franz_keldysh_absorption

photonforge.franz_keldysh_absorption(*, wavelengths, voltages, bandgap_wavelength, intrinsic_thickness, built_in_voltage=0.7, reduced_mass=0.02, absorption_scale=)[source]

Franz-Keldysh absorption table for a bulk electro-absorber.

An applied field tilts the bands, so photon-assisted tunnelling produces absorption below the bandgap. With \(\beta = (E_g - \hbar\omega)/\hbar\theta\) and electro-optic energy \(\hbar\theta = \left[q^2\hbar^2F^2/(2m_r)\right]^{1/3}\):

\[\alpha(\hbar\omega, F) = \alpha_s \frac{\sqrt{\hbar\theta}}{\hbar\omega} \left[\left|{\rm Ai}'(\beta)\right|^2 - \beta\left|{\rm Ai}(\beta)\right|^2\right]\]

The field follows from the reverse bias across the intrinsic region, \(F = (V_{bi} - V)/d\), so reverse bias is a negative voltage in the usual diode convention and the built-in field still applies at zero bias.

This is the mechanism in germanium-on-silicon modulators. Quantum-well devices are governed by the quantum-confined Stark effect instead and should be supplied as a measured table.

Parameters:

  • wavelengths (Sequence[Annotated[float, exclusiveMinimum=0, units='μm']]) – Wavelength grid, strictly increasing.

  • voltages (Sequence[Annotated[float, units='V']]) – Bias grid, strictly increasing. Reverse bias is negative.

  • bandgap_wavelength (Annotated[float, exclusiveMinimum=0, units='μm']) – Wavelength of the (direct) bandgap.

  • intrinsic_thickness (Annotated[float, exclusiveMinimum=0, units='μm']) – Thickness of the intrinsic region carrying the field.

  • built_in_voltage (Annotated[float, units='V']) – Junction built-in potential.

  • reduced_mass (Annotated[float, exclusiveMinimum=0]) – Reduced effective mass in units of the electron rest mass.

  • absorption_scale (Annotated[float, exclusiveMinimum=0, units='μm⁻¹']) – Calibration constant, absorbing the prefactor convention. Fit it to one measured absorption value.

Returns:

Absorption in 1/μm as an photonforge.Interpolator over voltage and wavelength, ready to pass as the absorption argument of AnalyticWaveguideModel or ElectroAbsorptionModTimeStepper.

Return type:

tuple[ndarray, ndarray, ndarray]

Note

Only the dependence on photon energy and field is physical here; the absolute scale is a fitting constant, as in every practical use of this expression. As \(F\to 0\) the expression reduces to the bulk \(\sqrt{\hbar\omega - E_g}\) edge.

Reference:

Chuang, S. L. (2009). Physics of Photonic Devices (2nd ed.), Wiley, section on the Franz-Keldysh effect.