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Similarly, Fresnel's sine law will apply to the p polarization instead of the s polarization, and his tangent law to the s polarization instead of the p polarization. This switch of polarizations has an analog in the old mechanical theory of light waves (see § History , above).
By Malus's definition, the plane of polarization of a ray was the plane of the ray and the optic axis if the ray was ordinary, or the perpendicular plane (containing the ray) if the ray was extraordinary. In Fresnel's model, the direction of vibration was normal to the plane of polarization.
In 1818, Fresnel [5] showed that Huygens's principle, together with his own principle of interference, could explain both the rectilinear propagation of light and also diffraction effects. To obtain agreement with experimental results, he had to include additional arbitrary assumptions about the phase and amplitude of the secondary waves, and ...
By Fresnel's sine law, r s is positive for all angles of incidence with a transmitted ray (since θ t > θ i for dense-to-rare incidence), giving a phase shift δ s of zero. But, by his tangent law, r p is negative for small angles (that is, near normal incidence), and changes sign at Brewster's angle , where θ i and θ t are complementary.
The Fresnel–Arago laws are three laws which summarise some of the more important properties of interference between light of different states of polarization. Augustin-Jean Fresnel and François Arago , both discovered the laws, which bear their name.
Fresnel's "plane of polarization", traditionally used in optics, is the plane containing the magnetic vectors (B & H) and the wave-normal. Malus's original "plane of polarization" was the plane containing the magnetic vectors and the ray. (In an isotropic medium, θ = 0 and Malus's plane merges with Fresnel's.)
When the two polarization states are relative to the direction of a surface (usually found with Fresnel reflection), they are usually termed s and p. This distinction between Cartesian and s–p polarization can be negligible in many cases, but it becomes significant for achieving high contrast and with wide angular spreads of the incident light.
The transfer-matrix method is based on the fact that, according to Maxwell's equations, there are simple continuity conditions for the electric field across boundaries from one medium to the next. If the field is known at the beginning of a layer, the field at the end of the layer can be derived from a simple matrix operation.