flex_rf.tidy3d.PML

Type: class │ Base(s): AbsorberSpec

Description

Specifies a standard PML along a single dimension.

Notes

1D Model Illustration

Consider a transformed wave equation in the $x$ dimension below _[1]:

\left( \left( \frac{1}{s(x)} \frac{\delta}{\delta x} \right)^2 - \frac{1}{c^2} \frac{\delta^2}{\delta t^2} \right) E = 0

where the wave stretch factor $s(x)$ depends on the PML boundary position in the $x$ dimension.

.. TODO what is x at 0?

s(x) = \left \{ \begin{array}{lr} 1, & \text{for } x < 0 \\ 1 - \frac{\sigma}{i \omega \epsilon_0}, & \text{for } x > 0 \end{array} \right \}

The wave equation can be solved and plotted accordingly as a function of the $x$ dimension.

E(x) = \left \{ \begin{array}{lr} e^{i(kx - \omega t)}, & \text{for } x < 0 \\ e^{i(kx - \omega t)} \times e^{-\frac{\sigma x}{c \epsilon_0}} & \text{for } x > 0 \end{array} \right \}

Hence, we see how this PML stretch factor induces frequency-independent exponential attentation and no reflection after the boundary at $x=0$ .

.. TODO make this image better

Usage Caveats

A perfectly matched layer (PML) is the most commonly used boundary condition in FDTD simulations to truncate a simulation domain and absorb outgoing radiation. However, many divergence issues are associated with the use of PML. One of the most common causes of a diverged simulation is structures inserted into PML at an angle.

.. TODO links to absorber boundaries

Incorporating a dispersive material into the PML can also cause simulation divergence in certain scenarios. If your simulation lacks any structures inserted into the PML at an angle, but includes dispersive material in PML, it is advisable to substitute a nondispersive material for the dispersive material. Alternatively, if dispersion is necessary, switching from the PML to Absorber can effectively address the issue.

The PML can effectively absorb outgoing radiation with minimum reflection as if the radiation just propagates into the free space. However, it’s important to keep in mind that the PML only absorbs propagating fields. For evanescent fields, the PML can act as an amplification medium and cause a simulation to diverge. In Tidy3D, a warning will appear if the distance between a structure is smaller than half of a wavelength to prevent evanescent fields from leaking into PML. In most cases, the evanescent field will naturally die off within half a wavelength, but in some instances, a larger distance may be required.

Example(s)

pml = PML(num_layers=10)

References

[1] W.C. Chew and W.H. Weedon, Microwave and Optical Tech. Lett., 7 (13), 599,1994; S. Johnson, arXiv 2108.05348, 2021

[2] Antonios Giannopoulos, IEEE Transactions on Antennas and Propagation, 56(9), 2995, 2008

Practical Advice

For best results, structures that intersect with the PML or simulation edges should extend all the way through using td.inf:

Structure(geometry=Box(size=(td.inf, width, height)), medium=core)

Structures that terminate inside PML can cause evanescent fields at the interface to be amplified by the absorber, potentially leading to simulation divergence.

Parameters

name [str | None] = None

Optional unique name for boundary.

num_layers [int] = 12

Number of layers of standard PML.

parameters [PMLParams] = DefaultPMLParameters

Parameters of the complex frequency-shifted absorption poles.

extrude_structures [bool] = True

Automatically extrude structures into the absorbing region (e.g., PML or adiabatic absorber). Any structure located within 2 cells of a simulation boundary will be extended through the full thickness of the PML/absorber. The extruded region is assigned the material properties of the structure 2 cells from the simulation boundary. Extrusion is performed along the direction normal to the PML/absorber surface.