How does the FDTD method work, and when should you use it? A guide to photonic simulation

What is the FDTD method?

FDTD, or Finite-Difference Time-Domain, is a finite-difference method in the time domain. Its foundation is Maxwell's equations, which describe the behaviour of electromagnetic fields. Unlike ray methods, in which light is represented by rays, FDTD describes the electric and magnetic fields directly.

This lets the program track how an electromagnetic wave:

• propagates through a structure;
• reflects off material boundaries;
• refracts;
• diffracts;
• interferes;
• scatters;
• changes its polarisation state;
• excites resonances.

OptiFDTD solves the full-vectorial form of the coupled Maxwell's equations in the time and space domain, so the method can be applied to complex geometries and structures in which wave effects matter.

Why does this matter?

In many photonic problems it is not enough to know where „the ray goes”. What also matters is the amplitude of the field, its phase, and how the electromagnetic fields interact with the structure. In such cases a full-wave model provides information that classical ray tracing does not.

How does FDTD simulation work?

The process can be represented, in simplified form, as: geometry → materials → source → grid → Maxwell's equations → successive time steps → E and H fields → analysis of results.

First the geometry of the device under study is defined, then material properties are assigned to the individual regions. Next you set:

• the light source, its position, wavelength or spectral range, and polarisation;
• the boundary conditions and the simulation region;
• the grid resolution;
• the detectors or observation planes.

The solver computes the evolution of the electromagnetic field over time. The result is not only the final transmission or reflection value, but also information about the spatial and temporal distribution of the fields. OptiFDTD provides, among others, time- and frequency-domain analysis, mode analysis, Poynting vector analysis, polarised-power analysis, and near-to-far-field transformation.

Maxwell's equations and the full-wave electromagnetic model

The most important feature of FDTD is that the method does not require the assumption that light can be treated as a set of rays. The starting point is Maxwell's equations, which in simplified terms describe the coupling between the electric field E and the magnetic field H.

In a classical FDTD solver the values of these fields are computed at successive points in space and successive instants in time. This makes it possible to reproduce wave phenomena without assuming in advance a particular mode of field propagation — the method simultaneously models propagation, scattering, diffraction, reflection, and polarisation.

The computational grid and the Yee cell

A computer does not solve Maxwell's equations for infinitely continuous space — the simulation region is divided into a large number of small cells. Classical FDTD uses the so-called Yee grid, in which the components of the electric and magnetic fields are staggered relative to each other. This allows an efficient numerical representation of the coupling between the fields.

In a 3D model the components Ex, Ey, Ez and Hx, Hy, Hz are taken into account. The more detailed the grid, the more cells have to be computed — which leads to one of the most important trade-offs in FDTD simulations: accuracy ↔ computation time ↔ memory. OptiFDTD supports, among others, a non-uniform grid, which lets you increase the resolution where it is actually needed.

How to choose the grid resolution?

There is no single universal grid resolution suitable for every model. You have to consider above all:

• the wavelength;
• the refractive index of the materials;
• the smallest significant geometry details;
• the required level of accuracy;
• the nature of the phenomenon analysed.

What matters is the wavelength inside the material, not just the wavelength in vacuum. Approximately, λ_mat = λ₀ / n, where λ₀ is the wavelength in vacuum, n is the refractive index, and λ_mat is the wavelength in the material. The higher the refractive index, the shorter the wavelength in the medium and the higher the resolution the model may require.

As a practical rule of thumb, roughly ten cells per wavelength are used, while also taking into account the effect of grid resolution on accuracy and numerical dispersion. This value should not, however, be treated as a universal guarantee of accuracy for every model.

Convergence test

It is worth running a convergence test in a project — for example three simulations: grid A, grid B with a higher resolution, and grid C with a still higher resolution. You then compare the parameter of interest. If the result changes less and less, that is an argument that the model is reaching adequate numerical convergence.

Time step and simulation stability

FDTD is a time-domain method, so besides discretising space you have to set the time step. It is linked to the cell size and the wave propagation speed — for the classical FDTD scheme a stability condition related to the Courant–Friedrichs–Lewy (CFL) criterion has to be met.

In practice this means that a smaller cell requires a correspondingly smaller time step. This is another reason why increasing the grid resolution can significantly raise the computational cost — the cell size should not be analysed in isolation from the time step.

Boundary conditions and PML

A numerical model has a finite size, whereas a wave can leave the structure under study. If the wave reached the artificial boundary of the computational region and reflected off it, it could disturb the result. That is why many simulations use a PML (Perfectly Matched Layer) — an absorbing layer whose job is to absorb the wave leaving the domain as effectively as possible.

OptiFDTD uses UPML (Uniaxial Perfectly Matched Layer). The program also supports, among others:

• PEC (Perfect Electric Conductor);
• PMC (Perfect Magnetic Conductor);
• PBC (Periodic Boundary Conditions).

The choice of boundary condition should follow from the physics of the problem. A periodic structure may require a different approach than an open component from which the wave should radiate freely into the surroundings.

Which phenomena can be analysed with FDTD?

The FDTD method is especially valuable when significant wave effects occur in a project. You can analyse, among others:

• Diffraction — how a wave propagates after passing through a diffractive structure.
• Interference — because the electromagnetic fields are modelled, amplitude and phase relationships can be taken into account.
• Scattering — analysis of scattering on structures and particles.
• Reflection and transmission — determining the parameters that describe the flow of energy through a structure.
• Polarisation — the full-vectorial nature of the solution lets you analyse the behaviour of different field components.
• Resonances — analysis of resonant structures such as micro-ring resonators and other photonic cavities.
• Nonlinear phenomena — OptiFDTD supports nonlinear material models, including those related to the Kerr effect and the Raman effect.

When is FDTD worth using?

The best criterion is not the size of the element itself, but the question: does the result depend on the full-wave behaviour of the electromagnetic field?

FDTD is worth considering when analysing:

• structures with dimensions comparable to the wavelength scale;
• diffractive elements;
• photonic crystals;
• resonators and photonic filters;
• microstructures;
• integrated-photonics components;
• systems with complex geometry;
• structures that require polarisation analysis;
• materials or devices in which nonlinear effects occur.

OptiFDTD is intended, among others, for simulating photonic crystals, optical filters and resonators, grating structures, diffractive micro-optics elements, plasmonics, and complex integrated-optics structures.

When is FDTD not the best choice?

A full-wave model does not always mean the best model. If you have a very long and relatively simple waveguide, running a full FDTD simulation of the whole structure may be unnecessarily expensive — a better solution is then BPM (Beam Propagation Method), e.g. OptiBPM.

If, on the other hand, you mainly want to determine the eigenmodes, the effective refractive index, or the modal field distribution, the natural first step is mode analysis, e.g. OptiMode.

That is why the proper workflow often looks like this: mode analysis → BPM → FDTD. Not all three stages are always necessary, but each of them corresponds to a different level of the problem.

OptiFDTD — software for FDTD simulation

OptiFDTD is Optiwave's solution for designing and analysing photonic components with the FDTD method. The program covers a geometry design environment, simulation setup, an FDTD solver, and analytical tools — including the Designer, Simulator, and Analyzer modules and tools for band and mode analysis.

Key capabilities:

• 2D and 3D simulations;
• a full-vectorial FDTD solver;
• a non-uniform grid;
• UPML and periodic boundary conditions;
• mode analysis;
• Poynting vector and polarised-power analysis;
• near-to-far-field transformation;
• time- and frequency-domain analysis;
• simulation automation and parameter sweeps.

The program also lets you import geometry from popular CAD formats, including DXF and GDSII, which makes the move from a layout design to a simulation model easier.

OptiOmega — FDTD and mode analysis for integrated photonics

In designing PIC (Photonic Integrated Circuits), running an FDTD simulation is often not the end of the process — you also need a way to pass the results to the next modelling level. In this context a useful solution is OptiOmega, which combines a VFD (Vector Finite Difference) mode solver with a three-dimensional FDTD solver.

The FDTD solver in OptiOmega uses GPU acceleration, and the environment provides, among others, automatic port detection, placement of input planes and monitors, and S-parameter extraction. This lets you build a more complete flow: mode analysis → FDTD simulation → parameter extraction → component model → circuit simulation.

OptiOmega can also work with other Optiwave tools, including OptiSystem and OptiSPICE, which lets you use the results of the component analysis in further modelling.

FDTD, BPM, and mode analysis — which method to choose?

In practice the choice of method should follow from the design question.

So it is not about choosing „the best program”, but the most appropriate physical model for the specific problem.

FDTD in photonic component design

In the photonics industry, FDTD simulation can be part of the whole design process. Example: a designer creates a photonic filter — first they define the material, geometry, wavelength, and input conditions, then run an FDTD simulation and analyse the field distribution, transmission, reflection, and spectral response.

If the result does not meet the requirements, the design parameters are changed — waveguide width, spacing between elements, resonator radius, structure period, fill factor, or material properties — and another simulation is run. With a larger number of parameters, the process can be partly automated with parameter sweeps and scripts; OptiFDTD offers automation and parametric-simulation tools.

From component simulation to a system-level model

One of the most important trends in photonic design is moving away from treating every element as an independent model — a component should be usable in a larger system. An example workflow:

• geometry — we create the component structure;
• mode analysis — we check which modes are guided;
• FDTD — we analyse the full-wave behaviour of the component;
• parameter extraction — we determine the characteristics needed for further modelling;
• reduced model — the component is represented by parameters appropriate for a higher simulation level;
• photonic system — the component is used in a larger circuit.

This approach reduces the need to run a full electromagnetic simulation of the whole system.

The most common mistakes in FDTD simulations

• „A denser grid is always better” — a denser grid may increase accuracy, but also the computational cost; the goal is sufficient resolution for the given problem, not the maximum number of cells.
• Choosing the grid based only on the vacuum wavelength — you have to account for the wavelength in the material and the smallest significant geometry details.
• Too small a domain — the boundary conditions can then affect the result.
• Incorrect boundary conditions — a wrong setting of PML, PBC, PEC, or PMC leads to results that do not correspond to the problem being studied.
• No convergence test — a single result from one grid configuration is not sufficient proof that the model is correct.
• Modelling every CAD detail — not every geometry element is electromagnetically significant; unnecessary details significantly raise the simulation cost.
• Treating FDTD as a black box — a good model requires understanding the source, the materials, the grid, the boundary conditions, the detectors, the simulation time, and the convergence criteria.

What does a professional FDTD workflow look like?

In industrial applications it is worth splitting the design into several levels:

• Defining the requirement — what the component is to achieve (e.g. a given transmission, bandwidth, reflection level, polarisation response).
• Choosing the physical model — mode analysis, BPM, FDTD, or a combination of these methods.
• Preparing the geometry — the significant elements of the device, not necessarily every construction detail.
• Defining the materials — properties relevant to the wavelength range.
• Choosing the grid — from the wavelength, materials, geometry, and required accuracy.
• Configuring the source — matching the real way the device is excited.
• Boundary conditions — chosen to fit the nature of the problem.
• Detectors — which quantities will be measured.
• Convergence test — the stability of the key results with respect to the numerical parameters.
• Parameter sweep — systematic optimisation of the geometry only after the basic model has been verified.

FDTD and industrial design

In industrial projects the greatest value of FDTD simulation is not the mere fact of running electromagnetic calculations, but shortening the path concept → model → verification → optimisation → prototype. The earlier a problem can be detected, the less costly it is to fix.

Simulation helps answer questions such as:

• Does the component geometry guide light as intended?
• Are there unwanted reflections?
• Where do losses arise?
• Does the structure produce the expected diffraction?
• Does the resonator operate in the right range?
• How does a change in geometry affect the spectral response?
• Does the component behave correctly for a given polarisation?

In more advanced projects, FDTD is one of the stages of the whole photonic design process, not an isolated research tool.

FDTD or ray tracing?

It is worth clearly separating these two approaches. Ray tracing describes light propagation using rays and is very effective for analysing large optical and illumination systems and complex optical paths. FDTD solves the full-wave electromagnetic problem and is intended for situations in which wave effects matter.

These are not methods to be compared as „better” and „worse” — they serve different problems. TracePro from Lambda Research is positioned primarily as an environment for non-sequential ray tracing, illumination, stray light analysis, and complex ray interactions with geometry. FDTD is more appropriate when the key subject of the analysis is the full electromagnetic field and its interaction with the structure. A broader comparison of methods → Methods for modelling optical phenomena.

Summary: when to use the FDTD method?

The FDTD method is especially useful when a project requires a full-wave electromagnetic analysis. Its strengths are a direct solution of Maxwell's equations, a full-vectorial representation of the fields, and analysis of propagation, diffraction, interference, reflection, transmission, scattering, polarisation, resonances, and selected nonlinear phenomena.

FDTD does, however, come at a price: the model has to be adequately discretised in space and time, and the computational cost can grow quickly as the domain and resolution increase. That is why the choice of method should proceed as follows: engineering problem → physical phenomenon → required model level → solver choice → verification → optimisation.

In the Optiwave ecosystem, this maps onto successive modelling levels: OptiMode → OptiBPM → OptiFDTD → OptiOmega → OptiSystem / OptiSPICE. In this way FDTD becomes not just a computational method, but part of a complete process of designing and verifying photonic devices. We set out the criteria for choosing a package in the guide How to choose optical simulation software?.