Application and Optimisation of the Spatial Phase Shifting ...

118 Comparison of noise in phase maps from TPS and SPS
5.4 Out-of-plane displacements
The sequence of tilts described in 5.2.2 was carried out for both phase-shifting methods; the results for
vertical fringes (varying N x ) are graphed in Fig. 5.5. The conversion factor from phase to displacement is
λ/713°, or equivalently, λ/(507 grey levels); this means that one wavelength of displacement gives rise to
almost two fringes in the sawtooth image. Hence, the maximal detectable σ d,max in the sawtooth images
(cf. Chapter 4.2) corresponds to 104λ/71374λ/5070.146 λ for the out-of-plane geometry.
0.14
σ d /λ
0.12
14
12
0.10
10
0.08
08
0.06
06
0.04
0.02
0.00
x
0 5 10 15
20 30 40 50
02
60 70 90 100
N
00
0 2 4 6 8 d s /d p 10
04
N x
0 5 10 15
20
60
30
70
40
90
50
100
0 2 4 6 8 d s /d p 10
Fig. 5.5: σ d for ESPI displacement measurements with SPS (left) and TPS (right) as a function of speckle size for
out-of-plane displacements. The parameter for each curve is N x , the number of vertical fringes per 1024
pixels, as indicated in the legend boxes.
In the interpretation of these plots, we will again have to bear in mind that we encounter both types of
speckle decorrelation here: (i), aperture-plane decorrelation, which progresses faster for small apertures
(large speckles) as we increase the tilt; (ii), sensor-plane decorrelation or speckle pattern displacement
due to object tilt, which leads to an increasing pixel position mismatch between initial and final speckle
pattern and affects the fringe quality more strongly for small speckles. It is true that the fringe quality
could be partly restored by re-positioning the images to compensate the shift of the speckle pattern, as
suggested in [Leh98]; but as this would frequently involve non-integer pixel shifts, we do not further
pursue this approach. Despite this minor flaw in the set-up, we will be able to carry out the intended
comparison.
Not surprisingly, the zero-displacement measurements with SPS turn out best with very large speckles,
since this minimises the problems for the phase calculation. But the high sensitivity to aperture-plane
decorrelation leads to a fast deterioration of the fringe quality as the tilt increases. Also, at N x =100, one
fringe would consist of only one speckle at d s = 10 d p , and this is clearly below the limit of 4 speckles
given in [Tan68]. For d s = 5 d p , which corresponds to 2 speckles per fringe when N x =100, we can already