Dispersion, aberrations, and ray-model limits

A ray diagram writes n as a constant of the material. Measure it and it is a function of wavelength. The reason is resonance: the bound electrons of a transparent glass have their absorption bands in the ultraviolet, and the closer a wavelength sits to a band, the harder the electrons are driven and the more the wave is retarded. Blue light, nearer the band, sees a larger n than red. Across the visible this is normal dispersion, dn/dλ < 0, and for BK7 crown the measured indices are 1.52238 at 486.1 nm, 1.51680 at 587.6 nm and 1.51432 at 656.3 nm — a spread of only 0.00806, and quite enough to matter. Cauchy's fit n = A + B/λ² captures it: forced through the F and C values it gives A = 1.5045 and B = 0.00422 μm², and then predicts n at 587.6 nm as 1.51674 against 1.51680 measured. Inside an absorption band the trend reverses, anomalous dispersion, and the two-term fit fails; that is why glass catalogues use the longer Sellmeier form.

Snell's law already holds the whole effect: sin θ₁ = n sin θ₂ with a different n for each colour returns a different θ₂ for each colour, so a single refracting surface fans white light out. A prism is the useful version because its two faces push the same way instead of cancelling. For a thin prism of apex angle A the deviation is δ = (n − 1)A, so the angular spread between the F and C lines is Δδ = (n_F − n_C)A, which is the same thing as δ_d/V_d. A 6.0° F2 flint wedge, with n_F = 1.63208 and n_C = 1.61503, deviates 486 nm by 3.79° and 656 nm by 3.69°: a 0.10° fan, six arcminutes, out of a piece of glass you could balance on a fingertip. A thick prism is handled at minimum deviation, where the ray crosses symmetrically and n = sin((A + δₘᵢₙ)/2)/sin(A/2) — the standard bench method for measuring an index to four decimal places, one colour at a time.

A thin lens has power P = (n − 1)(1/R₁ − 1/R₂), and of those factors only n depends on colour, so P is proportional to (n − 1) and the fractional spread in power is exactly (n_F − n_C)/(n_d − 1) = 1/V_d. That line is the whole of first-order chromatic aberration: the blue focus sits nearer the lens than the red one, by f_C − f_F ≈ f_d/V_d. A 100 mm BK7 singlet therefore splits its foci by 1.56 mm, an enormous distance when depth of focus is measured in tens of micrometres. That separation along the axis is longitudinal chromatic aberration. Put a screen halfway between the two foci and each cone has reopened to a diameter of about a/V_d, where a is the aperture radius: 0.195 mm for a = 12.5 mm, and that width is the transverse blur the split produces. Off axis a second form appears, because magnification depends on focal length too, so the colours land at different heights — the fringing you see at the corners of a cheap photograph but never at its centre.

Now set colour aside and use a single wavelength: a spherical surface still fails. The paraxial results were built on sin θ ≈ θ, and a wide beam breaks that approximation. For a concave mirror of radius R, a ray arriving parallel to the axis at height h crosses the axis exactly R − R/(2 cos θ) from the vertex, where sin θ = h/R. Take R = 240 mm. The paraxial focus is at 120.0 mm, while the ray at h = 48 mm crosses at 117.53 mm, 2.47 mm short; at h = 24 mm the shortfall is 0.605 mm, almost exactly a quarter, because expanding the exact expression leaves an error growing as h². The failing rays do not scatter at random either — their envelope is a caustic, the bright cusp you see on the surface of a coffee cup. Lenses misbehave the same way, and for a simple converging element the marginal rays always focus nearer than the paraxial ones, so the tightest spot sits just inside the paraxial focus.

Colour and shape need different cures. For colour, combine two glasses: thin lenses in contact add powers, P₁ + P₂ = P, and the pair is achromatic when P₁/V₁ + P₂/V₂ = 0. Because V is always positive, the two powers must carry opposite signs — a strong crown cemented to a weaker flint. For f = 200.0 mm from BK7 (V = 64.1) and F2 (V = 36.4) that gives P₁ = +11.57 D and P₂ = −6.57 D. The pair brings 486 nm and 656 nm to one focus, but the wavelengths between still miss it by a residual secondary spectrum of roughly f/2200, and removing that needs a third glass, an apochromat. Spherical aberration answers to geometry instead: bend the lens so the deviation is shared between its two surfaces, replace a surface with an asphere or a paraboloid, or simply close a stop, which bites hard on an h² law. And a mirror has no dispersion at all, which is one reason every large telescope objective is a mirror.

Every defect so far is geometric, so in principle every one can be designed away. Light does not allow it. An aperture of diameter D diffracts, and the image of a point through a perfect circular lens is an Airy disc whose first dark ring subtends θₘᵢₙ = 1.22 λ/D, a diameter of 2.44 λf/D in the focal plane. For f = 100 mm, D = 25 mm and λ = 550 nm that is 5.4 μm. Set it beside the 195 μm chromatic blur of the BK7 singlet of the same size: the singlet is aberration-limited by a factor of 36, so correcting it is worth the money. Once corrected, further polishing buys nothing and the lens is called diffraction-limited. The ray model is the λ → 0 limit of wave optics, trustworthy while every aperture and structure is large compared with λ; near a focus, at an edge, or in a slit a few wavelengths wide it stops being an approximation and starts being wrong.