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Analysis Technologies | Waveguide and Propagation Analysis
A method for planar photonic circuits that uses the vertical slab modes to reduce the structure to effective two-dimensional properties, then solves in-plane propagation with 2D FDTD.
In-plane orthogonal grid and time step
2D FDTD on effective 2D properties
Lumerical MODE
What the method is
varFDTD handles light propagation in planar integrated optics, from ridge waveguide systems through to complex geometries such as photonic crystals. What distinguishes it is that it assumes no propagation axis and can propagate in every in-plane direction.
The calculation proceeds in three stages. First the vertical slab modes of the core structure are obtained over the wavelength range of interest. Then the structure is meshed and, taking the slab mode profile into account, the layer stack at each point is replaced by effective two-dimensional properties, collapsing the vertical degree of freedom. Finally that two-dimensional structure is solved with 2D FDTD. The fields can be expanded back into three dimensions if required.
MODE provides several methods for that reduction: one based on a variational formulation and one based on the reciprocity theorem. Both produce dispersive effective materials that include material and waveguide dispersion, and fit them to a material model usable in a time-domain calculation. The method’s name comes from the variational formulation, but the reduction procedure is not limited to it.
How it works
At the center of the method is the approximation of fixing the vertical field profile in advance. A reference slab mode profile is fixed, and an integral weighted by that profile replaces the layer stack at each point with a single effective permittivity. The vertical degree of freedom is not solved, it is reduced away. The reduced two-dimensional structure is then solved on the same grid and time step as an ordinary 2D FDTD.
The method’s main assumption is that there is almost no coupling between the several slab modes supported. For a slab structure such as SOI that supports only two modes vertically, this is a very good assumption. Where vertical coupling or radiation matters, on the other hand, the assumption shows up as a systematic difference in the result. It does not produce results equivalent to 3D FDTD.
Strengths of this method
Handles propagation in every in-plane direction with no assumption about an optical axis. Structures where light circulates, such as ring resonators and photonic crystal cavities, are within scope.
Because three-dimensional geometry is reduced to effective two-dimensional properties, the cost stays at two-dimensional scale. Devices on the scale of hundreds of micrometers can be handled quickly.
The group index within the slab is handled correctly, and near-to-far-field transformation uses a function appropriate to propagation within a slab. Diffraction of guided modes can be studied without taking a large analysis region.
Where it fits
→ when you want to handle structures where light circulates in-plane, such as ring resonators and photonic crystals
→ when you want to evaluate a planar layout on the scale of hundreds of micrometers in a realistic time
→ when you want a spectrum over a wide wavelength range from a single run
→ when you want to sweep design parameters to see the trend and narrow down the geometry
→ when you want to look at diffraction within the slab, or how light spreads out of a waveguide
Structures where vertical coupling or radiation matters: because vertical coupling is treated as absent, coupling efficiency comes out too high. Vertical loss is not included either, so spectral peaks come out higher than in a three-dimensional calculation. In a ring resonator comparison, that difference is real and visible.
When final accuracy is required: this method suits the stage of studying the effect of design parameters, and optimization. For verifying results and securing final accuracy, use the slower three-dimensional time-domain calculation or eigenmode expansion.
Structures that are not planar: the method presumes a planar structure extruded in the vertical direction. It does not suit structures whose shape changes substantially in the vertical direction, or where coupling between slab modes cannot be neglected.
Length sweeps of long devices: for sweeping the length of a device whose shape changes gradually along the propagation direction, such as a taper, eigenmode expansion is more efficient because changing the length costs almost nothing extra.
Applications
Study free spectral range and resonance trends quickly and narrow down dimensions. Because vertical coupling is not included, confirm the final values with a three-dimensional calculation.
Evaluate the distribution and phase per wavelength, including diffraction in the slab region.
Study the band structure and propagation behavior of a line-defect waveguide.
Evaluate planar circuits at a scale awkward for three-dimensional time-domain calculation, at two-dimensional cost.
Inputs and outputs
INPUT
| Planar structure | A planar photonic circuit built from geometry extruded in the vertical direction |
|---|---|
| Slab mode reference | The coordinates of the reference point at which the effective index is obtained, and the polarization of the slab mode |
| Reduction method | The variational method, or the reciprocity-theorem method |
| Bandwidth handling | A setting that uses the center frequency only, and a broadband setting that produces dispersive effective materials |
| Mesh and boundary conditions | Grid density, the time step stability factor, and boundary conditions such as PML |
| Sources and monitors | The wavelength range and incident mode of the mode source, and the placement of the various monitors |
OUTPUT
| Transmission and reflection | Transmission and reflection spectra from frequency-domain power monitors |
|---|---|
| Field distribution | Field distributions and time waveforms from cross-sectional profile monitors and time monitors |
| Effective index | Effective index as a function of frequency, returned by the effective index monitor |
| Mode expansion results | The transmitted component into each mode, from a mode expansion monitor |
| Far field | The result of a near-to-far-field transformation appropriate to propagation within a slab |
How it works
Build the planar circuit from geometry extruded in the vertical direction and assign materials.
Place the analysis region, put the reference point at which the effective index is obtained on the core, and choose the slab mode polarization and the effective index method.
Decide whether to use the center frequency only or to produce dispersive effective materials, and set the grid density. Confirm that the effective materials produced fall within a physical range.
Take a run time long enough for the fields to decay sufficiently, then vary the grid and settings and confirm that the result does not move.
Frequency-domain monitor results are not correct if the run is cut short before the fields have decayed sufficiently. Allow a generous run time. A convergence check is needed both for this method and for the three-dimensional time-domain calculation. Once the design is narrowed down, verify with the three-dimensional calculation.
Comparison with related methods
| Method | Relationship | Main targets | When to use which |
|---|---|---|---|
| varFDTD (this method) | This method | Wide planar structures | Reduces the vertical direction to effective two-dimensional properties and solves in-plane propagation with 2D FDTD. No propagation axis is assumed. |
| FDTD | Complementary | Evaluation including vertical coupling and radiation | Discretizes the analysis region as a volume. Because it includes vertical loss, in the comparison example the peaks are lower and the resonant wavelengths differ. Used for final confirmation. |
| EME | Alternative | Long devices whose shape changes gradually along the propagation direction | Connects scattering matrices on a basis of cross-sectional modes. It is strong for length sweeps, while broadband results come more easily from this method: the two divide the work between them. |
| FDE | Upstream | The cross-sectional modes themselves | Solves the cross-sectional eigenvalue problem to obtain the modes. Used for deciding the incident mode and for checking the effective index. |
Supporting products
A commercial software environment for waveguide and optical coupling analysis. Alongside the 2.5D variational FDTD method, finite-difference eigenmode analysis and eigenmode expansion solvers sit in the same environment, so you can go from cross-sectional modes to propagation across a whole planar circuit without leaving it. See here for licensing, system requirements and deployment.
FAQ
Tell us the scale of the circuit and the characteristics you want to evaluate, and we will propose the method and product that suit.