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Analysis Technologies | Electromagnetic Analysis
An electromagnetic method that solves Maxwell’s equations directly in the time domain. Space is divided into a grid and the electric and magnetic fields are updated alternately, so a single run yields a broadband response.
Orthogonal grid (Yee cell)
Time domain, broadband
Lumerical FDTD
What the method is
The analysis region is divided into an orthogonal grid, with the components of the electric and magnetic fields placed at slightly different positions within each cell and updated alternately through time. It discretizes Maxwell’s curl equations in space and time and solves them directly, introducing no modeling approximation into propagation or scattering. The discretization itself is a source of error, though, so the accuracy you get depends on the settings.
Fourier transforming the fields obtained in the time domain gives frequency-domain quantities such as the complex Poynting vector, transmittance and reflectance. That is why a single run yields the response over a wide wavelength range.
The grid is generated automatically, with finer cells in high-index and strongly absorbing materials. The time step is tied to the smallest cell dimension, so a fine grid feeds straight into run time.
How it works
Electric and magnetic fields are not defined at the same point. They sit at offset positions within the cell and are updated half a time step apart. Both time and space derivatives are approximated by central differences, giving a scheme that is second-order accurate in space and time.
The update is explicit. The next values follow from the previous ones alone, so no large system of equations has to be solved. In exchange there is an upper bound on the time step: fail the CFL condition and the calculation diverges.
Accuracy does not come automatically from the method. The grid is generated according to the accuracy setting and the materials, with setting 1 giving 6 points per wavelength and each step after that adding 4. The run continues until the auto-shutoff condition is met, and PML is placed at open boundaries. For materials using measured (n,k) data, how well the dispersion model fits also affects the result.
Strengths of this method
One time-domain run gives the broadband response of the target. There is no need to repeat the calculation at different frequencies.
Because the formulation follows time, pulsed responses and nonlinear behavior are handled naturally. Application to dispersive, anisotropic and nonlinear media is documented.
Time advances as it does in nature, so you can follow directly how the fields build up and decay.
Where it fits
→ when the structure is not periodic in-plane
→ when you want a broadband spectral response from a single run
→ when you want S-parameters for a component excited with a guided mode
→ when you want scattering from an isolated object, separated from the incident field
Planar periodic structures: for a plane wave incident on a layered, in-plane periodic structure, RCWA takes far less computation time.
Problems that close within the layer stack: STACK is analytic and RCWA semi-analytic. For structures they can represent, they are generally more accurate than fully numerical FDTD.
Interfaces that do not follow the orthogonal grid: where the representation of curved or slanted interfaces decides the result, methods using an unstructured mesh and higher-order basis functions have the advantage in geometric fidelity.
Waveguide cross-sectional modes: if the aim is the modes of a structure uniform along the propagation direction, a method that solves the cross-sectional eigenvalue problem answers that question directly.
Applications
Obtain the spectrum and field distribution of the unit structure as the basis for phase design.
Project onto diffraction orders to evaluate the wavelength and angular dependence of diffraction efficiency.
Evaluate the field distribution and spectrum around fine pixel and emitter structures.
Excite waveguide crossings and grating couplers with a guided mode and obtain S-parameters.
Inputs and outputs
INPUT
| Structure | Combinations of parametric primitives, or geometry imported from external data |
|---|---|
| Material | Optical constants from the material database. Measured (n,k) data is used with a dispersion model fitted to it |
| Light source | Dipole, plane wave, beam, TFSF source, guided mode source, imported source |
| Analysis settings | Boundary conditions, mesh accuracy (1 to 8), maximum simulation time, auto-shutoff threshold |
OUTPUT
| Spectrum | Wavelength dependence of transmittance and reflectance from the fields recorded by monitors |
|---|---|
| Electromagnetic field distribution | Field distributions from frequency-domain and time monitors. These can also be recorded as movies |
| Far field and diffraction orders | Far-field projection from the near field, and projection onto diffraction orders for periodic structures |
| S-parameters | Component S-parameters from mode expansion monitors and the S-parameter matrix sweep |
How it works
Define the materials you will use, build the structure from primitives or by import, and assign materials to it. A dispersion model is fitted to measured data.
Set the analysis region and choose the mesh accuracy. Mesh accuracy corresponds to points per wavelength: 6 at setting 1, increasing by 4 for each step after that.
Place the sources and monitors, use PML at open boundaries, and where symmetry exists apply symmetric and antisymmetric boundaries to shrink the analysis region.
Run until nearly all the energy has left the analysis region, then evaluate the spectra and field distributions. Sweeps and optimization build on that.
A healthy run ends because the auto-shutoff condition was met, not because it hit the maximum simulation time. If a run ends with energy still in the analysis region, revisit the settings.
Comparison with related methods
| Method | Relationship | Main targets | When to use which |
|---|---|---|---|
| FDTD (this method) | This method | Arbitrary structures at wavelength scale | Solves the time-domain Maxwell equations on an orthogonal grid, giving a broadband response from a single run. |
| RCWA | Alternative | Layered structures with in-plane periodicity | A semi-analytic method: for the structures it can handle it takes far less computation time than FDTD and is generally more accurate. |
| STACK | Alternative | Reflection and transmission that close within a layer stack | An analytic method: for plane multilayer problems it is generally more accurate than fully numerical FDTD. |
| DGTD | Alternative | Structures with curved or slanted interfaces | Also time domain, but using an unstructured mesh of triangles and tetrahedra, so geometry is represented differently. Neither is always the better choice. |
| FDE | Complementary | Waveguide cross-sectional modes | A method that solves the cross-sectional eigenvalue problem on an orthogonal grid and returns modes. It answers a different question. |
Supporting products
A commercial software environment providing electromagnetic analysis by the FDTD method. FDTD, RCWA and STACK, three solvers of different character, sit in one design environment, with support for sweeps, optimization, yield analysis, and execution on HPC, GPU and cloud. See here for licensing, system requirements and deployment.
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