Methods of Calculations

There are two common computation approaches to studying dielectric structures such photonic crystals: time-domain and frequency domain. Each has its own advantages and disadvantages. Frequency-domain method does a direct computation of the eigenstates and eigenvalues of Maxwell's equations (band structure). Each field computed has a definite frequency. So, frequency-domain methods are much better suitable to calculating band structures, but couldn't applied directly to calculate time-dependence processes.

In contrast, time-domain techniques iterate Maxwell's equations in time. Computed fields defined in time (at each time step) but not have a defined frequency. Time-domain method is well-suited for simulations of wavepacket
propagation, transmission, etc. It also can be used to calculation of band structures making use the Fourier
transform. The advantage of this is that you get also band structure at once from a calculation of propagation of a single wavepacket.

Maxwell's equations for both electric and magnetic fields are iterated directly in real time and space using completely general coordinate system. Then time dependent information is Fourier transformed into the frequency domain. Finally, through analyzing of frequency domain information one could obtain a structure of photonic bands as well as
transmission and reflection coefficients and spectral characteristics of photonic density of states.

Method [1] is based on an extension of the finite difference time domain (FDTD) method, originally proposed by Yee [2] (Yee algorithm), also known as the Order-N method [3], which has recently become a popular way of calculating photonic band structures. It is a simple scheme for calculating Green's functions for photons propagating in complex structured dielectrics.

A.J.Ward, J.B.Pedry. Computer Physics Communication 128, 590-621, 2000.
K.S.Yee, IEEE Trans. on Antenna and Propagation 14, 302, 1966.
C.T.Chan, Q.L.Yu, and K.M.Ho, Phys.Rev. B 51, 16635, 1995.

Theory behind the Approaches

Most popular Freewares for Photonic Band Gap Calculations

Opto Electronics Research Group at the University of Glasgow
TMM Photonic Crystals & Virtual Crystals program by Andrew L. Reynolds

The Joannopoulos Research Group at MIT
The MIT Photonic-Bands package by Steven G. Johnson,

Imperial College , London ,Condensed Matter Theory Group
Order N electromagnetism program via a Finite Difference Time Domain scheme (A.J.Ward, J.B.Pendry)