PIC Directional Coupler Simulator: benchmark

1. Exact answer. For two flat silicon slabs side by side, whose coupling follows from an exact equation, FDE's error shrinks at each finer mesh level on all 24 comparisons, and its coupling length is within 0.98 % of the exact answer at Coarse.
2. Independent mode solver, straight couplers. On 18 values from straight silicon and silicon nitride couplers, FDE at its finest mesh differs from MPB by 0.813 % typically and 2.01 % at most, on the SOI Strip 220 pair in air with a 200 nm gap; see Limits. On 2 of the 18 it lands inside MPB's own range.
3. Full 3D simulation, a ring passing its bus. At ring radii of 10 and 5 µm, FDE's κ² at Coarse and at Standard lands inside Meep's range in both cases.
4. Where it is less accurate. When both waveguides bend into the coupler, FDE's κ² is too low: 59.5 to 59.8 % below Meep's value on the 3 µm S-bends and 18.5 to 18.6 % below on the 6 µm ones, at Standard. On such a design the tool cautions that the bends couple more than its estimate includes, by an amount measured in 3D on this same pair of waveguides; that is a correction, not a separate check. For a ring passing its bus, FDE's κ² is −8.63 % and −10.4 % from Meep's value at 10 and 5 µm: both below it, the same direction as the S-bend shortfall, but inside Meep's range of up to ±11 %, so this comparison is inconclusive there; Meep at a second resolution on one ring would tell. EIM has no model for both waveguides bending. 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.

Two flat silicon slabs side by side have a coupling that follows from an exact equation. We solve them with FDE at its three mesh levels and compare with the exact answer, computed to 23 significant figures. Table 1 lists the coupling length; Figure 1 shows how the error falls as the cells get smaller.

FDE's coupling length is within 0.98 % of the exact answer at Coarse, 0.25 % at Standard and 0.062 % at Fine; each halving of the cell size cuts the error about fourfold.

MPB is a free, open-source mode solver from MIT. We solved each straight pair with it at resolutions up to 200 pixels per µm. Its range is how much its answer still moves at its finest settings. Table 2 gives MPB's coupling length and FDE's difference from it at Coarse and Standard; the last column is the accuracy the tool itself prints on that design. Figure 2 plots both.

* larger than the accuracy the tool prints for that design

On the silicon pairs in SiO₂, FDE's coupling length is within 2.6 % of MPB's at Coarse and 0.84 % at Standard, at every gap from 120 to 640 nm. On the silicon nitride pairs in SiO₂ it is within 0.32 % and 0.054 %, inside MPB's range at Standard at 400 and 600 nm. In air the differences are larger: within 4.1 % and 2.1 % on the silicon pairs; on the silicon nitride pairs within 1.3 % at Coarse, and 0.53 % at 200 nm, the one gap that ran at Standard.

On four values, marked * in the table, the difference from MPB is larger than the accuracy the tool prints for that design; see Limits.

How fast the coupling changes with wavelength sets a coupler's bandwidth. On the silicon pair in SiO₂ with a 200 nm gap, FDE at Coarse gets it −0.819 % from MPB's value, and EIM −7.98 %.

Meep is a free, open-source electromagnetic simulator from MIT. We used it to simulate whole curved couplers in 3D at 50 pixels per µm, twice: once with the curved walls drawn on the grid as a staircase, once smoothed. Meep's value is the middle of the two; its range is half their spread. Both programs used the same fixed indices. Table 4 covers a ring passing a straight bus; Table 5, couplers whose two waveguides both bend.

At both radii FDE lands inside Meep's range at Coarse and at Standard, below Meep's value: −8.63 % at 10 µm and −10.4 % at 5 µm at Standard. Meep's own range there is up to ±11 % (two ways of drawing the curved walls), so this comparison cannot resolve FDE's κ² more finely than that.

Plane: where κ² is read, in µm from the closest point of the two waveguides toward the output ports.

FDE adds up the coupling along the gap as drawn, so it leaves out the extra coupling the bends themselves add: its κ² is 59.5 to 59.8 % below Meep's on the 3 µm S-bends and 18.5 to 18.6 % below on the 6 µm ones, at Standard. On such a design the tool cautions that the bends couple more than its estimate includes, by an amount measured in 3D on a reference pair of these same waveguides, Meep's run among them, and prints the κ² that extra coupling would give; the last column shows it.

EIM replaces each waveguide's cross-section by two one-dimensional layer problems and builds the pair from them. It runs instantly in your browser. Table 6 puts EIM and FDE (finest level run) against the same references, as a percentage of the reference value.

• EIM is for exploring: its coupling length is off by up to 43 % here, so set a length with FDE.
• Silicon pairs: EIM's coupling length is too short by 30.9 to 42.8 % on the pairs in SiO₂, more at wider gaps, and by 33.0 to 33.9 % on the pairs in air.
• Silicon nitride pairs: in SiO₂ it is too short by 12.2 to 24.3 %. In air its error changes sign with the gap: −7.08, +5.53, +23.2 % at 200, 400 and 600 nm, so no single correction fixes it.
• Ring passing a bus: EIM's κ² is more than double Meep's value: +113 % at 10 µm and +109 % at 5 µm.
• Wavelength dependence: EIM's slope of the coupling with wavelength is −7.98 % from MPB's, FDE's −0.819 %.
• Both waveguides bending: EIM has no model for this shape and prints no coupling.

• Two couplers did not run at Standard. The silicon nitride pairs in air with 400 and 600 nm gaps need a solve window at that mesh larger than our servers accept, and the tool refuses them the same way before starting. They ran at Coarse, as Table 2 shows. This is a limit on what runs, not on accuracy.
• Against the accuracy the tool prints. On four values with air above, FDE's coupling length differs from MPB's by more than the figure the tool prints for that design: the SOI Strip 220 pair (air above) at 200 nm, −4.07 % at Coarse against "within 4 %"; the SOI Strip 220 pair (air above) at 200 nm, −2.01 % at Standard against "within 1 %"; the SOI Strip 220 pair (air above) at 300 nm, −1.83 % at Standard against "within 1 %"; the SiN Strip 400 pair (air above) at 200 nm, −0.521 % at Standard against "within 0.4 %". That figure measures how far FDE's answer at a mesh level is from its own answer on finer meshes, measured on silicon and silicon nitride pairs with a 200 nm gap; it is not a comparison with another program. On each of these values MPB was still moving toward FDE between its two finest resolutions, and on the silicon pairs in air MPB's range is ±0.458 to ±0.650 %, against ±0.00941 to ±0.163 % on the pairs in SiO₂. With air above, the field jumps more sharply at the waveguides' top faces and across the gap, where any solver on a grid, MPB included, converges slowest, so MPB's range there is probably too narrow. That is our reading, not a measurement.
• Fine is not tested here. The tool offers Coarse, Standard and Fine; we ran Coarse and Standard on every coupler, and Coarse only for the wavelength slope.
• The references have their own ranges. Meep ran at one resolution, 50 pixels per µm, with two ways of drawing the curved walls; its range is their spread, not a resolution study. MPB's range is set from its two finest resolutions.
• Fixed indices. The silicon pairs in SiO₂, the wavelength slope, the rings and the S-bends used fixed indices (3.476 for silicon, 1.444 for SiO₂) in both programs. The design each one opens uses the tool's wavelength-dependent materials, which differ slightly.
• Shapes the tool cannot draw. The exact slabs ran on FDE's solver directly, not through the tool; they show the solver's accuracy, not a design you can open.
• Evidence from the rule that links coupling length to power transfer. The tool's model is built on that rule, so a check of it would pass by construction.
• The coupling length of a coupler you fabricate, closer than its gap tolerance allows.
• The coupling of a curved coupler more precisely than the 3D simulation's own range.
• S-bend shapes other than the two measured.

Every coupler 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, with fixed-index materials where the comparison used them. The exact slabs, which it cannot draw, ran on the same solver directly. We ran both engines on October 21, 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 23 significant figures. Every number on this page is read from one data file, whose SHA-256 fingerprint is d16f41d1…, and a separate program recomputes every comparison from its raw values before the page is published.

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