Ray-Optics Designer — benchmark

A paraboloid mirror of focal length 100 mm (curvature 1/200, conic constant −1) reflects 20 rays that arrive parallel to its axis, at heights of 1 to 45 mm and random azimuth: all 20 have to pass through the focus, at z = f.

Measured

7.9 × 10⁻¹⁵ mm (bar 10⁻¹¹)

Compared against

conic focal property (A)

An ellipsoid mirror with semi-axes a = 100 and b = 60 reflects a ray that leaves one focus: the ray has to arrive at the other focus, and the distance it travels between the two has to equal the long axis, 2a.

Measured

2.8 × 10⁻¹⁴ mm; miss at F₂ 1.9 × 10⁻¹⁶

Compared against

conic focal property (A)

A single spherical surface (radius 50, air into glass of index 1.5168, object at −200) is traced at three apertures two ways, by the engine’s vector formulas and by the classical trigonometric (L, U) trace with its own independent set of formulas: the two have to land in the same place.

Measured

2.3 × 10⁻¹³ mm (bar 10⁻¹⁰)

Compared against

the classical trigonometric (L, U) trace, an independent formula set (A)

A ray crosses a flat plate 10 mm thick of index 1.5 at 30°: it has to come out displaced sideways by exactly the closed form t·sin(θᵢ − θₜ)/cos θₜ.

Measured

4.4 × 10⁻¹⁶ mm

Compared against

closed form (A)

The Lagrange invariant H, a combination of ray height and angle that paraxial optics does not allow to change, is computed at every surface of the TRIPLET and has to come out the same at each one.

Measured

3.3 × 10⁻¹⁶ relative

Compared against

Welford (A)

Ten random skew rays are sent through the TRIPLET, and the skew invariant n(Ly − Mx), which an axially symmetric system cannot change, has to hold its value along each of them.

Measured

3.3 × 10⁻¹⁶

Compared against

axisymmetry (A)

Trace a skew ray through the TRIPLET test system, then walk the ray that comes out back through the same surfaces, re-solving each hit and each refraction: Snell’s law is reversible, so the returning ray has to come back to the point it was launched from.

Measured

1.2 × 10⁻¹⁶ mm

Compared against

reversibility of Snell (A)

The DOUBLET’s marginal ray is traced at a fraction ε of its full height, with ε running from 10⁻² to 10⁻³: third-order theory says the transverse error has to fall away as the third power of the height, and the power is fitted from the trace to see whether it does.

Measured

order 3.000 (Δ 2.2 × 10⁻⁵)

Compared against

Seidel theory (A)

A spherical surface has one pair of conjugate points that images perfectly at any aperture: with R = 50 and glass of index 1.5168, rays converging toward the first of them, the virtual object at R(n₁+n₂)/n₁, at heights of 2 to 30, all have to cross the axis at the second, R(n₁+n₂)/n₂ = 82.964135021.

Measured

spread 1.4 × 10⁻¹⁴ mm; crossing 0.0

Compared against

aplanatic points (A)

On ASPH-SINGLET, whose surface the engine has to find by iteration, two things have to hold: the point it settles on lies on the surface the sag formula describes, and the normal it returns is perpendicular to a tangent worked out separately by finite differences.

Measured

8.9 × 10⁻¹⁶ mm; normal 3.2 × 10⁻¹⁰ (finite-difference step)

Compared against

self-consistency (A)

A ray meets the boundary from glass of index 1.5168 into air at 60°, past the angle at which light can still get out: the engine has to raise an error that names total internal reflection rather than return a meaningless number.

Measured

raises, typed

Compared against

Snell (A)

Snell’s law is recomputed from the geometry the trace actually produced, at every surface and for ten skew rays: the residual |n₁ sin θᵢ − n₂ sin θₜ| has to be zero at each one.

Measured

8.3 × 10⁻¹⁷

Compared against

Snell (A)

The DOUBLET’s real-ray transverse aberration is fitted near the axis and its ρ³ coefficient extrapolated to ρ → 0, where it has to equal +S_I/(2n′u′), the same aberration reached through the third-order sums, sign included.

Measured

3.8 × 10⁻⁷ relative (ρ → 0 extrapolation)

Compared against

Welford + the pinned sign (A)

Two mirrors, radii −200 and −90, 60 mm apart: the engine’s own first-order focal length has to match the elementary power formula φ₁ + φ₂ − τφ₁φ₂, computed separately, at a separation where the answer on paper is exactly 900 mm.

Measured

899.9999999999998 mm from both routes, agreeing to the last bit (900.000000 at six figures).

Compared against

closed form (A)

A paraboloid has no spherical aberration at all, so for the Newtonian preset the engine’s third-order S_I, the spherical part and the aspheric increment together, has to come out zero.

Measured

0.0

Compared against

Welford ΔS (A)

The Cassegrain preset is built on the pair of conic foci that image perfectly, so the wavefront error the engine computes across its pupil has to be flat zero.

Measured

9.7 × 10⁻¹¹ waves

Compared against

conic conjugates (A)

A system symmetric about its stop and working at magnification m = −1 has no coma and no distortion, by symmetry: the engine’s S_II and S_V sums have to be zero.

Measured

≤ 1.7 × 10⁻¹⁸

Compared against

symmetry (A)

For a perfect circular pupil the answer is known in closed form and the engine has to reproduce it: a modulation transfer function (2/π)(arccos ν̄ − ν̄√(1 − ν̄²)), the first dark ring of the Airy pattern at 1.22 λf#, and a Strehl ratio of 1.

Measured

MTF rms 4.4 × 10⁻⁴; two-route MTF identity 5.6 × 10⁻¹⁷

Compared against

Born & Wolf (A)

A thin prism bends a ray by (n − 1) times its angle, so a powered singlet tilted 2° has to send its ray out at exactly that slope, tan((n − 1)θ).

Measured

exact

Compared against

thin-prism deviation (A)

A flat mirror at 45° folds the beam without changing the optics, so the folded doublet’s image coordinates and optical path lengths have to match the straight doublet’s.

Measured

≤ 4.3 × 10⁻¹⁴ mm

Compared against

unfold identity (A)

Light that bounces twice inside a flat plate makes a ghost image, and elementary optics puts its focus 2t/n upstream of the real one: on a plate 6 mm thick of index 1.5, the engine, an independently built ladder of the same bounces and the closed form all have to agree.

Measured

−8.000… mm; engine == independent ladder == closed form

Compared against

closed form (A)

On a fresh five-surface system, the flux the engine gives each of the 15 ghost pairs has to equal the product written out by hand for that exact sequence of reflections and transmissions, with every Fresnel reflectance taken from the indices as typed.

Measured

worst 3.6 × 10⁻¹⁶ relative

Compared against

event enumeration (A)

A pair of surfaces built entirely from numbers a computer stores exactly (R 64/32, t 64, n = 2) is exactly afocal: the quantity the engine tests afocality with has to be zero in floating point, and detuning the pair by 2⁻²⁰ has to give a focal length of exactly ∓2²¹.

Measured

C == 0.0; EFL = ∓2 097 152 exactly

Compared against

exact arithmetic (A)

The tolerancer tilts a plane plate rigidly by 2°, and the lateral shift it reports has to equal the closed form t·sin θ(1 − cos θ/√(n² − sin²θ)) = 1.189879 × 10⁻¹ mm.

Measured

9.9 × 10⁻¹⁶ mm

Compared against

closed form (A)

One element of the Cooke is decentered, and the chief-ray image shift the tolerancer reports has to match the first-order kick sum Δ·Σ B_k c_k(n′−n)/n′, with whatever is left over shrinking as the square of the decenter.

Measured

1.5 × 10⁻⁶ relative; order 2.00; −3.901452 mm per mm

Compared against

derived closed form (A)

A tolerance on the glass index alone has to move the focal length by exactly what the thick-lens closed form gives, and a budget with every tolerance set to zero has to return every trial at the nominal design, with no spread at all.

Measured

0.0 (59.285787 mm); spread 0.0 over all trials

Compared against

closed form / identity (A)

A merit function of two variables whose minimum can be written down is driven through the whole damped least-squares machinery, and it has to land on that minimum exactly.

Measured

0.0 at (1.25, −0.5)

Compared against

closed form (A)

The two mirror presets on the engine in your browser: rays arriving parallel to the axis, at the heights the presets were frozen with, have to cross the axis at one point. Cassegrain (R₁ −200 K −1, gap −70, R₂ −80 K −25/9): heights 4/12/20/25 mm, crossings 120.000 mm behind the secondary; sweep h = 4…25 mm in 1 mm steps. Newtonian: heights 5/20/35/50 mm.

Measured

Cassegrain span 5.0 × 10⁻¹³ mm (4 rays), 1.3 × 10⁻¹² mm (22 rays), worst distance from the paraxial focus 7.5 × 10⁻¹³ mm; Newtonian span 3.4 × 10⁻¹³ mm. R₂ = +80: span 0.35 mm, EFL 57.142857142857146 (panel prints 57.1429). The values frozen with the presets when they were built (measured then on the reference implementation) are 5.684 × 10⁻¹³ and 3.411 × 10⁻¹³.

Compared against

conic focal property (A)

The point where each ray meets each surface is compared, for four test systems at five pupil points each: SINGLET, DOUBLET and TRIPLET on axis and at 5° off axis (chief, zonal and skew rays), and ASPH-SINGLET on the Newton path.

Measured

worst 1.8 × 10⁻¹⁵ mm (bar 10⁻⁹)

Compared against

rayoptics real-ray trace

The DOUBLET’s focal length and back focal distance are computed from the same prescription on both sides and compared.

Measured

0.0 / 1.4 × 10⁻¹⁴ mm

Compared against

rayoptics first-order data

The DOUBLET’s five third-order sums S_I to S_V are compared, both sides started from the same marginal and chief rays.

Measured

6.0 × 10⁻¹⁵, normalization factor exactly 1

Compared against

rayoptics third-order sums

The optical path a ray accumulates through the DOUBLET, from the first surface to the last, is compared segment by segment.

Measured

3.6 × 10⁻¹⁵ mm

Compared against

rayoptics segment sums

For 20 glasses across SCHOTT, OHARA, CDGM and HOYA, in both the Sellmeier and the power-series dispersion forms, the index our catalog computes at the d, F and C lines is compared with the value the outside library computes from its own copy of the same vendor catalogs.

Measured

worst 2.22 × 10⁻¹⁶ (one ulp)

Compared against

opticalglass 2.0.2 (its own vendor catalogs)

The engine on our server, with nothing in between: its paraxial matrix elements D, A and C are compared on four afocal systems built for this check alone, a 2.5× Galilean, a Keplerian with an erector at M = +3, a cemented expander and a finite-conjugate collimator.

Measured

≤ 8.9 × 10⁻¹⁶ (39/39)

Compared against

rayoptics paraxial trace, separate process

The engine in your browser, with nothing in between: it writes each design as a .zmx file and the outside reader opens that file, so the two are compared surface by surface (curvature, thickness, conic, a model glass’s constant index, the toroid parameter) and then on focal length, over 5 fresh designs and 6 presets.

Measured

c/t/K 0.0 everywhere; constant-index n bit-exact; EFL worst 8.5 × 10⁻¹⁴ mm (81/81)

Compared against

rayoptics .zmx reader, separate process

The same trip in the other direction: a .zmx doublet fixture is read by our own parser, surface by surface, and the focal length of the system it produces is compared using the outside reader’s own resolved indices.

Measured

0.0

Compared against

rayoptics lens-file reader

Eighteen .seq files, 6 written for this check and 12 carried in from an open-source library under BSD-3, are read by our own parser and by two outside readers working separately, and the systems that come out have to agree field by field: surface count, curvature and radius, every gap including the folded shift, conic and asphere terms, the mirror flag, the medium’s type and name, the stop, the entrance pupil, the field points, the wavelengths, which of them is the reference, and the focal length computed at it.

Measured

curvature and radius 0.0 on every fixture; wavelengths exact; EFL ≤ 9.6 × 10⁻¹² mm (the file in inches), ≤ 3 × 10⁻¹⁴ elsewhere

Compared against

rayoptics 0.9.9 (377/377) and optiland 0.6.2 (439/439), each in a separate process

For every glass in the catalog, the index at the d line is recomputed from the dispersion coefficients we ship and compared with the maker’s own printed n_d in the same record, and the Abbe number V_d the same way.

Measured

n_d worst 5.457 × 10⁻⁶ (5.5 × 10⁻⁶ at two figures); V_d worst 8.8 × 10⁻³

Compared against

vendor catalog values as mirrored in RefractiveIndex.INFO (CC0), pinned commit; vintages SCHOTT 2017-01-20b, OHARA 2017-11-30, CDGM 2022-06, HOYA 2017-04-01

Limited by

Vendors print n_d to 5–6 decimals and V_d to 2; the residual is the vendor’s own fit-vs-print rounding, and the worst n_d case sits just above a five-decimal half-unit.

The three check values SCHOTT prints on its N-BK7 datasheet, n_d = 1.51680, n_e = 1.51872 and ν_d = 64.17, are compared with what the shipped record returns, which is the same record and the same coefficient set the thin-film simulator ships.

Measured

3.5 × 10⁻⁸ / 2.0 × 10⁻⁶ / 2.7 × 10⁻³

Compared against

SCHOTT N-BK7 datasheet (glass code 517642.251)

A Tessar typed in from US Patent 721,240 (P. Rudolph, Zeiss, 1902): the f = 1 table read from the scan at native resolution and scaled ×100, each radius taking the direction the patent prints, then traced to see whether the focal length comes back at the patent’s own.

Measured

EFL 99.25169308949212 mm at the patent’s D line (nominal 100, miss 0.75 mm); the rejected sign reading gives 18.6

Compared against

the patent (Google Patents scan, digit-verified 2026-08-13)

Limited by

Radii and thicknesses are printed to three decimals of f = 1; half a unit of each printed digit, propagated through the engine, moves EFL by ±2.0 mm worst-case aligned, ±0.9 mm root-sum-square.

A Cooke triplet typed in from US Patent 568,052 Series III (H. D. Taylor, 1896): the patent states its radii by which way each surface bulges, so each one is converted to a directional radius and scaled ×100, with the stop at the rear vertex of L2 as the parent patent’s text directs, and the focal length compared with the patent’s own.

Measured

EFL 98.26805838170866 mm (98.27 at the print’s two decimals; nominal 100, miss 1.73 mm: inside the worst case, 1.2× the root-sum-square); the un-converted reading gives 19.5; diaphragm ratio .131/.108 = 1.213 vs the f/6.5 : f/8 ratio 1.231

Compared against

the patent + US 540,122 p. 4 (the convention statement)

Limited by

The same propagation: ±3.1 mm worst-case aligned, ±1.5 mm root-sum-square; the two dominant sensitivities are R₁, 1.0 mm per 0.05 mm, and R₄, 0.8 mm.

A .seq file written by someone else, for a catalog achromat 25 mm in diameter and 100 mm in focal length made of N-BK7 and N-SF5, is read by our own parser and analyzed, and the focal length and center thickness that come out are compared with what its supplier publishes for that part.

Measured

EFL 100.0308127418172 mm (the two outside readers give 100.03081274181721 and 100.03081274181717); center thickness 8.50 exactly

Compared against

the supplier’s published product data, read at source 2026-09-02: EFL 100.00 mm ± 1 %, center thickness 8.50 mm

The building blocks of a trace, one at a time. Both engines are driven from the same frozen inputs and their results compared item by item: 19 traced results over 9 systems (a sphere, a Cassegrain form with a negative gap, a paraboloid, an ellipsoid, an even asphere found by iteration, two total-internal-reflection cases and 5 aperture-clip cases, one of them a ray landing exactly on the rim), 10 paraxial results, 3 first-order, 3 pupil, 3 afocal, the 231 points of a six-ring hexapolar grid, 21 fan coordinates and 1 transfer.

Measured

234/234 trace leaves ==; all paraxial/pupil/afocal/transfer leaves ==; hexapolar sin/cos ≤ 1 ulp (the browser’s and the server’s math libraries)

The engine in your browser is checked against the implementation it was built from, on five systems that are in no suite: a cemented triplet, a two-mirror Cassegrain form, a steep asphere at the edge of the domain (K = +0.5), a pair bracketing total internal reflection, and a real-catalog doublet at the F line, each engine using its own Sellmeier evaluation.

Measured

0.0 on every row except the asphere-Newton path: hits 1.8 × 10⁻¹⁵, directions 2.2 × 10⁻¹⁵, OPL 7.1 × 10⁻¹⁵

Everything the browser does after the trace (finding the entrance pupil, hexapolar spots, ray fans, Seidel sums) was written a second time from the specification alone, and the two versions are compared on three systems, one of them with an internal stop and one with a mirror.

Measured

worst 2.5 × 10⁻¹⁴

The same comparison on kinds of system the suites do not cover: toroids swept both ways, the aspheric Seidel increment of a conic mirror, a user-typed index table evaluated at its last point, a polychromatic point-spread function, and the wavefront of a telecentric system.

Measured

≤ 8.9 × 10⁻¹⁶ everywhere except telecentric OPD 2.4 × 10⁻¹¹ (reference-sphere conditioning, named)

The optimizer on our server is compared with the implementation it was built from on four problems built for this check, matching residuals, derivatives and every step taken along the way: recovering a Petzval form, a doublet whose air-gap bound is active, snapping a model glass onto a vendor other than SCHOTT, and an under-determined problem that asks only for a focal length.

Measured

0.0 exactly

The tolerancer on our server is compared with the implementation it was built from on six budgets built for this check: 19 rows covering all eight kinds of tolerance with a HOYA power-series glass; a uniform random draw, which never touches the error function; a plano-convex lens close enough to total internal reflection that 7 of its 300 wedge trials genuinely fail; budgets a two-mirror system has to refuse; and a run with the compensator switched off.

Measured

0.0 on every row; the two fixture comparisons show no difference anywhere.

Limited by

The error function is not one of the operations the arithmetic standard requires to be correct to the last bit, so the rows that go through it are held to a relative 10⁻¹⁰ rather than to exact equality, and the uniform rows are held exact. The 10⁻¹⁰ is the bar, not a measured difference.

The nine preset share links, each written by the tool’s own link writer and read back by its own reader to the same design, are loaded into the shipped engine, where the focal length and back focal distance they produce have to reproduce the values frozen when the presets were built, printed here to nine decimals.

Measured

EFL: seven presets within half a unit of the ninth decimal (worst Δ 4.9 × 10⁻¹⁰); Newtonian −500 exactly; Cassegrain 399.9999999999998 (Δ −2.3 × 10⁻¹³ mm; the panel prints 400.000). BFL: the same seven within 4.7 × 10⁻¹⁰, Cassegrain 119.99999999999991 (Δ −8.5 × 10⁻¹⁴), with one bookkeeping difference stated: the folded doublet’s frozen BFL (83.329670352) is the unfolded lens’s, measured from its last glass surface; the shipped engine measures from the fold mirror, 20 mm later, and reports 63.329670352158665 (Δ 1.6 × 10⁻¹⁰ against frozen − 20).

A plain design and the same design holding one configuration have to produce byte-identical requests in the browser and identical responses from the server, with one exception stated in the row: the wall clock.

Measured

identical (8/8, 7/7)

Exporting one design twice has to give byte-identical .zmx files, with no timestamp anywhere and the header carrying the attribution line; a fresh export has to read back as the same design; and the glass lines of an imported lineage file have to survive a re-export byte for byte.

Measured

byte-identical; preserved set exact (274/274)

Every run behind the published lens-design guide, 11 optimizations and 5 tolerance runs, is replayed on a later build of the engine, one that had gained a new kind of operand and multi-configuration support in the meantime.

Measured

16/16 digit-for-digit (e.g. R10 merit 0.030444656741558877; T4 yield 0.970, p90 0.05567831729257731)

The guide’s 2,000-trial yield simulation, started from seed 20260831, is run again on the engine build the guide was published with, and every digit has to come back the same.

Measured

yield 0.9795, p90 0.055674375803048404, baseline 0.04897491924768563: digit-identical

The same 2,000-trial budget is run from four different seeds, and the four yields have to sit inside the spread that chance alone allows for a sample this size.

Measured

passes 1959 / 1962 / 1962 / 1960 of 2000 = 97.95 / 98.10 / 98.10 / 98.00 %; pooled 7843/8000 = 0.980375; binomial SD at p = 0.98, n = 2000 is 0.0031 (0.31 %); observed spread 0.0015 (0.15 %)

A design optimized with an idealized model glass, then snapped to the nearest real catalog glass and re-optimized, has to end where a design that started from that catalog glass ends.

Measured

merit ratio snapped-and-reoptimized / from-scratch 0.9999999999995914 (1 − 4.1 × 10⁻¹³); the two curvature solutions within 3.9 × 10⁻¹¹ mm⁻¹

A frozen Monte Carlo run (a doublet, 13 rows of tolerances, seed 20260815, 300 trials, compensated) has to return the same mean 1.499370104 × 10⁻², median 1.498098526 × 10⁻² and first trial 1.403841665 × 10⁻² on the engine on our server as on the implementation it was built from.

Measured

observed 0.0

The same check as A27, listed here because it is the optimizer’s: a two-variable quadratic with a minimum that can be written down, driven through the whole damped least-squares path, has to land on that minimum exactly.

Measured

0.0 at (1.25, −0.5)

Every curvature of the Tessar patent geometry is perturbed by ±1 %, which makes the merit 52.3 times worse, and the optimizer is set to recover it: it gets there in 19 iterations and finishes at 0.038 times the merit of the patent geometry itself. There is nothing outside to compare this against, and the merit function is ours, so the row prints a recovery factor rather than agreement with anything.

Measured

×1386 recovery, 19 iterations

The convergence curves of the guide’s first two runs, 36 points and 28 points, each falling at every step from 0.869815991025691 and from 2.555107549821758, have to replay digit for digit.

Measured

digit-identical at replay