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.

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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.

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. Inclusion test: Require the inverse-even-power Cauchy dispersion form fitted to refractive-index data over a stated transparent normal-dispersion range. Exclusion test: Exclude Cauchy's functional equation f(x+y)=f(x)+f(y), Sellmeier resonance formulas, Abbe-number summaries, and coefficients used with an unstated or mismatched wavelength unit. Nearest boundary: Sellmeier equations model oscillator-like resonances and generally extrapolate more physically across wider transparent ranges; Cauchy is a local empirical series. Exit condition: The approximation should not be trusted near absorption bands, anomalous dispersion, ultraviolet or infrared regions beyond fit, or under changed temperature/composition without validation. Common misclassifications: 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. Nearest named distinctions: Cauchy's functional equation: Is the additive relation f(x+y)=f(x)+f(y). Sellmeier equation: Uses resonance denominators and different coefficients. Snell's law: Relates refraction angles given refractive indices. Group index: Describes pulse propagation and depends on dispersion derivative.

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.

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