.. _fefd-intro:

Finite-Element Frequency-Domain (FEFD) Electromagnetic Solver
==============================================================

.. contents::
   :local:
   :depth: 2
   :backlinks: top

**Based on:**

Jianming Jin,  
*"The Finite Element Method in Electromagnetics"*, 3rd Edition,  
Wiley Publishing, 2014. ISBN: 9781118571361


Tsuji, Yasuhide, and Masanori Koshiba,
*"Finite element method using port truncation by perfectly matched layer boundary conditions for optical waveguide discontinuity problems."*, 
Journal of lightwave technology 20, no. 3 (2002): 463-468.


Ivanova, R. Stoffer, and M. Hammer, 
*"A dimensionality reduction technique for 2D scattering problems in photonics,"*
J. Opt., vol. 12, no. 3, Art. no. 035502, Mar. 2010.


Introduction
---------------

This solver implements the **Finite-Element Frequency-Domain (FEFD)** method for solving Maxwell’s equations in 2D space and frequency domain, specifically optimized for the simulation of **integrated photonic devices**.

It supports **linear, isotropic, dispersive** materials and provides advanced features tailored to photonic circuit design, such as **automatic mode injection**, **port detection**, and **S-parameter extraction**.

Key Features
--------------

- **Frequency-domain Maxwell solver** using triangular discretization in 2D space.
- **Perfectly Matched Layer (PML)** boundary conditions for accurate open-region simulations.
- **GDSII Import**: Reads standard GDSII layout files to define device geometry.
- **Automatic Matching Mesh Generation**: Automatically generates a triangular mesh that precisely matches the passed geometry.
- **First and Second Order Element** Support: Choose between linear and quadratic elements for improved accuracy. 
- **2D and 2.5D simulation**: Simulate 2D geometries and collapse 3D ones into 2D with acceptable accuracy.
- **Port Identification & Mode Matching**:
  - Automatic port detection and classification.
  - Single-mode excitation using computed eigenmodes.
  - Mode solver integration for accurate injection and extraction.
- **S-Parameter Computation**:
  - Computes scattering parameters (S-matrix) from frequency-domain data.
  - Automatically exports results in standard `.snp` format.
- **HDF5 Output**:
  - All simulation data, field snapshots, and metadata are stored in `.h5` format for efficient post-processing.

Mathematical Foundation
-----------------------

The FEFD method relies on discretizing the 2D Helmholtz wave equation using triangular elements.

.. math::

   \begin{align}
   \nabla^2 u(x,y) + k^2(x,y)\,u(x,y) = -s(x,y),
   \end{align}

Where :math:`s` is the source, :math:`u` is either :math:`E_z` or :math:`H_z` for either TE or TM, respectively.

.. math::

   \begin{align}
   \nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2},
   \end{align}

and

.. math::

   \begin{align}
   k(x,y) = \frac{\omega}{c}\, n(x,y) = \omega \sqrt{\mu(x,y)\,\varepsilon(x,y)}.
   \end{align}

The 2.5D formulation is based on the method reported in:
"A dimensionality reduction technique for 2D scattering problems in photonics"

In which the field :math:`u` is decomposed into two orthogonal components :math:`Z(z)` and :math:`v(x,y)`.
Then, the variational formulation is applied in the :math:`z` direction by multiplying by a weight function :math:`Z'(z)`.
Then the resulting wave equation is integrated in the :math:`z` direction, resulting in a wave equation in :math:`x` and :math:`y`. 

Boundary Conditions
-------------------

This solver supports **Perfectly Matched Layer (PML)** absorbing boundary conditions, as described by Y. Tsuji and M. Koshiba in "Finite element method using port truncation by perfectly matched layer boundary conditions for optical waveguide discontinuity problems".
PML is used to prevent artificial reflections at simulation domain boundaries.


This is the core equation that defines the coordinate transformation in the frequency domain. For a coordinate direction :math:`i` (where :math:`i = x, y, \text{or } z`):

.. math::
      \begin{align}
      s_i(i, \omega) = \kappa_i(i) + \frac{\sigma_i(i)}{\alpha_i(i) + j\omega\epsilon_0}
      \end{align}
      
      
Where:

* :math:`\kappa_i` is the real coordinate stretching (usually :math:`\ge 1`).
* :math:`\sigma_i`: Conductivity profile (attenuation).
* :math:`\alpha_i`: Complex frequency shift (stability).
* :math:`\epsilon_0`: Vacuum permittivity.



Applications
------------

This solver is particularly suited for:

- Silicon photonics and integrated optical circuits
- Passive photonic devices (e.g., splitters, multiplexers, filters)
- Waveguide tapering, crossings, and junctions
- Irregularly shaped devices
- Dispersive and lossy materials
- Single frequency optimization
- High index contrast problems (First order elements are recommended for :math:`\delta n < 2` and second order for :math:`\delta n > 2`)

This solver is not suited for:

- Devices of thin layers (:math:`< 20 nm`) in the :math:`z` direction.