Thin-Film Coating Simulator — benchmark

MgF₂ 100 / Ag 15 / TiO₂ 60 on BK7, 400–899 nm.

Spot values you can check without running anything (B1a, 40°, s-polarised)

A 15 nm silver layer inside the stack at 40° and 70°, in both polarizations. Absorption (R + T = 0.977 at 40°, measured closure gap 2.0 × 10⁻²), oblique incidence and the p-polarised admittance are all live simultaneously. This is where sign conventions on r_p and the complex-cosine branch cut go wrong, and where they would be invisible at normal incidence.

MgF₂ 99.75 / BK7 1 mm incoherent / MgF₂ 99.75 in air, 350–849 nm.

This is the case a sophisticated skeptic actually probes. Our per-layer incoherent handling is Katsidis-Siapkas — a 2×2 intensity-matrix product between coherent blocks; the reference’s incoherent path is a different formulation of the same physics. That two independent incoherent formalisms agree to 1.1 × 10⁻¹⁶ on a coherent–incoherent–coherent sandwich is a much stronger statement than any coherent-only agreement, because the incoherent path is where implementations genuinely disagree.

SCHOTT’s published N-BK7 data sheet (glass code 517642.251), freely downloadable, prints n_d = 1.51680 (587.5618 nm), n_e = 1.51872 (546.074 nm) and ν_d = 64.17. Our Sellmeier coefficients come from the SCHOTT Zemax catalogue 2017-01-20b; the check values are an independent output of that fit, printed by the manufacturer.

Reference values are the manufacturer’s published check values for N-BK7 (SCHOTT AG data sheet, glass code 517642.251). Photizon values computed independently from the shipped Sellmeier fit. The three published values were read from the data sheet header and cross-checked against the RefractiveIndex.INFO SCHOTT page, which independently states n_d 1.5168 and V_d 64.17.

All five designs built from the paper’s own design formula and its published coefficients, then run through the shipped engine on the paper’s own grid: 400–900 nm at 0.5 nm, 1001 points, normal incidence, air incidence medium. Materials as the paper states them: n_H = 2.35, n_L = 1.45, substrate n_sub = 1.52, all non-dispersive and lossless. The formula coefficients are quarter-wave optical thickness units at a control wavelength of 500 nm, so a coefficient q on a layer of index n is a physical thickness q · 125 nm / n. A term whose coefficient is zero is omitted.

The design formula and its published coefficients

Outermost (air-side) layer first. Every number below is an input to the rebuild; the layer count beside each design is an output of it, and it is also the published one.

Design coefficients from Table 2, “Parameters of Five Wide Band Reflector Designs”, of A. V. Tikhonravov and M. K. Trubetskov, Applied Optics 51(30), 7319–7332 (2012), journal p. 7328, DOI: 10.1364/AO.51.007319. The printed table was read two independent ways — a text extraction and a 300 DPI visual render of the page — which agree on all 19 rows across all five designs. Photizon values computed independently.

Reference values (layer count, total optical thickness and minimum reflectance) are from Table 2 of A. V. Tikhonravov and M. K. Trubetskov, “Modern design tools and a new paradigm in optical coating design”, Applied Optics 51(30), 7319–7332 (2012), DOI: 10.1364/AO.51.007319. Photizon values computed independently with the shipped transfer-matrix engine on the paper’s own 0.5 nm, 1001-point grid over 400–900 nm.

Two internal consistency checks on the inputs

• Layer count is an arithmetic identity of the design formula: N = 7 + 2(m₁+m₂+m₃), minus one when c₄ = 0. Evaluated on the published m values this gives 73, 83, 85, 83, 82 — exactly the published layer counts, 5/5, with no fitting.
• Total optical thickness is 125 nm × Σ(coefficients), which reproduces the published totals to ≤ 0.031 % across all five designs — consistent with the 3-decimal rounding of the printed parameters, again with no fitting.

What the agreement is limited by, and the two open items

The published design parameters are printed to 3 decimals and the results to 2. Perturbing every coefficient uniformly within ±0.0005 — half of the last printed digit, 150 draws per design — moves R_min by sd 0.005 pp for design 1 and sd 0.002 pp for design 5, and moves R_avg by under 0.0001 pp. So a few hundredths of a percentage point is the agreement level this case can realistically reach, and any residual smaller than that is not evidence of anything.

These authors are the originators of needle optimisation and the principals behind the design package the coating industry actually uses. A design given by a closed-form expression with printed numeric parameters, checked against a printed table of numbers, needs no curve digitisation — which removes the single largest error source in any literature reproduction.

Macleod introduces this design by observing that an inner half-wave layer is an absentee at λ₀, so the two-layer stack’s peak reflectance in the middle of the low-reflectance region equals that of the single-layer coating. Raising the index of the central quarter-wave layer to 2.0 and then to 2.13 reduces the reflectance at λ₀ twice more. Four independently checkable claims in one paragraph. All designs are all-quarter-wave at λ₀, non-dispersive, lossless, normal incidence, air incidence medium.

Reference design and printed values from H. A. Macleod, Thin-Film Optical Filters, 4th ed., CRC Press / Taylor & Francis (2010), p. 146. Photizon values computed independently with the shipped transfer-matrix engine at normal incidence.

Two independent checks that this is a reproduction and not a coincidence

• Analytic identity. For an all-quarter-wave stack at λ₀, Y = n₁²n₃²/(n₂²n_s) and R = ((n₀−Y)/(n₀+Y))². Evaluated by hand this gives 0.01260079021463029, 0.0037647873741598423 and 2.370862680869589e-06 — the engine agrees to 6.9e-18, −2.6e-18 and 5.8e-19 respectively. That is a different computation from the transfer matrix, so it isolates the layer algebra.
• λ₀-invariance is bit-identical. Rebuilding the 0.38 % design at λ₀ = 400, 550, 1000 and 10 000 nm returns 0.0037647873741598397 at all four — the same float, not four close floats. A quarter-wave design’s on-resonance reflectance must not depend on the reference wavelength, and it does not.

Its first claim is the half-wave absentee theorem, which is already this page’s strongest case (nine lossless insertions returning a bit-identical number). This gives that invariance a textbook page number.

The only case on this page that tests the optimizer rather than the transfer-matrix algebra. The problem was chosen to sit inside both papers’ validity grids simultaneously: band 400–800 nm, air incidence, substrate glass n = 1.52, n_L = 1.45, n_H = 2.03, all non-dispersive, normal incidence. Merit: 101 uniform targets on 400–800 nm, R → 0, unpolarized. Optimizer: the shipped needle refinement (max 40 insertions, minimum layer 1.0 nm). Reporting grid: 0.1 nm, 4001 points.

Two independent published estimators of the average residual reflectance achievable by a two-material broadband AR, restated here in our own typography:

with, in the paper’s own terms: B the longest wavelength in the band divided by the shortest (valid 1.4 to 5.0); L the index of the outermost layer (valid 1.1 to 2.2); T the total optical thickness in waves at the geometric mean of the band edges (valid 1.0 to 9.0); D the highest minus the lowest index among all layers except the last (valid 0.4 to 2.8). Near-normal incidence only; the output is a percentage.

Estimator from Eq. (1) of R. R. Willey, “Predicting achievable design performance of broadband antireflection coatings”, Applied Optics 32(28), 5447–5451 (1993), p. 5448, DOI: 10.1364/AO.32.005447. Photizon values computed independently.

where M is the number of clusters in the design, and R_∞ and b are tabulated per band ratio and index ratio. The cluster optical thickness that turns a total optical thickness into a cluster count M is

Estimator and tabulated parameters from Eqs. (2)–(3) and Tables 2–3 of A. V. Tikhonravov, M. K. Trubetskov, T. V. Amotchkina and J. A. Dobrowolski, “Estimation of the average residual reflectance of broadband antireflection coatings”, Applied Optics 47(13), C124–C130 (2008), DOI: 10.1364/AO.47.00C124. Photizon values computed independently.

Three things this establishes. Neither envelope is violated — both formulas estimate the achievable optimum and our best design sits above both, as it must; a result below either would have been a red flag about the engine, the merit or our reading of the formula, and that is the falsifiable part. The two published estimators agree with each other to 0.96 % (0.26983 % vs 0.27241 %) — two different groups, fifteen years apart, one an empirical curve-fit and one a cluster-theory derivation, at the same operating point. And our shipped needle optimizer lands within about 10 % of both, which is a statement about the optimizer that no analytic case on this page can make.

Two assumptions, flagged rather than hidden

• The averaging measure. Neither paper states whether average residual reflectance is averaged uniformly in λ or in wavenumber. We used uniform in λ. For a design with sub-percent ripple across a 2:1 band this is a small effect, but it is unverified.
• Merit vs reported quantity. Our optimizer minimises RMS(R); both papers report the mean of R. The procedures match, but the optimised functional and the reported functional are not the same function.

Reproducing the cluster-thickness formula against its own published table

Before using Eq. (2) we ran it against all eighteen rows of the paper’s own table of cluster thicknesses. Seventeen of the eighteen agree within 0.51 nm (largest of those seventeen: 0.5011674418578878 nm), and one row differs by 8.276 nm. That summary is over the whole table, not over what is shown below.

7 of the 18 rows are shown here — the rows the argument turns on: the six that share ρ_HL = 1.62069 and so bracket the outlier on both sides, plus the row this page’s engine test uses. The outlier is one of the seven.

Printed cluster-thickness values from Table 2 of Tikhonravov, Trubetskov, Amotchkina and Dobrowolski, Applied Optics 47(13), C124–C130 (2008), DOI: 10.1364/AO.47.00C124 — 7 of the table’s 18 rows, being the rows this argument uses. The 17/18-within-0.51 nm statistic above is computed over all eighteen. Photizon values computed independently from Eq. (2) of the same paper.

The one row that does not reproduce, and why we print it

At 400–1100 nm with the index pair printed beside it (ρ_HL = 2.35/1.45 = 1.62069), Eq. (2) gives 633.7236490956041 nm against a printed 642 — a residual of −8.276 nm, while the other seventeen rows sit within 0.51 nm. The residual is internal to the published paper and we cannot resolve it from outside. Two explanations, neither asserted:

• (a) That cell was computed with a different index pair than the one printed beside it. Solving Eq. (2) = 642 at λ_u = 1100 gives ρ = 1.70174791563606, i.e. essentially exactly 1.7 = 2.465/1.45, for which Eq. (2) returns 641.8262507501745 — a −0.174 nm residual, dead-centre in the band the other rows occupy. A 1.45/2.465 row exists in both neighbouring blocks of the same table, so a one-column shift in a source spreadsheet is a mundane and specific mechanism.
• (b) Our reconstruction of Eq. (2) misses a ρ-dependence that only manifests at this point. The paper’s own data disfavours this heavily, and that can be quantified rather than asserted: ρ_HL = 1.62069 appears in six rows bracketing λ_u = 1100 on both sides, so a missing term would have to vanish at λ_u = 600, 700, 900, 1000 and 1500 and equal +8.276 nm at 1100 alone. No smooth function of (λ_u, ρ) behaves that way. We would not bet on this explanation — and we also cannot rule it out from outside, so both go on the record.

A second, weaker signal from the paper’s own data points the same way. For the five other ρ = 1.62069 rows, the experimental column minus the theory column is +26, +17, +2, +14 and +36 nm — all positive, mean +19.0. The 400–1100 row against the printed 642 gives −2 nm, outside that range; against Eq. (2)’s 633.724 it gives +6.276 nm, inside it. Five points is not a strong prior and we do not treat it as one.

We do not “fix” that row by adopting ρ = 1.7. Substituting our arithmetic for a publisher’s data is exactly what our data-integrity rule forbids, and no claim on this page depends on the row: the engine test above runs on the 400–800 row.

The tabulated parameters the estimate uses

Printed parameters from Table 3 of the same 2008 paper — 2 of the table’s 12 rows: the row the estimate above uses (first) and its neighbour in the index ratio. The ln b column is recomputed here from the printed b and reproduces the printed value 12/12 across the full table, which is what establishes that the b column was read correctly digit for digit. Photizon values computed independently.

Two independent published estimators of the best achievable broadband AR, from different groups fifteen years apart, agree with each other to 0.96 %, and our optimizer lands about 9 % above both. That is a claim about the tool, not about arithmetic.

A two-layer MgF₂ / Ta₂O₅ anti-reflection coating on BK7, target R = 0 at 532 nm, optimised by the shipped Levenberg-Marquardt refiner against a single-point merit function — a merit a user can build in the optimizer panel’s target table in three clicks.

/thin-film?layers=main/MgF2/Franta:123.0660934743829,main/Ta2O5/Gao:138.34811272910952&sub=BK7&range=400-899&target=532

Analytic, not “our own merit”. At a single wavelength the two-layer AR problem has a closed solution: setting the recursive Fresnel amplitude r = (r₀₁ + r₁e^2iβ₁)/(1 + r₀₁r₁e^2iβ₁) to zero is two real equations in (d₁, d₂). It was solved with an independent implementation — a recursive Fresnel / Airy recursion sharing no code with the engine’s characteristic-matrix path — at n_MgF₂(532) = 1.384210012165197 (the measured e-beam film entry main/MgF2/Franta), n_Ta₂O₅(532) = 2.16353 + 3.3 × 10⁻⁵i and n_BK7(532) = 1.5194725830654814. The solution is a two-branch family, periodic in each layer by its own half-wave thickness (MgF₂ 192.1673717587982 nm, Ta₂O₅ 122.94722051462193 nm):

…plus any integer number of half-waves added to either layer. This enumeration is the reference; it is a property of the physics, not of our optimizer.

Fifteen runs from fifteen starting points: five fixed — (100, 100), (150, 50), (50, 150), (200, 200) and (123.33, 138.00) nm — plus ten uniform-random in [20, 300]² nm from default_rng(20260804).

An optimizer is easy to fake: run it, print the merit it reached, declare success. Here the answer is known independently and is not unique, so the test is whether the optimizer lands on the solution manifold from arbitrary starting points — which catches a wrong analytic gradient, a sign error in the Jacobian, and a merit that is subtly not what it claims to be. Load the returned design and read R = 0.0000 % at 532 nm on the plot.