Application and Optimisation of the Spatial Phase Shifting ...

78 Electronic or Digital Speckle Pattern Interferometry
so that e.g. a speckle halo just fitting in the DFT's frequency plane, with ν max,s =ν N =1/(2 d p ), is seen to
come from a speckle pattern with d s =2.44 d p . In Fig. 3.20, we have d s =3 d p .
If we assume that a reference wave R of amplitude R is added as a point source in the centre of AS, which
~ ~ ~
is drawn in Fig. 3.21, the field on the CCD chip will be FT( S ⋅ TAS + Rδ ( 0, 0 )) = S * TAS
+ R . The intensity
on the sensor is ~ * ~ ~ S T R AS +
* *
2
, and its Fourier spectrum is FT( S ~ * T ~ ~
AS + R )
*
2
= ACF ( S ⋅ TAS)
+ R 2 δ ( 0, 0) +
( S ⋅TAS) * Rδ ( 0, 0 ) + ( S T ) * R δ( 0, 0 ) . The first term is again the speckle halo, the second term is a
⋅ AS
central peak due to the uniform reference wave; these are often called the self-interference terms. On
inspecting the mixed or cross-interference terms, we find that they reproduce the speckle field's
amplitudes, with an envelope that is the aperture function again. The convolution with the δ function of
the reference wave reproduces this distribution and multiplies it with R. The power spectrum of a speckle
interferogram therefore looks as in Fig. 3.21 on the right; again, the scale is logarithmic and d s =3 d p .
R
−ν N
0
DFT
glass fibre in tube
CCD
ν y
L
AS
z
ν N
-ν N 0 ν x ν N
Fig. 3.21: Left: Imaging geometry for ESPI: R, reference wave; other abbreviations as above in Fig. 3.20. Right:
power spectrum of speckle interferogram in log display; frequency plane as above.
The interference terms overlap in the centre and are point-symmetrical with respect to each other; hence
the shadow of the fibre guide, being an undesired but here instructive part of T AS , is visible in each of
them. The speckle halo is still the same as in Fig. 3.21, but the extent of the spectra or "bands" of the
* *
interference terms, ( S ⋅ TAS) * Rδ ( 0, 0 ) + ( S T ) * R δ ( 0, 0 ) , in the frequency plane is exactly half that
⋅ AS
of the speckle halo. This is the "doubling of speckle size" mentioned above. It occurs only when R is large
enough to suppress the speckle halo; the influence of R will play an important role later on.
As Fig. 3.21 indicates, the maximal spatial frequencies of the interference bands are given by interference
of R with the outermost rays passing the aperture. With
*
ν
max,i =
D 2λf
, (3.63)
where the subscript i stands for interference, we arrive at
d s
= 0.
61
ν
, (3.64)
max,i