Application and Optimisation of the Spatial Phase Shifting ...

5.3 Zero-displacement-gradient measurements 117
theoretically suitable to distribute the global phases uniformly over [0,2π) when the interferograms were
captured and stored at a fixed rate of 1/3 Hz. Fig. 5.4 shows results from this procedure for three different
speckle sizes and 120 measurements of σ d for each of them.
0.12
σ d / λ
0.10
0.08
0.06
0.04
0.02
0.00
d s /d p = 1.5
d s /d p = 3
d s /d p =10
-180 -90 0 90 180
∆ϕ/ deg
Fig. 5.4: Dependency of σ d as determined by SPS on the phase offset ∆ϕ for various speckle sizes and N x =N y =0, cf.
the error fringe profiles given in Fig. 3.39 and Fig. 3.40.
As in Chapter 3.4.6, the qualitative appearance of the graphs in Fig. 5.4 suggests that the underlying
phenomenon could mainly be a linear miscalibration of the phase shift: when we subtract one phase map
from another, the errors thus produced theoretically cancel at phase differences of ∆ϕ =0 and π, and add
up in between these values. In particular, this explanation seems reasonable because the smaller the
speckles, the higher their phase gradients in units of d p and thus the larger σ d . At ∆ϕ π, however, σ d
does not reach the minimum at ϕ 0 0 again, which tells us that there are other error sources than wrong
phase shift alone; this has been interpreted in Chapter 3.4.6.
The dependence of σ d on ∆ϕ is also found within displacement fringes (in which ∆ϕ progresses
deterministically from –π to π), so that the σ d which we assign to sawtooth images is in itself an average
over all ∆ϕ. Examples of this behaviour are the white curves in Fig. 3.39 and Fig. 3.40.
As can be seen from Fig. 5.4, the distribution of the ∆ϕ is still too irregular to permit a direct calculation
of the average; this effect does come from random phase fluctuations in the interferometer. Therefore it
was necessary to fit suitable functions (given in the figure as well) to the data points and to determine the
mean values of these instead. The values finally obtained constitute the entries for N x =N y =0 appearing in
the following plots.
With TPS, none of the described detours is necessary; the phase error does not depend on the global phase
offset, provided the phase shift is calibrated exactly enough. Consequently, one measurement with
N x =N y =0 suffices to determine the corresponding σ d . Furthermore, σ d is uniformly distributed in sawtooth
fringes from TPS, and there is no such thing as an error fringe profile in this case.