1. Design Realization lecture 25 John Canny 11/20/03

2. Last time  Improvisation: application to circuits and real- time programming.  Optics: physics of light.

3. This time  Reflection, Scattering  Refraction, TIR  Retro-reflection  Lenses

4. 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.

5. 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

6. 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.

7. 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

8. Refraction – wave representation  In transparent materials (plastic, glass), light propagates slower than in air.  At the boundary, wavefronts bend:

9. 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.

10. Refraction – Snell’s law  Incident and refracted rays satisfy:

11. Refraction – ray representation  In terms of rays, light bends toward the normal in the slower material.

12. 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.

13. Refractive index  Refractive index as a function of wavelength for glass and water

14. 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.

15. 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

16. Internal reflection  Across a refractive index drop, there is an angle beyond which ray exit is impossible:

17. 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.

18. 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.

19. 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.

20. Retro-reflection: Corner reflectors  In 3D, you need 3 mirrors to do this:  Analysis: each mirror inverts one of X,Y,Z

21. 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.    

22. 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.

23. 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.

24. 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

25. 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).

26. 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.

27. 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.

28. Lenses  When there are two focal points f 1, f 2 (sometimes called conjugates), then they satisfy:

29. Spherical Lenses  If the lens consists of spherical surfaces with radii r 1 and r 2, then the focal length satisfies 1/f = (  - 1) (1/r 1 - 1/r 2 )

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

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