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Saturday, March 19, 2011

Brewster's Law

Brewster's angle (also known as the polarization angle) is an angle of incidence at which light with a particular polarization is perfectly transmitted through a transparent dielectric surface, with no reflection. When unpolarized light is incident at this angle, the light that is reflected from the surface is therefore perfectly polarized. This special angle of incidence is named after the Scottish physicist, Sir David Brewster (1781–1868).









When light encounters a boundary between two media with different refractive indices, some of it is usually reflected as shown in the figure above. The fraction that is reflected is described by the Fresnel equations, and is dependent upon the incoming light's polarization and angle of incidence.
The Fresnel equations predict that light with the p polarization (electric field polarized in the same plane as the incident ray and the surface normal) will not be reflected if the angle of incidence is
\theta_\mathrm B = \arctan \left( \frac{n_2}{n_1} \right),
where n1 and n2 are the refractive indices of the two media. This equation is known as Brewster's law, and the angle defined by it is Brewster's angle.
The physical mechanism for this can be qualitatively understood from the manner in which electric dipoles in the media respond to p-polarized light. One can imagine that light incident on the surface is absorbed, and then reradiated by oscillating electric dipoles at the interface between the two media. The polarization of freely propagating light is always perpendicular to the direction in which the light is travelling. The dipoles that produce the transmitted (refracted) light oscillate in the polarization direction of that light. These same oscillating dipoles also generate the reflected light. However, dipoles do not radiate any energy in the direction of the dipole moment. Consequently, if the direction of the refracted light is perpendicular to the direction in which the light is predicted to be specularly reflected, the dipoles cannot create any reflected light.
With simple geometry this condition can be expressed as:
 \theta_1 + \theta_2 = 90^\circ,
where θ1 is the angle of incidence and θ2 is the angle of refraction.
Using Snell's law,
n_1 \sin \left( \theta_1 \right) =n_2 \sin \left( \theta_2 \right),
one can calculate the incident angle θ1 = θB at which no light is reflected:
n_1 \sin \left( \theta_\mathrm B \right) =n_2 \sin \left( 90^\circ - \theta_\mathrm B \right)=n_2 \cos \left( \theta_\mathrm B \right).
Solving for θB gives:
\theta_\mathrm B = \arctan \left( \frac{n_2}{n_1} \right) .


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