Exact boundary-value solution for an electromagnetic wave propagating in a linearly-varying index of refraction

TL;DR

This paper derives an exact spectral solution for electromagnetic wave propagation in a linearly-varying refractive index, analyzing Gaussian and speckled wavefields to understand focusing and caustic behaviors.

Contribution

It provides the first exact spectral boundary-value solution for such wave propagation, enabling detailed analysis of wave behavior in inhomogeneous media.

Findings

Oblique incidence causes rigid translation and focal shift for large Gaussian beams.

Hyperbolic umbilic caustic disappears when beam waist is sufficiently small.

Speckle pattern coupling depends on the parameter ta, with different regimes for large and small ta.

Abstract

The propagation of electromagnetic waves in a linearly-varying index of refraction is a fundamental problem in wave physics, being relevant in fusion science for describing certain wave-based heating and diagnostic schemes. Here, an exact solution is obtained for a given incoming wavefield specified on the boundary transverse to the direction of inhomogeneity by performing a spectral, rather than asymptotic, matching. Two case studies are then presented: a Gaussian beam at oblique incidence and a speckled wavefield at normal incidence. For the Gaussian beam, it is shown that when the waist $W$ is sufficiently large, oblique incidence manifests simply as rigid translation and focal shift of the corresponding diffraction pattern at normal incidence. The destruction of the hyperbolic umbilic caustic (corresponding to a critically focused beam) as $W$ is reduced is then demonstrated. The caustic disappears once $W \lesssim \delta_a \sqrt{L}$ ($L$ being the medium lengthscale normalized by the Airy skin depth $\delta_a$), at which point the wave behavior is increasingly described by Airy functions, but experiences less focusing as a result. To maximize the intensity of a launched Gaussian beam at a turning point, one should therefore minimize the imaginary part of the launched complex beam parameter while having the real part satisfy critical focusing. For the speckled wavefield, it is shown that the transverse speckle pattern only couples to the Airy longitudinal pattern when the coupling parameter $\eta = \sqrt{L}/f_\#$ is large, with $f_\#$ being the f-number of the launching aperture. When $\eta \ll 1$, a reduced description of the total wavefield can be obtained by simply multiplying the incoming speckle pattern with the Airy swelling.