Abstract

An unknown probability law p(x) can be efficiently estimated using a smoothness enforcing principle $I={\displaystyle \int dxp\text{'}{\left(x\right)}^2/p\left(x\right)}$ = minimum, called minimum Fisher information (MFI). An unknown diffraction pattern is an unknown probability law (on photon position x). Consider a best estimate of the centroid position in the pattern from observation of a single-photon position. It is known that the mean-square error e² in such an estimate must exceed the Cramer-Rao bound 1II,I above. Therefore, if we demand that e² be a maximum, we have constructed a physical rationale for the use of MFI for estimating p(x). To bring in the known refractive-index profile n(x) of the diffracting medium, demand as a constraint that the mean-square spatial phase ${\left(2\pi /\lambda \right)}^2{\displaystyle \int dx{p}^2\left(x\right)p\left(x\right)={\varphi }^2}$ be generally nonzero. Constructing an overall principle I — ϕ²= minimum, a variational problem in p(x) results. The Euler-Lagrange solution is the Helmholtz wave equation on a real amplitude function q(x), where p(x) = q(x)². Thus a diffraction pattern is seen to be nature’s way of maximally smearing out spatially the probability law on position for each photon forming the pattern.

© 1988 Optical Society of America