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 Band-gap 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 Martín-Moreno, AJ Ward, Computer Physics Commun. to appear (1995).

The Transfer Matrix

Start with tex2html_wrap_inline40 and tex2html_wrap_inline42 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, tex2html_wrap_inline46, is obtained as the eigenvalues of the transfer matrix, tex2html_wrap_inline48

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