# Lecture 12 - Light in Motion

## And Now from our Sponsor

1. Projects
2. Wednesday's class

# Using Colour in the Real World

### What does colour tell us?

• gives information about qualities of objects
• identifies objects

And it gives us this information at a distance.

### Capabilities of perception

• Search in parallel for a particular colour
• Rejecting unintersting objects is the key
• Ordinary rejection takes about 50-60 msec
• Fast (parallel) rejection takes about 3-5 msec
• Put together perceptual pieces of the same colour into wholes
• This seems to be an active filling-in process
• Label objects with colours
• Applying categorical judgements

### What other perceptual attributes are similar

• Combinations of colour, such as flags
• Visual textures
• Basic visual forms, like the difference between O and X
• Depth planes (controversial)

### What's important?

We are not very interested in the colour of light

• We are actually interested in the colour of the surfaces of objects

# Light moving through space

Waves

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

## Refraction

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

#### Chromatic aberration

n(x) is a weak function of wavelength

Prism

## Reflection

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

## The Eikonal

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)