Application and Optimisation of the Spatial Phase Shifting ...

68 Electronic or Digital Speckle Pattern Interferometry
for that arrangement occurs when they are 180° apart, i.e. at ν x =2ν 0x . Also, (3.18) measures ϕ O +π/2
instead of ϕ O (cf. Fig. 3.8). Therefore, in [Fre90a] we find S(n) and C(n) swapped, and the new S(n)
inverted, which cancels the offset. An example of how this works is given below in (3.53) and (3.54).
4
3
2
3.14
1.57
arg( S
~ ( νx))
arg( C
~ ( ν x ))
1
0
0 1
-1
S
-2
C
-3
S
-4
amp( ~ ( νx
))
amp( ~ ( ν x ))
amp( ~ ( νx))
⋅ 3
2
0
3 4 0 1 2 3 4
ν x /ν 0x
-1.57
-3.14
ν x /ν 0x
Fig. 3.9: Filter spectrum for 3-step-90° phase-sampling formula (3.18); left: amplitudes, right: phases.
As to be seen from Fig. 3.9, reliable operation of (3.18), i.e. validity of (3.41), is assured only within small
deviations of ν x from ν 0x : while dS
~ ~ ( ν ) / d ν | ν0
= 0, a maximum of dC ( ν ) / d ν occurs at ν0x . A low
x x x
influence of phase-shift errors would require both gradients to be equal or at least close to each other; then
the phase reconstruction would tolerate some miscalibration. The graphs shown in Fig. 3.9 are also
qualitatively valid for phase calculation with (3.17), and more generally with (3.15), since S(n) and C(n)
are just scaled to shift up or down that ν x which fulfils amp( S
~ ( ν )) = amp( C
~ ( ν )) . This is indicated by
the curve labelled " amp( S
~ ( ν )) ⋅ 3 ", which would suffice to change (3.18) to (3.17). The phase spectra
x
are indeed the same in either case.
With 3 phase steps of 90°, it is more common to use the representation (3.19), which formula has the
transfer properties depicted in Fig. 3.10; in this case, the amplitudes are equal for all ν x , while again
~ S ( ν ) and C
~ ( ν ) are in quadrature only at α=90° and –90°/sample; also, the inherent phase offset of -π/4
x
x
is clearly revealed by the graphs.
x
x
x
x
2
3.14
1.57
1
amp( S
~ ( νx))
amp( C
~ ( ν x ))
0
0 1 2 3 4
-1.57
ν x /ν 0x -3.14
0
0 1 2 3
ν x /ν 0x 4
arg( S
~ ( νx))
arg( C
~ ( ν x ))
Fig. 3.10: Filter spectrum for 3-step-90° phase-sampling formula (3.19); left: amplitudes, right: phases.
It is possible to balance amp( S
~ ( ν )) and amp( C
~ ( ν )) for α=120° as well, yet at the sacrifice of integer
coefficients. From (3.14), one can easily derive
x
x