pynoza 0.3

Computation of time-domain solutions of Maxwell's equations using the Cartesian multipole expansion

pynoza

Python implementation of time-domain solutions of Maxwell's equations using the cartesian multipole expansion. In homogeneous and isotropic media, these solutions can be computed thanks to an explicit knowledge of the Green’s function.

Currently, the current density must be time-separable, i.e., $\mathbf{J}(t,\mathbf{x})=h(t)\mathbf{j}(\mathbf{x})$

The documentation is available here.

The theory behind the code can be found in E. Le Boudec et al., “The time-domain Cartesian multipole expansion of electromagnetic fields,” Sci Rep, vol. 14, no. 1, p. 8084, Apr. 2024, doi: 10.1038/s41598-024-58570-1.

Installation

To install directly from git, use pip install "git+https://github.com/eliasleb/pynoza".

Tutorial

The Python >= 3.10 script tests/test_EPFL_logo.py shows some examples. A Jupyter notebook is also provided. The general procedure is the following:

1. Compute the electric time-domain current moments $C_\mathbf{\alpha}^{J_i}(t)=h^{'}(t)\iiint\mathbf{x}^\alpha j_i(\mathbf{x}) d^3\mathbf{x}$. Define a corresponding Python function that returns the space-dependence, whose signature must be moment(ind: tuple[int, int, int]) -> list[Number, Number, Number]. For a given multi-index (a1, a2, a3), the function must return moment(a1, a2, a3) = [j1, j2, j3] where ji is
   $\iiint\mathbf{x}^\alpha \frac{\partial j_i(t,\mathbf{x})}{\partial t}d^3\mathbf{x}$

   The charge moments can be computed automatically by calling the function pynoza.get_charge_moment.

   The time-dependence is kept separate, see later.

2. pynoza works in natural units, as does meep (see https://meep.readthedocs.io/en/latest/Introduction/#units-in-meep).

3. Define numpy arrays for the coordinates of interest $x_1,x_2,x_3,t$, for example

   x1 = np.array([0, ])
   x2 = x1.copy()
   x3 = np.array([wavelength, 2*wavelength, ])
   t = np.linspace(-T, T, 100)
   

4. Define a function for the time-dependence, for example, a sympy symbolic expression

   t_sym = sympy.Symbol("t", real=True)
   h_sym = sympy.cos(2*sympy.pi*f*t_sym)
   

   or an array

   t = np.linspace(0, T, dt)
   h = np.cos(2 * np.pi * f * t)
   

   In the latter case, it is up to you to ensure that the sampling time dt is small enough to compute the highest-order derivative. We use np.gradient to compute the derivative; see the relevant numpy documentation.

5. Create a pynoza.Solution object with the given medium light speed (in natural units) and given multipole expansion order.

6. Run the recurse method to initialize the Green’s function approximation (this gets slower for high orders), and the set_moments method to pass the current and charge moments that you computed above.

7. Run the compute_e_field method to compute the electric field (or compute_b_field for the magnetic field) from Green’s functions approximations and the current moments. Under the hood, this method will integrate and differentiate the time-dependent function to the needed order.

Complete example

See applications/demos/demo_EPFL_logo.py or tests/test_simple_example.py.