Electromagnetic Finite Difference Frequency Domain Solver

## Project description

![](img/dipole_dielectric_field.png)

# fdfdpy

This is a pure Python implementation of the finite difference frequency domain (FDFD) method. It makes use of scipy, numpy, matplotlib, and the MKL Pardiso solver. fdfdpy currently supports 2D geometries

## Installation

python setup.py install

## Structure

### Initialization

The Fdfd class is initialized as

simulation = Fdfd(omega, eps_r, dl, NPML, pol, L0)

- omega : the angular frequency in units of` 2 pi / seconds`
- eps_r : a numpy array specifying the relative permittivity distribution
- dl : the spatial grid size in units of L0
- NPML : defines number of PML grids [# on x borders, # on y borders]
- pol : polarization, one of {‘Hz’,’Ez’} where z is the transverse field.
- L0 : simulation length scale, default is 1e-6 meters (one micron)

Creating a new Fdfd object solves for:

- xrange : defines spatial domain in x [left-most position, right-most position] in units of L0
- yrange : defines spatial domain in y [bottom-most position, top-most position] in units of L0
- A : the Maxwell operator, which is used later to solve for the E&M fields.
- derivs : dictionary storing the derivative operators.

It also creates a relative permeability, mu_r, as numpy.ones(eps_r.shape) and a source src as numpy.zeros(eps_r.shape).

### Adding sources is exciting!

Sources can be added to the simulation either by manually editing the 2D src array inside of the simulation object,

simulation.src[10,20:30] = 1

or by adding modal sources, which are defined as planes within the 2D domain which launch a mode in their normal direction. Modal source definitions can be added to the simulation by

simulation.add_mode(neff, direction, center, width) simulation.setup_modes()

- neff : defines the effective index of the mode; this will be used as the eigenvalue guess
- direction : defines the normal direction of the plane, should be either ‘x’ or ‘y’
- center : defines the center coordinates for the plane in cell coordinates [xc, yc]
- width : defines the width of the plane in number of cells

Note that simulation.setup_modes() must always be called after adding mode(s) in order to populate simulation.src.

### Solving for the electromagnetic fields

Now, we have everything we need to solve the system for the electromagnetic fields, by running

fields = simulation.solve_fields(timing=False)

simulation.src is proportional to either the Jz or Mz source term, depending on whether pol is set to ‘Ez’ or ‘Hz’, respectively.

fields is a tuple containing (Ex, Ey, Hz) or (Hx, Hy, Ez) depending on the polarization.

### Setting a new permittivity

If you want to change the permittivity distribution, you may run

simulation.reset_eps(new_eps)

And this will reconstruct the system matrix and store it in FDFD. Note that simulation.setup_modes() should also be called if the permittivity changed within the plane of any of the modal sources.

### Plotting

Primary fields (Hz/Ez) can be visualized using the included helper functions:

simulation.plt_re(outline=True, cbar=True) simulation.plt_abs(outline=True, cbar=True)

These optionally outline the permittivity with contours and can be supplied with a matplotlib axis handle to plot into.

### Requirements

- numpy
- scipy
- matplotlib

To load the MKL solver:

git submodule update –init –recursive

### To Do

#### Whenever - [x] Modal source. - [x] More dope plotting methods. - [ ] xrange, yrange labels on plots. - [ ] set modal source amplitude (and normalization) - [ ] Add ability to run local jupyter notebooks running FDFD on parallel from hera. - [ ] Save the factorization of A in the Fdfd object to be reused later if one has the same A but a different b. - [ ] Allow the source term to have (Jx, Jy, Jz, Mx, My, Mz), which would be useful for adjoint stuff where the source is not necessarily along the z direction. - [ ] Clean up imports (e.g. import numpy as np to from numpy import abs, zeros, …)

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