Application and Optimisation of the Spatial Phase Shifting ...

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5 Comparison of noise in phase maps from TPS and SPS
Over the years, TPS has become a well-established technique that is confidently used in many
applications; SPS is far less frequently used in ESPI and seldom considered as an alternative despite its
ease of use and immunity to instabilities. And there are indeed reasons to doubt whether SPS can compete
with TPS in ESPI: the small aperture needed to generate speckles large enough for SPS leads to decreased
light efficiency, reduced spatial resolution, and also accelerates aperture-plane speckle decorrelation.
Moreover, the spatial intensity and phase variations of the speckle field obstruct an accurate phase
calculation, all the more as the number of available phase samples is very limited.
But also in TPS, where almost any error-compensating phase extraction with any number of intensity
readings could be employed, it is customary to use Carré's [Car66] or Schwider's [Schwi83] formula. This
is because not even the most sophisticated of formulae will help against speckle decorrelation and pixels
with too low modulation M I . Therefore, the uncertainty estimates have not changed much over the years;
they range from λ/15 [Nak85] to λ/30 [Rob86, Ker88] or even λ/50 [Vik91, vHaa94], depending on
whether correlation fringes or speckle phase maps are evaluated, and in the latter case, also on the fringe
density.
As yet, there are no corresponding data available for SPS, so that the decision which method to use
remains a matter of presumptions. The present chapter is intended as an attempt to fill this gap [Bur00a].
Although it must be borne in mind that the data presented here are, strictly speaking, only valid for the
interferometer and test object used, they do allow a comparison of TPS and SPS.
There are many parameters to be tested in such a study. The most essential ones are the phase shift and the
reference-to-object intensity ratio to use. Speckle size and shape can be expected to play a special role for
the fringe quality in SPS; and by varying the fringe densities, we will get an idea whether the reduced
spatial resolution of SPS matters in practice. Moreover, we will test the performance of TPS and SPS
under very low illumination levels to learn what restrictions the smaller aperture for SPS effects.
Although we will of course use imaging optics, we will determine the speckle size as if we were dealing
with objective speckles; this is owing to the slightly modified objective shown in Fig. 5.1. When we take
D as the diameter of the aperture and z as its distance to the camera chip, (2.43) remains perfectly valid,
although there is no "free" scattering after the lens anymore; but z is large enough for this simple
geometrical formula to function correctly, as was also confirmed by accurate measurements of the speckle
size as described in 3.3.1.
While the out-of-plane measurements can be carried out with the same interferometer geometry for both
methods, the symmetrical-illumination in-plane layout for TPS [Lee70] cannot be reproduced for SPS.
Therefore we will test two different approaches of in-plane displacement measurements with SPS to gain
a "three-dimensional" insight.