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2
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84927313655
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see also Photonic Band Gaps and Localization, edited by C.M. Soukoulis (Plenum, New York, 1993);
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3
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0000753914
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also numerous articles in
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and also numerous articles in J. Opt. Soc. Am. B. 10, 208 (1993).
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(1993)
J. Opt. Soc. Am. B.
, vol.10
, pp. 208
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7
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0001015766
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Measurement of a three-dimensional photonic band gap in a crystal structure made of dielectric rods
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(1994)
Physical Review B
, vol.50
, pp. 1945
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Ozbay, E.1
Abeyta, A.2
Tuttle, G.3
Tringides, M.4
Biswas, R.5
Chan, C.T.6
Soukoulis, C.M.7
Ho, K.M.8
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23
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84927313654
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In many cases, the Fourier transform is actually performed during the simulation to avoid storing the data on the disk.
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28
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84927313652
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In principle, the spatial derivatives can be calculated to the full precision allowed by the grid if we impose periodic boundary conditions by performing fast Fourier transform (FFT) from H(r) to H(k), and then obtain ktimesH(k) and inverse FFT back to real space to get nabla timesH. Since FFT scales like N ln(N), the algorithm is still basically linear. A FFT on a 64times64times64 grid can be accomplished in about 0.2 s on a 64 node N cube2 machine [D. Turner (private communication)]. Using FFT's to find nabla ×H takes six FFT's, plus some overhead to calculate k ×Hsub k (and the same for E), so that it takes approximately a few seconds for each time step, and a long simulation run is not impossible. The problem with FFT on parallel machines is that the code loses ``portability" since the FFT routines that are optimized on one class of machine perform poorly on others, and it is difficult to optimize the FFT routine if the dimensions are not powers of 2 on those machines.
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31
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84927313650
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There is no formal requirement that the E and H fields have to share the same grid. Instead of putting the E and the H fields on the same grid, we put them in a ``staggered" formation, with the H grid points occupying one sublattice and the E grid points occupying another sublattice. If the E and H fields are defined on the same grid points, the spatial derivatives for the finite difference equations are obtained from points that are 2d apart (d is the distance between adjacent grid points); while for a staggered grid of the same density the spatial derivatives are obtained from points that are at a distance d apart. The staggered grid is thus better in spatial resolution.
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32
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84927313648
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We should emphasize that the expansion in G space here is just a convenient way to impose the condition nabla ⋅ H=0. It is very different from the plane wave expansion method in which the number of G vectors used determines the accuracy and the computation time. In the present approach, the computation effort is not determined by the initial field. The number of G vectors used is usually small, and they are not intended to represent faithfully the eigenfunctions of the normal modes. All we need is to make sure that the initial field has nonzero projections onto the eigenstates we try to find. The G's here are the G's of the primitive unit cell, not the conventional cell.
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33
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84927313630
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For some k points, we have summed up all the grid points. There is no observable difference in the results when compared with 64 randomly selected points.
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43
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84927313628
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We note that Herring's formulas are derived for conductivities, but effective conductivity and effective dielectric constant satisfy the same equations.
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