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We have assumed no aperture, in which case P (r 1 ) = 1 for all r 1 .
It follows that
h(r O , r D ) = iλf exp −
ik (r O + r D )
2 ,
(3.275)
2f
where we have made use of the integral
∞
i
iβ
2
exp (iαx
2 ) J 0 (βx) x dx =
exp −
,
(α = 0).
0
2α
4α
(3.276)
The amplitude u D (z D ) in the diffraction plane is given by
2
1
r
u D (r D ) =
exp ik 2f +
D
· d
2 r O u O (r O )
iλf
2f
2
ikr
ik
· exp
O
exp − (r O + r D )
2 .
(3.277)
2f
2f
Expanding,
(r O + r D )
2 = r O
2 + r D
2 + 2 r O · r D .
(3.278)
Substituting into 3.277, it follows that the amplitude in the diffraction plane is given by
1
ikr O · r D
u D (r D ) =
d
2 r O u O (r O ) exp −
,
(3.279)
iλf
f
209
3.3. Diffraction
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ignoring the leading exponential phase factor, as this does not influence the intensity |u D |
2 . This is recognizable as a Fourier transform of the object, with the transform variable kr D /f . For this
reason, the diffraction plane z D is often referred to as the Fourier
plane.
Geometrically, each specific value of the ray slope in the object
plane is mapped into a unique position in the Fourier plane. This
enables one to directly obtain an intensity map of a diffraction
pattern. We notice that r D /f is the ray slope at the object. Equivalently, this is the tangent of the diffraction angle.