Abstract
We develop a theory of the magneto-optical and magnetoelectric properties of a topological insulator thin film in the presence of a quantizing external magnetic field. We find that low-frequency magneto-optical properties depend only on the sum of top and bottom surface Dirac-cone filling factors ${\ensuremath{\nu}}_{\text{T}}$ and ${\ensuremath{\nu}}_{\text{B}}$ whereas the low-frequency magnetoelectric response depends only on the difference. The Faraday rotation is quantized in integer multiples of the fine structure constant and the Kerr effect exhibits a $\ensuremath{\pi}/2$ rotation. Strongly enhanced cyclotron-resonance features appear at higher frequencies that are sensitive to the filling factors of both surfaces. When the product of the bulk conductivity and the film thickness in ${e}^{2}/h$ units is small compared to $\ensuremath{\alpha}$, magneto-optical properties are only weakly dependent on accidental doping in the interior of the film.
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