Design Realization lecture 25

John Canny

11/20/03

Last time

Improvisation: application to circuits and real-time programming.

Optics: physics of light.

This time

Reflection, Scattering Refraction, TIR Retro-reflection Lenses

Wavefronts and Rays

EM waves propagate normal to the wavefront surface, and vice-versa.

The ray description is most useful for describing the geometry of images.

Reflection

Most metals are excellent conductors. They reduce the E field to zero at the surface,

causing reflection. If I, R, N unit vectors:

IN = RN

I(N R) = 0

Ray-tracing

By tracing rays back from the viewer, we can estimate what a reflected object would look like. Follow at least two rays at extremes of the object.

Lambertian scattering

For most non-metallic objects, the apparent brightness depends on surface orientation relative to the light source but not the viewer.

i.e. brightness isproportional to IN

Refraction – wave representation

In transparent materials (plastic, glass), light propagates slower than in air.

At the boundary, wavefronts bend:

Refractive index

Refractive index measures how fast light propagates through a medium.

Such media must be poor conductors and are usually called dielectric media.

The refractive index of a dielectric medium is

where c is the speed of light in vacuum, and v is the speed in the medium. Note that > 1.

v

c

Refraction – Snell’s law

Incident and refracted rays satisfy:

rrii sinsin

Refraction – ray representation

In terms of rays, light bends toward the normal in the slower material.

Refraction in triangular prisms

For most media, refractive index varies with wavelength. This gives the familiar rainbow spectrum with white light in glass or water.

Refractive index

Refractive index as a function of wavelength for

glass and water

Refractive index

High-quality optical glass is engineered to have a constant refractive index across the visible spectrum.

Deviations are still possible. Such deviations are called chromatic aberration.

Refractive indices

Water is approximately 1.33 Normal glass and acrylic plastic is about 1.5 Polycarbonate is about 1.56 Highest optical plastic index is 1.66 Bismuth glass is over 2 Diamond is 2.42

Internal reflection

Across a refractive index drop, there is an angle beyond which ray exit is impossible:

Total internal reflection (TIR)

The critical angle is where the refracted ray would have 90 incidence.

The internal reflection angle is therefore:

For glass/acrylic, this is 42 For diamond, it is 24 - light will make many

internal reflections before leaving, creating the “fire” in the diamond.

/1arcsinT

Penta-prisms

Penta-prisms are used in SLR cameras to rotate an image without inverting it.

They are equivalent to two conventional mirrors, and cause a 90 rotation of the image, without inversion.

An even number of mirrors produce a non-inverted rotated image of the object.

Retro-reflection: Corner reflectors

In 2D, two mirrors at right angles will retro-reflect light rays, i.e. send them back in the direction they came from.

Retro-reflection: Corner reflectors

In 3D, you need 3 mirrors to do this:

Analysis: each mirror inverts one of X,Y,Z

Retro-reflection: TIR spheres

Consider a sphere and an incoming ray. Incoming and refracted ray angles are , . For the ray to hit the centerline, = 2. For retro-reflection, we

want = sin /sin For small angles, = 2

gives good results.

Retro-reflective sheets

Inexpensive retro-reflective tapes are available that use tiny corner reflectors or spheres embedded in clear plastic (3M Scotchlite)

They come in many colors, including black.

Retro-reflector gain

The retro-reflection response of a screen is normally rated in terms of gain.

Gain = ratio of peak reflected light energy to the energy reflected by a Lambertian surface.

Gains may be 1000 or more. Light source only needs 1/1000 of the light

energy to illuminate the screen, as long as the viewer is close enough to the source.

Application: personal displays

Each user has a personal projector (e.g. a PDA with a single lens in front of it), and projects on the same retro-reflective screen.

2

1

3

Application: Artificial backgrounds

Projector and camera along same optical axis, project scene onto actors and retro-reflective background.

Cameras sees background only on screen, not on the actors (3M received technical academy award for this in 1985).

Convex Lenses

A refractive disk with one or two convex spherical surfaces converges parallel light rays almost to a point.

The distance to this point is the focal length of the lens.

Lenses

If light comes from a point source that is further away than the focal length, it will focus to another point on the other side.

Lenses

When there are two focal points f1 , f2 (sometimes called conjugates), then they satisfy:

21

111

fff

Spherical Lenses

If the lens consists of spherical surfaces with radii r1 and r2, then the focal length satisfies

1/f = ( - 1) (1/r1 - 1/r2)

Spherical aberration

Spherical lenses cannot achieve perfect focus, and always have some aberration:

Spherical aberration

Compound lenses, comprising convex, concave or hybrid elements, are used to minimize aberration.