Snell's law is the simple formula used to calculate the refraction of light when travelling between two media of differing refractive index. It is named after one of its discoverers, Dutch mathematician Willebrord van Roijen Snell (1580-1626).

Overview

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In the diagram on the right, two media of refractive indices n₁ (on the left) and n₂ (on the right) meet at a surface or interface (vertical line). n₂ > n₁, and light has a slower phase velocity within the second medium.

A light ray PO in the leftmost medium strikes the interface at the point O. From point O, we project a straight line at right angles to the line of the interface; this is known as the normal to the surface (horizontal line). The angle between the normal and the light ray PO is known as the angle of incidence, θ₁.

The ray continues through the interface into the medium on the right; this is shown as the ray OQ. The angle it makes to the normal is known as the angle of refraction, θ₂.

Snell's law gives the relation between the angles θ₁ and θ₂:

$n\_1\backslash sin(\backslash theta\_1)\; =\; n\_2\backslash sin(\backslash theta\_2)\backslash ,$

Note that, for the case of θ₁ = 0° (i.e., a ray perpendicular to the interface) the solution is θ₂ = 0° regardless of the values of n₁ and n₂. In other words, a ray entering a medium perpendicular to the surface is never bent.

The above is also valid for light going from a dense to a less dense medium; the symmetry of Snell's law shows that the same ray paths are applicable in opposite direction.

A qualitative rule for determining the direction of refraction is that the ray in the denser medium is always closer to the normal. A handy way to remember this is to visualize the ray as a car crossing the boundary between asphalt (the less dense medium) and mud (the denser medium). Depending on the angle, either the left wheel or the right wheel of the car will cross into the new medium first, causing the car to swerve.

Snell's law is only generally true for isotropic media (such as glass). In anisotropic media such as some crystals, birefringence may split the refracted ray into two rays, the ordinary or o-ray which follows Snell's law, and the other extraordinary or e-ray which may not be co-planar with the incident ray.

Total internal reflection

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When moving from a dense to a less dense medium (i.e. n₁ > n₂), it is easily verified that the above equation has no solution when θ₁ exceeds a value known as the critical angle:

$\backslash theta\_\{\backslash mathrm\{crit\}\}\; =\; \backslash arcsin\; \backslash left(\; \backslash frac\{n\_2\}\{n\_1\}\; \backslash right)$

When θ₁ > θ_crit, no refracted ray appears, and the incident ray undergoes total internal reflection from the interface.

Vector form

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Given a normalized ray vector v and a normalized plane normal vector p, one can work out the normalized reflected and refracted rays: (note that the actual angles θ₁ and θ₂ are not worked out)

$\backslash cos\backslash theta\_1=\backslash mathbf\{v\}\backslash cdot\backslash mathbf\{p\}$

$\backslash cos\backslash theta\_2=\backslash sqrt\{1-\backslash left(\backslash frac\{n\_1\}\{n\_2\}\backslash right)^2\backslash left(1-\backslash left(\backslash cos\backslash theta\_1\backslash right)^2\backslash right)\}$

$\backslash mathbf\{v\}\_\{\backslash mathrm\{reflect\}\}=\backslash mathbf\{v\}-\backslash left(2\backslash cos\backslash theta\_1\backslash right)\backslash mathbf\{p\}$

$\backslash mathbf\{v\}\_\{\backslash mathrm\{refract\}\}=\backslash left(\backslash frac\{n\_1\}\{n\_2\}\backslash right)\backslash mathbf\{v\}+\backslash left(\backslash cos\backslash theta\_2-\backslash frac\{n\_1\}\{n\_2\}\backslash cos\backslash theta\_1\backslash right)\backslash mathbf\{p\}$

Note: v and p must be in the same direction

The cosines may be recycled and used in the Fresnel equations for working out the intensity of the resulting rays. During total internal reflection an evanescent wave is produced, which rapidly decays from the surface into the second medium. Conservation of energy is maintained by the circulation of energy across the boundary, averaging to zero net energy transmission.

Derivation

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Snell's law may be derived from Fermat's principle, which states that the light travels the path which takes the least time. By taking the derivative of the optical path length, the stationary point is found giving the path taken by the light. Alternatively, it can be derived using interference of all possible paths of light wave from source to observer - it results in destructive interference everywhere except extrema of phase (where interference is constructive) - which become actual paths. In a classic analogy by Feynman, the area of lower refractive index is replaced by a beach, the area of higher refractive index by the sea, and the fastest way for a rescuer on the beach to get to a drowning person in the sea is to run along a path that follows Snell's law.

History

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Snell's law was first discovered and described by Ibn Sahl in a manuscript written c.984 , who used it to work out the shapes of anaclastic lenses (lenses that focus light with no geometric aberrations). It was discovered again by Thomas Harriot in 1602 , who did not publish his work. In 1621, it was discovered yet again by Willebrord Snell, in a mathematically equivalent form, but unpublished during his lifetime. René Descartes independently derived the law in terms of sines in his 1637 treatise Discourse on Method, and used it to solve a range of optical problems. In French, Snell's Law is called "la loi de Descartes" or "loi de Snell-Descartes".

See also

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• Fresnel equations
• Reflection
• Refraction
• Total internal reflection
• Evanescent wave

References

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External links

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• Discovery of the law of refraction

Geometrical optics | Introductory physics | Eponymous laws

Snelliussches Brechungsgesetz | Ley de Snell | Lois de Snell-Descartes | חוק סנל | Wet van Snellius | スネルの法則 | Закон Снелла | Lomni zakon | Snells lag | 光的折射定律