Phase Estimation in Optical Interferometry

126 ◾ Kieran G. Larkin

Assuming that the local fringe structure induced by phase changes

faster than the amplitude, then a small region of the fringe pattern can be

expanded as a Taylor series of the phase Ψ:

ψ≈ψ+ψ+ψ

−=+ψ +ψ +ψ +−ψ+ψ+ψ(,)(,)

2

{exp[( )] exp[

()

]}

00 10 01

00 10 01 00 10 01

xy

gxyaxy

b

ixyi

xy

(4.11)

e forward and inverse Fourier transforms in 2-D are dened by

∫∫

== −π +

−∞

+∞

−∞

+∞

(,)F{(,)}(,)exp[ 2( )]Guvgxy gxyiux vy dxdy

(4.12a)

and

∫∫

== +π +

−

−∞

+∞

−∞

+∞

(,)F{(,)}(,)exp[ 2( )]

1

gxyGuv Guviux vy dudv (4.12b)

e (local) 2-D Fourier transform of these equations is simply two

Dirac delta side lobes:

−=−

=δ−− +ψ +δ ++ −ψ

F{ }

2

{( ,)exp[ ]( ,)exp[ ...