ELG5106 Fourier OpticsELG5106 Fourier Optics

Trevor Halltjhall@uottawa.ca

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

,exp

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

qpmqpmz

yq

z

ypqp

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

yikzikz

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.