Summary:
This thesis is titled “Extensions of the finite-difference time domain method in frequency dependent media”. It is a mix of computational physics and electromagnetism, using the appropriate programs and software we are able to apply the method (FDTD) to approximate the electromagnetic theory having as a final goal the simulation of media that have a frequency-dependent dielectric constant.
Initially, we refer to the basis of electromagnetic theory, we take a look into Maxwell’s equations as well as into the electric displacement field D and the auxiliary magnetic field Η, and then we examine the manner in which we derive all the formulas in one dimension from the standard three.
After that, we continue by establishing the finite-difference method by looking at Yee’s algorithm and also the mathematical theory of finite differences. Following that, we are able to create the repeating loop that will calculate the new values of all fields which means we can proceed with the modelling of an electromagnetic wave.
In order to do so, we have to transfer the loop from mathematical formulas to a programming language, in our case we will use MATLAB, and according to the initial conditions we ourselves insert, we will create the modelling of a certain E/M wave.
We begin with the modelling of a wave in empty space and then we take a look at an incident wave on a dielectric boundary. The waves we use are : sine, gaussian and wavepackets (sine waves multiplied by a gaussian wave). After we understand how the modelling of E/M waves works, we can move on to our final goal : extend the finite-difference time domain method in frequency dependent media.
Initially, we become more acquainted with the new theory, that refers to dielectric constants that are frequency dependent, and then we will examine various frequency-dependent media with greater detail. More specifically, we study Debye, Drude and Lorentz media. Also, we see the different methods that we can use to model these media and we will reach the conclusion that each media can be modelled more easily with a certain method, although it remains possible to model all media with every method. We look into the recursive convolution method, the auxiliary differential equation method as well as the Z-transform method.
Lastly, there is an appendix that contains programs written in MATLAB that one can use to run every simulation that is illustrated in this thesis.
Keywords:
FDTD, propagation of E/M waves, frequency dependent media, Maxwell's equations, dielectric, Yee's algorithm, repeating loop, free space, MATLAB, wavepacket, finite differences
