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

Ludek Klimes


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.


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.

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