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DIFFRACTIONDIFFRACTIONFourier Optics
2
Propagation between Planes in Free SpacePropagation between Planes in Free Space
3
x1
x2
y1
y2
x3
x3=0 x3=z
k
022 k
Plane Wave Expansion IPlane Wave Expansion I
213213221121
222
21
222
21213
222
21
22
21
2213
223
22
21
22
,exp,
: tosgeneralise ion thissuperpositby
,,
,,
)dependence exp(implicit wavegoing forward afor
exp,0
dkdkxkkkxkxkikka
kkkkkkikkk
kkkkkkkkk
ti
kkkkiak
x
k.x
Evanescent wave
4
Plane Wave Expansion IIPlane Wave Expansion II
2122112121
21213221121221
322112132121
21221121321
3
exp,,ˆ
where
,exp,ˆ2
1,
then
,,,,0,,,
:setting and
exp,0,,
:constant tivemultiplica awithin
ansformFourier tr inversean toreduces this0at that Noting
dkdkxkxkixxukku
dkdkzkkkykykikkuyyv
zxyxyxyyvxxxxxu
dkdkxkxkikkaxxx
x
5
Plane Wave Expansion IIIPlane Wave Expansion III
zkkikkkh
dkdkzkkkykykiyyh
dxdxxxuxyxyhyyv
dkdkdxdxzkkkxykxykixxuyyv
21321
212132211221
2121221121
212121322211121221
,exp,ˆ
,exp2
1,
,,,
,exp,2
1,
Explicity
Linear Shift Invariant System
Impulse Response /Point Spread Function
Spatial Frequency Response
6
Propagation as a filterPropagation as a filter
u v
022 k
1k
h2k
7
k0
1
unimodular phase function
exponential decay
Why is the angular spectrum of plane waves expansion rarely used?Why is the angular spectrum of plane waves expansion rarely used?
kz
qpqpi
qpqpm
dpdqmz
yq
z
ypi
kyyh
k
kd
k
kd
k
kkk
z
y
k
k
z
y
k
kikz
kyyh
dkdkzkkkykykiyyh
1,1
1,1
exp2
,
or
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2,
rewriten bemay
,exp2
1,
2222
2222
212
2
21
2121322112
2
21
212132211221
8
Oscillatory IntegralsOscillatory Integrals• We are left with the consideration of integrals of the form:
,,
exp
Ca
dppipaI
• If 0p
then the integrand is highly oscillatory and
,0I
• If 0*
pp
then there is a contribution from the integrand in the neighbourhood of the stationary point p*
9
Stationary Phase ConditionStationary Phase Condition
The stationary phase condition corresponds to a ray from source point to observation point ( recall shift invariance)
00;00
1;,,
21
2211
m
q
z
y
pm
p
z
y
p
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z
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1y
z
p
m3
2
3
1 ,k
k
m
q
k
k
m
p
10
Paraxial Approximation IParaxial Approximation IIn a paraxial system rays are inclined at small angles to the optical axis. One may then make the paraxial approximation:
2
2
2
1
2
2
2
1
2221
21
2222
2
1
2
1
2
1
2
11
2
1
2
11
,,
2
1
2
111
z
yq
z
yp
z
y
z
y
qpz
yq
z
yp
qpmz
yq
z
ypqp
qpqpm
z
11
Paraxial Approximation IIParaxial Approximation II
2
2
2
1
2
2
2
1
2
2
2
2
2
2
1222
2
2
2
2
1
2
2
2
12
2
2
2
21
2
1
2
11expexp
2
1
2
1
2
11exp
2
2
2
1
2
11expexp
2
2
1
2
11expexp
2
,exp2
,
z
y
z
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z
ik
z
y
z
yi
i
k
z
y
z
yidpdqqpi
k
z
y
z
yidpdq
z
yq
z
ypi
k
dpdqqpik
yyh
12
Fresnel DiffractionFresnel Diffraction
212211
2
2
2
121
2
1
2
1
21
2
22
2
1121
2121221121
exp2
1
2
11exp,
2
1
2
11exp
2
1
2
1
2
11exp,exp
2
1
,,,
dxdxyxyxz
ik
z
x
z
xikzxxu
z
y
z
yikz
z
ik
dxdxz
xy
z
xyikzxxuikz
z
ik
dxdxxxuxyxyhyyv
13
Up to a multiplicative quadratic phase factor (that is often neglected), the field at the observation plane is given by the Fourier transform of the field at the source plane multiplied by a quadratic phase factor.
Fraunhoffer DiffractionFraunhoffer Diffraction
2122112121 exp,, dxdxyxyxz
ikxxuyyv
14
If the source filed u has compact support (is zero outside some bounded aperture) and z is sufficiently large the variation of the quadratic phase factor over the support of u becomes negligible. The leading phase factor is also often neglected either because the region of interest in the observation plane subtends a sufficiently small angle with respect to the origin at the source plane or because it is the intensity only that is observed. The diffracted field distribution is then given by a Fourier transform of the field distribution in the source plane.
NotesNotes• The oscillatory integral representation of the impulse response of
this optical system can be evaluated asymptotically without recourse to the paraxial approximation using the method of stationary phase.
• The magnitude but not the phase of the leading multiplicative phase factors of the Fresnel and Faunhoffer diffraction integrals may be evaluated by appealing to energy conservation – the integral over the source and observation planes of the field intensity must be equal.
• The choice of outgoing plane waves in the plane wave spectrum ensures that all three diffraction integrals (plane wave expansion, Fresnel & Fraunhoffer formulae satisfy the Sommerfeld radiation condition at infinity.