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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.

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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

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