Index of Refraction Calculator

Background.

Light slows down when it enters glass, water or any other transparent material, and the refractive index says by how much. It is the ratio of light's speed in a vacuum to its speed in the medium: n = c / v. Glass with an index of 1.5 carries light at two-thirds of its vacuum speed; water, at about 1.33, at roughly three-quarters; diamond, near 2.42, at well under half.

That change of speed is what bends light at a boundary. A wavefront meeting glass at a slant touches the surface at one edge first, that edge slows before the rest, and the whole front swings round: the refraction described by Snell's law, n₁ sin θ₁ = n₂ sin θ₂. The same number governs how strongly a lens focuses, how much light a window reflects, and the critical angle beyond which light inside water or an optical fibre is reflected back instead of escaping.

It is also why spectacle wearers are offered high-index lenses. A thin lens bends light in proportion to n − 1 and to the curvature of its faces, so for the same prescription a material of index 1.67 needs only 0.5 ÷ 0.67, about three-quarters, of the curvature that a 1.5 lens needs. Flatter faces mean a thinner, lighter lens, especially at the edges of a strong prescription for short sight. The trade-off is that higher-index materials generally disperse colour more and reflect more light at each surface unless coated.

The calculation here runs in one direction: enter the speed of light in the medium and the index comes out. The vacuum speed is not a measurement but a fixed value. Since 1983 the metre has been defined so that c is exactly 299,792,458 m/s, and that exact figure is the numerator.

Two cautions. The index depends on wavelength, which is why a prism fans white light into colours, so a velocity for one colour gives the index for that colour only; handbook values are often quoted for yellow sodium light at 589 nm. And the speed entered must be the phase velocity, the speed of individual wave crests. Pulses travel at the group velocity, which in glass is somewhat slower, and dividing c by that gives the group index instead. Phase velocity can even exceed c, giving an index below 1, as it does for X-rays in most materials, without letting any signal outrun light.

What is index of refraction calculator?

The absolute refractive index, n, of a material is the speed of light in vacuum divided by the phase velocity of light in that material. It is a pure number with no units. For ordinary transparent materials at visible wavelengths it is above 1: about 1.0003 for air, 1.33 for water and around 1.5 for common glass. It changes with wavelength, with temperature and, for gases, with pressure, so a quoted index always belongs to particular conditions.

Light slows in a material because the material responds to it. The oscillating electric field of the wave sets the electrons in its atoms and molecules oscillating, those electrons radiate waves of their own, and the incoming and re-radiated waves add up to a single wave whose crests advance more slowly than c. Between the atoms nothing moves slower than light in a vacuum; the lower speed belongs to the combined wave. How strongly the electrons respond depends on frequency, which is why the index depends on colour.

That response is the material's permittivity at the frequency of the light, and for non-magnetic materials n = √εᵣ. The frequency matters enormously. Water's relative permittivity for slowly varying fields is about 80, which would suggest an index near 8.9, but at optical frequencies its molecules cannot turn fast enough to follow the field, and the index is about 1.33.

Tables usually give absolute indices, measured against vacuum, though some practical references quote values relative to air. Because air's own index is about 1.0003, the two differ only around the fourth decimal place: glass with an absolute index of 1.5 has an index of about 1.4996 relative to air.

How to use this calculator.

1. 01Enter the phase velocity of light in the material, in metres per second. Type the full number: 2.25 × 10⁸ m/s goes in as 225000000, and a speed given as 225,000 km/s needs multiplying by 1,000 first. If your source gives the speed as a fraction of c, you can skip the calculator: the index is the reciprocal, so 0.75c means n = 1.3333.
2. 02Use a velocity for the wavelength you care about. An index for red light differs slightly from one for blue light in the same material.
3. 03Make sure the speed is a phase velocity, not a pulse transit time. Timing a light pulse through a length of fibre gives the group velocity, and the result would then be the group index.
4. 04Read n. To go the other way, from a known index to the speed of light in the material, divide 299,792,458 by the index.

The formula.

n = c / v, where c = 299,792,458 m/s exactly and v is the phase velocity of light in the medium in m/s. The units cancel, leaving a dimensionless ratio. Because v sits in the denominator, the index moves inversely with it: a phase velocity 1% lower raises n by about 1%, and halving the speed doubles the index. The absolute indices feed straight into Snell's law, n₁ sin θ₁ = n₂ sin θ₂, and into the critical angle for light passing from a dense medium into air, arcsin(1 / n), which is about 48.6° for water with n = 1.333. The formula ignores absorption, which needs a complex index, and crystals whose index depends on polarisation and direction of travel.

Where the ratio comes from shows why it controls bending. In a time t, a wavefront crossing a boundary advances v₁t in the first medium and v₂t in the second; fitting both onto the same stretch of boundary gives sin θ₁ ÷ sin θ₂ = v₁ ÷ v₂, and replacing each speed with c ÷ n turns that into Snell's law. The same substitution gives travel time: light crossing a thickness L of material takes nL ÷ c, as though it had crossed a vacuum path n times longer. That product nL, the optical path length, is the quantity lens designers and interferometers actually work with.

Across the visible range a clear material's index usually changes smoothly with wavelength, and a simple empirical fit, Cauchy's equation n(λ) ≈ A + B ÷ λ², describes ordinary glass well, with the index rising toward the blue end. Near a wavelength the material absorbs, the curve bends sharply and the simple ratio of speeds stops being enough on its own.

A worked example.

Frequently asked questions.

References& sources.

1. [1]ISO 80000-7:2019, Quantities and units — Light and radiation (BIBLIOGRAPHIC/PAYWALL).
2. [2]OpenStax, University Physics Volume 3, section 1.1, The Propagation of Light.
3. [3]BIPM, The International System of Units (SI Brochure), 9th ed., version 3.01, coherent derived units and quantity equations.
4. [4]NIST Special Publication 811, Guide for the Use of the International System of Units, 2008 edition.
5. [5]NIST, 2022 CODATA speed of light in vacuum.