We describe a fast computational algorithm able to evaluate the Rayleigh-Sommerfeld diffraction formula, based on a special formulation of the convolution theorem and the fast Fourier transform. What is new in our approach compared to other algorithms is the use of a more general type of convolution with a scale parameter, which allows for independent sampling intervals in the input and output computation windows. Comparison between the calculations made using our algorithm and direct numeric integration show a very good agreement, while the computation speed is increased by orders of magnitude.

Our paper provides analytical expressions for the statistical errors related to statistical processing of digitally recorded Newton's rings interference patterns by least squares fitting. These results completes some of our previous papers concerned with Newton's rings fringe patterns processing, which well describe an iterative numerical algorithm that we commonly use for fringe processing3'4.

A self-calibrating algorithm for phase-shift interferometry is described that is able to cancel the effect of accidental relative tilts that may occur during phase stepping. The algorithm is able to retrieve both the phase steps and the tilts that accompany them. Only three phase-shifted interferograms are needed, and no other information about the intentional phase shifts or possible tilts has to be supplied. This purpose is achieved by division of the interferogram space into blocks on which a previously reported self-calibrating algorithm is applied and the actual values of the local phase shifts are calculated. The information thus obtained is used for extracting the global shift and tilt values. Further improvement in the results is achieved by means of a fitting routine.

We present a practical numerical method for processing the fringes obtained when two waves, with a quadratic phase difference function, interfere. This kind of fringe includes straight equispaced fringes and Newton's rings as particular cases. The numerical method we present is based on the discrete Fresnel ͑Fourier͒ transform of the data and has the same precision as least square fitting ͑LSF͒. Compared to the LSF method, this new method is better, as it is more efficient and does not require initial approximations for the fringe parameters to be determined.

A new statistical self-calibrating algorithm for phase-shift interferometry is presented. The algorithm can be applied to the case of three phase-shifted interferograms for which the assumption of a constant fringe contrast over the viewing field can be made. Self-calibration is achieved by minimizing the non-uniformity that appears in the map of the calculated contrast when incorrect phase shifts are supposed. No information on the actual phase shifts has to be supplied, the only input needed is the set of three interferograms. The residual errors of the algorithm are as low as in usual conditions.