Optical Dispersion

Equivalent forms

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

n^2 = 1 + \sum_i \frac{B_i\lambda^2}{\lambda^2 - C_i}

Two fitted constants reproduce the entire visible color-spread of a glass — the math behind every prism and the chromatic flaw in every cheap lens.

Symbol	Name	SI unit	Dimension	Range
$lambda$	Wavelength Vacuum wavelength of the light	nm	L	380 – 750
$A$	Cauchy A Asymptotic (long-wavelength) index	dimensionless	1	1.4 – 1.7
$B$	Cauchy B Dispersion coefficient	nm^2	L^2	1000 – 10000
$n$	Refractive indexoutput Wavelength-dependent index n(lambda)	dimensionless	1	1.4 – 1.8

Unit systems

SI:: lambda in nm (or m), n dimensionless
CGS:: lambda in cm
Imperial:: n dimensionless

Where it holds

Empirical fit valid across the visible/near-IR away from absorption bands. Near a resonance (anomalous dispersion, dn/dlambda > 0) the Sellmeier or full Lorentz model is required.

Discovery

Augustin-Louis Cauchy · 1836

Cauchy fit the measured index of glasses to an empirical inverse-power series in wavelength. Decades later Sellmeier grounded the behavior in resonance physics, but Cauchy's compact A + B/lambda^2 still serves the visible band.

Try this

Why does a prism fan white light into a rainbow?

Glass slows blue light more than red, so the refractive index depends on wavelength. Cauchy's formula captures this dispersion — the reason every color bends by a different angle.

Research status: stable

Common misconceptions

Dispersion is not the same as refraction: refraction bends all light, dispersion is the wavelength-dependence of that bend. Normal dispersion has n decreasing with lambda; anomalous dispersion (near absorption) reverses it.

Derivation

A Lorentz oscillator gives

n^2 = 1 + N e^2/(\varepsilon _0\,\mathrm{m}) * 1/(\omega _0^2 - \omega ^2)

.

Far below resonance, expanding in omega^2 (i.e.

1/lambda^2) yields

n \sim A + B/\lambda ^2

— the Cauchy form, with A and B set by the oscillator strength and resonance frequency.

Limiting cases

B -> 0⟶

n = A

constant — no dispersion, no rainbow

lambda near UV resonance⟶ Cauchy fails; full Sellmeier resonance term required

Snell's Law→How Rainbows Form→Refractive Index Definition→Group Velocity→