Application and Optimisation of the Spatial Phase Shifting ...

116 Comparison of noise in phase maps from TPS and SPS
for N x and N y . At lower speckle sizes, a higher bias noise is present (the curves start from higher values of
σ d F N x,Ny=0 ), but in turn, σ d increases more slowly with the fringe density. Apparently, a reduction of d s
effects an increase of σσ d
particularly for tilts about the x axis. Hence, there are most likely random inplane
object shifts of some µm, and subsequent speckle pattern shifts on the sensor, when the object is
tilted so as to produce horizontal fringes.
Therefore we will consider vertical fringes in most of the out-of-plane investigations; although the
performance was also checked with horizontal fringes and found to essentially agree with Fig. 5.3, we
would learn little from displaying those curves as well.
Since the tilts were adjusted by hand, there was also some fluctuation in the fringe densities given on the
abscissae of the plots. The error amounts to ¼ fringe for each "basic" displacement step of 5, 10, 20, 30
and 40 fringes; and for compositions of several of these (e.g. 100 fringes 10+20+30+40 fringes), the
deviation sometimes accumulated to 1 fringe, which still seems negligible for plotting. Also, there was
slight interaction between the axes, i.e. the fringes were rarely exactly vertical or horizontal; this deviation
remained within ¼ fringe per step as well and was not systematic. Although each curve for σ d
consists of only 12 data points, i.e. 12 different fringe densities, the values are linked to "curves" for the
sake of a better overview. This applies likewise to the σ d plots to follow, and will prove useful there.
Finally, in the TPS experiments, also the stability of the interferometer plays a role for the accuracy of
measurements. As mentioned before, I applied rather stringent a criterion to accept a phase-shifted frame
sequence. Since the laboratory was in the 1 st floor, with a railroad and a motorway nearby, it saved much
time to do these experiments with the least possible building vibration – whose maximal power was at
4.3 Hz –, i.e. between midnight and 4 a.m.
5.3 Zero-displacement-gradient measurements
Of the results of phase measurements that will be presented here, those with zero displacement gradient
are the most general ones, since they do not depend on the specific assembly’s parameters but should be
comparable for any set-up with only the speckle size as the relevant quantity. The way to obtain such
measurements is to leave the object untouched and to compare two nominally identical object states,
differing only by a controlled or random global phase offset ∆ϕ. Unfortunately, in SPS the measured σ d
depends strongly on ∆ϕ , which is due to the ample intensity and phase gradients in the object speckle
field; this has been discussed in detail in Chapter 3.4.4.
Therefore, the evaluation of zero-displacement measurement errors in SPS is quite an elaborate
procedure: one has to collect a set of phase maps with various ∆ϕ that suffices to reconstruct the
underlying continuous curve of σ d vs. ∆ϕ and then determine the mean of the errors. Since the
interferometer was fortunately too stable to produce phase drifts and fluctuations uniform in [0,2π), the
piezo-driven mirror assisted in generating the phase offsets. Of course, it has to move very slowly to
generate quasi-stable interferograms; I used an amplitude-modulated triangle waveform that was