Angle of Refraction Calculator (Snell's Law)
Enter the refractive indices of two media and the angle of incidence to find the angle of refraction via Snell's law. The calculator also finds the critical angle (when light travels from a denser to a less dense medium), the Brewster angle, and flags total internal reflection. Pick a material from the presets or type your own refractive index. Results update instantly as you type.
What is the angle of refraction?
When a light ray (or any wave) crosses the boundary between two media with different refractive indices, it changes direction. The angle of refraction is the angle between the transmitted ray and the normal (the perpendicular) to the surface on the far side. A ray arriving at 45 deg to the air-glass boundary will leave at about 28 deg - it bends toward the normal because glass is denser than air. The degree of bending depends entirely on the two refractive indices and the incoming angle, which is what Snell's law quantifies.
Snell's law - the formula
Snell's law states that n1 * sin(theta1) = n2 * sin(theta2), where n1 and n2 are the refractive indices of the first and second media, theta1 is the angle of incidence, and theta2 is the angle of refraction. To find the refracted angle you rearrange to theta2 = arcsin((n1 / n2) * sin(theta1)). The refractive index itself is the ratio of the speed of light in vacuum (c) to the speed in the medium (v), so n = c / v. A higher n means slower light and more bending.
Critical angle and total internal reflection
When light travels from a denser medium into a less dense one (n1 > n2), the refracted ray bends away from the normal. At a specific incidence angle called the critical angle, the refracted ray would graze along the surface at exactly 90 deg. Beyond the critical angle, no refracted ray can exist and all light reflects back: this is total internal reflection (TIR). The critical angle is theta_c = arcsin(n2 / n1). Optical fibres exploit TIR to guide light over long distances without loss - light bounces along the fibre core because the core has a slightly higher index than the cladding around it. This calculator highlights when TIR will occur and shows the critical angle whenever n1 > n2.
Brewster angle and polarization
At one particular angle of incidence called the Brewster angle (theta_B = arctan(n2 / n1)), the reflected beam is completely polarized: only the component whose electric field oscillates perpendicular to the plane of incidence reflects at all. The transmitted beam is partially polarized. Photographers use polarizing filters to cut reflections at Brewster's angle from water and glass surfaces. The Brewster angle always lies between 0 and 90 deg regardless of the media, so TIR and Brewster conditions never coincide at the same angle.
Refractive indices of common media (visible light, ~589 nm)
| Medium | Refractive index (n) | Notes |
|---|---|---|
| Vacuum | 1.000 000 | Exact, by definition |
| Air (standard) | 1.000 293 | Negligibly different from vacuum in most problems |
| Water (20 deg C) | 1.333 | Commonly rounded to 4/3 |
| Ice (0 deg C) | 1.309 | Slightly less than liquid water |
| Ethanol | 1.362 | Ethyl alcohol at 20 deg C |
| Fused silica | 1.458 | Pure SiO2 glass |
| Acrylic / PMMA | 1.490 | Plexiglass, Perspex |
| Crown glass | 1.520 | Common optical glass |
| Flint glass | 1.620 | Higher-dispersion optical glass |
| Diamond | 2.417 | Highest of common transparent solids |
Values at room temperature for the sodium D line (589 nm). Use these to fill n1 or n2.
Frequently asked questions
What is Snell's law?
Snell's law (also called the law of refraction) relates the angles of incidence and refraction to the refractive indices of the two media: n1 * sin(theta1) = n2 * sin(theta2). It was independently described by Willebrord Snell in 1621 and Rene Descartes in 1637. The law follows from the wave nature of light: when a wavefront crosses a boundary at an angle, the part that enters the new medium first changes speed, which tilts the whole wavefront and changes the ray direction.
Can the angle of refraction ever be 90 degrees?
Theoretically yes - if the incidence angle exactly equals the critical angle, the refracted ray emerges at 90 deg (parallel to the surface). In practice, at exactly the critical angle the refracted intensity drops to zero and total internal reflection takes over, so a 90-degree refracted ray is more of a mathematical limit than a useful beam.
Why does light bend toward the normal when entering glass from air?
Light travels slower in glass (n ~ 1.52) than in air (n ~ 1.0003). When the wavefront hits the boundary at an angle, the leading edge slows down while the trailing edge still moves at full air-speed, so the wavefront pivots toward the normal. The steeper the speed contrast (higher n2/n1 ratio), the more the ray bends.
What is total internal reflection and where is it used?
Total internal reflection occurs when light in a dense medium hits a less dense boundary at an angle greater than the critical angle. Because sin(theta2) would exceed 1, no real solution exists and all light reflects back. Optical fibres rely on TIR so that data-carrying light pulses stay inside the glass core over hundreds of kilometres. Diamonds are cut to exploit TIR and maximize the sparkle by reflecting light back up through the top facets rather than leaking it out through the sides.
How do I use material presets in this calculator?
Select a preset from the Medium 1 or Medium 2 dropdown. The calculator fills the corresponding refractive index field automatically. You can then override the number if you need a more precise or wavelength-specific value. The presets use standard values at ~589 nm (the sodium D line), which is the conventional reference wavelength for reporting refractive indices.
Does wavelength affect refraction?
Yes. The refractive index of most materials is slightly different for each wavelength of light, an effect called dispersion. Shorter (bluer) wavelengths typically have a higher index and bend more than longer (redder) wavelengths, which is why a prism splits white light into a spectrum. The values in this calculator are for 589 nm (yellow). For precise work, use the specific index at the wavelength you are working with.