#### The Evanescent Plane Wave Solutions of the Maxwell Equations Revisited. Snell Law and Fresnel Coefficients

###### Authors: Jean-Pierre Bérenger

###### Source: FERMAT, Volume 22, Article 2, Jul.-Aug., 2017

**Abstract**: In textbooks on electromagnetism, the plane wave solutions of the Maxwell equations in free space are in most cases limited to homogeneous waves where the field is uniform in any plane perpendicular to the propagation of the phase. However, non homogeneous plane waves where the field is evanescent in a direction perpendicular to the propagation also satisfy the Maxwell equations. This paper revisits such waves that significantly differ from the homogeneous waves. In particular the electromagnetic field is not transverse, it has a component in the direction of propagation of the phase. The equivalents of the Snell law and of the Fresnel coefficients are derived for the reflection and transmission of evanescent plane waves at interfaces, and it is shown that such waves can be generated in the computational domain of numerical methods that solve the Maxwell equations.

**Index Terms**: Electromagnetic theory, Electromagnetic fields, Electromagnetic propagation, Inhomogeneous waves, Evanescent waves, Plane waves, Maxwell equations.

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