Lecture 9

1. Lecture 9 Wave propagation. • Aims: • Huygen’s Principle: • Reflection and refraction. • Problems • Huygen’s-Fresnel principle • 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. This lecture

2. Huygens’ Principle • Remember the concept of wavefront - a surface of constant phase. • 1690 “Treatise on light”, Huygens. • “Every point of a primarywavefront behaves as the source of spherical, secondarywavelets, such that the primary wavefront at a later time is the envelope of these wavelets; the wavelets have the same frequency and velocity as the incoming wave” • Rectilinear propagation • Spherical propagation

3. Reflection and refraction • qr = qi • Result follows from the 2 right-angled triangles with same hypotenuse, both having one side of length vt. Thus qr = qi. • Snell’s Law

4. Huygen’s-Fresnel principle • Shortcomings It is easy to criticise Huygens: • No theoretical basis; • Why neglect parts of the wavelet other than those forming the envelope; • Why don’t wavelets propogate backwards; • It is no help in predicting amplitudes; etc... • None detract from its historical significnce and the fact that it works. • Fresnel (1818) (See handout). • He built in Young’s concept of interference. “Every unobstructed point of a wavefront … serves as a source of spherical secondary wavelets … The amplitude of the optical field at any point beyond is the superposition of all these wavelets (considering their amplitudes and relative phases)” • Note backward travelling wavelets tend to interfere destructively • Kirchoff (1824-1887) • Provided theoretical foundation by connecting the wave equation to a surface integral of spherical wavelets. • See Optics course, next term.