Fraunhofer Diffraction Wed Nov 20 2002 1 Kirchoff

Fraunhofer Diffraction Wed. Nov. 20, 2002 1

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 2

Geometry of single slit Have infinite screen with aperture A Let the hemisphere (radius R) and screen with aperture comprise the surface ( ) enclosing P. S Radiation from source, S, arrives at aperture with amplitude r’ ’ P r Since R E=0 on . R Also, E = 0 on side of screen facing V. 3

Fresnel-Kirchoff Formula n Thus E=0 everywhere on surface except the portion that is the aperture. Thus from (6) 4

Fresnel-Kirchoff Formula n n Now assume r, r’ >> ; then k/r >> 1/r 2 Then the second term in (7) drops out and we are left with, Fresnel Kirchoff diffraction formula 5

Obliquity factor Since we usually have ’ = - or n. r’=-1, the obliquity factor F( ) = ½ [1+cos ] n Also in most applications we will also assume that cos 1 ; and F( ) = 1 n For now however, keep F( ) n 6

Huygen’s principle n Amplitude at aperture due to source S is, n Now suppose each element of area d. A gives rise to a spherical wavelet with amplitude d. E = EAd. A n Then at P, n 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( ) n This is just a mathematical statement of Huygen’s principle. 7

Fraunhofer vs. Fresnel diffraction n 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) n If either S or P are close enough that wavefront curvature is not negligible, then we have Fresnel diffraction S P Hecht 10. 2 Hecht 10. 3 8

Fraunhofer vs. Fresnel Diffraction ’ r’ S r h’ d’ h d P 9

Fraunhofer Vs. Fresnel Diffraction Now calculate variation in (r+r’) in going from one side of aperture to the other. Call it 10

Fraunhofer diffraction limit n 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 11

Fraunhofer diffraction limit n n n 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 12

Fraunhofer, Fresnel limits n The near field, or Fresnel, limit is n See 10. 1. 2 of text 13

Fraunhofer diffraction Typical arrangement (or use laser as a source of plane waves) n Plane waves in, plane waves out n S screen f 1 f 2 14

Fraunhofer diffraction 1. 2. Obliquity factor Assume S on axis, so Assume small ( < 30 o), so Assume uniform illumination over aperture r’ >> so 3. 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 15

Fraunhofer diffraction n Then the magnitude of the electric field at P is, 16

Single slit Fraunhofer diffraction P y=b dy y r ro r = ro - ysin d. A = L dy where L ( very long slit) 17

Single slit Fraunhofer diffraction Fraunhofer single slit diffraction pattern 18