Plane-Wave Reflection and Transmission at Arbitrarily Oriented and Charged Planar Interfaces between Lossy Isotropic Media

A general algorithm for calculating the reflection and refraction of nonuniform plane waves from an arbitrarily oriented and charged planar interface between two lossy isotropic media is proposed based on the decomposition of the complex wave vector and complex wave numbers with respect to the unit normal vector of the interface. According to the complex vector analysis, the exact definition of the complex angles of incidence, reflection and refraction are presented and applied in the complex forms of Snell’s law and Fresnel equations to quickly and correctly calculate the complex wave vectors and the complex electric fields of the reflected and refracted waves at a charged interface where the surface charge densities are considered. Two practical examples are given to demonstrate the validity and effectiveness of the proposed methodology. The method may find applications in the fields of remote sensing, light ray-tracing and the interaction of electromagnetic waves with absorbing media.

💡 Research Summary

The paper presents a compact and rigorous algorithm for determining the reflection and transmission of non‑uniform (complex) plane waves at a planar interface that may be arbitrarily oriented and carry surface charge, when the two adjoining media are lossy isotropic dielectrics. The authors start from the well‑known Adler‑Chu‑Fano representation of a complex wave vector k = β + jα, where β is the phase‑propagation vector and α the attenuation vector. While previous works mainly treated the special case where β, α and the interface normal n̂ lie in the same plane, this study tackles the fully general situation in which the three vectors are non‑coplanar.

The key methodological step is the decomposition of k into a normal component kₙ and a tangential component kₜ with respect to the interface plane. By defining a complex angle θ through the relation

  sin θ = (kₜ·k)/|k|,

the authors obtain a compact expression for the normal and tangential parts:

  kₙ = |k| cos θ n̂,  kₜ = |k| sin θ t̂,

where t̂ is a complex unit vector lying in the interface plane. This definition of a complex incidence angle allows the direct use of a complex‑valued Snell’s law,

  n₁ sin θᵢ = n₂ sin θₜ,

where n₁ and n₂ are the complex refractive indices (√με) of the two media. The complex wave numbers are obtained from the dispersion relation k² = ω² με, with the complex permittivity ε = ε′ − jσ/ω and permeability μ = μ′ − jμ″/ω, thus incorporating both dielectric and magnetic losses.

The relationship between β, α and the material parameters is derived from the dispersion equation, yielding two real equations:

  β² − α² = Re(k₀²),  2 β α cos φ = Im(k₀²),

where φ is the angle between β and α. Solving these gives physically meaningful β and α for any prescribed loss tangent.

Surface charge on the interface is modeled as an equivalent surface conductivity σₛ. The usual boundary conditions for the tangential components of E and H are modified by a term σₛ Z₀ (Z₀ being the free‑space impedance). Consequently, the Fresnel coefficients for the two fundamental polarizations—parallel electric (PE) and parallel magnetic (PM)—acquire additional σₛ‑dependent terms. For example, the TE‑like (PE) reflection coefficient becomes

  R_PE =