Plane wave expansion method: Difference between revisions

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{{Short description|Technique in computational electromagnetism}}

{{Use American English|date=January 2019}}

'''Plane wave expansion method''' (PWE) refers to a computational technique in [[electromagnetics]] to solve the [[Maxwell's equations]] by formulating an [[eigenvalue]] problem out of the equation. This method is popular among the [[photonic crystal]] community as a method of solving for the [[band structure]] (dispersion relation) of specific photonic crystal geometries. PWE is traceable to the analytical formulations, and is useful in calculating modal solutions of Maxwell's equations over an inhomogeneous or periodic geometry.<ref>{{cite journal |last1=Andrianov |first1=Igor V. |last2=Danishevskyy |first2=Vladyslav V. |last3=Topol |first3=Heiko |last4=Rogerson |first4=Graham A. |title=Propagation of Floquet–Bloch shear waves in viscoelastic composites: analysis and comparison of interface/interphase models for imperfect bonding |journal=Acta Mechanica |date=25 November 2016 |volume=228 |page=1177-1196 |doi=10.1007/s00707-016-1765-4}}</ref> It is specifically tuned to solve problems in a time-harmonic forms, with [[Dispersion (optics)|non-dispersive]] media (a reformulation of the method named Inverse dispersion allows frequency-dependent refractive indices<ref >{{cite journal| title= Inverse dispersion method for calculation of complex photonic band diagram and PT symmetry |journal= Physical Review B|pages=165132 |year=2016|doi= 10.1103/PhysRevB.93.165132 | last1=Rybin |first1=Mikhail |last2=Limonov |first2=Mikhail |volume=93|issue= 16|arxiv=1707.02870 }}</ref>).

==Principles==

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The electric or magnetic fields are expanded for each field component in terms of the [[Fourier series]] components along the reciprocal lattice vector. Similarly, the dielectric permittivity (which is periodic along reciprocal lattice vector for photonic crystals) is also expanded through Fourier series components.

<math display="block">\frac{1}{\epsilon_r} = \sum_{m=-\infty}^{+\infty} K_m^{\epsilon_r} e^{-i \vecmathbf{G}. \veccdot \mathbf{r}}</math>

<math display="block">E(\omega,\vecmathbf{r}) = \sum_{n=-\infty}^{+\infty} K_n^{E_y} e^{-i\vecmathbf{G}. \cdot \vecmathbf{r}} e^{-i\vecmathbf{k} \cdot \vecmathbf{r}}</math>

with the Fourier series coefficients being the K numbers subscripted by m, n respectively, and the reciprocal lattice [[Vector (mathematics and physics)|vector]] given by <math>\vecmathbf{G}</math>. In real modeling, the range of components considered will be reduced to just <math>\pm N_\max</math> instead of the ideal, infinite wave.

Using these expansions in any of the curl-curl relations like,

<math display="block">\frac{1}{\epsilon(\vecmathbf{r})} \nabla \times \nabla \times E(\vecmathbf{r},\omega) = \left( \frac{\omega}{c} \right)^2 E(\vecmathbf{r},\omega)</math>

and simplifying under assumptions of a source free, linear, and non-dispersive region we obtain the [[eigenvalue]] relations which can be solved.

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For a y-polarized z-propagating electric wave, incident on a 1D-DBR periodic in only z-direction and homogeneous along x,y, with a lattice period of a. We then have the following simplified relations:

<math display="block">\frac{1}{\epsilon_r} = \sum_{m=-\infty}^{+\infty} K_m^{\epsilon_r} e^{-i \frac{2\pi m}{a}z}</math>

<math display="block">E(\omega,\vecmathbf{r}) = \sum_{n=-\infty}^{+\infty} K_n^{E_y} e^{-i\frac{2\pi n}{a}z} e^{-i \vecmathbf{k} \cdot \vecmathbf{r}}</math>

The constitutive eigenvalue equation we finally have to solve becomes,

<math display="block">\sum_n{\left( \frac{2\pi n}{a} + k_z \right)\left( \frac{2\pi m}{a} + k_z \right) K_{m-n}^{\epsilon_r} K_{n}^{E_y}} = \frac{\omega^2}{c^2} K_{m}^{E_y}</math>

This can be solved by building a matrix for the terms in the left hand side, and finding its eigen valueeigenvalue and vectors. The eigen valueseigenvalues correspond to the modal solutions, while the corresponding magnetic or electric fields themselves can be plotted using the Fourier expansions. The [[coefficients]] of the field harmonics are obtained from the specific eigen vectorseigenvectors.

The resulting band-structure obtained through the eigen modeseigenmodes of this structure are shown to the right.

===Example code===

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==Advantages==

PWE expansions are rigorous solutions. PWE is extremely well suited to the modal solution problem. Large size problems can be solved using iterative techniques like [[Conjugate gradient method]].

For both generalized and normal eigen valueeigenvalue problems, just a few band-index plots in the band-structure diagrams are required, usually lying on the [[brillouin zone edges]]. This corresponds to eigen modeseigenmodes solutions using iterative techniques, as opposed to diagonalization of the entire matrix.

The PWEM is highly efficient for calculating modes in periodic dielectric structures. Being a Fourier space method, it suffers from the [[Gibbs phenomenon]] and slow convergence in some configuration when fast Fourier factorization is not used. It is the method of choice for calculating the band structure of photonic crystals. It is not easy to understand at first, but it is easy to implement.