Spherical mirrors in the paraxial limit

Take a concave mirror of radius R with centre C and vertex V, and send in a ray parallel to the axis at height h. It strikes the surface at P, and the normal there is the radius CP, so the angle of incidence θ satisfies sin θ = h/R. The reflected ray leaves at angle θ on the other side of CP and crosses the axis at F′. Triangle CPF′ has equal base angles θ at C and at P — the first because CP makes angle θ with the axis, the second by the law of reflection — so it is isosceles, and dropping a perpendicular from F′ onto CP gives CF′ = R/(2 cos θ). Measured from the vertex, VF′ = R − R/(2 cos θ). Now take the limit. As h → 0, θ → 0, cos θ → 1 and VF′ → R/2: every near-axis parallel ray crosses at the same point, the paraxial focus F, and f = R/2 is exact as a limit. With R = 40.0 cm a ray at h = 4.0 cm has θ = 5.74°, cos θ = 0.99499 and VF′ = 40.0 − 20.10 = 19.90 cm, one part in two hundred from f = 20.0 cm. A convex mirror repeats the argument with C behind the surface: the reflected rays diverge as if from a point R/2 behind the vertex, and f is negative.

Put a point object O on the axis at distance s from V and follow a ray from O to a point P of the mirror at height h; it reflects to an image point I at distance s′. Let α, β and φ be the angles that OP, IP and the normal CP make with the axis. The ray makes equal angles θ with CP on either side, and the exterior-angle theorem applied to triangles OPC and OPI gives φ = α + θ and β = φ + θ, so α + β = 2φ with θ eliminated. That relation is exact. The paraxial step replaces each angle by its tangent measured from the vertex, α ≈ h/s, β ≈ h/s′, φ ≈ h/R, which drops the sagitta of the surface and terms of order θ³. Dividing by h gives 1/s + 1/s′ = 2/R, and the fact that h has cancelled is the whole point: every ray from O, whatever its height, arrives at the same I, which is what it means for I to be an image. Writing 2/R as 1/f turns this into the mirror equation. The plane mirror is the check: R → ∞ gives s′ = −s, an image as far behind the surface as the object is in front, exactly as the plane-mirror construction says.

State one convention and keep it. Measure every distance from the vertex. s is positive for a real object in front of the mirror. s′ is positive for a real image in front, negative for a virtual image behind. R and f are positive when the centre of curvature is in front, so concave mirrors have f > 0 and convex mirrors f < 0. Magnification is m = −s′/s, positive for upright, negative for inverted. Run a concave mirror with R = 40.0 cm, f = 20.0 cm, through three object positions. At s = 60.0 cm, 1/s′ = 1/20 − 1/60 = 1/30 cm⁻¹, so s′ = 30.0 cm and m = −0.500: real, inverted, reduced. At s = 30.0 cm, s′ = 60.0 cm and m = −2.00: real, inverted, enlarged. At s = 10.0 cm, inside the focus, 1/s′ = 1/20 − 1/10 = −1/20 cm⁻¹, so s′ = −20.0 cm and m = +2.00: the sign is the answer, a virtual upright image behind the mirror, which is a shaving mirror at work. A convex mirror with f = −20.0 cm and s = 30.0 cm gives 1/s′ = −1/20 − 1/30 = −1/12 cm⁻¹, s′ = −12.0 cm and m = +0.400: virtual, upright, reduced — and reduced for every real object, which is why it widens a field of view.

Three rays from the tip of the object can be drawn without a protractor. A ray parallel to the axis reflects through F on a concave mirror, or away from F on a convex one. A ray through F, or aimed at F behind a convex mirror, reflects parallel to the axis. A ray through C, or aimed at C, meets the surface along its own normal and returns on itself. Where two reflected rays cross is the image tip; where only their backward extensions cross, the image is virtual. Two habits keep the drawing honest. Bend the rays at a straight vertical line through V, not at the drawn arc: the rule that a parallel ray passes through F is a paraxial statement, and bending rays at a large drawn arc mixes the exact geometry with the approximate one and puts the crossing in the wrong place. And use the drawing for what it is good at — which side of the mirror, which way up, roughly how big — then take the numbers from 1/s + 1/s′ = 1/f and m = −s′/s. When the equation says s′ = −20 cm and the sketch shows the extensions crossing behind the mirror, the two agree and you can trust both.

Return to the exact crossing VF′ = R − R/(2 cos θ) and expand it. With sin θ = h/R, 1/cos θ = (1 − h²/R²)^(−1/2) = 1 + h²/(2R²) + 3h⁴/(8R⁴) + …, so the shortfall ΔL = f − VF′ = (R/2)(1/cos θ − 1) ≈ h²/(4R) + 3h⁴/(16R³). It is never negative: every non-paraxial ray crosses short of F, on the mirror's side. For R = 40.0 cm and h = 16.0 cm, sin θ = 0.400, cos θ = 0.9165, R/(2 cos θ) = 21.82 cm and VF′ = 18.18 cm, so ΔL = 1.82 cm — 9% of the focal length — while the leading term gives 1.60 cm and the h⁴ term lifts that to 1.79 cm. The reflected ray leaves the axis at angle 2θ = 47.2°, so at the paraxial focal plane it misses the axis by ΔL tan 2θ = 1.96 cm: the marginal zone paints a 4 cm blur where the paraxial model promised a point. This is spherical aberration. No sign convention removes it: it is not a bookkeeping error but the exact law of reflection disagreeing with its own small-angle version. The h² law says what to do: stop the aperture down, so the axial blur falls fourfold per halving of h, or use a paraboloid, which sends every parallel ray through one focus exactly.

A laboratory measures pairs (s, s′) by moving a screen until the image is sharp; the honest way to reduce them is to plot 1/s′ against 1/s. The mirror equation predicts a straight line of slope −1 with both intercepts at 1/f, so f = 20.0 cm appears as intercepts of 0.0500 cm⁻¹, and the two intercepts agreeing is a check that s and s′ were measured from the same reference point. Two failures show up separately. A slope that is not −1, or intercepts that disagree, point to a systematic offset — distances taken from the mirror's mount rather than its vertex, or a screen scale zeroed in the wrong place — and the paraxial model is not to blame. Points taken at large aperture that sit above the line, with the image sharpest nearer the mirror than predicted so that 1/s′ comes out too large, are spherical aberration: the marginal rays are pulling the best focus inward by an amount that grows as h² — h²/(4R) for a distant object, and nothing at all for an object at C, where every ray meets the mirror along its own normal. Mask the mirror to a small central aperture and those points drop back onto the line, which is the experimental meaning of taking the paraxial limit.