Robust Design of Two-Dimensional Optical Reference Signals Against Diffraction Effects

This paper presents two novel approaches to the design of two-dimensional (2-D) optical zero reference signals (ZRSs) that are robust against diffraction effects based on the parametric minimum cross-entropy (PMCE) method. Grating alignment systems require a 2-D optical ZRS to perform absolute measurements. A common method of acquiring 2-D optical ZRSs involves illuminating two identical superimposed 2-D zero reference codes (ZRCs). The output signal is the 2-D optical ZRS and can be represented as the autocorrelation of the 2-D ZRC transmittance. In ultrahigh-resolution systems, diffraction distorts the shadow of the first 2-D ZRC, degrading the autocorrelation and greatly reducing the amplitude of the 2-D optical ZRS. To improve the robustness of 2-D optical ZRSs against diffraction effects, this paper formulates two combinatorial optimization problems for the design of 2-D ZRCs with minimum diffraction effects: one of which is a maximization problem, and the other a minimization problem. Aiming at solving the two problems, this study proposes two schemes based on the PMCE method to search for an optimal 2-D ZRC. Simulation results reveal that there are 3.36-8.34% increases in the slope of the central peak of a 2-D optical ZRS and that there are 16.12-20.90% increases in the sum of the slope of the central peak and the effective signal amplitude of a 2-D optical ZRS, as compared with those obtained by the recently proposed cross-entropy method. The proposed PMCE-based schemes prove to search for 2-D ZRCs more effectively than existing solutions, while requiring less computational complexity.