Plane wave expansion method

Plane wave expansion method (PWE) refers to a computational technique in Electromagnetics, solve the Maxwells equations by formulating an Eigen value problem out of it. This method is a popular among 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 formualtions, and is useful to calculate modal solutions of Maxwells equations over an inhomogeneous or periodic geometry. It is specifically tuned to solving problems in a time-harmonic forms, with non-dispersive media.

Principles

Plane waves are solutions to the homogeneous Helmholtz equation, and form a basis to represent fields in the periodic media. PWE as applied to photonic crystals is described, primarily sourced from Dr. Danner's tutorial.

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 permitivitty (which is periodic along reciprocal lattice vector, for photonic crystals) is also expanded through fourier series components,

${\frac {1}{\epsilon _{r}}}=\sum _{m=-\infty }^{+\infty }K_{m}^{\epsilon _{r}}e^{-j{\vec {G}}.{\vec {r}}}$

$E(\omega ,{\vec {r}})=\sum _{n=-\infty }^{+\infty }K_{n}^{E_{y}}e^{-j{\vec {G}}.{\vec {r}}}e^{-j{\vec {k}}{\vec {r}}}$

with the fourier series coefficients being the K numbers subscripted by m, n respectively, and the reciprocal lattice vector given by ${\vec {G}}$. In real modeling the range of components considered will be reduced to just $\pm Nmax$ instead of the ideal, infinite wave.

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

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

and simplifying under assumptions of a source free, linear, and non-dispersive region we obtain the eigen value 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:

${\frac {1}{\epsilon _{r}}}=\sum _{m=-\infty }^{+\infty }K_{m}^{\epsilon _{r}}e^{-j{\frac {2\pi m}{a}}z}$

$E(\omega ,{\vec {r}})=\sum _{n=-\infty }^{+\infty }K_{n}^{E_{y}}e^{-j{\frac {2\pi n}{a}}z}e^{-j{\vec {k}}{\vec {r}}}$

$\sum _{n}{\left({\frac {2\pi n}{a}}+k_{z}\right)^{2}K_{m-n}^{\epsilon _{r}}K_{n}^{E_{y}}}={\frac {\omega ^{2}}{c^{2}}}K_{m}^{E_{y}}$

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.

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

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 value 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 modes solutions using iterative technqiues, as opposed to diagonalization of the entire matrix.

Disadvantages

Sometimes spurious modes appear. Large problems scaled as O(n³), with the number of the plane waves (n) used in the problem; this is both time consuming and complex in memory requirements as well.

Alternatives include order-N spectral method, and methods using FDTD which are simpler, and model transients.

See also

• Computational electromagnetics
• Finite-difference time-domain method
• Finite element method
• Maxwells equations