Application and Optimisation of the Spatial Phase Shifting ...

134 Improvements on SPS
intensity on adjacent pixels (x,y) and (x1,y), σ O0 ,O 1
, and of that of the imaging system's electronic
noise, σ e . * Hence we rewrite (6.1) as
σ
σ
2 O O + σ
2
0,
± 1 e
ϕ O
≅
2 ⋅
O
⋅ R
⋅
8
3
. (6.2)
In a speckle field, σ O0,O 1
depends on the degree of spatial coherence [Goo75], µ A (x 0 , x 1 ), of the points
(k,l) and (k1,l). For a circular aperture and d s = 3d p , we find µ A (x 0 , x 1 ) 0.81. Moreover, σ O0,O 1
is
conditioned on O 0 , which relationship is analytically known [Don79]. We can generalise (2.52) to read
( O )
2 2
0 ± 1
2 2 2 2
σ O , O = ( 1− µ A ) + 2 O µ A ( 1− µ A ) , (6.3)
and inserting (6.3) into (6.2), we can calculate σϕ O
, which is the same for both object states:
σϕ O =σϕ O,i =σϕ O,f
. For the phase difference ∆ϕ =ϕ O,f–ϕ O,i we therefore get σ ∆ ϕ = L2σϕ O
and from this the
corresponding quantity for the displacement, σ d , as a function of the beam ratio B = R/〈O〉. These data can
be compared with the experimental results.
Fig. 6.1 shows the performance of various evaluation methods for sawtooth images with N x =10, N y =0,
d s =3 d p and α x =120°/column from an out-of-plane configuration with SPS; for TPS, d s was set to 1 d p .
Curves in Fig. 6.1 that are not addressed here will be discussed later on.
The theoretical curve of σ d vs. B for SPS is the bold white line and matches the measured data reasonably
if we shift it vertically by adding a constant displacement deviation of σ d 0
= 0.05 λ. This is not an arbitrary
adjustment of data: since (6.2) does not account for spatial fluctuations of the phase ϕ O between adjacent
pixels, the predicted values of σ d will be too small. Of course, adding a constant σ d 0
relies on the simple
assumption that the influence of speckle phase gradients on σ d does not depend on B.
From the figure we see that TPS works well from B1 on, and σ d only starts to increase from B100 on,
where O is already weaker than the electronic noise. The quasi-constancy of σ d vs. B in TPS has also
been reported in [Hun97] for a beam ratio between 0.1B10. For SPS, σ d first decreases as the
reference wave gets stronger, and has its minimum around 30. With fading M I , the influence of electronic
noise grows and so does σ d . This behaviour agrees reasonably with our theoretical prediction.
Hence, in SPS a proper choice of the beam ratio is far more important than it is in TPS. Fortunately the
best SPS results turn up in a region of high beam ratio, which alleviates the problem of poor light
efficiency somewhat. Based on these results, for most of the investigations in Chapter 5 B was set to 10, at
which setting both SPS and TPS operate with near-optimum performance.
* With the imaging system used, a realistic value for σ n was ¡ 2.5 grey levels; this corresponds to a resolution of only 6-7 true
bits. With optimum intensity resolution, the usable beam ratios would have been even higher.