Application and Optimisation of the Spatial Phase Shifting ...

3.2 Phase-shifting ESPI 67
The spectral transfer properties of (3.48) are shown in Fig. 3.8: while the amplitudes of S
~ ( ν x ) and
~ C(
ν ) are seen to be the same throughout the frequency spectrum, the phases are in quadrature only at
x
ν x /ν 0x =1 and ν x /ν 0x =3, which corresponds to α=90° and 270°/d p (aliased as –90°/d p ), respectively. Also,
S'(x) will represent sin(ϕ O ) in the former and sin(–ϕ O ) in the latter case: if we reverse the phase shift, the
calculated phase must change its sign too. It can also be seen from the phase spectrum that (3.16)
measures ϕ O without offset: at ν 0x , arg( C
~ ( ν )) = 0 ° and arg( S
~ ( ν )) = − 90 ° , as (3.41) requires.
x
x
2
1
amp( S
~ ( νx
))
amp( C
~ ( ν x ))
3.14
1.57
-1
-2
ν x /ν 0x
-1.57
-3.14
arg( S
~ ( νx))
arg( C
~ ( ν x ))
0
0 1 2 3 4
ν x /ν 0x
0
0 1 2 3 4
Fig. 3.8: Filter spectrum for 4-step-90° phase-sampling formula (3.16); left: amplitudes, right: phases.
The zero transitions of amp( S
~ ( ν )) and amp( C
~ ( ν )) at ν x =nν N , n∈{0,1,2}, cause the phases to jump
x
x
by π; this corresponds to the "singular" cases of α=0°, 180°, 360°/d p , in which situations the differences of
phase-shifted intensity samples record only I b , with no intensity modulation, and a phase measurement is
impossible. The filter outputs then must vanish because of the requirement that I b be suppressed.
The spectral responses of simple sampling functions can sometimes be qualitatively understood without
Fourier analysis. For instance, a difference of two samples will be maximal in the average over all ϕ O
when they are 180° out of phase. This behaviour is reflected in Fig. 3.8: since in (3.16) the nominal phase
difference of the intensity samples in S(n) and C(n) is 180°, their responses peak at the nominal frequency
ν 0x . After this example, we now investigate the transfer properties of some phase-extraction methods that
recommend themselves for SPS because of their small number of samples.
3.2.2.3 Three-sample formulae
When we consider (3.18), we obtain
S
~ ( ν
~
C(
ν
x
x
⎛ π ν
) = 4sin
⎜
⎝ 4 ν 0
) = 4sin
2
x
x
⎛ π ν
⎜
⎝ 4 ν 0
⎞ ⎛π
ν
⎟cos
⎜
⎠ ⎝ 4 ν 0
x
x
⎞
⎟
⎠
⋅
x
x
⎞ ⎛ ⎛
⎜
1
⎟exp
iπ
⎜−
⎠ ⎝ ⎝ 2
⎛ ⎛
exp⎜i
π
⎜
⎝ ⎝
1
2
+
ν
ν
1
2
x
0x
ν
ν
x
0x
⎞⎞
⎟
⎟
⎠⎠
⎞⎞
⎟
⎟
⎠⎠
;
(3.49)
this time the phase factor associated with ν 0x is the same in both expressions, which means that the phases
always remain in quadrature; but in turn, the amplitudes depend on ν 0x as shown in Fig. 3.9. The samples
for C(n) are now nominally 90° apart, but by the argument used above, the maximum average response