# Validation report
Written by `photonoxide validate`; don't edit it by hand. CI fails when a case fails or this file is out of date. Values have six significant digits; a value that should be zero shows as "≤ tolerance" when it is within it.
| Case | Tier | What | Against | Measured | Expected | Tolerance | Result |
|---|---|---|---|---|---|---|---|
| `units/amplitude-convention` | analytic | The amplitude of a real signal $\operatorname{Re}(A e^{-i\omega t})$ is recovered with the kernel $e^{+i\omega t}$ (magnitude shown) | the $e^{-i\omega t}$ convention: $\frac{2}{T}\int_0^T \operatorname{Re}(A e^{-i\omega t})\thinspace e^{i\omega t}\thinspace dt = A$ over whole periods | 0.806226 | 0.806226 | 1e-9 | pass |
| `units/lossy-attenuation` | analytic | A wave in a medium with $\operatorname{Im}\varepsilon \gt 0$ decays over one wavelength by $\exp(-2\pi\kappa)$ (decay shown) | $e^{i n k_0 x}$ with $n = n' + i\kappa$, $\kappa \geq 0$ | 0.635495 | 0.635495 | 1e-12 | pass |
| `material/spline-line` | analytic | A natural cubic spline through points on a line is that line (largest deviation shown) | a line has zero second derivative, which the natural end conditions impose | ≤ 1e-13 | 0 | 1e-13 | pass |
| `material/silicon-li-table` | published | Silicon's index passes through Li's table at all 35 points (largest deviation shown) | H. H. Li, J. Phys. Chem. Ref. Data 9, 561 (1980), doi:10.1063/1.555624, Table 1, 293 K | ≤ 1e-12 | 0 | 1e-12 | pass |
| `material/silica-malitson-formula` | published | Silica's index equals Malitson's computed index at his 60 wavelengths (largest deviation shown) | I. H. Malitson, J. Opt. Soc. Am. 55, 1205 (1965), doi:10.1364/JOSA.55.001205, Table I, computed index to 6 decimals | ≤ 1e-6 | 0 | 1e-6 | pass |
| `material/silica-malitson-measured` | published | Silica's index matches the measured mean of three specimens at 60 wavelengths, to five decimals (largest deviation shown) | I. H. Malitson, J. Opt. Soc. Am. 55, 1205 (1965), doi:10.1364/JOSA.55.001205, Table I, computed index plus the C-D-G.E. residual | ≤ 1e-4 | 0 | 1e-4 | pass |
| `material/silica-leviton-table` | published | Fused silica's $n(\lambda, T)$ (Corning 7980) from the authors' Table 3 reproduces their Table 4: 16 wavelengths from 0.4 to 2.6 µm at 13 temperatures from 30 to 300 K (largest deviation shown) | D. B. Leviton, B. J. Frey, Proc. SPIE 6273, 62732K (2006), doi:10.1117/12.672853, Table 4, printed to 5 decimals | ≤ 6e-6 | 0 | 6e-6 | pass |
| `material/linbo3-zelmon-633` | published | Congruent lithium niobate (Zelmon, Table 1) at 633 nm against the $n_o = 2.2864$ and $n_e = 2.2022$ at which Jazbinšek and Zgonik give its tensors (larger deviation shown) | M. Jazbinšek, M. Zgonik, Appl. Phys. B 74, 407 (2002), doi:10.1007/s003400200818, Table 5's caption (their ref. 27); Zelmon's fit is within 2e-4 of its data | ≤ 2e-4 | 0 | 2e-4 | pass |
| `material/linbo3-zelmon-opo` | published | Congruent lithium niobate (Zelmon, $n_e$): the idler of a 1.064 µm-pumped PPLN OPO with a 30 µm grating, from $n_p/\lambda_p - n_s/\lambda_s - n_i/\lambda_i = 1/\Lambda$ (µm, shown) | D. E. Zelmon, D. L. Small, D. Jundt, J. Opt. Soc. Am. B 14, 3319 (1997), doi:10.1364/JOSAB.14.003319, Fig. 3, the predicted points: 3.35 µm, read off the plot to 0.03 | 3.35434 | 3.35000 | 3e-2 | pass |
| `material/linbo3-jundt-opo` | published | Congruent lithium niobate (Jundt, $n_e(\lambda, T)$): the idler of a 1.064 µm-pumped PPLN OPO at 250 °C with a 25.5 µm grating, expanded by Jundt's Eq. (3) (µm, shown) | D. H. Jundt, Opt. Lett. 22, 1553 (1997), doi:10.1364/OL.22.001553, Fig. 1, the 250 °C fit at 25.5 µm: 4.755 µm, read off the plot to 0.02 | 4.75506 | 4.75500 | 2e-2 | pass |
| `material/linbo3-shoji-miller` | published | Congruent lithium niobate: Miller's $\Delta_{33} = d_{33}/[(n_e^2(2\omega) - 1)(n_e^2(\omega) - 1)^2]$ from Shoji's $d_{33}$ and Zelmon's $n_e$ at the fundamentals 1.313, 1.064 and 0.852 µm (largest relative deviation shown) | I. Shoji et al., J. Opt. Soc. Am. B 14, 2268 (1997), doi:10.1364/JOSAB.14.002268, Tables 10 and 12 (3.92, 4.73 and 4.34, in units of 1e-13 m/V), Eq. (1); the paper's own indices differ from Zelmon's | ≤ 1e-2 | 0 | 1e-2 | pass |
| `material/linbo3-mgo-gayer-zelmon` | published | 5% MgO-doped lithium niobate at 21 °C: Gayer's $n_e$ (0.5 to 3 µm) and $n_o$ (0.5 to 1.62 µm) against Zelmon's Table 2 with its columns exchanged, as Gayer et al. find they must be (largest difference shown) | O. Gayer et al., Appl. Phys. B 91, 343 (2008), doi:10.1007/s00340-008-2998-2, Sec. 4.3.1: within 3.1e-4 ($n_e$) and 2.2e-4 ($n_o$); the tolerance adds Zelmon's own 2e-4 | ≤ 5.1e-4 | 0 | 5.1e-4 | pass |
| `material/gaas-gehrsitz-gap` | published | The direct gap of GaAs from Gehrsitz et al.'s Eq. (11) against the $E_0^2$ of their Table II at 298, 185 and 103 K (in µm⁻², largest deviation shown) | S. Gehrsitz et al., J. Appl. Phys. 87, 7825 (2000), doi:10.1063/1.373462, Table II, GaAs Fit 2, printed to 6 decimals | ≤ 2e-6 | 0 | 2e-6 | pass |
| `material/gaas-gehrsitz-n-inf` | published | GaAs: $n_\infty^2 = A + C_1/E_1^2 + C_0/E_0^2$ (Eq. (8)) from the temperature forms of Table II against its columns at 298, 185 and 103 K (largest deviation shown) | S. Gehrsitz et al., J. Appl. Phys. 87, 7825 (2000), doi:10.1063/1.373462, Table II, GaAs Fit 2; the temperature forms are themselves fits to the three columns | ≤ 1e-4 | 0 | 1e-4 | pass |
| `material/algaas-gehrsitz-samples` | published | AlGaAs: the analytic $n(x, \lambda)$ of Table IV against each of the nine samples' own fits (Table III) from 0.73 to 0.83 µm at 23 °C, in units of the sum of the two fits' $\Delta n_\mathrm{max}$ (largest shown) | S. Gehrsitz et al., J. Appl. Phys. 87, 7825 (2000), doi:10.1063/1.373462, Tables III and IV (quality of fit) | ≤ 1e0 | 0 | 1e0 | pass |
| `material/algaas-papatryfonos` | published | AlGaAs (Gehrsitz) against MBE layers measured by ellipsometry, $x$ from 0 to 0.452 at 825, 1300 and 1550 nm: the 17 points below the gap (largest relative deviation shown) | K. Papatryfonos et al., AIP Adv. 11, 025327 (2021), doi:10.1063/5.0039631, Table III; the paper reports differences of the order of 1% from the reference models | ≤ 1e-2 | 0 | 1e-2 | pass |
| `material/algaas-afromowitz-eta` | published | GaAs in Afromowitz's model: $\eta = \pi E_d / [2E_0^3(E_0^2 - E_\Gamma^2)]$ with $E_0 = 3.65$, $E_d = 36.1$ and $E_\Gamma = 1.424$ eV (shown) | M. A. Afromowitz, Solid State Commun. 15, 59 (1974), doi:10.1016/0038-1098(74)90014-3, p. 60: 0.1032 | 0.103246 | 0.103200 | 5e-5 | pass |
| `material/gaas-skauli-shg` | published | GaAs (Skauli, the Pikhtin form) at 21 °C: the first-order QPM periods $\Lambda/m = \lambda_\omega / 2(n_{2\omega} - n_\omega)$ of the seven measured SHG wavelengths inside its range (largest relative deviation shown) | T. Skauli et al., J. Appl. Phys. 94, 6447 (2003), doi:10.1063/1.1621740, Table I; a difference of indices is accurate to 0.2% of itself, and 61.2 µm is printed to 0.08% | ≤ 3e-3 | 0 | 3e-3 | pass |
| `material/gaas-skauli-dndt` | published | GaAs (Skauli, the Pikhtin form): $dn/dT$ at 1.5 µm and 22 °C (in units of 1e-4 per K, shown) | T. Skauli et al., J. Appl. Phys. 94, 6447 (2003), doi:10.1063/1.1621740, Sec. V: 2.33e-4 per K, from their fits | 2.32733 | 2.33000 | 5e-3 | pass |
| `material/ingap-tanaka-ueno` | published | InGaP (Tanaka's single oscillator, $E_0 = 3.39$ and $E_d = 28.07$ eV) at 1.579 µm, the index Ueno et al. take from it for their $d_{14}$ (shown) | Y. Ueno, V. Ricci, G. I. Stegeman, J. Opt. Soc. Am. B 14, 1428 (1997), doi:10.1364/JOSAB.14.001428, Table 2: 3.12, citing Tanaka et al. 1986 | 3.12244 | 3.12000 | 5e-3 | pass |
| `material/inp-pettit-turner-suzuki` | published | InP (Pettit and Turner, 298 K) at 1.064, 1.208, 1.306 and 1.50 µm against the indices Suzuki and Tada list beside their electro-optic coefficients (largest deviation shown) | N. Suzuki, K. Tada, Jpn. J. Appl. Phys. 23, 291 (1984), doi:10.1143/JJAP.23.291, Table I (3.29, 3.23, 3.20, 3.17); Pettit and Turner's fit is within 0.007 of their data | ≤ 7e-3 | 0 | 7e-3 | pass |
| `material/inp-suzuki-tada-voltages` | published | InP: the half-wave voltages $V_{\lambda/2} = \lambda_0 / 2n_0^3 r_{41}^T$ from the catalogue's $r_{41}^T$ against those Suzuki and Tada print, at four wavelengths (largest relative deviation shown) | N. Suzuki, K. Tada, Jpn. J. Appl. Phys. 23, 291 (1984), doi:10.1143/JJAP.23.291, Table I (11.4, 12.0, 12.9 and 14.4 kV); its indices and coefficients have three digits | ≤ 1.5e-2 | 0 | 1.5e-2 | pass |
| `material/inp-suzuki-tada-faust-henry` | published | InP: the Pockels nonlinearity $d_{41}^{EO} = -n_0^4 r_{41}^S / 4$ from the catalogue's $r_{41}^S$ at 1.064 µm, and its electronic and lattice parts through the Faust–Henry coefficient $C = -0.53$, against the 39, 83 and −44 pm/V Suzuki and Tada print (largest relative deviation shown) | N. Suzuki, K. Tada, Jpn. J. Appl. Phys. 23, 291 (1984), doi:10.1143/JJAP.23.291, Eqs. (6)–(7) and Table II (n₀ = 3.29); the printed values have two digits | ≤ 1.5e-2 | 0 | 1.5e-2 | pass |
| `material/aln-majkic-d33` | published | AlN: $d_{33}$ from the measured ratio $0.169\thinspace d_{33}(\mathrm{LiNbO_3})$ and the catalogue's $d_{33}$ of congruent lithium niobate at 1.064 µm (pm/V, shown) | A. Majkić et al., Phys. Status Solidi B 254, 1700077 (2017), doi:10.1002/pssb.201700077, p. 4 and Table 1: 4.3 pm/V at 1030 nm, from Shoji et al.'s 25.2 pm/V | 4.25880 | 4.30000 | 5e-2 | pass |
| `material/aln-rigler-se` | published | AlN, Al-polar and N-polar (Rigler et al. 2015, $n^2 = 1 + A\lambda^2/(\lambda^2 - B^2)$): $n_o$ and $n_e$ at 658 nm against the ellipsometric values their Table I lists beside the fit (largest deviation shown) | M. Rigler et al., Appl. Phys. Express 8, 042603 (2015), doi:10.7567/APEX.8.042603, Table I, the 658 nm SE column, to three decimals with A and B rounded | ≤ 1e-3 | 0 | 1e-3 | pass |
| `material/aln-rigler-maie` | published | AlN, Al-polar and N-polar (Rigler et al. 2015): $n_o$ and $n_e$ at 658 nm against the independent multi-angle ellipsometry of the same films (largest deviation shown) | M. Rigler et al., Appl. Phys. Express 8, 042603 (2015), doi:10.7567/APEX.8.042603, Table I, the 658 nm MAIE column, each to 0.01 | ≤ 1e-2 | 0 | 1e-2 | pass |
| `material/ingap-ferrini-table-consistency` | published | InGaP above the gap (Ferrini et al. 2002, Table 3): $\varepsilon_1 = n^2 - k^2$ and $\varepsilon_2 = 2nk$ from the table's $n$ and $k$ against its printed $\varepsilon$, all 37 rows (largest deviation shown) | R. Ferrini et al., Eur. Phys. J. B 27, 449 (2002), doi:10.1140/epjb/e2002-00177-x, Table 3; n and k have three decimals, which allows about 0.006 in the product | ≤ 1e-2 | 0 | 1e-2 | pass |
| `material/ingap-ferrini-table-knots` | published | InGaP above the gap: the catalogue's $n + ik$ at Ferrini et al.'s 37 photon energies against their Table 3 (largest deviation shown) | R. Ferrini et al., Eur. Phys. J. B 27, 449 (2002), doi:10.1140/epjb/e2002-00177-x, Table 3 (the second row printed 4.2 eV read as 4.1); a natural spline passes through its knots | ≤ 1e-12 | 0 | 1e-12 | pass |
| `mode/slab-te-book` | published | The TE mode of 220 nm of silicon (3.473) in oxide (1.444) at 1550 nm (effective index shown) | L. Chrostowski, M. Hochberg, Silicon Photonics Design (2015), doi:10.1017/CBO9781316084168, Section 3.2.2: 2.845 (3 decimals) | 2.84482 | 2.84500 | 5e-4 | pass |
| `mode/slab-tm-book` | published | The TM mode of 220 nm of silicon (3.473) in oxide (1.444) at 1550 nm (effective index shown) | L. Chrostowski, M. Hochberg, Silicon Photonics Design (2015), doi:10.1017/CBO9781316084168, Section 3.2.2: 2.051 (3 decimals) | 2.05110 | 2.05100 | 5e-4 | pass |
| `mode/vector-slab-limit-te` | analytic | The full-vector solver on the book's slab, uniform along one axis, at a 2.5 nm mesh: TE (error against the exact slab shown) | the exact slab (mode::slab); the scheme converges at second order, tested at 20, 10 and 5 nm | ≤ 1e-4 | 0 | 1e-4 | pass |
| `mode/vector-slab-limit-tm` | analytic | The full-vector solver on the book's slab, uniform along one axis, at a 2.5 nm mesh: TM (error against the exact slab shown) | the exact slab (mode::slab); the scheme converges at second order, tested at 20, 10 and 5 nm | ≤ 1e-4 | 0 | 1e-4 | pass |
| `mode/strip-book` | published | The TE-like mode of a $500 \times 220$ nm silicon strip in oxide at 1550 nm, at a 5 nm mesh (effective index shown) | L. Chrostowski, M. Hochberg, Silicon Photonics Design (2015), doi:10.1017/CBO9781316084168, Fig. 3.14: 2.443 (Lumerical MODE, 20 nm conformal mesh, accurate to about 1e-3 by its Fig. 3.9); ours converges at about first order at the convex corners (as on Hadley's corner problems below), 2.4435 at 2.5 nm | 2.44440 | 2.44300 | 3e-3 | pass |
| `mode/hadley-box-low` | published | Hadley's corner problem 1: a box, $\varepsilon = 2.25$ in a quarter of the $1 \times 1$ µm domain, at 1.5 µm, on an $80 \times 80$ grid (12.5 nm; effective index shown) | G. R. Hadley, J. Lightwave Technol. 20, 1219 (2002), doi:10.1109/JLT.2002.800371, Fig. 4: $1.27627404 \pm 10^{-8}$ (series expansion); ours converges at about first order at convex corners and 1.4-1.8 at concave ones | 1.27632 | 1.27627 | 1e-4 | pass |
| `mode/hadley-box-high` | published | Hadley's corner problem 2: a box, $\varepsilon = 8$ in a quarter of the $1 \times 1$ µm domain, at 1.5 µm, on an $80 \times 80$ grid (12.5 nm; effective index shown) | G. R. Hadley, J. Lightwave Technol. 20, 1219 (2002), doi:10.1109/JLT.2002.800371, Fig. 5: $2.65679692 \pm 10^{-8}$ (series expansion); ours converges at about first order at convex corners and 1.4-1.8 at concave ones | 2.65679 | 2.65680 | 1e-4 | pass |
| `mode/hadley-corner-low` | published | Hadley's corner problem 3: an impinged corner, $\varepsilon = 2.25$ in three quarters of the $1 \times 1$ µm domain, at 1.5 µm, on an $80 \times 80$ grid (12.5 nm; effective index shown) | G. R. Hadley, J. Lightwave Technol. 20, 1219 (2002), doi:10.1109/JLT.2002.800371, Fig. 6: $1.387926425 \pm 2 \times 10^{-9}$ (series expansion); ours converges at about first order at convex corners and 1.4-1.8 at concave ones | 1.38790 | 1.38793 | 1e-4 | pass |
| `mode/hadley-corner-high` | published | Hadley's corner problem 4: an impinged corner, $\varepsilon = 8$ in three quarters of the $1 \times 1$ µm domain, at 1.5 µm, on an $80 \times 80$ grid (12.5 nm; effective index shown) | G. R. Hadley, J. Lightwave Technol. 20, 1219 (2002), doi:10.1109/JLT.2002.800371, Fig. 7: $2.761465320 \pm 5 \times 10^{-9}$ (series expansion); ours converges at about first order at convex corners and 1.4-1.8 at concave ones | 2.76143 | 2.76147 | 1e-4 | pass |
| `mode/slab-group-index` | analytic | The group index of a TE slab (220 nm of 3.473 between 1.444 and air) at 1.55 µm, from differences of its effective index over $\pm 2$ nm (error shown) | Hellmann-Feynman, no material dispersion: $n_g = \langle\varepsilon\rangle/n_\text{eff}$, $\langle\varepsilon\rangle$ weighted by $E^2$ of the exact field | ≤ 1e-6 | 0 | 1e-6 | pass |
| `mode/strip-group-index-book` | published | The group index of a $500 \times 220$ nm strip at 1.55 µm, with the book's dispersive silicon and 1.444 oxide, on a $6.25 \times 5$ nm grid (group index shown) | L. Chrostowski, M. Hochberg, Silicon Photonics Design (2015), doi:10.1017/CBO9781316084168, Fig. 3.22b: about 4.18, read off the plot to $\pm 0.005$ (Lumerical MODE, 20 nm mesh); materials from its Listing 3.1 | 4.17290 | 4.18000 | 2e-2 | pass |
| `mode/multilayer-bound-chilwell` | published | The 8 bound modes (TE and TM 0-3) of a four-layer guide, 1.0 / 1.66, 1.53, 1.60, 1.66 (500 nm each) / 1.50 at 632.8 nm, exact (largest deviation shown) | J. Chilwell, I. Hodgkinson, J. Opt. Soc. Am. A 1, 742 (1984), doi:10.1364/JOSAA.1.000742, Table 3: effective indices to 6 decimals | ≤ 5e-7 | 0 | 5e-7 | pass |
| `mode/multilayer-leaky-chilwell` | published | The 5 TE leaky waves ($m = 4, \ldots, 8$) of the same guide, complex effective indices, exact (largest deviation of a real or imaginary part shown) | J. Chilwell, I. Hodgkinson, J. Opt. Soc. Am. A 1, 742 (1984), doi:10.1364/JOSAA.1.000742, Table 2: to 5 decimals; $m = 5$'s real part, 1.38250, is ours (1.3824892) plus 1.1e-5, one unit in the last place, the other nine our values rounded | ≤ 1.5e-5 | 0 | 1.5e-5 | pass |
| `mode/multilayer-power-chilwell` | published | The share of each bound mode's power in the cover, each film and the substrate, 48 percentages (largest deviation shown, in percentage points) | J. Chilwell, I. Hodgkinson, J. Opt. Soc. Am. A 1, 742 (1984), doi:10.1364/JOSAA.1.000742, Table 3: to 0.1 % | ≤ 6e-2 | 0 | 6e-2 | pass |
| `mode/multilayer-fresnel` | analytic | A plane wave's reflection coefficient at one interface, 1.0 to 1.5, TE and TM at 0, 20, 45 and 70 degrees, by the transfer matrices (largest difference in $r$ shown) | Fresnel's equations: $r_s = (n_1\cos\theta_1 - n_2\cos\theta_2)/(n_1\cos\theta_1 + n_2\cos\theta_2)$, $r_p = (n_2\cos\theta_1 - n_1\cos\theta_2)/(n_2\cos\theta_1 + n_1\cos\theta_2)$; J. Chilwell, I. Hodgkinson, J. Opt. Soc. Am. A 1, 742 (1984), doi:10.1364/JOSAA.1.000742, say Eq. 13 reduces to them | ≤ 1e-12 | 0 | 1e-12 | pass |
| `mode/multilayer-bragg` | analytic | The reflectance of 8 quarter-wave pairs, 2.3 / 1.38 on 1.52 at 550 nm, normal incidence, by the transfer matrices (shown) | the quarter-wave stack's closed form, $R = \left(\frac{1 - q}{1 + q}\right)^2$ with $q = (n_s/n_0)(n_H/n_L)^{2N}$, from its admittance | 0.999258 | 0.999258 | 1e-12 | pass |
| `mode/pml-soi-leakage-te` | analytic | The loss of 220 nm SOI's TE mode leaking through 0.5 µm of buried oxide into the substrate, full-vector with a PML (1 µm, strength 3), 2.5 nm grid (relative error in $\operatorname{Im} n_\text{eff}$ shown) | the exact leaky mode of the same stack by transfer matrices (mode::multilayer); PML by complex coordinate stretching, W. C. Chew et al., Microw. Opt. Technol. Lett. 15, 363 (1997) | ≤ 1e-3 | 0 | 1e-3 | pass |
| `mode/pml-soi-leakage-tm` | analytic | The same for the TM mode (relative error in $\operatorname{Im} n_\text{eff}$ shown) | the exact leaky mode of the same stack by transfer matrices (mode::multilayer); PML by complex coordinate stretching, W. C. Chew et al., Microw. Opt. Technol. Lett. 15, 363 (1997) | ≤ 1e-3 | 0 | 1e-3 | pass |
| `mode/pml-leaky-chilwell` | published | Chilwell and Hodgkinson's TE leaky waves $m = 4, \ldots, 7$, full-vector with a PML (2 µm, strength 5) in the substrate, 2.5 nm grid (largest deviation of a real or imaginary part shown) | J. Chilwell, I. Hodgkinson, J. Opt. Soc. Am. A 1, 742 (1984), doi:10.1364/JOSAA.1.000742, Table 2, to 5 decimals; $m = 8$ ($\operatorname{Re} n_\text{eff} = 1.00304$, just above the cover's 1.0) has a slowly decaying, inward-phased field in the cover and is checked only in the leaky_waves example, within 2e-4 | ≤ 5e-5 | 0 | 5e-5 | pass |
| `mode/eim-strip-book` | published | The effective index method on a $500 \times 220$ nm silicon strip (3.473 in 1.444) at 1550 nm, TE-like, exact slabs (effective index shown) | L. Chrostowski, M. Hochberg, Silicon Photonics Design (2015), doi:10.1017/CBO9781316084168, Section 3.2.5: 2.489, from the slab index rounded to 2.845 and a 10 nm 1D mesh (on that input the exact lateral slab gives 2.488558); the method: G. B. Hocker, W. K. Burns, Appl. Opt. 16, 113 (1977), doi:10.1364/AO.16.000113 | 2.48837 | 2.48900 | 1e-3 | pass |
| `mode/bend-slab-te` | analytic | A slab (2.845, 500 nm, in 1.444) bent at 1 µm, $E$ normal to the bend plane, full-vector on a conformally mapped 2.5 nm grid with a PML: effective index along the arc (error shown) | the exact bent slab (mode::bend: radial shooting matched to the outgoing Hankel function, D. Marcuse, Bell Syst. Tech. J. 50, 2551 (1971), doi:10.1002/j.1538-7305.1971.tb02620.x, Eq. 10); the map: M. Heiblum, J. H. Harris, IEEE J. Quantum Electron. 11, 75 (1975), doi:10.1109/JQE.1975.1068563, exact for this polarization; second order | ≤ 3e-5 | 0 | 3e-5 | pass |
| `mode/bend-slab-te-loss` | analytic | The same bend's radiation loss, $\operatorname{Im} n_\text{eff} = 9.29 \times 10^{-4}$ (relative error shown) | the exact bent slab (mode::bend); the PML starts at 2.5 µm, outside the bend's turning point | ≤ 2e-3 | 0 | 2e-3 | pass |
| `mode/bend-slab-tm` | analytic | The same bend with $E$ in the bend plane, where scaling an isotropic permittivity is an approximation (error shown) | the exact bent slab (mode::bend); the exact equivalent medium would be anisotropic in both permittivity and permeability; the error falls as the radius grows (1.3e-4 at 3 µm) | ≤ 2e-3 | 0 | 2e-3 | pass |
| `mode/bend-marcuse` | published | Marcuse's bending-loss formula against the exact loss of a slab (1.6 in 1.5, 1 µm, at 1 µm) bent at 120 µm (ratio minus one shown) | D. Marcuse, Bell Syst. Tech. J. 50, 2551 (1971), doi:10.1002/j.1538-7305.1971.tb02620.x, Eqs. 32-33, an approximation for large radii: its deviation falls as $1/R$, 0.14 at 80 µm, 0.084 at 120 µm and 0.061 at 160 µm | ≤ 1e-1 | 0 | 1e-1 | pass |
| `mode/leaky-wire-bienstman` | published | The leaky SOI wire benchmark ($500 \times 220$ nm Si 3.5 on 1 µm SiO₂ 1.45 on Si, air above, 1.55 µm), TE: $\operatorname{Re} n_\text{eff}$, Richardson-extrapolated from core grids of 5, 2.5 and 1.25 nm (order $\approx 0.67$, the corners'), with a PML in the substrate | P. Bienstman et al., Opt. Quantum Electron. 38, 731 (2006), doi:10.1007/s11082-006-9025-9, Table 6: 2.412372, from CAMFR and the aperiodic Fourier modal method (7 digits); raw errors +4.0e-3, +2.5e-3, +1.5e-3 | 2.41229 | 2.41237 | 2e-4 | pass |
| `mode/leaky-wire-bienstman-loss` | published | The same wire's substrate leakage, $\operatorname{Im} n_\text{eff} \times 10^8$, extrapolated alike | P. Bienstman et al., Opt. Quantum Electron. 38, 731 (2006), doi:10.1007/s11082-006-9025-9, Table 6: 2.9135 (CAMFR) and 2.91348 (aperiodic Fourier modal method); the raw results are 0.97, 0.98 and 0.99 of it | 2.92066 | 2.91350 | 6e-2 | pass |
| `mode/fields-butt-coupling` | analytic | The power a 220 nm silicon slab's TE mode launches into a 300 nm slab's (3.473 in 1.444, 1.55 µm), from the full-vector fields on a 5 nm grid (error shown) | the exact slab fields (mode::slab): for TE slabs $H$ is proportional to $E$, so the coupling is $(\int E_1 E_2)^2 / (\int E_1^2 \int E_2^2) = 0.994662$ | ≤ 5e-5 | 0 | 5e-5 | pass |
| `mode/marcatili-closed-form` | published | Marcatili's closed-form approximation against his transcendental equations, $E^x_{11}$ and $E^y_{11}$ of his guide $a = 2b$, $n_1/n_4 = 1.05$, where $(k_z^2 - k_4^2)/(k_1^2 - k_4^2) \geq 0.5$ (largest relative difference shown) | E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2071 (1969), doi:10.1002/j.1538-7305.1969.tb01166.x, p. 2083: 'within a few percent of the exact value' there; 4.1 % here | ≤ 5e-2 | 0 | 5e-2 | pass |
| `mode/marcatili-vector` | published | Marcatili's approximation (his transcendental equations) against the full-vector solver, $E^x_{11}$ of his guide $a = 2b$, $n_1/n_4 = 1.05$, at $(2b/\lambda)(n_1^2 - n_4^2)^{1/2} = 3$, far from cutoff (difference in the normalized constant shown) | E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2071 (1969), doi:10.1002/j.1538-7305.1969.tb01166.x, Eqs. 3, 6-7, 20-21; Fig. 6b's regime: 1e-4 apart at $B = 3$ and 4, 1.2e-3 at 1.5, 9e-3 at 1 near cutoff, where the corners Marcatili ignores hold field | ≤ 5e-4 | 0 | 5e-4 | pass |
| `mode/slab-fd-chilwell` | published | The 8 bound modes of Chilwell and Hodgkinson's four-layer guide by 1D finite differences on a 1 nm grid (largest deviation shown) | J. Chilwell, I. Hodgkinson, J. Opt. Soc. Am. A 1, 742 (1984), doi:10.1364/JOSAA.1.000742, Table 3: effective indices to 6 decimals; the scheme is second order (tested against the exact slab) | ≤ 2e-6 | 0 | 2e-6 | pass |
| `fdfd/slab-reflection-ez` | analytic | 2D FDFD, $E$ along $z$: the reflectance of 220 nm of silicon (3.476) on oxide (1.444) under air, 30 degrees, 1.55 µm, from the fluxes on a 2.5 nm grid (shown) | the exact stack by transfer matrices (mode::multilayer, J. Chilwell, I. Hodgkinson, J. Opt. Soc. Am. A 1, 742 (1984), doi:10.1364/JOSAA.1.000742, Eqs. 13-16, TE); second order: 2.1e-3, 5.5e-4, 1.4e-4, 3.5e-5 at 20, 10, 5, 2.5 nm | 0.0581346 | 0.0581696 | 5e-5 | pass |
| `fdfd/slab-reflection-hz` | analytic | The same with $H$ along $z$ (shown) | the exact stack by transfer matrices (TM); second order: 1.9e-3, 4.9e-4, 1.2e-4, 3.1e-5 at 20, 10, 5, 2.5 nm | 0.0268435 | 0.0268744 | 5e-5 | pass |
| `fdfd/flux-conservation` | analytic | 2D FDFD: the power through every row from the oxide through the silicon into the air, both polarizations at 0, 30 and 60 degrees, 10 nm grid (largest relative spread shown) | Poynting's theorem: no power is lost or made in a lossless region without sources; the scheme's own flux keeps this exactly | ≤ 1e-10 | 0 | 1e-10 | pass |
| `fdfd/pml-reflection` | analytic | 2D FDFD: what a 20-cell PML graded to $R = 10^{-8}$ ($m = 3$) sends back of a plane wave 17 degrees off its normal, in oxide on a 20 nm grid, both polarizations (largest amplitude shown) | W. Shin, S. Fan, J. Comput. Phys. 231, 3406 (2012), doi:10.1016/j.jcp.2012.01.013, Eqs. 2.7-2.9: graded for $R = 10^{-8}$ in vacuum at normal incidence; in oxide 17 degrees off, the round trip absorbs to $(10^{-8})^{1.38}$, an amplitude of 3e-6; measured 2.5e-6 | ≤ 1e-5 | 0 | 1e-5 | pass |
| `fdfd/port-mode-te` | analytic | 2D FDFD ports: the fundamental mode of a 220 nm silicon slab (3.476 in 1.444) at 1.55 µm, $E$ along $z$, solved on a port column of a 2.5 nm grid: effective index (shown) | the exact slab (mode::slab); the port's 1D operator is the 2D scheme's own, second order: 2.6e-3, 6.5e-4, 1.6e-4 at 10, 5, 2.5 nm | 2.84794 | 2.84778 | 2e-4 | pass |
| `fdfd/port-mode-tm` | analytic | The same with $H$ along $z$ (shown) | the exact slab (mode::slab); second order: 2.5e-3, 6.1e-4, 1.5e-4 at 10, 5, 2.5 nm | 2.05347 | 2.05332 | 2e-4 | pass |
| `fdfd/straight-guide` | analytic | 2D FDFD ports: a straight silicon slab between two ports 1.4 µm apart, both polarizations, 20 nm grid: largest of the magnitudes of $S_{11}$ and $S_{22}$ and of the errors of $S_{21}$ and $S_{12}$ against $\exp(i\beta L)$ (shown) | a uniform guide transmits its mode whole with phase $\beta L$; the port modes are the grid's own and the source is total-field/scattered-field (R. C. Rumpf, Prog. Electromagn. Res. B 36, 221 (2012), doi:10.2528/PIERB11092006, Eq. 55) | ≤ 1e-12 | 0 | 1e-12 | pass |
| `fdfd/reciprocity` | analytic | 2D FDFD ports: a slab stepping from 220 to 300 nm, both polarizations, 10 nm grid: $S_{21}$ against $S_{12}$ (largest relative difference shown) | Lorentz reciprocity: $S$ is symmetric for a reciprocal device; the scheme keeps it with the PMLs' stretches as weights and the modes normalized by the unconjugated Lorentz form | ≤ 1e-12 | 0 | 1e-12 | pass |
| `fdfd/step-reflection-te` | analytic | The same step's reflection of the 220 nm slab's TE mode, $E$ along $z$ (shown) | Fresnel's formula on the two modes' effective indices, $((n_1 - n_2)/(n_1 + n_2))^2 = 1.16503 \times 10^{-3}$ for 2.84742 and 3.04866: the TE modal impedance is the effective index; an approximation, good here to 0.07 % | 0.00116424 | 0.00116503 | 2e-5 | pass |
| `fdfd/adjoint-gradient-ez` | analytic | 2D FDFD, $E$ along $z$: the adjoint gradient of the power a silicon slab with a bump beside it delivers into its right port's mode, against fourth-order central finite differences ($\delta = 10^{-3}$) on a cell each in the bump, the core and the oxide (largest relative difference shown) | the adjoint variable method, G. Veronis, R. W. Dutton, S. Fan, Opt. Lett. 29, 2288 (2004), doi:10.1364/OL.29.002288, Eqs. 2-4: $\nabla F = -2\operatorname{Re}(\lambda^T \mathrm{d}A\thinspace u)$, $A^T\lambda = \mathrm{d}F/\mathrm{d}u$; the finite differences' own round-off is about $10^{-10}/\delta$ | ≤ 1e-6 | 0 | 1e-6 | pass |
| `fdfd/adjoint-gradient-hz` | analytic | The same with $H$ along $z$, where the permittivity enters through the faces' $1/\varepsilon$ (shown) | the adjoint variable method, as above; the faces' permittivity the mean of their two cells | ≤ 1e-6 | 0 | 1e-6 | pass |
| `mode/hadley-uniform-box` | analytic | Hadley's high-accuracy equations in a uniform region: the box of Hadley I, Fig. 5 ($n = 3.44$, $2 \times 2$ µm, 1.15 µm) on an $8 \times 8$ grid (250 nm; effective index shown) | exact, $\sqrt{\varepsilon - ((\pi/4)^2 + (\pi/2)^2)/k^2}$ for $H_y = \cos(\pi x/4) \sin(\pi y/2)$; the equations are G. R. Hadley, J. Lightwave Technol. 20, 1210 (2002), doi:10.1109/JLT.2002.800361, Eqs. 7-9; the standard scheme's error on this grid is 1.6e-4 | 3.42495 | 3.42495 | 1e-9 | pass |
| `mode/hadley-uniform-order` | analytic | The same box: the order of convergence of the effective index from $4 \times 4$ to $8 \times 8$ grids (shown to two decimals) | sixth order: Hadley I, Fig. 5, slope 6.03; errors 1.7e-8 and 2.4e-10 here (6.03 from $8 \times 8$ to $16 \times 16$, where 3.7e-12 nears round-off) | 6.12000 | 6.00000 | 2e-1 | pass |
| `mode/hadley-interface` | analytic | Hadley's interface equations: the two-dielectric box of Hadley I, Fig. 6 ($\varepsilon = 1$ over $\varepsilon = 11.8336$, 1.5 µm wide, 0.975 µm) on a 31.25 nm grid (effective index shown) | exact: separable, $H_y = \sin(\pi x/W)\thinspace Y(y)$ with $Y$ and $Y'$ continuous, so $k_b \tan(k_b L_b) + k_t \tan(k_t L_t) = 0$ with $k^2 = k_0^2 (\varepsilon - n_\text{eff}^2) - (\pi/W)^2$; the equations are Hadley I, Eqs. 20-24 and 43; the standard scheme's error on this grid is 5.4e-6 | 3.42099 | 3.42099 | 5e-9 | pass |
| `mode/hadley-interface-order` | analytic | The same box: the order of convergence of the effective index from 62.5 to 31.25 nm grids (shown to two decimals) | sixth order: Hadley I, Fig. 7 (fifth-order interface equations, diluted by a line of interface nodes); errors 9.7e-8 and 1.6e-9 here | 5.89000 | 6.00000 | 3e-1 | pass |
| `mode/hadley-interface-turned` | analytic | The same box turned on its side, so that $H_x$ is the component normal to the interface, at 62.5 nm (difference of the effective indices shown) | symmetry: Hadley derives the equations for a horizontal interface; a vertical one is the same with $x$ and $y$ exchanged | ≤ 1e-12 | 0 | 1e-12 | pass |
| `mode/hadley-corners-box-low` | published | Hadley's corner problem 1 (a box, $\varepsilon = 2.25$) by his high-accuracy equations on a $128 \times 128$ grid (7.8 nm; effective index shown) | G. R. Hadley, J. Lightwave Technol. 20, 1219 (2002), doi:10.1109/JLT.2002.800371, Fig. 4: $1.27627404 \pm 10^{-8}$ (series expansion); the corner equations are its Eqs. 50 and 52, their misprinted $\theta \sin\theta$ read as $\theta \sin 2\theta$ from Eq. 47; the standard scheme's error on this grid is 3.8e-5 | 1.27627 | 1.27627 | 1e-6 | pass |
| `mode/hadley-corners-box-high` | published | Hadley's corner problem 2 (a box, $\varepsilon = 8$) by his high-accuracy equations on a $128 \times 128$ grid (7.8 nm; effective index shown) | G. R. Hadley (2002), part II, Fig. 5: $2.65679692 \pm 10^{-8}$; the standard scheme's error on this grid is 9.0e-6 | 2.65680 | 2.65680 | 1e-6 | pass |
| `mode/hadley-corners-impinged-low` | published | Hadley's corner problem 3 (an impinged corner, $\varepsilon = 2.25$) by his high-accuracy equations on a $128 \times 128$ grid (7.8 nm; effective index shown) | G. R. Hadley (2002), part II, Fig. 6: $1.387926425 \pm 2 \times 10^{-9}$; the standard scheme's error on this grid is 1.4e-5 | 1.38793 | 1.38793 | 1e-6 | pass |
| `mode/hadley-corners-impinged-high` | published | Hadley's corner problem 4 (an impinged corner, $\varepsilon = 8$) by his high-accuracy equations on a $128 \times 128$ grid (7.8 nm; effective index shown) | G. R. Hadley (2002), part II, Fig. 7: $2.761465320 \pm 5 \times 10^{-9}$; the standard scheme's error on this grid is 1.6e-5 | 2.76147 | 2.76147 | 1e-6 | pass |
| `mode/hadley-corners-order` | published | Hadley's corner problem 1 by his equations: the order of convergence of the effective index from $32 \times 32$ to $128 \times 128$ grids (shown to two decimals) | G. R. Hadley (2002), part II, Section IV: second order for most cases (Figs. 8-11), where the standard scheme's is about first; errors 9.1e-6, 2.1e-6, 5.2e-7 here | 2.06000 | 2.00000 | 2.5e-1 | pass |
| `fdfd3d/film-reflection-te` | analytic | 3D FDFD, s (TE) polarized: the reflectance of 220 nm of silicon (3.476) on oxide (1.444) under air, 30 degrees from the normal in a plane 30 degrees from $x$, 1.55 µm, from the fluxes on a 2.5 nm grid (shown) | the exact stack by transfer matrices (mode::multilayer, J. Chilwell, I. Hodgkinson, J. Opt. Soc. Am. A 1, 742 (1984), doi:10.1364/JOSAA.1.000742, Eqs. 13-16, TE); second order: 2.8e-3, 7.3e-4, 1.9e-4, 4.6e-5 at 20, 10, 5, 2.5 nm | 0.0581234 | 0.0581696 | 6e-5 | pass |
| `fdfd3d/film-reflection-tm` | analytic | The same, p (TM) polarized (shown) | the exact stack by transfer matrices (TM); second order: 1.9e-3, 4.9e-4, 1.2e-4, 3.1e-5 at 20, 10, 5, 2.5 nm | 0.0268434 | 0.0268744 | 5e-5 | pass |
| `fdfd3d/flux-conservation` | analytic | 3D FDFD: the power through every plane from the oxide through the silicon into the air, both polarizations at 0, 30 and 60 degrees, 10 nm grid (largest relative spread shown) | Poynting's theorem: no power is lost or made in a lossless region without sources; the scheme's own flux (tangential $E$ averaged across the plane, $H$ on it) keeps this exactly | ≤ 1e-10 | 0 | 1e-10 | pass |
| `fdfd3d/pml-reflection` | analytic | 3D FDFD: what a 20-cell PML graded to $R = 10^{-8}$ ($m = 3$) sends back of a plane wave 17 degrees off its normal in a plane 30 degrees from $x$, in oxide on a 20 nm grid, both polarizations (largest amplitude shown) | W. Shin, S. Fan, J. Comput. Phys. 231, 3406 (2012), doi:10.1016/j.jcp.2012.01.013, Eqs. 2.5-2.9: graded for $R = 10^{-8}$ in vacuum at normal incidence; in oxide 17 degrees off, the round trip absorbs to $(10^{-8})^{1.38}$, an amplitude of 3e-6; measured 2.5e-6, as in 2D | ≤ 1e-5 | 0 | 1e-5 | pass |
| `fdfd3d/two-d-agreement` | analytic | 3D FDFD on a structure invariant along $z$ (a silicon rod in lossy oxide, Bloch-periodic in $x$ and $y$, 25 nm grid, one cell along $z$) against the 2D solver, $E$ along $z$ and $H$ along $z$ (largest field difference relative to the largest field shown) | with $\partial/\partial z = 0$ Maxwell's equations split into the two 2D polarizations (K. S. Yee, IEEE Trans. Antennas Propag. 14, 302 (1966), doi:10.1109/TAP.1966.1138693); on the same grid and averaging the two discrete systems are the same equations, one eliminating $H$ and the other $E$ | ≤ 1e-11 | 0 | 1e-11 | pass |
| `fdfd3d/port-mode-slab-te` | analytic | 3D FDFD ports: the fundamental TE mode of a 220 nm silicon slab (3.476 in 1.444) at 1.55 µm, uniform along $y$, solved on a port's plane normal to $x$ on a 2.5 nm grid: effective index (shown) | the exact slab (mode::slab); the port's eigenproblem is the 3D scheme's own on the plane, second order: 9.6e-4, 2.4e-4, 6.0e-5 at 10, 5, 2.5 nm | 2.84784 | 2.84778 | 1e-4 | pass |
| `fdfd3d/port-mode-slab-tm` | analytic | The same, the TM mode (shown) | the exact slab (mode::slab); second order: 2.5e-3, 6.1e-4, 1.5e-4 at 10, 5, 2.5 nm, the 2D ports' errors | 2.05347 | 2.05332 | 2e-4 | pass |
| `fdfd3d/port-mode-strip` | cross-code | 3D FDFD ports: the TE-like mode of a $500 \times 220$ nm silicon strip in oxide at 1.55 µm on a port's plane of a 20 nm grid, inside walls $2.02 \times 3.5$ µm: effective index (shown) | Hadley's high-accuracy equations (mode::hadley, G. R. Hadley, J. Lightwave Technol. 20, 1219 (2002), doi:10.1109/JLT.2002.800371) on 20, 10 and 5 nm grids, extrapolated: 2.445380; the port converges to it at second order (2.4e-3, 6.4e-4, 1.7e-4; its own limit 2.445387), and Fallahkhair et al.'s scheme (mode::vector) at about first, slowed by the corners | 2.44781 | 2.44538 | 3e-3 | pass |
| `fdfd3d/straight-strip` | analytic | 3D FDFD ports: a straight silicon strip ($500 \times 220$ nm in oxide) between two ports 0.25 µm apart, PMLs close around it, 50 nm grid: largest of the magnitudes of $S_{11}$ and $S_{22}$ and of the errors of $S_{21}$ and $S_{12}$ against $\exp(i\beta L)$ (shown) | a uniform guide transmits its mode whole with phase $\beta L$; the port modes are the grid's own and the source is total-field/scattered-field (R. C. Rumpf, Prog. Electromagn. Res. B 36, 221 (2012), doi:10.2528/PIERB11092006, Eq. 55) | ≤ 1e-12 | 0 | 1e-12 | pass |
| `fdfd3d/reciprocity` | analytic | 3D FDFD ports: a silicon strip stepping from 400 to 600 nm wide, off the grid's axis, PMLs close around it, 50 nm grid: $S_{21}$ against $S_{12}$ (relative difference shown) | Lorentz reciprocity: $S$ is symmetric for a reciprocal device; the scheme keeps it, with the PMLs' stretches as weights and the modes normalized by the unconjugated Lorentz form | ≤ 1e-12 | 0 | 1e-12 | pass |
| `fdfd3d/closed-guide-energy` | analytic | 3D FDFD ports: a silicon strip in a closed metal box of oxide ($0.6 \times 0.4$ µm), stepping from 300 to 400 nm wide, ports 1 µm from the step, 50 nm grid: the $S$-matrix between all the propagating modes on both sides (3 and 3), its distance from unitary (shown) | Poynting's theorem: a closed lossless guide loses no power, so $S^\dagger S = 1$ but for the power the evanescent modes carry across the ports' planes, which falls as $e^{-2\kappa d}$: 4e-4, 1e-4, 2.6e-5, 7e-6, 2e-6 for ports 0.2 to 1 µm from the step | ≤ 5e-6 | 0 | 5e-6 | pass |
| `fdfd3d/two-d-s-matrix` | analytic | 3D FDFD ports on a structure invariant along $z$ (a silicon slab stepping from 220 to 300 nm, one periodic cell along $z$, 20 nm grid) against the 2D solver's $S$-matrix, $E$ along $z$ and $H$ along $z$ (largest difference shown) | with $\partial/\partial z = 0$ the 3D scheme is the 2D one (fdfd3d/two-d-agreement); with $H$ along $z$ the two agree to the eigensolver's tolerance (2e-10), with $E$ along $z$ to 9e-9, the PMLs half a cell apart in the two grids; both take a mode's backward twin with the same tangential $E$, so the reflections agree in sign | ≤ 5e-8 | 0 | 5e-8 | pass |
| `fdfd3d/qmr-direct` | analytic | 3D FDFD by QMR, on the curl-curl operator and on Shin and Fan's ($s = -1$), to a relative residual of 1e-10, against the sparse direct solver: a silicon strip in oxide, $16^3$ cells of 40 nm, PMLs all round (largest field difference relative to the largest field shown) | the same system solved two ways: QMR, R. W. Freund, N. M. Nachtigal, Numer. Math. 60, 315 (1991), doi:10.1007/BF01385726, Algorithm 3.1 without look-ahead; Shin and Fan's operator, Opt. Express 21, 22578 (2013), doi:10.1364/OE.21.022578, Eq. 7, has the same solution; measured 1.1e-11 and 1.3e-10 | ≤ 1e-9 | 0 | 1e-9 | pass |
| `fdfd3d/qmr-plateau` | published | Shin and Fan's vacuum square (their Fig. 1: $50 \times 50$ cells of 2 nm, periodic, uniform along $z$, an $x$-polarized dipole at its centre, 1.55 µm), QMR on the curl-curl operator ($s = 0$): the relative residual where it stagnates, at iteration 20 (shown) | W. Shin, S. Fan, Opt. Express 21, 22578 (2013), doi:10.1364/OE.21.022578, Section 3 and Fig. 3: the residual's part in the near-null eigenspace, 0.707, holds the residual there initially (GMRES; QMR is GMRES for this real symmetric matrix) | 0.708633 | 0.707000 | 5e-3 | pass |
| `fdfd3d/qmr-iterations-curl-curl` | published | The same square, $s = 0$: QMR iterations to a relative residual of 1e-6 (shown) | W. Shin, S. Fan, Opt. Express 21, 22578 (2013), doi:10.1364/OE.21.022578, Fig. 3: the $s = 0$ curve crosses 1e-6 at about $m = 114$, read off the plot to $\pm 5$ | 114.000 | 114.000 | 5e0 | pass |
| `fdfd3d/qmr-iterations-shin-fan` | published | The same square, $s = -1$: QMR iterations to a relative residual of 1e-6 (shown) | W. Shin, S. Fan, Opt. Express 21, 22578 (2013), doi:10.1364/OE.21.022578, Fig. 3: the $s = -1$ curve crosses 1e-6 at about $m = 77$, read off the plot to $\pm 5$ | 79.0000 | 77.0000 | 5e0 | pass |
| `fdfd3d/pml-reflection-stretched` | analytic | 3D FDFD: what a 20-cell PML graded to $R = 10^{-8}$ ($m = 3$) and stretched as much as it absorbs, $s = 1 + (1 + i)\sigma$, sends back of a plane wave 17 degrees off its normal, in oxide on a 20 nm grid, both polarizations (largest amplitude shown) | W. C. Chew, W. H. Weedon, Microw. Opt. Technol. Lett. 7, 599 (1994), doi:10.1002/mop.4650071304: any stretch with $\operatorname{Im} s \gt 0$ absorbs without reflecting, so the real part changes only the discretization's reflection: 3.6e-6, against 2.5e-6 without it (fdfd3d/pml-reflection); thinner, it reflects more (10 cells of 10 nm: 2.3e-4 against 4e-5) | ≤ 1e-5 | 0 | 1e-5 | pass |
| `fdfd3d/qmr-ilu-direct` | analytic | 3D FDFD by QMR on Shin and Fan's operator preconditioned by its ILU(0), to a relative residual of 1e-10, against the sparse direct solver: a silicon strip in oxide, $24 \times 20 \times 16$ cells of 40 nm, stretched PMLs of 6 cells all round (largest field difference relative to the largest field shown) | the same system solved two ways; ILU(0) as in Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd ed., SIAM (2003), doi:10.1137/1.9780898718003, from the right, so the residual QMR stops on is the system's own; measured 2.4e-10, in 160 iterations against 548 without it | ≤ 1e-8 | 0 | 1e-8 | pass |
| `circuit/series-waveguides` | analytic | Circuits: two waveguides, 12.5 and 30.25 µm, in series are one of 42.75 µm ($n_\text{eff} = 2.4$, $n_g = 4.2$, 3 dB/cm), 1.54 to 1.56 µm (largest $\lvert \Delta S \rvert$ shown) | $e^{i\phi_1} e^{i\phi_2} = e^{i(\phi_1 + \phi_2)}$: a connection joins two reference planes with nothing between them; round-off of the 400 rad phases, about $10^{-13}$ | ≤ 1e-12 | 0 | 1e-12 | pass |
| `circuit/mzi-closed-form` | analytic | Circuits: a Mach-Zehnder interferometer of two lossless couplers ($\kappa^2$ = 0.5 and 0.3) and arms of 100 and 120 µm, 1.54 to 1.56 µm: its four transmissions (largest $\lvert \Delta S \rvert$ shown) | the product of the transfer matrices, $C \operatorname{diag}(t_1, t_2)\thinspace C$ with $C_{11} = C_{22} = r$ and $C_{12} = C_{21} = i\kappa$, $r^2 + \kappa^2 = 1$ | ≤ 1e-13 | 0 | 1e-13 | pass |
| `circuit/ring-all-pass-bogaerts` | published | Circuits: an all-pass ring, a coupler ($\kappa^2$ = 0.1 and 0.02) with one output fed back through 62.8 µm of waveguide, 1.54 to 1.56 µm: the through field (largest error shown) | W. Bogaerts et al., Laser Photonics Rev. 6, 47 (2012), doi:10.1002/lpor.201100017, Eq. 1: $e^{i(\pi + \phi)} (a - r e^{-i\phi}) / (1 - r a e^{i\phi})$, and its square, Eq. 2 | ≤ 1e-12 | 0 | 1e-12 | pass |
| `circuit/ring-add-drop-bogaerts` | published | Circuits: an add-drop ring, two couplers ($\kappa^2$ = 0.1 and 0.05) joined by two halves of a 62.8 µm ring, 1.54 to 1.56 µm: the through and drop powers (largest error shown) | W. Bogaerts et al., Laser Photonics Rev. 6, 47 (2012), doi:10.1002/lpor.201100017, Eqs. 5 and 6 | ≤ 1e-12 | 0 | 1e-12 | pass |
| `circuit/ring-fsr-bogaerts` | published | Circuits: the all-pass ring ($\kappa^2 = 0.1$, 62.8 µm, $n_g = 4.2$): the spacing of its two resonances either side of 1.55 µm, found as minima of the through power (nm, shown) | W. Bogaerts et al., Laser Photonics Rev. 6, 47 (2012), doi:10.1002/lpor.201100017, Eq. 9: $\lambda^2 / (n_g L)$ at the resonances' midpoint; first order, its error here $(\Delta\lambda / 2\lambda)^2$ relative, 8e-5 nm | 9.08149 | 9.08157 | 1e-4 | pass |
| `circuit/sub-network-growth` | analytic | Circuits: the sparse solve against sub-network growth on a netlist of 11 instances (two nested rings and an MZI, three reflective multiports, loops), 6 external ports, 1.54 to 1.56 µm (largest $\lvert \Delta S \rvert$ shown) | G. Filipsson, 11th European Microwave Conference, 700 (1981), doi:10.1109/EUMA.1981.332972, Eq. 6, one connection at a time: the same S to round-off | ≤ 1e-13 | 0 | 1e-13 | pass |
| `circuit/reciprocity` | analytic | Circuits: the same netlist, reciprocal parts: $S$ against $S^\mathsf{T}$ (largest $\lvert S_{qp} - S_{pq} \rvert$ shown) | a netlist of reciprocal components is reciprocal, $S = S^\mathsf{T}$ | ≤ 1e-13 | 0 | 1e-13 | pass |
| `circuit/unitarity` | analytic | Circuits: a lossless netlist (an add-drop ring, an all-pass ring and an MZI), 4 external ports, 1.54 to 1.56 µm: $S^\dagger S$ against $I$ (largest entry of the difference shown) | a netlist of lossless components is lossless, $S^\dagger S = I$ | ≤ 1e-13 | 0 | 1e-13 | pass |
| `circuit/adjoint-mzi` | analytic | Circuit adjoint: an MZI's bar power summed over five wavelengths, 1.549 to 1.551 µm, its gradient with respect to both couplers' $\kappa^2$ and both arms' lengths against fourth-order central finite differences of the whole circuit ($\delta$ = 1e-3 and 1e-4 µm; the largest difference relative to the largest component shown) | the adjoint variable method, G. Veronis, R. W. Dutton, S. Fan, Opt. Lett. 29, 2288 (2004), doi:10.1364/OL.29.002288, Eqs. 2-4, on the circuit's system $M = I - S_b \Gamma$: $\partial F/\partial \theta = 2 \operatorname{Re} \operatorname{tr}(\Lambda^\mathsf{T} \thinspace \partial_\theta S_b \thinspace A)$ with $M^\mathsf{T} \Lambda = E G$; the differences' own round-off and truncation, about 1e-10 | ≤ 1e-8 | 0 | 1e-8 | pass |
| `circuit/adjoint-ring` | analytic | Circuit adjoint: an add-drop ring ($\kappa^2$ = 0.1 and 0.05, 62.8 µm) on the flank of a resonance, 1.5521 µm: the gradient of $(\lvert S_{41} \rvert^2 - 0.5)^2$ with respect to its four parameters, the same way ($\delta$ = 1e-4 and 1e-5 µm; shown as above) | the adjoint variable method, as above; the differences' round-off, about 3e-11 | ≤ 1e-8 | 0 | 1e-8 | pass |
| `circuit/adjoint-mesh` | analytic | Circuit adjoint: a 4 × 4 mesh of six MZIs in Clements et al.'s rectangular arrangement, 24 parameters (every coupler's $\kappa^2$ and every phase): the gradient of $\sum_{qp} \lvert S_{qp} - T_{qp} \rvert^2$ for a complex target $T$, the same way ($\delta$ = 1e-3; shown as above) | the adjoint variable method, as above, with a sensitivity $G = \overline{S - T}$ that depends on the phases; the mesh is W. R. Clements et al., Optica 3, 1460 (2016), doi:10.1364/OPTICA.3.001460, Fig. 1(b); the differences' round-off, about 1e-11 | ≤ 1e-8 | 0 | 1e-8 | pass |
| `circuit/adjoint-nested` | analytic | Circuit adjoint: a netlist of circuits (an add-drop ring and an MZI as instances, a reflective 3-port in a loop, a waveguide), $\sum_\lambda \lvert S_{41} \rvert^2$ at three wavelengths, its 9 parameters through the inner circuits' own adjoints, the same way ($\delta$ = 1e-4 and 1e-5 µm; shown as above) | the adjoint variable method, as above; a circuit's derivatives are its Jacobian $\partial S/\partial \theta = Z^\mathsf{T} \partial_\theta S_b \thinspace A$, $M^\mathsf{T} Z = E$; the differences' round-off, about 5e-10 | ≤ 1e-8 | 0 | 1e-8 | pass |
| `circuit/adjoint-differences` | analytic | Circuit adjoint: the circuit solve's netlist of 11 instances, whose components give no derivatives, differentiated each by fourth-order differences of its S-matrix (step $\epsilon^{1/3} \max(\lvert \theta \rvert, 1)$): $\sum_\lambda \lvert S_{41} \rvert^2$ at three wavelengths, its 11 parameters, the same way ($\delta$ = 1e-4 and 1e-5 µm; shown as above) | the adjoint variable method, as above; the components' differences add about 1e-11, the circuit's differences' round-off about 5e-10 | ≤ 1e-8 | 0 | 1e-8 | pass |
| `circuit/component-differences` | analytic | A component's $\partial S/\partial \theta$ by fourth-order differences (central; one-sided inwards at a bound), against its closed form: a waveguide's length at 100 µm and at 0, a coupler's $\kappa^2$ at 0.3, a phase shifter at its bounds (the largest difference relative to the largest entry shown) | $\partial t/\partial L = t\thinspace(i 2\pi n/\lambda - \alpha \ln 10/20)$, $\partial r/\partial \kappa^2 = -1/2r$, $\partial \kappa/\partial \kappa^2 = 1/2\kappa$, $\partial e^{i\phi}/\partial \phi = i e^{i\phi}$; the round-off of a step of $6 \times 10^{-6} \max(\lvert \theta \rvert, 1)$, about 1e-11 | ≤ 1e-9 | 0 | 1e-9 | pass |
| `components/ring-all-pass-bogaerts` | published | Components: the all-pass ring's closed form ($\kappa^2$ = 0.1 and 0.02, 62.8 µm, $n_g = 4.2$, 3 dB/cm), 1.54 to 1.56 µm, against Bogaerts's through field and against its netlist, a coupler fed back through a waveguide, solved as a circuit (largest $\lvert \Delta S \rvert$ shown) | W. Bogaerts et al., Laser Photonics Rev. 6, 47 (2012), doi:10.1002/lpor.201100017, Eq. 1: $e^{i(\pi + \phi)} (a - r e^{-i\phi}) / (1 - r a e^{i\phi})$ | ≤ 1e-12 | 0 | 1e-12 | pass |
| `components/ring-add-drop-bogaerts` | published | Components: the add-drop ring's closed form ($\kappa_1^2 = 0.1$, $\kappa_2^2 = 0.05$, 62.8 µm), 1.54 to 1.56 µm, against Bogaerts's through and drop powers and against its netlist of two couplers and two half rings (largest difference shown) | W. Bogaerts et al., Laser Photonics Rev. 6, 47 (2012), doi:10.1002/lpor.201100017, Eqs. 5 and 6 | ≤ 1e-12 | 0 | 1e-12 | pass |
| `components/ring-extremes-bogaerts` | published | Components: the rings' through and drop powers on resonance ($\phi = 2\pi m$) and off it ($\phi = 2\pi m + \pi$), all-pass and add-drop, $\kappa^2$ = 0.05 and 0.2 (largest difference shown) | W. Bogaerts et al., Laser Photonics Rev. 6, 47 (2012), doi:10.1002/lpor.201100017, Eqs. 11 to 16, e.g. $R_\text{min} = (r - a)^2 / (1 - r a)^2$ | ≤ 1e-12 | 0 | 1e-12 | pass |
| `components/ring-linewidth-bogaerts` | published | Components: the full width at half maximum measured on the rings' spectra (all-pass through, add-drop drop; $\kappa^2$ = 0.02 and 0.05, 62.8 µm, 3 dB/cm) against Bogaerts's formulas (largest relative difference shown) | W. Bogaerts et al., Laser Photonics Rev. 6, 47 (2012), doi:10.1002/lpor.201100017, Eqs. 7 and 8, $(1 - ra)\lambda^2 / (\pi n_g L \sqrt{ra})$: a Lorentzian line, $\cos \phi \approx 1 - \phi^2 / 2$ across it, good to about $(1 - ra)^2$, 8e-4 at most here | ≤ 1e-3 | 0 | 1e-3 | pass |
| `components/ring-fsr-fdfd` | published | Components: the free spectral range of jobs/ring-fdfd.toml's all-pass ring (radius 2 µm, 2D FDFD by the effective index method, 25 nm grid), from its two resonances near 1.52 and 1.567 µm (nm shown) | W. Bogaerts et al., Laser Photonics Rev. 6, 47 (2012), doi:10.1002/lpor.201100017, Eq. 9, $\lambda^2 / (n_g L)$ at the resonances' midpoint, $n_g$ of the exact bent slab of the same plane (radial shooting, Marcuse 1971, doi:10.1002/j.1538-7305.1971.tb02620.x); the tolerance is the 25 nm grid's | 46.6964 | 46.9498 | 5e-1 | pass |
| `components/mzi-closed-form` | analytic | Components: a Mach-Zehnder interferometer netlist of two directional couplers (17 and 23 µm) and arms of 150 and 50 µm (3 dB/cm), 1.54 to 1.56 µm, against its transfer matrices (largest $\lvert \Delta S \rvert$ shown) | $C_2 \operatorname{diag}(t_\text{lower}, t_\text{upper})\thinspace C_1$, each coupler's $C$ its through and across fields | ≤ 1e-13 | 0 | 1e-13 | pass |
| `components/mzi-unitarity` | analytic | Components: the same interferometer without loss, 1.54 to 1.56 µm: $S^\dagger S$ against $I$ (largest entry of the difference shown) | a netlist of lossless components is lossless, $S^\dagger S = I$ | ≤ 1e-13 | 0 | 1e-13 | pass |
| `components/unitarity` | analytic | Components: the lossless waveguide, bend, coupler, directional coupler, all-pass and add-drop rings, 1.54 to 1.56 µm: $S^\dagger S$ against $I$ (largest entry of the difference shown) | a lossless component conserves power, $S^\dagger S = I$ | ≤ 1e-14 | 0 | 1e-14 | pass |
| `components/reciprocity` | analytic | Components: every first component, lossy, and the 1 x 2 and 2 x 2 MMIs, 1.54 to 1.56 µm: $S$ against $S^\mathsf{T}$ (largest $\lvert S_{qp} - S_{pq} \rvert$ shown) | reciprocity, $S = S^\mathsf{T}$, as the circuits' conventions state it (docs/design/components.md) | ≤ 1e-15 | 0 | 1e-15 | pass |
| `components/passivity` | analytic | Components: the same components: the largest singular value of $S$ (shown), at most 1 | a passive component gains no power: every singular value of $S$ at most 1, the lossless ones exactly 1 | 1.00000 | 1.00000 | 1e-12 | pass |
| `components/derivatives` | analytic | Components: the closed forms' exact parameter derivatives $\partial S / \partial \theta$ (by dual numbers) against central differences (largest difference relative to the derivative's largest entry shown) | $(S(\theta + h) - S(\theta - h)) / 2h$, the best of $h/\theta$ = 1e-5 to 1e-8: its truncation and round-off, about 1e-8 | ≤ 1e-7 | 0 | 1e-7 | pass |
| `components/directional-coupler-power` | published | Components: a directional coupler's power across and through from its supermodes ($\Delta n$ from $C = 0.04$ per µm), 5 to 100 µm long, 1.54 to 1.56 µm (largest difference shown) | L. Chrostowski, M. Hochberg, Silicon Photonics Design (2015), doi:10.1017/CBO9781316084168, Eqs. 4.1 to 4.3: $\sin^2(\pi \Delta n L / \lambda)$ and $\cos^2$; the round-off of phases up to 1000 rad, about 1e-13 | ≤ 1e-12 | 0 | 1e-12 | pass |
| `components/mmi-beat-length-soldano` | published | Components: the beat length $L_\pi$ of a 3 µm wide multimode section on 220 nm SOI at 1.55 µm, seen from above (TM across it), from its two lowest exact slab modes (µm shown) | L. B. Soldano, E. C. M. Pennings, J. Lightwave Technol. 13, 615 (1995), doi:10.1109/50.372474, Eq. 6, $4 n_r W_e^2 / 3\lambda_0$ with Eq. 4's effective width; the paraxial expansion's next term is 1 %, 0.23 µm, the tolerance about twice that | 22.6035 | 22.8115 | 5e-1 | pass |
| `components/mmi-fdfd` | analytic | Components: the 1 x 2 MMI of jobs/mmi-fdfd.toml (3 x 8.55 µm, 0.5 µm guides) at 1.55 µm by guided-mode propagation against 2D FDFD of the same plane (20 nm grid): the power in each output (the first shown) | L. B. Soldano, E. C. M. Pennings (1995), doi:10.1109/50.372474, Eqs. 8 to 12 with the exact slab modes, against the FDFD solver; what the guided modes leave out (radiation modes, the faces' reflections) sets the tolerance | 0.397813 | 0.380772 | 3e-2 | pass |
| `components/mzi-neff-dwivedi-470` | published | Components: measured Mach-Zehnder interferometers of a 470 x 211 nm silicon wire in oxide (the SEM's cross-section, a rectangle; Hadley's equations, about 10 nm grids), $\Delta L$ designed for the drawn 450 x 215 nm wire, two ideal splitters, the spectrum read as the paper reads it: $n_\text{eff}$ at 1550 nm from the $m$ = 15 interferometer's peak, $n_\text{eff} L = m \lambda$ (Eq. 3), carried to 1550 nm by $n_g$ | S. Dwivedi et al., J. Lightwave Technol. 33, 4471 (2015), doi:10.1109/JLT.2015.2476603, Table I, measured: 2.355 ± 0.002; the tolerance is their Eq. 5 with ±20 nm of width and ±5 nm of thickness (their Fig. 1's process), by the solver's derivatives, plus that uncertainty | 2.35831 | 2.35500 | 6.1e-2 | pass |
| `components/mzi-ng-dwivedi-470` | published | Components: measured Mach-Zehnder interferometers of a 470 x 211 nm silicon wire in oxide (the SEM's cross-section, a rectangle; Hadley's equations, about 10 nm grids), $\Delta L$ designed for the drawn 450 x 215 nm wire, two ideal splitters, the spectrum read as the paper reads it: $n_g$ at 1550 nm from the $M$ = 110 interferometer's peaks, $\lambda_1 \lambda_2 / ((\lambda_2 - \lambda_1) \Delta L)$ (Eq. 8), fitted by a line in $\lambda$ (Eq. 10) | S. Dwivedi et al., J. Lightwave Technol. 33, 4471 (2015), doi:10.1109/JLT.2015.2476603, Table I, measured: 4.2739 ± 0.0042; the tolerance is their Eq. 5 with ±20 nm of width and ±5 nm of thickness (their Fig. 1's process), by the solver's derivatives, plus that uncertainty | 4.23300 | 4.27390 | 4.9e-2 | pass |
| `components/mzi-neff-dwivedi-602` | published | Components: measured Mach-Zehnder interferometers of a 602 x 211 nm silicon wire in oxide (the SEM's cross-section, a rectangle; Hadley's equations, about 10 nm grids), $\Delta L$ designed for the drawn 600 x 215 nm wire, two ideal splitters, the spectrum read as the paper reads it: $n_\text{eff}$ at 1550 nm from the $m$ = 15 interferometer's peak, $n_\text{eff} L = m \lambda$ (Eq. 3), carried to 1550 nm by $n_g$ | S. Dwivedi et al., J. Lightwave Technol. 33, 4471 (2015), doi:10.1109/JLT.2015.2476603, Table I, measured: 2.534 ± 0.0035; the tolerance is their Eq. 5 with ±20 nm of width and ±5 nm of thickness (their Fig. 1's process), by the solver's derivatives, plus that uncertainty | 2.53496 | 2.53400 | 4.2e-2 | pass |
| `components/mzi-ng-dwivedi-602` | published | Components: measured Mach-Zehnder interferometers of a 602 x 211 nm silicon wire in oxide (the SEM's cross-section, a rectangle; Hadley's equations, about 10 nm grids), $\Delta L$ designed for the drawn 600 x 215 nm wire, two ideal splitters, the spectrum read as the paper reads it: $n_g$ at 1550 nm from the $M$ = 110 interferometer's peaks, $\lambda_1 \lambda_2 / ((\lambda_2 - \lambda_1) \Delta L)$ (Eq. 8), fitted by a line in $\lambda$ (Eq. 10) | S. Dwivedi et al., J. Lightwave Technol. 33, 4471 (2015), doi:10.1109/JLT.2015.2476603, Table I, measured: 4.0453 ± 0.0045; the tolerance is their Eq. 5 with ±20 nm of width and ±5 nm of thickness (their Fig. 1's process), by the solver's derivatives, plus that uncertainty | 4.03847 | 4.04530 | 3.2e-2 | pass |
| `components/mzi-neff-dwivedi-805` | published | Components: measured Mach-Zehnder interferometers of a 805 x 211 nm silicon wire in oxide (the SEM's cross-section, a rectangle; Hadley's equations, about 10 nm grids), $\Delta L$ designed for the drawn 800 x 215 nm wire, two ideal splitters, the spectrum read as the paper reads it: $n_\text{eff}$ at 1550 nm from the $m$ = 15 interferometer's peak, $n_\text{eff} L = m \lambda$ (Eq. 3), carried to 1550 nm by $n_g$ | S. Dwivedi et al., J. Lightwave Technol. 33, 4471 (2015), doi:10.1109/JLT.2015.2476603, Table I, measured: 2.67 ± 0.004; the tolerance is their Eq. 5 with ±20 nm of width and ±5 nm of thickness (their Fig. 1's process), by the solver's derivatives, plus that uncertainty | 2.65810 | 2.67000 | 3.1e-2 | pass |
| `components/mzi-ng-dwivedi-805` | published | Components: measured Mach-Zehnder interferometers of a 805 x 211 nm silicon wire in oxide (the SEM's cross-section, a rectangle; Hadley's equations, about 10 nm grids), $\Delta L$ designed for the drawn 800 x 215 nm wire, two ideal splitters, the spectrum read as the paper reads it: $n_g$ at 1550 nm from the $M$ = 110 interferometer's peaks, $\lambda_1 \lambda_2 / ((\lambda_2 - \lambda_1) \Delta L)$ (Eq. 8), fitted by a line in $\lambda$ (Eq. 10) | S. Dwivedi et al., J. Lightwave Technol. 33, 4471 (2015), doi:10.1109/JLT.2015.2476603, Table I, measured: 3.8902 ± 0.005; the tolerance is their Eq. 5 with ±20 nm of width and ±5 nm of thickness (their Fig. 1's process), by the solver's derivatives, plus that uncertainty | 3.88832 | 3.89020 | 2.1e-2 | pass |
| `compact/vf-paper-rms` | published | Vector fitting, the paper's test: an 18th-order response (2 real poles and 8 conjugate pairs, $d = 0.2$, $h = 2 \times 10^{-5}$) at 100 frequencies from 1 Hz to 100 kHz, fitted from the 20 complex starting poles of their Table 2 in one relocation, real model (RMS error shown) | B. Gustavsen, A. Semlyen, IEEE Trans. Power Deliv. 14, 1052 (1999), doi:10.1109/61.772353, Section 4.2 and Table 1: an RMS error of 3.8e-12, round-off on a response that peaks near 200; 1.4e-11 here | ≤ 1e-10 | 0 | 1e-10 | pass |
| `compact/vf-paper-poles` | published | The same fit's 18 poles against Table 1 (largest error relative to the pole's magnitude shown) | Gustavsen and Semlyen (1999), Table 3: errors up to 4e-10 Hz on poles of 4.5 to 90 kHz; the two surplus poles' residues are below 4.3e-7 Hz there, 3e-6 rad/s here | ≤ 1e-10 | 0 | 1e-10 | pass |
| `compact/vf-paper-residues` | published | The same fit's 18 residues against Table 1 (largest error relative to the residue's magnitude shown) | Gustavsen and Semlyen (1999), Table 3: errors up to 1e-7 Hz on residues of 3 to 94 kHz | ≤ 1e-9 | 0 | 1e-9 | pass |
| `compact/vf-paper-real-start` | published | The same response fitted from 20 real starting poles spread over the band, three relocations (RMS error shown) | Gustavsen and Semlyen (1999), Section 4.3 and Table 4: 7.1, 1.0e-11 and 4.2e-13 after one, two and three relocations; 20, 6e-12 and 2e-12 here | ≤ 1e-10 | 0 | 1e-10 | pass |
| `compact/vf-fast-equivalence` | analytic | Vector fitting's pole relocation with each response's own unknowns eliminated by QR, against the whole block least squares: 3 responses of an all-pass ring, 12 starting poles, 151 wavelengths (largest difference in the scaling function's coefficients, relative to the largest, shown) | D. Deschrijver, M. Mrozowski, T. Dhaene, D. De Zutter, IEEE Microw. Wireless Compon. Lett. 18, 383 (2008), doi:10.1109/LMWC.2008.922585, Eqs. 8, 10 and 11: the same least squares, reduced; 1.1e-11 here | ≤ 1e-8 | 0 | 1e-8 | pass |
| `compact/vf-ring-allpass` | analytic | An all-pass ring's through transmission ($r = 0.95$, $a = 0.98$, 10 µm radius, $n_g = 4.2$), 4 resonances from 1.53 to 1.57 µm at 401 wavelengths, fitted with 12 complex poles: largest $\lvert\Delta S\rvert$ at 1001 wavelengths (shown) | W. Bogaerts et al., Laser Photonics Rev. 6, 47 (2012), doi:10.1002/lpor.201100017, Eq. 1, with $n_\text{eff}$ first order in $\lambda$ (their Eq. 10), which makes the round trip's phase linear in $k = 2\pi/\lambda$ | ≤ 1e-8 | 0 | 1e-8 | pass |
| `compact/vf-ring-poles` | analytic | The same fit's poles against the ring's exact poles, where $r a e^{i\phi} = 1$ (largest error relative to the pole's real part, the resonance's half-width, shown) | Bogaerts et al. (2012), Eq. 1: $k_m = (2\pi m + C + i \ln ra)/(L n_g)$ with $C = L (n_g - n_0) k_0$, each with the residue $-(1/r - r)/(L n_g)$ in $s = -ik$ | ≤ 1e-8 | 0 | 1e-8 | pass |
| `compact/fdfd-ring-held-out` | analytic | A 2D FDFD spectrum (jobs/ring-fdfd.toml: an all-pass ring of 2 µm radius on 220 nm SOI by the effective index method, $H$ along $z$, at a 40 nm grid, 51 wavelengths from 1.50 to 1.60 µm), its 2-port S fitted with 12 poles on the 26 even-numbered wavelengths: largest $\lvert\Delta S_{qp}\rvert$ at the 25 between them (shown) | the solver's own S at the wavelengths left out, 4.5e-5 here; at its own wavelengths the fit is within 8.5e-7, and fitted to all 51 within 6.6e-7: on 25, 40 and 50 nm grids alike the solver's spectrum fits no closer than about 2e-7 | ≤ 2e-4 | 0 | 2e-4 | pass |
| `compact/fdfd-ring-passive` | analytic | The same ring's 12-pole model fitted to all 51 wavelengths: its largest singular value over 1001 wavelengths (shown; a passive device's is at most 1) | passivity: a device without gain gives out at most the power it takes in, so no singular value of its S exceeds 1 | 0.907740 | 1.00000 | 0e0 | pass |
| `compact/param-ring-coupling` | analytic | A parametric model of the all-pass ring over its self-coupling, $r$ from 0.90 to 0.97 (5 samples of 151 wavelengths, 12 basis poles, numerator and denominator polynomial of degree 2, all samples fitted at once): largest $\lvert\Delta S\rvert$ at the 4 values halfway between samples, 401 wavelengths each (shown) | Bogaerts et al. (2012), Eq. 1: $(r - a z)/(1 - r a z)$ is a ratio of functions linear in $r$, which a numerator and denominator of degree 1 represent exactly; the form is P. Triverio et al.'s, IEEE Trans. Adv. Packag. 32, 205 (2009), doi:10.1109/TADVP.2008.2007913, Eq. 6, solved as C. K. Sanathanan and J. Koerner do, IEEE Trans. Autom. Control 8, 56 (1963), doi:10.1109/TAC.1963.1105517, Eqs. 5-7; 2.3e-10 here | ≤ 1e-8 | 0 | 1e-8 | pass |
| `compact/param-ring-shift` | analytic | A parametric model of the ring over $n_\text{eff}$ at 1.55 µm from 2.39 to 2.41, which shifts its resonances by 0.8 of a free spectral range (13 samples, 16 basis poles, degree 6): largest $\lvert\Delta S\rvert$ at the 12 values halfway between samples (shown) | Bogaerts et al. (2012), Eq. 1; 1.9e-7 here; the 16 shared poles alone fit the samples only to 1.3, Triverio et al.'s piecewise-linear model fails between them (below), and fitting each sample alone and interpolating its poles missed by 0.2 to 0.8, measured while choosing the method | ≤ 5e-6 | 0 | 5e-6 | pass |
| `compact/param-ring-stable` | analytic | The same model made stable at 21 values (poles in the right half plane flipped, residues identified again over 801 wavelengths): largest $\lvert\Delta S\rvert$ against Eq. 1 (shown) | Bogaerts et al. (2012), Eq. 1; 1.2e-7 here; the flip is Gustavsen and Semlyen's (1999), Section 2; the polynomial model's own poles aren't constrained, and some fall in the right half plane, out of the band | ≤ 5e-6 | 0 | 5e-6 | pass |
| `compact/param-ring-piecewise` | published | Triverio et al.'s piecewise-linear model of the ring over its self-coupling, $r$ from 0.90 to 0.97 (5 samples, each fitted alone with 12 poles and rewritten on 12 basis poles spread over the band): largest $\lvert\Delta S\rvert$ at the 4 values halfway between samples (shown) | P. Triverio, S. Grivet-Talocia, M. S. Nakhla, IEEE Trans. Adv. Packag. 32, 205 (2009), doi:10.1109/TADVP.2008.2007913, Eqs. 8-9 and Theorem 1 (Eqs. 24-26), basis poles as their Section III-C; 8.6e-4 here, a straight line's error between samples; the same over the 0.8-free-spectral-range shift misses by 7, its poles crossing the axis | ≤ 2e-3 | 0 | 2e-3 | pass |
| `compact/param-ring-piecewise-stable` | analytic | The same model's uniform stability: the crossings of the imaginary axis found over the whole range, exactly, plus one if its poles sampled at 1001 values disagree (shown) | stable at the samples by construction (Triverio et al., Section III-D); between them, a rank-one change of the poles' matrix along each segment, whose crossings are the real zeros of a rational function, found as the generalized eigenvalues of its system pencil; Triverio et al.'s own test, linear matrix inequalities (their Theorem 2), needs a semidefinite solver | ≤ 0e0 | 0 | 0e0 | pass |
| `compact/param-ring-piecewise-crossings` | analytic | The test on the piecewise-linear model of the 0.8-free-spectral-range shift, 13 samples: of the crossings it finds (64 here), those the nearest pole's sign doesn't confirm, plus the steps of 2001 sampled values whose count of unstable poles changes with no crossing reported near (shown) | the poles at sampled values, from their own eigenvalues; poles within 1e-4 of the axis, relative to their size, are at the round-off of these eigenproblems (the rewritten coefficients reach 1e5 to 1e6, the ill-conditioning of Triverio et al.'s Table I) and not counted either way | ≤ 0e0 | 0 | 0e0 | pass |