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Design and Analysis Guides
PIC device design always involves a trade-off between accuracy and compute time. This article compares the principle and scope of FDTD and EME and, using a grating coupler and a tapered spot-size converter as examples, shows how to choose between the solvers on the basis of measured cost and accuracy.
Photonic Integrated Circuit (PIC) devices combine sub-wavelength nanostructures with large device geometries, and that combination is a serious problem for simulation tools: designers are often forced into an unwelcome trade-off between accuracy and compute time. Broadband, bidirectional or omnidirectional propagation adds further difficulty. This article shows how to address these challenges by combining several optical solvers.
PIC devices come in a wide range of shapes and sizes, and as things stand no single solver handles every PIC design optimally. Reliable, efficient virtual prototyping is best achieved by combining several solvers across the design process, from the initial concept through prototyping to final verification before manufacture.
Combining several solvers along a workflow lets designers concentrate on the design goal that matters at each stage of development, and makes the most efficient use of design time and compute resources.
| 1. Initial simulation model | ||
Design goal:
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Simulation requirement:
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Recommended solvers:
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| 2. Initial optimisation | ||
Design goal:
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Simulation requirement:
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Recommended solvers:
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| 3. Final optimisation and verification | ||
Design goal:
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Simulation requirement:
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Recommended solvers:
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Table 1: component design workflow from prototype to manufacture
The first step is to set up the initial simulation model. The solvers used at this stage generally include some approximations, so that you get sound qualitative insight across the wide design space being explored. Simulation accuracy is sometimes sacrificed at this stage in order to run many iterations quickly. Approximations may include using a 2D model rather than a 3D one, a coarser simulation mesh, or a less rigorous solver algorithm. Once a working simulation model is established, design optimisation can begin. Because optimisation requires many simulations, it is efficient to split it into two stages. The aim of the first optimisation stage is to narrow the parameter space substantially and reach an approximate optimum. The solver used at this stage must run fast while still producing reasonably accurate results and predicting the right trends. The final step in this workflow is final optimisation and verification. At this stage the simulation model must be optimised and the design ready to be fabricated if required. The number of simulations needed here should be kept to a minimum, and the most accurate solver should be used even if the cost per simulation is high.
Ansys Lumerical supports several optical solvers so that integrated optical components can be analysed accurately and efficiently. Mode analysis of waveguides and fibres with arbitrary cross-sections is handled by the finite-difference eigenmode (FDE) solver. Propagation analysis of geometries that vary laterally or longitudinally can be carried out with the finite-difference time-domain method (FDTD: Finite Difference Time Domain Method ) or eigenmode expansion (EME: Eigenmode Expansion).
Other methods can also be used to simulate light propagation in integrated optical components, such as the beam propagation method (BPM: Beam Propagation Method). BPM relies on the slowly varying envelope approximation (Slowly Varying Envelope Approximation), and standard BPM implementations are often unreliable for accurately modelling discontinuous geometries, large propagation angles and high index-contrast materials[1]. This article focuses on FDTDandEME. The sources of error in both methods are well understood, and both provide numerical solutions of Maxwell’s equations whose results converge to the correct answer.
FDTD is one of the most general and accurate methods for simulating light propagation in nanoscale components of arbitrary geometric complexity.FDTD scales very well with parallelisation, and a single simulation efficiently produces broadband results. For planar geometries, moreover, the 2.5D variational FDTD (varFDTD) method (Accuracy and speed of 2.5D varFDTD compared with 3D FDTD) 3D FDTD is an excellent alternative to 2D FDTD, achieving accuracy comparable to 3D FDTD at a compute time comparable to varFDTD is particularly useful in the early optimisation stage, where it offers a very efficient route to an approximate optimum[2].
A significant challenge in FDTD is the staircasing effect that arises from discretising curved surfaces on a cubic mesh. Time-domain methods can also struggle with dispersive materials over a broad wavelength range. The Ansys Lumerical FDTD implementation addresses these challenges. Multi-coefficient models, for example, allow broadband materials with arbitrary dispersion to be simulated, and conformal and graded meshes reduce the number of grid cells used without sacrificing accuracy.
Bidirectional EME is a well-established method for simulating long-distance propagation in waveguide and fibre devices.
It has the particular advantage that waveguide and fibre devices with a uniform cross-section, such as multimode interferometers and gratings, do not cost as much to compute as they would in FDTD even over long propagation distances.
Because EME includes all modes and polarisations in a single simulation, it is a very efficient method for extracting the complete S-matrix of a passive component.
The drawback of this frequency-domain method is that one simulation is needed for each frequency.
Basic EME requires that a continuously varying structure be approximated as a staircase in order to resolve geometric and material changes along the propagation direction.
Ansys Lumerical’s EME implementation addresses several of these challenges by adding extensions to the basic EME algorithm: the CVCS (continuously varying cross-sectional sub-cell) method, efficient multithreading and parallelisation (see the introduction to eigenmode expansion (EME) and its application to waveguide analysis).
FDTD is a computationally demanding method. Large PIC components require a large number of grid cells, which can make simulation time and memory requirements substantial.
These challenges can be mitigated to a degree by a highly optimised compute engine and efficient parallelisation. For 3D simulations where the component geometry exceeds 100 wavelengths in one or more dimensions, however, it may be worth considering a different algorithm for at least part of the analysis.
EME is a method well suited to simulating long-distance propagation, but when the device becomes large in the transverse direction, or the propagation angle relative to the propagation axis becomes very large, FDTD scales less favourably in compute time and memory.
FDTDandEME When comparing performance, how simulation time scales with device size is an important factor. Other factors can also affect solver performance substantially: how efficiently periodic structures are handled, the degree of parallelisation available, the trade-off between time-domain and frequency-domain simulation, and the effect of dispersion on the FDTD grid. As the examples below show, these trade-offs vary considerably with the particular application.
This example compares the performance of FDTDandEME on a grating coupler and a tapered spot-size converter.
Simulations are run on a standard 4-processor, 32-core workstation. The comparison covers all three stages of the design workflow: initial model, initial optimisation, and final optimisation and verification.
A grating coupler is used to couple light between a fibre and a photonic chip. That requires light to propagate out of plane, vertically. As with a waveguide splitter, evaluating the performance of a grating coupler requires the complete S-parameter matrix. Between a pair of input and output grating couplers, for example, back reflection often produces Fabry-Perot oscillations.
FDTD assumes no propagation direction, which makes it very well suited to a three-dimensional scattering problem such as a grating coupler. FDTD also has the advantage of computing broadband behaviour in a single simulation.EME is inherently bidirectional and can simulate propagation at large angles by using more modes, at the cost of increased simulation time and memory. To extract S-parameters, FDTD requires two simulations, one in-coupling and one out-coupling. For the response of a 2D grating coupler, 2D FDTD takes about 10 seconds across two simulations, while 2D EME takes about 30 seconds for a single wavelength, so there is no advantage in choosing FDTD over EME .
For a three-dimensional grating coupler of 40 mm x 20 mm x 2.5 mm, FDTD takes about 36 minutes in total.
| grating coupler | Lumerical 2D FDTD | Lumerical 2D EME |
|---|---|---|
| Simulation region size | 30μm×3.8μm | 30μm×3.8μm |
| Time to extract S at one wavelength | 10 seconds (two simulations) | 30s |
| Time to extract S at 100 wavelengths | 10 seconds (two simulations) | 3000s |
The spot-size converter geometry is based on reference [2], where a taper is used to transfer light from a single-mode silicon waveguide into a larger polymer waveguide. As with many taper designs, the aim is to find the shortest taper length that achieves this conversion with minimum loss.
For a continuously varying geometry, EME is ideal for scanning device length, since the full calculation does not have to be repeated for each length.FDTD requires a separate simulation for each taper length, and simulation time grows with the square of the taper length. Figure 4 shows how transmission varies for taper lengths from 5 um to 200 um.
EME produced results for 100 different taper lengths in under two minutes, whereas the FDTD simulations took more than six hours for 11 taper lengths.
Ansys provides a versatile, comprehensive design environment suited to any passive component, so that designers of photonic components and systems can adopt an efficient workflow for accurate design. The table below summarises the solvers available in Ansys Lumerical FDTD and Ansys Lumerical MODE .
| Eigenmode analysis | ||
| Solvers | Ansys tool | |
| FDE | Finite-element eigenmode solver | Lumerical MODE |
| Propagation analysis | ||
| Solvers | Ansys tool | |
| FDTD | 2D/3D Finite Difference Time Domain | Lumerical FDTD |
| varFDTD | 2.5D variational FDTD | Lumerical MODE |
| EME | Bidirectional eigenmode expansion | Lumerical MODE |
Table 3: MODE and FDTD share the same CAD environment, geometry and analysis tools, and let you select and run different solvers, so you can easily streamline the design process
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Integrated optical components come in a wide range of shapes and sizes, and no single simulation algorithm can model every type of component optimally. Even for the same component, however, combining several tools across the initial simulation model, initial optimisation and final optimisation stages often streamlines the design process and makes the workflow more effective.
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