Application and Optimisation of the Spatial Phase Shifting ...

62 Electronic or Digital Speckle Pattern Interferometry
signal I(x) with suitable "filter functions" of the frequency ν 0x , where generally ν 0x ν x is assumed. It is
however essential to note that we will later be concerned with the effects of ν x ≠ν 0x . To measure the
phase, we define the filter functions as
S( x) = − sin 2πν
x
0x
C( x) = cos2πν x ,
0x
(3.31)
and the multiplications yield the signals
I ( x) S( x)
M I ( x)
= − Ib
( x)
sin 2πν0x
x + sin ϕ + π ν − ν0 − sin ϕ + π ν + ν0
2
I ( x) C( x)
( ( ( x) 2 ( ) x) ( ( x) 2 ( ) x)
)
x x x x
M I ( x)
= Ib
( x) cos2πν0x
x + ( cos( ϕ( x) + 2π ( νx − ν0x ) x) + cos( ϕ( x) + 2π ( νx + ν 0x
) x)
) .
2
(3.32)
Both of the equations contain contributions from the pure carrier frequency and from difference and sum
frequencies. Since ν x ν 0x , the difference frequencies are low; in the ideal case, ν x –ν 0x =0, and the lowfrequency
contribution is determined by ϕ(x) alone. One can think of the fringes resulting from the
multiplication as a moiré effect [Wom84, Ara97, Kat97]. The second step of synchronous detection is to
remove, or "filter out", the high-frequency terms by integrating the product functions, which gives the socalled
"filter outputs". This integration, or filtering, is achieved by the cross-correlation functions
∞
∫
S'( x') = I ( x) S( x − x')
dx
−∞
∞
∫
C'( x') = I ( x) C( x − x')
dx
−∞
(3.33)
if we calculate them for x'=0. The "filter outputs" therefore are
∞
∫
S'( 0) = I ( x) S( x) dx ∝ sin ϕ( x)
−∞
∞
∫
C'( 0) = I ( x) C( x) dx ∝ cos ϕ ( x)
.
−∞
(3.34)
Using the central ordinate theorem [Bra87, p. 136] together with the convolution theorem [Bra87, p. 110],
we can replace I(x), S(x) and C(x) by their Fourier transforms * [Fre90a; Mal98, p.134] and rewrite (3.34) as
* To apply the convolution theorem, we must use S(x'–x) and C(x'–x) in (3.33), which changes the correlation into a convolution.
The sign change in (3.33) then simply leads to a complex conjugation in (3.35). This is possible since S(x) and C(x) are real
functions, which means that their Fourier transforms are Hermitian. This is, their real parts are even and remain unaffected by
the sign change, while their imaginary parts are odd and must be inverted after the integrations in (3.33), although their
contributions vanish anyway.