Higher-order contributions and nonperturbative effects in the nondegenerate nonlinear optical absorption of semiconductors using a two-band model

The semiconductor Bloch equations for a two-band model including inter- and intraband excitation are used to study the nonlinear absorption of single and multiple light pulses by direct-gap semiconductors. For a consistent analysis the contributions to the absorption originating from both the interband polarization and the intraband current need to be included. In the Bloch equation approach these contributions as well as different excitation pathways in terms of sequences of inter- and intraband excitations can be evaluated separately which allows for a transparent analysis, the identification of the dominant terms, and analyzing their dependence on the excitation conditions. In the perturbative regime we obtain analytical expressions for the multiphoton absorption coefficients for continuous-wave excitation. These results are shown to agree well with numerical results for short pulses and/or finite dephasing and relaxation times. We confirm the previously predicted strong enhancement of two-photon absorption for nondegenerate conditions for pulsed excitation. We discuss the dependencies of the multiphoton absorption on the light frequencies, initial band populations, and the time delay between the pulses. The frequency dependence of the two-photon absorption coefficient for nondegenerate excitation is evaluated perturbatively in third order. The higher-order contributions to the optical absorption include three- and four-photon absorption and show a rich frequency dependence including negative regions and dispersive line shapes. Nonperturbative solutions of the Bloch equations demonstrate a strongly nonmonotonous behavior of the intensity-dependent optical absorption for a single incident pulse and in a pump-probe setup.

Figure 1

Schematics of the perturbative pathways up to fifth order (after Ref. [22]). The full circles indicate paths contributing to the optical absorption of $(m+1)/2$ photons. The right half of the tree (intraband term for $m=1$) vanishes for an initially unexcited system. The transition dipole matrix element $\mathbf{d}$ enters only through its projection onto the polarization directions of the incident fields. We consider linearly parallel polarized pulses, i.e., ${\mathbf{e}}_1={\mathbf{e}}_2$ and thus introduce $\mu =\mathbf{d}·{\widehat{\mathbf{e}}}_1$. The power of $\mu$ of each absorption contribution is shown below $m=5$ circles.

Figure 2

(a) Absorption (arbitrary units) of a single Gaussian pulse of frequency ${\omega }_1$, peak amplitude ${\widehat{E}}_1$, and duration ${\tau }_1=50$ fs by a one-dimensional sample using the tight-binding model (12) with $\hslash {\omega }_{\mathrm{g}}=1.5$ eV, $\hslash {\omega }_{\mathrm{b}}=1.8$ eV. The dephasing time $T_2$ is infinite. The color scale does not include all data values, but the off-range regions $(>5)$ are small. (b) Solid lines are the same data as in (a), the additional dashed lines are for $T_2=200$ fs.

Figure 3

2PA coefficient for different frequency ratios ${\omega }_1/{\omega }_2$ (three-dimensional model). The frequency sum ${\omega }_1+{\omega }_2=1.55\mathrm{eV}={\omega }_{\mathrm{g}}+50$ meV is kept constant and ${\omega }_1$ is varied. Numerical results are shown by solid lines for ${\tau }_1={\tau }_2=50$ fs, $T_1\to \infty$, and $T_2=200$ fs. The dashed lines display the scaled analytical expressions given in the brackets, cf. Eq. (25).

Figure 4

2PA coefficient for varying band gap ${\omega }_{\mathrm{g}}$ in three dimensions. Numerical results (solid lines) are computed for two Gaussian pulses with ${\tau }_1={\tau }_2=50$ fs; ${\omega }_1=2{\omega }_2=1\mathrm{eV}/\hslash$. The dashed line is the fitted analytical expression from Eq. (27).

Figure 5

2PA coefficient for two Gaussian pulses as a function of an initial quasiequilibrium carrier density in three dimensions. The considered parameters are ${\omega }_1+{\omega }_2=1.7$ eV, ${\omega }_1/{\omega }_2=2, {\tau }_1={\tau }_2=50$ fs, $T_1\to \infty , T_2=200$ fs, and $T=300$ K. The total of all paths (shown by the black curve) agrees well with the factor $1-n_e-n_h$ as expected from Eq. (25). For $T\to 0$ this factor approaches the step function shown by the dotted line.

Figure 6

MPA spectra for $l=2,3,4$ corresponding to the orders $m=3,5,7$ for a one-dimensional tight-binding model (12) with ${\omega }_{\mathrm{g}}={\omega }_{\mathrm{b}}=1.5\mathrm{eV}/\hslash$. The probe frequency is fixed at ${\omega }_1=1.2\mathrm{eV}/\hslash$ and the pump frequency is varied. The pulse durations for solid lines are ${\tau }_1={\tau }_2=200$ fs and both dephasing and relaxation are switched off. The dashed lines are analytical results, Eq. (25) in (a) and Eq. (28) in (b). The dotted lines are approximations for the higher-order correction features according to Eq. (29) with $G\approx \text{const}.$ The vertical thin lines indicate MPA resonances with the band gap, i.e., ${\omega }_1+(l-1){\omega }_2={\omega }_{\mathrm{g}}$.

Figure 7

Transients obtained for the one-dimensional tight-binding model (12) with ${\omega }_{\mathrm{g}}={\omega }_{\mathrm{b}}=1.5$ eV/$\hslash$. (a) Absorption $\int dt{\mathbf{j}}^{(2l-1)}·{\mathbf{E}}_1$ of the probe pulse for $l=2,3,4$ as a function of the time delay between the probe and pump pulse. The parameters are $\hslash {\omega }_1=1.2$ eV, $\hslash {\omega }_2=0.31$ eV, ${\tau }_1={\tau }_2=50$ fs, and $T_2\to \infty$. The dashed lines correspond to Eq. (31). (b) Transients obtained for strong pump fields (peak amplitudes indicated). $\hslash {\omega }_1=1.1$ eV, $\hslash {\omega }_2=0.5$ eV, ${\tau }_1={\tau }_2=50$ fs, and $T_2=200$ fs.

Figure 8

Absorption of the weak probe pulse as a function of the pump field amplitude ($\hslash {\omega }_1=1.1$ eV, $\hslash {\omega }_2=0.5$ eV, ${\tau }_1={\tau }_2=50$ fs, $T_2\to \infty$). The black dotted line is the result from seventh-order perturbation theory. The yellow dotted line is the total pump absorption (on a different scale). The one-dimensional tight-binding model (12) with ${\omega }_{\mathrm{g}}={\omega }_{\mathrm{b}}=1.5\mathrm{eV}/\hslash$ was used.