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

238 5 Diffraction on

238 5 Diffraction on Apertures Consider now the geometric phenomenon of diffraction on a half-plane aperture, see Janeczko [1987], Keller [1978]. Let (a,b,x,y,u,v,w) ~ F(a,b,x,y,u,v,w) be the optical distance function from the wavefront {(x,y,z) : z = cp(x,y) = ~.1 x2 + ~2xY + ~.3y 2 + O3(x,y)} in the presence of the aperture {(a,b,c) : a>0, z = mb-1}, where m>0 and (a,b) ~ ~2 parametrize the aperture. If the incident ray goes from (x,y) = (0,0) to (a,b) -- (0,0) then the transformed rays from (a,b) = (0,0) to (u,v,w) are given by (0,u,v,w) = 0, Ob where F(b,x,y,u,v,w) -- F(0,b,x,y,u,v,w), see Fig. 5.1. Fig. 5.1 half- diffr Diffraction at an edge. The distance function F defined by leads to F(a,b,x,y,u,v,w) = ((x-a) 2 + (y-b) 2 + (~p(x,y) - mb + 1)2) ½ + ((u-a) 2 + (v-b) 2 + (w-mb+l)2) a/ m2u2+ v2(m2-1) - 2mv(l+w) = 0 and v + m(l+w) < 0. These conditions define a half-cone of diffracted rays, illustrated in Fig. 5.1, compare Keller [1978]. Let L be a source of light or transformed wavefront in (M,o~). Recall the geometric construction which allows us to define the caustic or wavefront evolution in V corresponding to L. Let E be the product symplectic manifold E = (M × T'V, n2*o~ V - nl*~0) where n 1, n 2 : M x T*V -~ M, T*V are the canonical projections. We can check that I~ = graph(riM) c_ E is a Lagrangian submanifold of E, see the start of §5. Thus there

exists a local generating Morse family, say 239 K:~ k x X x V 4 ~, (~t,~,q) ~, K(gt,~,q), where T*)( is an appropriate local cotangent bundle structure (special symplectic structure, Sniatycki and Tulczyjew [1972]) on (M,t0). The transformed system of rays forms a Lagrangian subvariety of if*V, co V) given as the image = (I~ o A)(L) c_ if*V, c0 V) where I(. A c E is a composition of symplectic relations, Weinstein [1981]. Let G:~ l x X. 1~ --)IR, ('~,x,~) ~ G(~,x,,~), x, ,~ ~ ~1 n be a generating family for A c_ $~, and let F:~m×x -~ IR (~.,x) ro F(~.,x) be a generating family for L. Then the transformed Lagrangian subvariety I~ c (T'V, co V) is generated by : IR k+~+m+2n x V -~ IR F(~,,'o,~t,x,~;q) = G(v,x,~) + K(~t,~,q) + F(~.,x) where ~k+l+m+2n is a parameter space. In optical arrangements the source of light is usually a smooth Lagrangian submanifold of (U,o~). Only after the transformation process through an optical instrument does it become singular. Definition Let L c (U,c0) be an initial source variety. We define its caustic by an optical instrument A c_ 9 to be a hypersurface of V formed by two components: 1) The singular values of ~V l ~. \ Sing i~, 2) r~ V (Sing l~). Here I~ = (I(. A) (L) and Sing I~ denotes the singular locus of I~. In reflection or refraction we do not go beyond the smooth category for L, so the associated caustics in transformed wavefronts L are those realizable by smooth generic sources. Thus in what follows we are mainly interested in caustics caused by diffraction, which enrich substantially the list of optical events and complete the correspondence between singularities of functions and groups generated by reflections, see Scherbak [1988]. Diffracted rays are produced, for example, when an incident ray hits the edge of an impenetrable screen (i.e. the edge of a boundary or interface). In this case the incident ray produces infinitely many diffracted rays, which make the same angle with the edge as does the incident ray. This is the case if both incident and diffracted rays lie in the same medium, otherwise the angles between the two rays and the plane normal to the edge are related by SHell's Law. Furthermore, the diffracted ray lies on the opposite side of the normal plane from the incident ray. The rays that are contained in the plane of the aperture are called rays at infinity.