Index of Refraction Calculator

What the index of refraction measures

Every clear material has a number that says how much it slows light down. That number is the index of refraction, written n, and the formula fits on one line: n = c / v. Here c is the speed of light in a vacuum, 299,792,458 m/s, and v is the speed inside the material.

A vacuum gets n = 1 because light moves at full speed there. Air comes in at 1.000293, water at 1.3330 and diamond at 2.417. A larger number means a slower beam and a sharper bend when the light crosses a surface.

Worked example. Light travels through water at 224,900,569 m/s. Divide the vacuum speed by that: 299,792,458 / 224,900,569 = 1.3330. So water's index is 1.3330, which means light moves 1.333 times slower in water than in a vacuum, or at 75.0 percent of the vacuum speed.

The number carries no units, since it is one speed divided by another, and it doubles as a ratio: light moves 1.333 times slower and its wavelength shrinks by the same factor.

One catch: n drifts with the color of the light. Labs quote it at a stated wavelength, and the usual choice is the yellow sodium D line at 589.3 nm, measured at 20 C. The NIST reference entry for liquid water gives 1.3330 under those conditions, and that is the figure used on this page.

Light speed and index in everyday materials

The table below puts five materials side by side. The speed column comes straight from n = c / v, so it costs nothing extra to fill in once the index is known.

Air sits so close to 1 that a beam crossing a room barely bends: the index differs from vacuum by 0.03 percent. Water and glass are where the effect turns visible, and diamond is in a class of its own, at 41.4 percent of the vacuum speed.

The glass figure is a typical value for crown glass, the ordinary optical glass used in windows and simple lenses. Different recipes land between about 1.50 and 1.54. The HyperPhysics index table from Georgia State University lists 1.52 for typical crown glass and 2.417 for diamond, both measured at 589 nm.

The vacuum speed is not something you look up: the metre is defined by fixing c at exactly 299,792,458 m/s, as the BIPM definition of the metre states. Every speed in the table is that exact number divided by a measured index, so the index carries all of the uncertainty.

Snell's law: why a straw looks broken

Light changes direction at a boundary because it changes speed. The rule that predicts the new direction is Snell's law: n1 sin(theta1) = n2 sin(theta2). Both angles are measured from the normal, the line perpendicular to the surface, not from the surface itself.

Worked example. A laser in air hits water at 40 degrees from the normal. The air index is 1.0003, so n1 sin(theta1) = 1.0003 x 0.6428 = 0.6430. Divide by water's 1.3330 and sin(theta2) = 0.4824. The inverse sine gives 28.8 degrees, so the beam leaves the surface steeper than it arrived.

A beam at 10 degrees barely moves, one at 60 degrees loses almost 20 degrees, and one at 80 degrees comes out at 47.6 degrees. The change grows as the beam gets closer to lying flat on the surface.

The direction rule is easy to keep straight. Going into the material with the higher index, the ray bends toward the normal. Coming out into the lower index, it bends away. A beam that arrives at 0 degrees, straight down the normal, does not bend at all, which is why a light aimed straight down into a pool still looks straight.

Total internal reflection and the critical angle

Turn the beam around. Light inside water heading for the surface speeds up as it leaves, so it bends away from the normal. Past a certain starting angle the math asks for a sine above 1, which has no solution, and the light cannot escape at all. It reflects back into the water as if the surface were a mirror.

That cutoff is the critical angle, found by setting the outgoing angle to 90 degrees: sin(thetac) = n2 / n1, with n1 the material the light starts in and n2 the material outside. The formula only works when n1 is the larger of the two. Light going from air into water never hits total internal reflection, no matter how flat the angle.

Worked example. Water to air: sin(thetac) = 1.0003 / 1.3330 = 0.7504, and the inverse sine is 48.6 degrees. Diamond to air: 1.0003 / 2.417 = 0.4139, which gives 24.4 degrees.

Diamond's small critical angle is the reason a cut stone sparkles. Light that enters through the top facet strikes the inside walls at angles well past 24.4 degrees, bounces several times, and leaves through the top toward your eye instead of leaking out the sides.

Fiber optic cables use the same trick with glass: light launched down the core meets the cladding past the critical angle and stays inside for miles. A swimmer looking up sees a bright circle of sky roughly 97 degrees wide, with mirror-like water outside it.

Pools, glasses, and windows: the shallow water effect

Stand at the edge of a pool and the bottom looks closer than it is. Light from the bottom bends away from the normal as it leaves the water, and your eye traces the ray back along a straight line that meets the surface at a nearer point. The result is an apparent depth you can work out with one division: apparent depth = real depth / n.

Worked example. A pool 2.4 m deep: 2.4 / 1.3330 = 1.80 m. The bottom looks 1.80 m down, about 60 cm closer than the true depth. In feet, an 8 ft pool looks 6.0 ft deep. The same shift is why a spearfisher aims below the fish rather than straight at it.

That simple division only holds when you look straight down. At an angle the shift depends on both angles, and the bottom appears to tilt as you move your head. The depth mode in the calculator uses the straight-down case and says so in the result.

Eyeglasses use the same rule at two curved surfaces: a material with n near 1.5 bends each ray, and the curve sets how strongly the rays converge. A higher index does the same job with a thinner lens, which is why strong prescriptions use high-index plastic.

Even a flat window pane shifts what you see: a 6 mm pane at 45 degrees moves the image sideways by about 2 mm.

Dispersion: why a prism splits white light

The index is not one fixed number per material. It rises as the wavelength gets shorter, a behavior called dispersion. Blue light near 450 nm sees a slightly higher index than red light near 650 nm, so blue bends more at every boundary it crosses.

That is what a prism does: white light enters, each color takes a slightly different path, and the colors fan out. Raindrops do the same job in the sky with one internal reflection added, which is how a rainbow forms.

Diamond's quoted 2.417 belongs to the 589 nm line. Across the visible range the value moves by a few percent, and that spread is what a jeweler calls fire: the stone throws different colors back at slightly different angles.

For everyday work the practical point is short. If a number matters to four decimals, use the index at the wavelength you care about, not a generic table value. A lens designed for infrared light will not focus red light at the same point, and that mismatch is called chromatic aberration. Camera lenses fix it by stacking glass elements with different dispersion.

Using the calculator and what it assumes

The calculator has three modes. Index from speed takes the speed of light inside a material and returns n. Snell's law takes two indexes and one angle and returns the angle on the other side, plus the critical angle when light is trying to leave the denser material. Apparent depth takes an index and a real depth and returns how deep it looks.

Every mode also reports the speed of light in the material and its share of c, so you can check the index against the table above.

Four assumptions sit behind the numbers. The boundary is flat and the two materials are uniform, so a curved lens surface or a mixture will not match. The light is a single wavelength, because n changes with color. The temperature is 20 C, which is where the standard water and glass values are quoted. Nothing absorbs or scatters the beam, which rules out foggy water and tinted glass.

Within those limits the math is exact: n = c / v and Snell's law are definitions and geometry, not fits to data.