CS781 - Colour for Computer Graphics - Winter 2009

Lecture 3

Light obeys the wave equation: Grad( Div Y ) = (c/n(x))^2 (d / dt )^2 Y

c: speed of light
n: index of refraction
Y amplitude of the electromagnetic field (light)
- Amplitudes add when solutions are combined
  - because the equation is linear
- Energy is amplitude squared.

Some things need explaining

If n(x) is constant then Y ( x, t ) = f( x +- (c/n) t) the equation is solved!
- If x +- (c/n(x)) t = constant then Y = constant.
- These are called wavefronts
- If n is constant each piece of the wavefront moves in a straight line
  - Huyghens construction
  - e.g. plane waves f = cos( \nu( x +- (c/n) t ) )
    1. Waves once when the argument goes through 2\pi
      e.g. Compare \nu( x - (c/n) t ) = 0 to \nu( x' - (c/n) t' ) = 2\pi
    2. Constant time (snapshot of wave at t = t' = 0).
      x = 0 and x' = 2\pi / \nu
      
      The wavelength: \lambda = 2\pi / \nu
    3. Constant location ( stand at x = x' = 0 and watch )
      t = 0 and t' = 2\pi n / \nu c
      
      The period: 1 / \omega = 2\pi n / \nu c
    4. Eliminate \nu: 1/ \omega = n \lambda / c; \omega = c / n \lambda; \lambda = c / n \omega
  - e.g. spherical waves
    1. Equally soluble, but only if you are as comfortable with the Laguerre function as you are with sin and cosine
In 3D replace x by k \dot x where |k| = 1
- k is the wavevector, which determines the direction of travel of the wavefront

If n(x) is not constant, the wavefront can change direction. The simplest case is a discontinuous change in n(x) on a plane.

n(x) is a weak function of wavelength

Prism

Rainbow on the CD

Bidirectional Reflectance Distribution Function

Rainbow in the air

Was anybody out, and looking up, on Friday night about midnight?

Sky

Fog

Suppose we try the substitution: k \ dot x --> S(x)

If the variation in n(x) is slow compared to the wavelength of the light then this equation satisfies
- del S(x) \dot del S(x) = n^2(x)

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