Impact of Feedback Delay on Closed-Loop Stability - System Control ...

JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 9, MAY 1, 2009 1095<strong>Impact</strong> <strong>of</strong> <strong>Feedback</strong> <strong>Delay</strong> on Closed-Loop Stabilityin Semiconductor Optical Amplifier Control CircuitsScott B. Kuntze, Baosen Zhang, Lacra Pavel, Member, OSA, andJ. Stewart Aitchison, Senior Member, IEEE, Fellow, OSAAbstract—Semiconductor optical amplifiers (SOAs) are attractivefor integrated photonic signal processing, but becausetheir response is so fast, delays in a controller feedback path canjeopardize performance and stability. Using state-space methods,we quantify the constraints imposed on feedback controllers byclosed-loop delay. We first derive a complete nonlinear state-spacecontrol model <strong>of</strong> a SOA with an equivalent circuit containingparasitics and dynamic impedance; the analytical state-spacemodel agrees well with a validated photonic-only control model.Using a linearized version <strong>of</strong> the model we demonstrate that timedelay in the feedback path can destabilize the SOA through phaseaccumulation. We then apply linear system theory to calculate thebest-case stable delay margin for a given controller norm, and finda potentially severe inverse relationship between delay marginand controller norm. Finally, guided by the delay–controllerrelationship we design a hybrid feedforward–feedback controllerto illustrate that good transient and steady-state regulation isobtained by carefully balancing the feedforward and feedbackcomponents. Our state-space modeling and design methods aregeneral and are easily adapted to the design and analysis <strong>of</strong> morecomplex photonic circuits.Index Terms—Equivalent circuits, feedforward systems, opticalcontrol, optical crosstalk, optical feedback, optimal control, semiconductoroptical amplifiers, state-space methods.I. MOTIVATION:QUANTIFYING OPTOELECTRONIC CONTROLLERDELAY CONSTRAINTSSEMICONDUCTOR optical amplifiers (SOAs) are versatileactive photonic devices that can be monolithicallyintegrated to provide complex photonic functions such assignal amplification [1], regeneration [2], switching [3], andfrequency conversion [4]. Precise output control is required asSOA-based optical signal processing scales to encompass morecomplex functions at greater levels <strong>of</strong> integration, and thusthere is growing interest in SOA control design and modeling[5]–[10].Controlling an SOA poses a special challenge: because theSOA responds so quickly (subnanosecond), a feedback controllermust respond on the same timescale. <strong>Delay</strong>s in the feedbackpath due to signal propagation, detection, processing andManuscript received October 18, 2007; revised May 16, 2008. Current versionpublished April 24, 2009.S. B. Kuntze and L. Pavel are with the Department <strong>of</strong> Electrical and ComputerEngineering, University <strong>of</strong> Toronto, Toronto, ON M5S 3G4, Canada (e-mail:scott.kuntze@utoronto.ca; pavel@control.toronto.edu).B. Zhang was with the Department <strong>of</strong> Electrical and Computer Engineering,University <strong>of</strong> Toronto, Toronto, ON M5S 3G4, Canada. He is now with the University<strong>of</strong> California, Berkeley CA 94720-1770 USA (e-mail: baosen_z@hotmail.com).J. S. Aitchison is with the Research Faculty <strong>of</strong> Applied Science and Engineering,University <strong>of</strong> Toronto, Toronto, ON M5S 3G4, Canada.Digital Object Identifier 10.1109/JLT.2008.928214Fig. 1. Electronic model signal flow. I (t) is the bias current (a controllableinput), I (t) is the current delivered into the SOA (a measurable output), andI(t) is the current through the SOA’s active region. The state and input power <strong>of</strong>the existing SOA model may influence the dynamic behavior <strong>of</strong> the equivalentcircuit.modulation can jeopardize the tight timing required for reliablecontrol because the SOA may respond to its inputs long beforethe corresponding control signal arrives. If the closed-loop delayis too large, the controller can actually destabilize the SOA,causing unpredictable output and possibly even damaging theSOA or other parts <strong>of</strong> the optoelectronic circuitry.In this paper, we quantify the constraints imposed on SOAfeedback control by nonzero delays in the closed-loop path:we show analytically that there is a inverse relationship betweenclosed-loop delay and controller strength. To overcomethe delay-imposed limitations on the feedback controller, wedesign a hybrid feedforward–feedback controller that benefitsfrom both fast transient response and accurate steady-state behavior.We proceed by first deriving in Section II an SOA state-spacemodel based on the photonic models <strong>of</strong> [5], [10], and [11].State-space models comprise sets <strong>of</strong> first-order ordinary differentialequations and are perfectly suited to control analysis anddesign, unlike cumbersome partial-differential equation modelsor SPICE-based models [6]. As shown in Fig. 1, our model addsan equivalent circuit to the front end <strong>of</strong> the SOA to model parasiticsthat can further delay the arrival <strong>of</strong> electronic controlsignals to the active region current. Furthermore, the resultingsource voltage is a useful measure <strong>of</strong> the SOA’s photonic response[12] that we leverage in the subsequent controller design.Linearized state-space models have many properties thatmake delay analysis and control design straightforward androbust, so we derive the corresponding SOA linear model inSection III.In Section IV, we illustrate that delay in the feedback pathcauses instability in an otherwise robust controller design; theeigenvalues <strong>of</strong> the linearized model direct us to the cause <strong>of</strong> the0733-8724/$25.00 © 2009 IEEE

1100 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 9, MAY 1, 2009The coefficients and relate the state and inputto the output . has rows and four columns;is square with dimension . Both are sparse, withdiagonal. If the electronic output is taken to be in Fig. 2,we have(25a)(25b)Fig. 4. <strong>Feedback</strong> <strong>of</strong> the SOA state N (t) into the drive current I (t) to suppresscrosstalk between optical channels P (t) as in [5].if the electronic output is instead,wehave(26a)(26b)Regardless, the remaining nonzero entries are given byand(27a)(27b)Fig. 5. Relative parametric phase portraits <strong>of</strong> the static augmented system(F;G; H;J ) under state feedback into the bias current from N (t) only; theflow <strong>of</strong> time is indicated by the arrows. The system—excited by a 13% stepmodulation in optical power for 2 ns—is unstable, spiraling ever outward untildevice failure.(28a)(28b)where again all the derivatives are found in the appendix and areevaluated at the equilibria .Referring back to Fig. 3, there is good qualitative agreementbetween the nonlinear (dashed line) and linear (solid line)equivalent circuit models. For input modulations typicallygreater than 20%, there is some qualitative separation betweenthe linear and nonlinear models; for a given application it isthe designer’s choice how much error is acceptable, althoughthe controller can be designed to switch between several precomputedoperating points to lessen discrepancies. We show inSection VI that a controller designed at a single operating pointworks well even with 100% optical modulations.IV. STATE FEEDBACK INTO THE DRIVE CURRENT ANDSYSTEM STABILITYState feedback provides the most effective control because thestate contains all the current information <strong>of</strong> the system. In thissection we isolate a source <strong>of</strong> closed-loop instability by comparingstate feedback controllers that drive the bias current withand without the equivalent circuit <strong>of</strong> Fig. 2.In [5], intrachannel crosstalk is suppressed by keeping the inversioncarrier density—and therefore the gain—constant. Thisgain control is realized by feeding back the deviation <strong>of</strong> theSOA’s state into the active region current .Forthe more general case <strong>of</strong> a parasitic network before the SOA asin Fig. 2, is no longer directly accessible and only maybe set by the user or controller. Using the same feedback schemeas in [5] with the parasitic equivalent circuit present leads toFig. 4. Analytically, this feedback is achieved by modifying thefirst equation in the nonlinear model (14) to include state feedbackthrough a constant controller ,(29)Using the exact SOA model <strong>of</strong> [5] augmented with the equivalentcircuit, the same operating point (150-mA bias, 1.6-mWtotal optical input power), and the feedback gain calculated forthe SOA-only model, now leads toan unstable closed-loop system, as illustrated by the outwardspreadingspirals in the parametric phase diagrams <strong>of</strong> Fig. 5.Under a 13% upward step modulation in optical power the components<strong>of</strong> the equivalent circuit state begin to oscillatewildly, and the magnitudes <strong>of</strong> these oscillations increase indefinitelyuntil some part <strong>of</strong> the system fails or is damaged. Alreadywithin 2 ns, 150 mA in Fig. 5 (left panel) and thus thenet current is flowing back to the source, which implies thecontroller calls for backwards through the diode.The phase diagrams <strong>of</strong> Fig. 5 indicate that the front end <strong>of</strong>the equivalent circuit is oscillating more heavily than the backend near the actual SOA—the swings in are roughly tentimes those <strong>of</strong> over the same duration—and so it appearsthat the path through the circuit is responsible for the instability.

KUNTZE et al.: IMPACT OF FEEDBACK DELAY ON CLOSED-LOOP STABILITY IN SOA CONTROL CIRCUITS 1101However, more insight is obtained from the linear model. First,we construct the closed-loop feedback system(30a)(30b)where is a sparse matrix with andwhere and have been partitionedover the controlled inputs (subscript ) and external inputs (subscript); note also that input vector now excludes any controlledinput and thus contains only external inputs. Examiningthe eigenvalues (poles) <strong>of</strong> the new system matrix ,Fig. 6. Optimal control schematic. SOA gain is regulated by full state feedbackthrough the minimum-cost K into an optical control channel and drive current.A. Least-Squares Optimal Control(31)reveals that the real parts <strong>of</strong> two circuit eigenvalues are positive(each), and are therefore the causes <strong>of</strong>the instability. The last eigenvalue belongs to the SOA and hasindeed been shifted further negative by the feedback as desired(open-loop value is). However, the unstablecircuit eigenvalues dominate and destabilize the entire system.Because the feedback gain remains identical between thesetwo models (with and without the front-end equivalent circuit),the closed-loop phase is responsible for the sudden instabilitywith the circuit. In particular, the delay caused by the parasiticfront-end has exceeded the stable delay margin with respect tothe SOA’s state . In Section V, we generalize and quantifythe effect <strong>of</strong> delay margin on the closed-loop stability for SOAsystems.The optimal controller is depicted in Fig. 6: the full stateis fed through the constant controller , and used to driveboth the input current and an auxiliary optical channel. The governing equations (30) are modified to includethe lumped delay that accounts for signal propagation, controllerlag, etc.,(32a)(32b)note again that and are partitioned over controlled and externalinputs and that contains only external inputs to thesystem. We note that for the time-domain simulations the nonlinearversion <strong>of</strong> this model is used [(6) and (14) with the appropriatedelayed feedback terms].The controller is optimal in the least-squares sense where acost associated with control effort is minimized; the cost isgiven byV. DELAY MARGIN FOR FEEDBACK STABILITYIn the previous section, our analysis led to the conclusion thatthe phase lag through the parasitic circuit caused closed-loopinstability; equivalently, the time delay for the controller actionexceeded the delay margin. <strong>Delay</strong> in the feedback loop is causedby other sources in addition to the phase lag <strong>of</strong> the SOA’s parasitics:propagation time through the SOA ( 10 ps for a 1-mmdevice); 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 employoptimal state feedback (Section V-A) and determine the maximumdelay margin for stability as a function <strong>of</strong> feedback gain(Section V-B). Although state feedback requires measurementor estimation <strong>of</strong> all the states—a difficult requirement to meet inpractice in real time—optimal control minimizes feedback gainand thus we resolve the best case. More practical closed-loopcontrol schemes such as output feedback have stricter delaymargins.(33)Here, is a constant positive-semidefinite matrix that penalizesexcursions in the state from equilibrium; similarly, is aconstant positive-definite matrix that penalizes effort <strong>of</strong> the controlledinputs. These penalty matrices are commonly chosen tobe diagonal so that an excursion in a particular variable penalizesonly that variable. Once the penalties are designed, the controlleris found by(34)where is a symmetric matrix found numerically via the algebraicRiccati equation(35)

1102 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 9, MAY 1, 2009For example, if we designthe optimal state feedback controller is(36)(37)This constant controller minimizes excursions in both the stateand controlled input.Because the SOA states , , and are more modelspecificcompared to the source voltage , we penalize thesestates more heavily by setting the form <strong>of</strong> to be(38)We choose the form <strong>of</strong> to have identical diagonal elementssince neither control input should be favoured a priori.B. <strong>Delay</strong> Margin <strong>of</strong> the <strong>Feedback</strong> ControllerWe model the delay using an -order Padé approximationgiven by [26](39)where is the delay and the Laplace frequency. This particularapproximation is strictly proper and thus suppresses transientsat the start <strong>of</strong> the delay period [27]. We found that there wasnegligible qualitative improvement beyond a -order delayin the feedback signals, and so that is the delay function we use.To measure the strength <strong>of</strong> the controller we use the Frobeniusnorm(40)because it is related to the magnitudes <strong>of</strong> the elements and istherefore representative <strong>of</strong> the combined efforts <strong>of</strong> the feedbackgains. Note that when is scalar, .For a given optimal controller design we calculate thecontroller and controller magnitude . To find the delaymargin associated with the controller magnitude, we employ theConstant Matrices Test [28, Th. 2.13], the first step <strong>of</strong> whichrequires composing two new matricesand(41)(42)where is the Kronecker tensor product. The closed-loopsystem is unconditionally stable if and only if at least one <strong>of</strong>two conditions is met: Either the generalized eigenvectors<strong>of</strong> do not intersect the unit circle <strong>of</strong> the complex plane,or else all the eigenvalues <strong>of</strong>are identically zer<strong>of</strong>or any generalized eigenvector that does intersect the unitcircle. Any controller that violates both these conditions hasconditional stability and an upper bound on the stable delaymargin [28]. For conditionally stable feedback controllers, thedelay margin’s upper bound can be calculated [28]. For each—that is, for each generalized eigenvector <strong>of</strong>that lies on the complex unit circle—we calculate suchthat is an eigenvalue <strong>of</strong> that lies on thepositive imaginary axis. The stable delay margin supremumis then given by [28](43)so that the system is stable for delays. Thus, atypical evaluation <strong>of</strong> delay margin starts with the design <strong>of</strong> theoptimal parameters and , proceeds with the calculation <strong>of</strong>the optimal feedback , and ends by finding , which is eitherinfinite or given by (43) according to the Constant MatricesTest.Fig. 7 shows a detailed analysis <strong>of</strong> the constraints on feedbackmagnitude imposed by finite delay margins in a two-channelsystem. Panel (a) illustrates the relationship between delaymargin and controller magnitude calculated with (43)using the optimal design (38) <strong>of</strong> the previous section for asequence <strong>of</strong> values <strong>of</strong> and diagonal matrices . The resultingsolid line in Fig. 7(a) separates unstable closed-loop systemresponse (above) from stable response (below). At very lowthe delay margin is essentially unbounded, while thedelay margin quickly drops for stronger controllers. The dashedlines are obtained by scaling each , and by a factor <strong>of</strong>10 to either reduce or increase the circuit delay from the solidline where 0.1 nH, 10 pF, for example; hencethe delay margin shifts depending on the reactive response <strong>of</strong>the circuit.For further analysis, we mark locations 1, 2, and 3 aboutthe solid line on Fig. 7(a). Panel (b) <strong>of</strong> Fig. 7 shows the stabilitycharacteristics <strong>of</strong> locations 1 and 2 <strong>of</strong> panel (a) with 20%modulation <strong>of</strong> the modulated “aggressor” channel (the “victim”channel remains idle at the input). Above the solid delay marginline at location 1 the phase portraits <strong>of</strong> the states spiral outwarduntil the system fails (note that the location 1 spirals are scaledby a factor <strong>of</strong> along both axes); at location 2, the system isstable and the states simply settle asymptotically at new equilibria.The performance at location 2 is further illustrated inpanel (c) where the controller magnitude is high: the aggressorchannel sees an improved gain pr<strong>of</strong>ile over time while crosstalkinto the idle victim channel is eliminated. By contrast, the performanceat the weaker controller at location 3 in panel (a) isshown in panel (d), and it is clear this controller is significantlyless effective despite the increase in delay margin.The implications <strong>of</strong> Fig. 7 are potentially severe: even for thisslower multi-quantum-well SOA, feedback control could be unfeasiblein electronic domain due to the delay margin constrainsthe controller—at location 2, the delay margin is on the order<strong>of</strong> mere picoseconds. Again, this particular result is for optimalstate feedback and represents the best case, whereas the delaymargin line shifts downwards for partial state or output feedbackschemes. Hence, to complete our investigation into the effects <strong>of</strong>

KUNTZE et al.: IMPACT OF FEEDBACK DELAY ON CLOSED-LOOP STABILITY IN SOA CONTROL CIRCUITS 1103Fig. 8. Hybrid feedforward–feedback control design. K is the constant feedforwardcontroller, while K (s) is the dynamic feedback controller with intrinsicdelay (including propagation delay <strong>of</strong> the SOA and optical circuitry).Fig. 7. Time delay margin analysis for the optimal controller depicted in Fig. 6using the parameters in Table I.time delay we examine the performance improvement <strong>of</strong> feedforwardcontrol.VI. HYBRID FEEDFORWARD–FEEDBACK CONTROLLERAs we have just seen in Section V, delay margin in the feedbackloop has the potential to restrict the norm and performance<strong>of</strong> a feedback controller, perhaps to the point where a sufficientlystrong and fast controller cannot be realized. A common solutionto overcoming the speed limitations <strong>of</strong> feedback in erbium-dopedfibre amplifier control is to employ feedforwardcontrol [29]. However, feedforward control has a significantlimitation itself in that the model and parameters <strong>of</strong> the SOAmust be well characterized and accurate because the controllercannot self-adjust based on the actual outputs <strong>of</strong> the system.Hence, we add a weak feedback controller to compensate forany errors induced by imperfect feedforward control. Moreover,the feedback is small enough to afford a delay margin largeenough for optoelectronic implementation.Fig. 8 illustrates the hybrid feedforward–feedback control design.The total power <strong>of</strong> the incoming data signals is sampledand fed through a constant controller ; if the implementation<strong>of</strong> has significant delay, the optical channels canbe relayed through an optical delay line such that feedforwardcontrol appears instantaneously at the input <strong>of</strong> the SOA. actuallyneeds to invert the input signal, and for this function asecond SOA with a continuous-wave input at frequency canbe used to generate by cross-gain modulation [30] withnegligible delay.For the feedback circuit, the source voltage is a convenientmeasure <strong>of</strong> the photonic state <strong>of</strong> the SOA (i.e., as carriersare consumed in the SOA active region, the SOA currentincreases and the result is reflected in the source voltage) and isreadily accessible. When steps up or down, there issome ripple in that we filter out using a first-order low-passR–C network with a cut<strong>of</strong>f frequency <strong>of</strong> 10 MHz. Although thelow-pass filter slows the leading edge <strong>of</strong> the feedback signal, ahigh-pass filter in parallel has little effect due to the relativelylow-frequency spectrum <strong>of</strong> ( is inherently low-pass filteredthrough the parasitic equivalent circuit from changes in theSOA active region).We have specifically designed this controller to avoid dealingwith phase relations between coherent signals. The feedback

1104 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 9, MAY 1, 2009Fig. 10. Eye diagrams <strong>of</strong> aggressor and victim channels under various controlschemes, modulated by 20-dB 2 0 1 pseudorandom bit sequences at 10 Gb/s;each vertical division is 1 mW and each cell is 100 ps long.Fig. 9. Comparison <strong>of</strong> feedforward, feedback, and hybrid controllers with+40% feedforward modeling error. On [0; 10] ns the aggressor channel doublesin power, while on [20; 30] ns it drops out completely. The best combined transientand steady-state performance in the outputs (a) and (b) is obtained with ahybrid controller: the feedforward component provides fast transient responsewhile the feedback component ensures steady-state accuracy. (c) shows thecomponents <strong>of</strong> the hybrid control including the 1-ns-delayed feedback.controller measures the terminal voltage <strong>of</strong> the SOAso that the subtraction is an incoherent operation (the subtractioncould be achieved by any differential amplifier in the electricaldomain, for example). The feedforward controller simplyneeds to invert the total optical power level at its input andscale the result. Hence, the controller provides an optical controlchannel governed by(44)To illustrate the operation <strong>of</strong> the hybrid feedforward–feedbackcontroller, we purposely introduce feedforward modeling errorby miscalculating by 40 , and introduce 1 ns <strong>of</strong> delay inthe feedback path in addition to the inherent delay <strong>of</strong> the filter.Real errors in could result from poor device characterization,device parameter drift with age or temperature, or changingASE at the input. Fig. 9 demonstrates that the best combinedtransient and steady-state response for both channels is obtainedwith the hybrid controller when the aggressor channel is modulated100 from 1 to 2 mW (over 0–10 ns) and 100 from1 to 0 mW (over 20–30 ns). Gain is enhanced in the aggressorchannel (a) while crosstalk into the idle victim channel (b) issuppressed most effectively with the hybrid control. Feedforward-onlycontrol introduces steady-state error, whereas feedback-onlycontrol suffers from very poor transient response dueto the closed-loop delay. Examining the components <strong>of</strong> the hybridcontrol signal in (c) shows that the calculation error in feedforwardcomponent is suppressed as the feedback correction arrives.The system response illustrated in Fig. 9 over the relativelylong time scales can be viewed as the envelope or average-powerresponse due to the various control schemes. Fig. 10 shows theeye diagrams for the same system as in Fig. 9, now modulatedby a pseudo-random bit sequence (PRBS) at 10 Gb/s anda depth <strong>of</strong> 20 dB ( 20 to 0 dBm), with no jitter and rise/falltimes that are essentially zero. As with Fig. 9, there are two datachannels and a control channel. The simulation takes place overthree PRBS sequences: only the victim channel is modulated forthe first two PRBS sequences, and the aggressor channel turnson for the final sequence. Furthermore, because the controllertakes time to settle after starting at , we do not plot the eyepatterns for the first PRBS sequence—this gives the appearancethat the controller has been running for a long time prior to theaggressor channel turning on.In Fig. 10, the open-loop eyes (first column) close comparedto perfect feedforward control (fourth column) due to downwardfluctuations in the population inversion density, whethercaused by interchannel crosstalk (i.e., an aggressor channel’sinput 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 populationinversion; in many cases the controller actually respondswith the wrong signal at the wrong delayed time and furthercloses the eyes. However, with the addition <strong>of</strong> fast feedforward

KUNTZE et al.: IMPACT OF FEEDBACK DELAY ON CLOSED-LOOP STABILITY IN SOA CONTROL CIRCUITS 1105(47)(48)(49)(50)(51)(52)(53)control (third column)—even with a feedforward gain miscalculatedby 40 as before—the eyes are reopened and the averagepower level approaches that <strong>of</strong> perfect (feedforward) control.Thus, the feedback controller works on smaller errors in thesignaland serves essentially to correct the averagepower or envelope over many bit periods.VII. CONCLUSIONSOA feedback control is challenging because the controllermust respond sufficiently quickly to the subnanosecond dynamics<strong>of</strong> the SOA. Signal detection, processing and routingcause time delay in the feedback path. In turn, time delay imposesan upper bound on the norm <strong>of</strong> the feedback controller:the greater the norm, the smaller the delay margin. Exceedingthe delay margin causes system instability that can damagethe SOA or surrounding optoelectronic circuitry. However,reducing the norm also reduces the controller performance.We calculated this delay–control trade<strong>of</strong>f by deriving astate-space SOA model that contains electronic dynamics,designing a set <strong>of</strong> best-case optimal state feedback controllers,and employing system stability theory. We have seen thatfeedback delay places an upper bound on the maximum controllerstrength for stable closed-loop operation. Equivalently,the stronger the feedback gain or the more parasitic the SOAequivalent circuit, the smaller the lumped time delay margin.Finally, we employed the delay–control relationship to guidethe design <strong>of</strong> a hybrid feedforward–feedback controller thatused relatively weak feedback only to correct steady-stateerrors due to feedforward–SOA mismatch.These state-space methods <strong>of</strong> model derivation, performanceanalysis, and controller design are entirely general, and can beapplied to design more sophisticated controllers and functionsfor active photonic circuitry.andAPPENDIXThe length-averaged optical powers are(45)(46)The derivative terms that appear in linear coefficients(22)–(28) are shown in (47)–(53) at the top <strong>of</strong> thepage, andwhere(54)(55)

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Ciaramella,“Cross-gain compression in semiconductor optical amplifiers,” J.Lightw. Technol., vol. 25, no. 3, pp. 915–921, Mar. 2007.Scott B. Kuntze received the B.Sc.Eng. degree in mathematics and engineeringfrom Queen’s University, Kingston, ON, Canada, in 2002 and the M.A.Sc. degreein electrical engineering (photonics) from the University <strong>of</strong> Toronto, ON,Canada, in 2004. He is currently working towards the Ph.D. degree in photonicsin the Department <strong>of</strong> Electrical and Computer Engineering at the University <strong>of</strong>Toronto.He spent internships working in the Advanced Technology Investments Departmentat Nortel Networks building next-generation photonic transceivers andswitches from 2000 to 2001, and in the High Performance Optical ComponentsDivision at Nortel studying semiconductor lasers using novel probingtechniques in 2002. His current research interests include the robust analysis,design, and control <strong>of</strong> active integrated photonic devices using control theory.Mr. Kuntze is the recipient <strong>of</strong> an NSERC Canadian Graduate Scholarship.Baosen Zhang received the Bachelor <strong>of</strong> Applied Science degree from the EngineeringScience program at the University <strong>of</strong> Toronto, ON, Canada, in June2008. He is currently working towards the Ph.D. degree in electrical engineeringat the University <strong>of</strong> California, Berkeley.Lacra Pavel received the Ph.D. degree in electrical engineering from Queen’sUniversity, Kingston, ON, Canada, in 1996, with a dissertation on nonlinearH-infinity control.She spent a year at the Institute for Aerospace Research (NRC), Ottawa,Canada, as an NSERC Postdoctoral Fellow. From 1998 to 2002, she workedin the optical communications industry at the frontier between systems control,signal processing, and photonics. She joined the University <strong>of</strong> Toronto, ON,Canada, in August 2002 as an Assistant Pr<strong>of</strong>essor in Electrical and ComputerEngineering Department. Her research interests include system control and optimizationin optical networks, game theory, robust and H-infinity optimal control,and real-time control and applications.

KUNTZE et al.: IMPACT OF FEEDBACK DELAY ON CLOSED-LOOP STABILITY IN SOA CONTROL CIRCUITS 1107Dr. Pavelis an Associate Editor, Member on the Program Committee <strong>of</strong> IEEEControl Applications Conference 2005; Associate Chair (Control) on the ProgramCommittee <strong>of</strong> IEEE Canadian Conference <strong>of</strong> Electrical and Computer Engineering2004. She is a member <strong>of</strong> CSS/ComSoc/LEOS and a member <strong>of</strong> theOptical Society <strong>of</strong> America (OSA).J. Stewart Aitchison (SM’00) received the B.Sc. (with first-class honors) andPh.D. degrees from the Physics Department, Heriot-Watt University, Edinburgh,U.K., in 1984 and 1987, respectively. His dissertation research was on opticalbistability in semiconductor waveguides.From 1988 to 1990, he was a Postdoctoral Member <strong>of</strong> Technical Staff at Bellcore,Red Bank, NJ. His research interests were in high nonlinearity glasses andspatial optical solitons. He then joined the Department <strong>of</strong> Electronics and ElectricalEngineering, University <strong>of</strong> Glasgow, U.K., in 1990 and was promoted toa personal chair as Pr<strong>of</strong>essor <strong>of</strong> Photonics in 1999. His research was focused onthe use <strong>of</strong> the half bandgap nonlinearity <strong>of</strong> III–V semiconductors for the realization<strong>of</strong> all-optical switching devices and the study <strong>of</strong> spatial soliton effects. Healso worked on the development <strong>of</strong> quasi-phase matching techniques in III–Vsemiconductors, monolithic integration, optical rectification, and planar silicatechnology. His research group developed novel optical biosensors, waveguidelasers, and photosensitive direct writing processes based around the use <strong>of</strong> flamehydrolysis deposited (FHD) silica. In 1996, he was the holder <strong>of</strong> a Royal Society<strong>of</strong> Edinburgh Personal Fellowship and carried out research on spatial solitonsas a visiting researcher at CREOL, University <strong>of</strong> Central Florida. Since 2001,he has held the Nortel Chair in Emerging Technology, in the Department <strong>of</strong>Electrical and Computer Engineering at the University <strong>of</strong> Toronto, ON, Canada.From 2004 to 2007, he was the Director <strong>of</strong> the Emerging Communications TechnologyInstitute at the University <strong>of</strong> Toronto. Since 2007, he has been Vice Dean<strong>of</strong> Research for the Faculty <strong>of</strong> Applied Science and Engineering, University <strong>of</strong>Toronto. His research interests cover all-optical switching and signal processing,optoelectronic integration, and optical bio-sensors. His research has resulted inseven patents, around 185 journal publications, and 200 conference publications.Dr. Aitchison is a Fellow <strong>of</strong> the Optical Society <strong>of</strong> America (OSA) and aFellow <strong>of</strong> the Institute <strong>of</strong> Physics London.