Application and Optimisation of the Spatial Phase Shifting ...

150 Improvements on SPS
The improvement in σ d by averaging phase measurements can be clearly seen; but a comparison with the
3+3 averaging formulae of Fig. 6.9 reveals that their performance is not being reached. Therefore our next
step will be to apply these as well in the averaging process, which can be done by extending the sampling
pixel cluster as already shown on the right side of Fig. 6.11.
To use 3+3 averaging formulae horizontally and vertically, we have the four different possibilities to use
pixels {2, 3, 4, 5}, {1, 3, 6, 8}, being the familiar horizontal and vertical calculations, and{1, 3, 4, 7},
{2, 3, 6, 7}. The latter combinations will still work in the presence of a constant speckle phase gradient
ϕ O,x , or ϕ O,y ; but since the pixels involved are not on a straight line, they would additionally impose ϕ O,x =
ϕ O,y , which cannot reasonably be inferred from Fig. 2.14. On the other hand, pixel 7 is spatially closer to
pixel 3, which is again our target point for all the calculations, and hence has greater spatial coherence
with respect to pixel 3 than pixels 5 or 8. Therefore we use
tan ϕ
O
to establish
( I − I ) ( I − I ) ( I − I ) ( I − I )
2 4 3 2 6 3 2 4 3 2 6 3
mod 2π
=
=
=
=
I − I − I + I I − I − I + I I − I − I + I I − I − I + I
ϕ
O
2 3 4 5
1 3 6 8
1 3 4 7
2 3 6 7
(6.15)
8I3 − 4I4 − 4I6
mod 2π
= arctan .
2I + 2I − 4I − 2I + I − 2I + 2I + I
(6.16)
1 2 3 4 5 6 7 8
There is also the possibility to inscribe four more pixel sequences in the shape of an L (plus appropriate
reflections and rotations) into the pixel cluster of Fig. 6.11; but again, this would merely double the terms
in (6.16) and we do not take them into account.
Since the definitions for the corresponding intensity-correction formulae are rather lengthy, I do not go
into detail here; the principle is already indicated in (6.12) and (6.13) where only the pixel indices have to
be inserted appropriately. It may suffice to note that again only the pixel sets {2, 3, 4, 5}, {1, 3, 6, 8},
{1, 3, 4, 7}, and {2, 3, 6, 7} need be used. Also in this case, it will be interesting to examine the twodimensional
frequency characteristics of these approaches experimentally with the help of bsc(ν x ,ν y ).
These are shown in Fig. 6.14, with the same input interferogram (and speckle pattern, d s =2.5 d p ) as for
Fig. 6.12 above.
In the plot for (6.16), the same circle structure of bsc(ν x ,ν y )=45° shows up as above; but thanks to the
correction of improper ν x and/or ν y , another line of correct phase determination appears at higher spatial
frequencies. This enlarges the region where FδϕF10° to cover almost the complete sidebands, which is
of course incidental for this particular speckle size. While the shape of bsc(ν x ,ν y )=45° looks
qualitatively different for the intensity–correcting formula, the stabilisation effect on the phase extraction
is almost the same.