Application and Optimisation of the Spatial Phase Shifting ...

154 Improvements on SPS
In classical interferometry, it is necessary to determine and remove the carrier frequency for wavefront
reconstruction [Nug85, dNic98, Li 98, Fer98]; in ESPI, this is fortunately done automatically by the image
subtraction of the initial from the final speckle phase map.
Let Fo(x,y)Fexp(iϕ O (x,y))=o(x,y) be the complex amplitude of the speckle field; then the speckle intensity
is O(x,y) = o(x,y)o*(x,y)= Fo(x,y)F 2 , which we assume to be unity. Adding a reference wave r(x,y), the
amplitude of the interferogram is i(x,y)=o(x,y)+r(x,y). By r(x,y)= LB r exp(i(2πν 0x x+2πν 0y y)),
r being the complex amplitude's unit, we adjust the beam intensity ratio to B, which is a real, positive
and spatially constant factor, and the spatial carrier frequencies to ν 0x andν 0y . The intensities in the
interferogram are then
( o r )( o r )
I( x, y) = ( x, y) + ( x, y) ( x, y) + ( x, y)
= O( x, y) + Br
*
2
*
+ o ( x, y) Br exp( i( 2πν x + 2πν y)) + o( x, y) Br exp( − i( 2πν x + 2πν y))
,
0x 0 y 0x 0y
(6.17)
which terms represent the speckle intensity, the reference intensity, and the complex representation of the
cosinusoidal interference term, respectively. The spectrum of this intensity distribution will be
~
( ( , )) = I ( ν x , ν y )
FT I x y
( x , ) B
~ 2
ν ν y r δ( ν x , ν y )
*( ν x ν y ) B δ( ν x ν x ν y ν y ) ( ν x ν y ) B δ( ν x ν x ν y ν y )
~
= O +
+ ~ o , * ~ r − , − + ~ o , * ~ r + , +
0 0 0 0
,
(6.18)
where * denotes convolution. This spectrum is a superposition of the speckle halo O ~ , the central peak
mostly due to the uniform reference illumination, proportional to B, and two sidebands in which o(x,y),
and therefore ϕ O , is encoded. Remembering the so-called sifting property of the δ-function [Bra87, p. 74],
we can account for the convolution by rewriting (6.18) as
~
I
( νx
, ν y )
~
( , ) ~ 2
( , ) ~ ~ *
O ν B B ( , ) B
~ ~
x ν y r δ νx ν y r o νx ν x ν y ν y r o( νx ν x , ν y ν y )
= + + − − + + +
0 0 0 0 .
(6.19)
As already explained in Chapter 3.3.1, the shape of the sidebands in the frequency plane is that of the
aperture, only now they are shifted by (ν x ,ν y ); see also [Vla94, p. 272]. The situation is depicted in Fig.
6.17 where the measured spectral power density ~ 2
( ν , ν ) of a speckle interferogram with ds =3 d p ,
I x y
α x =90°/column and α y =90°/row is shown in a logarithmic scale; nevertheless, the reference-wave peak
has been clipped to bring the details out more clearly.