Application and Optimisation of the Spatial Phase Shifting ...

126 Comparison of noise in phase maps from TPS and SPS
Fig. 5.12: Power spectra of interferograms from pure in-plane SPS set-up; left, d s = 3.6 d p ; right, d s = 6.0 d p . The
scaling is logarithmic and contrast-enhanced.
In contrast to the TPS set-up, where the in-plane sensitivity is obtained by symmetrical oblique
illumination, the SPS in-plane method relies on oblique viewing of the object. Unfortunately, the imaging
geometry is now quite different from all the assemblies presented before, and also, the viewing under
45° introduces a considerable perspective error. In principle, this could be corrected by use of prisms as
described in [Sir97b], but in order to valuate the configuration in its basic version, this was not done here.
Owing to the perspective and the altered imaging geometry, the field of view is 68.536.5 mm²; we will
have to take the greater image height into account when comparing fringe densities. (We continue
working with the familiar fringe counts because this keeps the quantity of "pixels per fringe" comparable.)
Moreover, the apparent height of the object (size in y-direction) changes with the x co-ordinate: it ranges
from 35 to 38 mm, so that the height statement is necessarily an average. Since the height changes have
opposite sign for the two viewing directions, there is also a position mismatch between the superposed
speckle images that is largest at the left and right edges of the field of view, and can vanish only on a
vertical line in its centre. This causes a slight sensitivity to displacement gradients, as in shearing ESPI,
but fortunately these do not affect displacements in x-direction. Furthermore, the quality of the mirror
prism bears some relevancy: a pyramidal shape error (i.e. the prism is a segment of a high three-sided
pyramid) will cause a rotation of the images against each other. Indeed, such an image rotation, of 2°,
was present, that added to the position mismatch caused by perspective.
The perspective error plays a role in so far as the fringes are not exactly localised on the object surface. In
white light-images of the object however, no significant defocusing was present over the width (size in x-
direction) of the image, which is due to the large depth of focus by the small apertures.
Since the aperture sizes D can be no larger than the separation of their centres, ∆x, we have
ds
λz
λz
360°
D ≤ ∆x
⇒ = ≥ =
122 . D ∆x
α , (5.1)
where zf is the distance of the aperture to the camera sensor. Hence, if we adjust α x to 120°/column
again, the smallest speckle size we can get is d s 3.7 d p . This can be seen in Fig. 5.13, where this entry is
x