Application and Optimisation of the Spatial Phase Shifting ...

152 Improvements on SPS
On the whole, it proves rewarding to use the full speckle extent for phase calculation, provided there is
sufficient object light to afford a circular aperture. For practical reasons, one would wish to decrease d s as
far as possible; but in SPS, this has effects that we have encountered in Chapter 5 before: for d s 3 d p , σ d
increases, regardless of the respective fringe density.
6.4 Reduction of speckle size
It seems worthwhile to see whether the methods to reduce σ d developed so far can assist in obtaining
"good" measurements from smaller speckles as well. Therefore we test two more speckle sizes, namely
2.5 and 2 pixels. The best phase calculation found in 6.3 was the average over four 3+3-sample phase
determinations for each pixel, where the intensity correction contributed only a small improvement.
Therefore we apply both (6.16) and its intensity-correcting extension to carry out these additional
measurements. The results are shown in Fig. 6.16, where the last two curves for d s =3 d p are repeated from
Fig. 6.15 for comparison.
0.10
σ d /λ
0.08
0.06
0.04
0.02
0.00
d s=2 d p, B=30
d s=2 d p, B= 3
d s=2.5 d p, B=30
d s=2.5 d p, B= 3
d s=3 d p, B=30
d s=3 d p, B= 3
0 20 40 60 80 N x 100
Fig. 6.16: σ d from ESPI displacement measurements as a function of N x , as calculated by (6.16) (black symbols)
and its intensity-correcting extension (white filled symbols), with various d s as listed in the legend box.
The results from d s =3 and 2.5 d p are very close together (except for d s = 2.5 d p , B=30 and low N x ), they
even cross each other sometimes, which means that the corresponding σ d match within the determination
uncertainty as explained in 5.2.2. This allows the conclusion that we may reduce the speckle size to 2.5 d p
at virtually no harm for the measurement's accuracy. Considering the curves for d s =2 d p , the beginning
increase of σ d vs. d s is clearly noticeable, especially at lower N x . Hence we can conclude that an optimal
adjustment of d s should be between 2.5 and 2 d p for SPS, which is anyhow sufficient to collect between
1.5 and 2 times more light than with the "standard" choice of 3 d p .
One could think up even smaller evaluation clusters to deal with small speckles and possibly enhance the
spatial resolution. Re-considering the arrangements of Fig. 6.11, it would be possible to use pixels
{1, 2, 3, 4, 6, 7} only, which still allows for two 3+3-sample calculations, or even {3, 4, 6, 7}, where it is
possible to average over two sets of 3 samples, {3, 4, 7} and {3, 6, 7}. But in both cases we have no