The 2.5D varFDTD solver accurately describes the propagation of light in planar integrated optical systems, from ridge waveguide-based systems to more complex geometries such as photonic crystals. The propagator allows for planar (omnidirectional) propagation without any assumptions about an optical axis, which allows for structures like ring resonators and photonic crystal cavities to be efficiently modeled – devices that have been traditionally treated with 3D FDTD. The varFDTD solver can model devices on the scale of hundreds of microns quickly.

## Solver physics

The varFDTD solver is based on collapsing a 3D geometry into a 2D set of effective indices that can be solved with 2D FDTD. This works best with waveguides made from planar structures, as the main assumption of this method is that there is little coupling between different supported slab modes. For many devices, such as SOI-based slab waveguide structures, that only support 2 vertical modes with different polarizations, this is an excellent assumption.

The calculation steps involve:

- Identification of the vertical slab modes of the core waveguide structure, over a desired range of wavelengths.
- Meshing of the structure and collapse of the 3rd dimension by calculation of the corresponding effective 2D indices (taking into account the vertical slab mode profile). There are currently two approaches to doing this:

### a) Variational

A variational procedure based on Hammer and Ivanova [1]. Here, the effective permittivities of the TE- and TM-like modes have the form:

$$\epsilon^{TE}_{eff}(x,y,\omega)=\left(\frac{\beta_r}{k}\right)^2+\frac{\int_z\left(\epsilon(x,y,z)-\epsilon(z,\omega)\right)\mid M(z,\omega)\mid^2dz}{\int_z\mid M(z,\omega)\mid^2dz}$$

$$\epsilon^{TM}_{eff}(x,y,\omega)=\left(\frac{\beta_r}{k}\right)^2\frac{\int_z\frac{1}{\epsilon_r}\mid M(z,\omega)\mid^2dz}{\int_z\frac{1}{\epsilon(x,y,z)}\mid M(z,\omega)\mid^2dz}+\frac{\int_z\left(\frac{1}{\epsilon_r}-\frac{1}{\epsilon(x,y,z)}\right)\mid \frac{\partial M(z,\omega)}{\partial z}\mid^2dz}{k^2\int_z\frac{1}{\epsilon(x,y,z)}\mid M(z,\omega)\mid^2dz}$$

where εr, M and βr are the 1-D reference permittivity profile, the associated guided slab mode and the propagation constant.

### b) Reciprocity Based

This is a procedure based on the reciprocity theorem, as described in Snyder and Love [2]:

$$n_{eff}(x,y,\omega)=\frac{\beta_r}{k}+\left(\frac{\epsilon_0}{\mu_0}\right)^\frac{1}{2}\frac{\int_z(\epsilon(x,y,z)-\epsilon_r(z,\omega))\mid{\bf{E}}(z,\omega)\mid^2 dz}{\int_z{\bf{P}}\cdot {\bf{n}}dz}$$

Note that in both cases, the generated effective materials are also dispersive, where the dispersion comes both from the original material properties (material dispersion) and the slab waveguide geometry (waveguide dispersion). These new materials are then fitted using Lumerical Solutions' multi-coefficient model into a time-domain form that can be used in the 2D FDTD simulation in step 3. Note that the effective index treatment may lead to generate materials that have properties that are unphysical (for example, having an artificial negative imaginary index). In this case, one has the option of restricting the range of generated indices to the min/max values defined by the physical material properties of the original materials. All of these settings can be found under the Effective index tab of the varFDTD simulation region.

- Simulation of the structure in 2D by FDTD.
- If desired, re-expansion of the fields into 3D.

For additional information on the varFDTD solver, see the Lumerical’s 2.5D FDTD Propagation Method whitepaper on our website.

## Meshing

The MODE Propagator uses a rectangular, Cartesian style mesh, like the one shown in the following screenshot. It's important to understand that the fundamental simulation quantities (material properties and geometrical information, electric and magnetic fields) are calculated at each mesh point. Obviously, using a smaller mesh allows for a more accurate representation of the device, but at a substantial cost. As the mesh becomes smaller, the simulation time and memory requirements will increase. MODE provides a number of features, including the conformal mesh algorithm, that allow you to obtain accurate results, even when using a relatively coarse mesh.

By default, the simulation mesh is automatically generated. To maintain accuracy, the meshing algorithm will create a smaller mesh in high index (to maintain a constant number of mesh points per wavelength) and highly absorbing (resolve penetration depths) materials. In some cases, it is also necessary to manually add additional meshing constraints. Usually, this involves forcing the mesh to be smaller near complex structures (often metal) where the fields are changing very rapidly.

Note: meshing time The general meshing algorithm can take a reasonable amount of time compared to the simulation because the mesh must be effectively completed in 3D. It is possible for the user to specify if the structure is composed of purely extruded structures - that is structures that are extruded along z with perfectly vertical sidewalls. In this case, the meshing can be accomplished much faster. |

## Units and normalization

Unless otherwise specified, all quantities are returned in SI units. Please see Units and Normalization for more information.

### Related publications

- Manfred Hammer and Olena V. Ivanova, MESA Institute for Nanotechnology, University of Twente, Enschede, The Netherlands

"Effective index approximation of photonic crystal slabs: a 2-to-1-D assessment", Optical and Quantum Electronics ,Volume 41, Number 4, 267-283, DOI: 10.1007/s11082-009-9349-3 - Allan W. Snyder and John D. Love, Optical Waveguide Theory. Chapman & Hall, London, England, 1983.

### Selected Application Examples

- Ring Resonator Getting Started Tutorial
- AWGs and Star Couplers
- Tapered Waveguide
- Non-linear Ring Modulator

### Video resources

- varFDTD introduction (YouTube.com)
- MODE Solutions overview (YouTube.com)