# 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 CI fails when a case fails or this file is out of date.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/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 |
| `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/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 |