Index of Refraction Calculator

Index of Refraction Calculator that finds n from light speed, wavelengths or measured ray angles, solves Snell’s law and Cauchy glass dispersion, and shows critical angle and loss.

Index of Refraction Calculator Modes

The Index of Refraction Calculator has ten modes in the Solve For menu. Each mode matches a different thing you can measure, from the speed of light inside a material to the angles of a ray you traced on paper in a lab.

Whatever you solve, the same results panel follows. It shows how fast light moves inside, the critical angle, how much light bounces off the surface and how the wavelength shrinks, plus a ray diagram, worked steps and a material list.

Light Speed and Wavelength Modes

Use “n from speed” when you know how fast light travels in the material. Type the speed in m/s, km/s, km/h, mi/s, mph or ft/s, and the tool divides the exact vacuum speed by it. The reverse mode, “Speed in medium”, takes an index or a preset material and returns the speed.

The wavelength mode is for optics problems that give the same light wave in two places. Enter the vacuum wavelength and the wavelength inside the material in nm, µm or Å. Since frequency does not change at a boundary, the ratio of the two wavelengths equals n.

Snell’s Law Modes

Four modes of the Index of Refraction Calculator use Snell’s law. You can solve for n₂, n₁, the angle of refraction θ₂ or the angle of incidence θ₁. Angles go in degrees or radians and are always measured from the normal, the line square to the surface, not from the surface itself.

The surrounding medium sits in its own drop-down with presets from vacuum to moissanite, including polycarbonate and polystyrene, or a custom value. In the other modes it moves into the Surrounding Medium accordion, set to air by default.

Depth, Critical Angle and Prism Modes

Three modes cover the classic lab methods. Enter a real and an apparent depth in mm, cm, m, in or ft, a critical angle, or a prism’s apex angle with its minimum deviation. Angles go in degrees or radians, and the step list shows every line of the working.

These methods measure the index relative to whatever surrounds the sample. The tool multiplies that relative value by the surrounding medium, air by default, and the hero line shows both. In a vacuum setting the two are the same number.

Cauchy Mode for Glass

The Cauchy mode in the Index of Refraction Calculator gives the index of a glass at any visible wavelength. Pick fused silica, BK7, K5, BaK4, BaF10 or SF10 to load its A and B coefficients, or choose Custom and type your own. The hero line also reports the glass’s Abbe number and how fast n drops per micrometer of wavelength.

Index of Refraction Formulas

School books usually teach two formulas for refractive index, the speed ratio and Snell’s law. The NCERT Class 12 ray optics chapter adds real and apparent depth and the critical angle, which gives the “three formulas” students search for. Cauchy’s formula covers how n changes with color.

Speed Ratio: n = c / v

The absolute index of refraction compares the speed of light in vacuum with its speed in the material. The calculator uses c = 299,792,458 m/s, the exact value fixed by the SI defining constants.

$$n = \frac{c}{v}$$

For a speed of 225,000 km/s, n = 299,792.458 ÷ 225,000 = 1.3324, which is close to water. Because n is one speed divided by another, the units cancel. The same rule written with wavelengths is n = λ₀ ÷ λ, so 589 nm in vacuum and 442 nm inside gives n = 1.3326.

$$n = \frac{\lambda_0}{\lambda}$$

Snell’s Law and the Relative Index

Snell’s law links the angles on both sides of a boundary. Here n₁ and θ₁ belong to the medium the light comes from, and n₂ and θ₂ to the medium it enters. Both angles are measured from the normal.

$$n_1 \sin\theta_1 = n_2 \sin\theta_2$$

The ratio of the two sines is the relative index of medium 2 with respect to medium 1, written ₁n₂ in Indian textbooks. It equals n₂ ÷ n₁, so you only get the absolute index when medium 1 is vacuum or, very nearly, air.

$${}_1n_2 = \frac{n_2}{n_1} = \frac{\sin\theta_1}{\sin\theta_2}$$

Real and Apparent Depth

Looking straight down into water, the bottom seems closer than it is. For near-vertical viewing, the index of the liquid equals the real depth divided by the apparent depth. A coin under 12 cm of water that looks 9 cm deep gives 12 ÷ 9 = 1.3333 relative to air. Times 1.000293 for air, the absolute index is 1.3337.

$$n = \frac{\text{real depth}}{\text{apparent depth}}$$

Critical Angle Form

Light going from a dense material into air is trapped once it hits the surface past the critical angle C. Against vacuum or air, the index follows directly from that angle. A stone with C = 24.4° gives 1 ÷ sin 24.4° = 2.4207 relative to air, or 2.4214 absolute, which points to diamond.

$$n = \frac{1}{\sin C}$$

Prism Minimum Deviation

A prism turns a ray least when the ray passes through it symmetrically. That smallest turn, the minimum deviation Dₘ, together with the apex angle A gives the index. It is the standard spectrometer method in school and university labs.

$$n = \frac{\sin\left(\frac{A + D_m}{2}\right)}{\sin\left(\frac{A}{2}\right)}$$

A 60° prism with a minimum deviation of 37.2° gives sin 48.6° ÷ sin 30° = 0.750111 ÷ 0.5 = 1.5002 relative to air. That matches crown glass, and the absolute value is 1.5007.

Cauchy’s Formula

Cauchy’s formula is an empirical fit for how a transparent material’s index falls as wavelength grows. The two-term version works well for glass across the visible band, with λ in micrometers and B in µm².

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

For BK7 with A = 1.5046 and B = 0.00420 µm², blue light at 486.1 nm gives n = 1.5046 + 0.0042 ÷ 0.2363 = 1.5224. Red light at 656.3 nm gives 1.5144. That gap of 0.0080 is why a prism spreads white light into colors.

How to Calculate Index of Refraction With Angles

Say a laser in air hits a plastic block at 45° and the refracted beam inside measures 28.0° from the normal. Open the Index of Refraction Calculator, choose “n₂ with Snell’s law”, leave n₁ at Air (1.000293), and type θ₁ = 45 and θ₂ = 28. Then press Calculate n₂.

The Step-by-Step Solution accordion lays the work out line by line. The relative index is sin 45° ÷ sin 28° = 0.707107 ÷ 0.469472 = 1.506172. Multiplied by 1.000293, that gives n₂ = 1.5066, and the hero line marks it as a match for crown glass.

To find the angle of refraction instead, switch to “Angle of refraction (θ₂)”. Light from air into water at 30° gives sin θ₂ = 1.000293 × 0.5 ÷ 1.333 = 0.3752, so θ₂ = 22.04°. The hero also shows the ray turns 7.96° toward the normal.

If the light starts in the denser medium, sin θ₂ can come out above 1. The tool then reports total internal reflection instead of an angle, and the ray diagram draws the full reflected ray with no refracted ray below the boundary.

What the Result Cards Mean

All four cards use the index you solved for and the surrounding medium. The examples below assume water at 20 °C in air, so you can compare them with your own run of the Index of Refraction Calculator.

Speed Inside the Material

The first card shows light’s speed as a share of c. In water it is 75.02% of c, light needs 4.446 ns to cross 1 m, and it falls 1.111 µs behind a vacuum beam over each kilometer. In fused silica the delay grows to 1.529 µs per kilometer.

Critical Angle and Escape Cone

The second card gives the critical angle, 48.63° for water against air, and the direction in which total internal reflection can happen. A fish looking up sees the whole sky squeezed into a cone about 97° wide, twice that angle, known as Snell’s window.

The escape cone row shows what share of light from a source inside the material heads into that cone at one flat surface. For water it is 16.95%. For diamond it falls to 4.48%, which is why a well-cut diamond holds light in and sends it back out through the top.

Reflection Head-On

The third card uses the Fresnel equation for light hitting the surface straight on. Water in air reflects 2.04%, window glass about 4.3% and diamond 17.2%. The row below it adds the loss across both faces of a flat window, counting light that bounces between them. For uncoated window glass that is 8.24%.

$$R_0 = \left(\frac{n_2 – n_1}{n_2 + n_1}\right)^2$$

The same card lists the Brewster angle, where reflected light is fully polarized. It is 53.12° for water and 56.65° for glass with n = 1.52. The Brewster’s Angle Calculator goes further with polarizer setups and reflectance at any angle.

Wavelength Inside

The fourth card divides the vacuum wavelength by n. Yellow sodium light at 589.3 nm becomes 442.1 nm inside water, while its frequency stays at 508.7 THz. In the wavelength mode it shows how much shorter the wave gets inside, and in the Cauchy mode it uses your own wavelength. Both add photon energy in eV.

If you only know the color of a laser from its diffraction pattern, the Diffraction Grating Calculator turns the order angle into a vacuum wavelength. That number then goes into the wavelength or Cauchy mode here.

Index of Refraction List

The list inside the Index of Refraction Calculator highlights the material closest to your result. The table here gives measured values at the 589 nm sodium line, with the speed share and the two figures that matter most at a boundary with air. For birefringent crystals with two values, the speed, reflection and angle come from the lower one.

The values come from the RefractiveIndex.INFO database, which traces each one to a published measurement. Water uses Daimon and Masumura, fused silica uses Malitson, the two glasses use the SCHOTT catalog data, and moissanite uses the pair of indices in GIA’s study of synthetic moissanite.

Sapphire, rutile and moissanite split light into two rays with different indices, so each shows a range. The gap is small for sapphire but large for rutile, which is why facet edges look doubled through a rutile crystal. Glass, water and diamond have one index in every direction.

Index of Refraction of Air

The air preset in the Index of Refraction Calculator is 1.000293, the standard figure for dry air at 0 °C and 1 atm. At 15 °C the Ciddor equation gives 1.000277, because warmer air at the same pressure is thinner. Choose Custom and type that value for room conditions.

For most lab problems, air and vacuum give results that differ only in the fourth decimal of an angle. Over long paths the small index matters. Air near the ground bends light enough to lift distant objects into view, the effect behind the refraction setting in the Distance to Horizon Calculator.

The same bending changes how much of a far ship or building hides below the curve. The Earth Curvature Calculator handles that drop with and without refraction, using an effective Earth radius instead of an index.

IOR Values for 3D Rendering

IOR is simply short for index of refraction. Physically based render engines, including Blender’s Principled BSDF, take it as an input and use it for two things, how much a clear material bends light and how strongly any material reflects at a straight-on view.

To calculate IOR from a reflectance you already have, run the Fresnel formula above backwards. Going forward, enter the IOR in the “Speed in medium” mode of the Index of Refraction Calculator, and the Reflected Head-On card gives the matching F0 value that many shader guides ask for.

The plastics come from Sultanova and co-workers, the skin layers from Ding et al., and blood from Liu et al. Ceramic has no single IOR. Alumina ceramic is the same compound as sapphire, while a glossy glaze reflects like the glass layer it is.

Gold needs two numbers. In Johnson and Christy’s data, its real index is only 0.27 at 589 nm and the extinction part is 2.95. Light barely enters the metal, so about 89% reflects. Metal shaders take n and k, not a single IOR.

Glass Dispersion and Cauchy Coefficients

Optical glass makers give their index at the helium d line, 587.6 nm, and an Abbe number that rates color spread. A high Abbe number means low dispersion. The A and B below are fitted to the maker’s dispersion data from 400 to 700 nm, so the Cauchy mode lands on the catalog index.

The index and Abbe columns are from the SCHOTT optical glass data sheets, and fused silica is from Malitson. The two-term fit stays within 0.0002 of the catalog for silica, BK7, K5 and BaK4, and within 0.0005 for BaF10. SF10 disperses more steeply, so its fit can drift by up to 0.0025 at the ends of the visible band.

BaK4 is the barium crown used in many binocular prisms. Its higher index gives a smaller critical angle than BK7, about 39.6° against 41.3°, so more off-axis light stays trapped in the prism. That keeps the exit pupil round, which helps when ranging with the Binoculars Range Calculator and a mil reticle.

Index also sets lens strength. A higher-index glass bends light more at each surface, so a lens reaches the same power with flatter curves. The Diopter Calculator converts that power into focal length for glasses and simple lenses.

Common Refractive Index Questions

These answers cover what students and lab users ask most about refractive index. More tools for lenses, light and waves are on the Optics Calculators hub, next to the Index of Refraction Calculator.

What is the unit of index of refraction?

Index of refraction has no unit. It is a ratio of two speeds, or of two wavelengths, so the units cancel. A value of 1.5 means the same thing in any unit system, and you can enter speeds in mph or km/s in the calculator and still get the same n.

What does a refractive index of 1.33 mean?

An index of 1.33 means light travels at 1 ÷ 1.33, or about 75% of its vacuum speed, roughly 225,000 km/s. It is the value for water. Light entering it from air bends toward the normal, and light leaving it is trapped past a critical angle of about 48.6°.

What does a refractive index of 2.42 mean?

A refractive index near 2.42 is diamond, measured at 2.417 at 589 nm. Light moves at about 41% of c, near 124,000 km/s. The critical angle is only 24.4°, and each surface reflects 17.2% head-on, which together give diamond its brilliance.

Which stone has the highest refractive index?

Among the gem materials in the list, rutile is highest, with 2.909 on its extraordinary ray. Moissanite follows at 2.648 to 2.691, then diamond at 2.417. Diamond still sits well above corundum, the mineral of ruby and sapphire, at about 1.76 to 1.77.

Diamond is almost pure carbon, but most natural stones carry traces of nitrogen, which gemologists use to sort diamond types. Glass and plastics above about 1.6 are usually called high-index materials, while most clear liquids and plastics fall between 1.33 and 1.6.

Whose refractive index is 3:2?

An index of 3:2, or 1.5, is the textbook value for ordinary glass. Light then moves at two thirds of c, about 199,862 km/s. Its critical angle against vacuum is 41.8°, and real crown glasses sit close by, such as BK7 at 1.5168 and window glass at 1.5233.

What is 1n2 in refractive index?

₁n₂ is the refractive index of medium 2 relative to medium 1. It equals n₂ ÷ n₁, or sin i ÷ sin r for light going from 1 into 2. For glass with n = 1.5 and water with n = 1.33, the index of glass relative to water is 1.5 ÷ 1.33 = 1.13. The Snell’s law step list prints this ratio as its own line.

What are the first and second laws of refraction?

The first law says the incident ray, the refracted ray and the normal at the point of incidence all lie in one plane. The second law is Snell’s law, which says sin i ÷ sin r stays constant for a given pair of media and a given color of light.

What are the two types of refraction?

Light either bends toward the normal, when it enters a slower, optically denser medium, or away from it, when it enters a faster one. Crystals such as rutile and calcite also show double refraction, where one ray splits into two with different indices.

How is refractive index measured?

Labs mostly use a refractometer, which finds the critical angle where light stops passing through a liquid or solid against a glass prism. Students often use a glass slab and pins with Snell’s law, a travelling microscope for real and apparent depth, or a prism on a spectrometer at minimum deviation. Each of those readings has its own mode in the calculator.

What causes the index of refraction?

Light’s electric field makes the electrons in a material oscillate. They send out their own small waves, which combine with the original wave and delay its phase. Denser, more polarizable materials delay it more, which shows up as a higher index and a slower wave.

Can a refractive index be less than 1?

Yes, for the phase index. X-rays in most solids have n just below 1, which is why X-ray mirrors only work at grazing angles. Crystal diffraction, worked out in the Bragg’s Law Calculator, is the usual way to steer them. Gold at 589 nm has a real index near 0.27. No signal travels faster than c in either case.