Optimum lens aperture in phase-shifting speckle interferometric setups for maximum accuracy of phase measurement

Negative exponentially distributed intensities of speckle fields seem unfavorable in terms of precision metrology, if interferometric setups are involved with a saturable photodetector and an analog-to-digital converter that imposes a finite resolution. By spatial integration, extended detector apertures modify the intensity distribution toward a less awkward function. However, because the detector aperture also integrates over points of rapidly changing speckle phases, this is done at the expense of a lower modulation of measured intensity during phase shift. An optimum set of parameters is calculated here, consisting of values for the lens aperture, the mean speckle intensity, and the beam ratio. The remaining phase-measurement error assumes its minimum of 10.6 mrad when the space-bandwidth product of the lens-detector system (thus concerning the lens aperture) is 0.31, the mean speckle intensity is 1/11 of the saturation intensity, and the reference intensity is four times higher than the mean speckle intensity. The 90 degrees phase-shift algorithms with either three, four, or five frames turned out to be quite powerful, even with interference signals of rather poor modulation. Not needing a very small lens aperture is interesting, because stopping down the lens is a trade-off with the limited power of the laser.