Chapter 38: Diffraction

 

Learning Goals

  1. Understand why diffraction occurs.
  2. Understand diffraction from narrow slits and circular holes.
  3. Understand how diffraction relates to optical resolution.
  4. Understand diffraction gratings.
  5. Understand polarization.

 

Diffraction

Diffraction occurs whenever light waves (or any other 3D wave, for that matter) passes through apertures or around obstacles.

 

Fraunhofer Diffraction

In this case, the diffracting aperture is illuminated with plane waves, and the diffraction pattern is observed on a screen far from the diffracting aperture. If these assumptions are not valid, the Fresnel diffraction theory must be used. We will always use Fraunhofer diffraction in Physics 222.

 

Single Slit Aperture

In this case, the diffracting aperture is a single long, narrow slit of width a. The resulting Fraunhofer diffraction pattern has intensity minima located by

 

 where m is the diffraction order and l is the wavelength.

 

Circular Aperture

In this case, the diffraction pattern has a central maximum whose angular width (determined by the radius of the first dark fringe) is given by

 

qmin = 1.22l/D

where D is the diameter of the circular aperture. An example of such an aperture is the input aperture of a telescope, hence this simple relation can be used to estimate the resolution of optical instruments. In this regard, the Rayleigh Criterion says that two distant objects can just be resolved if their diffraction patterns are separated by at least qmin .

 

Diffraction Grating

This is a very useful aperture which consists of a great many parallel, evenly spaced slits. The resulting diffraction pattern is characterized by principle maxima located by

 

d sin(q) = ml

where d is the slit spacing and m is the order number. The resolving power of the grating is given by

 

R = Nm

where N is the number of slits illuminated and m is the diffraction order.

 

Polarization

The polarization plane of an electromagnetic wave is determined by the plane that electric field oscillates in. Reflection from shiny surfaces is partially polarized; this effect is greatest when the angle of reflection is equal to Brewster’s angle:

 

qp = tan-1(n)

where n is the index of reflection of the reflecting surface. This is the principle behind polaroid sunglasses.