Lens design bench

Design. Start from a preset — an equiconvex singlet, a plano-convex lens both ways round, the Cooke triplet, a cemented achromat, a Newtonian’s paraboloidal mirror (and the same mirror as a sphere), a magnifier and a Ramsden eyepiece — or edit the surfaces. Each row is one surface:

Set the entrance pupil diameter and the largest field angle (the object is at infinity). The layout is drawn to scale with real rays at 0, 0.7 and 1 of the field. Below it: the effective, back and front focal lengths, the principal planes, the f-number and NA, the paraxial image height, the longitudinal colour and the on-axis RMS spot against the Airy disc, and the reduced ABCD system matrix. ⇄ Reverse turns the lens round. Export the prescription as CSV or JSON, copy it as a table, or import a JSON prescription. ⛶ Full screen (or F) gives the layout — or, on the Spots tab, the spot diagrams — the whole screen; ◫ Panels hides the prescription.

Aberrations. The five Seidel sums S_I–S_V and the two chromatic sums, surface by surface and in total (as a table and a bar chart), the wavefront coefficients W040, W131, W222 and W220 in waves, distortion in per cent, longitudinal spherical aberration, longitudinal colour (F − C), the secondary spectrum and the Petzval radius, and a plot of the paraxial focus against wavelength.

Spots & MTF. Spot diagrams at three fields in the F, d and C lines with the Airy disc and RMS radius; tangential and sagittal ray fans; and a geometric MTF estimate against the diffraction limit.

Bending. Spherical aberration and coma of a thin singlet against its shape factor B = (c₁ + c₂)/(c₁ − c₂), with the shapes for least spherical aberration and zero coma, a picture of the lens and the thick lens’s real-ray spherical aberration. Drag the slider.

Achromat. Pick a crown and a flint, the focal length and aperture: the designer splits the power for zero primary colour, corrects the thick cemented doublet exactly (F and C foci coincide), bends it for zero spherical aberration and shows the residual secondary spectrum against a singlet’s colour. Open in the design tab to study it further.

Glasses. The catalogue on a glass map (n_d against V_d) with n_d, V_d, partial dispersion and the source of each Sellmeier formula.

Sequential, rotationally symmetric, object at infinity. Paraxial rays use the y–nu method (n′u′ = nu − yφ, φ = (n′ − n)c) and the reduced ABCD matrix; a mirror is a surface with n′ = −n. Real rays are traced in 3-D through conic surfaces c(x² + y² + (1 + k)z²) = 2z with the vector form of Snell’s law, aimed paraxially at the entrance pupil. The Seidel sums follow Welford (Aberrations of Optical Systems, 1986) in Hopkins' notation; the thin-lens bending formula is Welford’s, also in Smith (Modern Optical Engineering) and Kingslake (Lens Design Fundamentals). Wavelengths are the Fraunhofer F (486.13 nm), d (587.56 nm) and C (656.27 nm) lines.

Glass data: SCHOTT optical glass data sheets (N-BK7, N-SK16, F2, N-SF11); I. H. Malitson, JOSA 55, 1205 (1965) for fused silica and Appl. Opt. 2, 1103 (1963) for CaF₂; PMMA (Szczurowski 2013) and polycarbonate (Sultanova et al., Acta Phys. Pol. A 116, 585, 2009) as tabulated at refractiveindex.info. The one-term polycarbonate fit gives V_d ≈ 28, a little below the ≈ 30 often quoted; plastics vary by grade.

The Cooke triplet is the classic f = 50 mm f/5 ±20° design in N-SK16 and F2 distributed as the Zemax sample "Cooke 40 degree field". The plano-convex lens is like Thorlabs LA1509 (R = 51.5 mm, 3.6 mm thick, f = 100 mm).

The tests check: Sellmeier indices against the catalogues (N-BK7 n_d = 1.5168, V_d = 64.17, and every glass’s n_d); the thick-lens EFL, BFL and FFL against the lensmaker’s equation; the triplet’s 50 mm f/5; real rays against paraxial rays in the paraxial limit, on and off axis; a paraboloid’s perfect axial focus; a spherical mirror’s S_I = 2y⁴/R³ (zero for a paraboloid) and zero coma and astigmatism with the stop at its centre of curvature (the Schmidt principle); the thin-lens Petzval radius −nf; third-order LSA, longitudinal colour and distortion against real and paraxial traces; Welford’s thin-lens S_I and S_II against surface-by-surface sums; least spherical aberration at B = 2(n² − 1)/(n + 2) ≈ 0.74; a plano-convex lens with its flat side first having about four times the spherical aberration; and the achromat’s zero F − C colour, exact focal length, zero spherical aberration and a secondary spectrum close to −f(P₁ − P₂)/(V₁ − V₂).

Related: telescopes and optics (thin-lens ray bench, lensmaker’s equation, Snell’s law), lasers and fibre optics (Gaussian beams, ABCD trains, focusing), depth of field and lenses, astrophotography, spectroscopy.

Share link copies a link with the prescription and the other tabs' settings (the prescription is packed into ?s=). Links can be written by hand; add the tab after #: design, aberrations, spots, bending, achromat or glasses.