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SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

Definition 236 Let A c_

Definition 236 Let A c_ (M x 1~, ~x2"o3 - ~l*EO) be the canonical variety of an optical system. Then L is a Lagrangian variety of incident rays, the incident wavefront; and the transformed system of rays, the transformed wavefront, is the image of L under the symplectic relation A, namely A(L) c (I~,~). We show now that although the canonical variety A for reflection or refraction is the graph of a symplectomorphism, it defines completely the focusing properties of the system, i.e. the formation of new caustics during the optical transformation. Example 4.1 Reflection of a parallel beam of rays. The beam of parallel rays is given in (M,c0) by L = {(r,s) : r = (3}. Consider a mirror t ~ (cp(t), t), where cp(0) = ~p'(0) = 0, cp"(0) ~ 0. By reflection in this mirror the canonical variety A endows L with a focusing property, and produces the well known caustic (Fig. 4.5) (p(p'(l + (1),2)2 / A(L): ~,~) = ( 2cp' , t --:-kcp'2+l cp/z 1 L -- f + x v Fig.4.5 Caustic by reflection. (-3t 2 -41- +O(t3),8t3+ 0(4)) O(3),t) Remark Local genericity of the wavefront produced by L is preserved during the process of reflection because the canonical variety is the graph of a symplectomorphism, Guillemin and Sternberg [1984]. This may not be so in diffraction processes, where A is no longer the graph of a symplectomorphism, Janeczko [1987].

237 The Billiard Map as an Optical Instrument We return to the example of §1 and reinterpret it within our general formalism. Let ~ be a smooth compact convex region in ~ n, with boundary )~. Let X, X' be open domains in X situated, for example, as in Fig. 4.6. Fig. 4.6 Notation for the billiard map on a domain ft. Let ~ be a completely transparent system. Then the canonical variety A of the system, A c_ (MxlV[, rc2"o3-~1"~ ) is the graph of the identity mapping M~ ~'= ~ ~/vl. In local coordinates on X, X this identity mapping takes the following form: To the oriented line g ~ M corresponds (x, ~) ~ T*X such that x = ~ n X and is the unique element of T*xX such that I1~11 < 1 and (~,v> --- (e,v> for all v ~ TxX. Here e is the unique unit element of (IRn) * corresponding to ~, see (4.1, 4.2). The subset {(x,~) ~ T*X : x ~ X, ~ ~ T'X, ~11 < I} forms a chart on (M,¢0). The subordinate ray ~' -- ~ in the canonical variety A defines the point x' = ~ n X' and the covector ~' ~ T*x,X' such that

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