Fraunhofer Diffraction

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# Fraunhofer Diffraction - PowerPoint PPT Presentation

Fraunhofer Diffraction. Wed. Nov. 20, 2002. Kirchoff integral theorem. This gives the value of disturbance at P in terms of values on surface  enclosing P. It represents the basic equation of scalar diffraction theory. Geometry of single slit. Have infinite screen with aperture A.

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## Fraunhofer Diffraction

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### Fraunhofer Diffraction

Wed. Nov. 20, 2002

Kirchoff integral theorem

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

It represents the basic equation of scalar diffraction theory

Geometry of single slit

Have infinite screen with aperture A

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

P

S

r

r’

’

Radiation from source, S, arrives at aperture with amplitude

Since R 

E=0 on .

R

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

Fresnel-Kirchoff Formula
• Thus E=0 everywhere on surface except the portion that is the aperture. Thus from (6)
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,

Fresnel Kirchoff diffraction formula

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()
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.
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

Fraunhofer vs. Fresnel diffraction

S

P

Hecht 10.2

Hecht 10.3

Fraunhofer vs. Fresnel Diffraction

’

r’

r

h’

h

d’

S

P

d

Fraunhofer Vs. Fresnel Diffraction

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

Fraunhofer diffraction limit

sin’

sin

• Now, first term = path difference for plane waves

’

sin’≈ h’/d’

sin ≈ h/d

sin’ + sin =  ( h’/d + h/d )

Second term = measure of curvature of wavefront

Fraunhofer Diffraction 

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,

Fraunhofer or far field limit

Fraunhofer, Fresnel limits
• The near field, or Fresnel, limit is
• See 10.1.2 of text
Fraunhofer diffraction
• Typical arrangement (or use laser as a source of plane waves)
• Plane waves in, plane waves out

screen

S

f1

f2

Fraunhofer diffraction
• Obliquity factor

Assume S on axis, so

Assume  small ( < 30o), so

• Assume uniform illumination over aperture

r’ >>  so is constant over the aperture

• Dimensions of aperture << r

r will not vary much in denominator for calculation of amplitude at any point P

consider r = constant in denominator

Fraunhofer diffraction
• Then the magnitude of the electric field at P is,
Single slit Fraunhofer diffraction

P

y = b

r

dy

ro

y

r = ro - ysin

dA = L dy

where L   ( very long slit)

Single slit Fraunhofer diffraction

Fraunhofer single slit diffraction pattern