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Snell’s law calculator

Refracted angle, critical angle, total internal reflection and Brewster angle, from n₁·sin θ₁ = n₂·sin θ₂.

Parameters

Air ≈ 1.000, water ≈ 1.33, common glass ≈ 1.5.
Measured from the normal to the surface, from 0° (normal incidence) to 90°.

Result

Refracted angle

28.13°

Calculation
sin θ₂ = n₁·sin θ₁ ÷ n₂ = 1 × sin(45°) ÷ 1.5 → θ₂ = 28.13°
Nn₁ = 1n₂ = 1.545°28.1°
θ₁ = 45°
Ray deviation
16.87°
Notes (3)

Playback only moves the θ₁ slider from 0° to 90° at ten degrees per second; every position is fully recalculated.

No critical angle, since n₁ ≤ n₂: total internal reflection only exists toward a less refractive medium (glass → air).

The ray is refracted at 28.13°: it bends toward the normal (denser medium).

Calculation, assumptions and sources

OverviewThree refraction calculations: the refracted angle (n₁·sin θ₁ = n₂·sin θ₂), with the bending direction and total internal reflection past the critical angle; the critical angle, which only exists toward a less dense medium; the Brewster angle, where the reflection is fully polarised.

Three refraction calculations: the refracted angle (n₁·sin θ₁ = n₂·sin θ₂), with the bending direction and total internal reflection past the critical angle; the critical angle, which only exists toward a less dense medium; the Brewster angle, where the reflection is fully polarised.

Method & formulasn₁ · sin θ₁ = n₂ · sin θ₂
n₁ · sin θ₁ = n₂ · sin θ₂critical angle: θc = arcsin(n₂/n₁), if n₁ > n₂Brewster: θB = arctan(n₂/n₁)

Angles are measured from the normal to the surface: a ray at 30° to the water surface arrives at 60° from the normal. Toward a denser medium (n₂ > n₁) the ray bends toward the normal; toward a less dense one it bends away. Beyond the critical angle, sin θ₂ would exceed 1 and there is no refracted ray (total internal reflection). At the Brewster angle, the reflected and refracted rays are perpendicular.

Refractive index (n)
the ratio of the speed of light in vacuum to its speed in the medium. It depends on wavelength (dispersion).
Critical angle (θc)
the incidence beyond which refraction becomes impossible, from the denser medium to the less dense one.
Brewster angle (θB)
the incidence where the component polarised in the plane of incidence is no longer reflected: the reflected light is fully polarised.
Validity domainDispersion is not modelled: the usual indices are given for sodium yellow.

Dispersion is not modelled: the usual indices are given for sodium yellow. Interfaces are plane, between homogeneous, transparent, isotropic media (no index gradient, no absorption, no birefringence). Reflected and transmitted intensities (Fresnel coefficients) are not computed.