Application and Optimisation of the Spatial Phase Shifting ...

Appendix D: Alternative error-compensating formulae 203
n y
/n 02
3
n y
/n 02
3
0|4
0|4
1
1
2
2
2 3 4|0 1 2 n x
/n 0
2 3 4|0 1 2 n x
/n 0
Fig. D.5: bsc(ν x ,ν y ) for (D.1) (left) and (D.2) (right). Black lines: frequency co-ordinates leading to correct phase
calculation, bsc(ν x ,ν y )= 45°; white outlines: areas of -10°¡δϕ¡10°.
As to be seen, both formulae are capable of calculating ϕ O with FδϕF10° in a very wide range of (ν x ,ν y );
(D.1) exhibits a slightly worse phase calculation at very high spatial frequencies, so that we can expect to
find small performance differences in phase measurements. When the interferograms already used for 6.3-
6.5 were re-evaluated with the formulae presented here, the resulting σ d (N x ) were as graphed in Fig. D.6.
0.10
s d /l
0.08
0.06
0.04
5-sample averaging 30/90/150°, B=30
0.02
5-sample averaging 90°, B=30
0.00
0 20 40 60 80 100
Fig. D.6: σ d from ESPI displacement measurements as a function of N x , obtained with (D.1) (white filled symbols)
and (D.1) (black symbols). Input interferograms were from the same tilt series as in 6.3.
The plots show that the performance of (D.1) and (D.2) is indeed very similar * ; by comparison with the
results in terms of σ d in 6.3-6.5, it can be seen that they are also well suited for phase evaluation in SPS
and yield a performance similar to that obtained by (6.16) and its intensity-correcting version, and by the
FTM. However a correction for speckle intensity cannot readily be incorporated in these formulae.
Moreover, a careful comparison of σ d at higher N x with that in previous results shows that now the spatial
extent of the phase-sampling window contributes significantly to the smoothing of phase maps. In this
N x
* The same applies to formulae with zero error for α=45/90/135° and α=60/90/120°, which were tested as well.