Refractive index (n)

The refractive index $n$ of a material is the ratio of the speed of light in vacuum to its speed in the material, $n=c\mathrm{/}v$, so light crosses the material at $c\mathrm{/}n$: about 225,000 km/s in water ($n=1.333$) and 200,000 km/s in a glass with $n=1.5$. It is a pure number, exactly 1 in vacuum and 1.0003 in air. Common transparent solids range from about 1.4 (fused silica, fluoride crystals) to 2.42 (diamond) in the visible, and the semiconductors used in infrared photonics reach 3.2 to 4.3. The index decides how much light bends at a surface (Snell's law), how much of it reflects there (Fresnel equations), how long the wavelength is inside the material, and how tightly a waveguide can confine light. It changes with wavelength, which is dispersion, and with temperature.

Definition and formulas

The index defined by the speed of the wave crests, the phase velocity $v_p$, is

$$n=\frac{c}{v_p}\mathrm{.}$$

The frequency of light does not change when it enters a material, so its wavelength shrinks to ${\lambda }_0\mathrm{/}n$: light of 1550 nm vacuum wavelength has a wavelength of 446 nm inside silicon. At a boundary between media of index $n_1$ and $n_2$, the directions of the incident and refracted rays obey Snell's law, $n_1\sin {\theta }_1=n_2\sin {\theta }_2$, and at normal incidence a fraction

$$R={\left(\frac{n_1-n_2}{n_1+n_2}\right)}^2$$

of the power reflects: 4.2% from each face of N-BK7 glass in air and 30.6% from silicon, which is why lenses and silicon photonic facets carry anti-reflection coatings.

For a non-magnetic material the index is the square root of the relative permittivity at the optical frequency, $n=\sqrt{{\epsilon }_r}$. The static permittivity does not apply: water's is about 80, which would give 9, while its optical index is 1.33, because the rotation of the water molecules that dominates the static value cannot follow a field oscillating at hundreds of terahertz.

In an absorbing material the index is complex, $\tilde{n}=n+i\kappa$, where the extinction coefficient $\kappa$ sets the intensity absorption coefficient

$$\alpha =\frac{4\pi \kappa }{{\lambda }_0}\mathrm{.}$$

At 1550 nm, $\kappa ={10}^{-4}$ already gives $\alpha =8.1{\text{cm}}^{-1}$, a loss of 35 dB/cm; transparent optical materials have $\kappa$ many orders of magnitude smaller.

Refractive index of common materials

At the sodium D line (589 nm), near 20 °C:

At 1550 nm, at room temperature:

Crystals such as sapphire and lithium niobate have two or three indices, depending on the polarization and direction of the light relative to the crystal axes; that difference is birefringence.

Phase index and group index

A pulse, or the envelope of any modulated signal, travels at the group velocity $c\mathrm{/}{n}_g$, set by the group index

$$n_g=n-\lambda \frac{dn}{d\lambda }\mathrm{.}$$

In fused silica at 1550 nm, $n=1.444$ and $n_g=1.463$; in standard single-mode fiber the effective group index is about 1.468, which is why light in fiber takes 4.9 µs to cover a kilometer (speed of light in fiber). In a waveguide the relevant indices are those of the guided mode: a silicon strip waveguide has an effective index of about 2.4 and a group index of about 4.3 at 1550 nm, because the mode's confinement changes quickly with wavelength. The group index sets delays, the free spectral range of resonators and the spacing of laser modes; the phase index sets the phase accumulated over a length, and with it resonance wavelengths and phase matching.

Dispersion

In a transparent material the index falls as the wavelength rises, which is normal dispersion: N-BK7 has $n=1.522$ at 486 nm, 1.517 at 588 nm, 1.514 at 656 nm and 1.501 at 1550 nm, and a prism bends blue light more than red. The Sellmeier equation fits the index across a material's transparent range:

$$n^2(\lambda )=1+\sum _i\frac{B_i{\lambda }^2}{{\lambda }^2-C_i}\mathrm{.}$$

For fused silica (Malitson, $\lambda$ in µm) the coefficients are $B=0.6961663$, $0.4079426$, $0.8974794$ and $C={0.0684043}^2$, ${0.1162414}^2$, ${9.896161}^2$ µm${}^2$, which give 1.4585 at 587.6 nm and 1.4440 at 1550 nm. The Refractive Index Calculator evaluates the published Sellmeier equations of fused silica, N-BK7, N-SF11, calcium fluoride, sapphire and silicon across each fit's stated wavelength range, with the group index and dispersion. For visible optics, the spread is summarized by the Abbe number $V_d=(n_d-1)\mathrm{/}(n_F-n_C)$: 64.2 for N-BK7, 67.8 for fused silica and 55.3 for diamond, whose large index spread, $n_F-n_C=0.026$ or three times that of N-BK7, produces the colors of a cut stone. The slope $dn\mathrm{/}d\lambda$ determines chromatic dispersion and chromatic aberration. Close to an absorption band the index changes steeply and can rise with wavelength inside the band, which is anomalous dispersion; the Kramers–Kronig relations tie the index at every wavelength to the absorption spectrum.

Temperature and other effects

The thermo-optic coefficient $dn\mathrm{/}dT$ describes the change with temperature. Values near 1550 nm:

Silicon's large coefficient shifts the resonances of silicon ring resonators by about 80 pm/K and makes thermo-optic phase shifters the most common tuning element on silicon photonic chips; polymer claddings, with the opposite sign, are used to cancel part of that sensitivity. Thermo-optic coefficients gives the measurements behind these numbers.

The index also changes with carrier density (the plasma dispersion effect in semiconductors), with an applied electric field (the electro-optic effect), with optical intensity (the Kerr effect) and with mechanical strain (the photoelastic effect). These effects are the basis of optical modulators and switches.

Measuring the refractive index

An Abbe refractometer finds the index of a liquid or a polished solid at 589 nm from the critical angle at a glass prism, to about the fourth decimal place. The most accurate method for bulk glass is minimum deviation: a prism of apex angle $A$ bends light least at the angle $D_{\text{min}}$, and

$$n=\frac{\sin ((A+D_{\text{min}})\mathrm{/}2)}{\sin (A\mathrm{/}2)};$$

a 60° prism of N-BK7 gives $D_{\text{min}}=38.65\mathrm{°}$ at 587.6 nm. Ellipsometry and prism couplers measure thin films, and a prism coupler also gives the mode indices of a planar waveguide. For a waveguide or fiber, the free spectral range of a resonator or interferometer built from it gives the group index directly. The methods, from prism minimum deviation to ellipsometry, are compared in How to Measure Refractive Index.

Common questions

What is the refractive index of water?

1.333 at 589 nm and 20 °C. It falls slightly as the temperature or the wavelength rises, to about 1.318 at 1550 nm.

What is the refractive index of glass?

It depends on the glass: 1.458 for fused silica, 1.517 for the common crown glass N-BK7 and about 1.52 for window glass, and dense flint and lanthanum glasses reach about 1.9 (all at 589 nm).

Can the refractive index be less than 1?

The phase index can. X-rays in matter see an index slightly below 1, so do radio waves in the ionosphere, and the real part of the index of silver and gold in the visible is well below 1. The phase velocity then exceeds $c$, but information travels at the signal velocity, which never does. Engineered metamaterials can even have a negative index over a narrow band.

Why is the index different for each color?

The index comes from the response of the material's electrons, which is strongest near their resonances in the ultraviolet. Blue light is closer to those resonances than red light, so the material responds more strongly and its index is higher.

References: M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999). I. H. Malitson, "Interspecimen comparison of the refractive index of fused silica," Journal of the Optical Society of America 55, 1205 (1965). H. H. Li, "Refractive index of silicon and germanium and its wavelength and temperature derivatives," Journal of Physical and Chemical Reference Data 9, 561 (1980). P. E. Ciddor, "Refractive index of air: new equations for the visible and near infrared," Applied Optics 35, 1566 (1996). SCHOTT, N-BK7 optical glass data sheet. M. N. Polyanskiy, "Refractiveindex.info database of optical constants," Scientific Data 11, 94 (2024).