Diffraction See Chapter 10 of Hecht

Diffraction Send light through apertures, slits or gratings Predict intensity distribution of light Further examples of interference.

Huygens – Fresnel Principle Every unobstructed point of a wavefront serves as a source of spherical wavelets. Resulting amplitude is the superposition of all of these waves.

Fraunhofer Diffraction – Far Field Fresnel – Near Field Kirchhoff – Derivation of diffraction from wave equation In the limit l -> 0 recover geometric optics

Line of point sources (pinholes), all in phase with same amplitude q d=distance between sources Note that:

Single Slit & Fraunhofer y = D/2 P r y R q x y = -D/2 Using Huygens, treat slit (length D) as a line of point radiators. Point radiator at y is a distance r from observation point P; R is distance from slit center to P.

Single Slit, other dimension. Slit width of b P q x z Same derivation as before:

Double Slit P q x z y = D/2 y Slits of width b, separated by distance a. z y = -D/2

Line of point sources (pinholes), all in phase with same amplitude q d=distance between sources Note that:

Circular Aperture (of radius a) Source is at an angle b and at radius r. R z q y b f x

Diffraction Gratings Used to separate light of different wavelengths with high resolution. Diffraction grating applications: measuring atomic spectra in laboratory instruments and telescopes. A large number of parallel, closely spaced slits constitutes a diffraction grating. The condition for maximum intensity is the same as that for the double slit or multiple slits, but with a large number of slits the intensity maximum is very sharp and narrow, providing the high resolution for spectroscopic applications.

Transmission Grating

Example of a transmission grating.

Use Fraunhofer to model a transmission grating of N-slits N-slits, b – wide, separated by distance a. P q x a b

Reflection Grating

Angular Dispersion How the diffraction angle changes with wavelength. The effective width of a spectral line? Phase difference between minima (zeros) is:

Resolving Power Transmission Reflection

A hole in a opaque screen A hole in a opaque screen. Model hole as a distribution of point-like spherical radiators; integrate over hole. If that were true, and how nature works, then not only would light propagate forward through the hole, but light would also propagate back towards source of light. Light is not observed propagating back to source!? Solution – Kirchhoff & Fresnel

Kirchhoff’s Diffraction Theory Fraunhofer diffraction: The source wavefront is assumed to be planar, the different elements of the wavefront have a constant phase difference. In the Fresnel diffraction the curvature of the wavefront is included, the relative phase is not constant..

r r P r0 Source r0 q

Semi-Infinite Opaque Screen Screen from y No screen for x z

y r r r0 r0 x z

Babinet’s Principle A hole in a screen versus A small opaque screen, which would fill hole above

If hole filled in, no radiation out Two cases produce the same field, but 180o out of phase. => Same intensity pattern