Abstract

For several problems in optics it is necessary to estimate the magnitude of an optical field from a photon-limited measurement of its intensity (squared magnitude). An example is wavefront sensing from one or more measured intensities using an iterative transform algorithm such as the Gerchberg–Saxton algorithm. The maximum-likelihood estimate of the intensity and the minimum mean-squared error estimate of the intensity is simply the measured photon-limited intensity. The square root of the measured intensity is the maximum likelihood estimate of the magnitude, but it has a negative bias in that the square root of the intensity systematically underestimates the magnitude. We show that there exists no unbiased estimator for the magnitude. Other estimators, such as the square root of (1 + intensity), where intensity is in terms of number of detected photons, perform better than the square root estimator, as do some Bayesian estimators that use an assumed prior probability distribution for the intensity. The Cramer–Rao lower bound on the mean-squared error in estimating the magnitude is 1/4, and all of the above estimators have a mean-squared error close to 1/4 when the intensity is large but significantly differ from 1/4 when the intensity is small. The analysis is also given for intensity measurements with a background bias.

© 1992 Optical Society of America