= Refractive index
{title2=$n$}
{wiki}
= Index of refraction
{synonym}
The <refractive index> compares the <phase velocity> of <light> in a medium with the <vacuum> <speed of light> $c$. For a transparent, uniform medium with <isotropy> and a positive-index propagation branch,
$$
n(\omega)=\frac{c}{v_{\rm ph}(\omega)}=\frac{ck(\omega)}{\omega}.
$$
Here $\omega>0$ is the <angular frequency>, $k>0$ is the <wavenumber>, and $v_{\rm ph}=\omega/k$ is the speed of a surface of constant <wave phase>. The <refractive index> is a <dimensionless quantity>; it is a property of a specified material response at a specified <frequency>, rather than a universal number attached to the word “glass”. More general absorbing or negative-index media require the conventions discussed in <refractive index and electromagnetic response>. <RP Photonics Encyclopedia by Rüdiger Paschotta/Refractive index>[RP Photonics Encyclopedia, Rüdiger Paschotta, Refractive Index: What is a Refractive Index?]{ref}
For an original model example, choose $n=3/2$ at <vacuum> <wavelength> $\lambda_0=600\,\mathrm{nm}$. The <phase velocity> is $v_{\rm ph}=2c/3\simeq1.999\times10^8\,\mathrm{m/s}$, the <frequency> is $f=c/\lambda_0\simeq4.997\times10^{14}\,\mathrm{Hz}$, and the medium <wavelength> is $400\,\mathrm{nm}$. These are stipulated model values, not measured properties of glass or water. A value at this single <frequency> does not determine a pulse's <group velocity>.
The usefulness of <refractive index> is that one measured response connects several different observations: <Snell's law> predicts <refraction>, <optical path length> predicts accumulated <wave phase>, and changes with <frequency> describe <optical dispersion>. Even at <normal incidence>, where a <light ray> does not bend, a changed <refractive index> changes <wavelength> and <wave phase>. The following comparison isolates that effect; the oblique geometry is developed in <wavefront>.
\Image[/refractive-index-phase-spacing.png]
{title=Chosen <refractive index> $n=3/2$: shorter <wavelength>, slower <wavefronts>, unchanged interface <frequency>}
{description=The top panel compares equal normalized <electric field> components at one time on either side of a stationary interface at <normal incidence>. The lower panel tracks constant-<wave phase> crests in space and time. Curves represent incident and transmitted components, with their amplitudes normalized separately; reflected <light> and interface amplitude matching are omitted. The chosen single-<frequency> model contains no measured material data.}
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