Strongly nonresonant four-wave mixing in semiconductors

When semiconductors are optically excited above or slightly below the band gap, the linear and nonlinear responses originate predominantly from the interband polarization. Here, we demonstrate that intraband excitations, i.e., rapidly oscillating currents that originate from the electric-field-induced acceleration of electrons and holes, contribute strongly to transient four-wave mixing when it is performed with center frequencies near half the band gap frequency. Within a two-band model we show that the presence of several pathways arising from different combinations of inter- and intraband excitations and their interference give rise to characteristic signatures in time- and spectrally resolved signals. Our approach is based on the semiconductor Bloch equations and includes the dynamics of off-resonant electron-hole excitations on a microscopic level. The predicted significant broadening and structure appearing in the four-wave-mixing spectra are in good qualitative agreement with experimental results.

Figure 1

(a) FWM signal in the time domain for $\hslash {\omega }_L=0.8\mathrm{eV}$ (corresponding to ${\lambda }_L=1550\mathrm{nm}$). The black solid and dashed lines show the square of the pulse envelope, i.e., ${\left[\widehat{E}\left(t\right)\right]}^2$, and its third power ${\left[\widehat{E}\left(t\right)\right]}^6$, respectively. (b) FWM signal in the frequency domain for different $\hslash {\omega }_L$ (thick colored lines). The black solid and dashed lines show the squared Fourier transforms of $E\left(t\right)$ and ${\left[E\left(t\right)\right]}^3$, i.e., ${\left|E\left(\omega \right)\right|}^2$ and ${\left|E\left(\omega \right)\right|}^6$, respectively. In both (a) and (b) we use $\overline{t}=50\mathrm{fs}$ and $\hslash {\omega }_{cv}=1.5\mathrm{eV}$, which is close to the band gap of GaAs.

Figure 2

(a) FWM intensity in the time domain $|{\mathbf{J}}_p\left(t\right)+{\mathbf{J}}_n{\left(t\right)|}^2$ along with individual contributions. (b) Real part of the density contribution $J_n\equiv \widehat{\mathbf{e}}·{\mathbf{J}}_n$ [compare Eq. (6)] to the FWM directed third-order current density. The total $J_n$ is a sum of two contributions, $J_{n,p^{\left(2\right)}}$ and $J_{n,n^{\left(2\right)}}$, which are also shown. In (b) the black solid and dashed lines correspond to $\widehat{E}\left(t\right)$ and ${\left[\widehat{E}\left(t\right)\right]}^3$, respectively. In (a) these envelopes are squared and shown by the same line types. The parameters are ${\lambda }_L=1550\mathrm{nm}$ ($\hslash {\omega }_L\simeq 0.8\mathrm{eV}$), $\overline{t}=55\mathrm{fs}, T_1=\infty , T_2=150\mathrm{fs}, \hslash {\omega }_{\mathbf{k}=0}=1.58\mathrm{eV}, m_e^{*}=0.1m_0, m_h^{*}=0.5m_0, \left|\mathbf{d}\right|=3e\AA$; that is, the material parameters closely correspond to GaAs.

Figure 3

FWM intensity in the frequency domain $|\omega {\mathbf{J}}_p\left(\omega \right)+\omega {\mathbf{J}}_n{\left(\omega \right)|}^2$. The black solid and dashed lines correspond to the squared Fourier transform of $E\left(t\right)$ and ${\left[E\left(t\right)\right]}^3$, i.e., ${\left|E\left(\omega \right)\right|}^2$ and ${\left|E\left(\omega \right)\right|}^6$, respectively. (a) The FWM signal for ${\lambda }_L=1550\mathrm{nm}$ along with individual contributions. The parameters are the same as in Fig. 2. (b) The right spectrum is the same as in (a); for the left one we changed only the excitation wavelength to ${\lambda }_L=1400\mathrm{nm}$ (both spectra are normalized). This roughly matches the excitation conditions used in the experiment (compare Fig. 4).

Figure 4

Normalized transmission and FWM spectra of the CdTe and GaAs samples at a temperature of 10 K. The FWM spectra have been integrated over a time delay of 125 fs around the zero time delay between the two excitation pulses and are subtracted by the uniform stray light of the excitation pulses. While the directly transmitted pulses (dotted lines) represent the spectrum of the respective excitation pulses, the FWM signals (solid lines) show a distinct spectral broadening.

Figure 5

Contour plot of the self-diffracted light in the direction of $2{\mathbf{k}}_2-{\mathbf{k}}_1$ of a 500-$\mu \mathrm{m}$-thick GaAs sample. The intensity of the detected light is reflected by the color scale. For a time delay of the two excitation pulses close to 0 ps a pronounced FWM signal is observed, while for larger time delays between both pulses only comparatively weak signals of stray light are detected. The FWM signal is substantially broadened compared to the stray light of the excitation pulses.

Figure 6

(a) Snapshot of the transient density grating $\mathrm{\Delta }{n}^{\left(2\right)}$ at $t=10$ fs generated by the interference of two incident plane-wave pulses which propagate in the ${\mathbf{k}}_1$ and ${\mathbf{k}}_2$ directions, respectively. The sample medium is described as a small slab with a homogeneous density of two-level systems $({10}^{24}/{\mathrm{cm}}^3)$ with transition dipoles $\left|d\right|=0.624\mathrm{e}\AA$. The transition energy of the two-level systems is $\hslash {\omega }_{cv}=1.6\mathrm{eV}$; the incident pulses have a frequency of ${\omega }_L=0.6{\omega }_{cv}$, have a Gaussian temporal envelope with $\overline{t}=55\mathrm{fs}$, and have maxima of their envelopes at $x=0.3875\mu \mathrm{m}$ for $t=0$. (b) Snapshot of the energy flux density of the third-order electromagnetic field at 200 fs. At this time the diffracted pulses have traveled about $60\mu \mathrm{m}$. The FWM direction of $2{\mathbf{k}}_2-{\mathbf{k}}_1={\mathbf{k}}_2+({\mathbf{k}}_2-{\mathbf{k}}_1)$ is indicated by the red arrow. In addition, a second pulse diffracted into the direction of the first pulse ${\mathbf{k}}_1={\mathbf{k}}_2-({\mathbf{k}}_2-{\mathbf{k}}_1)$, which corresponds to a pump-probe signal, is visible. The indicated thickness of the sample (gray) is not to scale.