Application and Optimisation of the Spatial Phase Shifting ...

3.4 Spatial phase shifting 79
which allows for a very convenient determination of the speckle size from the power spectrum of the
interferogram. When the interference bands are centred on each other (i.e. the source point of R is placed
in the exact centre of AS), the edges of the frequency plane are reached when d s =1.22 d p , but using its
corners, we can accurately determine speckle sizes down to 0.86 d p .
The advantage of using interferogram power spectra gets clear when we consider Fig. 3.22: determining
the speckle size from this image is very easy, while it is problematic to apply the autocorrelation
technique for so small a speckle size.
Fig. 3.22: Power spectrum of interferogram with d s =1 d p ; spatial frequency axes as in Fig. 3.20.
Finally, if the source point of R is not in the plane of AS, the δ function above will broaden; then, on
convolution with ST AS , the sharp edges of the cross-interference spectra will smear out. This behaviour
provides us with a very accurate means to match the curvatures of the two wave fields.
3.4 Spatial phase shifting
An elegant way to get rid of the problems associated with inter-frame temporal parameter fluctuations is
to acquire the phase-shifted data simultaneously. Since the phase shift then has to take place in space
instead of time, this approach is quite generally called spatial phase shifting (SPS) [Schwi90, Tak90b,
Kuj93, Vla94]; the underlying principle has been known for a long time [Lei62]. With SPS, phasemeasuring
methods gain access to unstable environments and transient events. For very rapid phenomena,
the use of pulsed illumination represents an effective way to suppress even the intra-frame fluctuations
and freeze virtually anything. In principle, it gets possible to track the object phase at the frame rate of the
camera, with the additional benefit that any frame of the series can be appointed the new reference image.
The increased temporal resolution of this approach has, however, to be paid for in terms of spatial
resolution, since it is of course necessary to spatially separate the I n . In analogy to TPS, we can distinguish
between phase stepping and phase ramping. The former is implemented by generating several images of
the same object and recording them simultaneously on several sensors, or different parts of the same
sensor. The necessary phase shift between the images can be generated by polarisation optics [Smy84,
Kuj93, vHaa94], diffraction gratings [Kwo84, Kuj88], CGHs [Bar99] or combinations of these [Kra98,
Kem99, Het00]. For the phase retrieval to work properly, the I n must be aligned with subpixel accuracy