SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)
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.
Lecture Notes in Mathematics Editor
Editors Mark Roberts Ian Stewart Ma
Co~e~s P.J. Aston, Scaling laws and
Scaling Laws and Bifurcation P.J.As
where = It follows immediately from
esults for unitary representations
and T are orthogonal representation
which is also a subgroup of F. Clea
Proof 11 The equivalent result that
13 such that the Equlvariant Branch
]5 since h is orthogonal. As b, e,
17 and from Lemma 4.2, hj : X ~k --
]9 in a bifurcating branch of solut
21 Duncan, K. and Eilbeck, 3. C. (1
23 the symmetry arises naturally fr
25 the traction problem in nonlinea
27 First we dispose of the case k =
29 The symbols 0 ..... 4 indicate r
31 In terms of the radially project
S I --U 33 A Figure 4 .. • . ".
35 k z 4 I For k s 6 the ideas abov
37 [GG] Golubitsky, M. and Guillemi
39 small equivariant perturbations
41 It may happen that an obstructio
, 43 0(3) {yo} {xO'Yo, Y2+Y-2} D 2
45 stability requires order 5 (Golu
3.1. Phase portrait in FixfD2~.Z ~
49 (11) kl=Z,l+- "-c+ c2~2 2 anothe
51 When c=0, the eigenvalue in the
53 Remark 2. This heteroclinic cycl
55 Armbruster et al. [1988]. This w
57 Notes: 1) each picture shows pro
~< o( 59 i o( o(" ~X1r IX~£~ Figur
¢* 1 61 T ........... "--7 y¢, --
Boundary Conditions as Symmetry Con
65 We illustrate this point in the
U T(u) 67 m=2 -~ ................ 0
69 be the reduced bifurcation equat
71 3 The Couette-Taylor Experiment
73 00 u(x,y) = v(x,y) = ~-x (x,y) =
75 (dS) F of S at F has an eigenval
77 experimental geometry. 'Upper bo
79 M.G.M. Gomes [1989]. Steady-stat
81 To describe the results, we supp
83 The author's work on these quest
85 certain elements of G may interc
87 isolated singularity at 0. We sa
in particular, Zp~ = 1 + ~ iii) Vp
91 Similarly we let X(G) denote the
93 F I Fix(G') x ~ : Fix(G') x $t -
Example 4.5: 95 Let G = Z/mE act on
97 Thus, we see that X q = 1 + 4V.
99 Q(F1) = Cx,~/I(F1) = Cx,~/I(F 0)
Corollary 7 101 For i) we know by t
103 generator for m A, the module o
105 D3 On the number of branches fo
On a Codimension-four Bifurcation O
109 Setting x = u3 and introducing
111 not matter. We choose b > 0. Th
113 SLot x SLs Figure 2: Phase port
SLs(H} H.(SL s) SNs I SN~ (~6) 115
iii 1 SLo/" SNo ,y X 117 i 11 21 .
119 extent the dynamics is influenc
121 [12] J. Guckenhelmer, SIAM J. M
123 produces a continuous function.
is generated by 125 V(A~) = {f : f(
and the abstract integral equation
129 (iii). X® and X+ are finite di
131 Definition 6.1 Let E and F be B
133 6.3 Contractions on embedded Ba
135 Theorem 6.13 (Center Manifold)
137 The AIE (6.14) is equivalent to
and 139 (q,p) = foh d-"~)p(-r) (7.8
[Cha71] [Die87] [Dui76] [DvG84] [Ha
143 Much is already known about the
145 We refer to ,g = {Su[y E a} as
147 ttemark 3.1 It follows by our m
149 Definition 4.1 Let 11 be a clos
151 Remark 4.3 It follows from Theo
(a) f(=) = ~(=)x, au x E GIH. (b) f
155 Since ] and ~ are smooth, so ar
157 of X at all points z E a. Neces
159 In this section we wish to desc
161 A straightforward application o
8.1 Poincard maps 163 We review the
165 Proposition 8.3 Let P. be a rel
On the Bifurcations of Subharmonics
169 and so we give in a fourth part
171 of KerL, the elements of B are
173 From the fixed-point subspaces
175 and the following curve is the
q even q=3 a g 177 s~llx-ag odd Fig
q=3 S 179 s=lk-aa q ~ - - q>5 ~ a ~
3.2 dimB--1 181 We are now in the s
183 Puting them back into (14) and
4.2 The recognition problem 185 Thi
is always real. such that is real,
291 I. The fixed point (2.9) remain
References 293 [i] R. W. Lucky (196
295 -- ba+~+~t h For small values o
h =0 XX ~0 (h~ 1, h~L 2) H = (0, O)
299 4 Description of the proof of t
Versal Deformations of Infinitesima
303 form. (2) By dropping the sympl
305 Jij q + L~j=l Ji]~j )' we have
307 For (0) n, n--even,~= 1, set Ix
D% I -% ,,~ Fig. 1 309 Each oblique
311 I -- T/ Fig. 2 7~-form Now, def
313 b'st = r (-1)s-t[; s t' case(c)
H i [X22"FILXl 2] ~ [(2x2x4-x32) -F
H 2Re[+2AZlZ2+Z2Z2 +#zlT" l ] 2E--R
319 unfolding H(g) of a Hamiltonian
ADDRESSES OF CONTRIBUTORS D.Armbrus