Branch-point-tolerant least-squares phase reconstructor

Abstract

In reconstructing images from coherently illuminated objects, the far-field two-dimensional phase function in the entrance pupil plane (measurement plane) of a coherent imaging system has to contend with the presence of point discontinuities or branch points. The general class of least-squares phase reconstructors that use phase gradients (or phase differences) on a grid of points in the entrance pupil measurement plane fails to correctly determine the two-dimensional pupil phase when branch points are present. The phase estimation error results from the fact that the phase gradient or phase difference inputs to the least-squares reconstructor are being wrapped by the measurement system into the principal-value range [-π, π]. Recently, the existence and the determination of a hidden phase term was presented that accounts for branch-point effects [ D. L. Fried , J. Opt. Soc. Am. A15, 2759 (1998)]. Fried’s hidden phase term is used in this study to present a branch-point-tolerant least-squares phase reconstructor for estimation of the two-dimensional measurement-plane phase function. Simulations of three different types of coherently illuminated object demonstrate the utility of this approach. The sensitivity of the reconstruction method to additive Gaussian noise is also presented.