Photonic band structure, or photonic bandgaps, of a diamond
lattice of spheres * diamond sphere structures * pyrochlore structures
Photonic Band Structure of a Diamond Lattice
Indisputably, one of the hallmarks in the study of
photonic band gap structures is an article
by Ho, Chan, and Soukoulis . It opened the way
for a fabrication of the first photonic structure
with a complete photonic band gap (CPBG)  and advanced
the field considerably. Its main conclusions
were that, regarding a CPBG, the diamond structure
(for geometrical arrangement of spheres in the diamond lattice see
fares much better
than a simple face-centered-cubic (fcc) one. Namely:
However, the bulk photonic KKR method  yielded
results quantitatively different from the earlier plane-wave calculations
of Ho, Chan, and Soukoulis .
These results have been first discussed in my comment.
Additionally, you can also download a few
data ASCII files, as obtained by the bulk photonic KKR method , and two
PDF figures (courtesy of Mischa Megens), which show a comparison of the results
obtained by the bulk photonic KKR method against those obtained
MIT Photonic-Bands (MPB) package. Full account of this story
you will find in Ref. , which also includes
metallo-dielectric spheres and zinc-blende structures.
Here a brief summary is provided.
the threshold value of the dielectric contrast eps to open a CPBG
is 4 (8.2 for an fcc structure ),
- ii) a CPBG opens between the 2nd and 3rd bands
(the 8th-9th bands for an fcc structure), and, consequently,
is much more stable against disorder, and
- iii) a CPBG
is significantly larger (15% and 5%, for the respective
diamond and fcc closed packed lattice of spheres with a dielectric
The KKR method is based on the
first-principle multiple-scattering theory and turns out
to be the most efficient in dealing with both highly
dispersive materials and highly symmetric scatterers.
The size of a secular matrix in the bulk photonic KKR method [3,4] is
2*N*(lmax+1)**2 x 2*N*(lmax+1)**2,
where 2 stands for two independent polarizations,
N is the number of atoms in the lattice unit cell, and lmax
is the cutoff on the orbital angular momentum used in the expansions
into spherical waves. Convergence is tested with increasing the cutoff value of lmax.
Computational times with and without dispersion of the dielectric
contrast are the same. I have run calculations till lmax=9.
You can download here the
(lmax=9) data for
the two lowest photonic bands, calculated with lmax=9 (in units [c/A],
where c is the speed of light in vacuum and A is the lattice constant of the
conventional cubic unit cell of the diamond lattice).
In the first column, there are labels of (between any pair of special
points, equidistantly spaced) points on the characteristic path used
in Fig. 1 of the comment.
The special points carry a label 1+7*n, where n is an integer.
The second column of the (lmax=9) data contains
eigenvalues (zeros of the photonic KKR determinant).
I have used such a high value of lmax to maintain precision
of eigenvalues within 0.001 in the whole
frequency range. In order to have an idea of the
convergence properties of the photonic KKR method,
you can download here the respective (lmax=2) data
and (lmax=8) data for
the two lowest photonic bands, calculated with the respective lmax=2 and lmax=8.
You will find that, already for lmax=2 (i.e., with the secular matrix
of the size 36 x 36), the lowest two bands are determined
within 6% of the converged values (see figure).
The (lmax=8) data
are within 0.001 of the (lmax=9) data
For a comparison of the low frequency dispersion with the Garnett formula [5,6]
see comment and Ref. .
For those who would like to reproduce these results,
the cutoff on the number of direct lattice vectors used in the Ewald summation
was set to NRMAX=343, whereas the cutoff on the number of
reciprocal lattice vectors used in the
Ewald summation was set to NKMAX=1400.
The value of the Ewald parameter in the photonic KKR method  was taken to be Q=1.6
(in the notation of Ref. ). This choice of the Ewald parameter
ensured that the two cutoffs NRMAX and NKMAX have never been exceeded
in my calculations and yet ensured that the structure constants have been
determined within 5 digits of accuracy in the whole
In the comment, it has been discussed that
results computed in early days of the plane-wave expansion method have often
turned out imprecise, an example being Ref. . Indeed, for a
two-dimensional square lattice of air circular rods with a filling fraction
f=67% in a dielectric material of refractive index n=4.25,
Villeneuve and Piche predicted a 2D complete band gap of 11%
(see Figs. 1b and 2 of Ref. ). However, the 2D bulk photonic KKR method 
predicts that the complete band gap actually disappears
That the results of Villeneuve and Piche  are imprecise, has been, privately,
confirmed by the authors. This can also be verified
independently, using a transfer-matrix of Pendry and McKinnon .
Seemingly, when using the plane-wave expansion method, one has to exercise
extreme care when handling systems with a stepwise changes in the dielectric
contrast, which is a generic case of photonic crystals. The reason is an
unaviodable presence of the Gibbs instability, resulting in a poor convergence of
the Fourier transform of the hard-sphere dielectric function, as has been nicely discussed
in Ref.  (see, for instance, Fig. 2 there). Even up-to-date variants of
the plane-wave expansion method, in particular the
MPB package of Steven G. Johnson, still needs an extrapolation
to infinitely many plane waves to yield converged results.
Recently, using the MPB package, Mischa
Megens performed such an extrapolation of the plane wave method results
obtained with N**3 plane waves, where N=16 and
32. A linear extrapolation of the
N=16 and N=32 frequencies as a function of
1/N to 1/N --> 0 yielded result almost identical to that of
the photonic KKR method but for the two lowest bands (see Fig.
here). The band edges of the two lowest bands are
cca 1.5% lower than those calculated by the photonic KKR method. A convergence
of the 2nd-3rd gap width as a function of N is presented
here. The last two figures are courtesy of
Mischa Megens who also kindly provided his control file, which you
can download here.
Prof. C. Soukoulis has
agreed on the validity of my calculations.
A follow-up 
has been recently published, which deals with photonic
band structure calculations for diamond and pyrochlore
crystals and which compares the KKR results against those obtained by
the MIT-package. Please, see for details
Some additional information you can find in the PhD thesis of Esther C. M. Vermolen
on Manipulation of colloidal crystallization (cca 9.4 MB) from 2008.
- K. M. Ho, C. T. Chan, and C. M. Soukoulis, Existence of a photonic gap
in periodic dielectric structures,
Phys. Rev. Lett. 65, 3152-3155 (1990).
- E. Yablonovitch, T. J. Gmitter, and K. M. Leung, Photonic band structure:
The face-centered-cubic case employing nonspherical atoms,
Phys. Rev. Lett. 67, 2295-2298 (1991).
- A. Moroz and C. Sommers,
Photonic band gaps of
three-dimensional face-centered cubic lattices,
Phys.: Condens. Matter 11,
[story behind this article]
- A. Moroz, Metallo-dielectric diamond and zinc-blende photonic
Phys. Rev. B 66, 115109 (2002).
- J. C. Maxwell Garnett, Colours in metal glasses and in metallic films,
Phil. Trans. R. Soc. London 203, 385-420 (1904).
- A. Moroz, A simple formula for the L-gap width of a
face-centered-cubic photonic crystal,
Opt. A: Pure Appl. Opt. 1,
- A. R. Williams, J. F. Janak, and V. L. Moruzzi,
Exact Korringa-Kohn-Rostoker energy-band method with the speed
of empirical pseudopotential methods,
Phys. Rev. B 6, 4509-4517 (1972).
- P. R. Villeneuve and M. Piche, Photonic band gaps in two-dimensional
square lattices: Square and circular rods,
Phys. Rev. B 46, 4973-4975 (1992).
- H. van der Lem and A. Moroz, Towards two-dimensional complete
photonic-bandgap structures below infrared wavelengths,
Opt. A: Pure Appl. Opt. 2, 395-399 (2000).
- J. B. Pendry and A. MacKinnon, Calculation of photon dispersion relations,
Phys. Rev. Lett. 69, 2772-2775 (1993).
- H. S. Sozuer, J. W. Haus, and R. Inguva, Photonic bands: Convergence problems
with the plane-wave method,
Phys. Rev. B 45, 13962-13972 (1992).
- E. C. M. Vermolen, J. H. J. Thijssen, A. Moroz, M. Megens, and A. van Blaaderen,
Photonic band structure calculations for diamond and pyrochlore crystals,
Optics Express 17(9), 6952-6961 (2009).
My personal webpage
Full list of my scientific publications
List of my publications
Top-10 of my most frequently cited articles
Selected Links on Photonics,
Photonic Crystals, Numerical Codes, Free Software
Alexander Moroz, August 1, 2001 (last updated on April 1, 2011)