Theoretical expression for the correlation signal of nonlinear joint-transform correlators

Abstract

The nonlinear joint-transform correlator is obtained by placing a nonlinear spatial light modulator at the Fourier plane of the conventional joint-transform correlator. Here the error-function limiter is used to model the linear and the nonlinear effect of the spatial light modulator. The theoretical expression for the correlation signal of the nonlinear joint-transform correlator is thus derived. It is shown that every part of the interference intensity can be used to construct the correlation signal of the nonlinear joint-transform correlator. In addition, the correlation signal of the nonlinear joint-transform correlator is the convolution of the conventional correlation signal and a nonlinear signal. Since not all of the light energy is used to construct a given order of correlation signal, the light use should not be 100%. The nonlinear joint-transform correlator becomes bipolar when the nonlinearity is a hard limiter. The theoretical expression for the correlation signal of the bipolar joint-transform correlator is also derived. The hard-limiting effect converts the correlation signal at the Fourier plane from an amplitude-modulated signal into a pulse-width modulated signal. An equation for deciding the optimal threshold value to maximize the autocorrelation peak of the bipolar joint-transform correlator is also obtained. This optimal threshold value is shown to be invariant to the separation between the input and the reference signal.