On Solutions of the Maxwell’s Equations from the Viewpoint of Geometrical Optics

Traditionally ideas of geometrical optics apply to research of the approximate solutions corresponding to high-frequency limit, but it is known that, e.g. jumps of solutions of the equations of Maxwell satisfy to Huygens’s law also. In the article we indicate the class of exact solutions of the Maxwell’s equations for which the approach of the geometrical optics can be still used. We consider solutions of the Maxwell’s equations with which it is possible to associate orthogonal system of coordinates of (x1,x2,x3) so that the directions of vectors ⃗E and ⃗e2 and also ⃗H and ⃗e3 are coincided. Conditions on Lamé coefficients of this system of coordinates are found: euh1 doesn’t depend on x2 and x3 and logarithmic derivatives eh1h3 h2 and μh1h2 h3 with respect to x1 don’t depend on x2 and x3 respectively. The first condition means that x1-lines are rays of geometrical optics and it gives a reason to call such systems of coordinates as ray systems how it is accepted in geometrical optics. Thus the solution of the Maxwell’s equations can be described as a wave extending along a ray, that is as the solution of the two-dimensional hyperbolic equation. Necessary and sufficient conditions are found for association of such coordinates systems with the solution of the equations of Maxwell: the directions of vectors ⃗E and ⃗H don’t change over time, they are orthogonal each other and consist in involution, that is ⃗E × ⃗ H,rot ⃗E × ⃗ H = 0.

eometrical optics, Maxwell’s equations, rays, wavefront, Fermat’s principle