1 Waves 10 Lecture 10 Wave propagation. D Aims: ëFraunhofer 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.

2 Waves 10 Fraunhofer diffraction D Diffraction. ëPropagation of partly obstructed waves. > Apertures, obstructions etc... D Diffraction régimes. ëIn the immediate vicinity of the obstruction: > Large angles and no approximations > Full solution required. ëIntermediate distances (near field) > Small angles, spherical waves, > Fresnel diffraction. ëLarge distances (far field) > Small angles, and plane waves, > Fraunhofer diffraction. ë(More formal definitions will come in the Optics course)

3 Waves 10 Young’s slits D Fraunhofer conditions ëFor us this means an incident plane wave and observation at infinity. D Two narrow apertures (2 point sources) ëEach slit is a source of secondary wavelets ëFull derivation (not in handout) is…. Applying “cos rule” to top triangle gives

4 Waves 10 2-slit diffraction ëSimilarly for bottom ray ëResultant is a superposition of 2 wavelets  The term expi(kR-  t) will occur in all expressions. We ignore it - only relative phases are important. Where s = sin .

5 Waves 10 cos-squared fringes ëWe observe intensity cos-squared fringes. ëSpacing inversely proportional to separation of the slits. D Amplitude-phase diagrams. Spacing of maxima Slit 1 Slit 2 Resultant

6 Waves 10 Three slits  Three slits, spacing d.  Primary maxima separated by /d, as before. ëOne secondary maximum.

7 Waves 10 N slits and diffraction gratings  N slits, each separated by d. ëA geometric progression, which sums to  Intensity in primary maxima  N 2  In the limit as N goes to infinity, primary maxima become  -functions. A diffraction grating. Spacing, as before N-2, secondary maxima

8 Waves 10 Single broad slit  Slit of width t..  Slit of width t. Incident plane wave. ëSummation of discrete sources becomes an integral. /t

9 Waves 10 Generalisation to any aperture D Aperture function  The amplitude distribution across an aperture can take any form a(y). This is the aperture function. ëThe Fraunhofer diffraction pattern is  putting ks=K gives a Fourier integral ëThe Fraunhofer diffraction pattern is the Fourier Transform of the aperture function. ëDiffraction from complicated apertures can often be simplified using the convolution theorem.  Example: 2-slits of finite width Convolution of 2  -functions with a single broad slit. FT(f*g)  FT(f).FT(g) Cos fringes sinc function