Physics 54. Lenses and Mirrors. And now for the sequence of events, in no particular order. Dan Rather

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1 Physics 54 Lenses and Mirrors And now or the seuence o events, in no articular order. Dan Rather Overview We will now study transmission o light energy in the ray aroximation, which assumes that the energy travels in straight lines excet when there is relection, reraction., or intercetion by an obstacle. This aroximation works well when the sizes o aertures and obstacles are large comared to the wavelength o the light. Our main interest will be in ormation o images by lenses and mirrors. We will also study the use o these in simle otical instruments. The lenses and mirrors will generally be assumed to have either lane or sherical suraces. This simliies the geometry. For the most art, we will urther assume that the rays o light make small angles with the symmetry axis o the device. This araxial ray aroximation allows derivation o simle ormulas or locating and describing images. Finally, we will assume that the indices o reraction o lenses are indeendent o the wavelength o light, ignoring the eects o disersion. Some "aberrations" that arise rom violations o these aroximations will be discussed briely. Mirrors We consider mirrors made o a conducting material (so the relection is essentially 100%) in the shae o art o a shere. I the mirror surace is concave, the mirror is called converging or "ositive" (or reasons to be made clear); i the surace is convex, the mirror is called diverging or "negative". onsider irst a concave mirror, shown in a side view. The line along the shere's diameter is the symmetry axis. We consider two incident rays, arallel to the axis and close to it, so that the angles o incidence and relection are small.! Point is the center o the sherical surace, which has radius R. Ater relection the two rays cross each other at a oint on the axis. This is the ocal oint o the mirror. Its distance rom the mirror (the ocal length) is obtained Axis! " d PHY 54 1 Lenses and Mirrors

2 by some simle geometric arguments. The two right triangles with oosite side d give (using the araxial ray assumtion that both angles are small)! " tan! = d/r, # " tan # = d/ The angle o relection is eual to the angle o incidence (α), so β = 2α. Thus we ind a simle ormula or : Focal Length o a Mirror = R / 2 In this case the arallel rays converge to a ocus, which is why the mirror is called converging. Its radius R and its ocal length are assigned ositive values in this case, which is why it is also called a ositive mirror. Next consider a convex mirror, as shown. The center o the shere is on the side oosite to that where the light iminges and is relected. The relected rays diverge as though they had come rom a oint behind the mirror. This ocal oint is virtual. Light does not actually come rom this aarent source behind the mirror, but the brain o an observer viewing the relected rays will interret them as though they originated rom that oint. Axis Our sense o where an object is located comes rom the caacity o our brain to roject back diverging rays to their source, whether that source is real or virtual. The same argument used above leads to the same ormula or in terms o R. But because o the virtual nature o this ocal oint, is deined to be negative, and R is corresondingly negative. This mirror is called negative. Because it diverges arallel incident rays, it is also called diverging. In the sign convention widely used in geometric otics, ositive distances reresent "real" things, while negative distances reresent "virtual" things. For mirrors, centers o curvature and ocal oints in ront o the mirror are real and R and are ositive; centers o curvature and ocal oints behind the mirror are "virtual" and R and are negative.!! d " PHY 54 2 Lenses and Mirrors

3 Image Formation by Mirrors Images are o two tyes: Real images. Rays rom a oint in the object are converged by the otical system at a oint in sace, which is the corresonding oint in the real image. Virtual images. Rays rom a oint in the object are diverged by the otical system as though they had emanated rom a oint in sace, which is the corresonding oint in the virtual image. To locate the image oint ormed by a mirror, one uses two or three rincial rays: A ray rom the object oint, assing through or toward the center o curvature. This ray strikes the mirror at normal incidence and is relected straight back. A ray arallel to the symmetry axis. For a ositive mirror, this ray is relected through the real ocal oint. For a negative mirror it is relected away rom the virtual ocal oint. A third rincial ray, assing through or toward the ocal oint, and emerging arallel to the axis, can also be used. We consider a converging mirror, with an object (reresented by the uright arrow) located at a distance rom the mirror greater than the ocal length. The two rincial rays rom the ti o the object converge to orm the ti o the image (reresented by the small! inverted arrow). Since the rays do actually " converge, this is a real image. The distance o the object rom the mirror (the object distance) is denoted by ; the distance o the image rom the mirror (the image distance) is denoted by. Dierent textbook authors use dierent notations or these two distances. None are standard. Using the right triangles ormed with the two small angles shown, one inds ater a short calculation that Image Location Formula = 1 Let the height o the object be h and that o the image h'. Then one inds that h'/h = /. This ratio gives the "magniication" o the image relative to the object. It is customary to deine the magniication with a negative sign to denote the act that a real image is inverted relative to the object. Thus we have PHY 54 3 Lenses and Mirrors

4 Lateral Magniication m =! In the case shown, < so the image is smaller than the object. (This is always the case when the image distance is greater than 2.) Suose the inverted arrow were the object. Then the rays would be the same, excet reversed in direction. The uright arrow would then become the image. In this case, where the object distance is between and 2, the image distance is greater than 2. The image is real, inverted (relative to the object) and enlarged. This is an examle o the useulness o the "rincile o reversibility", which says that reversing the directions o all rays gives another ossible otical situation. Things are dierent i the object is closer to the mirror than the ocal oint. Shown is such a case. The rays diverge ater relection as though they had come rom the ti o the dashed arrow. This is the virtual image. The image distance is negative, because the image is behind the mirror. The image is erect (relative to the object) and enlarged. The details can be calculated using the ormulas given above. An object laced exactly at the ocal oint o a ositive mirror results in relected rays that are all arallel to the axis. The relector mirror in a searchlight is an examle o this. onversely, an object at essentially ininite distance will roduce an image at the ocal oint. The relector mirrors in astronomical telescoes are examles. A negative mirror always gives a virtual image o a real object. Shown is a case. The virtual image is erect and reduced. The details can be calculated rom the ormulas, but one must remember that is negative. Surveillance mirrors in shos are o this tye, as are the outside right mirrors in modern automobiles. Note that the image is always closer to the mirror than the object. Thin Lenses Lenses are systems made o a transarent material; their urose is to maniulate light by reraction, usually to orm images. PHY 54 4 Lenses and Mirrors

5 Our analysis will be restricted to lenses with sherical suraces, and with thickness that are small comared to the radii o curvature o the suraces. These are thin lenses. One deines ocal oints or lenses in a way similar to that or mirrors. I, ater assing through the lens, incident rays arallel to the axis are converged at a oint, then that oint is the ocal oint, and we have a converging or ositive lens; its ocal length (the distance rom the lens to the ocal oint) is ositive. I, ater assing through the lens, the arallel rays diverge as though coming rom a oint on the same side o the lens as the incident light, then we have a diverging or negative lens; the ocal length is negative. For araxial rays one can show, using the law o reraction and small angle aroximations, that the ocal length is given by the ollowing ormula: Lens Maker s Formula 1 = " n % $! 1 # n ' 0 & ( " 1! 1 % $ # R 1 R ' 2 & Here n is the index o reraction o the substance rom which the lens is made (usually glass or lastic), n 0 is the index o reraction o the transarent medium on either side o the lens (usually air, or which n 0 = 1 ). R 1 and R 2 are the radii o the two lens suraces, or which there are sign conventions. As one imagines light entering the lens, R 1 is the radius o the irst surace encountered, while R 2 is the radius o the other surace. These numbers are ositive i the suraces are convex (as the light iminges on them); they are negative i the suraces are concave. For ordinary glass lenses in air, the ollowing shows tyical tyes > 0 < 0 > 0 R 1 > 0 R 2 < 0 R 1 < 0 R 2 > 0 R 2 > R 1 > 0 One sees a simle rule here: lenses that are thicker in the middle have ositive ocal length; those that are thinner in the middle have negative ocal length. PHY 54 5 Lenses and Mirrors

6 Image Formation with Lenses The rocedure or locating images with lenses is similar to that or mirrors. The commonly used rincial rays are: A ray rom the object oint to the center o the lens, where the two suraces are arallel. This ray asses through essentially without delection. A ray rom the object oint arallel to the axis. This is reracted through the ocal oint or a ositive lens, or away rom it or a negative lens. A ray assing through or toward a ocal oint emerges arallel to the axis. This is another rincial ray. Shown is a real image ormed by a ositive lens, with object beyond the ocal oint. The analysis (in the araxial ray aroximation) gives the same ormulas or location o the image (and or lateral magniication) as we had or mirrors. In the case shown above, the object distance is greater than 2 so the image distance is less than 2, and the image is real, inverted and reduced. (To orm a real and enlarged image, the object distance must be between and 2.) Most cameras orm real, reduced images o objects more distant than 2. Slide rojectors roduce real and enlarged images on a distant screen, o objects between and 2. As with the ositive mirror, an object laced closer to the lens than will orm a virtual image. The rays are as shown, orming an uright, enlarged virtual image. PHY 54 6 Lenses and Mirrors

7 The image is erect and enlarged. An examle o this is roduction o an enlarged virtual image by a magniying glass. Like negative mirrors, negative lenses roduce only virtual images o real objects. The images are erect and reduced, located between the object and the lens, as the ollowing diagram shows. In this case, both and are negative. Aberrations The ormulas we have discussed are simle because o the aroximations we made. Deviations rom them are to be exected, and do occur. They are called aberrations. Some o the aberrations have to do with inadeuacy o the aroximations used in deriving our simle ormulas. Three o the most common roblems are these: I the actual system has incident rays that are not araxial, i.e., whose distance rom the axis is not small comared to the radii o suraces o lenses and mirrors, then our claim that all rays arallel to the axis will be brought to a single ocal oint is not valid. The resulting blurring o images is called sherical aberration. It can be reduced by utting a small aerture next to the lens, ermitting only araxial rays to enter. O course this limits the amount o light admitted, and hence the brightness o the image. It also increases the eects o diraction, as we will see later. I the otical system is not really axially symmetric, we have an aberration called astigmatism. This is a common deect o the eye. In the case o lenses, we have ignored the slight variation o the indices o reraction with wavelength (disersion). As a conseuence o disersion, light waves o dierent wavelengths have dierent ocal oints. The resulting blurring o images is called chromatic aberration. Proessional otical systems oten use multi-element systems to correct or aberrations. PHY 54 7 Lenses and Mirrors