Plane wave expansion method: Difference between revisions

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{{Short description|Technique in computational electromagnetism}}

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

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{{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. It is specifically tuned to solve problems in a time-harmonic forms, with [[Dispersion (optics)|non-dispersive]] media.

{{Cleanup bare URLs|date=August 2022}}

{{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. It is specifically tuned to solve problems in a time-harmonic forms, with [[Dispersion (optics)|non-dispersive]] media.

==Principles==

{{Dubious|date=August 2009}}

[[Plane wave]]s are solutions to the homogeneous [[Helmholtz equation]], and form a basis to represent fields in the periodic media. PWE as applied to photonic crystals as described is primarily sourced from Dr. Danner's tutorial.<ref>{{Cite web |last=Danner |first=Aaron J. |date=2011-01-31 |title=An introduction to the plane wave expansion method for calculating photonic crystal band diagrams |url=https://www.ece.nus.edu.sg/stfpage/eleadj/planewave.htm |url-status=live |archive-url=https://web.archive.org/web/20220615161702/https://www.ece.nus.edu.sg/stfpage/eleadj/planewave.htm |archive-date=2022-06-15 |access-date=2022-09-29 |website=Aaron Danner - NUS}}</ref>

from Dr. Danner's tutorial.<ref>http://www.ece.nus.edu.sg/stfpage/eleadj/planewave.htm</ref>

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.

E(<math display="block">\omega,\vecfrac{r1}{\epsilon_r}) = \sum_{nm=-\infty}^{+\infty} K_nK_m^{E_y\epsilon_r} e^{-i \vec{G}.\vec{r}} e^{-i\vec{k}\vec{r}}</math>

:<math>

<math display="block">E(\frac{1}{omega,\epsilon_rvec{r}) = \sum_{mn=-\infty}^{+\infty} K_mK_n^{\epsilon_rE_y} e^{-i \vec{G}.\vec{r}} e^{-i\vec{k}\vec{r}}</math>

</math>

:<math>

E(\omega,\vec{r}) = \sum_{n=-\infty}^{+\infty} K_n^{E_y} e^{-i\vec{G}.\vec{r}} e^{-i\vec{k}\vec{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>\vec{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(\vec{r})} \nabla \times \nabla \times E(\vec{r},\omega) = \left( \frac{\omega}{c} \right)^2 E(\vec{r},\omega)</math>

:<math>

\frac{1}{\epsilon(\vec{r})} \nabla \times \nabla \times E(\vec{r},\omega) = \left( \frac{\omega}{c} \right)^2 E(\vec{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.

==Example for 1D case==

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:

[[Image:Photonic Crystal 1D DBR aircore epsr12point25 DbyA0point8.png|thumb|right|Band structure of a 1D Photonic Crystal, DBR air-core calculated using plane wave expansion technique with 101 planewaves, for d/a=0.8, and dielectric contrast of 12.250.]]

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>

<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,\vec{r}) = \sum_{n=-\infty}^{+\infty} K_n^{E_y} e^{-i\frac{2\pi n}{a}z} e^{-i\vec{k}\vec{r}}</math>

</math>

:<math>

E(\omega,\vec{r}) = \sum_{n=-\infty}^{+\infty} K_n^{E_y} e^{-i\frac{2\pi n}{a}z} e^{-i\vec{k}\vec{r}}

</math>

The constitutive eigeneigenvalue 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>

:<math>

\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 value and vectors. The eigen values 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 vectors.