Optical inner-product implementation of neural networks models.

The Hopfield neural network model is described in terms of inner product, i.e., as a matched filtering step followed by a pattern synthesis step. Optical implementation by two cascaded coherent filtering setups with holographic matched filters is described. Suitable encoding of information in the form of bipolar (positive or negative) amplitudes in one hologram and of non-negative amplitudes in the other allows one to deal only with non-negative quantities in the input and output planes, thereby avoiding use of multiple channels and coherent detection. The performance of this scheme is analytically evaluated. The above coding can moreover be adapted to the cases of nonzero average memorized states and to the higher-order models associated to Hopfield's algorithm. Numerical simulations and experimental results are presented to illustrate the analysis.