Solution of linear and nonlinear equations by optical methods

A confocal Fabry-Perot processor, with coherent image amplification provided by a photorefractive BaTiO(3) crystal in the feedback path, is analyzed and implemented to perform the iterative algorithm based on the relation B(-1) = (I - A)(-1) = (infinity)Sigma(k=0) A(k), where B is the matrix to be inverted and I is the identity matrix. Both A and B are large size matrices. When the feedback loop contains a coherent matrix-vector multiplier (AX) and the input vector is sequentially scanned from one element to another, the columns of B(-1) can be sequentially generated at the output. The photorefractive BaTiO(3) amplifier provides loss compensation and coherence restoration of the feedback signal, thereby increasing the effective number of iterations in the algorithm. Thus it becomes possible to use this technique to implement slowly (as well as rapidly) converging algorithms. Experimental verification of the matrix inversion algorithm is presented, along with an analysis of possible real-time operations.

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