fraunhofer diffraction l.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
Fraunhofer Diffraction PowerPoint Presentation
Download Presentation
Fraunhofer Diffraction

Loading in 2 Seconds...

play fullscreen
1 / 18

Fraunhofer Diffraction - PowerPoint PPT Presentation


  • 442 Views
  • Uploaded on

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.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Fraunhofer Diffraction' - Mia_John


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
fraunhofer diffraction

Fraunhofer Diffraction

Wed. Nov. 20, 2002

kirchoff integral theorem
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
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
Fresnel-Kirchoff Formula
  • Thus E=0 everywhere on surface except the portion that is the aperture. Thus from (6)
fresnel kirchoff formula5
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
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
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.
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

Fraunhofer vs. Fresnel diffraction

S

P

Hecht 10.2

Hecht 10.3

fraunhofer vs fresnel diffraction9
Fraunhofer vs. Fresnel Diffraction

’

r’

r

h’

h

d’

S

P

d

fraunhofer vs fresnel diffraction10
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
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 limit12
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
Fraunhofer, Fresnel limits
  • The near field, or Fresnel, limit is
  • See 10.1.2 of text
fraunhofer diffraction14
Fraunhofer diffraction
  • Typical arrangement (or use laser as a source of plane waves)
  • Plane waves in, plane waves out

screen

S

f1

f2

fraunhofer diffraction15
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 diffraction16
Fraunhofer diffraction
  • Then the magnitude of the electric field at P is,
single slit fraunhofer diffraction
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 diffraction18
Single slit Fraunhofer diffraction

Fraunhofer single slit diffraction pattern