Plane wave expansion method: Difference between revisions

Content deleted Content added
No edit summary
No edit summary
Line 10:
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.
 
Line 24:
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===
Line 102:
==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.