Application and Optimisation of the Spatial Phase Shifting ...

6.2 Modified phase reconstruction formulae 141
temporal resolution in other measurements. Moreover, (6.4) assumes R and O to be fully interferent,
which is not the case when depolarising objects are being tested. In this case, one must accept that the
treatment overestimates M I (which is related to the square roots), or re-polarise the waves appropriately.
A comparison of (3.19) and (6.6) with stable speckle patterns is given in Fig. 6.1, which shows σ d from
phase calculations without (black squares) and with intensity correction (black squares filled white) as a
function of B. The data leading to the curves were the very same set of interferograms in both cases. For
the intensity correction, I used both the initial and final speckle patterns for the respective object states.
The figure shows that (6.6) is indeed able to keep σ d almost constant for 1B10. When we compare the
best σ d of either evaluation series, the improvement by the intensity correction amounts to 3%. This is
quite small a difference and it may seldom be worthwhile to record extra speckle images to make use of it.
Moreover, it will not help against the most likely problem in SPS, namely too low speckle intensity.
With increasing B, i.e. fading O, the performance of (6.6) quickly worsens. This is because speckle
intensity readings of zero are obviously not permissible in (6.4): the phase calculation will not function for
points of the speckle image that are digitised to zero. But as B is increased, as desirable from a practical
point of view, exactly this will occur more and more frequently. Then (6.4) breaks down on a fraction of
image pixels that grows larger as the speckle pattern gets darker.
In practice, one can circumvent this by simply replacing the zeros under the square roots by a non-zero
value (for simplicity, a factor of one); this introduces some arbitrariness in the calculation and is justified
only by the observation that this ad hoc remedy is better than none in this case, and that (6.5) and (6.6)
then become their standard versions (3.18) and (3.17) also for O –1 =O 0 =O 1 =0. Therefore the advantage
gained by the modified calculation must vanish as the O n approach each other. This is also shown in Fig.
6.1: the modified intensity-correcting formula overriding zero readouts for the O n (black curve, white
squares) links smoothly to the curve without error correction; from B50 on, both curves are very nearly
the same. Therefore the σ d from the intensity-correcting formula are not plotted anymore for B160, all
the more as using (6.4) would only lead to superfluous computational effort for higher B.
The data shown pertain to the depolarised speckle patterns which the test object generates directly; no
substantial improvement was found when the intensity correction was applied to speckle patterns
exclusively co-polarised with the reference light. This shows that the subtraction of the speckle
background, taking place in the D n , is more important than the exact M I ; also, the background subtraction
is justified for any polarisation state.
To check the preliminary results of Fig. 6.1, I carried out several tilt series with α x = 90°/sample and
B ∈ {3, 10, 30, 100, 300}. As seen before, this quasi-geometric series of B values is sufficient to find the
best performance of either method. Fig. 6.6 presents an overview of the best results for d s =3 d p .