Choosing a Photonics Simulation Method: Mode Solver, EME, BPM, FDTD

Scope

This article compares the numerical methods used to design photonic devices and helps pick the cheapest one that answers a given question. It covers eigenmode solvers, eigenmode expansion (EME), the beam propagation method (BPM), the finite-difference time-domain method (FDTD), and circuit-level models that assemble device results into larger systems. The same device often uses several: a mode solver for the waveguide cross-section, EME for a long taper, FDTD for a grating coupler, and a circuit model for the whole chip.

In short: use a mode solver whenever the structure does not change along the propagation direction; EME for long structures made of a few sections that change slowly or in steps; BPM for long, gently varying paraxial structures in low-contrast materials; FDTD when light scatters, reflects strongly, radiates out of plane, or the structure is resonant, small and complicated; and circuit models above the device level.

At a glance

Eigenmode solvers

A mode solver takes a waveguide's cross-section and finds the field patterns that propagate unchanged, each with an effective index. Finite-difference and finite-element solvers are the common implementations. Because the problem is two-dimensional, a solve takes seconds, and sweeping width, height, wavelength or temperature is cheap. From the modes follow the group index and dispersion (by solving at several wavelengths), the confinement in each material, the overlap with a fiber mode, and, with a conformal transformation or a cylindrical solver, the modes and radiation loss of bends. The Waveguide Mode Explorer on this site solves the slab case exactly.

A mode solver is the first step for almost every integrated device, and many design questions (single-mode width, bend loss, phase-matching condition, coupling length of two parallel guides) are answered by it alone.

Eigenmode expansion (EME)

EME divides a device into sections along the propagation direction, solves for a set of modes in each, and joins neighbouring sections with overlap integrals into a scattering matrix. Within a uniform section, propagation is exact and costs nothing whatever the length; only changes in cross-section cost computation. That makes EME efficient for long devices built from a few distinct sections, or with a slow change that can be cut into steps: spot-size converters and inverse tapers hundreds of micrometres long, multimode interference couplers, directional couplers, and periodic gratings, where one period's matrix is reused.

It is bidirectional, so it captures reflections, and a length sweep of a taper costs little more than a single run. Its limits are the mode set: light that radiates into modes not included (strong scattering, out-of-plane radiation) is lost from the calculation, so the number of modes must be increased until the result stops changing.

Beam propagation method (BPM)

BPM advances the field step by step along the propagation axis, assuming that it varies slowly in that direction compared with the wavelength. It is fast for long structures and was the workhorse for low-index-contrast devices such as silica planar lightwave circuits and fiber components. Standard BPM assumes near-paraxial propagation and neglects back-reflection; wide-angle and bidirectional variants relax these assumptions at added cost. In high-contrast platforms such as silicon photonics, EME and FDTD are usually preferred.

FDTD

FDTD discretizes space and time and steps Maxwell's equations directly, with absorbing boundaries (perfectly matched layers) around the domain. It makes no assumption about the structure, so it handles scattering, strong reflection, out-of-plane radiation, sub-wavelength features and resonances, and one run with a short pulse gives the broadband response.

The cost is set by the mesh and the time step. The mesh must resolve the wavelength in the highest-index material and the smallest feature; a common starting point is a tenth to a twentieth of the wavelength in the material. The time step is tied to the mesh by the Courant condition, $\Delta t \le \Delta x/(c\sqrt{3})$ on a uniform cubic mesh.

Worked example. A 3D region 10 × 5 × 2 μm with a uniform 20 nm mesh has 12.5 million cells; storing the six field components in single precision takes 300 MB, before material and monitor data. The Courant limit is 3.85 × 10⁻¹⁷ s, so simulating 1 ps takes about 26,000 steps. Halving the mesh multiplies the cell count by 8 and the step count by 2, sixteen times the work, which is why non-uniform meshes that are fine only near small features are standard.

Resonant devices are the expensive case: a high-Q ring stores light for a long time, and the run must continue until the field decays or the spectrum must be extracted by fitting. For such devices FDTD is usually applied to the coupling region only, with the ring completed in a circuit model. Effective-index or 2.5D FDTD collapses the vertical dimension of a planar device into an effective slab index, making runs far quicker at some cost in accuracy, useful for early layout of in-plane devices.

Circuit-level models

Once each component is described by its S-parameters (transmission and reflection between ports as functions of wavelength), a circuit simulator connects them with waveguide sections of known effective and group index and loss. A ring resonator, for example, is a coupler (from FDTD or EME) plus a waveguide loop (from a mode solver), and its full spectrum follows in milliseconds. Mach-Zehnder interferometers, lattice filters, arrayed waveguide gratings and whole transceiver circuits are designed this way, with Monte Carlo runs over fabrication variation.

Which method for which device

Common errors

Mesh not converged. A result that changes when the mesh is refined is not yet a result. Refine until the quantity of interest (effective index to four decimals, transmission to 0.01 dB) stops changing.

Boundaries too close. Absorbing layers placed where the mode field is still significant absorb guided light or reflect it. Leave at least a mode width of cladding between the waveguide and the boundary.

Too few modes in EME. Increase the number of modes until transmission converges, especially in tapers and MMIs.

Material data at the wrong wavelength. Dispersive materials need their dispersion in the model for broadband runs; a single index gives the wrong group index.

Two-dimensional results used as three-dimensional ones. A 2D grating coupler or effective-index simulation fixes trends and periods but not absolute efficiency; confirm the final design in 3D.

References: A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (3rd ed., Artech House, 2005); K. Okamoto, Fundamentals of Optical Waveguides (2nd ed., Academic Press, 2006); L. Chrostowski and M. Hochberg, Silicon Photonics Design (Cambridge University Press, 2015).