Geometrical optics. Consider a thin biconvex lens with two spherical faces. AVThis is a

Chapter 2, Problem 108

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Geometrical optics. Consider a thin biconvex lens with two spherical faces. AVThis is a good model for the lens of the human eye and for the lenses used in many optical instruments, such as reading glasses, cameras, microscopes, and telescopes, j The line through the centers of the spheres defining the two faces is called the optical axis of the lens.Optical axisCenter of sphere defining the left face/Center of sphere defining the right faceIn this exercise, we learn how we can track the path OE a ray of light as it passes through the lens, provided that the following conditions are satisfied: The ray lies in a plane with the optical axis. The angle the ray makes with the optical axis is si d uwi small To keep track of the ray, we introduce two reference planes perpendicular to the optical axis, to the left and to the right of the lens.plane planeWe can characterize the incoming ray by its slope m and its intercept x with the left reference plane. Likewise, we characterize the outgoing ray by slope n and intercept y.We want to know how the outgoing ray depends on the incoming ray, that is, we are interested in the transformationT: I p2. JC y > m nWe will see that T can be approximated by a linear transformation provided that m is small, as we assumed. To study this transformation, we divide the path of the ray into three segments, as shown in the following figure: We have introduced two auxiliary reference planes, directly to the left and to the right of the lens. Our transformationX y > m ncan now be represented as the composite of three simpler transformations:V w y > > m n nFrom the definition of the slope of a line we get the relations v = jc + Lm and y = w + Rn.It would lead us too far into physics to derive a formula for the transformationV W y m nhere.12 Under the assumptions we have made, the transformation is well approximated byW 1 0" V n -k 1 mfor some positive constant k (this formula implies that w = v). x y The transformation > is m n matrix product is represented by the1 L 0 11 - Rk -kL + R - k L R 1 -kLFocusing parallel rays. Consider the lens in the human eye, with the retina as the right reference plane. In an adult, the distance R is about 0.025 meters (about 1 inch). The ciliary muscles allow you to vary the shape of the lens and thus the lens constant k, within a certain range. What value of k enables you to focus parallel incoming rays, as shown in the figure? This value of k will allow you to see a distant object clearly. (The customary unit of measurement for k is 1 diopter = r ^ . ) b. What value of k enables you to read this text from a distance of L = 0.3 meters? Consider the following figure (which is not to scale):c. The telescope. An astronomical telescope consists of two lenses with the same optical axis.Find the matrix of the transformationX y m nin terms of k\, k2, and D. For given values of k\ and &2, how do you choose D so that parallel incoming rays are converted into parallel outgoing rays? What is the relationship between D and the focal lengths of the two lenses, \/k\ and i/k2?