Application and Optimisation of the Spatial Phase Shifting ...

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6 Improvements on SPS
The comparison of TPS and SPS has shown that TPS yields lower measurement errors especially in the
region of low fringe densities. Since it is generally more preferable to record several sawtooth images
with few fringes than one image with many fringes [Flo93, Her96], we shall therefore explore some ways
to reduce the σ d associated with SPS in this chapter. First of all, the beam ratio in the interferograms is
shown to be of great importance; but there are also possibilities to reduce the measurement error by phase
calculation formulae tailored for SPS. And lastly, we employ the "single-frame" measurement capability
of SPS to introduce some improvements.
6.1 Optimisation of experimental parameters
6.1.1 Beam ratio
Although the best intensity ratio of reference to object wave, B, has been thoroughly investigated [Sle86,
Leh95, Maa97] in order to maximise the interferometric modulation, it has also been stated that the least
permissible M I can be set quite low, e.g. at some 8 grey levels or even less [Dör82, Ker88, Vro91, Hac00].
Consequently, phase shifting in ESPI yields reasonable results for quite a large range of B. In what
concerns TPS, we can expect the errors to remain approximately constant as long as M I is beyond its
lower threshold. With growing intensity of the reference wave, the modulation drops and electronic noise
and digitisation errors gradually gain the upper hand over the signal.
For SPS however, the speckle character of the object wave constitutes an error source that depends on the
object intensity: the intensity readouts I n (cf. (3.12)) from a set of adjacent pixels should have equal I b and
M I if the phase calculation is to function correctly; but the brighter the speckles are, the greater become
their intensity gradients and the worse is the mismatch of the interferometric parameters on adjacent
pixels. It is clear that the absolute intensity errors drop when the beam ratio is increased; but this is of no
consequence for the measurement, because the modulation goes down as well. An improvement comes
about only by a decrease of the relative intensity errors, and it has been shown in a simple form in
[Bur99a] that this is indeed the consequence of a brighter reference wave.
To describe the phenomenon, we first need to know how statistical intensity fluctuations are propagated to
phase errors σ ϕ ο by the phase calculation. Assuming a standard deviation of σ I for the intensity readings,
this relationship is described by Eq. (12) of [Bot97] in a general form for 3-bucket formulae. For
α x =120°/sample, it reads
σ
ϕ O
σ I
= ⋅
2
M I
8
3 , (6.1)
where σ ϕ ο is the standard deviation of the calculated phase averaged over all ϕ O , and σ I that of the
interferogram intensities. In a simple approximation, σ I is composed mainly of the standard deviation of