## Transformation of paraxial matrices
at a general interface between two general media

**Ludek Klimes**
### Summary

Paraxial matrices are the derivatives
of the phase-space coordinates of rays
with respect to the initial conditions
for Hamilton's equations of rays.
In smooth media, the paraxial matrices
satisfy the Hamiltonian equations of geodesic deviation,
also called the paraxial ray tracing equations
or the dynamic ray tracing equations.

We derive the explicit equations for transforming
these paraxial matrices
at a general smooth interface between two general media.
The transformation equations are applicable
to both real-valued and complex-valued paraxial matrices.
The equations are expressed in terms of
a general Hamiltonian function and are applicable
to the transformation of paraxial matrices
in both isotropic and anisotropic media.
The interface is specified by an implicit equation.
No local coordinates are needed for the transformation.

### Keywords

Ray theory, Hamilton-Jacobi equation, Hamilton's equations,
geodesic deviation, paraxial rays, paraxial matrices,
reflection or refraction at curved interfaces,
anisotropy, heterogeneous media,
paraxial approximation, Gaussian beams, wave propagation.

### Whole paper

The paper is available in
PDF (113 kB).

In: Seismic Waves in Complex 3-D Structures, Report 20,
pp. 115-126, Dep. Geophys., Charles Univ., Prague, 2010.