Application and Optimisation of the Spatial Phase Shifting ...

2.3 Second-order speckle statistics 43
of speckle size to pixel size. Hence, Fig. 2.27 gives the expected standard deviation of the error in many
ESPI phase measurements, where ϕ 1 (x,y) is the initial and ϕ 2 (x,y) the final speckle phase distribution.
2.3.3.3 Interaction of intensities and phases
As already pointed out in [Don79], (2.49) is not separable into a product of marginal pdf's, which means
that all of the involved quantities are mutually dependent. Hence we have dropped some information by
eliminating intensities or phases from (2.49). Also, in 2.2 we have found that high speckle intensities are
associated with low phase gradients, and vice versa. Therefore, we will now consider the interaction of
intensities and phases more generally. This has been done in [Don79] as well; I list the results for
completeness here and also give a simple qualitative interpretation for π that I think has not been
mentioned before.
Since we are again interested in phase differences between two points instead of absolute phases, our
relative phase variable will be useful again. Then, we can investigate two general cases: (i), what
influence do I 1 and have on I 2 , and (ii) what does do when we constrain I 1 and I 2 ? The pdf of I 2
conditioned on the other quantities is [Don79]
( | I , ϕ , ϕ )
p I
2 1 1 2
where we have abbreviated
( , I , ϕ , ϕ )
p I1 2 1 2
= =
p I
1
exp
( ) 2
⎜ −
1, ϕ1,
ϕ 2
2 I ( 1−
µ A ) ⎜
⎝
I ( 1−
µ A )
⎛
⎜
I
− 2 µ cosϑ
I I
2 A 1 2
⎞
⎟
⎟ D
⎟
⎠
( δ)
, (2.56)
δ = µ cosϑ
A
2
( 1−
µ A )
( )
2 −1
D( δ ) = 1+ πδ exp( δ )( 1+
erf δ ) .
I
I
1
(2.57)
In contrast to the calculations in [Don79], we use Fµ Α F everywhere; since appears as an argument of a
cosine only, we can constrain 0 π and still explore -1Fµ Α Fcos 1; thus, we need not deal with the
ambiguity of µ A cos as was done in [Don79]. Unfortunately, it is still confusing to go through the many
possible ways of plotting (2.56) with various fixed and running variables, so that we will resort to simpler
functions. Indeed, it turns out that the statistical quantities
2 2
( A )( D )
I
I2 | I1, ϑ =
2 1 − µ 3 − ( δ ) + 2 δ
I
σI
| I1, ϑ = ( A ) ( ) D D
2 1 − µ 6 − 1 − 2 δ ( δ ) − ( δ ) + 8 δ
2
2 2 2 2
(2.58)
will provide sufficient insight. There are still three parameters to vary, namely |µ A |, , and I 1 ; unlike
[Don79], we do not use the composite parameter I 1 cos 2 /I, but only I 1 , which will allow us a direct
interpretation and to see the effect of more clearly. We normalise I to unity and investigate