Indisputably, one of the hallmarks in the study of photonic band gap structures is an article by Ho, Chan, and Soukoulis [1]. It opened the way for a fabrication of the first photonic structure with a complete photonic band gap (CPBG) [2] 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 here) fares much better than a simple face-centered-cubic (fcc) one. Namely:

**i)**the threshold value of the dielectric contrast eps to open a CPBG is 4 (8.2 for an fcc structure [3]),**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 contrast eps=12.96).

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

where

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 [3] was taken to be *Q=1.6*
(in the notation of Ref. [7]). 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
frequency range.

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. [8]. 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. [8]). However, the 2D bulk photonic KKR method [9]
predicts that the complete band gap actually disappears
(see figure).
That the results of Villeneuve and Piche [8] are imprecise, has been, privately,
confirmed by the authors. This can also be verified
independently, using a transfer-matrix of Pendry and McKinnon [10].
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. [11] (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 [12]
has been recently published, which deals with photonic
band structure calculations for __ diamond__ and

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,*

J. Phys.: Condens. Matter**11**, 997-1008 (1999). [physics/9807057] [story behind this article] - A. Moroz,
*Metallo-dielectric diamond and zinc-blende photonic structures*,

Phys. Rev. B**66**, 115109 (2002). [cond-mat/0209188] [pdf] - 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,*

J. Opt. A: Pure Appl. Opt.**1**, 471-475 (1999). [physics/9903022] - 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,*

J. 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).

Alexander Moroz, August 1, 2001 (last updated on April 1, 2011)