Application and Optimisation of the Spatial Phase Shifting ...

72 Electronic or Digital Speckle Pattern Interferometry
a quasi-sinusoidal dependence on ϕ O ; however it has been shown [Lóp00] that this dependence
approaches a sawtooth profile when the detuning error is large.
0.14
δ ϕ O /rad
0.07
ϕ O /rad
0
0 1.57 3.14 4.71 6.28
Fig. 3.14: Deviation δϕ O of calculated phase from true phase ϕ O when α=95° instead of 90°. Arrows: alteration of
phase measurements due to δϕ O (see 3.4.6).
There is a simple intuitive way to understand these phenomena: when the sample spacing is incorrect,
errors periodical in ϕ O will arise in the sine and cosine terms of the phase-sampling formulae; their
relative phase lag introduces a double(2ν 0x )- and a zero-frequency (offset) error [Lar92c] in their quotient,
which then propagates into the calculated phase.
The fact that δϕ O δ(ϕ O +90°) allows for a very simple approach of error suppression. If the nominal
phase shift is set to α=90°/sample, we can use (3.19) and construct two consecutive phase measurements
with an offset of 90°,
I2 − I1
N0
sin ϕ'
O
ϕ' O mod 2π
= arctan : = =
0
I0 − I1
D0
cos ϕ'
O
I3 − I2
sin ( ϕ' O + 90°
)
(3.52)
ϕ'
O mod 2π
= arctan =
1
I − I cos ( ϕ' + 90° )
,
1 2
where we abbreviate ϕ O –45° by ϕ' O , cf. (3.19). In these two sampling sequences, we have δϕ' O0
δϕ' O1
,
which allows us to cancel the error by averaging the results. But for this to function, we must modify the
second formula to yield ϕ' O instead of ϕ' O +90°:
O
ϕ'
O1
'
mod 2π
=
arctan
I
I
− I
− I
2 1
3 2
sin ϕ'
O N1
= : = ,
cos ϕ'
D
(3.53)
O
1
where we have used
( ϕ 90 )
( ϕ )
sin ϕ = − cos + °
cos ϕ = sin + 90 ° .
(3.54)
Then, when constructing the phase average, it is better to average the N n and D n terms before executing
the arctangent operation, as opposed to averaging ϕ' O 0 and ϕ' O1
after separate arctangent operations. This
can be justified theoretically and has been done in [Hun97]; to understand the basic idea, it is very helpful