A Lagrangian Formulation For Optical Backpropagation Training In Kerr-Type Optical Networks

Steven R. Skinner Electrical Engineering Wichita State University Wichita, KS 67260-0044 Elizabeth C. Behrman Physics Department Wichita State University Wichita, KS 67260-0032 A training method based on a form of continuous spatially distributed optical error back-propagation is presented for an all optical network composed of nondiscrete neurons and weighted interconnections. The all optical network is feed-forward and is composed of thin layers of a Kerrtype self focusing/defocusing nonlinear optical material. The training method is derived from a Lagrangian formulation of the constrained minimization of the network error at the output. This leads to a formulation that describes training as a calculation of the distributed error of the optical signal at the output which is then reflected back through the device to assign a spatially distributed error to the internal layers. This error is then used to modify the internal weighting values. Results from several computer simulations of the training are presented, and a simple optical table demonstration of the network is discussed. 772 Elizabeth C. Behrman 1 KERR TYPE MATERIALS Kerr-type optical networks utilize thin layers of Kerr-type nonlinear materials, in which the index of refraction can vary within the material and depends on the amount of light striking the material at a given location. The material index of refraction can be described by: n(x)=no+nzI(x), where 110 is the linear index of refraction, ~ is the nonlinear coefficient, and I(x) is the irradiance of a applied optical field as a function of position x across the material layer (Armstrong, 1962). This means that a beam of light (a signal beam carrying information perhaps) passing through a layer of Kerr-type material can be steered or controlled by another beam of light which applies a spatially varying pattern of intensity onto the Kerr-type material. Steering of light with a glass lens (having constance index of refraction) is done by varying the thickness of the lens (the amount of material present) as a function of position. Thus the Kerr effect can be loosely thought of as a glass lens whose geometry and therefore focusing ability could be dynamically controlled as a function of position across the lens. Steering in the Kerr material is accomplished by a gradient or change in the material index of refraction which is created by a gradient in applied light intensity. This is illustrated by the simple experiment in Figure 1 where a small weak probe beam is steered away from a straight path by the intensity gradient of a more powerful pump beam.