.. _fdtd-intro:

Finite-Difference Time-Domain (FDTD) Electromagnetic Solver
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.. contents::
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**Based on:**

A. Z. Elsherbeni, V. Demir,  
*"The Finite-Difference Time-Domain Method for Electromagnetics with MATLAB® Simulations"*, 2nd Edition,  
SciTech Publishing, 2015. ISBN: 9781613531754

Susan C. Hagness, Allen Taflove,  
*"Computational Electrodynamics: The Finite-Difference Time-Domain Method"*, 3rd Edition,  
Artech House, 2005. ISBN: 9781580538329

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

This solver implements the **Finite-Difference Time-Domain (FDTD)** method for solving Maxwell’s equations in time and space, specifically optimized for the simulation of **integrated photonic devices**. The implementation is grounded in the foundational works by Taflove, Hagness, Elsherbeni, and Demir, and adapts their rigorous mathematical formulation to a highly automated and scalable simulation environment.

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

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

- **Time-domain Maxwell solver** using second-order central difference in both time and space.
- **Perfectly Matched Layer (CPML)** boundary conditions for accurate open-region simulations (as per Elsherbeni and Demir).
- **GDSII/STL Import**: Reads standard GDSII or STL layout files to define device geometry.
- **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 time-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 FDTD method discretizes Maxwell’s curl equations using a **Yee grid**:

.. math::

   \begin{align}
   \frac{\partial \mathbf{B}}{\partial t} &= -\nabla \times \mathbf{E} \\
   \frac{\partial \mathbf{D}}{\partial t} &= \nabla \times \mathbf{H}
   \end{align}

In linear, non-dispersive, and isotropic media, we use:

.. math::

   \mathbf{D} = \varepsilon \mathbf{E}, \quad \mathbf{B} = \mu \mathbf{H}

Time stepping is performed using a leapfrog scheme:

- Update magnetic field :math:`\mathbf{H}` at half-integer time steps.
- Update electric field :math:`\mathbf{E}` at integer time steps.

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

This solver supports **Convolutional Perfectly Matched Layer (CPML)** absorbing boundary conditions, as described in Chapter 8 of Elsherbeni & Demir. CPML is used to prevent artificial reflections at simulation domain boundaries.

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