# OPTICS OF

# COMPLEX SYSTEMS

## Finite Difference Time Domain

Finite Difference Time Domain is a simulation tool to study electromagnetic wave propagation in complex geometries. The core of FDTD simulation is based Maxwell's equations, discretised in time and space using finite differences. For each time step , one computes the magnetic field from the electric field distribution at the previous time step, and then, knowing the magnetic field distribution, updates the electric field distribution for the next time step.This methods allows then to compute, for each point of the space of interest, the time-dependance of the electromagnetic field.

This simulation tool then allows one to have the time-dependant response of the system for a given excitation, as well as te spatial

distribution of the electromagnetic fields. The frequency response of the system is accessible by Fourier transforming the results of the simulations.

One of the advantages of this simulation tools is that the frequency response of the system over the whole frequency range of interest

is accessible from one single simulations.

These simulations are used to compute reflections and transmission spectra of 3D photonic crystal, to compute the modes of 2D photonic crystal cavities, or the localised modes in 2D random systems.

In order to run this type of simulations, we have set a cluser of 10 processor, on which is installed the MEEP program, from MIT. We also have installed Crystal Wave from PhotonD.

## Plane-wave expansion

Plane wave expansion methods is, at the opposite of FDTD, a frequency-domain method. We solve here Maxwell's equations expressed as an eigenvalue problem. This method is used mainly to solve the band structure of photonic crystal, and compute bandgaps, group velocity, dispersion and photonic density of states.To run this kind of simulations we have installed on our cluster the MPB progam developed at MIT.

## Monte carlo methods

If one is not interested in the peculiarities of light transport due to its wave nature, it is possible to study the system of interest by mean of Monte Carlo Simulations.In Monte Carlo Simulations, one follows the trajectory of a light packet multiply scattered inside the system. The principle of the simulation is based on the generation of pseudo-random number, which allow to simulate the random scattering process inside the studied material. Monte Carlo simulations are perfectly adaptated to simulate the physics of transport in complex disordered systems.

To run this type of simulations, we have different home-made Monte Carlo code, that can be used to study a great variety of systems.

## References

The finite difference time domain method for electromagnetics - K.S. Kuntz, R.J. Luebbers, CRC Press (1993).

**Photonic
Crystals: Molding the Flow of Light**
- J.D. Joannopoulos,
R.D. Meade, J.N. Winn, Princeton
University Press (1995).