Dispersion and chromatic effects

Model a bound electron as a mass on a spring with natural angular frequency ω₀, driven by the light's field E₀ cos ωt. Its steady-state amplitude is (eE₀/mₑ)/(ω₀² − ω²), so the dipole moment per unit volume, and with it the susceptibility n² − 1, carries the factor 1/(ω₀² − ω²). Summing over the resonances of the material with their strengths fⱼ gives the Lorentz form n² − 1 = (Ne²/ε₀mₑ) Σ fⱼ/(ω₀ⱼ² − ω²). For glass the dominant ω₀ lies in the ultraviolet, near λ₀ = 100 nm, so every visible frequency sits below it. Raise ω towards ω₀ and the denominator shrinks: n rises with frequency, falls with wavelength, dn/dλ < 0. That is what normal dispersion means, and it is why the numbers for BK7 run 1.5143 at 656 nm, 1.5168 at 588 nm, 1.5224 at 486 nm and 1.5302 at 405 nm. Check the limit: far below every resonance, ω → 0, n² − 1 → (Ne²/ε₀mₑ) Σ fⱼ/ω₀ⱼ², a constant, and the index goes flat, which is why a glass has one static dielectric constant but no one refractive index.

Keep one resonance and write the Lorentz result in wavelengths: n² − 1 = Cλ²/(λ² − λ₀²), the one-term Sellmeier form. For λ ≫ λ₀ expand the denominator, n² − 1 ≈ C(1 + λ₀²/λ²), and the square root of a slowly varying quantity gives Cauchy's law n ≈ A + B/λ² with A = √(1 + C) and B = Cλ₀²/(2A). Two measured indices fix the two constants. For BK7, n_d = 1.5168 at 0.5876 μm and n_F = 1.5224 at 0.4861 μm give B = 0.00420 μm² and A = 1.5046, and the fit then predicts n_C = 1.5144 against the measured 1.5143. Differentiate once: dn/dλ = −2B/λ³, which is −0.0414 μm⁻¹ at 0.5876 μm, but −0.0731 μm⁻¹ at the F line and only −0.0297 μm⁻¹ at C. The blue end of any spectrum is stretched about 2.5 times more than the red end by the same glass. The Abbe number V_d = (n_d − 1)/(n_F − n_C) packages the whole visible slope into one figure: 64 for BK7, 28 for the dense flint SF10, whose n_F − n_C = 0.0256 is three times larger.

A prism of apex A set at minimum deviation obeys sin[(A + δ)/2] = n sin(A/2). Only A is fixed, so differentiate with respect to n: cos[(A + δ)/2] (dδ/2) = sin(A/2) dn, giving dδ/dn = 2 sin(A/2)/cos[(A + δ)/2]. Then dδ/dλ = (dδ/dn)(dn/dλ): the first factor is geometry, the second is the glass. For an equilateral BK7 prism, δ_d = 38.65°, cos 49.32° = 0.652 and dδ/dn = 1.534 rad per unit index; with dn/dλ = −0.0414 μm⁻¹ the angular dispersion is −0.0635 rad μm⁻¹, or 0.22 arcmin per nanometre, and the whole C-to-F fan is 1.534 × 0.0081 = 0.0124 rad = 0.71°, blue deviated most. Two checks. In the thin-prism limit sin(A/2) → A/2 and the cosine → 1, so dδ/dn → A and the fan is simply A Δn; the 60° prism beats that by 1.534/1.047 = 1.47 because its rays leave the second face steeply. And the sign is right: larger n, larger δ, so under normal dispersion the shorter wavelength is bent more.

Trace a ray into a spherical drop at incidence i, refracting to r with sin i = n sin r, reflecting once inside and leaving. The total deviation is D = 180° + 2i − 4r, and it has a minimum: dD/di = 2 − 4(dr/di) = 0 needs dr/di = ½, and differentiating Snell's law, cos i = n cos r (dr/di), turns that into 4cos² i = n² cos² r = n² − sin² i, so cos² i = (n² − 1)/3. Near this Descartes ray many incidence angles share almost the same D, so the returned light piles up there and the eye sees a bright ring at θ = 180° − Dₘᵢₙ = 4r − 2i from the antisolar point. For red, n = 1.3312 gives i = 59.5°, r = 40.3°, θ = 42.3°; for violet, n = 1.3435 gives 40.6°. Since dθ/dn ≈ −146° per unit index, larger n means a smaller ring: red outside, violet inside, 1.8° wide. Two internal reflections give D = 360° + 2i − 6r, the condition cos² i = (n² − 1)/8, and θ = 180° + 2i − 6r: 50.4° for red, 53.6° for violet. Now dθ/dn ≈ +263° per unit index, so the secondary bow is reversed, red inside, and 3.2° wide. Between 42.3° and 50.4° no Descartes ray of either family lands: Alexander's dark band.

Put the damping back: the oscillator's denominator gains a term −iγω, so n² acquires an imaginary part, absorption, and its real part no longer diverges at ω₀. Instead n rises steeply just below the resonance, drops through it across a width set by γ, and comes out below its distant value. Inside that band dn/dλ is positive: anomalous dispersion, red bent more than blue. Just above the band n < 1, so the phase velocity exceeds c; no signal does, because energy travels at the group velocity, which stays below c. Cauchy's single 1/λ² term cannot describe any of this, which is why glass catalogues use the multi-term Sellmeier form. BK7's three resonances sit at 77 nm, 141 nm and 10.2 μm: the ultraviolet pair set the visible slope, and the infrared one pulls n down faster than Cauchy predicts beyond 1 μm. The figure below drags a single resonance from the ultraviolet into the visible so you can watch the F and C indices cross.

Every imaging equation to come takes n as a constant. The lensmaker's equation of topic 4.07 makes a thin lens's power 1/f proportional to (n − 1), so between the F and C lines Δf/f = −Δn/(n − 1) = −1/V_d: a BK7 lens of focal length 200 mm brings blue light to a focus 200/64 = 3.1 mm nearer than red, and a flint lens does more than twice as badly. That spread is longitudinal chromatic aberration, a direct consequence of everything above, not a defect of manufacture. Naming it is enough for this unit: a single quoted n is a narrow-band approximation whose error is Δn across the band you actually use. Quantifying the cure, a crown-flint doublet whose two powers cancel each other's colour, belongs with the multi-element instruments of topic 4.09.