Theory of the two-photon Franz-Keldysh effect and electric-field-induced bichromatic coherent control

TL;DR

This work develops a complete independent-particle theory for the two-photon Franz–Keldysh effect in semiconductors under a uniform DC field, extending earlier one-photon FKE formulations. Using a 14-band ${\bf k}\cdot{\bf p}$ model for GaAs and nonperturbative DC-field treatment, it computes two-photon absorption amplitudes and bichromatic interference terms via field-modified interband couplings and velocity matrix elements. The results show field-enabled TPA below the gap, Franz–Keldysh oscillations above the gap with strong polarization dependence, and DC-field–driven, sign-dependent odd FKE tensor elements due to inversion-symmetry breaking, along with DC-field–enabled bichromatic population control. These findings establish a detailed baseline for nonlinear electro-optics in semiconductors and inform strategies for DC-field-tunable multi-photon devices and frequency-comb control schemes.

Abstract

The effect of a constant electric field on two-photon absorption in a semiconductor is calculated using an independent-particle theory. The theoretical framework is an extension of a theory of the one-photon Franz-Keldysh effect [Wahlstrand and Sipe, Phys. Rev. B 82, 075206 (2010)]. The theory includes the effect of the constant field, including field-induced coupling between closely spaced bands, in the electronic wavefunctions and calculates optical absorption perturbatively. Numerical calculations are performed using a 14-band $\mathbf{k} \cdot\mathbf{p}$ band structure model for GaAs. For all nonzero tensor elements, field-enabled two-photon absorption (TPA) below the band gap and Franz-Keldysh oscillations in the TPA spectrum are predicted, with a generally larger effect in tensor elements with more components parallel to the constant electric field direction. Some tensor elements that are zero in the absence of a field become nonzero in the presence of the constant electric field and depend on its sign. Notably, these elements are linear in the electric field to lowest order and may be substantial away from band structure critical points at room temperature and/or with a non-uniform field. Electric-field-induced changes in the carrier injection rate due to interference between one- and two-photon absorption are also calculated. The electric field enables this bichromatic coherent control process for polarization configurations where it is normally forbidden, and also modifies the spectrum of the process for configurations where it is allowed by crystal symmetry.

Figures (6)

• Figure 1: Two-photon absorption spectra [Eq. (\ref{['beta']})] in GaAs calculated using the 14-band ${\bf k} \cdot {\bf p}$ model without a field, and with a DC field pointing along $[001]$, for ${\bf E}_{\mathrm{opt}} \parallel {\bf E}_{\mathrm{dc}}$ (solid) and ${\bf E}_{\mathrm{opt}} \parallel [100] \perp {\bf E}_{\mathrm{dc}}$ (dashed). The curves at each field strength are displaced vertically by 2.5 cm/GW.
• Figure 2: Elements of $\mathrm{Im}[\chi^{(3)}]$ calculated using the 14-band ${\bf k} \cdot {\bf p}$ model for $E_{\mathrm{dc}} = 44$ kV/cm along [001]. Dotted lines indicate $\hbar \omega = E_g/2$ and $\hbar\omega = E_{so}/2$. (a) Elements related to the even FKE. The solid lines are curves for the $zzzz$, $xzxz$, and $xxzz$ elements, which have at least one component pointing along the DC field direction. The dashed lines are for the $xxxx$, $xyxy$, and $xxyy$ elements, which have all components perpendicular to the DC field. (b) Elements related to the odd FKE: $xyzz$, $xxxy$, and $zxyz$.
• Figure 3: Even two-photon electroabsorption spectra [Eq. (\ref{['evenFKE']})] calculated using the 14-band ${\bf k} \cdot {\bf p}$ model in GaAs. All calculations used $E_{\mathrm{dc}} = 44$ kV/cm. Dotted lines indicate $\hbar \omega = E_g/2$ and $\hbar\omega = E_{so}/2$. (a) Spectra for the optical field polarized parallel to the DC field direction. (b) Spectra for the optical field polarized perpendicular to the DC field.
• Figure 4: Odd two-photon electroabsorption spectra [Eq. (\ref{['oddFKE']})] calculated using the 14-band ${\bf k} \cdot {\bf p}$ model in GaAs. All calculations used $E_{\mathrm{dc}} = 44$ kV/cm. Dotted lines indicate $\hbar \omega = E_g/2$ and $\hbar\omega = E_{so}/2$.
• Figure 5: Calculation of bichromatic interference control for a DC field along [001]. Dotted lines indicate $\hbar \omega = E_g/2$ and $\hbar\omega = E_{so}/2$. (a) Field-induced elements $zzz$, $xzx$, and $xxz$ for a DC field of 44 kV/cm. (b) Population control element $xyz$ with no DC field (dashed line) and at $E_{\mathrm{dc}} = 44$ kV/cm.