Transfer matrix method tool (tmm-tool)

This tool implements the transfer matrix method (TMM) to describe the propagation of electromagnetic (EM) waves in stacked layered materials at normal incidence. It uses a 4x4 matrix formalism which can account for the propagation of the EM fields with both x and y components. The EM waves are described using the Jones vector formalism. This allows the user to study the behaviour of the system with respect to the polarisation of light. Also, any layer can be arbitrary rotated by a given angle, thus allowing to study complex responses such as linear and/or circular dichroism.

Installation

Preferred installation: pip

The preferred installation method is by using pip. This will allow the package to be installed in Windows, Unix and MacOS. In the release page, simply download the .zip file and extract it at your favorite location. Then move into the folder and run

pip install .

This will install two things:

• the tmm-tool python package
• and the tmm-tool shell command.

If you are using anaconda as you python environment, you can check the succesfull installation of the package by typing conda list.

To uninstall type pip uninstall tmm-tool.

Stand-alone installation

If you are not confortable in using pip and prefer not to install the system-wide tmm-tool shell command, simply copy-paste the source and run the tool from within the base folder.

Requirements

This tool requires:

• python >=3.5
• numpy
• matplotlib

Usage

There are two ways of using tmm-tool:

• as a shell command via ~$ tmm-tool <my_input_file>
• as a python package via from tmm.material import Material

Shell command usage

Invoking the tmm-tool from the command line is the easiest way to start using the tool. Simply log into the terminal and type

tmm-tool <my_input_file> (for the pip installation case), or

python tmm-tool.py <my_input_file> (stand-alone installation, from the base directory).

The tool requires a file <my_input_file> in json format. An example of this is given in input.txt and a thorough description of the json (key, value) pairs is given in here.

The software will (in order):

• initialise the optical constants of the selected materials
• initialise the amplitudes of the incident electric fields
• construct the heterostructures (layers) given in input
• compute the total reflection, transmission, absorption coefficients (and more)
• output the data to the destination folders

At the moment, the command line code only computes the total transmission, reflection or absorption of the system. In other words, the code computes the transfer matrix $T = T_{if}$ between the initial (i) and final (f) layers. If the user wants to compute the coefficients for the intermidiate layers (e.g. as in a echo removal calculation), the python package (see below) should be used. This feature will be implemented in the next versions.

Python package usage

To use tmm-tool as a package, you simply need to import the Material class

from tmm.material import Material

For a very basic usage you can copy the following snippet

from tmm.material import Material

materials = {"air" : 1.0, "sapphire" : [3.07, 0.07] } # optical constants
efield    = {"left_pol" : [1.0, "left"]}              # Jones vector
freqs     = [0.0, 6.0, 1000]                          # frequency in eV
geometry  = [ ["air", "sapphire", "air"],             # material arrangement
              [100.0],                                # sapphire thickness
              [0.0, 0.0, 0.0] ]                       # angles of every single layer (rad)

# the calculation starts
mat = Material(materials, efield, freqs)
mat.setGeometry(*geometry)
mat.TMM()
mat.calculateCoeff()
mat.plotResults('left_pol')

Output

The tool computes the complex electric field amplitudes (incident, reflected and transmitted) and the real electric field intensities (transmitted, reflected and absorbed). Also, it computes the phase difference between the x and y component of the electric field (incident, reflected and transmitted). The package provides the plotting routine mat.plotResults(). For example, left circular polarised light going through a 100nm thick stack of $\pi/2$ twisted carbon nanotubes produces the following results

Roadmap

The project is stable enough to deserve a release and offers the basic capabilities of the transfer matrix method. However, several improvements are on the making and will include:

• a more efficient and versatile input file for scripting;
• a routine to perform the removal of echos between layers;
• a routine to calculate linear and circular dichroism;
• the possibility of using different unit of measures (wavelength vs. frequency vs. energy);
• an integration with the RefractiveIndex database, which provides the necessary optical constants.

Support

For any problems, questions or suggestions, please contact me at tenobaldi@gmail.com.

Authors and acknowledgment

The development of this tool is proudly powered by me. Also, please consider citing the relevant literature if you are going to use this tool:

• Phys. Rev. B 107, 144407 (2023)
• Preprint arXiv:2206.00895 (2022)
• Phys. Rev. B 104, 155437 (2021)
• J Infrared Milli Terahz Waves 42, 1142–1152 (2021)

License

This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.

This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with this program. If not, see https://www.gnu.org/licenses/.