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

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 y_i being affected only by the corresponding input, u_i, 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 R_C 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: u_i is the input and, G_i(x)=a_iγ_i(z-z_tr,i)  is the gain function; a_i is the differential gain, γ_i is the confinement factor;  z_tr,i is the transparency carrier density;  ω_c,i is the carrier frequency; and α_i  is the waveguide loss.  The signals y_i  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 u_i(t) is integrated and its integral y_i(t) appears at the output without inter-modal modulation. Simulation results for a pair of typical inputs are shown in Figure 1.

References:

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4. S. B. Kuntze and L. Pavel, Controlling a semiconductor amplifier using a state space model, IEEE J. Quantum Electron., 43 (2007) 123-129.