Quasiequilibrium time-domain susceptibility of semiconductor quantum wells

We analyze the time-domain optical response of quantum well (QW) media in the quasiequilibrium approximation. The resulting macroscopic polarization can be expressed as a convolution integral that permits a simple and efficient numerical implementation for its use in dynamical modeling. As a practical example, the resulting polarization is used in conjunction with a travelling-wave model (TWM) for analyzing saturable absorption as a pulse propagates in a waveguide that incorporates an unpumped QW in its core. Frequency-selective saturable absorption and pulse distortion are observed.

Figure 1

Analytical (black) and effective imaginary parts susceptibilities (red dashes) obtained from numerical convolution for fields at different frequencies $\omega$ for ${\mathrm{\gamma ̃}}_{\perp }=100$ and $b=10$. Left (right) panel corresponds to $N=3N_t$ ($N=0$).

Figure 2

Same as Fig. 1 but extending the frequency interval to twice the Nyquist frequency. Aliasing in the numerical convolution clearly manifests as a periodization of the susceptibility.

Figure 3

Left: Imaginary part of the susceptibility obtained from Eq. (15) for $b=10$ (black line), $b=50$ (red dashes), and $b=100$ (green dots). The parameters are ${\mathrm{\gamma ̃}}_{\perp }=100$ and $N=3N_t$. Right: Zoom of the normalized gain spectrum close to the gap region. In this spectral range the three curves almost coincide in spite of the different values of $b$.

Figure 4

Same as in Fig. 3 but for the real part of the susceptibility. Notice that, in the spectral range of interest around the bandgap (right panel), the three curves are simply shifted upward as $b$ increases.

Figure 5

Optical spectra of a transparent waveguide [black (upper)] and of a waveguide with an unpumped QW in its core [red (lower)].

Figure 6

Optical spectrum of a $800$ fs FWHM gaussian pulse after single propagation in the slab without QW (black line) and with QW (red dashes). For clarity, both spectra are normalized to the maximal spectral density of the pulse obtained without QW.

Figure 7

Time domain intensity profile of a $1$ ps FWHM pulse exiting the saturable absorber (black line) and carrier density at the output facet (red dots). From top to bottom, the pulse energy is $e=5.5$ fJ and $e=55$ fJ, respectively. The pulse peak powers are normalized to unity for clarity. Notice that only the low energy pulse remains symmetric and that the more energetic the pulse, the longer the propagation time of the peak. On the bottom panel, substantial absorption occurs that significantly increases the carrier density close to the output facet.