[Paper Review] Two-Scale Approach to an Asymptotic Solution of Maxwell Equations in Layered Periodic Media

Maria V. Perel, Mikhail S. Sidorenko|arXiv (Cornell University)|Oct 31, 2015

Photonic Crystals and Applications10 references3 citations

TL;DR

This paper develops a two-scale asymptotic method to solve Maxwell's equations in three-dimensional layered periodic media near a stationary point of the dispersion surface. It shows that the leading-order solution consists of two Floquet-Bloch modes with slowly varying envelopes, where the envelopes satisfy hyperbolic or elliptic equations depending on the nature of the stationary point, enabling the description of localized or undistorted wave beams.

ABSTRACT

An asymptotic investigation of monochromatic electromagnetic fields in a layered periodic medium is carried out under the assumption that the wave frequency is close to the frequency of a stationary point of the dispersion surface. We find solutions of Maxwell equations by the method of two-scale asymptotic expansions. We establish that the principal order of the expansion of a solution dependent on three spatial coordinates is the sum of two differently polarized Floquet-Bloch solutions, each of which is multiplied by a slowly varying envelope function. We derive that the envelope functions satisfy a system of differential equations with constant coefficients. In new variables, it is reduced to a system of two independent equations, both of them are either hyperbolic or elliptic, depending on the type of the stationary point. The envelope functions are independent only in the planar case. Some consequences are discussed.

Motivation & Objective

• To mathematically analyze monochromatic electromagnetic fields in 3D layered periodic media near a stationary point of the dispersion surface.
• To extend the two-scale asymptotic method to Maxwell’s equations in matrix form, applicable to non-divergence form equations.
• To derive a system of differential equations for envelope functions that govern the slow modulation of Floquet-Bloch modes.
• To characterize the type (hyperbolic or elliptic) of the envelope equation based on the nature of the stationary point in the dispersion surface.
• To establish conditions under which the envelope functions remain independent and the solution remains in the required function space.

Proposed method

• Employ the two-scale asymptotic expansion method, introducing fast (periodic) and slow (macroscopic) spatial variables.
• Express Maxwell’s equations in matrix form as proposed by Felsen and Marcuvitz, enabling systematic asymptotic analysis.
• Decompose the solution into a sum of two Floquet-Bloch modes, each modulated by a slowly varying envelope function.
• Derive a system of differential equations with constant coefficients for the envelope functions by matching terms in the asymptotic expansion.
• Transform the system into two independent equations—either hyperbolic or elliptic—depending on the curvature of the dispersion surface at the stationary point.
• Use functional analysis and symmetry arguments to ensure the existence and continuity of solutions, particularly by compensating jumps in the particular solution via homogeneous Floquet-Bloch solutions.

Experimental results

Research questions

• RQ1 How can the two-scale asymptotic method be adapted to Maxwell’s equations in non-divergence form for layered periodic media?
• RQ2 What is the structure of the leading-order solution to Maxwell’s equations near a stationary point of the dispersion surface in 3D?
• RQ3 Under what conditions do the envelope functions of the two polarized Floquet-Bloch modes decouple into independent hyperbolic or elliptic equations?
• RQ4 How can the solution be constructed to remain continuous and belong to the required function space when the particular solution exhibits jumps at periodic boundaries?
• RQ5 What determines whether the resulting wave dynamics are hyperbolic (wave-like) or elliptic (localized) in the effective envelope description?

Key findings

• The principal-order solution is a superposition of two differently polarized Floquet-Bloch modes, each modulated by a slowly varying envelope function.
• The envelope functions satisfy a system of differential equations with constant coefficients, which decouples into two independent equations.
• The type of the envelope equation—hyperbolic or elliptic—is determined by the nature of the stationary point on the dispersion surface.
• The envelope functions are independent only in the planar case, where the wavevector dependence simplifies.
• The solution can be constructed to be continuous across the period by adjusting the coefficients of the homogeneous Floquet-Bloch solutions to cancel boundary jumps in the particular solution.
• The method confirms the existence of undistorted beam propagation in layered media, consistent with earlier numerical and physical studies, but now with rigorous mathematical justification in 3D.