Science calculator

Snell's law and critical angle.

Solve a refraction angle or refractive index, or find the critical angle for total internal reflection. Angles are measured from the normal.

Relationship

Incident index n₁

Incident angle θ₁

Transmitted index n₂

Refracted angle θ₂

Refracted angle θ₂Solved

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n₁ sin(θ₁) = n₂ sin(θ₂)

Critical-angle unit

Display precision

n_{1} \sin (θ_{1}) = n_{2} \sin (θ_{2})

\sin (θ_{c}) = \frac{n_{2}}{n_{1}}

n₁, n₂ refractive indices · θ₁, θ₂ angles from the normal · θ_c critical angle in the higher-index medium

Assumptions

Flat boundary between homogeneous, isotropic media; indices are positive phase refractive indices at the stated wavelength.
Every angle is measured from the surface normal, not from the surface.
The critical angle exists only for travel from a higher index n₁ to a lower index n₂.
At incidence above θc the model reports no transmitted ray: total internal reflection occurs in the ideal lossless case.

How it works

Snell's law conserves the wave-vector component parallel to a boundary. A ray bends towards the normal on entering a higher-index medium and away on entering a lower-index medium.

Leave one Snell quantity blank to rearrange the relation. If the required sine exceeds one, no real refracted angle exists and the ray is totally internally reflected.

The critical angle is reached when the transmitted ray runs along the boundary at 90°. It is defined only from higher to lower refractive index.

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