Application and Optimisation of the Spatial Phase Shifting ...

140 Improvements on SPS
6.2 Modified phase reconstruction formulae
Besides the optimisation of optical parameters, it is of course desirable to utilise some of today's
knowledge about phase-sampling methods to tailor phase calculation methods specially for SPS. As
mentioned before, the most stringent restriction for error suppression is the small number of sampling
points available in the speckles that we even wish to make as small as possible. In the following
paragraphs, we explore possibilities to construct few-sample formulae with reasonable rejection of errors
due to speckle intensity and phase gradients, and eventually we test the combination of these approaches
in out-of-plane ESPI deformation measurements.
6.2.1 Consideration of speckle intensity gradients
One possible way to reduce the phase errors induced by the fluctuations of the object wave's intensity has
already been shown in 6.1.1. As we have seen in Chapter 2.2.3.1, it would be very difficult to account for
the I x statistics of a speckle field in SPS: the assumptions that we could model by a modified three-sample
formula would be too crude in the case of speckle intensity.
There is however an exact method of compensating the errors due to intensity fluctuations; it relies on an
additional measurement of the speckle intensity alone before or during the displacement observation. In
the linear equation system constituted by (3.68), we usually assume O, and hence I b and M I , to be constant
in all the equations. If however each equation gets its own O n from a speckle intensity image stored
beforehand, it is still possible to solve for ϕ O , as long as we use three phase steps of (-α, 0, α). Details of
this procedure are outlined in Appendix C; with D n I n –O n , we arrive at [Bot97]
ϕ
O
which is for α = 90°:
mod 2π
= arctan
α( )
( O1 D−1 − D0 − O−1 D0 − D1
)
O ( D − D ) + cos O ( D − D ) + O ( D − D )
0 1 −1 −1 0 1 1 −1 0
sin α ( ) ( )
, (6.4)
ϕ
O
mod 2π
= arctan
O ( D − D )
0 1 −1
O ( D − D ) − O ( D − D )
1 −1 0 −1 0 1
(6.5)
and for α = 120°:
ϕ
O
2
mod 2π
= arctan
O0 ( D1 − D−1) − O−1 ( D0 − D1 ) − O1 ( D−1 − D0
)
. (6.6)
3
( O1 ( D−1 − D0 ) − O−1 ( D0 − D1
))
Of course, these formulae collapse to their standard versions (3.18) and (3.17) when O –1 =O 0 =O 1 .
A disadvantage of this method is the necessity to record speckle images before and, if decorrelation
occurs, also during the measurement. This will rule out highly dynamic phenomena and reduce the