Sphere, asphere, and the lens that does not exist
Three 1-inch lenses of the same 25 mm focal length under one 20 mm monochromatic bundle: a perfect point, a 7.5 mm smear, and a point again.
Open in the canvas →Background
The thin-lens construction every optics course starts with sends every ray through a single focal point. It is an approximation, and the approximation is paraxial — it assumes rays stay close enough to the axis that sin θ ≈ θ. Push light out to the rim of a real lens and the neglected terms arrive.
A spherical surface is the shape that is easy to make, not the shape that is right. Grinding two glass blanks against each other with abrasive between them naturally produces spheres, because the sphere is the only surface that slides on itself in every direction — which is why almost every lens ever made has been spherical. But a sphere curves away from the axis faster than focusing requires. A ray striking it far from the axis meets a surface that is too steeply tilted, is refracted too strongly, and crosses the axis before the paraxial focus. The further out the ray, the worse the error: it grows as the cube of ray height, which is why spherical aberration is invisible near the axis and brutal at the edge, and why stopping a lens down cures it so dramatically.
The consequence is not a blurrier focus. It is that there is no focus at all — no plane anywhere along the axis where the light comes to a point. The rays instead form a caustic, the bright cusped envelope you can see in a coffee cup. The best available plane is the circle of least confusion, sitting well inside the paraxial focus, and it is a disc rather than a point.
An asphere breaks the manufacturing constraint to fix the optical one. Adding a conic constant k to the sag equation keeps the curvature at the vertex — so the paraxial focal length is untouched — while flattening the surface progressively away from the axis, by exactly the amount needed to stop over-bending the marginal rays. One number, chosen correctly, removes almost the whole error.
What this setup demonstrates
Three lanes, each starting identically: a monochromatic 587.6 nm point source at the front focus of a collimator, producing a 20 mm bundle of parallel rays. Every lens under test is 1 inch across with a 25 mm focal length, so all three are working at f/1.25 and any difference between them is shape alone.
- The ideal thin lens puts every ray height through one point. Longitudinal spread: 0.000 mm. This is the construction, not a lens that can be built.
- The N-BK7 spherical singlet focuses its paraxial rays at 26.4 mm and its rim rays at 19.6 mm — 6.8 mm apart. At the paraxial plane, where the screen sits, the bundle is spread over 7.5 mm. Even at its tightest, 5.2 mm short of that plane, the circle of least confusion is still 1.7 mm across.
- The N-BK7 asphere, same power, same aperture, with k₁ = −0.55 on its front face: longitudinal spread 0.046 mm and a 0.02 mm spot. That is 147 times better than the sphere longitudinally, and about 74 times tighter than the sphere's best plane.
Two things are worth doing by hand. Select the spherical singlet and drag its aperture down: the blur collapses far faster than the aperture does, because the transverse error goes as the cube of ray height — this is why a slow lens is easy and a fast one is hard, and why photographers stop down. Then select the asphere and sweep k₁ away from −0.55: the focal length readout does not move, because the conic constant does not touch vertex curvature, but the rim rays swing through the focus and back out again.
Notice also that the three screens are not at the same distance. Equal focal length does not mean equal back focal distance: the asphere's is shorter because its principal planes sit differently. Focal length is measured from the principal plane, not from the glass.
A 2D meridional trace shows only the aberrations that live in that plane. Spherical aberration and defocus do; coma, astigmatism and field curvature need the third dimension or a real off-axis field, so the asphere here is being judged on the one job this model can actually check. A real aspheric condenser is corrected for one conjugate and one wavelength, and the conic that fixes an infinite conjugate is not the conic that fixes a finite one.
All three lenses are f/1.25 so the effect is unmistakable at a glance; a normal f/8 singlet would show a blur too small to see at this scale. The tracer samples nine rays, so the caustic is drawn as a handful of distinct crossings rather than the continuous envelope it really is. Nothing here is manufactured: no surface figure error, no roughness, no centring tolerance, and no coating — a real asphere is considerably harder to make than these numbers suggest, which is the whole reason spherical lenses dominated for four centuries.