Plane wave expansion method: Difference between revisions

Revision as of 03:47, 26 April 2014

Revision as of 13:04, 1 October 2014

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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 [[fourierFourier 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>

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</math>

with the fourierFourier 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 Nmax</math> instead of the ideal, infinite wave.

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

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</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 fourierFourier 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.

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If implemented correctly, spurious solutions are avoided. It is less efficient when index contrast is high or when metals are incorporated. It cannot be used for scattering analysis.

Being a Fourier-space method, Gibb'sGibbs phenomenon affects the methodsmethod's accuracy. This is particularly problematic for devices with high dielectric contrast.

==See also==