Introduction
With the suggestion by Yablonovitch that a periodic array
of holes could give some very interesting properties to an otherwise
dull dielectric, theorists have been challenged to predict these
properties. The challenge is made more urgent by the experimental
difficulties of manufacturing materials containing micron scale
structure in three dimensions.
Theorists have been prompted by the
analogy with electrons in periodic solids which have inspired a host of
different techniques and an extensive literature. In case you wondered
why Condensed Matter Theorists are tangling with Optics:
here is the
reason! Maxwell's equations present their own unique challenges and
calculation of photon band structure is not a trivial extension of
electron band structure Central to our own approach has been the
concept of a transfer matrix (see the panel below) which
is embodied in
a set of computer codes, known as OPAL
(Optical Program ALgorithm)
which calculate the transfer matrix for a given system once the
permittivity and permeability have been specified as functions of
position and frequency. OPAL can be used to calculate photonic band
structure, transmission and reflection coefficients, and dispersion
surfaces.

Some References:
 Photonic Bandgap Crystals
E Yablonovitch, J. Phys. [Condensed Matter] 5 2443 (1993).
 Calculation of Photon Dispersion Relations
JB Pendry, A MacKinnon,
Phys. Rev. Lett. 69 2772 (1992).
 Photonic Band Structures
JB. Pendry. J. Mod. Optics 41 209 (1993).
 A Program for Calculating
Photonic Band Structures and Transmission Coefficients of Complex
Structures
PM Bell, JB Pendry, L Mart¡nMoreno, AJ Ward, Computer
Physics Commun. to appear (1995).

The Transfer Matrix
Start with and at one side
of the unit cell and use our equations to calculate the fields at the
far side of the cell at z=a,
and impose the Bloch condition,
thus the band structure, , is
obtained as the eigenvalues of the transfer matrix,
Commercial Break
The OPAL codes are available from the Computer Physics
Communications library at Queen's University, Belfast, for a small
charge to cover postage expenses. Details of how to obtain them can be
had from a current volume of Computer Physics Commun. available in the
Condensed Matter Theory Library. Alternatively, the OPAL codes can be
obtained by emailing your request to
a.j.ward@ic.ac.uk
