Magnification, stops, mirrors More geometric optics D. Craig 2005-02-25 Transverse magnification Refer to figure 5.22. By convention, distances above the optical axis are taken positive, those below, negative. So in the figure y o > 0 and y i < 0. This an inverted image. If y i > 0 when y o > 0, the image is erect or right-side-up. The lateral or transverse magnification is M T y i y o = s i Positive M T means an erect image, negative means inverted. s i and are positive for real objects and images, so all real images formed by a single thin lens will be inverted.. Important note M T is the transverse magnification for the image formed by a single lens. Do not confuse it with angular magnification, which is the usual magnification referred to in describing telescopes, microscopes, etc. See p. 220-22, eq. 5.84, etc. for this. 2 Newtonian form for thin lenses This is the form x o x i = f 2 where x o, x i are the distances from the respective focal points to the object and image. Be careful of signs. The object and image must be on opposite sidef their respective focal points.
3 Combining thin lenses The basic procedure here is: Find the image position for the first lens. Use the first lens image abject for the second, find the image for the pair. Continue as necessary. This is best shown with examples. See figure 5.28 (close lenses) and 5.30 (well-separated lenses) The vergence method noted earlier in the class is also a good way to approach this conceptually, and can be used for thick lenses by considering the dioptric power of each surface (see below). 3. Thin lenses in contact If thin enough on the scale of the focal lengths that we can ignore the separation of the centers, then the focal length f of two in contact is f = f + f 2 /f iften referred to as the Dioptric power D = /f of a lens and then D = D + D 2. This is common practice among opticians. Dioptric power is commonly measured in inverse meters (diopters). 4 Aperture and field stops Aperture stop (A. S.): any element which determines the amount of light reaching the image. Can be the edge of a lens, inserted diaphragm, etc. Field stop (F. S.): the element limiting the size or angular extent of the object that can be imaged by the system determines the field of view. See fig. 5.33 for a simple example. 5 Entrance and exit pupils Pupils are imagef the aperture stop. Entrance pupil: image of the A. S. seen from an axial object point through all lenses proceeding the stop. Determines the cone of light entering a system. 2
Exit pupil: image of the A. S. seen from an axial point on the image plane through any intermediate lenses. Determines the cone of light leaving a system. These are shown in figs. 5.34 36. E x p marks the center of the exit pupil, E n p the center of the entrance pupil. 5. Practicalities If it s unclear which element is the aperture stop, each element must be imaged by the remaining elements to the left till you get the smallest entrance pupil. Separated lenseften restrict the cone of rayff-axis leads to vignetting: the dimming of edge of field of view. See fig. 5.37. Large exit pupil easy to look through and find images by eye. Eye relief. Large entrance pupil wide FOV, bright image. 6 f/#, F-number Recall that the transverse magnification M T f/x o. So the area of an image will go as f 2. If the entrance pupil has diameter D, then the brightness at the image plane (D/f) 2. D/f is the relative aperture, and its inverse f/# = f D is the f-number. Sometimes called the speed of a lens in photography (lower being faster). See figs. 5.38, 39. 7 Mirrors and sign conventions Consult tables 5.4 and 5.5 for the sign conventions for mirrors. Many are reversed because for example, with propagation from the left, real images appear to the left of the vertex for concave mirrors, which focus similarly to convex lenses. When you look in a plane (flat) mirror at yourself, you see a virtual image, as it is behind the mirror. 7. Planar mirrors These are pretty familiar, but examine figures 5.40 43 carefully. seeing the rays traced to understand appearances in plane mirrors It is worth 3
7.2 Aspherical mirrors Most important are paraboloidal mirrors (fig. 5.45), which bring parallel rays to a focus if concave. Spherical mirrors can approximate them well for certain conditions that we ll examine. Shallow parabolic surfaces can be made by grinding a spherical surface randomly and then correcting the outer parts by more systematic grinding. A single paraboloidal mirror can act as a telescope with an eyepiece, or directly image onto a detector. Construction is relatively straightforward for radio and microwave dishes. Hyperboloid and ellipsoid mirrors produce perfect imagery between conjugate foci, and are often useful in compound systems. Manufacture is difficult, but increasingly common. 7.3 Spherical mirrors If the optical axis is along x and y is distance from the axis, a cross-section of a sphere of radius R is (see fig. 5.49) y 2 + (x R) 2 = R 2. This can be solved for x and expanded in a binomial series: x = y2 2R + y4 2 2!R 3 + 3y6 2 3 3!R 5 + Compare this with a formula for a parabola in the same situation: y 2 = 4fx. If we let 4f = 2R, i.e. f = R/2, the first term in the series is the parabolic part, the rest the deviation from the parabola. The deviation in x from a perfect parabola for a spherical mirror of radius R is thus: x = y4 8R 3 + y6 6R 5 + So, close to the axis, (y R) this deviation will be insignificant. In the paraxial region parabolic and spherical mirrors are indistinguishable. What s a practical criterion? In telescope building one typically decides that a spherical mirror of say, about f/8 will be just as good as a parabolic if x λ 8 for outer partf the mirror. These values are just an example. The general idea is that a deviation of a small fraction of the wavelength may be ignorable. This principle is used in construction of long-focus telescope mirrors, etc. 4
7.4 Mirror formulas (spherical) If, s i are the object and image distances, and R is the radiuf curvature of a mirror (fig. 5.50), then + s i = 2 R. Concave mirrors have R < 0, convex R > 0. We also find by letting or s i that the focal length is given by + s i = f = 2 R. So, f = R 2, which is simpler than lenses! 7.5 Useful examples in the figures Figures 5.52 and 5.53 show how to trace rays appropriately with spherical mirrors. 5.54 shows a real image from a concave mirror and a virtual image from a convex mirror. 5.55 shows the various possibilities for a concave mirror, which should be compared directly to fig. 5.24 (thin positive lens). This example shows how the sign conventions switch around for mirrors vs. lenses. 5