Sellmeier and Cauchy Dispersion Equations

No transparent material has a single refractive index. Pass white light through a prism and it fans into colour because $n$ is larger for blue than for red — the index is a function of wavelength, and that function is dispersion. Away from the material's absorption bands the variation is smooth, monotonic, and astonishingly reproducible: a glass boule cast in Jena and one cast a century later share the same $n(\lambda)$ curve to five decimals. That reproducibility lets practical optics compress a material's colour response into a handful of tabulated coefficients, evaluated by one of two workhorse formulas. The elder is the Cauchy equation (1836), a power series $n=A+B/\lambda^2+\cdots$ that Cauchy fitted to prism data before anyone knew what an atom was. The physically grounded successor is the Sellmeier equation (1871), a sum of resonance terms $n^2=1+\sum_j B_j\lambda^2/(\lambda^2-C_j)$ that we recall here is nothing but the Lorentz oscillator model of aq-optics-lorentz-oscillator-model stripped of its damping and rewritten in wavelength. Each Sellmeier term is one bound-electron resonance seen from far away, in the transparent window where absorption is negligible and $n$ is real. aq-optics-refractive-index already previewed the Sellmeier form when it drew the BK7 dispersion curve; this article derives it, fits it, and shows exactly how Cauchy sits inside it. Two payoffs organize what follows. First, Sellmeier is the master formula: with three resonance pairs it reproduces Schott N-BK7's indices $n_d=1.51680$ , $n_F=1.52238$ , $n_C=1.51432$ and its Abbe number $V_d=64.17$ — the numbers a lens designer actually uses. Second, Cauchy is Sellmeier's leading Taylor term below the nearest resonance: a two-term fit to the yellow and red lines predicts the blue line to $\pm1.8\times10^{-4}$ , good enough for the visible and useless near a band. Because both give $n(\lambda)$ analytically, every derived quantity — the normal-dispersion slope $\mathrm{d}n/\mathrm{d}\lambda<0$ of aq-optics-normal-anomalous-dispersion, the group index, and the pulse-broadening $\mathrm{d}^2n/\mathrm{d}\lambda^2$ of the article that follows — comes by differentiation.

Key takeaways

Real transparent media disperse: $n=n(\lambda)$ , larger for blue than red, and it is captured by compact tabulated formulas valid across the transparency window.
The Sellmeier equation $n^2(\lambda)=1+\sum_j B_j\lambda^2/(\lambda^2-C_j)$ is the undamped, wavelength-form of the Lorentz oscillator sum (recalled by name): each term is one transparent resonance.
The Cauchy equation $n=A+B/\lambda^2+C/\lambda^4+\cdots$ is the leading Taylor expansion of Sellmeier for $\lambda$ well below the nearest resonance — simple, but it fails near an absorption band.
Sellmeier reproduces Schott N-BK7 to five decimals: $n_d=1.51680$ , $n_F=1.52238$ , $n_C=1.51432$ , Abbe number $V_d=(n_d-1)/(n_F-n_C)=64.17$ .
A two-term Cauchy fit ( $A=1.5043$ , $B=0.004311\ \mu\mathrm{m}^2$ ) to the yellow d and red C lines predicts the blue F line as $1.52255$ , an error of only $+1.8\times10^{-4}$ against Sellmeier.
Both give normal dispersion $\mathrm{d}n/\mathrm{d}\lambda<0$ throughout the window and diverge unphysically as $\lambda\to\sqrt{C_j}$ — the edge of the band, where damping must be restored.
Key idea: Sellmeier is the Lorentz oscillator model in the transparency window — damping dropped, frequency traded for wavelength — $n^2=1+\sum_j B_j\lambda^2/(\lambda^2-C_j)$ ; Cauchy is its leading $1/\lambda^2$ Taylor term, exact enough in the visible and wrong near a resonance.

The core insight

Sellmeier is the Lorentz oscillator model in the transparency window with damping dropped and frequency traded for wavelength: $n^2=1+\sum_j B_j\lambda^2/(\lambda^2-C_j)$ , each term one bound-electron resonance. Cauchy, $n=A+B/\lambda^2+\cdots$ , is its leading Taylor expansion for $\lambda\gg\sqrt{C}$ — accurate across the visible, divergent near a band.

Where this lives today

Sellmeier coefficients are the quiet infrastructure of every optical design suite. Lens-design software stores each catalogue glass as its Sellmeier triplet and evaluates equation (1) millions of times per optimization, pairing high- $V_d$ crowns with low- $V_d$ flints so their dispersions cancel while their focusing powers add — the achromatic doublets that keep colour fringes off an image. In fibre optics the material dispersion of fused silica comes straight from its Malitson Sellmeier fit; differentiating it twice gives the material $\mathrm{d}^2n/\mathrm{d}\lambda^2$ that the next article turns into pulse broadening, and the material zero near $1.27\ \mu\mathrm{m}$ that fibre designers shift with waveguide dispersion toward the $1.55\ \mu\mathrm{m}$ loss minimum. Nonlinear optics leans on it hardest: a phase-matching angle for second-harmonic generation in BBO or KTP means solving $n(\omega)=n(2\omega)$ from one Sellmeier equation per crystal axis, accurate to the fourth decimal or the conversion fails. Thin-film metrology and ellipsometry run the inverse problem — measured $n(\lambda)$ at several wavelengths is fitted back to a Cauchy or Sellmeier form whose coefficients fingerprint an unknown glass or polymer.

The physical picture

Think of every optical material as a stack of resonators. Deep in the ultraviolet its bound electrons have a natural frequency; far in the infrared its ions and molecular bonds have theirs. Ordinary glass therefore sits in a transparency window pinned between an electronic resonance near $0.1\ \mu\mathrm{m}$ and a vibrational resonance near $9\ \mu\mathrm{m}$ . Visible light lies on the long-wavelength tail of the UV band and the short-wavelength tail of the IR band, and both tails pull the index the same way: up toward the ultraviolet. Blue, being closer to the UV resonance, is polarized more strongly and slowed more than red — the entire content of "blue bends more". Sellmeier is just the arithmetic of adding those tails. Each resonance contributes a term $B_j\lambda^2/(\lambda^2-C_j)$ that blows up when $\lambda$ approaches its resonance wavelength $\sqrt{C_j}$ and flattens far away; summing a UV term, a mid term, and an IR term reproduces the gentle rise of $n$ across the whole visible window. The first figure is that construction.

Figure

Sellmeier as a sum of resonance tails: a UV electronic band (~0.1 µm) and an IR vibrational band (~9 µm) bracket the transparency window; in the visible, n rides the falling tail of the UV resonance, so it rises toward the blue (dn/dλ < 0, normal dispersion). Each Sellmeier term Bⱼλ²/(λ²−Cⱼ) is one such tail.

Cauchy versus Sellmeier is the difference between a Taylor polynomial and the function it approximates. Cauchy keeps only the leading $1/\lambda^2$ behaviour of the nearest tail — superb far from every pole, blind to its existence. Sellmeier keeps the pole explicit, so it stays accurate right up to the edge of the window. The second figure makes that quantitative on a real glass: the BK7 Sellmeier curve with the three Fraunhofer reference points, and the two-term Cauchy fit laid over it — coincident across the visible to a part in $10^4$ , then peeling upward as the blue edge approaches the unseen UV resonance.

Figure

The punchline — Schott N-BK7: the Sellmeier curve (solid) carries the reference indices n_F = 1.52238, n_d = 1.51680, n_C = 1.51432 and Abbe number V_d = 64.17. A two-term Cauchy fit (dashed, A = 1.5043, B = 0.004311 µm²) to the d and C lines tracks it to ±1.8×10⁻⁴ at F, then runs ~4×10⁻⁴ high by 400 nm as the UV resonance nears.

The core equations

The Sellmeier equation is the standard vehicle for tabulating a transparent material's dispersion. It is written in wavelength, conventionally micrometres, and each of its terms represents one effective resonance of the underlying bound charges; the coefficients are fitted once per material and printed in every glass catalogue.

n^2(\lambda)=1+\sum_{j}\frac{B_j\,\lambda^2}{\lambda^2-C_j} \qquad (1)

where: $n$ — real refractive index in the transparency window; $\lambda$ — vacuum wavelength (µm by convention); $B_j$ — dimensionless oscillator strength of resonance $j$ ; $C_j=\lambda_j^2$ — the square of the resonance wavelength, in $\mu\mathrm{m}^2$ . As $\lambda\to\infty$ , $n^2\to1+\sum_j B_j$ ; as $\lambda\to\sqrt{C_j}$ , the $j$ -th term diverges — the edge of the window. The Cauchy equation predates it by a generation and is a bare power series in $1/\lambda^2$ . It carries no explicit resonance, which is both its convenience and its limitation: it interpolates beautifully inside the visible and cannot know where the bands lie.

n(\lambda)=A+\frac{B}{\lambda^2}+\frac{C}{\lambda^4}+\cdots \qquad (2)

where: $A$ — the long-wavelength asymptotic index; $B$ — first dispersion coefficient (units $\mu\mathrm{m}^2$ ); $C$ — second dispersion coefficient (units $\mu\mathrm{m}^4$ ). For visible glass the first two terms already fit to $\sim10^{-4}$ ; $A$ sets the overall index and $B$ the slope that fans a prism. The physical origin of Sellmeier is the driven-damped oscillator susceptibility owned by aq-optics-lorentz-oscillator-model. In the transparency window the frequency $\omega$ lies between resonances, the damping $\gamma_j$ is negligible, and the imaginary (absorptive) part vanishes, leaving a real sum of poles.

n^2(\omega)=1+\sum_j\frac{f_j\,\omega_p^2}{\omega_j^2-\omega^2-i\gamma_j\omega}\;\xrightarrow{\ \gamma_j\to0\ }\;1+\sum_j\frac{f_j\,\omega_p^2}{\omega_j^2-\omega^2} \qquad (3)

where: $\omega_j$ — resonance angular frequency of oscillator $j$ ; $\omega_p=\sqrt{Ne^2/\varepsilon_0 m}$ — plasma frequency; $f_j$ — oscillator strength, $\sum_j f_j=1$ ; $\gamma_j$ — damping rate, dropped in the window. Substituting $\omega=2\pi c/\lambda$ and collecting constants turns this into equation (1). A single number summarizes a glass's dispersion for lens design: the Abbe number, built from the index at three standard Fraunhofer lines — the blue F ( $486.1$ nm), yellow d ( $587.6$ nm), and red C ( $656.3$ nm).

V_d=\frac{n_d-1}{n_F-n_C} \qquad (4)

where: $n_d-1$ — the refracting power at the yellow line; $n_F-n_C$ — the blue-to-red index spread (mean dispersion). A large $V_d$ means low dispersion (crown glass); a small $V_d$ means high dispersion (flint). BK7 gives $V_d=64.17$ . Differentiating Sellmeier gives the dispersion slope in closed form — the quantity that fixes normal versus anomalous dispersion (recall aq-optics-normal-anomalous-dispersion) and feeds the group index and pulse-broadening of the following articles.

\frac{\mathrm{d}n}{\mathrm{d}\lambda}=-\frac{1}{n}\sum_j\frac{B_j\,C_j\,\lambda}{(\lambda^2-C_j)^2} \qquad (5)

where: every $B_j,C_j>0$ and $\lambda>\sqrt{C_j}$ in the window, so each term is negative — $\mathrm{d}n/\mathrm{d}\lambda<0$ throughout, i.e. normal dispersion. For BK7 at the d-line this evaluates to $\approx-0.0418\ \mu\mathrm{m}^{-1}$ .

Derivation ASellmeier as the transparent limit of the Lorentz oscillator sum

The goal is to obtain the standard Sellmeier form directly from the microscopic response, showing that its coefficients $B_j$ and $C_j$ are not free parameters but combinations of oscillator strengths, plasma frequency, and resonance wavelengths — so that a purely empirical fitting formula is revealed as classical electron theory viewed from inside the transparency window. We recall the driven-oscillator susceptibility by name from aq-optics-lorentz-oscillator-model, drop the damping that is negligible between bands, and change variable from frequency to wavelength.

Step 1. From aq-optics-lorentz-oscillator-model, a bound electron driven at frequency $\omega$ gives the multi-resonance susceptibility $\chi(\omega)=\sum_j f_j\omega_p^2/(\omega_j^2-\omega^2-i\gamma_j\omega)$ , so that $n^2=1+\chi$ . Why: the index is never postulated — it is the collective response of the material's oscillators, and the sum over $j$ is the sum over its distinct electronic and vibrational modes.

Step 2. Restrict to the transparency window, where $\omega$ lies between all resonances and the damping terms $\gamma_j\omega$ are tiny compared with $|\omega_j^2-\omega^2|$ . Dropping them makes each term real, so $n^2(\omega)=1+\sum_j f_j\omega_p^2/(\omega_j^2-\omega^2)$ . Why: it is exactly the absence of absorption that lets $n$ be a real, smooth function of colour — Sellmeier is the model with its imaginary part removed, valid only where the glass is clear.

Step 3. Trade frequency for wavelength with $\omega=2\pi c/\lambda$ and $\omega_j=2\pi c/\lambda_j$ . Then $\omega_j^2-\omega^2=(2\pi c)^2(\lambda^{-2}_{j}-\lambda^{-2})$ , and multiplying numerator and denominator of each term by $\lambda^2\lambda_j^2/(2\pi c)^2$ converts $1/(\omega_j^2-\omega^2)$ into $\lambda^2\lambda_j^2/[(2\pi c)^2(\lambda^2-\lambda_j^2)]$ . Why: catalogues tabulate wavelength, not frequency, and this algebraic change is the only content separating equation (3) from equation (1).

Step 4. Collect the constants. Each term becomes $B_j\lambda^2/(\lambda^2-C_j)$ with $C_j=\lambda_j^2$ and $B_j=f_j\omega_p^2\lambda_j^2/(2\pi c)^2$ , giving $n^2(\lambda)=1+\sum_j B_j\lambda^2/(\lambda^2-C_j)$ . Why: this identifies the "empirical" Sellmeier coefficients with physical quantities — $C_j$ is a squared resonance wavelength and $B_j$ measures how many electrons that resonance commands.

Step 5. Check the limits. As $\lambda\to\infty$ , $n^2\to1+\sum_j B_j$ , the static optical permittivity; as $\lambda\to\sqrt{C_j}$ from above, the $j$ -th denominator $\to0^+$ and $n^2\to+\infty$ , the (unphysical, undamped) signature of an absorption band. Why: the two limits are exactly the Lorentz behaviour, confirming the reduction preserved the physics it was meant to. *Verified: the $\gamma\to0$ reduction, the $\omega\to\lambda$ change of variable, and both limits reproduce equation (1) in verify_aq-optics-sellmeier-cauchy-equations.py.*

Go deeper · Derivation B · 2 minCauchy as the leading Taylor term of Sellmeier below a resonance

Derivation BCauchy as the leading Taylor term of Sellmeier below a resonance

The aim is to prove that the Cauchy series is not an independent law but the first two terms of a Taylor expansion of Sellmeier for wavelengths far above the nearest resonance, which explains at a stroke why Cauchy works in the visible, why it fails near a band, and why its coefficient $A$ is close to but not equal to unity. We take a single-resonance Sellmeier term, expand it in the small parameter $C/\lambda^2$ , take the square root, and match term by term to $n=A+B/\lambda^2$ .

Step 1. Take one dominant (UV) term, $n^2=1+B_s\lambda^2/(\lambda^2-C_s)$ , and write it as $n^2=1+B_s/(1-C_s/\lambda^2)$ by dividing through by $\lambda^2$ . Why: the ratio $u=C_s/\lambda^2$ is the natural small parameter — "how close are we to the resonance" — and below the band it is much less than one.

Step 2. Expand the geometric factor for $u\ll1$ : $(1-u)^{-1}=1+u+u^2+\cdots$ , so $n^2=(1+B_s)+B_sC_s/\lambda^2+B_sC_s^2/\lambda^4+\cdots$ . Why: this is the pivotal step — a pole expanded far from itself becomes a power series in $1/\lambda^2$ , which is precisely the shape of Cauchy's equation for $n^2$ .

Step 3. Take the square root and re-expand: $n=\sqrt{1+B_s}\,\bigl(1+\tfrac{B_sC_s}{2(1+B_s)\lambda^2}+\cdots\bigr)$ , so that $A=\sqrt{1+B_s}$ and $B=B_sC_s/2\sqrt{1+B_s}$ . Why: the square root is why $A$ is not simply $1$ — it is the long-wavelength index $\sqrt{1+B_s}$ — and why Cauchy's $B$ ties together an oscillator strength and a resonance position.

Step 4. Read off the failure mode. The expansion is a geometric series in $C_s/\lambda^2$ , so it converges only for $\lambda>\sqrt{C_s}$ and worsens as $\lambda\to\sqrt{C_s}$ ; truncating at $1/\lambda^2$ therefore grows steadily wrong toward the blue and diverges from the true (pole-keeping) Sellmeier near the band. Why: this is the honest boundary of Cauchy — a Taylor polynomial cannot reproduce a pole, so it must be abandoned before the resonance.

Step 5. A real multi-resonance glass folds all its bands into effective $A,B$ , so a two-line fit gives $A\approx1.50$ rather than the single-term $\sqrt{1+B_s}$ ; the form $A+B/\lambda^2$ is what Sellmeier guarantees, and the numbers come from the fit (worked example 2). Why: it separates the structural claim — Cauchy is Sellmeier's leading term — from the empirical one — the coefficients are fitted, not derived from one pole. *Verified: sympy expands the single-term Sellmeier in $u=1/\lambda^2$ about $u=0$ to $n^2=(1+B)+BC/\lambda^2+O(\lambda^{-4})$ in verify_aq-optics-sellmeier-cauchy-equations.py.*

Limiting cases

Case	Result	Meaning
Far below all resonances ( $\lambda\gg\sqrt{C_j}$ )	$n\approx A+B/\lambda^2+\cdots$	each Sellmeier term Taylor-expands into the Cauchy series; the visible regime of ordinary glass
Long wavelength ( $\lambda\to\infty$ )	$n^2\to1+\sum_j B_j$	the static optical permittivity relative to the highest band; $n$ flattens to a constant
Approaching a band ( $\lambda\to\sqrt{C_j}^{+}$ )	$n^2\to+\infty$	the lossless formula diverges; real damping caps $n$ and the material absorbs — Sellmeier invalid here
Single dominant term	$n^2=1+B\lambda^2/(\lambda^2-C)$	the minimal model for a narrow band ruled by one resonance
Below vs on the resonance	$\mathrm{d}n/\mathrm{d}\lambda<0$ vs sign reversal	normal dispersion in the window; anomalous dispersion only inside the band (recall aq-optics-normal-anomalous-dispersion)

Common trap“Sellmeier and Cauchy coefficients are just numbers you can plug wavelengths into.”

They are numbers tied to a unit convention and a validity range, and ignoring either produces silently wrong indices. Sellmeier's $C_j$ are almost always quoted with $\lambda$ in micrometres, so $C_j$ carries units of $\mu\mathrm{m}^2$ ; feed in $\lambda=550$ (nanometres) instead of $0.55$ and every term collapses to $\approx B_j$ , returning the long-wavelength limit $n^2\to1+\sum_j B_j$ — a badly wrong number, not an error message. Equally, the coefficients are fitted only across a stated window (for BK7, roughly $0.3$ – $2.5\ \mu\mathrm{m}$ ); extrapolate past it, or across a resonance, and the lossless formula diverges where the real material simply absorbs. And the fitted $C_j$ are effective resonance positions that lump several true lines together, not exact spectroscopic line centres. Cauchy inherits all of this and adds one more trap: having no pole, it gives no warning toward the ultraviolet — it just drifts high, as the punchline figure shows.

Worked examples

1 · Reproducing BK7 from its Sellmeier coefficients (confidence builder)

Schott N-BK7 has Sellmeier coefficients $B=[1.03961212,\,0.231792344,\,1.01046945]$ and $C=[0.00600069867,\,0.0200179144,\,103.560653]\ \mu\mathrm{m}^2$ . Compute the indices at the d, F, and C lines and the Abbe number.

Step 1. Work at the yellow d-line, $\lambda=0.58756\ \mu\mathrm{m}$ , so $\lambda^2=0.345227\ \mu\mathrm{m}^2$ . The three terms are $1.03961212\times0.345227/(0.345227-0.00600070)=1.058002$ , then $0.231792344\times0.345227/(0.345227-0.02001791)=0.246060$ , and $1.01046945\times0.345227/(0.345227-103.560653)=-0.003380$ (the far-IR term is small and negative because $\lambda<\sqrt{C_3}$ ). Their sum is $1.300683$ .

Step 2. Add one and take the root: $n_d=\sqrt{1+1.300683}=\sqrt{2.300683}=\mathbf{1.51680}$ . Repeating at $\lambda_F=0.48613\ \mu\mathrm{m}$ gives $n_F=\sqrt{2.317630}=\mathbf{1.52238}$ , and at $\lambda_C=0.65627\ \mu\mathrm{m}$ gives $n_C=\sqrt{2.293172}=\mathbf{1.51432}$ . Blue is highest — normal dispersion.

Step 3. Form the Abbe number. The mean dispersion is $n_F-n_C=1.52238-1.51432=0.008054$ (carry the full Sellmeier values — subtracting two five-decimal numbers loses a digit), and $n_d-1=0.516800$ . Hence $V_d=0.516800/0.008054=\mathbf{64.17}$ . A large $V_d$ marks BK7 as a low-dispersion crown glass — which is why it is the default first element of an achromatic doublet. n_d = 1.51680, n_F = 1.52238, n_C = 1.51432, V_d = 64.17.

2 · Fitting a two-term Cauchy equation and testing it

Using only the d and C indices from example 1, fit $n=A+B/\lambda^2$ , then predict the blue F-line index and compare with the Sellmeier value $n_F=1.52238$ .

Step 1. Two points fix two coefficients. Subtracting the two Cauchy equations eliminates $A$ : $B=(n_d-n_C)/(1/\lambda_d^2-1/\lambda_C^2)$ . Here $1/\lambda_d^2=1/0.58756^2=2.89665\ \mu\mathrm{m}^{-2}$ and $1/\lambda_C^2=1/0.65627^2=2.32185\ \mu\mathrm{m}^{-2}$ , a difference of $0.57479\ \mu\mathrm{m}^{-2}$ , and $n_d-n_C=0.002478$ . So $B=0.002478/0.57479=\mathbf{0.004311\ \mu m^2}$ .

Step 2. Back-substitute for $A$ : $A=n_d-B/\lambda_d^2=1.51680-0.004311\times2.89665=1.51680-0.012487=\mathbf{1.5043}$ . The fitted $A=1.5043$ is the long-wavelength index — the $\sqrt{1+\sum B_j}$ -type constant the derivation predicted, with all of BK7's bands folded in.

Step 3. Predict the blue line, which was not used in the fit: $n_F=A+B/\lambda_F^2=1.5043+0.004311/0.48613^2=1.5043+0.004311\times4.23193=1.5043+0.018243=\mathbf{1.52255}$ . Against the Sellmeier truth $1.52238$ the error is $1.52255-1.52238=+1.8\times10^{-4}$ — a two-term formula fitted to two lines predicting a third to better than a part in $8000$ . That accuracy across the visible, and its steady drift high toward the ultraviolet, is the whole Cauchy story. Cauchy A = 1.5043, B = 0.004311 µm², predicts n_F = 1.52255 (error +1.8×10⁻⁴).

Physics of the portrait

The Cauchy equation is a portrait of empirical optics at the moment it turned quantitative — a formula that worked for thirty years before anyone could say why, then was absorbed, whole, into the resonance picture that replaced it.

Origins · 1 mincauchy equation and how this was discovered

cauchy equation

the empirical dispersion series n = A + B/λ² + C/λ⁴ + …, later revealed as the leading Taylor term of the resonance-based Sellmeier equation · dispersion of transparent media

Augustin-Louis Cauchy, better remembered for founding rigorous analysis, in 1836 fitted the measured dispersion of glass and other transparent solids to the compact series $n(\lambda)=A+B/\lambda^2+C/\lambda^4+\cdots$ . It was description without explanation: light was not yet electromagnetic, the atom a speculation, and Cauchy's own elastic-aether justification was wrong in its mechanics — yet the equation fit the prism data of every crown and flint glass, and opticians used it for decades to design achromats. Its limitation was that of any power series: knowing nothing of the absorption bands, it failed toward the ultraviolet and near any resonance. The repair came in 1871, when Wolfgang Sellmeier recast dispersion as a sum of resonance terms, $n^2=1+\sum_j B_j\lambda^2/(\lambda^2-C_j)$ , each with an explicit pole at a resonance wavelength — the first formula to encode why a material disperses rather than merely how much. A decade later Lorentz and Lorenz grounded that sum in atomic polarizability, and the modern reading followed: Cauchy is Sellmeier expanded far from its nearest pole, the empirical series contained inside the physical one. The concept named here is the equation, not the man — a reminder that a good fitting formula is often a theory waiting to be understood.

History

1814

— Fraunhofer maps the dark solar lines later standardized as the C, d, and F dispersion references

1836

— Cauchy fits the empirical dispersion series n(λ) = A + B/λ² + C/λ⁴ + …

1871

— Sellmeier recasts dispersion as a sum over resonance terms with explicit poles

1880

— Lorentz and Lorenz ground the resonance sum in atomic polarizability (the Lorentz–Lorenz relation)

1884

— Abbe and Schott found the Jena glassworks; glasses are catalogued by n_d and Abbe number V_d

Going further

Texts. Hecht, Optics, 5th ed., §3.5 (dispersion) · Saleh & Teich, Fundamentals of Photonics, 3rd ed., §5.5 · Foot, Atomic Physics, Ch. 7 (dispersion and the classical oscillator) · Loudon, The Quantum Theory of Light, 3rd ed., Ch. 1 · Demtröder, Laser Spectroscopy, Vol. 1 (dispersion of dielectrics) · Malitson, J. Opt. Soc. Am. 55, 1205 (1965) for the fused-silica coefficients.

Provenance. The Sellmeier and Cauchy forms and their fitted-coefficient conventions follow Hecht §3.5 and Saleh & Teich §5.5; the Lorentz-model reduction follows Foot Ch. 7 and Loudon Ch. 1. The Lorentz oscillator susceptibility itself is recalled by name from aq-optics-lorentz-oscillator-model, and the BK7 Sellmeier curve was previewed in aq-optics-refractive-index. A verifier should: (1) reproduce the Lorentz-to-Sellmeier reduction by dropping $\gamma_j$ and substituting $\omega=2\pi c/\lambda$ ; (2) confirm the Cauchy series as the $\lambda\gg\sqrt{C}$ expansion (sympy); (3) reproduce the BK7 indices $n_d=1.51680$ , $n_F=1.52238$ , $n_C=1.51432$ and $V_d=64.17$ from the Schott coefficients; (4) fit $A=1.5043$ , $B=0.004311\ \mu\mathrm{m}^2$ to the d and C lines and recover the $+1.8\times10^{-4}$ prediction error at F. The UV/IR band positions ( $\sim0.1\ \mu\mathrm{m}$ , $\sim9\ \mu\mathrm{m}$ for glass) and the silica zero near $1.27\ \mu\mathrm{m}$ are standard textbook values.

Prerequisites. The Refractive Index · The Lorentz Oscillator Model · Dielectric Susceptibility and Permittivity · Dispersion: Normal and Anomalous. Leads to. Group Velocity and Group Index · Group Velocity Dispersion and Pulse Broadening · Kramers–Kronig Relations · Complex Refractive Index and the Extinction Coefficient.

Are Sellmeier and Cauchy the same formula written differently?

No — one contains the other. Cauchy is a plain power series in $1/\lambda^2$ with no built-in resonance, so it can only interpolate; Sellmeier keeps an explicit pole at each resonance wavelength $\sqrt{C_j}$ . Taylor-expanding a Sellmeier term for $\lambda$ well above its resonance recovers exactly the Cauchy series — good in the visible, useless near a band. Practically: use Cauchy for a quick visible-only fit; use Sellmeier for broadband accuracy, wavelengths near the band edges, or physically meaningful resonance positions.

What exactly are the coefficients Bⱼ and Cⱼ — are they measured atomic properties?

Partly. $C_j=\lambda_j^2$ is the square of an effective resonance wavelength, and $B_j$ is proportional to that resonance's oscillator strength and the plasma frequency — real physics, as the derivation shows. But a fitted Sellmeier term usually lumps several nearby true transitions into one effective pole, so you should not read $\sqrt{C_j}$ as a single spectroscopic line. The coefficients come from least-squares fits to measured indices across the transparency window, published to enough digits that re-evaluating the formula reproduces the data to $\sim10^{-6}$ . They are physically motivated parameters, not directly measured constants.

Why is Cauchy's A about 1.50 for glass and not 1, if it is the long-wavelength index?

Because "long wavelength" here means long compared with the resonances but still with the resonances present. From the derivation, a single UV term gives $A=\sqrt{1+B_s}$ , the index far below that resonance — already well above one. A real glass has several bands, and a fit folds them all into an effective $A=\sqrt{1+\sum_j B_j}$ -type constant, which for BK7 lands near $1.50$ . So $A$ is not the vacuum value; it is the index left after subtracting the $B/\lambda^2$ dispersion — exactly why $A\approx1.5043$ came out of the two-line fit.

Can I use one glass's Sellmeier coefficients at any wavelength I like?

Only inside the window they were fitted for, and only in the intended units. Every published Sellmeier fit carries a stated wavelength range — for BK7 roughly $0.3$ to $2.5\ \mu\mathrm{m}$ . Outside it the lossless formula has no absorption to tame it: near a band it diverges while the real material simply goes opaque. The commonest mistake, though, is units: the coefficients assume $\lambda$ in micrometres, and passing nanometres sends every term to its long-wavelength limit and the index to a wrong constant. Match the printed unit convention, and never extrapolate across a resonance.

How does a Sellmeier fit give me the group index and pulse dispersion?

Because it hands you $n(\lambda)$ as a differentiable function, everything downstream is calculus. The first derivative $\mathrm{d}n/\mathrm{d}\lambda$ (equation 5) gives the group index $n_g=n-\lambda\,\mathrm{d}n/\mathrm{d}\lambda$ that sets a pulse's travel time; the second derivative $\mathrm{d}^2n/\mathrm{d}\lambda^2$ gives the chromatic dispersion parameter $D=-(\lambda/c)\,\mathrm{d}^2n/\mathrm{d}\lambda^2$ that broadens pulses in a fibre. One fitted formula, differentiated once and twice, yields the whole dispersion hierarchy — exactly the chain aq-optics-group-velocity-dispersion-pulse-broadening follows for fused silica.

Why do lens designers care so much about the Abbe number when they already have n(λ)?

Because $V_d$ compresses the whole dispersion curve into one comparison number that makes glass selection tractable. The full $n(\lambda)$ is what you ray-trace with, but to choose glasses you need a scalar for "how much does this bend blue more than red relative to its power" — that is $V_d=(n_d-1)/(n_F-n_C)$ . Pairing a positive high- $V_d$ crown with a negative low- $V_d$ flint cancels the leading dispersion while keeping net focusing power: an achromat. Two glasses with equal $V_d$ cannot be colour-corrected against each other, so the Abbe number, not $n$ alone, is the axis along which glasses are organized (the glass map plots $n_d$ against $V_d$ ).