Symplectic Singularities and Optical Diffraction S.Janeczko and Ian Stewart ABSTRACT Singularities of symplectic mappings are important in mathematical physics; for example in optics they determine the geometry of caustics. Here we survey the structure of symplectic singularities and extend the results from mappings to symplectic relations, by making use of Lagrangian varieties (which may have singularities) in place of Lagrangian manifolds. We explain how these ideas apply to classical ray-optical diffraction: the highly singular geometry in physical space turns out to be the projection of well-behaved geometry in phase space. In particular we classify generic caustics by diffraction in a half-line aperture, and discuss diffraction at a circular obstacle. Introduction "Everything in the world is Lagrangian" Alan Weinstein [1981] p.5 Symplectic geometry, the modern version of Hamilton's formalism, has become important in several areas of mathematical physics, notably optics, mechanics, and thermodynamics. In all of these applications, singularities of symplectic mappings play an important role. For example in optics they determine the caustics. In this paper we survey (and on occasion extend) the known results on the structure of symplectic singularities. For definiteness we usually emphasize the optical setting. § 1 provides some simple motivation for the occurrence of symplectic structure in mechanics and optics, by considering billiards: the trajectory of a particle or light ray reflected in a smooth curve. We show that the mapping that takes the state of the particle at one collision with the boundary to the state at the next collision is, in suitable coordinates, area-preserving. Since the system is two-dimensional, this is equivalent to the mapping being symplectic. In §2 we introduce basic concepts of symplectic geometry, such as symplectic manifolds, phase space, Darboux coordinates, Lagrangian submanifolds, and in particular Lagrangian varieties, which are like Lagrangian manifolds but permit the occurrence of singularities. In §3 we introduce a 'universal' phase space for symplectic geometry, discuss the unifying notions of a symplectic image and a symplectic relation, and describe the classification of stable Lagrangian projections. As an example, we show in §4 how to formulate the properties of an optical
221 instrument in terms of composed symplectic relations. We describe examples including refraction in an inhomogeneous medium and reflection in a curve in the plane. In both cases the symplectic relation is the graph of a diffeomorphism. We also relate the ideas to § 1 by analysing the biUard map as an optical instrument. In §5 we observe that the notion of symplectic relation permits the method to be carried over to classical diffraction, where the mapping is in general not a diffeomorphism but is a symplectic relation. This corresponds to the classical approach to diffraction in which a single incident ray gives rise to an entire family of transmitted rays. The point is that this highly singular geometry in the rays results from the projection into physical space of a well-behaved structure in phase space. We use this approach to analyse diffraction at an aperture, and in particular classify generic caustics by diffraction in a half-line aperture. The theory is extended in §6 to diffraction at a smooth obstacle, and we classify canonical varieties of generic obstacle curves in the plane. We state a theorem of Scherbak classifying generic symplectic images for surfaces in R3. Appendix 1 discusses the formulation of classical mechanics in universal phase space and the notion of a generalized mechanical system, with an example from geometrical optics. Appendix 2 considers in more detail diffraction at a circular obstacle, for which the geometry is surprisingly subtle. In particular the complete caustic, including the caustic by diffraction, involves discontinuities in the third derivative. Acknowledgements This research was supported in part by a grant from the Science and Engineering Research Council of the UK and by MPI Bonn. It was completed during the Warwick Symposium on Singularity Theory and its Appplications, 1988-89, which was also funded by SERC. 1. Symplectic Billiards The underlying symplectic structure of both optics and mechanics can be illustrated using a single example: billiards. Suppose that C is a piecewise smooth (closed) convex curve. Consider a particle A moving inside C under the action of no forces, and undergoing perfectly elastic collisions with the boundary. Or equivalently consider A as describing a light ray and C as a perfect mirror. In either case the collision with the boundary will be as shown in Figure 1.1. A segment of the trajectory is specified by two quantities: the initial point on C and the angle 0 between the segment and the tangent. We choose the following coordinates: q = arc-length along C from some chosen reference point, p = cos 0. Then (p,q) ~ (-1,1) x C = P, the phase space. Suppose that A starts from (q,p) and makes a single bounce, emerging with coordinates (q',p'), as in Fiigure 1.2. What is the relation between (p',q') and (p,q)? Clearly there exists some function q~ such that
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