Understanding Lenses: Formation and Functions

1. Chapter 9Optics(Section 4)

2. 9.4 Lenses and Images • In Section 9.2, we described how curved mirrors are used in astronomical telescopes and other devices to redirect light rays in useful ways. • Microscopes, binoculars, cameras, and many other optical instruments use specially shaped pieces of glass called lenses to alter the paths of light rays. • As was the case with mirrors, the key to making glass or other transparent substances redirect light is the use of curved surfaces (interfaces) rather than flat (planar) ones.

3. 9.4 Lenses and Images • Suppose we grind a block of glass so that one end takes the shape of a segment of a sphere as shown in cross section in the figure.

4. 9.4 Lenses and Images • Let parallel rays strike the convex spherical surface at various points above and below the line of symmetry (called the optical axis) of the system. • If one applies the law of refraction at each point to determine the angle of refraction of each ray, the results shown in the figure. • In particular, rays traveling along the optical axis emerge from the interface undeviated.

5. 9.4 Lenses and Images • Rays entering the glass at points successively above or below the optical axis are deviated ever more strongly toward the optical axis. • The result is to cause the initially parallel bundle of rays to gradually converge together—to become focused—into a small region behind the interface. • This point is called the focal point and is labeled F in the figure.

6. 9.4 Lenses and Images • The figure shows the behavior of parallel rays refracted across a spherical interface that, instead of bowing outward, curves inward. • In this case, the emergent rays diverge outward as though they had originated from a point F’to the left of the interface.

7. 9.4 Lenses and Images • The ability to either bring together or spread apart light rays is the basic characteristic of lenses, be they camera lenses, telescope lenses, or the lenses in human eyes.

8. 9.4 Lenses and Images • In most devices the light rays must enter and then leave the optical element (lens) that redirects them. • Common lenses have two refracting surfaces instead of one, with one surface typically in the shape of a segment of a sphere and the second either spherical as well or flat (planar).

9. 9.4 Lenses and Images • The effect on parallel light rays passing through both surfaces is similar to that in the previous examples with one refracting surface. • A converging lens causes parallel light rays to converge to a point, called the focal point of the lens.

10. 9.4 Lenses and Images • The distance from the lens to the focal point is called the focal length of the lens. • A more sharply curved lens has a shorter focal length. • Conversely, if a tiny source of light is placed at the focal point, the rays that pass through the converging lens will emerge parallel to each other. This is the principle of reversibility again.

11. 9.4 Lenses and Images • A diverging lens causes parallel light rays to diverge after passing through it. • These emergent rays appear to be radiating from a point on the other side of the lens. • This point is the focal point of the diverging lens.

12. 9.4 Lenses and Images • The distance from the lens to the focal point is again called the focal length, but for a diverging lens it is given as a negative number, –15 centimeters, for example. • If we reverse the process and send rays converging toward the focal point into the lens, they emerge parallel.

13. 9.4 Lenses and Images • For both types of lenses there are two focal points, one on each side. • Clearly, if parallel light rays enter a converging lens from the right side in the figure, they will converge to the focal point to the left of the lens.

14. 9.4 Lenses and Images • Whether a lens is diverging or converging can be determined quite easily: • If it is thicker at the center than at the edges, it is a converging lens; • if it is thinner at the center, it is a diverging lens.

15. 9.4 Lenses and ImagesImage Formation • The main use of lenses is to form images of things. • First, let’s consider the basics of image formation when a symmetric converging lens is used. • Our eyes, most cameras (both still and video), slide projectors, movie projectors, and overhead projectors all form images this way.

16. 9.4 Lenses and ImagesImage Formation • The figure illustrates how light radiating from an arrow, called the object, forms an image on the other side of the lens. • One practical way of demonstrating this would be to point a flashlight at the arrow so that light would reflect off the arrow and pass through the lens. • The image could be projected onto a piece of white paper placed at the proper location to the right of the lens.

17. 9.4 Lenses and ImagesImage Formation • Although each point on the object has countless light rays spreading out from it in all directions, it is simpler to consider only three particular rays from a single point—the arrow’s tip. • These rays are called the principal rays. 1. The ray that is initially parallel to the optical axis passes through the focal point (F) on the other side of the lens.

18. 9.4 Lenses and ImagesImage Formation 2. The ray that passes through the focal point (F’) on the same side of the lens as the object emerges parallel to the optical axis. 3. The ray that goes exactly through the center of the lens is undeviated because the two interfaces it encounters are parallel.

19. 9.4 Lenses and ImagesImage Formation • Note that the image is not at the focal point of the lens. Only parallel incident light rays converge to this point (assuming an ideal lens).

20. 9.4 Lenses and ImagesImage Formation • We could draw principal rays from each point on the object, and they would converge to the corresponding point on the image. • This kind of image formation occurs when you take a photograph or view a slide on a projection screen.

21. 9.4 Lenses and ImagesImage Formation • In the latter case, light radiating from each point on the slide converges to a point on the image on the screen. • Note that the image is inverted (upside down). • That’s why you load slides in the tray or carousel upside down if you want their images to be right side up.

22. 9.4 Lenses and ImagesImage Formation • The distance between the object and the lens is called the object distance, represented by s, and the distance between the image and the lens is called the image distance, p. • By convention (with the light traveling from left to right), s is positive when the object is to the left of the lens, and p is positive when the image is to the right of the lens.

23. 9.4 Lenses and ImagesImage Formation • If we place the object at a different point on the optical axis, the image would also be formed at a different point. In other words, if s changes, then so does p. • Using a lens with a different focal length for fixed s would also cause p to change. • For example, the image would be closer to the lens if the focal length were shorter.

24. 9.4 Lenses and ImagesImage Formation • The following equation, known as the lens formula, relates the image distance p to the focal length f and the object distance s.

25. 9.4 Lenses and ImagesExample 9.3 • In a slide projector, a slide is positioned 0.102 meter from a converging lens that has a focal length of 0.1 meter. • At what distance from the lens must the screen be placed so that the image of the slide will be in focus?

26. 9.4 Lenses and ImagesExample 9.3 • The screen needs to be placed a distance p from the lens, where p is the image distance for the given focal length and object distance. So:

27. 9.4 Lenses and ImagesExample 9.3 • If the slide-to-lens distance is increased to 0.105 meter (about a 3% change), the distance to the screen (p) would have to be reduced to 2.1 meters (more than a factor of 2.4 reduction in distance). • If the lens is replaced by one that has a shorter focal length, the distance to the screen would have to be reduced as well.

28. 9.4 Lenses and ImagesImage Formation • The images formed in the manner just described are called real images. • Such images can be projected onto a screen. • We see the image on the screen because the light striking the screen is diffusely reflected to our eyes.

29. 9.4 Lenses and ImagesImage Formation • A simple magnifying glass is a converging lens, but the image that it forms under normal use is not a real image—it can’t be projected onto a screen. • We see the image by looking into the lens, just as we see a mirror image by looking into the mirror. • This type of image is called a virtual image, • similar to the image formed by a concave mirror

30. 9.4 Lenses and ImagesImage Formation • The figure shows how an image is formed in a magnifying glass. • In this case, the object is between the focal point F’ and the lens, so the object distance s is less than the focal length f of the lens.

31. 9.4 Lenses and ImagesImage Formation • Note that the image is enlarged and that it is upright. • It is also on the same side of the lens as the object, which means that p is negative.

32. 9.4 Lenses and ImagesExample 9.4 • A converging lens with focal length 10 centimeters is used as a magnifying glass. • When the object is a page of fine print 8 centimeters from the lens, where is the image?

33. 9.4 Lenses and ImagesExample 9.4 • The negative value for p indicates that the image is on the same side of the lens as the object. • Therefore, it is a virtual image and must be viewed by looking through the lens.

34. 9.4 Lenses and ImagesImage Formation • A virtual image is also formed when you look at an object through a diverging lens. • In this case, the image is smaller than the object, as it is with a convex mirror.

35. 9.4 Lenses and ImagesMagnification • One of the most useful properties of lenses is they can be used to produce images that are enlarged (larger than the original object) or reduced (smaller than the original object). • In either case, the magnification, M, of a particular configuration is the height of the image divided by the height of the object.

36. 9.4 Lenses and ImagesMagnification • If the image is twice the height of the object, the magnification is 2. • If the image is upright, the magnification is positive. • If the image is inverted, the magnification is negative (because the image height is negative).

37. 9.4 Lenses and ImagesMagnification • The magnification that one gets with a particular lens changes if the object distance is changed. • Because of the simple geometry, the magnification also can be written in an alternate, equivalent form as minus the image distance divided by the object distance.

38. 9.4 Lenses and ImagesMagnification • From this we can conclude that: • If p is positive (image is to the right of the lens and real), M is negative; the image is inverted.

39. 9.4 Lenses and ImagesMagnification • If p is negative (image is to the left of the lens and virtual), M is positive; the image is upright.

40. 9.4 Lenses and ImagesExample 9.5 • Compute the magnification for the projector in Example 9.3 and for the magnifying glass in Example 9.4. • In the first case in Example 9.3, s = 0.102 meters and p = 5.1 meters. • Therefore:

41. 9.4 Lenses and ImagesExample 9.5 • The image is 50 times as tall as the object, but it is inverted (because M is negative). • A slide that is 35 millimeters tall has an image on the screen that is 1,750 millimeters (1.75 meters) tall. • When s is 0.105 meters, p is 2.1 meters, and the magnification is –20.

42. 9.4 Lenses and ImagesMagnification • In Example 9.4, s = 8 cm and p = –40 cm. Consequently: • The image of the print seen in the magnifying glass is five times as large as the original, and it is upright (because M is positive).

43. 9.4 Lenses and ImagesMagnification • Telescopes and microscopes can be constructed by using two or more lenses together. • The figure shows a simple telescope consisting of two converging lenses.

44. 9.4 Lenses and ImagesMagnification • The real image formed by lens 1 becomes the object for lens 2. • The light that could be projected onto a screen to form the image for lens 1 simply passes on into lens 2.

45. 9.4 Lenses and ImagesMagnification • In essence, lens 2 acts as a magnifying glass and forms a virtual image of the object. • In this telescope, the image is magnified but inverted. • Replacing lens 2 with a diverging lens yields a telescope that produces an upright image.

46. 9.4 Lenses and ImagesAberrations • In real life, lenses do not form perfect images. • Suppose we carefully apply the law of refraction to a number of light rays, all initially parallel to the optical axis, as they pass through a real lens that has a surface shaped like a segment of a sphere.

47. 9.4 Lenses and ImagesAberrations • We would find that the lens exhibits the same flaw we saw in Section 9.2 with spherically shaped curved mirrors: spherical aberration. • Rays striking the lens at different points do not cross the optical axis at the same place.

48. 9.4 Lenses and ImagesAberrations • In other words, there is no single focal point. • This causes images formed by such lenses to be somewhat blurred. • Lens aberrations of this type can be corrected, but this process is complicated and often necessitates the use of several simple lenses in combination.

49. 9.4 Lenses and ImagesAberrations • Another type of aberration shared by all simple lenses even when used under ideal conditions is chromatic aberration. • A lens affected by chromatic aberration, when illuminated with white light, produces a sequence of more or less overlapping images, varying in size and color.

50. 9.4 Lenses and ImagesAberrations • If the lens is focused in the yellow-green portion of the EM spectrum where the eye is most sensitive, then all the other colored images are superimposed and out of focus, giving rise to a whitish blur or fuzzy overlay. • For a converging lens, the blue images would form closer to the lens than the yellow-green images, whereas the reddish images would be brought to a focus farther from the lens than the yellow-green ones.