Wave propagation.

1
Lecture 9

• Wave propagation.
• Aims
• Huygens Principle
• Reflection and refraction.
• Problems
• Huygens-Fresnel principle
• Fraunhofer diffraction (waves in the far
  field).
• Youngs 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 primary wavefront behaves as
  the source of spherical, secondary wavelets, 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.
• Snells Law

4
Huygens-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 dont 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 Youngs 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.