Multiple-Input Multiple-Output Optical Integrator Design by Non-Interacting Control via Dynamic Extension

Abstract

Semiconductor optical amplifiers (SOAs) have recently attracted much attention as the potential building blocks for active photonic circuits. They are being used in many applications of optical signal processing in addition to amplification of the optical signal. Their use as wavelength converters and optical logic gates has been demonstrated [1,2]. In this paper we propose an SOA based design of a multiple-input multiple-output integrator for optical signals. The design is based on non-interacting state feedback control through dynamic extension. If a nonlinear dynamical system takes the form of Equation (1) where, z is in Rn , f, g and h are n-dimensional smooth vector fields in the neighborhood of z0 in Rn , and it also has a well-defined vector relative degree in the neighborhood of z0, it is well known in the nonlinear control literature that the system can be transformed into a chain of integrators by state feedback [3]. Moreover, the effect of such feedback will result in the outputs which are completely decoupled from each other, with each output yi being affected only by the corresponding input, ui, hence it is named non-interacting control. This control method is commonly used in nonlinear multiple-input multiple-output systems. From the physical governing equations of SOA, the state space model for the device was developed by Knutze et. al, in [4] in the form of Equation (2), where z is the charge carrier density and RA, RB and RC are, respectively, linear, bimolecular and Auger recombination coefficients; I is the electric current, q is the charge of an electron; and V and L are, respectively, the volume and the length of SOA. For each optical channel: ui is the input and, Gi(x)=aiγi(z-ztr,i) is the gain function; ai is the differential gain, γi is the confinement factor; ztr,i is the transparency carrier density; ωc,i is the carrier frequency; and αi is the waveguide loss. The signals yi are the optical outputs. Equation (2) does not take the form of (1) because it has the inputs directly appearing in the outputs. Thus, non-interacting control method cannot be applied directly. Therefore, the dynamics of the system are extended by renaming z, u1 and u2 as z1, z2 and z3, respectively, and the external inputs: v1 and v2are used to control the dynamics of these extended states. Now the state space model becomes Equation (3). Since Equation (3) takes the form of Equation (1), and it was verified that the system has a well-defined vector relative degree 1 in a large range of operating points, the non-interacting feedback laws are derived and the state feedback is applied. After applying feedback, the closed loop device is simulated in Matlab Simulink and it was observed that each optical input signal ui(t) is integrated and its integral yi(t) appears at the output without inter-modal modulation. Simulation results for a pair of typical inputs are shown in Figure 1.