Physics 1230: Light and Color Ray tracing for lenses

Transcription

1 Physics 1230: Light and Color Ray tracing for lenses

2 simple lens applet

3 Lenses and Images Lenses are used in - eyeglasses magnifying glasses cameras telescopes eye lens

4 LAW OF REFRACTION Light going from a substance of small n to a substance of large n is bent TOWARD the normal Light going from a substance of large n to a substance of small n is bent AWAY from the normal normal to surface incident rays glass or water refracted rays diagram works in either direction

5 How do rays behave at spherical convex interfaces? Convex glass surfaces are focusing Concave glass surfaces are defocusing C F glass convex interface Rays parallel to the axis are deflected through the focus

6 How do rays behave at spherical convex interfaces? Convex glass surfaces are focusing F C F glass convex interface Rays parallel to the axis are deflected through the focus

7 How do rays behave at spherical convex interfaces? Convex glass surfaces are focusing C F glass convex interface Rays from the focus are made parallel to the axis

8 How do rays behave at spherical interfaces? Concave glass surfaces are defocusing F C F concave interface glass Rays parallel to the axis are deflected as if they emerged from the focus

9 How do rays behave at spherical convex interfaces? Concave glass surfaces are defocusing F C F glass concave interface Rays parallel to the axis are deflected as if they came from the focus

10 Thin Lenses

11 Ray Tracing for Thin Lenses Unlike a mirror there are two focal points of a lens, one on either side RULE #1: A ray through the center of the lens is not bent RULE #2: A ray parallel to the axis is deflected through F (or as if it came from F) RULE #3: A ray from F is deflected parallel to the axis (this rule is the reverse of #2) Notice how we approximate what should be 2 bends with 1 bend

12 Lens Simulations

13 Why are Lenses curved anyway? Why not just take two prisms stacked one above the other? Parallel rays parallel to the axis are imaged at the focus of the lens The Arabian physicist and mathematician Ibn Sahl (c.940 c.1000) used what is now known as Snell's Law to calculate the shape of lenses

14 Review: Ray Tracing for Thin Lenses Where will the images be? Which ray tracing rules can we apply?

15 Lenses - examples of Ray Tracing Parallel rays parallel to the axis are imaged at the focus Parallel rays parallel to each other but not parallel to the axis are focused in the focal plane

16 Images To locate the image - 1. Draw at least 2 rays: here is one object F F

17 Images To locate the image - 1. Draw at least 2 rays: here is a second object F F

18 Images To locate the image - 1. Draw at least 2 rays: here is a third Where is the image? Is it upright or inverted? object F F

19 Concept question Where is the base of the arrow in the image? A. Above the axis B. Below the axis C. On the axis D. There is no image object F F

20 Concept question Answer: Notice that with a convex lens, the image is flipped object F F image height S i

21 Images in THIN Lenses What rules do rays 1, 2 and 3 obey for thin lenses?

22 Imaging with lenses Concept question - if the person walks towards the lens, his image will become A. Bigger B. Smaller C. Stay the same

23 Imaging with lenses Concept question - if the person walks towards the lens, his image will become A. Bigger B. Smaller C. Stay the same

24 Imaging with lenses Concept question - if the person walks towards the lens, his image will become A. Bigger B. Smaller C. Stay the same D. Upright E. Inverted

25 Imaging with convex lenses photography case: real images (can be projected on film) real images real images magnifying glass case: virtual image real image of tree converginglenses/index.html virtual image of man

26 Virtual images in thin convex lenses - magnifying glass image object F F thin convex lens when the object is closer to the lens than the focal point, we see a magnified, virtual image beyond it

27 Virtual images in thin convex lenses - magnifying glass Object simple magnifying glass applet

28 More Ray Tracing in Lenses: Concave Lens

29 Virtual images in thin concave lenses demagnifying glass object F image F thin concave lens Is there one more ray that you can draw here? diverging lens applet

30 Magnification Magnification = Size of image = Image distance Size of object Object distance

31 Magnification Magnification = Size of image = Image distance Size of object Object distance From similar triangles S o = - S i => S i = - X i X o X i S o X o NOTE: Negative magnification means that the image is upside down object height S o image height S i x o x i

32 Magnification Magnification = Size of image = Image distance Size of object Object distance object height S o F image height S i x o x i

33 Finding images: the Thin Lens formula Say that I know F and the distance of the object from the lens Can I find out where the image will be? Yes from - 1/x o + 1/x i = 1/F => 1/x i = 1/F - 1/x o where x i = distance to image, x o = distance to object, and F = focal length

34 Example of finding images Say that F = 0.5m, and x o = 0.8m 1/x o + 1/x i = 1/F => 1/x i = 1/0.5m - 1/0.8m = 2/m /m = 0.75/m = 3/4m => x i = 1.33m

35 Raindrop lenses A water droplet can act as a convex lens. Can you see the inverted image of the house in this picture?

36 Icicle drop lens A water droplet can act as a convex lens. Can you see the inverted image of the tree in this picture?

37 A water droplet and a blade of glass can act as a retroreflector. Only one incident ray is shown, and only a few of the many diffusely reflected rays due to this one incident ray are shown. Raindrop retroreflectors

38 Fresnel lenses For lighthouses and stage lighting (spotlights) for example, to be efficient in collecting and focusing the light, one wants a lens that is large and that has a short focal length This suggests a very thick lens - but these are heavy, lossy, and are difficult to make and maintain So use a Fresnel lens instead (developed by Augustine Fresnel ) In a fresnel lens, most of the glass is removed.

39 Uses of Fresnel lenses Overhead projectors Theatre spotlights Focusing of x-rays Lighthouses making fire with a Fresnel lens

40 More uses of Fresnel lenses Traffic lights Aircraft carriers plane landing on aircraft carrier

41 Aberrations - Chromatic F blue F red Chromatic aberration happens because blue light is bent more than red light It causes color separation to appear at the edges of images F Chromatic aberration can be eliminated by using two different kinds of glass in a lens so that the effect cancels in the two different lenses

42 Aberrations - Chromatic

43 Example of Chromatic Aberration

44 Lens #1 Lens #2 Adding lenses F 1 F 2 F total Where is the combined focal length of two convex lenses A: Closer to the lens than either focus? B: Between the two focii? C: Further from the lens than either focus?

45 Formula for adding lenses F To combine lenses, add their DIOPTERS (not their focal lengths) where diopter (D) = 1/F, and F is the focal length So D total = D 1 + D 2 or 1/F total = 1/F 1 + 1/F 2

46 Example of adding lenses F Question: If F 1 is 25 cm, and F 2 is 1 m, what is F total? Answer: D 1 = 4D and D 2 = 1D => D total = D 1 + D 2 = 5D => F total = 1/5 m = 20cm

47 Another example of adding lenses Question: If F 1 is 0.5m, and F 2 is -1m, what is F total? Answer: D 1 = 2D and D 2 = -1D => D total = D 1 + D 2 = 1D => F total = 1 m

48 Spherical Aberrations in Mirrors and Lenses Spherical aberration happens because a sphere does not have a perfect focus for light rays Rays at the edge focus closer than rays from the center Spherical aberration can be eliminated by - 1. using a parabolic mirror (headlights or telescopes) 2. using an aperture to stop the edge rays

49 Spherical Aberrations in Lenses Spherical aberration happens because a sphere does not have a perfect focus for light rays A perfect lens (top) focuses all incoming rays to a single point on the optic axis, but a lens ground with spherical surfaces (bottom) focuses different rays to different points along the optic axis, depending on the radial position of each incoming ray. Rays at the edge focus closer than rays from the center

50 Aberrations - Spherical

51 Concept Question: Spherical Aberration Hubble Space Telescope Main Mirror When spherical aberration is present, what part of the image is blurred? A. Center B. Edges C. All

52 Off-Axis Aberrations Spherical and chromatic aberrations are on-axis aberrations i.e., these aberrations happen to rays that are close to the axis Off-axis aberrations include CURVATURE OF FIELD, COMA, ASTIGMATISM, and DISTORTION Curvature of field aberration CURVATURE OF FIELD: Here the image of a flat object does not lie in a plane. The center is in focus while the extremes are blurred. This aberration can be compensated for by curving the viewing screen (e.g. large TV screens or drive-in theatres)

53 Off-Axis Aberrations COMA: Coma is a modification of spherical aberration for offaxis objects. The blur due to spherical aberration moves to one side when the object is off axis, so that the image seems to have a tail (like a comet, from which this aberration is named!) Coma aberration

54 Off-Axis Aberrations: Barrel and Pincushion Distortions Barrel aberration DISTORTION: Barrel and pincushion aberrations distort the image. These aberrations can be limited by using stops to eliminate the extreme rays. Pincushion aberration