Plane wave expansion method
Encyclopedia
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 solving problems in a time-harmonic forms, with non-dispersive
media.
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.
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.
with the fourier series coefficients being the K numbers subscripted by m, n respectively, and the reciprocal lattice vector
given by
. In real modeling, the range of components considered will be reduced to just 
instead of the ideal, infinite wave.
Using these expansions in any of the curl-curl relations like,
and simplifying under assumptions of a source free, linear, and non-dispersive region we obtain the eigen value relations which can be solved.
The constitutive eigen equation we finally have to solve becomes,
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.
or GNU Octave
to compute the same band structure,
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 techniques, 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.
Alternatives include Order-N spectral method, and methods using Finite-difference time-domain (FDTD) which are simpler, and model transients.
See also
Maxwell's equations
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...
by formulating an eigenvalue problem out of the equation. This method is popular among the photonic crystal
Photonic crystal
Photonic crystals are periodic optical nanostructures that are designed to affect the motion of photons in a similar way that periodicity of a semiconductor crystal affects the motion of electrons...
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 solving problems in a time-harmonic forms, with non-dispersive
Dispersion (optics)
In optics, dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency, or alternatively when the group velocity depends on the frequency.Media having such a property are termed dispersive media...
media.
Principles
Plane wavePlane wave
In the physics of wave propagation, a plane wave is a constant-frequency wave whose wavefronts are infinite parallel planes of constant peak-to-peak amplitude normal to the phase velocity vector....
s are solutions to the homogeneous Helmholtz equation
Helmholtz equation
The Helmholtz equation, named for Hermann von Helmholtz, is the elliptic partial differential equation\nabla^2 A + k^2 A = 0where ∇2 is the Laplacian, k is the wavenumber, and A is the amplitude.-Motivation and uses:...
, 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.
The electric or magnetic fields are expanded for each field component in terms of the fourier series
Fourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
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.
with the fourier series coefficients being the K numbers subscripted by m, n respectively, and the reciprocal lattice vector
Vector (mathematics and physics)
In mathematics and physics, a vector is an element of a vector space. If n is a non negative integer and K is either the field of the real numbers or the field of the complex number, then K^n is naturally endowed with a structure of vector space, where K^n is the set of the ordered sequences of n...
given by
. In real modeling, the range of components considered will be reduced to just instead of the ideal, infinite wave.Using these expansions in any of the curl-curl relations like,
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:The constitutive eigen equation we finally have to solve becomes,
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.
Example code
We can use the following code in MatlabMATLAB
MATLAB is a numerical computing environment and fourth-generation programming language. Developed by MathWorks, MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages,...
or GNU Octave
GNU Octave
GNU Octave is a high-level language, primarily intended for numerical computations. It provides a convenient command-line interface for solving linear and nonlinear problems numerically, and for performing other numerical experiments using a language that is mostly compatible with MATLAB...
to compute the same band structure,
%
% solve the DBR photonic band structure for a simple
% 1D DBR. air-spacing d, periodicity a, i.e, a > d,
% we assume an infinite stack of 1D alternating eps_r|air layers
% y-polarized, z-directed plane wave incident on the stack
% periodic in the z-direction;
%
%parameters
d=8; %air gap
a=10; %total periodicity
d_over_a = d/a;
eps_r =12.2500; %dielectric constant, like GaAs,
% max F.S coefs for representing E field, and Eps(r), are
Mmax=50;
% Q matrix is non-symmetric in this case, Qij != Qji
% Qmn = (2*pi*n + Kz)^2*Km-n
% Kn = delta_n / eps_r + (1 - 1/eps_r)(d/a)sinc(pi.n.d/a)
% here n runs from -Mmax to + Mmax,
freqs=[];
for Kz=-pi/a:pi/(10*a):+pi/a
Q=zeros(2*Mmax + 1);
for x=1:2*Mmax+1
for y=1:2*Mmax+1
X=x-Mmax;
Y=y-Mmax;
kn=(1 -1/eps_r)*d_over_a.*sinc((X-Y).*d_over_a) + ((X-Y)0)*1/eps_r;
Q(x,y)=(2*pi*Y/a + Kz).^2*kn;
end
end
fprintf('Kz = %g\n',Kz)
omega_c=eig(Q);
omega_c=sort(sqrt(omega_c));%important step.
freqs=[freqs; omega_c.'];
end
close
figure
hold on
idx=1;
for idx=1:length(-pi/a:pi/(10*a):+pi/a)
plot(-pi/a:pi/(10*a):+pi/a,freqs(:,idx),'.-')
end
hold off
xlabel('Kz')
ylabel('omega/c')
title(sprintf('PBG of 1D DBR with d/a=%g, Epsr=%g',d/a,eps_r))
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
Conjugate gradient method
In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is symmetric and positive-definite. The conjugate gradient method is an iterative method, so it can be applied to sparse systems that are too...
.
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 techniques, 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.
Alternatives include Order-N spectral method, and methods using Finite-difference time-domain (FDTD) which are simpler, and model transients.
See also
- Photonic crystalPhotonic crystalPhotonic crystals are periodic optical nanostructures that are designed to affect the motion of photons in a similar way that periodicity of a semiconductor crystal affects the motion of electrons...
- Computational electromagneticsComputational electromagneticsComputational electromagnetics, computational electrodynamics or electromagnetic modeling is the process of modeling the interaction of electromagnetic fields with physical objects and the environment....
- Finite-difference time-domain methodFinite-difference time-domain methodFinite-difference time-domain is one of the primary available computational electrodynamics modeling techniques. Since it is a time-domain method, FDTD solutions can cover a wide frequency range with a single simulation run, and treat nonlinear material properties in a natural way.The FDTD method...
- Finite element methodFinite element methodThe finite element method is a numerical technique for finding approximate solutions of partial differential equations as well as integral equations...
- Maxwell's equationsMaxwell's equationsMaxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...