Refraction

1. Refraction Snell’s Law: n1 sin(1) = n2sin(2) where n1 = c/v1and v1 = [1/]1/2 and  = Ko where K = K(). This means that n = n(), and this means that -transmitted depends on the frequency. Thus different colors will be bent slightly differently. This is called DISPERSION.

2. Refraction and Dispersion • DISPERSION • rainbows & prisms break white light into component colors

3. Refraction and Dispersion • DISPERSION • rainbows & prisms break white light into component colors • signal degeneration (except thru vacuum): • signal pulses are Fourier series of frequencies • and each frequency travels at different speed • so pulse shape disperses

4. Refraction and Thin Lenses Can use refraction to try to control rays of light to go where we want them to go. Let’s see if we can FOCUS light.

5. Refraction and Thin Lenses What kind of shape do we need to focus light from a point source to a point? lens with some shape for front & back point source of light screen s’ = image distance s = object distance

6. Refraction and Thin Lenses Let’s try a simple (easy to make) shape: SPHERICAL. Play with the lens that is handed out Does it act like a magnifying glass?

7. Refraction and Thin Lenses Let’s try a simple (easy to make) shape: SPHERICAL. Play with the lens that is handed out Does it act like a magnifying glass? Does it focus light from the night light?

8. Refraction and Thin Lenses Let’s try a simple (easy to make) shape: SPHERICAL Play with the lens that is handed out Does it act like a magnifying glass? Does it focus light from the night light? Does the image distance depend on the shape of the lens? (trade with your neighbor to get a different shaped lens)

9. Refraction and Thin Lenses Spherical shape is specified by a radius. The smaller the sphere (smaller the radius), the more curved is the surface! R1 R2 R R

10. Refraction and Thin Lenses In the computer assignment Thin Lenses (Vol. 5, #2), in the Introduction, the following relation is derived if we make a couple of assumptions: Assumptions: THIN lenses SMALL ANGLES

11. Refraction and Thin Lenses This is called the THIN LENS equation: NOTE: • all the parameters on the left (n,R1,R2) refer to how the lens is MADE; • all the parameters on the right (s,s’) refer to how the lens is USED.

12. Refraction and Thin Lenses Therefore, we break the THIN LENS equation: (nglass – nair) 1 1 1 1 nair R1 R2 s s’ Into the LENS MAKERS equation and the LENS USERS equation: (nglass – nair) 1 1 1 11 1 nair R1 R2ffs s’ where f is a distance called the focal length. * { } + = + * { } = & = + +

13. Refraction and the Lensmakers Eq. Note that the order of R1 and R2 does NOT matter - which means that it doesn’t matter which side of the lens is front and which is back.

14. Refraction and the Lensmakers Eq. QUESTION: Can the focal length, f, benegative?

15. Refraction and the Lensmakers Eq. Can the focal length, f, be negative? Can the radii, R1 & R2, be negative?

16. R1>0 The NORMAL (convex) lens above is considered to have R1 and R2 both positive, and so f must be positive. R2>0

17. R1<0 This (concave) lens is considered to have R1 and R2 both negative, and so f must be negative! R2<0

18. Refraction and the Lensmakers Eq. What about these two shapes? Is f positive or negative? HINT: which radius in each case is bigger?

19. LEFT: |R1|< |R2|, R1 wins, R1>0 so f>0 (lens is thicker in middle) • RIGHT: |R1| > |R2|, R2 wins, R2<0 so f<0 (lens is thinner in middle)

20. Negative focal lengths Positive focal length lenses focus light – that is, they bend the rays so that the rays tend to converge. Negative focal length lenses tend to make the light diverge rather than converge. Often a negative focal length lens is used when one lens converges the light too much – the negative lens then makes the light converge less quickly. The same formulas work for negative lenses as for positive lenses. We just have to be careful with the signs of s and s’. The computer homework on Thin Lenses talks about negative focal length lenses and one of their uses: in eye glasses.

21. Designing Lenses If we specify the focal length for a lens, then we have three parameters to specify: nglass, R1, and R2 . Our design will entail choosing one of the two possible designs: for positive lenses either or . For the first case, , we could make the choice of R1 = R2= R. This simplifies the above equation to become: . For the second case, , R1>0 and R2<0, we need to be careful. We can’t choose |R1| = |R2 | since this will give 1/f = 0. We also have to make sure that |R1| < |R2 | to keep our shape.

22. Designing Lenses For negative lenses, we again have two choices for shapes similar to the choices for positive lenses. For the first case, R1<0 and R2<0, we could make the choice of R1 = R2= R. This simplifies the above equation to become: , where both fand R are negative. For the second case, R1>0 and R2<0, we again need to be careful. We can’t choose |R1| = |R2 | since this will give 1/f = 0. In this case, have to make sure that |R1| > |R2 | to keep our shape.

23. Designing Lenses: Examples If we are to design a lens to have a positive focal length of 30 mm, then we can use the symmetrical design so that R1=R2=R. We need to choose one more parameter, so we can solve for the remaining one. Let’s choose nglass = 1.67. Our equation now becomes: {(1.67 – 1)/1}*(2/R) = 1/(30 mm). We can solve this for R: R = (30 mm)*(0.67)*2 = 40.2 mm.

24. Designing Lenses: Examples Let’s now choose a harder example: that of a negative focal length contact lens. Here we want R1>0 and R2<0, and we want |R1| > |R2 | . Let’s say we want a focal length of -55 mm, and we choose a clear plastic of index of refraction 1.67. We need to choose either R1 or R2. Let’s choose R2 = -100 mm. Our equation becomes: (1.67 – 1)*{1/R1 + 1/(-100mm)} = 1/(-55mm) , or (.67)/R1– (.0067/mm) = -.0182/mm, or (.67)/R1 = -.0115/mm, or R1/(.67) = - 1mm/.0115 , or R1 = - .67mm/.0115 = -58.35 mm. But this can’t work since we wanted R1>0 for a contact lens design. We need to choose |R2| to be a smaller number (< .67* |f|) so that R1 can be positive. If we choose R2 = -30mm, then we get R1 = 161.4 mm.

25. Refraction and the Lens-users Eq. f > 0 s > 0 AND s > f s’> 0 AND s’> f f f s s’ Example: 1 / 5 cm = 1 / 10 cm + 1 / 10 cm

26. Refraction and the Lens-users Eq. as s gets bigger, s’gets smaller (but still need s’ > f) f f s s’ Example: 1 / 5 cm = 1 / 30 cm + 1 / 6 cm

27. Refraction and the Lens-users Eq. as s approaches infinity s’approaches f f f s s’ Example: 1 / 5 cm = 1 / 200 cm + 1 / 5.128 cm

28. Refraction and the Lens-users Eq. f > 0 s > 0 AND s > f s’> 0 AND s’> f f f s s’ Example: 1 / 5 cm = 1 / 10 cm + 1 / 10 cm

29. Refraction and the Lens-users Eq. as s gets smaller, s’gets larger f f s s’ Example: 1 / 5 cm = 1 / 6 cm + 1 / 30 cm

30. Refraction and the Lens-users Eq. as s approaches f, s’approaches infinity f f s s’ Example: 1 / 5 cm = 1 / 5.1 cm + 1 / 255 cm

31. Refraction and the Lens-users Eq. Before we see what happens when s gets smaller than f, let’s use what we already know to see how the lens will work.

32. Refraction and the Lens-users Eq. • Any ray that goes through the focal point on its way to the lens, will come out parallel to the optical axis. (ray 1) f f ray 1

33. Refraction and the Lens-users Eq. • Any ray that goes through the focal point on its way from the lens, must go into the lens parallel to the optical axis. (ray 2) f f ray 1 ray 2

34. Refraction and the Lens-users Eq. • Any ray that goes through the center of the lens must go essentially undeflected. (ray 3) object image ray 1 f f ray 3 ray 2

35. Refraction and the Lens-users Eq. • Note that a real image is formed. • Note that the image is up-side-down. object image ray 1 f f ray 3 ray 2

36. Refraction and the Lens-users Eq. • By looking at ray 3 alone, we can see by similar triangles that M = h’/h = -s’/s. object h s’ image h’<0 f s f ray 3 note h’ isup-side-down and so h’ < 0

37. Refraction and the Lens-users Eq. This is the situation when the lens is used in a camera or a projector. Image is REAL. object image ray 1 f f ray 3 ray 2

38. Refraction and the Lens-users Eq. What happens when the object distance, s, changes? object h s’ image ray 1 s h’ f f ray 3 ray 2

39. Refraction and the Lens-users Eq. Notice that as s gets bigger, s’ gets closer to f and |h’| gets smaller. object s’ h image h’ ray 1 f s f ray 3 ray 2

40. Focusing To focus a camera, we need to change s’ as s changes. To focus a projector, we need to change s as s’ changes. We do this by screwing the lens closer or further from the film (image) or slide (object). But what about the eye? How do we focus on objects that are close and then further away with our eyes? Do we screw our eyes in and out like the lens on a camera or projector?

41. Focusing But what about the eye? How do we focus on objects that are close and then further away with our eyes? Do we screw our eyes in and out like the lens on a camera or projector? - NO, instead our eyes CHANGE SHAPE and hence change f as s changes, keeping s’ the same!

42. Refraction and the Lens-users Eq. Let’s now look at the situation where s< f (but s is still positive): s f f Example: 1 / 5 cm = 1 / 4 cm + 1 / s’

43. Refraction and the Lens-users Eq. We can still use our three rays. Ray one goes through the focal point on the left side. ray 1 s f f

44. Refraction and the Lens-users Eq. Ray two goes through the focal point on the right side (and parallel to the axis on the left). ray 1 s f f ray 2

45. Refraction and the Lens-users Eq. Ray three goes through the center of the lens essentially undeflected. ray 1 h’ s f f ray 2 s’ ray 3

46. Refraction and the Lens-users Eq. Notice that: s’is on the “wrong” side, which means that s’ < 0 , and that |s’| > |s| so f > 0. ray 1 h’ s f f ray 2 s’ ray 3 Example: 1 / 5 cm = 1 / 4 cm + 1 / -20 cm

47. Refraction and the Lens-users Eq. Notice that: h’ right-side-up and so h’ > 0, M = h’/h = -s’/s . M > 0 (s’ < 0 but -s’ > 0). h’ s f f s’ ray 3

48. Refraction and the Lens-users Eq. This is the situation when the lens is used as a magnifying glass! Image is VIRTUAL. ray 1 h’ s f f ray 2 s’ ray 3

49. Refraction and the Lens-users Eq. The same lens can be used as: • a camera lens: s >> f, s > s’, M < 0, |M| < 1 • a projector lens: s > f, s’ > s, M < 0, |M| > 1 • a magnifying glass: s < f, s’ < 0, M > 0, M > 1

50. Refraction and the Lens-users Eq. Notes on using a lens as a magnifying glass: • hold lens very near your eye • want IMAGE at best viewing distance which has the nominal value of 25 cm so that s’ = -25 cm.