Index of Refraction Calculator
Enter two media and an angle to solve Snell's law instantly. Choose a solve mode to find the refraction angle, incidence angle, refractive index of either medium, or the critical angle for total internal reflection. Select from 18 material presets or enter a custom refractive index. The phase velocity of light in each medium is shown alongside the step-by-step working.
Formula
Worked example
A ray of light travels from air (n1 = 1.000293) and strikes the surface of crown glass (n2 = 1.52) at a 30-degree angle of incidence. sin(30 deg) = 0.5000. sin(theta2) = (1.000293 / 1.52) * 0.5000 = 0.3290. theta2 = arcsin(0.3290) = 19.21 degrees. The ray bends toward the normal because it moves into a denser medium. Since n1 < n2 there is no critical angle in this direction.
What is the index of refraction and Snell's law?
The index of refraction (refractive index), written as n, measures how much slower light travels through a material compared with a vacuum. It is defined as n = c / v, where c is the speed of light in vacuum (about 299,792,458 m/s) and v is the phase velocity in the material. A higher n means the material is optically denser and light slows down more. Air has n = 1.000293 (nearly 1), water has n = 1.333, and diamond has n = 2.417, so light moves about 2.4 times faster in vacuum than in diamond. Snell's law links the refractive indices of two media to the angles a light ray makes with the normal at the boundary: n1 * sin(theta1) = n2 * sin(theta2). When light crosses into a denser medium (n2 > n1) it bends toward the normal; when it crosses into a less dense medium it bends away.
How to use this calculator
Select what you want to solve for using the 'Solve for' dropdown: the angle of refraction (theta2), the angle of incidence (theta1), or either refractive index (n1 or n2). Pick a preset material from the medium dropdowns or choose Custom and type in a refractive index directly. Enter the known angle(s) in degrees. The result updates instantly, including the critical angle for total internal reflection when n1 > n2, the phase velocity of light in both media, and a step-by-step breakdown of the arithmetic. The chart shows how the refraction angle changes across all possible incidence angles for the chosen material pair, with the total internal reflection zone highlighted if it exists.
Critical angle and total internal reflection
When light travels from a denser medium into a less dense medium (n1 > n2), there is a maximum incidence angle called the critical angle, beyond which no refracted ray can exit. At this angle sin(theta2) would equal 1, giving theta_c = arcsin(n2 / n1). For incidence angles larger than theta_c, all light is reflected back into the first medium, a phenomenon called total internal reflection (TIR). This is the operating principle of optical fibers: a glass core (n ~ 1.46) is surrounded by a cladding with a slightly lower refractive index, so any light ray that enters within the acceptance cone undergoes TIR and travels the length of the fiber with minimal loss. Diamonds are also cut to exploit TIR, with facet angles chosen so most light entering the top is totally reflected internally before exiting toward the viewer.
Dispersion: why n depends on wavelength
The refractive index values in this calculator and in the reference table are measured at 589 nm, the standard yellow sodium D-line. In reality, n varies with wavelength, a property called dispersion. Glass has a slightly higher n for violet light than for red light, which is why a prism separates white light into a spectrum. Dispersion is quantified by the Abbe number V = (n_D - 1) / (n_F - n_C), where n_D, n_F and n_C are the refractive indices at 589 nm, 486 nm, and 656 nm. Low Abbe numbers indicate high dispersion (flint glasses, V around 30-40); high Abbe numbers indicate low dispersion (crown glasses, V around 60-70). The Cauchy and Sellmeier equations are empirical models that describe how n varies continuously with wavelength for a given material.
Index of refraction for common materials
| Material | Phase state | Refractive index (n) | Speed of light (m/s) |
|---|---|---|---|
| Vacuum | N/A | 1 | 299792458 |
| Air | Gas | 1.000293 | 299704644 |
| Water Ice | Solid | 1.31 | 228848441 |
| Water (20 C) | Liquid | 1.333 | 224870558 |
| Acetone | Liquid | 1.36 | 220435631 |
| Ethanol | Liquid | 1.361 | 220273665 |
| Kerosene | Liquid | 1.39 | 215679466 |
| Corn Oil | Liquid | 1.47 | 203940447 |
| Glycerol | Liquid | 1.473 | 203524411 |
| Acrylic (PMMA) | Solid | 1.491 | 201066036 |
| Crown Glass | Solid | 1.52 | 197231880 |
| Plate Glass | Solid | 1.52 | 197231880 |
| Sodium Chloride | Solid | 1.544 | 194164676 |
| Polycarbonate | Solid | 1.6 | 187370286 |
| Flint Glass | Solid | 1.62 | 185056455 |
| Sapphire | Solid | 1.77 | 169372572 |
| Cubic Zirconia | Solid | 2.17 | 138123483 |
| Diamond | Solid | 2.417 | 124015088 |
| Silicon | Solid | 3.45 | 86896362 |
| Germanium | Solid | 4.05 | 74022828 |
Values measured for yellow sodium light at 589 nm (standard D-line). Values vary with wavelength.
Frequently asked questions
What is the index of refraction?
The index of refraction (refractive index) of a material is the ratio of the speed of light in vacuum (c) to the speed of light in that material (v): n = c / v. Because light can only slow down inside a material (not speed up), n is always 1 or greater. Vacuum has n = 1 exactly; air has n = 1.000293, essentially 1 for most purposes; water has n = 1.333; and diamond has n = 2.417. A higher n means the material is optically denser.
What is Snell's law?
Snell's law describes how light bends (refracts) when it crosses the boundary between two materials with different refractive indices. The formula is n1 * sin(theta1) = n2 * sin(theta2), where n1 and n2 are the refractive indices, and theta1 and theta2 are the angles the ray makes with the normal to the surface in each medium. All four values are linked: if you know any three, you can solve for the fourth.
What is the critical angle?
The critical angle is the angle of incidence (measured from the normal) at which the refracted ray just grazes along the boundary (refraction angle = 90 degrees). It only exists when light travels from a denser medium to a less dense medium (n1 > n2), and equals arcsin(n2 / n1). Any incidence angle greater than the critical angle results in total internal reflection: no light passes into the second medium.
What is total internal reflection and where is it used?
Total internal reflection (TIR) occurs when a light ray travelling in a denser medium hits a boundary at an angle greater than the critical angle. All light is reflected back rather than transmitted. TIR is the fundamental principle of optical fiber communications, where glass or plastic fibers guide light over long distances with very low loss. It also produces the sparkle of gemstones, the bright appearance of road reflectors, and the operation of periscope prisms.
Why does light bend when it changes medium?
Light bends because its phase velocity changes at the boundary. Imagine a column of soldiers marching in formation at an angle toward a muddy field where they must slow down. The soldiers entering the mud first slow down while those still on hard ground continue at speed, causing the column to pivot - the direction changes. The same geometry applies to the wavefronts of light: when one side of a wavefront slows (in the denser medium) while the other side is still in the first medium, the wavefront tilts and the beam direction changes. Snell's law is the quantitative expression of this geometry.
Does the refractive index depend on the wavelength of light?
Yes. This property is called dispersion. In most transparent materials, shorter wavelengths (violet, blue) are slowed more than longer wavelengths (red), giving them a higher refractive index. The values in this calculator are for yellow sodium light at 589 nm, the conventional reference wavelength. For applications where wavelength accuracy matters (lens design, fiber optics, spectroscopy), you must look up or measure n at the specific wavelength you are working with, using the Sellmeier or Cauchy dispersion equations for the material.
How is the refractive index measured experimentally?
Several techniques exist depending on precision required. The simplest is Snell's law itself: measure the incidence and refraction angles at an air-glass interface with a protractor and solve for n. Higher-precision methods include minimum-deviation through a prism (measuring the angle where a prism bends a monochromatic beam least), interferometry (counting fringes as a path length changes), and ellipsometry (measuring how polarization state changes on reflection). Critical-angle refractometers such as the Abbe refractometer measure n directly by finding the angle at which light undergoes total internal reflection off the unknown sample.