[Paper Review] Orthogonal-Phase-Velocity Propagation of Electromagnetic Plane Waves

Tom G. Mackay, Akhlesh Lakhtakia|ArXiv.org|Nov 24, 2005

Metamaterials and Metasurfaces Applications18 references5 citations

TL;DR

This paper proposes orthogonal-phase-velocity (OPV) propagation of electromagnetic plane waves in isotropic, nondissipative dielectric-magnetic media moving relative to an inertial frame. Using the Minkowski constitutive relations derived via Lorentz transformation, it demonstrates that the phase velocity and time-averaged Poynting vector can be mutually orthogonal under specific conditions—particularly when the product of relative permittivity and permeability is less than unity—extending beyond conventional positive- and negative-phase-velocity regimes.

ABSTRACT

In an isotropic, homogeneous, nondissipative, dielectric-magnetic medium that is simply moving with respect to an inertial reference frame, planewave solutions of the Maxwell curl postulates can be such that the phase velocity and the time-averaged Poynting vector are mutually orthogonal. Orthogonal-phase-velocity propagation thus adds to the conventional positive-phase-velocity propagation and the recently discovered negative-phase-velocity propagation that is associated with the phenomenon of negative refraction.

Motivation & Objective

• To investigate whether electromagnetic plane waves can propagate with phase velocity orthogonal to the time-averaged Poynting vector in moving media.
• To extend the understanding of wave propagation beyond positive- and negative-phase-velocity regimes by introducing orthogonal-phase-velocity (OPV) materials.
• To analyze the conditions under which OPV behavior emerges in isotropic, nondissipative dielectric-magnetic media via relativistic transformations.
• To determine the dependence of OPV occurrence on relative velocity, wavevector direction, and material parameters (εrμr < 1).

Proposed method

• Derives Minkowski constitutive relations from Lorentz transformation of electromagnetic phasors between inertial frames Σ (observer frame) and Σ′ (co-moving frame).
• Applies the Lorentz transformation to the constitutive relations in the co-moving frame (εr, μr) to obtain effective constitutive relations in the observer frame.
• Solves the Maxwell curl postulates in the observer frame to determine the wavenumber k and phase velocity direction, using the wavevector k and relative velocity v.
• Evaluates the scalar W = k·(k + ξβ(k₀ - kβcosθ)cosθ) to classify wave behavior: W > 0 (positive-phase-velocity), W < 0 (negative-phase-velocity), W = 0 (orthogonal-phase-velocity).
• Analyzes the discriminant Δ = [ξβcosθ]² - 4(1 - ξβ²cos²θ) to determine whether the wavenumber k is real or complex, indicating propagating or evanescent waves.
• Performs numerical simulations for varying β (relative speed), θ (angle between k and v), and εrμr to map regions of OPV, PPV, and NPV behavior.

Experimental results

Research questions

• RQ1 Can electromagnetic plane waves propagate in a moving isotropic dielectric-magnetic medium such that the phase velocity and time-averaged Poynting vector are orthogonal?
• RQ2 What are the conditions on material parameters (εr, μr) and relative motion (β, θ) for orthogonal-phase-velocity propagation to occur?
• RQ3 How does the OPV regime compare in extent and symmetry to the previously known negative-phase-velocity regime?
• RQ4 What role does the product εrμr play in determining whether OPV behavior is possible, and what happens as εrμr approaches unity?
• RQ5 Can OPV behavior be approximately realized in weakly dissipative materials, despite inherent losses?

Key findings

• Orthogonal-phase-velocity (OPV) propagation occurs when the product of relative permittivity and permeability is less than unity (0 < εrμr < 1), leading to a null scalar W = 0 between phase velocity and Poynting vector.
• The OPV region occupies a substantial portion of the βθ plane for small εrμr, and it is symmetric about θ = π/2, unlike the NPV region which is confined to θ > π/2.
• As εrμr increases toward unity, the OPV region shrinks and vanishes in the limit εrμr → 1, where only positive-phase-velocity propagation remains.
• For εrμr > 1, OPV does not occur; the phase velocity is either positive or negative, with the NPV region expanding as εrμr increases.
• When Δ < 0, the wavenumber becomes complex (k = kR + ikI), indicating evanescent waves, with kI ≠ 0 for certain β and θ configurations, but W = 0 still defines the OPV condition.
• Numerical results confirm that OPV can be nearly satisfied in weakly dissipative materials, suggesting potential for experimental realization despite inherent losses.