Snell's law

Snell's law (also known as the Snell–Descartes law, and the law of refraction) is a formula used to describe the relationship between the angles of incidence and refraction, when referring to light or other waves passing through a boundary between two different isotropic media, such as water, glass, or air.

In optics, the law is used in ray tracing to compute the angles of transmission or refraction, and in experimental optics to find the refractive index of a material. The law is also satisfied in meta-materials, which allow light to be bent "backward" at a negative angle of refraction with a negative refractive index.

The law states that, for a given pair of media, the ratio of the sines of angle of incidence

(

θ

1

)

{\displaystyle \left(\theta _{1}\right)}

and angle of refraction

(

θ

2

)

{\displaystyle \left(\theta _{2}\right)}

is equal to the refractive index of the second medium with regard to the first (

n

2

,

1

{\displaystyle n_{2,1}}

) which is equal to the ratio of the refractive indices

(

n

2

n

1

)

{\displaystyle \left({\tfrac {n_{2}}{n_{1}}}\right)}

of the two media, or equivalently, to the ratio of the phase velocities

(

v

1

v

2

)

{\displaystyle \left({\tfrac {v_{1}}{v_{2}}}\right)}

in the two media.

sin

⁡

θ

1

sin

⁡

θ

2

=

n

2

,

1

=

n

2

n

1

=

v

1

v

2

{\displaystyle {\frac {\sin \theta _{1}}{\sin \theta _{2}}}=n_{2,1}={\frac {n_{2}}{n_{1}}}={\frac {v_{1}}{v_{2}}}}

The law follows from Fermat's principle of least time, which in turn follows from the propagation of light as waves.

History

Ptolemy, in Alexandria, Egypt, had found a relationship regarding refraction angles, but it was inaccurate for angles that were not small. Ptolemy was confident he had found an accurate empirical law, partially as a result of slightly altering his data to fit theory (see: confirmation bias).

The law was eventually named after Snell, although it was first discovered by the Persian scientist Ibn Sahl, at Baghdad court in 984. In the manuscript On Burning Mirrors and Lenses, Sahl used the law to derive lens shapes that focus light with no geometric aberration.

Ibn al-Haytham, in his Book of Optics (1021), came close to rediscovering the law of refraction, but he did not take this step.