Physics 142 Lenses and Mirrors Page 1 Lenses and Mirrors Now or the sequence o events, in no particular order. Dan Rather Overview: making use o the laws o relection and reraction We will now study ormation o images by lenses and mirrors, and the use o these in simple optical instruments. For this purpose the ray approximation is suicient. A lens is a device that uses reraction to bend light rays; a mirror uses relection. The purpose in bending the rays, in most cases, is to orm an image o an object: an optical replica o the object. The lenses and mirrors we discuss will be assumed to have either plane or spherical suraces, which simpliies the geometry. For the most part, we will urther assume that the rays that are used make small angles with the symmetry axis o the device. This is the paraxial ray approximation and it allows derivation o simple ormulas or locating and describing images. Finally, we will assume that the indices o reraction o lenses are independent o the wavelength o light, ignoring the eects o dispersion. Some "aberrations" that arise rom violations o these approximations will be discussed. Mirrors: ocal points, real and virtual We consider mirrors made o a smooth conducting material (so the relectivity is close to 1) in the shape o a section o a sphere. I the mirror surace is concave (as seen rom the direction rom which the light comes), the mirror is called converging or "positive" (or reasons to be made clear); i the surace is convex, the mirror is called diverging or "negative". Shown in side view is a concave mirror, made o part o a sphere o radius R. The dotted line passing through the sphere s center C is the symmetry axis. Consider two rays incident rom the let as shown, parallel to the axis and close to it, so the angles involved in the relection are small. Ater relection the two rays cross each other at a point on the axis. This is the ocal point o the mirror. Its distance rom the mirror (the ocal length) is obtained by simple geometric arguments. The two right triangles with opposite side d give (using the paraxial ray assumption that both angles are small) tan = d/r, β tan β = d/ We see rom the drawing that β = 2. This gives us a simple ormula or : Axis C β d
Physics 142 Lenses and Mirrors Page 2 Focal length o a spherical mirror = R/2 In this case the parallel rays converge at the ocal point, which is why the mirror is called converging. Its radius R and its ocal length are assigned positive values in this case, which is why it is also called a positive mirror. Next consider a convex mirror, as shown. The center o the sphere is on the side opposite to that where the light impinges and is relected. The relected rays (on the let side o the mirror) Axis d β diverge as though they had come rom a point behind the mirror. This is the ocal point, but it is virtual. The relected light does not actually come rom this point, but an observer whose eyes and brain process the inormation rom the relected rays will interpret them as i they originated rom that point, because the rays would be the same i they had. Our sense o where an object is located comes rom the capacity o our brain to project back diverging rays to their source, whether that source is real or virtual. It is only because o other inormation we possess that we can decide whether the source is real or virtual. The same geometric argument used in the converging case leads to the same ormula or in terms o R, but here by convention we assign to both quantities negative numbers. This mirror is called negative. Because it diverges parallel incident rays, it is also called diverging. In the sign convention we use, positive distances represent "real" things, while negative distances represent "virtual" things. For mirrors, centers o curvature and ocal points in ront o the mirror are real and R and are positive; those points behind the mirror are "virtual" and R and are negative. Image ormation by mirrors: location and magniication Images are o two types: Real images. Rays rom a point in the object are converged by the optical system at a point in space, which is the corresponding point in the real image. Virtual images. Rays rom a point in the object are diverged by the optical system as though they had emanated rom a point in space, which is the corresponding point in the virtual image. To locate the image point ormed by a mirror, one uses principal rays: 1. A ray rom the object point, passing through (or toward) the center o curvature. This ray strikes the mirror at normal incidence and is relected straight back. 2. A ray parallel to the symmetry axis. For a positive mirror, this ray is relected through the real ocal point. For a negative mirror it is relected away rom the virtual ocal point. 3. A ray passing through (or toward) the ocal point, and relected back parallel to the axis. Any two o these are suicient to locate images. C
Physics 142 Lenses and Mirrors Page 3 We consider a converging mirror, with an object (represented by the blue arrow) located at a distance p rom the mirror, greater than the ocal length. The two principal rays rom the tip o the object converge to orm the tip o the image (represented by the small red arrow) located at distance q rom the mirror. Since the rays do actually converge there, this is a real image. p C q β We are using paraxial rays and small angle approximations, so we represent the mirror by a vertical straight line and draw the rays as i they impinge on that line. To make the drawing clear, the rays shown are deinitely not paraxial. Let the height o the object be h and that o the image h. Then rom the various similar right triangles in the drawing we ind ater a short calculation: Image location ormula 1 p + 1 q = 1 We also ind that h'/h = q/p. This ratio gives the lateral magniication o the image relative to the object. It is customary to deine this magniication with a negative sign to denote the act that a real image is inverted relative to the object. Thus we have Lateral magniication m = q p In the case shown, q < p so the image is smaller than the object. One can show rom the image location ormula that i is positive and p > 2 then q is positive and q < p, as in the case here. Now suppose the red arrow were the object. Then the principal rays would be the same, except reversed in direction. The green arrow would become the image. This is an example o the "principle o reversibility", which says that reversing the directions o all the rays gives another possible optical situation. In that case the object distance is between and 2, or which the image distance is greater than 2. The image is real, inverted (relative to the object) and enlarged. Things are dierent i the object is closer to the mirror than the ocal point. Shown is such a case. The rays diverge ater relection as though they had come rom the tip o the dashed arrow behind the mirror. This is a virtual image. The image distance q is now negative. The image is upright (relative to the object) and enlarged. The details can be calculated using the two ormulas given above, which are still valid. An object placed exactly at the ocal point o a positive mirror results in relected rays that are all parallel to the axis. The relector mirror in a searchlight is an example o this. C
Physics 142 Lenses and Mirrors Page 4 Conversely, an object at essentially ininite distance will produce an image at the ocal point. The relector mirrors in astronomical telescopes are examples. A negative mirror gives a virtual image o a real object, no matter where it is placed. Shown is a case. The virtual image is upright and reduced. The details can be calculated rom the ormulas, but one must remember that is negative. The image is always closer to the mirror than the object. Surveillance mirrors in shops are o this type, as are the outside right mirrors in modern automobiles. Thin lenses A lens is a device made o a transparent material with index o reraction dierent rom that o the medium rom which the light impinges. Focal points and images are produced by reraction at the suraces. Our analysis will be restricted to lenses with spherical suraces, and with thickness that are small compared to the radii o curvature o the suraces. These are called thin lenses. One deines ocal points or lenses in a way similar to that or mirrors. I, ater passing through the lens, incident rays parallel to the axis are converged at a point, then that point is the ocal point, and we have a converging or positive lens; its ocal length (the distance rom the lens to the ocal point) is positive. I, ater passing through the lens, the parallel rays diverge as though coming rom a point on the same side o the lens as the incident light, then we have a diverging or negative lens. The ocal point is where the diverging rays appear to have come rom, and the ocal length is negative. For paraxial rays one can show, using the law o reraction and small angle approximations, that the ocal length is given by the ollowing ormula: C Lens maker s ormula 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 plastic), and n 0 is the index o reraction o the transparent medium on either side o the lens (i it is air then n 0 1 ). R 1 and R 2 are the radii o the two lens suraces, or which there are sign conventions. For the ormula as given, the sign conventions are as ollows: As one ollows the light through the lens, one encounters the irst surace and then the second; i either surace is convex (bulging toward the incoming light) its radius is positive, otherwise it is negative. Various texts may use dierent conventions, and thereore have dierent signs in the ormula.
Physics 142 Lenses and Mirrors Page 5 For ordinary glass lenses in air, the ollowing are typical shapes in cross section. The signs are as ollows: 1. R 1 > 0, R 2 < 0, > 0. 2. R 2 > R 1 > 0, > 0. 3. R 1 > R 2 > 0, < 0. 4. R 1 < 0, R 2 > 0, < 0. 1 2 3 4 One sees a simple rule here: lenses that are thicker in the middle have positive ocal length; those that are thinner in the middle have negative ocal length. Shapes 2 and 3 are typical o corrective eyeglass lenses, 2 to correct arsightedness or enhance accommodation, 3 to correct nearsightedness. For cosmetic reasons shape 3 with its convex ront surace is used; in cases o extreme nearsightedness shape 4 may be required. Image ormation with lenses The procedure or locating images with lenses is similar to that or mirrors. The commonly used principal rays are: 1. A ray rom the object point to the center o the lens, where the two suraces are parallel. This ray passes essentially straight through. 2. A ray rom the object point parallel to the symmetry axis. This is reracted through the ocal point or a positive lens, or away rom it or a negative lens. 3. A ray passing through or toward a ocal point emerges parallel to the axis. Shown is a real image ormed by a positive lens, with object beyond the ocal point. p q The analysis (in the paraxial ray approximation) gives the same ormulas or location o the image and 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 an enlarged real image, the object distance must be between and 2. Most cameras orm real, reduced images o objects more distant than 2. Slide projectors orm on a distant screen a real, inverted and enlarged image o a slide between and 2.
Physics 142 Lenses and Mirrors Page 6 As with a positive mirror, an object placed closer to the lens than orms an upright, enlarged virtual image. The rays are as shown. The image distance is negative, so the image is on the same side o the lens as the object. An example is the enlarged virtual image produced by a magniying glass, to be discussed in the next notes. Like a negative mirror, a negative lens produces only virtual images o real objects, no matter where they are placed. The images are erect and reduced, located closer to the lens than either the object or the ocal point, as the diagram shows. In this case, both and q are negative. Aberrations The ormulas we have discussed are simple because o the approximations we made. Deviations rom them are to be expected in practice. They are called aberrations. Three o the most common are these: Spherical aberration. I the actual system has incident rays that are not paraxial, i.e., whose distance rom the axis is not small compared to the radii o suraces o lenses and mirrors, then our claim that all rays parallel to the axis will be brought to a single ocal point is not valid. The resulting blurring o images is called spherical aberration. It can be reduced by putting a small aperture next to the lens, permitting only paraxial rays to enter. O course this limits the amount o light admitted, and hence the brightness o the image. It also increases the importance o diraction eects, as we will see later. Astigmatism. I the optical system is not really axially symmetric, we have an aberration called astigmatism. This is a common deect o the eye. Chromatic aberration. In the case o lenses, we have ignored the slight variation o the indices o reraction with wavelength (dispersion). As a consequence o dispersion, light waves o dierent wavelengths have dierent ocal points. The resulting blurring o images is called chromatic aberration.