Design Realization lecture 25

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 sourcebut not the viewer. • i.e. brightness isproportional 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 iswhere 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,  = 2gives 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. 3 2 1 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.

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 f1 , f2 (sometimes called conjugates), then they satisfy:

29. 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)

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.