Suppressing Interchannel Crosstalk in a Multichannel Semiconductor Optical Amplifier Using Optoelectronic State-space Gain Control Scott B. Kuntze, Lacra Pavel, J. Stewart Aitchison Department of Electrical and Computer Engineering, University of Toronto, 10 King’s College Rd., Toronto, Ontario, Canada M5S 3G4 Email: [email protected]

Abstract: Interchannel crosstalk is suppressed in multichannel semiconductor optical amplifiers (SOAs) using nonlinear state-space control methods. Optical input power is related to source voltage, which is used regulate the gain with an optical control channel. Our model opens the way to robust optoelectronic control of SOAs.

As SOA-based photonic circuits become more complex, it becomes increasingly important to regulate their behaviour for the circuits and receivers that follow downstream. For example, electronic feedback has been used to control an SOA in 3R regeneration schemes [1]. We have designed analytical gain controllers that measure output light from an SOA and drive an optical control channel [2], but limited chip space may prohibit post-amplifier optoelectronic detection and signal routing. A gain controller that could interpret the gain state of the SOA by electronic means would offer photonic circuit designers greater flexibility, so in this paper we derive and test the necessary control model. The equivalent circuit between a current source and an SOA is shown in Fig. 1(a), adapted from Ref. [3]; this particular circuit captures the essential shunt capacitances, series and source resistances, and series inductance of a typical parasitic network. The drive current Is (t) can be set by the user or control system, while the active region current ¯ depends on the diode and stimulated emisison currents. I(t) The controller design uses source voltage vp (t) to monitor the amount of optical input power and regulates the SOA gain by keeping the input optical power constant, thereby decoupling the data channels. A possible implementation of the filtering controller is shown in Fig. 1(a): a comparator buffers the input vp (t) and performs the comparison with nominal set-point vp0 . An R–C circuit then suppresses the ringing, and the result is sent to a laser with the appropriate gain. ¯ as a function of the optical inputs P in (t), we turn to the input–output In order to obtain the active region current I(t) state-space model for the SOA alone [2]. The photonic output relation is Pout,i (t) = Pin,i (t)e[gi (N,t)−αi ]L ,

i ∈ {0, 1, . . . , m}.

(1)

where N (t) is the average inversion level over the whole device. 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 use linear circuit analysis to write all the time derivatives as a series of coupled ordinary first-order differential equations. The complete nonlinear state update equation is then obtained by assembling the linear circuit model with the average inversion level rate equation [2], and is given by     vp     v˙ p 1/Cp   −1/RinCp −1/Cp 0   ip + Is (2a) = 0 1/Lp −Rp /Lp −1/Lp i˙ p vs   vs nkB T ip N − + ln +1 (2b) v˙ s = Cs Rs Cs qRs Cs Ne   −1   nkB T qg0 (e[g0 −α0 ]L − 1)(Pin,0 + kωc x) N vs nkB T Csc ˙ − ln + 1 − qV R − + qV N= q(N + Ne ) Rs qRs Ne ~w0 (g0 − α0 ) (2c)  m X qgi (e[gi −αi ]L − 1)Pin,i − . ~wi (gi − αi ) i=1
(2d) x(t) ˙ = − ωc x(t) + vp (t) − vp0 .

20

(i)

outputs [mW]

15

open-loop feedback

chnl 1

10 chnl 2 5

chnl 3

controller Rf

0

driver Pin,0 (t) Cf

carrier density 18 -3 [x10 cm ]

vp0

3.65 3.6 3.55 3.5 3.45 3.4

vp Is

Rin

Lp Cp

SOA Rp v Rs s Cs

vd Csc

open-loop feedback

(iii) chnl 1

1.2

chnl 2

0.8

chnl 3

0.4 0

I¯

control [mW]

ip

inputs [mW]

1.6

(ii)

1.6

(iv)

1.2

chnl 0

0.8 0.4 0 0

1

2

3

4

5

6

7

time [ns]

(a) SOA equivalent circuit with filtered output feedback controller.

(b) System response to channel add/drop.

Figure 1: (a) SOA equivalent circuit model with parasitic network (adapted from Ref. [3]) with 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). (b) Comparison of nonlinear system closed-loop (solid line) to open-loop (dashed line) responses for channel add/drop. The controller (iv) responds to adding/dropping of channels (iii) to keep the inversion carrier density as constant as possible (ii), thereby suppressing interchannel crosstalk in the output (i). 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). Eq. (2d) is the state of the integrating filter, 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 shown in Fig. 1(a). Figure 1(b) shows the closed-loop system response for a controller with cutoff frequency ωc = 2π(5 × 107 ) rad/s and gain k = 3. The optical control channel is set to a nominal value of 1 mW, and the bias current is set to 150 mA. The remaining values are taken from Ref. [2] with gain gi (t) = Γi ai [N (t) − Ntr,i ] and recombination R(t) = RA N (t) + RB N 2 (t) + RC N 3 (t). When Channel 2 turns off at t = 1 ns (iii), the controller increases Channel 0 to compensate (iv), resulting in relatively constant inversion level (ii) and suppressed crosstalk into Channel 1 at the output (i), as well as improved steady-state gain. The controller reacts appropriately when Channel 2 returns at t = 3 ns, and when Channel 3 comes online at t = 5 ns. At the leading edge of each step input, the controller reacts somewhat slowly due to the low-pass filtering, so there is a design trade-off between transient time and ringing, but the overall response is good once the controller settles (in this case, 1 ns). The response could be further improved by using a higher-order low-pass filter or by introducing a derivative component to the feedback. Our analytical state-space design methods employed here are entirely general, can be used with any linear or linearized equivalent circuit models, and can be used to design more complex SOA control schemes. References [1] H. Wessing, B. Sorensen, B. Lavigne, E. Balmefrezol, and O. Leclerc, “Combining control electronics with SOA to equalize packet-to-packet power variations for optical 3R regeneration in optical networks at 10 Gbit/s,” Proc. Optical Fiber Comm. Conf., vol. 1, p. WD2, 2004. [2] 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, no. 2, pp. 123–129, Feb 2007. [3] R. S. Tucker, “High-speed modulation of semiconductor lasers,” IEEE J. Lightwave Technol., vol. LT-3, no. 6, pp. 1180–1192, Dec 1985.