Application and Optimisation of the Spatial Phase Shifting ...

148 Improvements on SPS
To use the intensity correction, we re-define our auxiliary quantities, the K n :
( tan ϕ )
O
( tan ϕ )
O
1
2
=
=
O3 ( D4 − D2
)
O ( D − D ) − O ( D − D ) : =
4 2 3 2 3 4
O3 ( D6 − D1
)
O ( D D ) O ( D D ) : =
− − − K
6 1 3 1 3 6
K2
K − K
3 1
K5
− K
6 4
(6.12)
and obtain
ϕ
O
mod π = arctan
K2 + K5
− K + K − K + K
1 3 4 6
,
(6.13)
which method of averaging is correct for this purpose, since both expressions should yield the same
phase. (In this case, the inclusion of two more quotients from pixels {1, 3, 4} and {2, 3, 6} is not
equivalent to a doubling of the terms; but on doing so, the reduction of σ d is minimal.) The spectral
transfer properties of (6.11) and (6.13) are now genuinely two–dimensional, so that we can rewrite (3.73) as
~ ~ ~ ~
( ( , ) ( , ) ) arg ( ( , ) ( , ) )
bsc( νx , νy ) = arg I ~ ( νx , νy ) ⋅ C ~ ( νx , νy ) + I νx νy ⋅ S νx νy − I νx νy ⋅C
νx νy
⎛
~
S ( νx
, νy
) ⎞
= arg
⎜1+
~ ⎝ C ( νx
, νy
) ⎟
⎠
(6.14)
and examine the course of bsc(ν x ,ν y ) experimentally by the now familiar 2-D representation. This is done
in Fig. 6.12. Both maps of bsc(ν x ,ν y ) are calculated from the same input interferogram, only (6.13)
processes also the previously stored speckle pattern O(x,y). Since α x =α y =90°/sample, one can use
d s =2.5 d p (cf. 3.4.4), whereby each signal band fills approximately one quadrant of the spatial frequency
plane. The left-hand image in Fig. 6.12 visualises this arrangement.
2
2
2
ν y /n 0
ν y /ν 0
2 3 4|0 1 2
3
3
0|4
1
0|4
1
2
2 3 4|0 1 2
ν x /ν 0
2
ν y /ν 0
2 3 4|0 1 2
3
0|4
1
ν x /ν 0
2
n x
/n 0
Fig. 6.12: Left: power spectrum of input interferogram (displayed in contrast-enhanced log scale); centre:
bsc(ν x ,ν y ) for (6.11); right: bsc(ν x ,ν y ) for (6.13). Black lines: frequency co-ordinates leading to correct
phase calculation, bsc(ν x ,ν y )= 45°; white outlines: areas of –10°¡δϕ¡10°.
In these images, the behaviour of bsc(ν x ,ν y ) on the line given by ν x =ν y corresponds to the onedimensional
cases we have considered before. The black lines indicate correct phase calculation