PIC Waveguide Mode Solver: benchmark

1. Exact answers. On 16 modes of shapes whose answer is known exactly, FDE's effective index stays within the accuracy the tool prints at every mesh level: 48 of 48 comparisons.
2. Independent mode solver. On 19 values from common silicon, silicon nitride and lithium niobate waveguides, FDE at its finest mesh differs from MPB by 0.00935 % typically and 1.71 % at most, for how a pulse spreads on a silicon nitride strip in air; see Limits. On 6 of the 19 it lands inside MPB's own range.
3. Full-wave simulation of bends. On silicon strip and slot waveguides bent to radii from 1 to 10 µm, FDE at its finest mesh lands inside Meep's range on 8 of 9 values. The one outside is the loss of the bent slot waveguide at 2 µm, +1.67 % from Meep's value where Meep's own range is ±0.423 %; at 4.67 µm it lands inside, +1.22 % against ±1.75 %; see Limits.
4. Where the free engine falls short. Outside flat layers, where it is exact by construction, EIM lands outside the reference on all 23 values, furthest for loss in a bend and for how a pulse spreads; on a silicon strip in SiO₂ it gets the sign of the dispersion wrong. When EIM is enough

On every value where both engines give a number, FDE at its finest mesh is at least as close to the reference as EIM.

An independent program's answer also shifts a little with its own settings. Its range is that shift at its finest settings, and the record prints every difference, inside it or outside it.

Some waveguides have answers that follow from an exact equation: flat slabs, a slab bent into an arc, and a slab that leaks into a substrate. We solve each one with FDE at its three mesh levels and compare with the exact answer, computed to 24 significant figures. For each material the tool prints how accurate its effective index is at each level. Figure 1 puts that printed accuracy, dashed, over the error we measure here.

Each figure is how far FDE's answer at that level is from its own answer on finer meshes, measured on waveguides of that material and rounded up to one significant figure. It is not a comparison with another program; the sections below are. For lithium niobate, see the note below. No shape on this page is made of InGaAsP, so its figure is not tested here.

Lithium niobate. Its printed figure at each level is the larger of two measurements: the tool's lithium niobate rib preset, and the flat lithium niobate slab in the exact-answer section. At Fine, checking the slab against a finer mesh needs more memory than a solve may use, so its Fine figure is measured against the value its Standard and Fine results extrapolate to, assuming the error shrinks as the square of the cell size.

On these slabs FDE's group index is within 0.038 % of the exact answer at Fine and within 0.67 % at Coarse; its dispersion is within 0.19 % at Fine and 2.6 % at Coarse.

MPB is a free, open-source mode solver from MIT. We solved the tool's preset waveguides with it at resolutions up to 400 pixels per µm. Its range is how much its answer still moves at its finest settings. Table 4 lists the effective index, group index and dispersion β₂; Figure 2 shows FDE's difference from MPB at Coarse and at the finest level we ran.

At the finest mesh we ran, FDE's effective index is within 9.7 × 10⁻⁴ of MPB's value on every silicon, silicon nitride and lithium niobate waveguide here; at Coarse, the default, within 0.0072. Its group index at Fine is within 3.1 × 10⁻⁴ of MPB's value, and its dispersion within 0.66 %, except on the silicon nitride strip in air.

On the silicon nitride strip in air, the dispersion lands near MPB at Coarse and Standard; at Fine it moves away from both; see Limits. On the silicon nitride strip in SiO₂, FDE's dispersion moves toward MPB at each finer level and ends −0.651 % from MPB's value at Fine.

Which modes each engine calls guided

On the tool's 6 silicon and silicon nitride presets, each engine lists the modes it finds; MPB says which are guided. The words in brackets are the tool's own: FDE prints "Not resolved" when a mode is too close to the cutoff for the mesh to decide, and EIM marks a mode "near cutoff" or "not confined" when its simple model cannot decide.

A silicon waveguide on a thin SiO₂ buffer leaks light into the silicon wafer below. The exact answer exists for a flat slab: a 220 nm silicon layer, a SiO₂ buffer 0.3, 0.5 and 0.7 µm thick, and a silicon substrate, with air or SiO₂ above. FDE's loss at Fine is within 0.25 % of the exact answer at every point, and within 4.1 % at Coarse.

Meep is a free, open-source electromagnetic simulator from MIT. We used it to simulate a whole ring of each waveguide, using the ring's rotational symmetry to solve the full wave equations, and read the effective index and loss of the bent mode from the ring's resonance. Each radius resonates at its own wavelength, so each comparison is made at that wavelength, with the same constant material indices in both programs.

Meep's range at 1 µm, and for the slot's effective index, is how much its answer changed between its two finest resolutions; at 2.5, 5 and 10 µm it is how far an extrapolation from two resolutions moved its finest answer. For the slot's loss it is the sum of three measured differences, listed in the appendix.

On the slot, FDE's effective index agrees with Meep within Meep's own last refinement step. Its loss at Fine lands inside Meep's range at 4.67 µm, +1.22 % against ±1.75 %, and outside it at 2 µm, +1.67 % against ±0.423 %; see Limits. Most of the earlier difference came from how the reference turned the ring's decay into a loss (see the appendix); the difference that remains at 2 µm is not attributed to a cause.

On the strip, FDE at Standard and Fine lands inside Meep's range at every radius, for the effective index and for the loss at 1 µm. At Coarse, the default, its effective index sits 0.00170 to 0.00217 below Meep's value, just outside it.

The strip's loss is compared at 1 µm only. No TM bend is compared on this page.

EIM replaces the two-dimensional cross-section by two one-dimensional layer problems. That is exact for a flat layer and approximate for everything else, and it runs instantly in your browser. Table 9 puts EIM and FDE (finest level) against the same references, as a percentage of the reference value.

• Effective index: EIM is within 0.90 % of MPB on the silicon nitride strips. On silicon strips and ribs it is 1.51 to 3.97 % high, and on the slot 10.6 % high.
• Group index: EIM is 1.02 % high and 1.31 % low on the two silicon nitride strips, and 3.06 to 3.94 % low on silicon.
• Dispersion (β₂): do not use EIM's value. It is 65.7 % low on the silicon nitride strip in air and 58.8 % too small in magnitude on the silicon strip in air, and on the silicon strip in SiO₂ it has the wrong sign (+1.29 ps²/m against MPB's −0.618). The tool marks that value whenever it shows it.
• Bend loss: EIM's bend formula misses the exact bent slab by −53.3 % at 10 µm, −16.8 % at 5 µm and +7.47 % at 3 µm, and Meep's ring by +894 % on the 1 µm strip and by +274 % and −62.2 % on the slot. Use FDE for a bend loss you rely on.
• Lithium niobate: EIM cannot model the crystal's two indices and gives no value.

• Near cutoff. A mode at the edge of being guided is the hardest case. On a thin silicon strip at 1310 nm (Thin SOI 1310), FDE at Coarse lists the TM mode as "Not resolved"; at Standard and Fine it is no longer guided, and MPB agrees it is not guided. The tool withholds a mode it cannot place within its printed accuracy.
• Differences we have not explained: the remaining loss difference of the bent slot waveguide at 2 µm; the dispersion of the silicon nitride strip in air at Fine, and of the same strip in SiO₂, which ends −0.651 % from MPB's value.
• Against MPB at Fine. On two silicon waveguides, FDE's effective index at Fine differs from MPB's value by more than the 6e-4 the tool prints for silicon at Fine: +8.77 × 10⁻⁴ on SOI Strip 220's TM mode and +9.69 × 10⁻⁴ on the slot. At Coarse and Standard both sit inside the printed figures. Both are where the field jumps most sharply at a material boundary (the TM mode at the strip's top face, the slot mode across the slot's walls), which is where any solver on a grid, MPB included, converges slowest, so MPB's own range there is probably too narrow. That is our reading, not a measurement.
• Shapes the tool cannot draw. The flat slabs in the exact-answer and leakage sections ran on FDE's solver directly, not through the tool; their results show the solver's accuracy, not a design you can open.
• References have their own ranges. MPB's and Meep's ranges are set from their own resolution steps; on the slot's loss, Meep's range also adds the other measured differences listed in the appendix. At 2.5, 5 and 10 µm the strip's Meep range comes from extrapolating two resolutions, with no third to confirm it.
• The loss of a waveguide you fabricate. Real waveguides lose light to sidewall roughness, which nothing here models.
• The loss of a bend you fabricate, including the junctions where it meets a straight waveguide.
• An effective index more precise than your waveguide's fabrication tolerance allows.
• Accuracy on shapes other than the ones on this page.

Every waveguide the tool can draw ran through the same path a design takes in the tool: the same input checks, the same solver and the same mesh rule. The exact shapes it cannot draw ran on the same solver directly. We ran both engines on October 18, 2026, and ran MPB 1.12.0 and Meep 1.34.0 ourselves at the resolutions each table names; the exact answers were computed with separate code, to 24 significant figures. Every number on this page is read from one data file, whose SHA-256 fingerprint is 934bee1f…, and a separate program recomputes every comparison from its raw values before the page is published.

Download the data file (JSON)