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# Faraday Rotation

Excerpt from Field Guide to Polarization

When a polarized beam propagates through a block of glass that is subjected to a very strong magnetic field, the direction of the beam's propagation is parallel to the direction of the magnetic field and the polarization ellipse rotates. Materials that exhibit this behavior are called Faraday, magneto-optical media, or, more commonly, Faraday rotators. The rotation angle is given by

where *V* is Verdet's constant, *H* is magnetic field intensity, and *l* is the propagation distance. Faraday rotation is described by a Mueller rotation matrix that is identical to that of optical activity except the magnetic field intensity is directional: *H* is +*H* when propagation is left to right, and -*H* when the propagation is right to left. From the relation the rotation angle becomes

-θ on the return path. The configuration for propagation of a beam that is reflected back through a magneto-optical medium is shown.

The Mueller matrix for this configuration is

The above matrix is the Mueller matrix of a pseudorotation matrix; that is, in a single trip the rotation angle doubles and the ellipticity is reversed.

If an additional reflector is placed before the Faraday medium (the medium is now in an optical cavity,) then for *N* trips the Mueller rotation matrix becomes

By slightly tilting the reflecting surface on the right side of the optical cavity, the effect of allowing the beam to make *N* passes can be used to measure the small rotation angle θ.

For propagation in a Faraday medium the field can be expressed as a superposition of two circularly polarized waves propagating with different wave numbers:

where *k*_{0} is the propagation constant in free space and *n _{L}*and

*n*are the refractive indices associated with each of the circular field components, respectively. The circular birefringence is then defined to be

_{R}where ß is a parameter and *n* is the mean refractive index. The refractive indices are also called dextrorotary and levo-rotatory or, simply, *R-* and *L- *rotatory, and indicates that LCP and RCP waves propagate with different phase velocities.

**Citation:**

E. Collett, *Field Guide to Polarization*, SPIE Press, Bellingham, WA (2005).

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