Application and Optimisation of the Spatial Phase Shifting ...

3.2 Phase-shifting ESPI 73
to think of adding weighted and unweighted phasors, respectively, as detailed in [Stroe96]. Therefore, the
N and D terms are averaged according to [Schwi83, Har87, Schwi93]
which results in
ϕ '
O
ϕ '
O
N
mod 2π
= arctan
D
+ N
+ D
0 1
, (3.55)
0 1
( I − I )
2sin
ϕ'
O
mod 2π
= arctan = arctan
2cos
ϕ'
I − I − I + I
O
2 2 1
; (3.56)
0 1 2 3
and this is the formula given in [Schwi93], subsequently referred to as 3+3 averaging formula. Its transfer
properties are shown in Fig. 3.15.
4
3.14
3
2
1.57
arg( S
~ ( νx))
arg( C
~ ( ν x ))
1
0
0 1 2 3 4
-1
ν x /ν 0x
0
0 1 2 3 4
ν x /ν 0x
-3.14
-2
-3
-4
amp( S
~ ( νx))
amp( C
~ ( ν x ))
Fig. 3.15: Filter spectrum for 3+3-step-90° phase-sampling formula (3.56); left: amplitudes, right: phases.
As in (3.19), the offset of the reconstructed phase is –45°; but the phases of S
~ ( ν x ) and C
~ ( ν x )
~ ~
are in quadrature for all ν x , and also the gradients of S ( ν x ) and C(
ν x ) are matched:
~ ~
dS ( ν ) / d ν | = dC ( ν ) / d ν | . This assures stable performance for a larger range of deviations,
x x ν
x x ν
0x
0x
because S
~ ( ν ) and C
~ ( ν ) are nearly equal for a broader range of ν x . In [Ser97b], an iterative search for
x
x
smallest miscalibration sensitivity showed that (3.56) is an almost optimal solution. The offset-free
version of the 3+3 formula, also given in [Schwi93], is
ϕ
O
mod 2π
-1.57
− I0 + 3I1 − I2 − I3
= arctan ; (3.57)
I + I − 3I + I
0 1 2 3
this formula shows equal amplitudes for S ~ ( ν ) and C ~ ( ν ) , similar to (3.19), but much better quadrature
stability than (3.19), as to be seen in Fig. 3.16.
5
3.14
4
1.57
3
2
0
0 1 2
ν x /ν 0x 3 4
amp( S
~ ( ν ))
1
x
amp( C
~ ( ν x ))
0
0 1 2 3 4
-1.57
ν x /ν 0x -3.14
arg( S
~ ( νx))
arg( C
~ ( ν x ))
Fig. 3.16: Filter spectrum for 3+3-step-90° phase-sampling formula (3.57); left: amplitudes, right: phases.