ELG5106 Fourier Optics

Trevor Halltjhall@uottawa.ca

DIFFRACTIONFourier Optics

2

Propagation between Planes in Free Space

3

x1

x2

y1

y2

x3

x3=0 x3=z

k

022 k

Plane 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

tikkkkiak

x

k.x

Evanescent wave

4

Plane 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 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 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?

kzqpqpi

qpqpm

dpdqmzyq

zypikyyh

kkd

kkd

kkkk

zy

kk

zy

kkikzkyyh

dkdkzkkkykykiyyh

1,1

1,1

exp2

,

or

,exp2

,

rewriten bemay

,exp2

1,

2222

2222

212

2

21

2121322112

2

21

212132211221

8

Oscillatory 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 Condition

The stationary phase condition corresponds to a ray from source point to observation point ( recall shift invariance)

00;00

1;,,

21

2211

mq

zy

pmp

zy

p

qpmqpmzyq

zypqp

1y

z

p

m3

2

3

1 ,kk

mq

kk

mp

10

Paraxial Approximation IIn a paraxial system rays are inclined at small angles to the optical axis. One may then make the paraxial approximation:

22

21

22

21

2221

21

2222

21

21

21

211

21

211

,,

21

2111

zyq

zyp

zy

zy

qpzyq

zyp

qpmzyq

zypqp

qpqpm

z

11

Paraxial Approximation II

22

21

22

21

2

2

2

22

2122

2

2

22

21

22

21

2

2

2

2

21

21

211expexp

21

21

211exp2

2

21

211expexp

2

21

211expexp

2

,exp2

,

zy

zyikzikz

zik

zy

zyi

ik

zy

zyidpdqqpik

zy

zyidpdq

zyq

zypik

dpdqqpikyyh

12

Fresnel Diffraction

212211

22

21

21

21

21

21

222

211

21

2121221121

exp21

211exp,

21

211exp

21

21

211exp,exp

21

,,,

dxdxyxyxzik

zx

zxikzxxu

zy

zyikz

zik

dxdxzxy

zxyikzxxuikz

zik

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 Diffraction

2122112121 exp,, dxdxyxyxzikxxuyyv

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.

Notes• 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.