Derivations and Formula
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 (though it should be noted that the result does not show light taking the least time path, but rather one that is stationary with respect to small variations as there are cases where light actually takes the greatest time path, as 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.
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.
Another way to derive Snell’s Law involves an application of the general boundary conditions of Maxwell equations for electromagnetic radiation.
Yet another way to derive Snell's law is based on translation symmetry considerations. For example, a homogeneous surface perpendicular to the z direction can not change the transverse momentum. Since the propagation vector is proportional to the photon's momentum, the transverse propagation direction must remain the same in both regions. Assuming without loss of generality a plane of incidence in the plane . Using the well known dependence of the wave number on the refractive index of the medium, we derive Snell's law immediately.
where is the wavenumber in vacuum. Note that no surface is truly homogeneous, in the least at the atomic scale. Yet full translational symmetry is an excellent approximation whenever the region is homogeneous on the scale of the light wavelength.
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