Mode solver

A mode solver is a program that takes the cross-section of a waveguide or fiber, assumed uniform along the direction of propagation $z$, and finds the field patterns that propagate without changing shape, together with their propagation constants $\beta$. Each solution is a guided mode (or a leaky one) with an effective index $n_\text{eff} = \beta/k_0$. Mathematically the solver turns Maxwell's equations on the cross-section into a matrix eigenvalue problem: the eigenvalues are $\beta^2$ and the eigenvectors are the transverse fields. A full-vectorial solve of a silicon strip takes seconds to a minute on a desktop computer, so mode solving is the routine first step in waveguide and fiber design.

Methods

• Analytic solutions. The slab waveguide and the round step-index fiber have exact eigenvalue equations, solved by root finding. They serve as references for checking numerical solvers.
• Effective index method. Two slab problems solved in turn; good for trends. For a 500 × 220 nm silicon strip it gives 2.49 against 2.45 from a full vectorial solver.
• Finite difference (FDE). The cross-section is sampled on a rectangular grid and the derivatives replaced by differences. Robust, but it staircases curved or slanted boundaries.
• Finite element (FEM). A triangular mesh follows arbitrary shapes such as sloped sidewalls and microstructured fibers, refined locally where fields change fastest.
• Film mode matching. The cross-section is divided into vertical slices, each a multilayer slab whose modes are matched at the slice boundaries; fast and accurate for layered rectangular structures.

Commercial and open-source solvers implement all of these. Scalar variants suffice for low-contrast glass guides; silicon and nitride need full-vectorial solvers.

Outputs

From the mode at one wavelength come $n_\text{eff}$, the field profile, the effective area used in nonlinear calculations, and the confinement factor in any region. A complex refractive index or absorbing boundary gives the modal loss as the imaginary part of $n_\text{eff}$. Solving at several wavelengths gives the group index, $n_g = n_\text{eff} - \lambda\, dn_\text{eff}/d\lambda$, and the dispersion. Solvers in cylindrical coordinates, or with a conformal transformation of the index profile, give the modes and radiation loss of bends.

Convergence and a worked check

Two numerical choices set the accuracy: the mesh and the computational window. For the TE modes of a slab the problem reduces to the one-dimensional scalar equation

$$\frac{d^2E_y}{dx^2} + k_0^2 n^2(x)\,E_y = \beta^2 E_y,$$

which makes a convenient test. For 220 nm of silicon ($n$ = 3.476) clad by silica ($n$ = 1.444) at 1550 nm, the analytic TE0 effective index is 2.8478. A second-order finite-difference solve in a 4 µm window with zero-field walls, with the core edges falling on cell boundaries, gives 2.8519 with 20 nm cells, 2.8488 with 10 nm, 2.8480 with 5 nm and 2.8478 with 1 nm; the error falls about fourfold for each halving of the cell, as expected for a second-order scheme. With 1 nm cells, shrinking the window to 1.0 µm gives 2.8475 and to 0.6 µm gives 2.8289, because the TE0 field decays into the oxide over about 0.10 µm and the walls clip its tails. Modes closer to cutoff, TM modes, and bends need wider windows.

The walls themselves matter. Hard (zero-field) boundaries reflect, so leaky modes and bend radiation appear as spurious standing waves. A perfectly matched layer (PML), an artificial absorbing region at the window edge, lets radiation leave, which is required for computing bend loss, substrate leakage through a thin buried oxide, or the loss of a leaky rib mode. A PML placed too close absorbs the guided field itself, which shows up as a loss that changes with PML position.

Pitfalls

• Spurious modes. Solvers also return cladding and PML modes; guided modes have most of their power near the core and an $n_\text{eff}$ above the cladding index.
• Polarization labels. In a channel waveguide modes are hybrid, so the labels TE-like and TM-like refer to the dominant field component, and conventions differ between tools. The TE and TM polarization entry gives the slab definitions. For the 220 nm slab the TM0 index is 2.053, far from TE0, while near a square cross-section the two cross and mix.
• Material dispersion. Bulk indices at a single wavelength give a correct $n_\text{eff}$ but a wrong group index; the material dispersion of silicon and silica must be included when sweeping wavelength.
• Geometry. Measured waveguides have sloped sidewalls and widths that differ from the drawn layout by tens of nanometers. A process design kit usually states the cross-section to model.

Common questions

What is the difference between a mode solver and FDTD?

A mode solver treats only a cross-section that is uniform along $z$ and returns its modes directly, in seconds. FDTD simulates the full three-dimensional structure in time and handles anything that varies along the propagation direction, at far greater cost. The article on choosing a photonics simulation method compares them.

How fine should the mesh be?

Fine enough that $n_\text{eff}$ stops changing at the precision needed. For high-contrast silicon waveguides cells of 5–10 nm near the core are typical; a convergence test that halves the cell size once is the practical check.

References: K. Okamoto, Fundamentals of Optical Waveguides, 2nd ed. (Academic Press, 2006); L. Chrostowski and M. Hochberg, Silicon Photonics Design (Cambridge University Press, 2015); Z. Zhu and T. G. Brown, "Full-vectorial finite-difference analysis of microstructured optical fibers," Optics Express 10, 853 (2002); A. B. Fallahkhair, K. S. Li, and T. E. Murphy, "Vector finite difference modesolver for anisotropic dielectric waveguides," Journal of Lightwave Technology 26, 1423 (2008).