Application and Optimisation of the Spatial Phase Shifting ...

6.2 Modified phase reconstruction formulae 145
0.12
σ d /λ
0.10
0.08
0.06
0.04
0.02
0.00
3-sample 90°, B=30
3-sample 90° with intensity correction, B=3
3+3-sample averaging 90°, B=30
3+3-sample averaging 90° with intensity correction, B=3
0 20 40 60 80 N x 100
Fig. 6.9: Overview of σ d from ESPI displacement measurements as a function of N x , obtained with various phase
calculation formulae from two series of interferograms: without intensity correction, B=30 (black
symbols); with intensity correction, B=3 (black symbols filled white); 3-sample formulae, triangles; 3+3-
averaging formulae, squares.
The summary presented in Fig. 6.9 allows some conclusions: (i) the use of a 3+3 averaging scheme alone
is definitely a better choice than an intensity-error compensating formula alone. (ii) The combination of
both error-reduction methods leads to the lowest overall error; at d s =3 d p , σ d remains below λ/20 up to
N x =20. (iii) Introducing the intensity-error correction effects indeed a slightly greater improvement in σ d
when a phase-shift error elimination is already present, and vice versa. (In other words, the lower two
curves are farther apart than the upper two.) This confirms the initial presumption that motivated this
subsection: the two methods profit from each other if used together. However, as mentioned before, in
most practical cases it will suffice to set B30 and to do without the small benefit of the intensity
correction, all the more since this speeds up the calculations considerably and even makes them accessible
to the use of look-up tables.
Finally, it may be worth noting that a 3+3 averaging formula according to [Bur98a]
ϕ ' mod π = arctan
O
K2 + K5
− K + K − K + K
1 3 4 6
(6.9)
is error-compensating only by averaging, but does not eliminate the cyclical errors shown in Fig. 3.39,
since it constitutes the average over ϕ O and ϕ O +90°. Hence, (6.9) will do little more for error reduction
than the intensity correction without averaging, which means that even the pure phase-shift error
compensation of (3.56) would perform better. This was in fact found in [Bur98a], where (6.9) was used
instead of (6.8), and emphasises the relevance of an optimal composition of the averaging formula.