Application and Optimisation of the Spatial Phase Shifting ...

3.3 Temporal phase shifting 75
Again, we find errors increasing symmetrically on both sides of ν 0x when α is nominally 90°; the key to
error suppression is the vanishing slope of bsc(ν 0x ). The same is true for α=120°; but as above in Fig.
3.13, we find a steep increase of errors for ν x >ν 0x , simply because ν 0x is not centred between ν 0x =0 and
ν 0x =2ν N and hence the bsc(ν x ) curve cannot be symmetrical.
Generally, the error compensation cancels the oscillating error only; the zero-order error (phase offset)
persists, as can also be seen from the phase spectra of (3.56) and (3.58): while the difference of
arg( S
~ ( ν )) and arg( C
~ ( ν )) remains constant, the reconstructed phase will depend on the phase-shift
x
x
deviation, as gets obvious from the progression of arg( S
~ ( ν )) and arg( C
~ ( ν )) with ν x . Hence, in ESPI
the correct absolute phase difference ∆ϕ is only obtained when the phase-shift error is the same in both
sets of samples. In TPS, this is generally not the case, but as long as the error is spatially uniform, the
determination of phase gradients will not suffer: a fringe offset in the sawtooth image is irrelevant. In
SPS, the offsets fluctuate locally with the speckle phase gradients; but since the speckle field is supposed
to remain correlated during the measurement, the errors cancel on subtraction of the speckle phase maps.
As mentioned above, these theoretical considerations do not account for the spatial coherence present or
not present within the sampling pixel window. For instance, a 3+3 formula need not automatically reduce
the measurement errors, because its error compensation might be superseded by low spatial correlation of
the sampling points. Therefore we will subject also the compensating formulae to an experimental check
in 3.4.5.
x
x
3.3 Temporal phase shifting
Many of the peculiarities of TPS have already been treated implicitly in 3.2.1.1, so that we now address
only two more subjects: first, we consider the loss of modulation associated with phase ramping instead of
stepping, and second, we take a look at the power spectrum of a speckle interferogram and consider a very
simple method to determine the average speckle size.
While the phase-shifted interferograms are recorded sequentially in time, the different α n are adjusted by
means of a phase shifter such as a mirror on a piezoelectric crystal in the reference arm. While it is
possible to set the α n statically, i.e. no change takes place during the exposure of each frame, it is more
convenient and has become popular to shift the phase linearly during the recording sequence, so that each
measurement becomes an integral over a phase interval. This changes (3.12) to
αn+
α 2
αn−
α
2
α
2 ;
1
' '
In = ⋅ ∫ Ib + M I ⋅ cos( ϕO + αn
+ α ) dα
α
2sin( )
= Ib + M I ⋅ ⋅ cos( ϕO + αn)
α
(3.59)
the additional factor is 0.9 when α=90°, and 0.83 for α=120°, so that the overall effect of the ramping
approach is a slight decrease in the modulation of the data; however, the measured ϕ O remains the same