Application and Optimisation of the Spatial Phase Shifting ...

60 Electronic or Digital Speckle Pattern Interferometry
3.2.1.3 Complex-division method
Both of the methods discussed thus far have in common that they require one phase calculation and one
subtraction, and differ in the order of these operations. There are however also methods to calculate ϕ O in
only one computation step. They require two complete phase-shifted data sets and combine the steps of
phase calculation and difference formation in one formula. Examples of such calculations have been
given before [Ste85, Ste90, Fac93, Hun93a, Sal96]; however the somewhat laborious derivation of the
formulae can be generalised and greatly simplified when treated by the formalism of complex division
[Bur98b]. As mentioned above, the numerator in phase-shifting formulae should correspond to the sine
and the denominator to the cosine of the phase angle to be found, so that we can switch to complex
notation and write:
ϕ
ϕ
O,
i
O,
f
ϕO,
i
mod 2π
= arctan sin
= arg( cosϕO, i + i sin ϕO,
i ): = arg( zi
)
cosϕO,
i
ϕO,
f
π arctan sin
. (3.24)
mod 2 = = arg( cosϕO, f + i sin ϕO,
f ): = arg( z f )
cosϕ
Now ∆ϕ can be determined according to
where
O,
f
⎛ z f ⎞
∆ϕ mod 2π = ϕO, f − ϕO, i = arg( z f ) − arg( zi
) = arg ⎜ ⎟ mod 2π
,
⎝ z
(3.25)
⎠
i
z
z
f
i
sinϕ sinϕ + cosϕ cosϕ
=
sinϕ cosϕ − sinϕ cosϕ
+ i
cos ϕ + sin ϕ
O, i O, f O, i O,
f O, f O, i O, i O,
f
cos
2 ϕO, i + sin
2 ϕO,
i
2 O, i
2
O,
i
. (3.26)
Eventually we combine these expressions to get
⎛ z f ⎞ sinϕO, f cosϕO, i − sinϕO, i cosϕO,
f
∆ϕ mod 2π
= arg⎜
⎟ = arctan mod 2π
,
⎝ z ⎠ cosϕ cosϕ + sinϕ sinϕ
(3.27)
i
O, i O, f O, i O,
f
which provides a generally valid instruction on how to compose the expressions of phase-shifting
formulae; of course, the same result follows from the trigonometric relationship for the difference of
arctangents [Cre94]. Now we can instantly establish one-step calculations; for instance, from (3.16),
and (3.17) changes to
( I3 f − I1f )( I0i − I2i ) − ( I3i − I1i )( I0 f − I2
f )
∆ϕ mod 2π
= arctan ,
( I − I )( I − I ) + ( I − I )( I − I )
(3.28)
0i 2i 0 f 2 f 3i 1i 3 f 1f
∆ϕ mod 2 π = arctan
( I2 f − I1f )( 2I0i − I1i − I2i ) − ( I2i − I1i )( 2I0 f − I1f − I2
f )
3
.
( 2I − I − I )( 2I − I − I ) + 3( I − I )( I − I ) (3.29)
0i 1i 2i 0 f 1f 2 f 2i 1i 2 f 1f