franz_keldysh_absorption

franz_keldysh_absorption(
    *,
    wavelengths: collections.abc.Sequence[
        typing.Annotated[float, exclusiveMinimum=0, units='μm']
    ],
    voltages: collections.abc.Sequence[typing.Annotated[float, units='V']],
    bandgap_wavelength: Annotated[float, exclusiveMinimum=0, units='μm'],
    intrinsic_thickness: Annotated[float, exclusiveMinimum=0, units='μm'],
    built_in_voltage: Annotated[float, units='V'] = 0.7,
    reduced_mass: Annotated[float, exclusiveMinimum=0] = 0.02,
    absorption_scale: Annotated[
        float, exclusiveMinimum=0, units='μm⁻¹'
    ] = 1.0,
) -> tuple[numpy.ndarray, numpy.ndarray, numpy.ndarray]

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.

Args

• wavelengths: Wavelength grid, strictly increasing.
• voltages: Bias grid, strictly increasing. Reverse bias is negative.
• bandgap_wavelength: Wavelength of the (direct) bandgap.
• intrinsic_thickness: Thickness of the intrinsic region carrying the field.
• built_in_voltage: Junction built-in potential.
• reduced_mass: Reduced effective mass in units of the electron rest mass.
• absorption_scale: Calibration constant, absorbing the prefactor convention. Fit it to one measured absorption value.

Returns

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

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.

References

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