for these delays, we delay the data channels entering SOA 1 as well: we set a delay equal to the full lumped delay of SOA 2, plus the delay to the peak SOA 2’s inherent photonic response time, as illustrated by Fig. 7.5, where a simple step is input into SOA 2 at t = 0 ns. Finally, putting the three design pieces together—feedforward bias of SOA 2, feedback, and delay offset into SOA 1—we run the same PRBS scheme as we did in Fig. 6.10
Chapter 7 Incoherent optoelectronic control of a semiconductor optical amplifier
uncontrolled
controlled
100 ps
100 ps
133
aggressor [mW]
3 2 1
victim [mW]
0 3 2 1 0
Figure 7.6: Eye diagrams for 28 −1 PRBS sequences at 10 Gb/s for SOA-controlled SOA. The eyes are clearer and better defined for the controlled case to the right. back in Chapter 6: three 28 − 1 PRBS sequences at 10 Gb/s are input into the victim channel, while the aggressor channel turns on only for the final sequence. Figure 7.6 shows the eye diagram result where the controlled eye is more open and concentrated around the two power levels compared to the uncontrolled case.
7.4
Conclusion: a novel SOA-based SOA controller and design techniques
In this chapter we used the models developed in the rest of the dissertation to design an SOA-based SOA controller that uses both feedforward control through cross-gain modulation and feedback control through electronic monitoring and bias adjustment. In particular, we developed a novel design technique that employed the linearized models to set the cross-gain modulation correctly. We also used transfer function analysis to dictate the type of feedback needed to suppress steady-state errors. A remaining issue is that the primary SOA is saturated and so the data channels see small input–output gain and receive relatively low amplification. There are straightfor-
134
Section 7.4 Conclusion: a novel SOA-based SOA controller and design techniques
ward solutions: either put another SOA in the feedforward path before the controller SOA to amplify the controller’s effect, or chain another SOA after the primary SOA for post-amplification. The latter solution is preferred since placing an SOA in the feedforward path is more complicated because it affects the controller, whereas simple chaining after SOA 1 partitions the functions where the first controlled SOA 1 rejects disturbances and cleans the data channels, while the second SOA in the data path provides gain.
Chapter 8 Contributions and conclusions If active photonic integrated circuits are to deliver complex, robust functionality, a rigorous system analysis is needed. Although SOAs—the leading candidates for linear and nonlinear active functionality—have been modelled physically and optimized heuristically, no comprehensive, systematic evaluation existed before the present work. And although SOAs have been controlled with electronic and optoelectronic means, there was little theoretical evidence of robust SOA control until the work in this dissertation. Over the course of the last five chapters we developed a comprehensive state-space modelling framework for SOA-based photonic circuits. We evaluated the limits of analytical modelling and explored in depth the assumptions required to create an analytical state-space model from the governing equations. We then applied the linearized models to design robust integrated photonic circuitry in several novel ways.
8.1
Original contributions
In Chapter 3 (along with Appendix C) we derived a linear time-invariant state-space model for multi-quantum-well SOAs from the governing rate equations, and used the model to design controllers that suppressed interchannel crosstalk and improved transient response with either electronic or optical feedback. The novel contributions were 1. Specification of the form of the propagation partial differential equation required to obtain an explicit input–output relation; 2. A nonlinear state-space formulation of the core photonic dynamics of an SOA, including input–output dynamics of both optical power and phase; 135
136
Section 8.1 Original contributions
3. The time-invariant coefficients of the linearized state-space mode; 4. SOA observer design and demonstration; 5. Cross-talk suppression designs using state and output feedback control; and 6. An optical phase controller that does not measure phase directly, but rather estimates phase with an observer that employs the SOA state-space model. A selection of these results was published in Ref. [127]. In Chapter 4 we derived a SOA state-space model with gain compression from a generalized state-space model using polynomial gain compression. The compressed model fit pump–probe experiments with greater accuracy than the noncompressed model. We employed the model to design an output power regulator that maintained a relatively constant output power despite disturbances at the input. The novel contributions were 7. A solution to the propagation partial differential equation with polynomial gain compression in the single-channel or total-power case; 8. A formulation of the state update ordinary differential equation using the propagation solution with polynomial gain compression; 9. A comparison and validation of compressed and noncompressed state-space models with experimental carrier dynamic data; 10. An innovative simulation method for pump–probe experiments that fully leverages state-space models; 11. A SOA regulator design that operates in single-channel (average power) mode; and 12. A comparison between compressed and noncompressed controllers using the analytical model and numerical solutions. The salient points of these results were published in Ref. [242]. In Chapter 5 we derived a general state-space framework for modelling dynamic equivalent circuits that augmented the existing purely-photonic model of the previous chapters. The electronic model was an interconnection of the linear time-invariant equivalent circuit and the nonlinear SOA. The augmented model agreed well in comparison to the SOA-only model with expected differences (lower inversion level, ringing under bias current modulation). A designer could implement any linear or linearized equivalent circuit
Chapter 8 Contributions and conclusions
137
and produce a state-space model tailored to any physical implementation by using our derivation as a step-by-step guide. The novel contribution was 13. A state-space formulation of the electronic equivalent circuit intrinsically coupled to the compressed photonic model. The results were published in Refs. [243–245]. In Chapter 6 we used the complete linearized optoelectronic SOA model to demonstrated closed-loop instability caused by delay in the feedback path. We employed an optimal controller design to quantify the constraints imposed on SOA feedback control by nonzero delays in the closed-loop path. We demonstrated analytically that there is an inverse relationship between closed-loop delay and controller strength. To overcome the delay-imposed limitations on the feedback controller, we designed a hybrid feedforward– feedback controller that provided both fast transient response and error-free steady-state settling. The novel contributions were 14. An illustration of unstable design brought about by ignoring the electronic parasitics, and an investigation of stability via eigenvalue analysis of the linearized state-space model; 15. The calculation of the relationship between lumped delay margin and controller strength using the linearized state-space models; 16. An order-n implementation of the Constant Matrices Test over least-square optimal control designs; and 17. A hybrid feedforward–feedback SOA controller that takes into account the restrictive delay margin and that employs dynamic feedback to improve steady-state response. These results were published in Refs. [238, 243]. Finally, in Chapter 7 we applied our models to the design of an innovative SOA– SOA controller. A controller SOA provided both fast feedforward control via cross-gain modulation, and stability-restricted slower feedback control via bias modulation. Our design method used several innovative applications of the linearized state-space models. The novel contributions were 18. An innovative SOA controller design that uses a second SOA to control the first, using cross-gain modulation for feedforward control and electronic feedback control;
138
Section 8.2 Future directions
19. A new design technique for biasing cross-gain modulation by using the linearized models at steady-state; and 20. A transfer function method for determining the type of electronic needed given the SOA parasitics in the feedback path. These contributions together satisfy the seven Goals we outlined in Chapter 1: we developed new SOA state-space models that accurately cover the full optoelectronic operation, we applied the models to proof-of-concept control designs, and we evaluated performance issues such as feedback delay and filter design in the context of real SOAbased photonic circuits. We employed step functions for the majority of the proof-of-concept tests and validations because the rise-times at the simulated temporal resolutions are far more severe than those at even 100 Gb/s data rates. Thus, these stepped pulses have essentially instantaneous rise-times and represent the most taxing inputs for the models. However, these models responded just as accurately under the sec2 (t) inputs used in the pump– probe simulations in Chapter 4, as well as the pseudo-random inputs of Chapters 6 and 7. Furthermore, these models respond accurately to truly stochastic and noisy inputs as reported in Ref. [238].
8.2
Future directions
This dissertation has brought control theory to the field of integrated photonics, but it has only hinted at what is possible. There are several immediate extensions possible from the present work. The addition of a second photonic state to the models could capture the ultrafast dynamics such as the initial spikes present in the experimental pump–probe responses of Chapter 4. With extra states it may be possible to capture so-called nonlinear optical effects such as four-wave mixing by assigning a state to track the conversion of one mode to the next. With the observer design in Chapter 3 we have shown that estimation of the carrier density is possible with only input–output measurements and an approximate linear model. We could therefore probe the inner workings of photonic devices using system observers to post-process input–output experiments on real devices. Furthermore, there is an entire sub-field of control theory devoted to model identification (see for example
Chapter 8 Contributions and conclusions
139
Refs. [246, 247]) that determines linear coefficients from input–output data and some basic knowledge about the system dimension. Such diagnostic tools could help SOA and diode laser designers evaluate internal device performance. Monitoring circuits could also be implemented in the field to detect an SOA with drifting physical parameters that could fail. For these applications with sampled data, discrete-time models would be advantageous and it is worth noting that all our linearized models can be converted from continuous to discrete time [174]. Control formulations developed to model networks [112] could be adapted for large integrated circuits, thus treating interconnections of SOAs as in Fig. 1.3 and other components as miniature optical networks. These formulations are powerful and allow global control criteria to be set, analyzed, and achieved robustly. We could further employ stochastic state-space modelling for noisy and bursty traffic travelling through photonic circuits [238], and evaluate overall system performance improvements (e.g. optical signal-to-noise ratios, bit error rates, quality factors, etc.) Effects such as cross-phase modulation and polarization-mode dispersion tend to appear as random intensity fluctuations, and so stochastic modelling could aid in their suppression. Integrated receiver design in particular would benefit from controllers designed with stochastic methods.
8.3
Perspective
We have seen that photonic signal processing is a key enabling technology for faster computing and a faster Internet. The current trend in photonic signal processing is clearly toward integrated solutions. The semiconductor optical amplifier is a highly versatile, highly integrable device that has been demonstrated in a variety of important functions. As SOAs are integrated together on a large scale, it becomes more difficult to guarantee that their interplay will produce the desired output. With control theory we can analyze and design SOA control systems with a robust theoretical approach. Robustness is not always the primary goal in system or subsystem design, however, because there is often a robustness–performance trade-off. For example, consider the delay-margin analysis of Chapter 6: in certain cases it may be desirable to operate a feedback controller in an unstable regime to gain either controller strength or speed at the expense of the other. In other cases such as in logic gate design, positive feedback is often used to control switching threshold at the inputs (see for example [248] pp. 1085–
140
Section 8.3 Perspective
1086). Ultimately though, a designer must fully understand the rules of system stability and robustness before breaking them to achieve specific performance trade-offs, and this present work provides those rules in quantitative and analytical forms. In this dissertation, we brought control theory and integrated photonics together, deriving the necessary device models and demonstrating robust integrated circuit design and analysis. For designers primarily interested in all-optical design and analysis, the state-space model derived in Chapter 4 is sufficient. Designs that leverage some aspect of the SOA’s electronic demonstrate need the equivalent circuit model derived in Chapter 5. These state-space methods of model derivation, performance analysis, and controller design are entirely general, and can be applied to design more sophisticated controllers and functions for active photonic circuitry.
Appendix A Single-input–single-output output feedback Let the controller be a continuous time-invariant SISO linear system (Ak , bk , ck , dk ). Let σ measure a subset of outputs and process them into a single signal as shown in Fig. A.1; for example, averaging of the m data channels is achieved by taking i 1 h σ= 1 1 ··· 1 . m
(A.1)
Finally, let r(t) be a reference input into the controller.
A.1
Output relation and controlled input
The controller output is given by yk = ck xk + dk (σy + r). u−j (t) uj (t)
(A.2) y(t)
(A, b, c, D) σ k(t)
r(t)
Figure A.1: SISO PID implementation for constant power. 141
142
Section A.2 Output relation and controlled input
Let dj be the j th column of D, and let D−j be D with dj removed. Let uj (t) be the j th element of u(t), and let u−j (t) be u(t) with uj (t) removed. Partitioning the plant output by the j th element gives y =cx + D −j u−j + dj (uj + yk )
(A.3a)
=cx + Du + dj yk
(A.3b)
=cx + Du + dj (ck xk + dk σy + dk r)
(A.3c)
=cx + Du + dj ck xk + dk dj σy + dk dj r " # " # h i x h i u = c dj ck + D dk dj + dk dj σy. xk r
(A.3d) (A.3e)
Let S , (I − dk dj σ)−1 ,
(A.4)
and solve for y to get h i y = S c dj ck
"
x
xk
#
" # h i u + S D dk dj , r
(A.5)
which is the output relation written compactly. Using the output relation (A.5) we can write the controller output explicitly as " # " # h i x h i u yk =ck xk + dk σ S c dj ck + dk r + S D dk dj xk r " # " # h i x h i u = dk σSc (1 + dk σSdj )ck + dk σSD dk (1 + dk σSdj ) . xk r
(A.6a)
(A.6b)
Let p , 1 + dk σSdj , then h i yk = dk σSc pck
"
x
xk
#
" # h i u + dk σSD dk p r
(A.7)
(A.8)
is the controller output written compactly. The controlled input into the plant is simply uj (t) + yk (t).
Chapter A Single-input–single-output output feedback
A.2
143
State update equations
The controller state update equation is given by x˙ k =Ak xk + bk (σy + r)
(A.9a)
h i = bk σSc Ak + bk σSdj ck
"
x
xk
#
h + bk σSD bk p
" # i u r
.
(A.9b)
The plant state update equation is given by x˙ =Ax + b−j u−j + bj (uj + yk )
(A.10a)
=Ax + bu + bj yk h
(A.10b)
= A + bj dk σSc bj pck
i
"
x
xk
#
h + b + bj dk σSD bj dk p
" # i u r
.
(A.10c)
Combining states into an augmented system yields "
x˙
x˙ k
#
=
" A + bj dk σSc bk σSc
A.3
bj pck Ak + bk σSdj ck
#"
# " #" # b + bj dk σSD bj dk p u + . (A.11) xk bk σSD bk p r x
Closed-loop transfer function
Let σ = (0, 1, . . . , 1). We have state equation sX(x) = AX(s) +
m+2 X
bi Ui (s).
(A.12)
i=1
We don’t care about the electronic output Y1 (s) and consider only the optical outputs: Yσ (s) =cσ X(s) + =cσ (s − A) =
m+2 X
Dii Ui (s)
i=2 m+2 X −1
bi Ui (s) +
i=1 m+2 X
cσ b1 U1 (s) + s−A
i=2
(A.13a) m+2 X
Dii Ui (s)
(A.13b)
i=2
Dii s − (ADii − cσ bi ) Ui (s) s−A
(A.13c)
144
Section A.3 Closed-loop transfer function
Let Tσ1 (s) , and Tσi (s) ,
cσ b1 s−A
Dii s − (ADii − cσ bi ) , s−A
so we can write Yσ (s) = Gσ1 (s)U1 (s) +
m+2 X i=2
(A.14)
i = 2, . . . , m + 2
Tσi (s)Ui (s).
(A.15)
(A.16)
Appendix B State observer (estimator) design B.1
Observer system
We can calculate the state of the actual system by running a model in parallel given the same inputs for both the system and model. If we know the parameters of the actual ˆ = b) system exactly, we need only the parts of the model that generate the state (Aˆ = A, b are needed to generate the estimated state xˆ(t) perfectly, xˆ(t) = x(t) ∀ t ≥ 0, as shown in Fig. B.1. If the model (A, b, c, D) is not known exactly or if a parameter in one of the coefficients ˆ fails to estimate the state and the estimate may ˆ b) drifts over time, simply running (A, ˆ ˆc, D) ˆ b, ˆ (for example, Aˆ = A±20%), drift unbounded. If we estimate the model to be (A, then we can compare outputs from both the actual and estimated models and correct the state estimation by routing in the comparison difference (weighted and summed by SOA D U (s)
X(s) b
model
ˆ b
(s − A)
−1
ˆ −1 (s − A)
Y (s) c
ˆ X(s)
ˆ = b) for X(s) ˆ Figure B.1: Simple full state observer that requires (Aˆ = A, b = X(s) ∀ s. 145
146
Section B.1 Observer system
SOA D U (s)
X(s) (s − A)−1
b
Y (s) c
ˆℓ Yˆ (s)
ˆ X(s)
ˆ b
s
−1
ˆ c
Aˆ ˆ D
model
Figure B.2: Full state observer.
vector ℓ) as illustrated in Fig. B.2. The model linear system with state estimate is simply ˆ ˆx + bu xˆ˙ = Aˆ
(B.1)
When the comparative feedback is employed via constant ℓ we get ˆ + ˆℓ(y − c ˆx + bu ˆ ˆxˆ − Du) xˆ˙ = Aˆ
(B.2a)
ˆ − ˆℓD)u ˆ = (Aˆ − ˆℓˆ c)ˆ x + (b + ˆℓy
(B.2b)
ˆ − ˆℓD ˆ + ˆℓD)u. = (Aˆ − ˆℓˆ c)ˆ x + ˆℓcx + (b
(B.2c)
Augmenting the original linear system we find the full system–observer to be # #" # " " # " b x A 0 x˙ u. + = ˆ − ˆℓ(D ˆℓc Aˆ − ˆℓˆ ˆ − D) b c xˆ xˆ˙
(B.3)
The error in the estimate is defined by x˜ , x − xˆ
(B.4)
Chapter B State observer (estimator) design
147
and so the error dynamics are x˜˙ = (A − ℓc)˜ x
(B.5)
s − (A − ℓc) = 0.
(B.6)
with characteristic equation
Observer design consists of choosing ℓ so that x˜ → 0 quickly and is stable i.e. choose the error dynamics to be faster than the plant (further into the negative real line). For a pole designed at s = ℓ′ A, we solve ℓc = (1 − ℓ′ )A
(B.7)
(ℓ is not unique). Arbitrarily setting all components equal we have ℓi =
(1 − ℓ′ )A P , j cj
(B.8)
which can then be plugged into Eq. (B.3).
B.2
Observer-based feedback control
If we now feed the estimated state back into the main system as in Fig. B.3 we get state update " # " #" # " # ˆ x˙ A −bk x b = + u ˆℓc Aˆ − ˆℓ(ˆ ˆ ˆ − ˆℓ(D ˆ − D) xˆ˙ c + D k) xˆ b
(B.9)
" # h i x ˆ y = c −D k + Du. xˆ
(B.10)
and output
148
Section B.2 Observer-based feedback control
SOA D U (s)
Y (s)
X(s) b
(s − A)−1
c
ˆℓ ˆ k Yˆ (s) ˆ b
s−1 Aˆ
model
ˆ c ˆ X(s)
ˆ D
Figure B.3: Constant estimated-state feedback with full state observer.
Appendix C Methods for deriving general state-space models C.1
Two core photonic states
¯ (t) with inputs I(t) ¯ and The rate equation (3.5b) is a nonlinear differential equation in N P (0, t); however, it is coupled to the length-averaged optical power through the Σi P¯i (t) ¯ (t)) is an unknown function in time, we make it a term. Because Σi P¯i (t) (specifically P second state variable so that ¯ x1 (t) = N(t) and
m X αi ¯ x2 (t) = Pi (t). ~ωi i=0
(C.1)
(C.2)
Substituting the propagation equation (3.1) into the photon rate equation ∂P (z, t) ∂P (z, t) = vg ∂t ∂z
(C.3)
and integrating with respect to z and normalizing by L gives dP¯i (t) vg,i [Γi ai (N¯ (t)−Ntr,i )−αi ]L = (e − 1)Pi(0, t), dt L
(C.4)
Although the expression (C.4) appears to only account for photons gained or lost through ¯ the facets, it does indeed contain gain from carrier injection (via N(t)) and cavity loss 149
150
Section C.1 Two core photonic states
(via α). Multiplying each side by αi /~ωi and summing each side over the optical channels we have
m
m
d X αi ¯ 1 X vg,i αi [Γi ai (N¯ (t)−Ntr,i )−αi ]L Pi (t) = (e − 1)Pi (0, t). dt i=0 ~ωi L i=0 ~ωi
(C.5)
This now provides a second first-order differential equation coupled back to equation (3.5b).
C.1.1
Linearization
Ultimately, we seek a linear system ˙ x(t) = Ax(t) + Bu(t)
(C.6a)
y(t) = Cx(t) + Du(t).
(C.6b)
x˙ = f (x, u),
(C.7)
Write
where f (x, u) = and write
" − xτc1 −
P e[Γiai (x1 −Ntr,i )−αi ]L −1 uelec − m uopt,i i=0 qV ~ωi V Pm vg,i αi (e[Γi ai (x1 −Ntr,i )−αi ]L −1) uopt,i i=0 ~ωi L x2 A
+
y = h(x, u), where
#
,
(C.8)
(C.9)
uelec /qV e[Γ0 a0 (x1 −Ntr,i )−α0 ]L u opt,0 h(x, u) = . .. . e[Γm am (x1 −Ntr,i )−αm ]Luopt,m
(C.10)
Choosing an equilibrium (operating) point (x0 , u0 ), linearized coefficients can be computed in a straightforward manner. Defining εi , e[Γi ai (x1,0 −Ntr,i )−αi ]L
(C.11)
Chapter C Methods for deriving general state-space models
151
we calculate # " P Γi ai εi −1 −τc−1 − m u −A opt,i,0 i=0 ~ω A , A = Dxf (x0 , u0 ) = Pm vg,i αi Γi ai εii u 0 opt,i,0 i=0 ~ωi
(C.12)
where uopt,i,0 is the ith -channel equilibrium point, B T = Duf (x0 , u0 ) =
" 1/qV
1−ε0 ~ω0 V vg,0 α0 (ε0 −1) ~ω0 L
0
and
··· ···
0
0
Γ a Lε u 0 0 0 opt,0,0 C = Dxh(x0 , u0 ) = .. . Γm am Lεm uopt,m,0 1 0 D = Duh(x0 , u0 ) =
1−εm ~ωm V vg,m αm (εm −1) ~ωm L
#
,
0 , .. . 0
(C.14)
0
0 elsewhere . .. . 0 elsewhere εm ε0
(C.13)
(C.15)
Together, these four matrices with equilibrium points constitute a time-invariant linear system, (A, B, C, D).
C.1.2
Equilibrium points: lasing threshold condition
Equilibrium points (x0 , u0 ) occur when Eqs. (3.5b) and (C.5) are in the steady-state, so from Eq. (3.5b) we get m
x1,0 x2,0 uelec,0 X e[Γi ai (x1,0 −Ntr,i )−αi ]L − 1 − + − uopt,i,0 0=− τc A qV ~ωi V i=0
(C.16)
and from Eq. (C.5) we get m X vg,i αi (e[Γi ai (x1,0 −Ntr,i )−αi ]L − 1) i=0
~ωi L
uopt,i,0 = 0.
(C.17)
152
Section C.1 Two core photonic states
Consider Eq. (C.17) in the general multichannel situation. Clearly it is satisfied if uopt,i,0 = 0 ∀i
(C.18)
e[Γi ai (x1,0 −Ntr,i )−αi ]L − 1 = 0 ∀i αi ⇒ x1,0 = + Ntr,i ∀i. Γi ai
(C.19a)
or
(C.19b)
Condition (C.18) corresponds to all optical channels being completely dark and is the trivial solution. Condition (C.19b) is the laser threshold condition and merits further consideration. Equation (C.19b) is the carrier threshold condition at which gain is exactly equal to loss [146]. While it is unlikely that all channels across the gain spectrum reach threshold at the same carrier density, this equilibrium condition is possible if a central channel is chosen to be at threshold exactly, and the remaining channels are sufficiently near threshold. Equation (C.17) may also be satisfied if the sum contains positive and negative terms that cancel one another. Although all the parameters are greater or equal to zero for every channel, the ith term may be negative if e[Γi ai (x1,0 −Ntr,i )−αi ]L − 1 < 0 ⇐⇒ x1,0 <
αi + Ntr,i , Γi ai
(C.20)
and so it may be possible that the sum is zero at equilibrium without conditions (C.18) or (C.19) if some channels are below threshold. Regardless, the problem here is that the photon rate equation’s equilibrium imposes a threshold condition that clamps the gain to the loss. While this phenomenon occurs in lasers, it is absent in laser amplifiers and so the photon state equation (C.5) cannot be used.
C.1.3
Linear gain conclusion
¯ (t), although necessary because it is an unknown function We conclude that isolating P ¯ t)/dt, imposes the lasing threshold condition on the SOA and a straightforward in dN(z, linear control model is not directly possible. Setting N(z, t) = N(t) as in the main
Chapter C Methods for deriving general state-space models
153
text alleviates the need for a second rate equation in P (z, t) and also allows nonlinear recombination forms.
C.2 C.2.1
gi N (z, t)
Gain generalization
If gi N(z, t) , we can draw generalizations from the analysis above. From the propagation equation (3.1), a generalized input/output relation is produced by collecting like terms and integrating on [0, L], 1
Pi (L, t) = Pi (0, t)e[ L where
RL 0
gi (N (z,t))∂z−αi ]L
1 g¯i (t) , L
Z
0
= Pi (0, t)e[¯gi (t)−αi ]L,
(C.21)
L
gi N(z, t) ∂z
(C.22)
is the average gain over the length of the amplifier. As before, rearranging the propagation equation (3.1), substituting it into the carrier rate equation (3.5b) integrating with respect to z on [0, L], and using (C.21) yields a generalized average carrier rate equation, m ¯ ¯ ¯ (t) dN(t) 1 X 1 Pi (0, t)(e[¯gi(t)−αi ]L − 1) I(t) N ¯ = − − + αi Pi (t) . dt qV τc A i=0 ~ωi L
(C.23)
Letting x3 (t) , g¯i (t)
(C.24)
the average gain can be encapsulated in a state variable for any set of gains gi N(z, t) provided x˙ 3 (t) is available explicitly.
If the gain is linear as per Eq. (3.6), 1 x3 (t) = L
Z
L 0
Γi ai N(z, t) − Ntr ∂z = Γi ai (x1 (t) − Ntr )
(C.25)
and x˙ 3 (t) = Γi ai x˙ 1 (t), which are linearly dependent with x1 (t) and x˙ 1 (t) and thus superfluous.
(C.26)
154
Section C.3 Separable N, N(z, t) = Nz (z)Nt (t)
If the gain is logarithmic as per Eq. (3.7), x3 (t) = Γi ai and
1 L
Z
L
ln N(z, t)∂z − ln Ntr
0
Γi ai d x˙ 3 (t) = L dt
Z
(C.27)
L
ln N(z, t)∂z.
(C.28)
0
The spatial integral of ln N(z, t) is not available explicitly and therefore neither is x˙ 3 (t). We could encapsulate the integral by redefining x1 , 1 x1 (t) , L
Z
L
ln N(z, t)∂z,
(C.29)
0
but this only hides the problem temporarily as x˙ 1 (t) is not available in a usable form: N˙ (z, t) ∂z (C.30a) 0 N(z, t) Z L Z L Z L m I(z, t) 1 ∂Pi (z, t) Pi (z, t) 1 1 1 X 1 = ∂z − − ∂z + αi ∂z , qV L 0 N(z, t) τc V i=0 ~ωi 0 N(z, t) ∂z 0 N(z, t)
1 x˙ 1 (t) = L
Z
L
(C.30b)
for N(z, t) 6= 0. The first term inside the sum is not Pi (L, t) − Pi (0, t) due to the factor of 1/N(z, t). This result contains many products of indefinite functions of z that cannot be assigned to state variables because definite rate equations are not available. Note also that we cannot simply take x1 , N because
C.3
d dt
ln N is not available.
Separable N , N (z, t) = Nz (z)Nt (t)
In the preceding analysis for polynomial compressive gain, the light output depends on an indefinite spatial integral of the carrier concentration N(z, t), an inexplicit quantity. Separating N(z, t) with respect to time and space might aid analytical development, but the separation must be carried through from the beginning. Let the carrier concentration be separable in space and time, N(z, t) = Nz (z)Nt (t)
(C.31)
Chapter C Methods for deriving general state-space models
155
The carrier rate equation (3.5b) becomes dNt (t) I(z, t) Nt (t) = − dt qV Nz (z) τc m X gi (z, t)P (z, t) 1 Pi (z, t). − ANz (z) i=0 ~ωi
(C.32)
In general, gain g(z, t) is not separable as a result of Eq. (C.31). For linear gain, gi = Γi ai (Nz Nt − Ntr,i ) = Γi ai Nz Nt − Γi ai Ntr,i ,
(C.33)
which is not separable due to the constant term; for logarithmic gain, gi = Γi ai ln(Nz Nt /Ntr,i ) = Γi ai (ln Nz + ln Nt − ln Ntr,i),
(C.34)
which is also not separable geometrically (it is arithmetically separable). The same is true for more complex compressive gain forms. For the propagation equation solution (4.19) with polynomial compressive gain, the indefinite integral in the denominator becomes Z
[Γa(Nz Nt − Ntr ) − α]∂z = ΓaNt
Z
Nz dz − ΓaNtr z − αz.
(C.35)
R
Nz dz that is
So now, rather than one large indefinite integral, there remains only unknown explicitly.
If Nz can be approximated or characterized generally over the length of the SOA, the indefinite integral can be solved. If the SOA can be treated as a lumped element with respect to the average optical pulse length, L ≪ τpulse , then Nz can be considered a R constant over z and Nz dz = Nz z.
C.4
Finite difference decomposition in space
Because there are algebraic difficulties in the continuous calculus of the preceding cases, we discretize the governing equations and attempt to derive forms suitable for state-space modelling. Take n discrete finite differences in z only; time remains continuous. Discretize space in the leap-frog mesh shown in Fig. C.1: carrier concentrations occupy inter locations
156
Section C.4 Finite difference decomposition in space N, I
P
i∆z
(i + 12 )∆z
N, I
P
(i + 1)∆z (i + 23 )∆z
Figure C.1: Finite difference space mesh. while optical powers occupy half-integer locations; the distance between two like quantities is ∆z.
C.4.1
Propagation equation
The propagation equation is written Pi+1/2,j (t) − Pi−1/2,j (t) = [gi,j (t) − αj ]Pi,j (t), ∆z
(C.36)
where i is the spatial index and j the optical channel index. The lefthand side results in a quantity at i∆z, but Pi on the righthand side does not exist, so an average must be taken about i∆z. Furthermore, part of the gain depends on N (which does exist at i∆z); in the nonlinear gain functions considered above, the gain is separable into carrier- and photondependent parts g N and g P , respectively, which either exist at a given spatial location or can be averaged. Combining these elements together, the propagation equation at time t becomes P P gi+1/2,j + gi−1/2,j Pi+1/2,j − Pi−1/2,j Pi+1/2,j + Pi−1/2,j N − αj = gi,j . ∆z 2 2
(C.37)
In anticipation of the carrier rate equation, the propagation equation is rearranged to get N gi,j
P P gi+1/2,j + gi−1/2,j
2
Pi+1/2,j + Pi−1/2,j 2 Pi+1/2,j − Pi−1/2,j Pi+1/2,j + Pi−1/2,j = . + αj ∆z 2
(C.38)
If the gain is independent on optical power, we can isolate the most advanced spatial step Pi+1/2,j (t), yielding Pi+1/2,j (t) =
N 2/∆z + gi,j (t) − αj Pi−1/2,j (t). N 2/∆z − gi,j (t) + αj
(C.39)
Chapter C Methods for deriving general state-space models
157
The overall output can be found in terms of the input, Pn+1/2,j (t) = P1/2,j (t)
n N Y 2/∆z + gi,j (t) − αj i=1
N 2/∆z − gi,j (t) + αj
.
(C.40)
If the gain is dependent on optical (either polynomially or rationally), the most advanced spatial step at time t has the quadratic form
Pi+1/2,j
q f1 giN , Pi−1/2,j ± f2 giN , Pi−1/2,j , = f3 giN
(C.41)
where f1,2,3 are rational functions of gain and power. The system is no longer affine and computationally cumbersome to recurse back to the input optical power. Still, using finite-difference permits a closed-form input/output relation that is not available the in continuous-space analysis above.
C.4.2
Carrier rate equation
Turning to the carrier rate equation, we have at time t m
dNi 1 X gi,j Ii Ni − Pi,j = − dt qV τc A j=0 ~ωj m Ni 1 X 1 Pi+1/2,j − Pi−1/2,j Ii − − = qV τc A j=0 ~ωj ∆z
(C.42a) + αj
Pi+1/2,j + Pi−1/2,j 2
,
(C.42b)
where the last line is obtained by substitution of (C.38). Integrating over z along the length of the SOA now takes on the form of a Riemann sum with cell length ∆z. Summing and normalizing we have n
n
n
d 1X 1 X 1 X Ni ∆z = Ii ∆z − Ni ∆z dt L i=1 qV L i=1 τc L i=1 m n 1 X ∆z X Pi+1/2,j − Pi−1/2,j − A j=0 ~ωj L i=1 ∆z Pi+1/2,j + Pi−1/2,j , + αj 2
(C.43)
158
Section C.4 Finite difference decomposition in space
noting that the summations Σj and Σi commute with each other. The first term under the double summation is a telescoping sum in i for which all the inner terms cancel, n X
(Pi+1/2 − Pi−1/2 ) = Pn+1/2 − P1/2 .
(C.44)
i=1
For the rest of the terms, define spatial average quantities in the expected way: n X ¯ (t) = ∆z N Ni (t), L i=1
and
n X ¯ = ∆z Ii (t), I(t) L i=1 n n ∆z X ∆z X Pi+1/2 (t) + Pi−1/2 (t) ¯ P (t) = Pi (t) = . L i=1 L i=1 2
(C.45)
(C.46)
(C.47)
Then the carrier rate equation for any gain becomes
m m ¯ ¯ (t) ¯ 1 X Pn+1/2,j (t) − P1/2,j (t) 1 X αj ¯ I(t) N dN(t) − + Pj (t). = − dt qV τc V j=0 ~ωj A j=0 ~ωj
(C.48)
The input/output relations (C.40) and (C.41) can be substituted in directly for the output Pn+1/2,j (t). Now considering linear gain dependence on carrier density, gi,j (t) = Γj aj Ni (t) − Ntr,j ,
(C.49)
the input/output relation (C.40) at time t becomes Pn+1/2,j
n Y 2/∆z + Γj aj Ni − Ntr,j − αj = P1/2,j . 2/∆z − Γ a N − N + α j j i tr,j j i=1
(C.50)
¯ (t) and so it Unlike the continuous case, the input/output relation does not contain N becomes unclear what the carrier-related state should be. Considering logarithmic gain dependence on carrier density, we have gi,j (t) = Γj aj ln Ni (t)/Ntr,j ,
(C.51)
Chapter C Methods for deriving general state-space models
159
which was problematic in the spatially continuous analysis above. The input/output relation (C.40) at time t becomes Pn+1/2,j
n Y 2/∆z + Γj aj ln Ni /Ntr,j − αj . = P1/2,j 2/∆z − Γj aj ln Ni /Ntr,j + αj i=1
(C.52)
This leads to the same problem as before: even if we define states ln Ni (t), we do not have corresponding (and necessary) state rate equations.
C.4.3
Optical power rate equation
Finally, the photon rate equation is obtained by averaging the carrier concentration at time t in space, dPi+1/2,j = vg [gi+1/2,j − αj ]Pi+1/2,j dt N N gi+1,j + gi,j P gi+1/2,j − αj Pi+1/2,j . = vg 2
C.4.4
(C.53a) (C.53b)
Finite difference conclusion
Although finite-difference provides closed-form input/output relations for cases with gains in the form g(N, P ) ∝ N +NP +P and g(N, P ) ∝ N/P (albeit prohibitively cumbersome ones), it fails for the simpler g(N) ∝ ln N form for the same reasons as in the spatiallycontinuous case in the sections above.
160
Section C.4 Finite difference decomposition in space
Appendix D Equivalent circuit model derivatives The derivative terms that appear in linear coefficients (F , G, H, J) Eqs. (5.21)–(5.27) are ∂Pout Pin ∂g = −GL −GL ∂N Ge + Pin gǫ(1 − e ) ∂N −GL G (GL − 1)e − e−GL Pinǫ(eGL + gL) × 1+ Ge−GL + Pin gǫ(1 − e−GL ) ∂Pout G Pin gǫ(e−GL − 1) = 1+ ∂Pin Ge−GL + Pin gǫ(1 − e−GL ) Ge−GL + Pin gǫ(1 − e−GL ) e−GL (GL + Pin ǫ(1 − eGL − gL) − 1 1 ∂ P¯ = − + L − ∂N Ge−GL + Pin gǫ(1 − e−GL ) G 1 1 ∂g Pingǫ GL − ln 1 + (e − 1) g G gǫL ∂N 1 − e−GL ∂ P¯ = ∂Pin GLe−GL + PingLǫ(1 − e−GL ) ∂ϕ 1 ∂ζ (ζϕ + G) ∂g ∂Qout ζ = +ϕ + 1− ∂N 2ǫg ∂N ∂N g ∂N ∂Qout ζ ∂ϕ = ∂Qin 2ǫg ∂Qin ¯ ∂Q 2ξ 1 ∂ϕ 1 ξ ∂ζ 1 ∂g ∂ξ − = − + +L ∂N 2ǫgL ξ 2 − ζ 2 ∂N ζ ∂N ϕ − 1 ϕ + 1 ∂N ∂N (ξ/ζ − 1)(ξ/ζ + 1) ∂g 1 ln + GL − 2 2ǫg L (ϕ − 1)(ϕ + 1) ∂N 161
(D.1)
(D.2)
(D.3)
(D.4) (D.5) (D.6)
(D.7)
162
Section D.0
and
¯ 1 2ξ 1 ∂ϕ 1 ∂Q ∂ξ = − , + ∂Qin 2ǫgL [ξ 2 − ζ 2 ] ∂Qin ϕ − 1 ϕ + 1 ∂Qin
(D.8)
L ∂ζ ζL ξ 1 ∂ζ ∂ξ 2 + sech arctanh + ξ −ζ ξ 2 − ζ 2 ∂N ∂N 2 ∂N ζ 2
(D.9)
where ∂ϕ = ∂N
ξ ζL ∂ϕ ζ ∂ξ 2 + = 2 sech arctanh ∂Qin [ζ − ξ 2 ] ∂Qin ζ 2 1 ∂ζ ∂Rsp ∂g p = 2ǫg + (G + 2ǫRsp ) ∂N ∂N ∂N 4ǫgRsp + G2 ∂g ∂ξ = (2ǫQin − 1) ∂N ∂N ∂ξ = 2ǫg ∂Qin
and
∂Rsp = ∂N
g−
∂g Ng +N N − Ntr ∂N
c . 2neff L(N − Ntr )
(D.10) (D.11) (D.12) (D.13)
(D.14)
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