The code in this repository aims to solve two specific eigenvalue problems commonly encountered in engineering electromagnetic (including nanophotonic) waveguides and resonators. The code was originally written in MATLAB and has been recently verified to work in Octave.
The repository also serves as a code accompanying our paper First-Principles Full-Vectorial Eigenfrequency Computations for Axially Symmetric Resonators and includes code to reproduce the figures in the paper.
Eigenvalue problems (or computations) in electromagnetic engineering are encountered in two forms: eigenmode and eigenfrequency. Both forms have the same parent eigenvalue equation, which we call the Master equation. The Master equation is derived from time-harmonic Maxwell's equations and its detailed derivation is shown in the appendix section of the paper. An eigenmode problem results when the operating wavelength is fixed and we solve the Master equation for the allowed mode index(es) at that wavelength. Conversely, an eigenfrequency problems results when we fix the mode index and solve the master equation for the allowed resonant frequencies in the photonic structure.
Mathematically, the structure of the full vectorial electromagnetic eigenvalue problem is:
In the eigenmode problem the frequency ω (or wavelength) is known, we move it to the left hand side and solve for the mode index ν. This situation applies to the computation of the propagating modes of the waveguides or optical fibers. When the mode index ν is known, we move that to the left hand side and solve for the resonant frequency ω. This situation applies to the computation of the resonant frequencies of microring, microdisk and microsphere resonators. In both cases the solution of the eigenvalue problem yields the eigenvalue itself (mode index or the resonant frequency) and the eigenvectors (the modal or resonant fields).
Full vectorial refers to including the cross terms in the LHS matrix. The two field components are coupled due to the cross terms. Semi vectorial refers to excluding the cross terms (and hence the coupling) and computing the mode indices or resonant frequencies and their associated fields using only the diagonal terms. Semivectorial solution reduces the computation by one fourth at the expense of loss of accuracy. For large structures with linear materials, semivectorial solution may be enough. For small components with high refractive indices a full vectorial solution may be needed. The present code implements only a full vectorial solution. If a faster, semivectorial solution is desired, it is straightforward to modify the code.
To install, simply clone or download the code. The core mathematics is
implemented in the files contained in /lib/cyl and /lib/cart. The
apps directory shows how to simulate commonly encountered waveguide
structures. We have included examples for rectangular waveguide on a
substrate, an optical fiber, a microring resonator, a microdisk
resonator and a microsphere resonator.
You can run these examples either via bash commandline, Octave
console, or via the Octave GUI. For example, to run the microdisk example
(microdisk.m) via the commandline, type
# Make sure you're in the `apps` directory
$ cd apps
$ octave microdisk.m
To run the same example via Octave console, start Octave
# Make sure you're in the `apps` directory
$ cd apps
$ octave
This will open the octave console. You can type the name of the
desired application microdisk and wait for the computation to
finish.
To run the same example via Octave GUI, start Octave GUI
# Make sure you're in the `apps` directory
$ cd apps
$ octave --gui
This will open the octave console. You can type the name of the
desired application microdisk and wait for the computation to
finish.
One advantage of using the console or the GUI is that the computed output stays in the memory and you can explore it further, for example by plotting the fields or creating additional info for your report or publication.
The code includes basic field plotting facilities. The main function
for plotting the fields is plot_modes() That function accepts the
input and output structures (IN, OUT), the list of solution to be
plotted (e.g. [1 5 9 20]) and the plotting scale (lin or log).
Code in the paper folder reproduces the figures in the publication.
