Phase-shifting speckle interferometry with unresolved speckles: A theoretical investigation

Abstract

In phase-shifting speckle interferometry, the varying intensity of speckle fields always causes some pixels to be invalid. It is either saturation or low modulation that prevents a reliable phase from being calculated. Proceeding from a criterion that considers a point to be valid as soon as it does not saturate and its modulation exceeds a minimum threshold, the efficiency of a measurement is defined as the fraction of valid points. In order to calculate this efficiency, the statistical distribution of the background intensity and the modulation is derived for an arbitrary number n of speckles per pixel, for both the speckle and the smooth reference wave setup. Optimum wavefield intensities leading to a maximum efficiency are determined. The results contradict the generally accepted statement that satisfying measurements require well resolved speckles: The maximum efficiency, which is expected to decrease as n increases, proves to remain satisfactory for relatively high values of n. For low yet reasonable modulation thresholds, the efficiency of the speckle reference wave setup is even improved when a pixel integrates over several speckles.