Application and Optimisation of the Spatial Phase Shifting ...

176 Summary
method to search for a phase-shifting formula with high phase-shift error resistance, the question arose
whether it would be better to optimise the amplitude or the phase spectrum of the phase extraction
formula for low phase-measurement errors. To settle the question, a simple auxiliary function was
introduced which is invariant under the various optimisations and thus showed that nothing is to be gained
by simply representing a formula in different ways. This behaviour was confirmed in SPS experiments.
Another valuable means of characterising the spatial phase evaluation is the dependence of the phasemeasurement
error on the phase to be measured. It was elucidated how the phase reconstruction generates
periodic errors in the sawtooth fringes and systematic biases for the phase calculation, and what role the
choice of the phase-calculation formula plays for this effect.
The quest for a reliable performance figure of ESPI phase measurements, which is indispensable to carry
out comparisons and quantify improvements, has led to the creation of a standardised noise quantification
method that fits an ideal data set to a real one and delivers the standard deviation of the remaining phase
differences. The displacements to use this method were standardised as well. The advantages of the fitting
method have been demonstrated by confrontation with various other methods of generating reference data.
The noise quantification tool was then extensively used to compare the performance of SPS with that of
TPS in various measuring geometries, where a simple phase-shifting scheme was used under stable
laboratory conditions. A multi-purpose interferometer allowed to carry out this comparison under the best
possible constancy of experimental parameters. By varying quantities like fringe densities, speckle size
and shape, and object illumination intensity, characteristic behaviours of SPS and TPS were explored. It
was found that TPS offers advantages for in-plane measurements and under severe shortage of laser
power; for out-of-plane configurations, the difference was found to vanish with increasing fringe density.
As an extension of the performance study with standard experimental parameter settings and data
processing, several ways to improve the SPS technique have been implemented and tested. A very
important result is the finding that the role of the beam ratio is decisive in SPS but has far less impact on
TPS. Then, in agreement with the hints from theoretical considerations on spatial phase sampling, a phase
shift of 90° per sample was found to yield better measurements than 120° per sample.
Based on the conclusion from the investigation of speckle statistics, a formula was established that can
make use of a separate recording of the speckle pattern alone to correct phase-calculation errors
introduced by speckle intensity gradients. The performance thus gained is however almost reached by
uncorrected measurements when the beam ratio is set to its optimum, which was around 30 for the
experimental set-up used.
To compensate measurement errors by speckle phase gradients, a simple averaging formula was used and
seen to bring about a relevant improvement. This improvement can only partly be ascribed to the
cancellation of phase-shift errors; also the enlargement of the spatial sampling window from 3 to 4 pixels
plays a role.