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The classical problem of phase unwrapping in two dimensions, that of how to create a path-independent unwrapped map, is extended to the case of a three-dimensional phase distribution. Whereas in two dimensions the path dependence problem arises from isolated phase singularity points, in three dimensions the phase singularities are shown to form closed loops in space. A closed path that links one such loop will cross a nonzero number of phase discontinuities. In two dimensions, path independence is achieved when branch-cut lines are placed between singular points of opposite sign; an equivalent path-independent algorithm for three dimensions is developed that places branch-cut surfaces so as to prevent unwrapping through the phase singularity loops. The placing of the cuts is determined uniquely by the phase data, which contrasts with the two-dimensional case for which there are many possible ways in which to pair up the singular points. The performance of the new algorithm is demonstrated on three-dimensional phase data from a high-speed phase-shifting speckle pattern interferometer.

We present results from numerical simulations of a dynamic phase-shifting speckle interferometer used in the presence of mechanical vibrations. The simulation is based on a detailed mathematical model of the system, which is used to predict the expected frequency response of the rms measurement error, in the time-varying phase difference maps, as a result of vibration. The performance of different phase-shifting algorithms is studied over a range of vibrational frequencies. Phase-difference evaluation is performed by means of temporal phase shifting and temporal phase unwrapping. It is demonstrated that longer sampling windows and higher framing rates are preferred in order to reduce the phase-change error that is due to vibration. A numerical criterion for an upper limit on the length of time window for the phase-shifting algorithm is also proposed. The numerical results are finally compared with experimental data, acquired with a phase-shifting speckle interferometer of 1000 frames/s.

We describe a phase-shifting out-of-plane speckle interferometer operating at 1 kHz for studying dynamic events. The system is based on a Pockels cell that is synchronized to a high-speed video camera to ensure that the phase shifting occurs between frames. Phase extraction is performed by use of a standard four-frame algorithm, and temporal phase unwrapping allows sequences of several hundred absolute (rather than relative) displacement maps to be obtained fully automatically. The maximum theoretical surface velocity of 67 microm s(-1) is a factor of 40 greater than can be achieved with a speckle interferometer based on a conventional video camera. We test the system using a target that is displaced with constant speed in a direction normal to its surface by means of a piezoelectric transducer. The system's performance in a practical situation is illustrated with measurements on a thin plate undergoing out-of-plane deformation.

A simple geometrical model was developed for calculation of the contrast of a polychromatic image-plane speckle pattern from a source of light with high spatial coherence. It is based on counting the number of independent speckle patterns that contribute to a given point in the image plane. This results in a simple equation for the contrast as a function of imaging geometry; relative orientation of the projection direction, observation direction, and specimen normal; bandwidth of the light source; and surface roughness. Its validity was established by comparison with an exact solution: rms errors in the calculated contrast were only 0.033 over a wide range of parameter values likely to be encountered in practice.