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I find magneto-optical effects fascinating, and especially the Faraday effect. But most sources only give a phenomenological description, while I want a deeper explanation of its mechanism. Is there a simple model that can explain the formula $\beta = \mathcal{V}Bd$? No need for a precise calculation of the Verdet constant.

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It is indeed a topic that is discussed in many books but only a few give a rigorous mathematical description of the phenomena.

For stringency in non-linear optics topics I always trust HARTMANN ROMER: Theoretical Optics, An Introduction. 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

A book which is also mathematically rigorous is BOYD, ROBERT W: Nonlinear Optics, Third Edition. Academic Press, 2008

The simplest way/model to explain the Faraday effect and get the $\beta=\mathcal{V}BL$ result is to solve the dynamic problem of a classical electron moving in a non-conducting substance under the action of an applied magnetic field $\vec B$ in the $z$ direction:

$$ m\frac{{{d^2}\vec r}}{{d{t^2}}} + K\vec r = - e\vec E - e\left( {\frac{{d\vec r}}{{dt}} \times \vec B} \right) $$

Solving for the rotational coordinates $R_{\pm}=x\pm iy$ one finds that the introduction of the applied magnetic field will break the linear isotropy of the substance by splitting the refractive index into two distinct parts, $n_{-}$ and $n_{+}$, that spin in opposing directions. This will end up resulting in a Faraday rotation of the plane of polarization of incident waves of the form:

$$ \beta = \frac{\pi }{\lambda }L\left( {{n_ - } - {n_ + }} \right) \simeq \left[ {\frac{{{\omega _p}^2}}{{{n_0}c}}\frac{e}{m}\left( {\frac{{{\omega ^2}}}{{{\omega _L}^4 - 4{\omega _0}^2{\omega _L}^2}}} \right)} \right]{B_z}L = \mathcal{V}B_{z}L $$

There are many details to this derivation that you can look up in the books I've mentioned. I hope you find these references useful.

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