Design Realization
lecture 25
John Canny
11/20/03
Last time
 Improvisation: application to circuits and realtime programming.
 Optics: physics of light.
This time
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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:
IN = RN
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 is
proportional to IN
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
c

v
where c is the speed of light in vacuum, and v
is the speed in the medium. Note that  > 1.
Refraction – Snell’s law
 Incident and refracted rays satisfy:
i sin i r sin  r
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
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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:
T  arcsin 1 /  
 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.
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 noninverted rotated image
of the object.
Retro-reflection: Corner reflectors
 In 2D, two mirrors at right angles will retroreflect 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
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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
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:
1 1 1
 
f
f1 f 2
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.