JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 9, MAY 1, 2009

1095

Impact of Feedback Delay on Closed-Loop Stability in Semiconductor Optical Amplifier Control Circuits Scott B. Kuntze, Baosen Zhang, Lacra Pavel, Member, OSA, and J. Stewart Aitchison, Senior Member, IEEE, Fellow, OSA

Abstract—Semiconductor optical amplifiers (SOAs) are attractive for integrated photonic signal processing, but because their response is so fast, delays in a controller feedback path can jeopardize performance and stability. Using state-space methods, we quantify the constraints imposed on feedback controllers by closed-loop delay. We first derive a complete nonlinear state-space control model of a SOA with an equivalent circuit containing parasitics and dynamic impedance; the analytical state-space model agrees well with a validated photonic-only control model. Using a linearized version of the model 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. Finally, guided by the delay–controller relationship we design a hybrid feedforward–feedback controller to illustrate that good transient and steady-state regulation is obtained by carefully balancing the feedforward and feedback components. Our state-space modeling and design methods are general and are easily adapted to the design and analysis of more complex photonic circuits. Index Terms—Equivalent circuits, feedforward systems, optical control, optical crosstalk, optical feedback, optimal control, semiconductor optical amplifiers, state-space methods.

I. MOTIVATION: QUANTIFYING OPTOELECTRONIC CONTROLLER DELAY CONSTRAINTS

S

EMICONDUCTOR optical amplifiers (SOAs) are versatile active photonic devices that can be monolithically integrated to provide complex photonic functions such as signal amplification [1], regeneration [2], switching [3], and frequency conversion [4]. Precise output control is required as SOA-based optical signal processing scales to encompass more complex functions at greater levels of integration, and thus there is growing interest in SOA control design and modeling [5]–[10]. Controlling an SOA poses a special challenge: because the SOA responds so quickly (subnanosecond), a feedback controller must respond on the same timescale. Delays in the feedback path due to signal propagation, detection, processing and Manuscript received October 18, 2007; revised May 16, 2008. Current version published April 24, 2009. S. B. Kuntze and L. Pavel are with the Department of Electrical and Computer Engineering, University of Toronto, Toronto, ON M5S 3G4, Canada (e-mail: [email protected]; [email protected]). B. Zhang was with the Department of Electrical and Computer Engineering, University of Toronto, Toronto, ON M5S 3G4, Canada. He is now with the University of California, Berkeley CA 94720-1770 USA (e-mail: [email protected]). J. S. Aitchison is with the Research Faculty of Applied Science and Engineering, University of Toronto, Toronto, ON M5S 3G4, Canada. Digital Object Identifier 10.1109/JLT.2008.928214

Fig. 1. Electronic model signal flow. I (t) is the bias current (a controllable input), I (t) is the current delivered into the SOA (a measurable output), and I(t) is the current through the SOA’s active region. The state and input power of the existing SOA model may influence the dynamic behavior of the equivalent circuit.

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 this paper, we quantify the constraints imposed on SOA feedback control by nonzero delays in the closed-loop path: we show analytically that there is a inverse relationship between closed-loop delay and controller strength. To overcome the delay-imposed limitations on the feedback controller, we design a hybrid feedforward–feedback controller that benefits from both fast transient response and accurate steady-state behavior. We proceed by first deriving in Section II an SOA state-space model based on the photonic models of [5], [10], and [11]. State-space models comprise sets of first-order ordinary differential equations and are perfectly suited to control analysis and design, unlike cumbersome partial-differential equation models or SPICE-based models [6]. As shown in Fig. 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 [12] that we leverage in the subsequent controller design. Linearized state-space models have many properties that make delay analysis and control design straightforward and robust, so we derive the corresponding SOA linear model in Section III. In Section IV, we illustrate that delay in the feedback path causes instability in an otherwise robust controller design; the eigenvalues of the linearized model direct us to the cause of the

0733-8724/$25.00 © 2009 IEEE Authorized licensed use limited to: IEEE Xplore. Downloaded on April 22, 2009 at 09:12 from IEEE Xplore. Restrictions apply.

1096

JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 9, MAY 1, 2009

TABLE I 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

Fig. 2. SOA model with parasitic network, adapted from [13]. The active region current I(t) may differ significantly from the drive current I (t) and is a function of the optical inputs and population inversion density. All other components are defined in Table I.

delay. In Section V, 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 behavior. In fact, the constraint on feedback norm may be so severe as to render the controller ineffective. Hence, guided by the delay–controller relationship, we design a hybrid feedforward–feedback in Section VI. This control configuration has fast feedforward transient response and just a small amount of feedback to correct any feedforward modeling errors despite a relatively large feedback delay.

II. NONLINEAR STATE-SPACE EQUIVALENT CIRCUIT MODEL Existing SOA state-space models [5], [10], [11] 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. 2, adapted from [13]–[15] with components defined in Table I. This particular circuit captures the essential shunt capacitances, series and source resistances, and series inductance of a typical diode parasitic network; further parasitic elements [16], leakage pathways [13], and further active region complexity [17], [18] are added using the same approach. In Fig. 2, the drive current is set by the user or a control system, while the active region current depends on the diode current (through the diode voltage ) and stimulated current (which is drawn as carriers are used up in stimulated emission). The current between the source and the SOA is dependent on both and and is either measured directly with a series ammeter or calculated from if 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 components (Section II-A). We then relate the active region current to the optical powers incident on the SOA, (Section II-B). To find the SOA’s impedance, we relate the diode voltage to and , and recast in terms of the SOA inversion carrier density (Section II-C). Finally, we obtain the complete nonlinear state-space model by rewriting the resulting equations as coupled ordinary differential equations (Section II-D).

A. State-Space Realization of the Equivalent Circuit For the equivalent circuit shown in Fig. 2, convenient choices for states are the capacitor voltages , , and , and the inductor current . 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. In matrix form, we get

(1)

The active region current is treated as an input in (1), but it is actually a nonlinear function of optical inputs and

Authorized licensed use limited to: IEEE Xplore. Downloaded on April 22, 2009 at 09:12 from IEEE Xplore. Restrictions apply.

KUNTZE et al.: IMPACT OF FEEDBACK DELAY ON CLOSED-LOOP STABILITY IN SOA CONTROL CIRCUITS

junction voltage ; we obtain space model of the SOA.

from the photonic state-

B. SOA Active Region Current In order to obtain as a function of the optical inputs , we turn to the governing equations for a SOA [20]: the inversion carrier density rate equation

1097

. where the compressed gain has the form This photonic component of the model was fully validated by pump–probe experiments in [10]. ASE is modeled as a collection of discrete signals, and each ASE “channel” has its own entry in the input vector. Typically, every data channel is paired with an ASE channel that keeps track of the corresponding ASE power, while one or more out-of-band ASE channels account for the depletion of the carrier density due to the remaining ASE spectrum [11]. The output ASE powers are given by integrating (4)

(7) where we define for convenience [10] (2) (8a) the set of signal propagation equations

(8b) (8c)

(3)

(8d) and the set of amplified spontaneous emission (ASE) propagation equations [11]

(4)

and The corresponding length-averaged powers are given in the Appendix. Finally, the current drawn by the SOA’s active region is found by rearranging the length-averaged inversion carrier density (5)

where all parameters are defined in Table I, and where the sums data and ASE channels, respectively. of (2) are over the Following the procedure outlined in [10], substituting the propagation equations (3) and (4) into (2) and averaging over the device length, we obtain

(9)

(5) This length-averaged rate equation and the solutions to (3) and (4) constitute the photonic state-space model of the SOA. For notational convenience, we omit the overbar for unless we need to distinguish from . The associated input–output relations equations are derived similarly by length-integration (3)

The terms in order are the currents due to diffusion capacitance, recombination, and stimulated emission. is nonlinear in [through ] and , and linear in . C. SOA Current–Voltage Relationship We now have the current through the SOA’s active region , but need the SOA’s junction voltage to complete the circuit model. Hence, we relate the junction voltage to the inversion carrier density ,

(10) In order to obtain a closed-form solution we must impose the condition that exists with

(6) Authorized licensed use limited to: IEEE Xplore. Downloaded on April 22, 2009 at 09:12 from IEEE Xplore. Restrictions apply.

(11)

1098

JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 9, MAY 1, 2009

Additionally, we need the time derivative

(12) The function can be the Boltzmann approximation [21], [22] or another Fermi function approximation [23] provided (11) yields an explicit closed-form relation between and . Using (10) and (12) in the last line of the circuit equations (1) given in (9) results in with the expression for

(13) Hence, we have replaced the circuit state of the SOA. inversion carrier density

in (1) with the

D. Nonlinear 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. 2. in terms of , , and in (13), Solving for making the substitution of (10) into in (1), and assembling the entire system gives

(14a)

Fig. 3. Comparison of open-loop system responses to 20% step inputs: (a) optical output; (b) inversion carrier density; (c) optical input; and (d) bias current. The nonlinear SOA model without the circuit is (5) and (6), the nonlinear SOA model with the circuit in Fig. 2 is (14) and (6), and the linear SOA model with the circuit is (21).

(dotted–dashed line) from [10] by direct numerical integration of state equations (14) using the Livermore solver for ordinary differential equations [24]; 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

(14b) (15) while recombination is polynomial (16) The Boltzmann relation is employed for the junction voltage (14c) (17) The first matrix equation is a linear, time-invariant subsystem for the linear equivalent circuit, coupled to the nonlinear photonic rate equation of through the middle equation in . The photonic output relation remains unchanged by the variable substitutions and is given by (6). Fig. 3 compares the nonlinear (dashed line) equivalent circuit model with the previously verified nonlinear SOA-only model

where is the average inversion level at thermal equilibrium over the whole device (usually on the order of cm in InP/InGaAsP SOAs [22]). Values for the model parameters are given in Table I and are representative of previous studies in the literature [10], [15], [22]; the remaining circuit values are 1 nH, 10 pF and 1 pF and are chosen here to

Authorized licensed use limited to: IEEE Xplore. Downloaded on April 22, 2009 at 09:12 from IEEE Xplore. Restrictions apply.

KUNTZE et al.: IMPACT OF FEEDBACK DELAY ON CLOSED-LOOP STABILITY IN SOA CONTROL CIRCUITS

be conservative to emphasize the effects of the equivalent ciris set to 70 mA [Fig. 3(d)] cuit. The nominal drive current and the nominal optical power of the single channel is set to 1 mW [Fig. 3(c)]. Due to the 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 inversion carrier density [Fig. 3(b)] and output power [Fig. 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 (6) and (14) and photoniconly model (6) and (5) are qualitatively similar. Hence, for alloptical control design the simpler model (6) and (5) from [10] 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.

1099

the poles of the system’s transfer function and therefore predict the nature of the system’s evolution over time. Computing the Jacobian [25] of (14) with respect to the states, we get

(22)

where

(23a) (23b)

III. LINEARIZED STATE-SPACE EQUIVALENT CIRCUIT MODEL For notational convenience, let the system state of (6) and (14) be denoted by

(18) let the input be denoted by

(19) (23c) and let the output be

or

(20)

The form of the linearized state-space model corresponding to (6) and (14) is then

All derivatives in (23) and subsequent coefficients are lengthy and are listed in the Appendix. relates the inputs (bias current, data channels plus one optional optical control channel, and ASE channels) to the state derivative , and has four rows and columns as a result. The Jacobian of (14) with respect to the inputs yields a sparse matrix with the nonzero elements given by

(21a) (21b) (24a) which we denote by the ordered set of the constant coefficient matrices . These linear system coefficients are obtained by first finding the equilibrium points for with a given set of inputs ; plotting against shows there is a single equilibrium point for a given operating point . All the subsequent linearizations are evaluated at and . The constant coefficient matrix relates the system state to its time derivative ; the eigenvalues of correspond to Authorized licensed use limited to: IEEE Xplore. Downloaded on April 22, 2009 at 09:12 from IEEE Xplore. Restrictions apply.

(24b)

(24c)

1100

JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 9, MAY 1, 2009

The coefficients and relate the state and input to the output . has rows and four columns; is square with dimension . Both are sparse, with diagonal. If the electronic output is taken to be in Fig. 2, we have

(25a) (25b) if the electronic output is instead

Nt P t

Fig. 4. Feedback of the SOA state ( ) into the drive current ( ) as in [5]. crosstalk between optical channels

I (t) to suppress

, we have

(26a) (26b) Regardless, the remaining nonzero entries are given by

(27a) (27b)

Fig. 5. Relative parametric phase portraits of the static augmented system ( ) under state feedback into the bias current from ( ) 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.

F ;GG;H H ;JJ

Nt

and

(28a) (28b) where again all the derivatives are found in the appendix and are . evaluated at the equilibria Referring back to Fig. 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 Section VI that a controller designed at a single operating point works well even with 100% optical modulations. IV. STATE FEEDBACK INTO THE DRIVE CURRENT AND SYSTEM STABILITY State feedback provides the most effective control because the state contains all the current information of the system. In this section we isolate a source of closed-loop instability by comparing state feedback controllers that drive the bias current with and without the equivalent circuit of Fig. 2. In [5], intrachannel crosstalk is 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 into the active region current . For

the more general case of a parasitic network before the SOA as is no longer directly accessible and only may in Fig. 2, be set by the user or controller. Using the same feedback scheme as in [5] with the parasitic equivalent circuit present leads to Fig. 4. Analytically, this feedback is achieved by modifying the first equation in the nonlinear model (14) to include state feedback through a constant controller ,

(29) Using the exact SOA model of [5] 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 , now leads to the SOA-only model an unstable closed-loop system, as illustrated by the outwardspreading spirals in the parametric phase diagrams of Fig. 5. Under a 13% upward step modulation in optical power the combegin to oscillate ponents of the equivalent circuit state wildly, and the magnitudes of these oscillations increase indefinitely until some part of the system fails or is damaged. Already within 2 ns, 150 mA in Fig. 5 (left panel) and thus the net current is flowing back to the source, which implies the controller calls for backwards through the diode. The phase diagrams of Fig. 5 indicate that the front end of the equivalent circuit is oscillating more heavily than the back are roughly ten end near the actual SOA—the swings in times those of over the same duration—and so it appears that the path through the circuit is responsible for the instability.

Authorized licensed use limited to: IEEE Xplore. Downloaded on April 22, 2009 at 09:12 from IEEE Xplore. Restrictions apply.

KUNTZE et al.: IMPACT OF FEEDBACK DELAY ON CLOSED-LOOP STABILITY IN SOA CONTROL CIRCUITS

1101

However, more insight is obtained from the linear model. First, we construct the closed-loop feedback system

(30a) (30b) where is a sparse matrix with and where and have been partitioned over the controlled inputs (subscript ) and external inputs (subnow excludes any conscript ); note also that input vector trolled input and thus contains only external inputs. Examining the eigenvalues (poles) of the new system matrix ,

Fig. 6. Optimal control schematic. SOA gain is regulated by full state feedback through the minimum-cost into an optical control channel and drive current.

K

A. Least-Squares Optimal Control

(31)

reveals that the real parts of two circuit eigenvalues are positive (each ), 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 ). However, the unstable (open-loop value is circuit eigenvalues dominate and destabilize the entire system. Because the feedback gain 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 front-end has exceeded the stable delay margin with respect to the SOA’s state . In Section V, we generalize and quantify the effect of delay margin on the closed-loop stability for SOA systems.

The optimal controller is depicted in Fig. 6: the full state is fed through the constant controller , and used to drive and an auxiliary optical channel both the input current . The governing equations (30) are modified to include the lumped delay that accounts for signal propagation, controller lag, etc.,

(32a) (32b) note again that and are partitioned over controlled and external inputs and that contains only external inputs to the system. We note that for the time-domain simulations the nonlinear version of this model is used [(6) and (14) with the appropriate delayed feedback terms]. The controller is optimal in the least-squares sense where a cost associated with control effort is minimized; the cost is given by

V. 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 (Section V-A) and determine the maximum delay margin for stability as a function of feedback gain (Section V-B). 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.

(33) Here, is a constant positive-semidefinite matrix that penalizes excursions in the state from equilibrium; similarly, is a constant positive-definite matrix 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 is found by

(34) where is a symmetric matrix found numerically via the algebraic Riccati equation

Authorized licensed use limited to: IEEE Xplore. Downloaded on April 22, 2009 at 09:12 from IEEE Xplore. Restrictions apply.

(35)

1102

JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 9, MAY 1, 2009

For example, if we design

(36) the optimal state feedback controller is

(37) This constant controller minimizes excursions in both the state and controlled input. are more modelBecause the SOA states , , and specific compared to the source voltage , we penalize these states more heavily by setting the form of to be

(38)

We choose the form of to have identical diagonal elements since neither control input should be favoured a priori. B. Delay Margin of the Feedback Controller We model the delay using an imation given by [26]

-order Padé approx-

(39) where is the delay and the Laplace frequency. This particular approximation is strictly proper and thus suppresses transients at the start of the delay period [27]. We found that there was -order delay negligible qualitative improvement beyond a in the feedback signals, and so that is the delay function we use. To measure the strength of the controller we use the Frobenius norm (40) because it is related to the magnitudes of the elements and is therefore representative of the combined efforts of the feedback gains. Note that when is scalar, . we calculate the For a given optimal controller design controller and controller magnitude . To find the delay margin associated with the controller magnitude, we employ the Constant Matrices Test [28, Th. 2.13], the first step of which requires composing two new matrices (41) and (42)

is the Kronecker tensor product. The closed-loop where system is unconditionally stable if and only if at least one of two conditions is met: Either the generalized eigenvectors of do not intersect the unit circle of the complex plane, are identically zero or else all the eigenvalues of for any generalized eigenvector that does intersect the unit circle. Any controller that violates both these conditions has conditional stability and an upper bound on the stable delay margin [28]. For conditionally stable feedback controllers, the delay margin’s upper bound can be calculated [28]. For each —that is, for each generalized eigenvector of that lies on the complex unit circle—we calculate such that is an eigenvalue of that lies on the positive imaginary axis. The stable delay margin supremum is then given by [28] (43) . Thus, a so that the system is stable for delays typical evaluation of delay margin starts with the design of the optimal parameters and , proceeds with the calculation of the optimal feedback , and ends by finding , which is either infinite or given by (43) according to the Constant Matrices Test. Fig. 7 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 calculated with (43) using the optimal design (38) of the previous section for a sequence of values of and diagonal matrices . The resulting solid line in Fig. 7(a) separates unstable closed-loop system response (above) from stable response (below). At very low the delay margin is essentially unbounded, while the delay margin quickly drops for stronger controllers. The dashed lines are obtained by scaling each , and by a factor of 10 to either reduce or increase the circuit delay from the solid line where 0.1 nH, 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. 7(a). Panel (b) of Fig. 7 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 along both axes); at location 2, the system is by a factor of 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 (d), and it is clear this controller is significantly less effective despite the increase in delay margin. The implications of Fig. 7 are potentially severe: even for this slower multi-quantum-well 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

Authorized licensed use limited to: IEEE Xplore. Downloaded on April 22, 2009 at 09:12 from IEEE Xplore. Restrictions apply.

KUNTZE et al.: IMPACT OF FEEDBACK DELAY ON CLOSED-LOOP STABILITY IN SOA CONTROL CIRCUITS

1103

K

Fig. 8. Hybrid feedforward–feedback control design. is the constant feedforward controller, while (s) is the dynamic feedback controller with intrinsic delay  (including propagation delay of the SOA and optical circuitry).

K

Fig. 7. Time delay margin analysis for the optimal controller depicted in Fig. 6 using the parameters in Table I.

time delay we examine the performance improvement of feedforward control. VI. HYBRID FEEDFORWARD–FEEDBACK CONTROLLER As we have just seen in Section V, 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 [29]. 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. Fig. 8 illustrates the hybrid feedforward–feedback control design. The total power of the incoming data signals is sampled ; if the implementaand fed through a constant controller tion of has significant delay, the optical channels can be relayed through an optical delay line such that feedforward accontrol appears instantaneously at the input of the SOA. tually needs to invert the input signal, and for this function a second SOA with a continuous-wave input at frequency can be used to generate by cross-gain modulation [30] with negligible delay. For the feedback circuit, the source voltage is a convenient measure of the photonic state of the SOA (i.e., as carriers are consumed in the SOA active region, the SOA current increases and the result is reflected in the source voltage) and is readily accessible. When steps up or down, there is some ripple in that we filter out using a first-order low-pass R–C network with a cutoff frequency of 10 MHz. 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 ( is inherently low-pass filtered through the parasitic equivalent circuit from changes in the SOA active region). We have specifically designed this controller to avoid dealing with phase relations between coherent signals. The feedback

Authorized licensed use limited to: IEEE Xplore. Downloaded on April 22, 2009 at 09:12 from IEEE Xplore. Restrictions apply.

1104

JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 9, MAY 1, 2009

Fig. 10. Eye diagrams of aggressor and victim channels under various control 1 pseudorandom bit sequences at 10 Gb/s; schemes, modulated by 20-dB 2 each vertical division is 1 mW and each cell is 100 ps long.

0

Fig. 9. Comparison of feedforward, feedback, and hybrid controllers with +40% feedforward modeling 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.

controller measures the terminal voltage of the SOA 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 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

(44) To illustrate the operation of the hybrid feedforward–feedback controller, we purposely introduce feedforward modeling error by miscalculating by 40 , and introduce 1 ns of delay in the feedback path in addition to the inherent delay of the filter. Real errors in could result from poor device characterization, device parameter drift with age or temperature, or changing ASE at the input. Fig. 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. 9 over the relatively long time scales can be viewed as the envelope or average-power response due to the various control schemes. Fig. 10 shows the eye diagrams for the same system as in Fig. 9, now modulated pseudo-random bit sequence (PRBS) at 10 Gb/s and by a a depth of 20 dB ( 20 to 0 dBm), with no jitter and rise/fall times that are essentially zero. As with Fig. 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 controller , we do not plot the eye takes time to settle after starting at 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. In Fig. 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

Authorized licensed use limited to: IEEE Xplore. Downloaded on April 22, 2009 at 09:12 from IEEE Xplore. Restrictions apply.

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 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 and serves essentially to correct the average signal power or envelope over many bit periods.

APPENDIX The length-averaged optical powers are

(45)

VII. CONCLUSION SOA feedback control is challenging because the controller must respond sufficiently quickly to the subnanosecond 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 tradeoff by deriving a state-space SOA model that contains electronic dynamics, designing a set of best-case optimal state feedback controllers, 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 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. 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.

and

(46) The derivative terms that appear in linear coefficients (22)–(28) are shown in (47)–(53) at the top of the page, and

(54) where

Authorized licensed use limited to: IEEE Xplore. Downloaded on April 22, 2009 at 09:12 from IEEE Xplore. Restrictions apply.

(55)

1106

JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 9, MAY 1, 2009

(56)

(57) (58) (59)

(60)

REFERENCES [1] K. D. LaViolette, “The use of semiconductor-optical-amplifiers for long optical links in the CATV upstream optical network,” IEEE Photon. Technol. Lett., vol. 10, pp. 1165–1167, Aug. 1998. [2] D. Wolfson, A. Kloch, T. Fjelde, C. Janz, B. Dagens, and M. Renaud, “40-Gb/s all-optical wavelength conversion, regeneration, and demultiplexing in an SOA-based all-active mach zehnder interferometer,” IEEE Photon. Technol. Lett., vol. 12, pp. 332–334, Mar. 2000. [3] H. J. S. Dorren, X. Yang, A. K. Mishra, Z. Li, H. Ju, H. d. Waardt, G.-D. Khoe, T. Simoyama, H. Ishikawa, H. Kawashima, and T. Hasama, “All-optical logic based on ultrafast gain and index dynamics in a semiconductor optical amplifier,” IEEE J. Sel. Topics Quantum Electron., vol. 10, pp. 1079–1092, Sep./Oct. 2004. [4] S. Diez, C. Schmidt, R. Ludwig, H. G. Weber, K. Obermann, S. Kindt, I. Koltchanov, and K. Petermann, “Four-wave mixing in semiconductor optical amplifiers for frequency conversion and fast optical switching,” IEEE J. Sel. Topics Quantum Electron., vol. 3, pp. 1131–1145, Oct. 1997. [5] S. B. Kuntze, L. Pavel, and J. S. Aitchison, “Controlling a semiconductor optical amplifier using a state-space model,” IEEE J. Quantum Electron., vol. 43, pp. 123–129, Feb. 2007. [6] J. Gurfinkel, D. Sadot, and M. Glick, “Dynamic control analysis for semiconductor optical amplifier dynamics in optical network applications,” Opt. Engr., vol. 46, no. 3, pp. 035004–035004, Mar. 2007. [7] Y. Li, C. Wu, S. Fu, P. Shum, Y. Gong, and L. Zhang, “Power equalization for SOA-based dual-loop optical buffer by optical control pulse optimization,” J. Lightw. Technol., vol. 43, no. 6, pp. 508–516, Jun. 2007. [8] C. Michie, A. E. Kelly, I. Armstrong, I. Andonovic, and C. Tombling, “An adjustable gain-clamped semiconductor optical amplifier (AGCSOA),” J. Lightw. Technol., vol. 25, no. 6, pp. 1–8, Jun. 2007. [9] F. Tabatabai and H. S. Al-Raweshidy, “Feedforward linearization technique for reducing nonlinearity in semiconductor optical amplifier,” J. Lightw. Technol., vol. 25, no. 9, pp. 1–8, Sep. 2007. [10] S. B. Kuntze, A. J. Zilkie, L. Pavel, and J. Stewart Aitchison, “Nonlinear state-space model of semiconductor optical amplifiers with gain compression for system design and analysis,” J. Lightw. Technol., vol. 26, no. 14, pp. 2274–2281, Jul. 2008. [11] W. Mathlouthi, P. Lemieux, M. Salsi, A. Vannucci, A. Bononi, and L. A. Rusch, “Fast and efficient dynamic WDM semiconductor optical amplifier model,” J. Lightw. Technol., vol. 24, no. 11, pp. 4353–4365, Nov. 2006. [12] A. Wonfor, S. Yu, R. V. Penty, and I. H. White, “Novel constant output power control of a semiconductor optical amplifier based switch,” Proc. CLEO, pp. 43–43, 2001. [13] R. S. Tucker, “High-speed modulation of semiconductor lasers,” J. Lightw. Technol., vol. LT-3, no. 6, pp. 1180–1192, Dec. 1985. [14] R. S. Tucker and D. J. Pope, “Microwave circuit models of semiconductor injection lasers,” IEEE Trans. Microw. Theory Technol., vol. MTT-31, no. 3, pp. 289–294, Mar. 1983. [15] R. S. Tucker and I. P. Kaminow, “High-frequency characteristics of directly modulated InGaAsP ridge waveguide and buried heterostructure lasers,” J. Lightw. Technol., vol. LT-2, no. 4, pp. 385–393, Aug. 1984. [16] W. I. Way, “Large signal nonlinear distortion prediction for a singlemode laser diode under microwave intensity modulation,” J. Lightw. Technol., vol. LT-5, no. 5, pp. 305–315, Mar. 1987.

[17] M. F. Lu, J. S. Deng, C. Juang, M. J. Jou, and B. J. Lee, “Equivalent circuit model of quantum-well lasers,” IEEE J. Quantum Electron., vol. 31, pp. 1418–1422, Aug. 1995. [18] J. Gao, B. Gao, and C. Liang, “Large signal model of quantum-well lasers for SPICE,” Microw. Opt. Technol. Lett., vol. 39, no. 4, pp. 295–298, Nov. 2003. [19] L. A. Coldren and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits. New York: Wiley, 1995. [20] J. Mørk, A. Mecozzi, and G. Eisenstein, “The modulation response of a semiconductor laser amplifier,” IEEE J. Sel. Topics Quantum Electron., vol. 5, pp. 851–860, May/Jun. 1999. [21] S. Iezekiel, C. M. Snowden, and M. J. Howes, “Nonlinear circuit analysis of harmonic and intermodulation distortions in laser diodes under microwave direct modulation,” IEEE Trans. Microw. Theory Technol., vol. 38, no. 12, pp. 1906–1915, Dec. 1990. [22] S. A. Javro and S. M. Kang, “Transforming tucker’s linearized laser rate equations to a form that has a single solution regime,” J. Lightw. Technol., vol. 13, no. 9, pp. 1899–1904, Sep. 1995. [23] W. B. Joyce and R. W. Dixon, “Analytic approximations for the Fermi energy of an ideal Fermi gas,” Appl. Phys. Lett., vol. 31, no. 5, pp. 354–356, Sep. 1977. [24] K. Radhakrishnan and A. C. Hindmarsh, “Description and use of LSODE, The Livermore Solver for ordinary differential equations,” NASA/Lawrence Livermore National Laboratory, Livermore, CA, Tech. Rep. UCRL-ID—113855, 1993. [25] Z. Stanislaw, Systems and Control. New York: Oxford Univ. Press, 2003. [26] S. Skogestad and I. Postlethwaite, Multivariable Feedback Control, 2nd ed. New York: Wiley, 2005. [27] M. Vajta, “Some remarks on Padé-approximations,” in Proc. 3rd TEMPUS—INTCOM Symp. Intelligent Systems in Control and Measurement, Sep. 9–14, 2000, pp. 53–58. [28] K. Gu, J. Chen, and V. Kharitonov, Stability of Time-Delay Systems. Boston, MA: Birkhauser, 2003. [29] L. Pavel, “Control design for transient power and spectral control in optical communication networks,” in Proc. IEEE Conf. Control Applications, Jun. 2003, vol. 1, pp. 415–422. [30] G. Contestabile, R. Proietti, N. Calabretta, and E. 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 engineering from Queen’s University, Kingston, ON, Canada, in 2002 and the M.A.Sc. degree in electrical engineering (photonics) from the University of Toronto, ON, Canada, in 2004. He is currently working towards the Ph.D. degree in photonics in the Department of Electrical and Computer Engineering at the University of Toronto. He spent internships working in the Advanced Technology Investments Department at Nortel Networks building next-generation photonic transceivers and switches from 2000 to 2001, and in the High Performance Optical Components Division at Nortel studying semiconductor lasers using novel probing techniques in 2002. His current research interests include the robust analysis, design, and control of active integrated photonic devices using control theory. Mr. Kuntze is the recipient of an NSERC Canadian Graduate Scholarship.

Baosen Zhang received the Bachelor of Applied Science degree from the Engineering Science program at the University of Toronto, ON, Canada, in June 2008. He is currently working towards the Ph.D. degree in electrical engineering at the University of California, Berkeley.

Lacra Pavel received the Ph.D. degree in electrical engineering from Queen’s University, Kingston, ON, Canada, in 1996, with a dissertation on nonlinear H-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 worked in the optical communications industry at the frontier between systems control, signal processing, and photonics. She joined the University of Toronto, ON, Canada, in August 2002 as an Assistant Professor in Electrical and Computer Engineering Department. Her research interests include system control and optimization in optical networks, game theory, robust and H-infinity optimal control, and real-time control and applications.

Authorized licensed use limited to: IEEE Xplore. Downloaded on April 22, 2009 at 09:12 from IEEE Xplore. Restrictions apply.

KUNTZE et al.: IMPACT OF FEEDBACK DELAY ON CLOSED-LOOP STABILITY IN SOA CONTROL CIRCUITS

Dr. Pavelis an Associate Editor, Member on the Program Committee of IEEE Control Applications Conference 2005; Associate Chair (Control) on the Program Committee of IEEE Canadian Conference of Electrical and Computer Engineering 2004. She is a member of CSS/ComSoc/LEOS and a member of the Optical Society of America (OSA).

J. Stewart Aitchison (SM’00) received the B.Sc. (with first-class honors) and Ph.D. degrees from the Physics Department, Heriot-Watt University, Edinburgh, U.K., in 1984 and 1987, respectively. His dissertation research was on optical bistability in semiconductor waveguides. From 1988 to 1990, he was a Postdoctoral Member of Technical Staff at Bellcore, Red Bank, NJ. His research interests were in high nonlinearity glasses and spatial optical solitons. He then joined the Department of Electronics and Electrical Engineering, University of Glasgow, U.K., in 1990 and was promoted to a personal chair as Professor of Photonics in 1999. His research was focused on the use of the half bandgap nonlinearity of III–V semiconductors for the realization of all-optical switching devices and the study of spatial soliton effects. He

1107

also worked on the development of quasi-phase matching techniques in III–V semiconductors, monolithic integration, optical rectification, and planar silica technology. His research group developed novel optical biosensors, waveguide lasers, and photosensitive direct writing processes based around the use of flame hydrolysis deposited (FHD) silica. In 1996, he was the holder of a Royal Society of Edinburgh Personal Fellowship and carried out research on spatial solitons as a visiting researcher at CREOL, University of Central Florida. Since 2001, he has held the Nortel Chair in Emerging Technology, in the Department of Electrical and Computer Engineering at the University of Toronto, ON, Canada. From 2004 to 2007, he was the Director of the Emerging Communications Technology Institute at the University of Toronto. Since 2007, he has been Vice Dean of Research for the Faculty of Applied Science and Engineering, University of Toronto. His research interests cover all-optical switching and signal processing, optoelectronic integration, and optical bio-sensors. His research has resulted in seven patents, around 185 journal publications, and 200 conference publications. Dr. Aitchison is a Fellow of the Optical Society of America (OSA) and a Fellow of the Institute of Physics London.

Authorized licensed use limited to: IEEE Xplore. Downloaded on April 22, 2009 at 09:12 from IEEE Xplore. Restrictions apply.