Dispersion and the Cauchy Equation

Dispersion is the variation of a material's refractive index with wavelength. It is the reason a prism separates white light into a spectrum, and the reason rainbows exist.

Why Dispersion Occurs

In a dielectric material, the refractive index arises from the interaction of light with bound electrons. The electrons have natural resonance frequencies; light near these frequencies is affected more strongly. In the visible range (away from resonances), shorter wavelengths (blue/violet) experience a higher refractive index than longer wavelengths (red) — this is called normal dispersion.

The Cauchy Equation

An empirical formula that fits the refractive index of many transparent materials across the visible spectrum:

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

Where $\lambda$ is the wavelength in nm and $A$, $B$ are material constants:

The Cauchy equation shows that $n$ increases as $\lambda$ decreases — blue light bends more than red light, explaining prismatic dispersion.

Dispersion: Rate of Change of n

The dispersion $dn/d\lambda$ tells us how rapidly the refractive index changes with wavelength:

$\frac{dn}{d\lambda} = -\frac{2B}{\lambda^3}$

The negative sign confirms that $n$ decreases as $\lambda$ increases (normal dispersion). A larger magnitude of $dn/d\lambda$ means stronger dispersion — colors separate more.

Example: Crown glass at $\lambda = 550\,\text{nm}$:

$n(550) = 1.5 + \frac{5000}{550^2} \approx 1.5165$

$\frac{dn}{d\lambda}\bigg|_{550} = -\frac{2 \times 5000}{550^3} \approx -6.01 \times 10^{-5}\,\text{nm}^{-1}$

Your Task

Implement the Cauchy equation and its derivative. Wavelengths are in nm; $B$ has units of nm².