Application and Optimisation of the Spatial Phase Shifting ...

3.4 Spatial phase shifting 81
3.4.1 Geometrical description of spatial phase shift
The lateral offset ∆x of the reference wave´s origin generates a quasi-linear geometric path and hence
phase difference between the object wave O and the reference wave R over the sensor. Fig. 3.24 sketches
the principle.
sensor
y
∆z
∆x
source point of R
source point of O
∆r
y
>λ
λ
0
-λ
∆z
r R
∆x
x
x
r O
Fig. 3.24: Left: incidence of two spherical waves with origins displaced by ∆x; right: construction of
corresponding pathlength differences ∆r = r O – r R .
While a phase shift in the sensor's y-direction may be added by a displacement ∆y, this case is still onedimensional
in the appropriate co-ordinate system. Hence it is sufficient to consider the phase difference
α(x), given by
π ⎛
2
2
2 x
x
α( x)
= − = ⎛ ∆ ⎞
⎜ x + y z x y z
λ ⎝ ⎠
⎟ + − ⎛
⎜
⎝
− ⎞
⎞
2 2
∆
2 2
r r
⎜
⎟ + +
O R
+ ∆
∆
⎝ 2 2 ⎠
⎟
(3.65)
⎠
where x = 0 is defined to be the y axis in the middle between the waves´ source points. This is not
generally the central sensor column: since the centre of the aperture should lie on the optical axis over the
centre of the sensor, the object wave´s origin cannot be shifted from there. However, y = 0 does lie on the
central row of the sensor. The corresponding phase gradient in x-direction,
⎛
⎜
∆ x
π
x
αx x y 2 ⎜ +
( , ) =
2
⎜
λ
2
⎜ ⎛ ∆ x⎞
⎜ ⎜ x + ⎟ + y
⎝ ⎝ 2 ⎠
2 2
+ ∆ z
−
∆ x
x −
2
2
⎛ ∆ x⎞
⎜ x − ⎟ + y + ∆ z
⎝ 2 ⎠
2 2
⎞
⎟
⎟
⎟ , (3.66)
⎟
⎟
⎠
is quasi-constant when ∆z is much larger than everything else, which is quite reasonable to assume when
using common imaging optics. Then ∆z will be on the cm scale, whilst the other quantities are on the mm
scale. It turns out that the y co-ordinate also has a weak influence on α x ; hence the carrier fringes are not
exactly straight. In fact, they have hyperbolic shape, which also follows from the definition of a hyperbola
as the set of points for which Fr O F–Fr R F is constant. Fig. 3.25 depicts the situation for an average nominal
phase gradient of α x (x,y) = 120° per sensor column and the optical configuration of Fig. 5.1. The spatial
dimensions refer to sensor of the camera that was used throughout the work to follow, ADIMEC MX12P
with 1024768 pixels of size (7.5 µm) 2 .