1

Diffraction

2

Diffraction theory (10.4 Hecht)

We will first develop a formalism that will describe the propagation of a wave – that is develop a mathematical description of Huygen’s principle

We know this

Wavefront UU

What is UU here?

3

Diffraction theory Consider two well behaved functions U1’, U2’ that

are solutions of the wave equation. Let U1’ = U1e-it ; U2’ = U2e-it

Thus U1 , U2 are the spatial part of the functions and since,

we have,

0

10'1

'

22

22

12

12

2

22

2

2

22

UkUUkU

eUv

Ut

U

vU ti

4

Green’s theorem

Consider the product U1 grad U2 = U1U2

Using Gauss’ Theorem

Where S = surface enclosing V

Thus,

SV

SdUUdVUU

2121

dSn

USdU

22

normaloutwardndSnSd ˆˆ

5

Green’s Theorem Now expand left hand side,

Do the same for U2U1 and subtract from (2), gives Green’s theorem

V S

dSn

UUdVUUUU 2212

2121

V S

dSn

UU

n

UUdVUUUU 31

22

112

222

1

6

Green’s theorem

Now for functions satisfying the wave equation (1), i.e.

Consequently,

since the LHS of (3) = 0

12

12

22

22 UkUUkU

4012

21

dSn

UU

n

UU

S

7

Green’s theorem applied to spherical wave propagation Let the disturbance at t=0 be,

where r is measured from point P in V and U1 = “Green’s function”

Since there is a singularity at the point P, draw a small sphere P, of radius , around P (with P at centre)

Then integrate over +P, and take limit as 0

r

eUU

ikr

o11

8

Spherical Wave propagation

522

2

22

dr

e

rU

r

U

r

e

dSr

e

nU

n

U

r

e

P r

ikrikr

ikrikr

n̂

P

n̂

Thus (4) can be written,Thus (4) can be written,

9

Spherical Wave propagation

In (5), an element of area on P is defined in terms of solid angle

and we have used

Now consider first term on RHS of (5)

dddS

d sin2

rn

1022

0 limlim Idr

Ue

r

ik

10

Kirchoff’s Integral Theorem

Now U2 = continuous function and thus the derivative is bounded (assume)

Its maximum value in V = C Then since eik 1 as 0 we have,

The second term on the RHS of (5)

rU

2

0limlim 010 CdI

dr

e

rUI

r

ikr2

2020 limlim

11

Kirchoff’s Integral Theorem

dUikU

deik

U

dr

e

rUI

ik

r

ikr

220

2220

22020

lim

1lim

limlim

Now as 0 U2(r) UP (its value at P)

and, PPP UdUdU 4

Now designate the disturbance U as an electric field E

12

Kirchoff integral theorem

64 dSn

E

r

e

r

e

nEE

ikrikr

P

This gives the value of disturbance at P in terms of values on surface enclosing P.

It represents the basic equation of It represents the basic equation of scalar diffraction scalar diffraction theorytheory

13

Geometry of single slit

n̂

n̂

R

S P

’

Have infinite screen with aperture A

Radiation from source, S, arrives at aperture with amplitude

'

'

r

eEE

ikr

o

Let the hemisphere (radius R) and screen with aperture comprise the surface () enclosing P.

Since R

E=0 on .

Also, E = 0 on side of screen facing V.

r’r

14

Fresnel-Kirchoff Formula Thus E=0 everywhere on surface except the

portion that is the aperture. Thus from (6)

dSr

e

nr

e

r

e

nr

eE

ikrikrikrikr

P

''

4''

)7(1

ˆˆ

ˆˆ

..

,'

'ˆˆ'ˆˆˆˆ

2ikr

ikrikr

err

ikrn

r

e

rrn

r

e

nge

Thusr

rnnn

andr

rnnn

15

Fresnel-Kirchoff Formula

Now assume r, r’ >> ; then k/r >> 1/r2

Then the second term in (7) drops out and we are left with,

'coscos2

1'ˆˆˆˆ

2

1

'

,

'ˆˆˆˆ'

4

'

'

rnrnF

dSFrr

eiEE

or

dSrnrnikrr

eEE

aperture

rriko

P

aperture

rrik

oP

Fresnel Kirchoff Fresnel Kirchoff diffraction formuladiffraction formula

16

Obliquity factor

Since we usually have ’ = - or n.r’=-1, the obliquity factor

F() = ½ [1+cos ] Also in most applications we will also

assume that cos 1 ; and F() = 1 For now however, keep F()

17

Huygen’s principle

Amplitude at aperture due to source S is,

Now suppose each element of area dA gives rise to a spherical wavelet with amplitude dE = EAdA

Then at P,

Then equation (6) says that the total disturbance at P is just proportional to the sum of all the wavelets weighted by the obliquity factor F()

This is just a mathematical statement of Huygen’s principle.

'

'

r

eEE

ikr

oA

r

edAEdE

ikr

AP

FdEP

18

Fraunhofer vs. Fresnel diffraction

In Fraunhofer diffraction, both incident and diffracted waves may be considered to be plane (i.e. both S and P are a large distance away)

If either S or P are close enough that wavefront curvature is not negligible, then we have Fresnel diffraction

P

S

Hecht 10.2Hecht 10.2 Hecht 10.3Hecht 10.3

19

Fraunhofer vs. Fresnel Diffraction

S

P

d’

d

’

hh’

r’ r

20

Fraunhofer Vs. Fresnel Diffraction

2

2222

22222222

'

11

2

1

'

'

2

11

'

'

2

11'

2

11

'

'

2

11'

'''

ddd

h

d

h

d

hd

d

hd

d

hd

d

hd

hdhdhdhd

Now calculate variation in (r+r’) in going from one side of aperture to the other. Call it

21

Fraunhofer diffraction limit

Now, first term = path difference for plane waves

’

sinsin’

sin’≈ h’/d’

sin ≈ h/d

sin’ + sin = ( h’/d + h/d )

Second term = measure of curvature of wavefront

Fraunhofer Diffraction

21

'

1

2

1

dd

22

Fraunhofer diffraction limit

If aperture is a square - X The same relation holds in azimuthal plane and 2

~ measure of the area of the aperture Then we have the Fraunhofer diffraction if,

apertureofaread

or

d

,

2

Fraunhofer or far field limit

23

Fraunhofer, Fresnel limits

The near field, or Fresnel, limit is

See 10.1.2 of text

2

d

24

Fraunhofer diffraction Typical arrangement (or use laser as a

source of plane waves) Plane waves in, plane waves out

S

f1 f2

screen

25

Fraunhofer diffraction

1. Obliquity factorAssume S on axis, so Assume small ( < 30o), so

2. Assume uniform illumination over aperture

r’ >> so is constant over the aperture

3. Dimensions of aperture << rr will not vary much in denominator for calculation of amplitude at any point Pconsider r = constant in denominator

1'ˆˆ rn1ˆˆ rn

'

'

r

eikr

26

Fraunhofer diffraction

Then the magnitude of the electric field at P is,

aperture

ikrikr

oP dSe

rr

eikEE

'2

'

27

Single slit Fraunhofer diffraction

y = b

y

dy

P

ro

r

r = ro - ysin

dA = L dy

where L ( very long slit)

28

Single slit Fraunhofer diffraction

'2sin

2

,

sin

_______________

'

sin

rr

eikEC

kb

where

ebCeE

dyeeCE

dAeCE

ikro

iikrP

ikyb

o

ikrP

ikrP

o

o

2

2sin

oII

Fraunhofer single slit diffraction pattern

2bCIo