Lecture 10 Aims: Wave propagation. Fraunhofer diffraction (waves in the “far field”). Young’s double slits Three slits N slits and diffraction gratings A single broad slit General formula - Fourier transform. 1 Waves 10 Fraunhofer diffraction Diffraction. Propagation of partly obstructed waves. Apertures, obstructions etc... Diffraction régimes. In the immediate vicinity of the obstruction: Large angles and no approximations Full solution required. Intermediate distances (near field) Small angles, spherical waves, Fresnel diffraction. Large distances (far field) Small angles, and plane waves, Fraunhofer diffraction. (More formal definitions will come in the Optics course) 2 Waves 10 Young’s slits Fraunhofer conditions For us this means an incident plane wave and observation at infinity. Two narrow apertures (2 point sources) Each slit is a source of secondary wavelets Full derivation (not in handout) is…. p A exp ikr1 t A exp ikr2 t Applying “cos rule” to top triangle gives 2 d 2 2 d r1 R 2 R sin 2 2 1/ 2 d r1 R1 sin R d d R 1 sin R sin 2 2R 3 Waves 10 2-slit diffraction Similarly for bottom ray d d r2 R1 sin R sin 2 2R Resultant is a superposition of 2 wavelets kd kd i ( kR sin t ) i ( kR sin t ) 2 2 p Ae Ae kd i kd sin i sin i ( kR t ) 2 2 Ae e e The term expi(kR-t) will occur in all expressions. We ignore it - only relative phases are important. p A e ikd sin / 2 Ae ikd sin / 2 2 A cos(kd sin / 2) 2 A cos(kds / 2) Where s = sin. 4 Waves 10 cos-squared fringes We observe intensity 2 I p cos 2 kds / 2 cos-squared fringes. Spacing of maxima Spacing inversely proportional to separation of the slits. Amplitude-phase diagrams. Resultant Slit 1 Slit 2 5 Waves 10 Three slits Three slits, spacing d. p A e ikds ei 0 eikds A1 2 cos kds I 1 2 cos kds2 Primary maxima separated by l/d, as before. One secondary maximum. 6 Waves 10 N slits and diffraction gratings N slits, each separated by d. p A ei 0 eikds ei 2kds .......... . ei ( N 1) kds A geometric progression, which sums to p A(1 eiNkds ) /(1 eikds ) A A eikds / 2 e ikds / 2 eikds / 2 eiNkds / 2 e iNkds / 2 eiNkds / 2 eiNkds / 2 sin( Nkds / 2) eikds / 2 sin( kds / 2) I sin 2 ( Nkds / 2) / sin 2 (kds / 2) Spacing, as before N-2, secondary maxima Intensity in primary maxima a N2 In the limit as N goes to infinity, primary maxima become d-functions. A diffraction grating. 7 Waves 10 Single broad slit Slit of width t. Incident plane wave. Summation of discrete sources becomes an integral. p e iky sin dS t/2 p A e ikysdy t / 2 A p e ikts / 2 eikts / 2 (iks) 2A sin( kts / 2) At sinc( kts / 2) ks l/t 8 Waves 10 Generalisation to any aperture Aperture function The amplitude distribution across an aperture can take any form a(y). This is the aperture function. The Fraunhofer diffraction pattern is p a( y)e ikysdy putting ks=K gives a Fourier integral p a( y)e iKydy The Fraunhofer diffraction pattern is the Fourier Transform of the aperture function. Diffraction from complicated apertures can often be simplified using the convolution theorem. Example: 2-slits of finite width Convolution of 2 d-functions with a single broad slit. FT(f*g) a FT(f).FT(g) Cos fringes sinc function 9 Waves 10