Application and Optimisation of the Spatial Phase Shifting ...

5.2 Preliminary investigations 115
generally higher noise level, the accuracy of the SPS measurements shows a less pronounced dependence
on the phase shift. Moreover, the 90° formula performs slightly better, in contrast to the theoretical
findings in [Bot97], but in agreement with the more speckle-specific investigations in Chapter 3.2.2. The
slight difference in performance does however not appear to discourage using 120° also for SPS in this
study, and we will do so to maintain comparability, but will come back to α=90° in SPS in Chapter 6.1.2.
For higher fringe densities, TPS and SPS deliver similar performance; this is partly due to the
aforementioned fact that significant speckle displacement occurs for larger tilts, which contributes the
larger part to speckle decorrelation when the speckles are small.
5.2.2 Reproducibility of the σ d values
While the fitting algorithm described in Chapter 4.2 yields a very reliable average of phase errors in one
sawtooth image, this tells us nothing about whether we will get the same error in a second experiment.
This deserves particular attention because the test object had not been specially made: it was a large
mirror mount onto which a rotation stage was fitted with the aluminium plate on it. The out-of-plane tilts
were generated by manual setting via the fine-thread screws of the mirror mount, and the in-plane
rotations by manual setting of the rotation stage via a reduction gear. While the latter yielded excellent
reproducibility of the measured displacement errors, the former showed some fluctuations, which had to
be investigated in more detail to learn how reliable the σ d measurements are. Fig. 5.3 shows the results for
a set of SPS experiments. For each of the speckle sizes 1.5, 3, and 6 d p , the tilt sequence was repeated 10
times; the averages σ d with their respective standard deviations σσ d
are given in the figure. This was
done for both vertical and horizontal sawtooth fringes.
0.12
σ d ¡ /λ
Tilt about y axis
12
Tilt about x axis
0.1
1
0.08
08
0.06
0.04
d s =1.5 d p
d s =3 d p
04
0 20 40 60 80 N x 100
06
d s =1.5 d p
d s =3 d p
d s =6 d p
0 20 40 60 80 N y 100
Fig. 5.3: Reproducibility of measurements of σ d vs. N x and N y for out-of-plane tilts. Left: tilt about y-axis, vertical
fringes; right: tilt about x-axis, horizontal fringes. Note that the ordinates begin at σ d = 0.04 λ to expand
the error bars.
For a speckle size of 6 d p , the reproducibility is excellent. At low fringe densities, the spatial phase
measurement works well because of the low intensity and phase gradients in the speckles; but as the tilt
increases, aperture-plane decorrelation impairs the accuracy. For d s =6 d p , the σ d curves are very similar