Singlet vs achromat — axial colour
Two f = 40 mm prescriptions under the same 400–750 nm beam: one glass spreads the spectrum along the axis, two glasses fold it back.
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A lens works because glass slows light, and it slows every colour by a different amount. The refractive index of any ordinary glass falls as wavelength rises — blue light sees a denser medium than red — and since the power of a thin lens is (n − 1) times its curvature, a single positive lens is simply stronger in the blue. Blue focuses short, red focuses long, and the focal length becomes a function of colour. That is axial, or longitudinal, chromatic aberration.
Nothing about the shape can remove it. Bending a singlet — changing the two radii while holding the power — is the classical cure for spherical aberration and does nothing at all for colour, because colour comes from the material, not the geometry. One glass has one dispersion curve, and the lens is stuck with it.
The escape is to use two glasses with different dispersions. Put a positive crown element in contact with a negative flint element of much stronger dispersion, and choose the two powers so that φ₁/V₁ + φ₂/V₂ = 0, where V is the Abbe number — about 64 for N-BK7 and 26 for N-SF11. The colour errors then cancel to first order while the powers still add to something useful. The flint gives back less power than it removes colour, which is exactly the trade an achromat is.
What this setup demonstrates
Both lanes are identical apart from the prescription: the same 400–750 nm supercontinuum, the same 20 mm beam, the same 1-inch clear aperture, and the same 40 mm focal length at the d line. Anything that differs downstream is the glass.
The uncorrected N-BK7 singlet spreads its focus over 1.39 mm between 400 and 750 nm. At the d-line focal plane, where the screen sits, that leaves a coloured blur about 0.39 mm across — blue core, red skirt, and no plane anywhere that is sharp in every colour at once. The cemented N-BK7 + N-SF11 achromat, at the same focal length, brings that to 0.15 mm of focus spread and a 0.04 mm spot: about ten times tighter.
None of that is a flag or a display effect. Every sampled wavelength refracts through the drawn faces with its own catalogue index, and the separation is whatever those interactions produce. Select either group to read its surface table and its axial-colour figure, or edit a row to make a custom copy and watch the correction break.
The residual in the corrected lane is worth looking at rather than ignoring. Two glasses can bring exactly two wavelengths to a common focus; everything between and beyond them lands slightly differently, and that leftover bow is the secondary spectrum. Removing it needs a third glass with anomalous dispersion — which is what separates an apochromat from an achromat, and most of what you pay for in one.
This is a 2D meridional geometric trace: it shows longitudinal colour, not diffraction-limited spot size, lateral colour, off-axis aberration, coatings, or manufacturing tolerance. The catalogue glasses use visible-band Cauchy fits rather than full Sellmeier data, so the residual figures are indicative rather than a design-grade prediction. The doublet here is solved for this aperture in this model; it is a teaching prescription, not a catalogue part.
Both lenses are deliberately fast — f/2 on a 20 mm beam — which makes the colour easy to see but also means each lane carries spherical aberration of its own. The spot sizes quoted are the combined blur, not colour alone.