How to Calculate the Angle of Refraction
When light passes from one transparent medium into another, it changes speed. This speed change causes the ray to bend at the boundary between the two media. The relationship between the incident angle and the refracted angle is given by Snell's Law:
n1 sin(θ1) = n2 sin(θ2)
Where n1 is the refractive index of the first medium, n2 is the refractive index of the second medium, theta1 is the angle of incidence (measured from the normal to the surface), and theta2 is the angle of refraction (also measured from the normal). Rearranging to solve for the angle of refraction:
θ2 = arcsin( n1 × sin(θ1) / n2 )
Worked Example
Light travels from air (n1 = 1.000) and strikes a water surface at an angle of incidence of 30 degrees (measured from the normal).
- n1 x sin(30 deg) = 1.000 x 0.5000 = 0.5000
- sin(theta2) = 0.5000 / 1.333 = 0.3751
- theta2 = arcsin(0.3751) = 22.03 degrees
The refracted ray travels at 22.03 degrees from the normal inside the water, bending towards the normal because water is optically denser than air (n2 > n1).
Refractive Indices of Common Materials
| Material | Refractive Index (n) | Notes |
|---|
| Vacuum | 1.000 (exact) | Speed of light is exactly c |
| Air (at STP) | 1.000293 | Treated as 1.000 for most calculations |
| Water at 20 degrees C | 1.333 | At 589 nm (sodium D-line) |
| Acrylic (PMMA) | 1.490 | Common in lenses and displays |
| Crown Glass | 1.520 | Standard optical glass |
| Flint Glass | 1.620 | Higher dispersion than crown glass |
| Diamond | 2.417 | High n causes strong sparkle (TIR inside gem) |
Total Internal Reflection and the Critical Angle
When light travels from a denser medium (higher n) to a less dense medium (lower n), there is a maximum angle of incidence beyond which no refracted ray exists and all light is reflected back into the denser medium. This is total internal reflection (TIR). The critical angle is given by:
θc = arcsin( n2 / n1 ) (only defined when n1 > n2)
For example, for glass (n = 1.520) to air (n = 1.000), the critical angle is arcsin(1/1.520) = 41.1 degrees. Any ray inside the glass hitting the surface at more than 41.1 degrees from the normal will be totally reflected. This principle is fundamental to the operation of optical fibres, prism binoculars, retroreflectors, and diamond cutting.
Angles Are Always Measured From the Normal
A common source of error is measuring angles from the surface rather than from the normal. The normal is an imaginary line perpendicular to the boundary surface at the point of incidence. All angles in Snell's Law must be measured from this normal line. An angle of incidence of 0 degrees means the ray hits the surface dead-on (perpendicular) and passes straight through with no bending, regardless of the refractive indices.
Dispersion: Refractive Index Varies with Wavelength
The refractive index of most materials is not exactly constant. It varies slightly with the wavelength (colour) of light, a phenomenon called dispersion. This is why a glass prism splits white light into a rainbow. The refractive indices listed in this calculator are standard values for the sodium D-line at 589 nm (yellow-orange). For precise optical design work, use wavelength-specific values from material datasheets.
Related Calculators
Sources and method: Snell's Law (Willebrord Snellius, 1621; published by Rene Descartes, 1637). Refractive index values from CRC Handbook of Chemistry and Physics (standard sodium D-line, 589 nm). Critical angle derived from Snell's Law with theta2 = 90 degrees. This calculator applies to electromagnetic radiation (light) in isotropic, homogeneous media. Results assume monochromatic light at 589 nm unless otherwise noted.
This calculator provides results based on the standard Snell's Law formula. Real optical systems may be affected by surface coatings, material impurities, wavelength-dependent dispersion, temperature, and polarisation. For precision optical engineering, use manufacturer-supplied refractive index data and ray-tracing software.
