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.

- Amplitudes add when solutions are combined

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 ) )
- 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

- Constant time (snapshot of wave at t = t' = 0).
x = 0 and x' = 2\pi / \nu

The wavelength: \lambda = 2\pi / \nu

- 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

- Eliminate \nu: 1/ \omega = n \lambda / c; \omega = c / n \lambda; \lambda = c / n \omega

- Waves once when the argument goes through 2\pi
- e.g. spherical waves
- 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.

- construction of wavefronts: planes on each side of the discontinuity
- On the boundary x \dot normal = 1.
- Geometric construction

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?

- ring around the moon.

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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