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Programs to calculate the photonic band structure of crystals with spherical atoms
(copyright by R. Sprik)

The following programs calculate the photonic band structure of structures with a single spherical 'atom' per unit cell. The spheres must be non overlapping. Examples of such systems are photonic band gap materials based on colloidal crystals and air spheres in a dielectric matrix [1].

The programs use the well trusted plane wave expansion of the Maxwell equations in electric field or in magnetic field form [2,3] . Usually about 700 plane waves give convergence of the answers within a few percent.

The code is in FORTRAN 77 and uses some routines from the IMSL numerical library [4] for diagonalization etc. The programs have been compiled and tested on Pentium machines under Windows 95 and Windows NT 4.0 with Microsoft Powerstation 4.0 professional. With minor changes, the programs have also been compiled and run on an IBM workstation under AIX 3.5, and CRAY YMP [5] . A parallel version for the IBM SP2 [6] is available upon request. Although the programs have been tested with various inputs, there may still be bugs left. Checking the sanity of the output is always advised.

The programs and files are distributed for general non-commercial use to calculate the photonic band structure in colloidal type photonic band gap. When you use the programs in your work, due reference will be appreciated. Sent comments to: sprik@phys.uva.nl.

1. Files
2. The input file: 'pbs.inp'
3. The output file
4. The examples
5. References

1. Files:
All files are collected in one zipped file that can be downloaded and decompressed. Listed are the included files and a short description.

pbs.htm this file

Source code
pbse\pbse.f Electric field form of the problem
pbse\pbse.inc include file with parameters and definitions
pbsh\pbsh.f Magnetic field form of the problem
pbsh\pbsh.inc include file with parameters and definitions

Input files for 'pbse' or 'pbsh'
pbse\pbs1.inp, pbsh\pbs1.inp Close packed fcc photonic crystal of silica colloidal spheres in water
pbse\pbs2.inp, pbsh\pbs2.inp Close packed fcc photonic crystal of air spheres in silicon

Executables for Windows 95/NT 4.0 made with Microsoft Fortran Powerstation 4.0 professional
The executables can handle a maximum of 700 plane waves.
pbse\pbse.exe executable of pbse.f,
pbsh\pbsh.exe executable of pbsh.f

Log and output files of examples
pbse\e1.log, pbse\e1.out
pbse\e2.log, pbse\e2.out
pbsh\h1.log, pbsh\h1.out
pbsh\h2.log, pbsh2\h2.out

Batch jobs to produce the output examples
pbse\pbse.bat
pbsh\pbsh.bat

2. The input file: 'pbs.inp'

1 ax ay az real space lattice vectors for the crystal unit cell
2 bx by bz
3 cx cy cz
4 epsb epsa fill dielectric constant of matrix (epsb) and atom (epsa)
5 neig, nreci, Gcut number of eigen values to calculate ordered from low to high,
number of plane waves = (2*nreci+1)**3 grid around the origin will we selected with magnitude < Gcut
6 NRec, NRecInt
NREC = number of trajects in k-space
NrecInt = number of points per interval
7 RKx, Rky, RKz traject points in reciprocal unit vectors
8 … continued NRec lines

note:
Both the E-field and the H-field method use the z-direction as a preferential direction.
In the E-field method:
The electric field in the z direction is eliminated from the equations to reduce the size of the matrix that has to be diagonalized.
In the H-field method:
The z-direction is used to determine the aligning of the triad vector.
As a consequence, a zero component in the traject vectors should be avoided. It may lead to unpredictable behavior and error messages.

3. The output:

The program generates as output:

1) Logfile
Information is written to standard output file to verify the input parameters and to keep track of the progress of the program.
In the examples the log files are stored in files with extension '*.log'.

2) 'pbs.out'
An ascii output file with the results of the photonic band structure.
- The first two columns give an index for the k-vector and an index for the eigenvalue at that k-vector.
These indices are useful to make plots of the band structure using a spreadsheet or plot program.
- Three columns with the kx, ky and kz value respectively.
- Last column gives the eigenvalue divided by the refractive index of the host.

4. The examples:

Two examples are included in the distribution to test the performance. Each example is evaluated using the electric field and the magnetic field representation.

Example 1: FCC crystal of close-packed silica spheres in water

The bandstructure still looks a lot like the 'free photon band structure' in a background with an effective refractive index [1] . Only near the Brillouin zone stop band effects develop.

Example 2: FCC crystal of close-packed air spheres in silicium

This is one of the tougher problems to solve with plane wave methods. The high contrast of the refractive index and the sharp transition at the edge of the sphere results in difficulties with the expansion of the dielectric constants in reciprocal wave space [7] . Eeven with 1000 planewaves, a difference remains between what the E-field and the H-field method finds. It helps to use the numerical inverse of epsilon(k) to expand 1/epsilon(k), instead of the analytical expansion of 1/exp(k) (see discussion in [8] ).

5. References:

[1] W.L. Vos, R. Sprik, A. van Blaaderen, A. Imhof, A. Lagendijk, G.H. Wegdam, 'Influence of optical band structures on the diffraction of photonic colloidal crystals', Phys. Rev. B53, 16231 (1996). J. E.G. J. Wijnhoven and W.L. Vos. 'Preparation of Photonic Crystals Made of Air Spheres in Titania', Science 281, 802 (1998). See scientific output for a 'pdf' reprint.
[2] See e.g. J.D. Joannopoulos, R.D. Meade, J.N. Winn, 'Photonic Crystals; Molding the Flow of Light', Princeton University Press (1995).
[3] 'Photonic Band Gap Materials', ed. C.M. Soukoulis, NATO ASI Vol.E:315, Kluwer (1996).
[4] The following IMSL routines are used:
DLINDS Inversion of a positive definite symmetric matrix
DEVLRG Eigenvalues of a general real matrix
These routines can easily be replaced by equivalent routines from other libraries e.g. NAG: F01ABF, F02AFF
[5] Developed on the CRAY YMP supported by the National Computing Facility
[6] Developed on the IBM SP2 machine installed at SARA in Amsterdam
[7] H.S. Sozuer, J.W. Haus, R. Inguva, 'Photonic bands: Convergence problems with the plane-wave method', Phys. Rev. B45, 13962 (1992).
[8] K. Busch, S. John, 'Photonic band gap formation in certain self-organizing systems', Phys. Rev. E58, 3896 (1998).

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