1) The Effect
Im going to be using the Faraday effect to measure the magnetic field. The Faraday effect is the rotation of the polarization plane of light due to the application of a magnetic field in the direction of light propagation. Faraday rotation is then the turning of the polarization by the magnetic field. There are two Faraday rotations: there is the linear one and the nonlinear one.
Linear faraday roation is using the Zeeman shift caused by the magnetic field to change the indexes of refraction in a material and thus turn the polarization. A Zeeman shift is the change in resonant frequencies of the atomic transition. If the energy in an atomic transition is h*f (no magnetic field) then with a magnetic field that energy would change by g*u*B, where g is the gyrometric ration and u*B is the Bohr Magneton, or the applied magnetic field.
It would be much easier to understand this topic if one though of light's standard polarization as a circular one. That means that the two vectors are going in circles. Light thats linearly polarized would mean the superposition of there two vectors, and its direction of that would be the point where the 2 circular vectors meet. For clarity we can call a liner polarized light where the two circular vectors meet in the positive y axis.
What the Zeeman shift does is lower the index of refraction of one of the vectors and raises the index of refraction for the other. This would mean that one of the vectors would be going slower than the other and would mean that after they come out of the sample they are not going to meet at the top of the y axis anymore but somewhere else, thus turning the polarization of the light. The angle by which the polarization is turned is linear with small magnitutes of magnetic fields, peaks at h*f=g*u*B and then falls with increasing magnetic field.
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