Generalized Franz-Keldysh theory of electromodulation

We present a detailed generalization of the Franz-Keldysh theory of electromodulation which takes into account the presence of a dc surface electric field, ${\mathit{scrE}}_{\mathrm{dc}}^{\mathit{s}}$. We demonstrate rigorously that when the ac modulating electric field is small compared to ${\mathit{scrE}}_{\mathrm{dc}}^{\mathit{s}}$, the period of the Franz-Keldysh oscillations yields ${\mathit{scrE}}_{\mathrm{dc}}^{\mathit{s}}$. This result is independent of the relation between the width of the space-charge region and the penetration depth of light. The effects of broadening also are considered.