5 years ago

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

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

232 dimX = 4: A1,

232 dimX = 4: A1, A2,A3±,A4,D4±,A5±,D5 +. + + dim X = 5: A 1, A 2, A3±, A 4, D4±, A5±, D5±, A 6, D6-, E6-. Illustrations of the corresponding caustics - critical values of the Lagrange projections in ~13 - can be found in Arnold etal. [19851. In general, the situation is much more complicated. Functional moduli appear, and the normal forms for generic germs of Lagrangian submanifolds are known only for dim X < 10, see Zakalyukin [1976]. The theory of singularities of Lagrange projections L qT*X-~ X with singular L was recently formulated in Givental [1988], where their use in general symplectic geometry is also described. 4 Action of a General Optical Instrument Let V ~- ~3 be the configuration space (with refractive index V ~ 1) of geometrical optics. The associated phase space (M,0~), i.e. the space of rays, is given by the standard symplectic reduction ~M : H'1(0) ~ M ~- T*S 2 , where the hypersurface H1(0) is described by the Hamiltonian H: T*V ~ ~i, H(p,q) = ½(llplt 2-1). In standard (p,q)-coordinates on (T'V, ~V) and Darboux coordinates (r,s) on (M,c0) we can write and (r,s) = ~M I H-I(0) (P2,P3 ; ql,q2,q3) qlP2 qlP3 = (P2,P3 ; q2 2~ ,q3 2 ~ ) (4.1) co V 1 H-l(0) = X'MOO. In the chart U = {(p,q) : Pl > 0} on M, to each point (r,s) ~ M n U we can uniquely associate the corresponding ray r 2 (ql,q2,q3) = (0,Sl,S2) + t (1, rl , / By (4.2) we can translate concrete optical problems into questions about the Lagrangian varieties of the phase space (M,m) and conversely. Phase Space for the General Optical System Let (P,to), ~,03) be symplectic manifolds of optical rays in homogeneous media, i.e. open subsets of the phase space of all rays in gl 3. Usually these manifolds respectively denote the incident rays and the transformed rays (by the optical instrument), Janeczko [1987] and Luneburg [1964]; see Fig. 4.2.

233 Definition The phase space of an optical instrument is the symplectic manifold $> = (P x 1 g , x2"03 - ~1"¢0), where x 1, x2 : P x P -~ P, P are the canonical projections. Fig. 4.1 Coordinates on the space of rays. Fig. 4.2 (P,0~) S q2 incident ...,...-< A general optical system. qt ~f ,,¢.s transforme~,, J ,7o) ~ geodesics ~ ~ qt in (W,g) The process of optical transformation, say reflection, refraction or diffraction, of the incident rays is governed by the corresponding Lagrangian subvariety of (symplectic relation from (P,o~) to (P,oS)) called the canonical variety of the system, Janeczko [1987]. Refraction in an lnhomogeneous Optical Medium We assume the following refraction coefficient in ~ 3: See Fig. 4.2. Here fl for q ~ ~I3\W ~(q) = ~'~(q) for q ~ W W = {q~ ~13 :q'l

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