1

Fraunhofer Diffraction

Wed. Nov. 20, 2002

2

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

3

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

4

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

5

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

6

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()

7

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

8

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

9

Fraunhofer vs. Fresnel Diffraction

S

P

d’

d

’

hh’

r’ r

10

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

11

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

12

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

13

Fraunhofer, Fresnel limits

The near field, or Fresnel, limit is

See 10.1.2 of text

2

d

14

Fraunhofer diffraction Typical arrangement (or use laser as a

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

S

f1 f2

screen

15

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

16

Fraunhofer diffraction

Then the magnitude of the electric field at P is,

aperture

ikrikr

oP dSe

rr

eikEE

'2

'

17

Single slit Fraunhofer diffraction

y = b

y

dy

P

ro

r

r = ro - ysin

dA = L dy

where L ( very long slit)

18

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