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\title{APPLICATIONS OF COMPLEX RAY THEORY TO PROBLEMS IN DIFFRACTION}
\author{Dr S. J. Chapman, Mr J. M. H. Lawry, Miss V. H. Saward}{Mathematical
Institute, Oxford University\\24--29 St Giles, Oxford OX1 3LB, United Kingdom}

The ray treatment of the linear wave equation, which is asymptotically
valid for high frequencies, results in the well-known eikonal equation
and the recursive system of transport equations. These equations are
nonlinear but have solutions in the form of rays, curves in space
(straight lines in the case of homogeneous media) along which signals
are propagated. This converts a partial differential equation problem
with given boundary data into ordinary differential equation problems
along each ray for phase and amplitude.

There are many instances where the solutions of the ray equations give
plane wave fields with complex phase which in turn propagate along
complex-valued trajectories.  This occurs in cases involving
dissipative media, incident beams, edge waves of various kinds, and
fluid/solid acoustic interactions, despite real eikonal boundary data
being imposed.

It is necessary to consider the analytic continuation of the problem
into a complex domain. In a homogeneous medium, the complex rays are
straight lines in the complex co-ordinate space. If we think of this
as a real four- or six-dimensional co\"{o}rdinate space, then the ray is
a two-dimensional plane. Complex rays define real observable rays
where the complex ray intersects the real co-ordinate space, and
whether or not this happens for a given complex ray may be analysed in
geometric terms.

The analytic continuation of a boundary does not divide the complex
space into two separate pieces (as it does in real space) so, treated
na\"{\i}vely, complex rays are not stopped by boundaries and pass
straight through obstacles. New rules must be formulated to decide
where rays cease to exist (e.g.\ on the dark side of a caustic or
shadow boundary). The new rules are provided by the Stokes phenomenon,
which can be used to distinguish which complex phases are physically
acceptable. Stokes lines for an analytic function of a single complex
variable are straight lines: for a wavefield they are curved
multidimensional surfaces, at which ray fields are born or
disappear. They may intersect real space, for example in the vicinity
of cusps of caustics and near points of total internal reflection.

The great theoretical advantage of considering the full
complexification of ray problems is that problems with beams and
inhomogeneous waves can be considered to be merely different aspects
of problems with point sources and plane waves respectively.

We outline the theory of complex rays and give simple examples of line
sources near plane interfaces separating media of differing refractive
indices. The problem of total internal reflection (source in slow
medium) and the reverse problem (source in fast medium) are compared
and contrasted.

A second problem considered is that of scattering of a plane wave or a
line source by a smooth cylinder: although the solution to the problem
of a circular cylinder is known both exactly and asymptotically for a
real incident wave, considering the analytic continuation of the ray
field for complex rays reveals a rich structure near the point of
tangency where the creeping wavefield is generated and enables the
corresponding problem with an inhomogeneous plane wave or a Gaussian
beam (for which the tangential ray is a complex one) to be examined
more easily.

A third class of problems to be presented is those involving
scattering of inhomogeneous plane waves and Gaussian beams by a
half{}line with particular emphasis on the location of shadow boundaries
by means of Stokes phenomenon. The problem of a creeping ray field
incident on a half-line is also considered, which can be used to
calculate the diffracted field generated when the creeping field on a
boundary in shadow meets a discontinuity.



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