Snell's Law Refraction Calculator

Why light bends at a boundary

Light changes direction at an interface because it changes speed. The refractive index n of a material is defined as n = c/v, the vacuum speed of light divided by its phase speed in that material, so water at n = 1.333 slows light to 2.249 × 10⁸ m/s and diamond at n = 2.417 slows it to 1.240 × 10⁸ m/s. Nothing exceeds c; the wave simply propagates more slowly through matter because the oscillating charges in the medium re-radiate with a phase lag.

The direction change follows from the wavefront staying continuous across the boundary. If one edge of a wavefront enters the slow medium before the other, that edge advances less in the same time, and the whole front pivots — the same reason a marching band wheels when the left file enters mud first. Working the geometry through gives n₁ sinθ₁ = n₂ sinθ₂.

Fermat's principle gives the same result from a completely different direction: of all possible paths between two points, light takes the one with stationary optical path length. Minimise the travel time across the boundary and Snell's law falls out as the condition. That equivalence — a local wave condition and a global extremum condition producing the same law — is one of the more elegant results in classical optics, and it is why Snell's law also describes seismic waves and sonar refracting through layered media.

The law is named for Willebrord Snellius, who found it in 1621, though Ibn Sahl described the same relationship in Baghdad in 984 and Descartes published it independently in 1637. In French-language work it is still la loi de Descartes.

The three consequences you can read straight off the formula

Direction of bend. Rearranged, sinθ₂ = (n₁/n₂) sinθ₁. When n₂ > n₁ — going into the denser medium — the ratio is less than one, so θ₂ < θ₁ and the ray bends towards the normal. Going the other way it bends away. This is why a straw in a glass looks broken at the surface and why a pool always looks shallower than it is.

Total internal reflection. Going from dense to rare, sinθ₂ can exceed 1, which no real angle satisfies. The largest incidence angle that still produces a refracted ray is the critical angle θ_c = arcsin(n₂/n₁). Above it, every photon reflects back, with no transmission loss at all — the only perfectly efficient mirror in optics. Optical fibres, prismatic binoculars and the sparkle of a brilliant-cut diamond all depend on it, and diamond's very small critical angle of 24.4° is precisely why so much light entering a diamond bounces around inside before leaving through the crown.

Brewster's angle. At θ_B = arctan(n₂/n₁) the reflected and refracted rays are exactly 90° apart, and the reflected light is completely polarised perpendicular to the plane of incidence. That is roughly 53° for air-to-water and 56° for air-to-glass, which is why polarised sunglasses cut glare off a lake or a windscreen so effectively. Brewster's angle always exists in both directions, unlike the critical angle.

One thing Snell's law does not tell you is how much light is transmitted versus reflected. That requires the Fresnel equations, which take the same angles as input and return reflectance for each polarisation. At normal incidence into glass roughly 4% reflects at each surface, which is why lens coatings exist.

Judging your answer and choosing the right index

Three quick checks catch nearly every error. First, if n₂ > n₁, the refracted angle must be smaller than the incident angle; if it is larger, you have the media the wrong way round. Second, the refracted angle can never exceed the critical angle in the dense-to-rare direction and never exceeds arcsin(n₁/n₂) in the rare-to-dense direction — at grazing incidence into water the refracted ray still only reaches 48.6°. Third, at 0° the answer must be 0°, whatever the indices.

Choosing the index deserves more care than most people give it. Refractive index depends on wavelength, a phenomenon called dispersion, and published values are usually quoted at the sodium D line, 589.3 nm. For BK7 crown glass the index runs from 1.5224 at 486.1 nm (blue) to 1.5143 at 656.3 nm (red) — a spread of 0.0081 that seems small but is exactly what splits white light into a spectrum through a prism and what causes chromatic aberration in a lens. If your application is broadband, one index is not enough; you need the Abbe number or a full Sellmeier fit.

Index also depends on temperature, pressure and, for solutions, concentration. That last dependence is the basis of the refractometer: measuring the critical angle of a sugar solution gives its Brix value directly, and the same instrument reads salinity, urine specific gravity and engine coolant strength. If you are reading an index off an instrument, check what wavelength and temperature it is calibrated for.

Refractive index, light speed and critical angle to air

Critical angles are arcsin(1/n) evaluated for each row; speeds are 299,792,458 ÷ n. Index values vary slightly between sources and with wavelength, temperature and glass melt.

Mistakes that produce a wrong refraction angle

• Measuring the angle from the surface instead of the normal. The two differ by 90°, and using the wrong one turns sines into cosines. A ray at 20° from the surface is at 70° from the normal.
• Swapping n₁ and n₂. The formula is not symmetric in its effect. Take the same glass at n = 1.500 both times: air-to-glass at 30° gives sinθ₂ = 0.5/1.5 = 0.3333 and so 19.47°, while glass-to-air at 30° gives sinθ₂ = 1.5 × 0.5 = 0.75 and so 48.59°. Always identify which medium the ray starts in.
• Expecting a refracted ray past the critical angle. Above θ_c there is no solution because the sine would exceed 1. That is a physical result, not a calculation failure.
• Using one index for white light. Dispersion means blue and red bend differently. For prism, rainbow or lens-design work you need the index at each wavelength.
• Assuming the frequency changes. Wavelength and speed both drop by a factor of n; frequency is fixed by the source and is continuous across the boundary.
• Forgetting the second surface. Light through a parallel-sided window refracts twice and emerges parallel to its original direction, merely displaced sideways. Only the net effect matters for a windowpane.
• Treating total internal reflection as lossless in a real fibre. The reflection itself is perfect, but the evanescent field extends into the cladding, so bends, contamination and absorption all cost signal.

Where Snell's law shows up beyond a glass block

Optical fibre is the largest commercial application. A step-index fibre has a core index a fraction of a percent above the cladding, so rays inside a narrow cone stay trapped by total internal reflection over kilometres. The acceptance cone is described by the numerical aperture, NA = √(n_core² − n_clad²), which is Snell's law applied at the input face.

Atmospheric refraction is the same law applied to a continuously varying index. Air density falls with altitude, so index falls too, and light from a low sun curves gently — enough that the Sun is visibly above the horizon when it is geometrically already below it, by roughly its own diameter at sunset. Mirages, the flattened setting Sun and the green flash are all refraction in an index gradient.

Seismology uses the identical relation with sound speeds in place of light speeds: sinθ₁/v₁ = sinθ₂/v₂. Refraction of P-waves at the core–mantle boundary produced the shadow zone that revealed the Earth has a liquid outer core, and marine sonar refracts through thermoclines by exactly the same arithmetic — a problem closely tied to the relationship between wave speed, frequency and wavelength and to the Doppler shift that gives a moving target away.

Once a ray has been refracted into a lens, the next question is where it forms an image, which is the job of the thin lens equation; the lensmaker's equation connects the two by deriving focal length from the same refractive index and the surface curvatures. Underwater, the index of the water is what makes a diving mask necessary, and the pressure a diver reads at depth follows from a separate hydrostatic calculation that shares nothing with optics except the medium.