Cauchy's Equation¶

An empirical optical dispersion formula expressing a transparent material's refractive index as A + B/λ² + C/λ⁴ + … over a fitted normal-dispersion wavelength range.

Version: v1 · 2026-09-28 · History
Domain-specific #: 8368
Domain group: Natural Sciences
Origin domain: Physics
Subdomains: Optics, Optical Dispersion → Physics
Aliases: Cauchy Equation, Cauchy's Transmission Equation

Core Idea¶

Cauchy's equation is a compact empirical description of normal optical dispersion. Refractive index is fitted as a constant plus inverse even powers of wavelength, often using only two terms in the visible region.

Its coefficients are inseparable from material, temperature, units, truncation, and fit interval. Mathematical simplicity should not be mistaken for a universal light–matter theory or license to extrapolate through resonances.

How would you explain it like I'm…

The Color-Bending Recipe

When light goes into glass, different colors bend by slightly different amounts, which is how a prism makes a rainbow. Cauchy's equation is a simple recipe that fits how much each color bends for one kind of glass. It works well for ordinary colors, but it's a fitted recipe, not a rule for everything.

The Color-Bending Formula

Clear materials like glass bend light, and the amount of bending, called the refractive index, changes a little with the color (wavelength) of the light. Cauchy's equation is a simple formula that fits this: a fixed number plus smaller correction terms that shrink as the wavelength gets longer. Often just two terms are enough for visible light. The numbers in the formula come from measurements of one material at certain conditions, so they don't automatically work for other materials, temperatures, or kinds of light far outside the measured range.

Empirical Dispersion Fit

Cauchy's equation is an empirical formula describing normal dispersion — the way a transparent material's refractive index n changes with wavelength λ. It writes n as a constant plus terms in inverse even powers of wavelength: n(λ) = A + B/λ² + C/λ⁴ + …, and in the visible range often only A and B are used. The coefficients are fitted to data, so they depend on the material, temperature, the units used for λ, how many terms are kept, and the wavelength range of the fit. Because it is a curve fit rather than a physical theory of how light interacts with matter, it should not be extended across resonances where the material absorbs strongly and dispersion behaves differently.

Cauchy's equation is a compact empirical description of normal optical dispersion: the refractive index is fitted as n(λ) = A + B/λ² + C/λ⁴ + ..., a constant plus inverse even powers of wavelength, with two terms often adequate across the visible region. It is a curve fit, not a derivation from a physical model of light-matter interaction, so its coefficients have meaning only together with the material, temperature, units, truncation order, and fitting interval. Because it describes only the smooth normal-dispersion regime, it cannot be extrapolated through absorption resonances, where the index behaves anomalously. Its mathematical simplicity is a convenience and should not be mistaken for generality.

Structural Signature¶

Sig role-phrases:

Transparent material — Provides the medium and its measured dispersion. It is physical carrier. Counterfactual: Coefficients do not transfer between materials.
Vacuum wavelength λ — Supplies the independent variable in the declared unit. It is spectral input. Counterfactual: Using frequency or in-medium wavelength without conversion breaks coefficients.
Refractive index n — Provides the phase-index response being fitted. It is output. Counterfactual: Group index requires differentiation or separate treatment.
Coefficients A,B,C,… — Encode an empirical fit under one unit convention. It is model parameters. Counterfactual: Their dimensions depend on wavelength units.
Truncation order — Selects two-term or higher approximation complexity. It is model form. Counterfactual: More terms can overfit limited data.
Validity interval — Bounds normal dispersion away from resonances. It is domain. Counterfactual: Extrapolation can be physically implausible.

What It Is Not¶

It is not Cauchy's additive functional equation.
It is not a first-principles resonance model.
Coefficients are not unit-free.
A visible-range fit is not automatically valid in ultraviolet or infrared.
Closest near-miss. Sellmeier equations model oscillator-like resonances and generally extrapolate more physically across wider transparent ranges; Cauchy is a local empirical series.

Scope of Application¶

Optical design. Interpolates refractive index over validated bands.
Material characterization. Summarizes measured normal dispersion.
Ray tracing. Provides a computationally simple wavelength-dependent index.
Model comparison. Contrasts empirical Cauchy and resonance-based Sellmeier forms.

Clarity¶

Report material and composition, temperature, pressure where relevant, phase versus group index, vacuum wavelength unit, coefficient units, term count, fit range, measurement method, fitting loss, residuals, uncertainty, and extrapolation prohibition.

Manages Complexity¶

A short series compresses a spectral response into reusable coefficients, but unit powers and domain boundaries make silent misuse easy. The simplest model can be the best local description and a poor global one simultaneously.

Abstract Reasoning¶

Collect refractive-index data under fixed material and environmental conditions.
Choose vacuum-wavelength units and a normal-dispersion interval away from absorption.
Fit the minimum justified inverse-power terms.
Inspect residuals, coefficient uncertainty, and held-out wavelengths.
Use only within validated range or replace it with a more physical dispersion model.

Knowledge Transfer¶

Inverse-power fitting transfers across transparent materials, but coefficients and intervals never do. Model-selection logic transfers farther than the optical formula itself.

Examples¶

Canonical¶

Visible-range refractive-index measurements for one glass are least-squares fitted to n=A+B/λ² using vacuum λ in micrometres, with residuals and spectral bounds reported.

Mapped back: material → one glass; input → vacuum wavelength in μm; model → two-term Cauchy; evidence → measured n; scope → visible fit interval.

Applied / In Practice¶

Applying tabulated A and B quoted for micrometres to wavelengths entered in nanometres gives a numerical curve but not the intended Cauchy model.

Mapped back: coefficients → μm convention; input → nm; unit conversion → absent; verdict → invalid application.

Structural Tensions¶

T1 — Simple Fit versus Physical Extrapolation. Few parameters reproduce smooth visible dispersion while lacking resonance structure needed outside the window.

Diagnostic: Is the task interpolation or extrapolation?

T2 — Higher-Order Accuracy versus Parameter Stability. More inverse powers can reduce residuals while amplifying collinearity and edge behavior.

Diagnostic: Does held-out or uncertainty analysis justify added terms?

Structural–Framed Character¶

Cauchy's Equation is structural as an inverse-even-power dispersion fit and framed by material-specific optical calibration.

Structural Core vs. Domain Accent¶

The general pattern is empirical series approximation. Optics adds refractive index, vacuum wavelength, normal dispersion, absorption boundaries, and material coefficients.

This entry presupposes Constraint.

Approved optical-model root. No current parent entails this specific inverse-power refractive-index law.
Related — Sellmeier equation, refractive index, dispersion, Abbe number, and group index. They are an alternate model, target property, phenomenon, summary, and derived quantity.

Relationships to Other Abstractions¶

Current abstraction Cauchy's Equation Domain-specific

Parents (1) — more general patterns this builds on

Cauchy's Equation presupposes Constraint Prime

Cauchy's Equation presupposes Constraint: the parent's defining role is necessary to the child's frozen mechanism or criterion.

Hierarchy path (1) — routes to 1 parentless root

Cauchy's Equation → Constraint

Neighborhood in Abstraction Space¶

Cauchy's Equation sits in a crowded region of the domain-specific corpus (36^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Optical & Astrophysical Phenomena (25 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With¶

Cauchy's functional equation. Tell: Is the additive relation f(x+y)=f(x)+f(y).
Sellmeier equation. Tell: Uses resonance denominators and different coefficients.
Snell's law. Tell: Relates refraction angles given refractive indices.
Group index. Tell: Describes pulse propagation and depends on dispersion derivative.

References¶

Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Cauchy%27s_equation (revision 1340963965).
Preserved source candidate: https://gallica.bnf.fr/ark:/12148/bpt6k901948/f159.item
Preserved source candidate: http://www.astro.rug.nl/~sctrager/teaching/OA/old/2018/Atmosphere.pdf

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.