Derivation from Fermat's principle

Snell's law can 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. (There are situations of light violating Fermat's principle by not taking the least time path, as in reflection in a (spherical) mirror.) In a classic analogy, 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.

Light from medium 1, point Q, enters medium 2, refraction occurs, and finally the light reaches point P.

As shown in the figure to the right, assume the refractive index of medium 1 and medium 2 are ${\displaystyle n_{1}}$ and ${\displaystyle n_{2}}$ respectively. Light enters medium 2 from medium 1 via point O.

${\displaystyle \theta _{1}}$ is the angle of incidence, ${\displaystyle \theta _{2}}$ is the angle of refraction with respect to the normal.

The phase velocities of light in medium 1 and medium 2 are

${\displaystyle v_{1}=c/n_{1}}$ and

${\displaystyle v_{2}=c/n_{2}}$ respectively.

${\displaystyle c}$ is the speed of light in vacuum.

Let T be the time required for the light to travel from point Q through point O to point P.

${\displaystyle T={\frac {\sqrt {x^{2}+a^{2}}}{v_{1}}}+{\frac {\sqrt {b^{2}+(l-x)^{2}}}{v_{2}}}={\frac {\sqrt {x^{2}+a^{2}}}{v_{1}}}+{\frac {\sqrt {b^{2}+l^{2}-2lx+x^{2}}}{v_{2}}}}$

where a, b, l and x are as denoted in the right-hand figure, x being the varying parameter.

To minimize it, one can differentiate :

${\displaystyle {\frac {dT}{dx}}={\frac {x}{v_{1}{\sqrt {x^{2}+a^{2}}}}}+{\frac {-(l-x)}{v_{2}{\sqrt {(l-x)^{2}+b^{2}}}}}=0}$ (stationary point)

Note that ${\displaystyle {\frac {x}{\sqrt {x^{2}+a^{2}}}}=\sin \theta _{1}}$

and ${\displaystyle {\frac {l-x}{\sqrt {(l-x)^{2}+b^{2}}}}=\sin \theta _{2}}$

Therefore,

${\displaystyle {\frac {dT}{dx}}={\frac {\sin \theta _{1}}{v_{1}}}-{\frac {\sin \theta _{2}}{v_{2}}}=0}$

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

${\displaystyle {\frac {n_{1}\sin \theta _{1}}{c}}={\frac {n_{2}\sin \theta _{2}}{c}}}$

${\displaystyle n_{1}\sin \theta _{1}=n_{2}\sin \theta _{2}}$

Derivation from Huygens's principle

Alternatively, Snell's law 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.

Derivation from Maxwell's equations

Another way to derive Snell's Law involves an application of the general boundary conditions of Maxwell equations for electromagnetic radiation and induction.

Derivation from conservation of energy and momentum

Yet another way to derive Snell's law is based on translation symmetry considerations.^[20] For example, a homogeneous surface perpendicular to the z direction cannot change the transverse momentum. Since the propagation vector ${\displaystyle {\vec {k}}}$ is proportional to the photon's momentum, the transverse propagation direction ${\displaystyle (k_{x},k_{y},0)}$ must remain the same in both regions. Assume without loss of generality a plane of incidence in the ${\displaystyle z,x}$ plane ${\displaystyle k_{x{\text{Region}}_{1}}=k_{x{\text{Region}}_{2}}}$. Using the well known dependence of the wavenumber on the refractive index of the medium, we derive Snell's law immediately.

${\displaystyle k_{x{\text{Region}}_{1}}=k_{x{\text{Region}}_{2}}\,}$

${\displaystyle n_{1}k_{0}\sin \theta _{1}=n_{2}k_{0}\sin \theta _{2}\,}$

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

where ${\displaystyle k_{0}={\frac {2\pi }{\lambda _{0}}}={\frac {\omega }{c}}}$ is the wavenumber in vacuum. Although no surface is truly homogeneous at the atomic scale, full translational symmetry is an excellent approximation whenever the region is homogeneous on the scale of the light wavelength.

Vector form

Given a normalized light vector ${\displaystyle {\vec {l}}}$ (pointing from the light source toward the surface) and a normalized plane normal vector ${\displaystyle {\vec {n}}}$, one can work out the normalized reflected and refracted rays, via the cosines of the angle of incidence ${\displaystyle \theta _{1}}$ and angle of refraction ${\displaystyle \theta _{2}}$, without explicitly using the sine values or any trigonometric functions or angles:^[21]

${\displaystyle \cos \theta _{1}=-{\vec {n}}\cdot {\vec {l}}}$

Note: ${\displaystyle \cos \theta _{1}}$ must be positive, which it will be if ${\displaystyle {\vec {n}}}$ is the normal vector that points from the surface toward the side where the light is coming from, the region with index ${\displaystyle n_{1}}$. If ${\displaystyle \cos \theta _{1}}$ is negative, then ${\displaystyle {\vec {n}}}$ points to the side without the light, so start over with ${\displaystyle {\vec {n}}}$ replaced by its negative.

${\displaystyle {\vec {v}}_{\mathrm {reflect} }={\vec {l}}+2\cos \theta _{1}{\vec {n}}}$

This reflected direction vector points back toward the side of the surface where the light came from.

Now apply Snell's law to the ratio of sines to derive the formula for the refracted ray's direction vector:

${\displaystyle \sin \theta _{2}=\left({\frac {n_{1}}{n_{2}}}\right)\sin \theta _{1}=\left({\frac {n_{1}}{n_{2}}}\right){\sqrt {1-\left(\cos \theta _{1}\right)^{2}}}}$

${\displaystyle \cos \theta _{2}={\sqrt {1-(\sin \theta _{2})^{2}}}={\sqrt {1-\left({\frac {n_{1}}{n_{2}}}\right)^{2}\left(1-\left(\cos \theta _{1}\right)^{2}\right)}}}$

${\displaystyle {\vec {v}}_{\mathrm {refract} }=\left({\frac {n_{1}}{n_{2}}}\right){\vec {l}}+\left({\frac {n_{1}}{n_{2}}}\cos \theta _{1}-\cos \theta _{2}\right){\vec {n}}}$

The formula may appear simpler in terms of renamed simple values ${\displaystyle r=n_{1}/n_{2}}$ and ${\displaystyle c=-{\vec {n}}\cdot {\vec {l}}}$, avoiding any appearance of trig function names or angle names:

${\displaystyle {\vec {v}}_{\mathrm {refract} }=r{\vec {l}}+\left(rc-{\sqrt {1-r^{2}\left(1-c^{2}\right)}}\right){\vec {n}}}$

Example:

${\displaystyle {\vec {l}}=\{0.707107,-0.707107\},~{\vec {n}}=\{0,1\},~r={\frac {n_{1}}{n_{2}}}=0.9}$

${\displaystyle c=\cos \theta _{1}=0.707107,~{\sqrt {1-r^{2}\left(1-c^{2}\right)}}=\cos \theta _{2}=0.771362}$

${\displaystyle {\vec {v}}_{\mathrm {reflect} }=\{0.707107,0.707107\},~{\vec {v}}_{\mathrm {refract} }=\{0.636396,-0.771362\}}$

The cosine values may be saved and used in the Fresnel equations for working out the intensity of the resulting rays.

Total internal reflection is indicated by a negative radicand in the equation for ${\displaystyle \cos \theta _{2}}$, which can only happen for rays crossing into a less-dense medium (${\displaystyle n_{2}<n_{1}}$).