Spherical-surface refraction & the lensmaker's equation

Put an object point O on the axis a distance s in front of the vertex V of a spherical surface of radius R that separates index n₁, where the light starts, from n₂. A ray leaves O at angle α to the axis, strikes the surface at height h, refracts, and crosses the axis at the image point I a distance s′ beyond V, arriving at angle β. The normal at the strike point P passes through the centre of curvature C and makes angle φ with the axis. Fix the convention now and keep it for the lens: s > 0 for a real object on the incoming side, s′ > 0 for a real image on the outgoing side, and R > 0 when C lies on the outgoing side, so a face that bulges toward the incoming light has R > 0. Exterior-angle bookkeeping on the two triangles OPC and PCI then gives the angle of incidence θ₁ = α + φ and the angle of refraction θ₂ = φ − β. Nothing has been approximated yet; the geometry is exact.

Snell's law, n₁ sin θ₁ = n₂ sin θ₂, is not linear, and a non-linear law cannot map points to points. The paraxial limit makes it linear: for small angles sin θ ≈ tan θ ≈ θ, and each angle is a height over a distance, α ≈ h/s, β ≈ h/s′, φ ≈ h/R, because P sits so close to V that the axial distances are s, s′ and R to first order. Substituting, n₁(h/s + h/R) = n₂(h/R − h/s′). Every term carries one factor of h, so h cancels and n₁/s + n₂/s′ = (n₂ − n₁)/R. That cancellation is the whole content of the word image: a ray leaving O at any small angle reaches the same I, and it does so only because the sines were replaced by angles. Numbers: a glass rod (n₂ = 1.50) with a convex end of R = 5.0 cm and an object 30 cm away in air gives 1/30 + 1.50/s′ = 0.50/5.0, so 1.50/s′ = 0.1000 − 0.0333 = 0.0667 cm⁻¹ and s′ = 22.5 cm inside the glass, a real image. The ray through the vertex obeys Snell's law with the same linearisation and no geometry, and it gives the magnification m = −n₁s′/(n₂s) = −22.5/(1.50 × 30) = −0.50.

P = (n₂ − n₁)/R is the surface's power, in dioptres when R is in metres, and it is fixed by the surface alone, whatever the object does. Its sign is the product of two signs. A convex face (R > 0) into a denser medium (n₂ > n₁) has P > 0 and converges; the same R > 0 into a rarer medium, water into an air bubble, has P < 0 and diverges. Send light out of a glass sphere instead and both signs flip together, C now on the incoming side so R < 0 and n₂ < n₁, and P is positive again: a surface converges or diverges regardless of which way the light crosses it. Now run the parameters to their limits. n₂ → n₁: the right side vanishes, s′ = −s, and the image sits on the object, as it must with no surface there. R → ∞: the face is flat, s′ = −(n₂/n₁)s, a virtual image at a reduced distance with m = +1; a coin 1.00 m under water (n₁ = 1.33) seen from air appears at 0.75 m, the apparent-depth result, its size unchanged. s → ∞: the image lands at f₂ = n₂R/(n₂ − n₁), the surface's image-side focal length, while a parallel bundle sent the other way focuses at n₁R/(n₂ − n₁): one surface has two focal lengths, in the ratio n₂/n₁. The cornea shows how strong one surface can be: (1.376 − 1)/0.0078 m ≈ 48 D from a single air-to-tissue boundary of radius 7.8 mm, most of the eye's roughly 60 D.

A lens is two surfaces in a row, so image sequentially. At the first face the light enters the glass: nₘ/s₁ + n_L/s₁′ = (n_L − nₘ)/R₁. Its image at s₁′ is the object for the second face, but measured from the second vertex, a distance d further on: s₂ = d − s₁′. If the first image lies beyond the second face, s₂ is negative, a virtual object, and the convention absorbs it with no special case. At the second face the light leaves the glass, so the indices swap roles: n_L/s₂ + nₘ/s₂′ = (nₘ − n_L)/R₂. Now let d → 0, so s₂ = −s₁′, and add the two equations. The n_L/s₁′ terms cancel, leaving nₘ/s₁ + nₘ/s₂′ = (n_L − nₘ)(1/R₁ − 1/R₂). Divide by nₘ: 1/s + 1/s′ = (n_L/nₘ − 1)(1/R₁ − 1/R₂). The right side is the same for every object distance, so it earns a name, 1/f. For a biconvex lens with R₁ = +20 cm and R₂ = −20 cm (its centre lies on the incoming side) of n_L = 1.52 in air, 1/f = 0.52 × (1/20 + 1/20) = 0.052 cm⁻¹ and f = 19.2 cm. Turn the lens round and the two radii swap and change sign together, so f is the same from either side.

The prefactor is n_L/nₘ − 1, and (n − 1) is what it becomes at nₘ = 1. Submerge the lens and the glass is unchanged; what changes is the index step at each face, so both surface powers P₁ = (n_L − nₘ)/R₁ and P₂ = (nₘ − n_L)/R₂ shrink. The 19.2 cm lens under water, nₘ = 1.333: 1.52/1.333 − 1 = 0.140, 1/f = 0.140 × 0.10 = 0.0140 cm⁻¹, f = 71.3 cm, 3.7 times longer. The medium enters twice. The surfaces contribute P₁ + P₂ = (n_L − nₘ)(1/R₁ − 1/R₂), and 1/f = (P₁ + P₂)/nₘ, so the lens is weaker both because each step is smaller and because a given power buys less bending in a denser medium. Match the indices, nₘ = n_L, and the lens vanishes, which is why a glass rod in oil of the same index cannot be seen. Go past that, nₘ > n_L, and the sign flips: a biconvex pocket of air in water has prefactor 1/1.333 − 1 = −0.25 and diverges. The unaided eye under water loses its 48 D cornea to a 5.5 D one, (1.376 − 1.333)/0.0078 m, more than the lens inside the eye can recover: hence goggles, which put air back in front of the cornea.

Two approximations were made and each can be costed. Thickness: keeping d in the sequential trace gives, for a lens in air, 1/f = (n − 1)[1/R₁ − 1/R₂ + (n − 1)d/(nR₁R₂)]. For the 19.2 cm lens with d = 1.0 cm the extra term is 0.52 × 1.0/(1.52 × 20 × (−20)) = −8.6 × 10⁻⁴ cm⁻¹ against the thin part's 0.100 cm⁻¹: f moves to 19.4 cm, under 1%, and is now measured from principal planes inside the glass rather than from the vertices. Paraxiality: keep the sines. A parallel ray at height h meeting a single convex surface has sin φ = h/R exactly, sin θ₂ = (n₁/n₂) sin φ, and the sine rule in triangle CPF puts its axis crossing at VF = R + R sin θ₂/sin(φ − θ₂). Expand for small φ and the leading term is n₂R/(n₂ − n₁), the paraxial f₂; the next term is negative and grows as h², so marginal rays focus short. For a glass surface (n₂ = 1.52) of R = 5.0 cm the paraxial focus is 14.6 cm from V; a ray at h = 2.0 cm crosses at 14.1 cm, 3.6% short. That gap is longitudinal spherical aberration, and it is why a fast lens is never one spherical surface.