Application and Optimisation of the Spatial Phase Shifting ...

66 Electronic or Digital Speckle Pattern Interferometry
Z( xk
) = I ( xk
) ⊗ ( C( n) + iS( n))
0 0
,
with ϕ( x) mod 2π
= arg( Z( x ))
(3.45)
where arg(•) is the polar angle of a complex number; this corresponds to a notation c n =b n +ia n in (3.13)
and is the starting point for the description of phase-shifting formulae by complex polynomials [Sur96].
To illustrate the significance of the facts compiled thus far, we rewrite (3.16) as
k0
⎛ 3P0 x ⎞ P
x
I⎜
⎟ − I
⎛ ⎝ ⎠ ⎝ ⎜
0 ⎞
⎟
4 4 ⎠
ϕO
mod 2 π = arctan
P
x
I( ) − I
⎛ ,
⎝ ⎜
0 ⎞
0 ⎟
2 ⎠
(3.46)
where P 0x =4 d p is the period of the carrier fringes, and α=2π/P 0x = 90°/d p . (This denotes the phase shift
per pixel, not the phase gradient in °/m.) The filter functions are
⎛ 3P0 x ⎞ ⎛ P0
S( x)
= δ ⎜ x − ⎟ −δ
⎜ x −
⎝ 4 ⎠ ⎝ 4
⎛ P0
C( x)
= δ( x)
− δ⎜
x −
⎝ 2
x
x
⎞
⎟
⎠
⎞
⎟
⎠
(3.47)
and the corresponding spectra read
~ ⎛ 3 ⎞ ⎛ 1 ⎞ ⎛ π ν
S ( x ) exp i x exp i x sin
x
⎞
ν = ⎜ π ν ⎟ − ⎜ π ν ⎟ = ⎜ exp iπ
⎝ ν ⎠ ⎝ ν ⎠ ⎝ ν ⎠
⎟ ⎛ ⎛ 1
2
2
2
⎜ ⎜
4
4
2 ⎝ ⎝ 2
+
0x
0x
0x
0x
~ ⎛ 1 ⎞ ⎛ π ν
C ( x ) exp i x sin
x
⎞
ν
exp
x
ν = − ⎜ π ν ⎟ = ⎜
iπ
⎝ ν ⎠ ⎝ ν ⎠
⎟ ⎛ ⎛
⎜
⎝
− 1
1 2
2
⎜
2
2 ⎝ 2 + 2ν
0x
0x
0x
ν
ν
x
⎞⎞
⎟⎟
⎠⎠
⎞⎞
⎟⎟
⎠⎠
(3.48)
with ν 0x =1/P 0x . In these expressions, the sine terms represent the amplitudes and the exponentials
represent the phases of the filter spectra, so that the behaviour of S
~ ( ν ) and C
~ ( ν ) can be read off
directly. Whenever we get a pure phase term, it is possible to plot the rest of the expressions as real
amplitudes, which we will denote by amp(). For more complicated formulae, it is not always possible to
arrive at separable expressions; but once S
~ ( ν ) and C
~ ( ν ) are established, one can obtain at least their
x
moduli and arguments separately.
This now gives us a means to explore the transfer characteristics of phase-shifting formulae by plotting
their spectra. Extending the common practice of plotting only the amplitude spectra, we will consider the
phase spectra as well. In all our spectra plots that follow, the frequencies will be normalised by ν 0x and the
range of frequencies will be from 0 to 2ν N , where ν N is the Nyqvist frequency 1/(2 d p ), corresponding to
α=180°/d p . Consequently, when ν 0x = 90°/d p , 2ν N = 4ν 0x ; and for ν 0x = 120°/d p , 2ν N = 3ν 0x . The
ordinates of the amplitude plots are dimensionless and scale with the a n and b n in the underlying sampling
functions; the phases are shown in radians.
x
x
x