We perform high-intensity pulse propagation experiments in semiconductors. On a free-exciton resonance, we demonstrate coherent Self-Induced Transmission. Tuning the laser towards higher energy, thus exciting continuum states, the degree of transmission is reduced. The pulse breakup disappears. Increasing the pulse intensity by several orders of magnitude, pulse breakup can be observed again.
© 1999 Optical Society of America
Coherent pulse propagation effects in atomic two-level systems have been the subject of intense investigations for more than 30 years. Following the McCall-Hahn area theorem [1, 2], which defines the area Θ of the electric field envelope E(t) by the temporal integral over the Rabi frequency Ω(t)= 2π dcvE(t)/h (with dcv being the dipole matrix element of the transition) as
coherent pulse propagation effects such as Self-Induced Transparency (SIT) have been experimentally observed by Gibbs and Slusher  in Rb vapor and Krieger and Toschek  in a Ne discharge. In SIT, a strong laser pulse is tuned to a resonance of an absorber which can be as long as several Beer absorption lengths. The Bloch vector describing the state of the two-level system is coherently driven around the Bloch sphere by the external light field with envelope E(t). This so-called Rabi flopping manifests itself in strong oscillations in the transmitted light intensity. Besides this coherent pulse breakup, further signatures of SIT include a high degree of transmission which can reach up to 100% despite the fact that one propagates through an absorbing medium. In case of coherent self-focusing, the transmission through a certain area can even be higher than unity . This high degree of transmission is only possible in the case of pulse areas 2π and larger. In this case the Bloch vector performs a full revolution around the Bloch sphere and leaves the system in its initial state. The medium behaves in this case as if it were transparent for the pulse. However, the detailed calculations  show that the transmitted temporal pulse has a sech2 shape. Additionally, the leading edge of the pulse is steepened, and a characteristic delay between incoming and transmitted pulse due to the coherent propagation in the SIT regime can take place . Even pulse compression can occur in the case that the input pulse area is different from integer multiples of 2π [8, 9].
A crucial condition for the observation of SIT is the coherence between the electric field and the two-level system. Any incoherent processes, such as atom-atom scattering or decay into other states will destroy the necessary coherence and render the observation of SIT obsolete. Additionally, complications such as degeneracy of levels [10, 11] and inhomogeneous broadening might result in deviations from the simple pulse breakup picture. Recently, Eberly has derived a modified area theorem for short optical pulses and inhomogeneous broadening . Therefore, one should always bear in mind to keep the pulse duration or at least the observation interval below the dephasing time T2.
Soon after the discovery of SIT in atomic systems, it has been assumed by several researchers [13–20] that the bound electron-hole pair in semiconductors, the so-called exciton, might be considered to give a two-level resonance well suited for SIT-like pulse propagation in semiconductors. The exciton could be bound to an impurity in the semiconductor crystal, for example a donor. The wavefunction of such a donor-bound exciton is strongly localized and the system behaves indeed as an ideal two-level system. Following this insight, Jütte and coworkers have discovered SIT in semiconductors using donor-bound excitons in CdS . In contrast, the wavefunctions of free excitons (which are Wannier excitons) are quite delocalized and therefore different excitons interact easily with each other.
In 1992, theorists had proven analytically using the Semiconductor-Bloch equations  that exact SIT, yielding 100% transmission and sech2 pulses, cannot occur on the free exciton resonance in semiconductors . This is due to the fact that large light fields mean high photon flux, giving a large exciton density in the semiconductor. Therefore, excitation-induced dephasing caused by the large interaction between the exciton wavefunctions (i.e., exciton-exciton scattering) will decrease the dephasing time, obscuring the coherent pulse breakup effects. The dephasing times in unexcited semiconductors are on the order of several picoseconds and can become as short as a few hundred femtoseconds at higher intensities . However, theory predicted extremely long-distance propagation even on the absorbing resonance and pulse breakup due to carrier-density Rabi flopping. Furthermore, the theorists calculated that the pulse area at which the large degree of transmission and the pulse reshaping would set in is not 2π, as in the atomic systems, but around π due to the renormalization of the Rabi frequency . The renormalized Rabi frequency in the semiconductor reads
where the second term is the renormalization due to the polarization (the ‘local field’) induced in the semiconductor. V∣k-q∣ is the interaction potential between different k-states of the excitonic polarization. The second term in the sum is for subpicosecond pulses and for semiconductors with a bandgap in the 1.5–2 eV region about as large as the first term, thus effectively doubling the Rabi frequency for a given electric field.
These effects, mainly due to many-body interaction between the fundamental excitations, are the main difference between ideal two-level systems and semiconductors. Over the last few years, the well known coherent transients such as Free Induction Decay, Photon Echo and Rabi Flopping have been rediscovered in semiconductors [26–28]. Pulse breakup due to escape from adiabatic following was found by Harten and coworkers , but in this case the pulse was tuned well below the excitonic resonance. Due to the fact that exact SIT was proven impossible in semiconductors, we still tried to find the predicted coherent long-distance propagation and the pulse reshaping and breakup associated with it. Using CdSe and subpicosecond pulses, we succeeded nicely and called the phenomenon Self-Induced Transmission in Semiconductors . The laser was tuned exactly to resonance in this experiment, but in this article, we want to consider also effects that arise when tuning the laser into the continuum.
We used pulses with a duration of 50–80 fs around 680 nm with pulse energies of about 100 nJ from an home-built optical parametric amplifier  pumped by a regenerative Ti:sapphire amplifier (COHERENT REGA) at a repetition rate of 200 kHz. Careful alignment and characterization using FROG  made sure that the pulses had only a rather small chirp, which is important for the experiment. The pulses were focused onto CdSe samples in a cryostat (T=8 K), and the transmitted beam was imaged onto a pinhole, cutting out the central part of the beam in order to investigate regions of constant intensity. Part (c) of figure 1 shows the experimental setup: After propagation through the sample, the pulses are time-resolved by cross correlation with 50 fs pulses in a 1-mm-thick beta-barium-borate crystal. The transmitted spectra were recorded simultaneously. Strained hot-wall epitaxial samples [33, 34] with αL=1.7 and 6.8 were escpecially tailored for this experiment (α is the absorption coefficient and L is the sample length). The large strain due to the thermal expansion and lattice mismatch with the BaF2 substrate caused an A-B exciton splitting of up to 72 meV. Figures 1a and 1b show the linear absorption spectra. Both samples show substantial inhomogeneous broadening due to strain relaxation.
In order to model the results of the pulse propagation experiment, the semiconductor Maxwell-Bloch equations have been solved. Details can be found in reference . Approximations have to be made because the nonlinear polarization of the semiconductor many-body system is too difficult to be calculated exactly. In the present case, we applied the slowly varying envelope approximation of the field, whereas the material equations include mean-field and correlation effects (diagonal and nondiagonal dephasing as well as nonlinear polarization scattering) in the second-order Born approximation.
We propagated relatively short pulses (180 fs duration) through the thin sample. Owing to the short propagation distance, the pulse breakup is not really distinct, rather a pulse reshaping can be observed , which is well described even quantitatively by the theory. At an input intensity around 12 MW/cm2, corresponding to an area of about π, the transmitted pulse has the same shape as the input pulse. Sending the 180 fs pulses through the thicker sample with αL=6.8 does not lead to the expected nice pulse breakup behavior . This is due to the fact that the pulse spectrum is broader than the inhomogeneous broadening of the exciton line, therefore leading to a situation similar to ‘sharp-line SIT’ , exhibiting wobbling and leapfrogging in the transmitted temporal pulse shapes. Sharp-line SIT had been theoretically discussed by Crisp . Pulse breakup for 0π pulses had been reported by Rothenberg et al. , and 0π pulse propagation in the extreme sharp-line limit was investigated by Matusovsky and coworkers . They propagated low-intensity 60 fs pulses through several thousand absorption lengths of Cs vapor and found good agreement between observations and theory when including resonant and off-resonant excitation. Aavikso et al. discovered optical precursors in a GaAs semiconductor , a phenomenon closely related to 0π propagation. Reducing the spectral pulse width and therefore using 900 fs pulses solved our problem with the missing pulse breakup, and the spectral pulse width was well within the inhomogeneous linewidth of the exciton. Figure 2 shows the transmitted temporal and spectral pulse shapes, and the agreement between theory and experiment is quite good. The most distinct feature is the multiple pulse breakup. The number of pulses increases nicely with pulse area.
Now we want to concentrate on high-intensity pulse propagation in the continuum. Keeping the situation similar to figure 2, now using an intensity around 70 MW/cm2, but tuning the laser 15 meV above the exciton resonance into the continuum states, we observe a transmitted temporal pulse shape as shown in figure 3: the peak transmission is reduced by at least a factor of three, the pulse does no longer show breakup, the transmitted pulse duration is only on the order of 400 fs (which is substantially shorter than the 900 fs input duration), and the leading edge occurs about 250–300 fs later than in the resonant case. The most likely explanation for this behavior is the highly increased scattering rate for free electrons and holes, leading to a much reduced dephasing time T2, and therefore destroying the coherent pulse breakup. Measurements of T2 on resonance and in the continuum using pump-probe spectroscopy and four-wave mixing [40, 41] showed that the dephasing time on resonance can range in the several hundred femtosecond range, whereas dephasing times in the continuum are in the sub-100 femtosecond range. The importance of the homogeneous broadening of the transition for pulse propagation was also pointed out by Miklaszewski et al.  The detuning dependence is not easily measured because a spectrally narrow picosecond laser pulse would be necessary to perform these experiments, being unable to resolve dephasing times shorter than the pulse width. An additional reason for the vanishing breakup in the on- and off-resonance behavior might be the unequal dipole moments of the different continuum states. Matusovsky and coworkers discussed in Ref.  that the sensitivity to pulse breakup is reduced in this case.
Additional experiments using 300 fs pulses have been performed tuning below, on, and above the resonance into the continuum. However, due to the larger spectral width and probably due to the unequal dipole moments and the multiplicity of the levels , no pulse breakup did occur. However, for intensities below 2π, a velocity slowdown of the pulses when tuning into the continuum occured around the resonance. At intensities around 2π, a clear dip in the detuning-dependent velocity was visible . This dip was interpreted by theorists treating the exciton in the semiconductor in a two-level fashion as evidence for SIT [19, 20]. We wanted to check whether under the high-intensity / short-pulse conditions pulse breakup in the continuum would be possible. Recent results from other groups provided significant promise: Fürst and coworkers  observed Rabi flopping using sub-100 fs pulses in the continuum of GaAs, and Joschko et al.  showed that coherences in the continuum of GaAs can be retained if sub-20 fs pulses are used.
Figure 4 shows our results of propagating 62 fs pulses through the thick CdSe sample with increasing intensities from 13 GW/cm2 to 70 GW/cm2. At low intensity, only a small shoulder is visible at the trailing edge. At higher intensity, a long tail develops, and also a small preshoulder is visible. At the highest intensity, clearly a second pulse is visible. Whether this breakup is due to coherent Rabi flopping of free carriers in the continuum is unclear. However, recent theoretical calculations for a thin sample by Banyai and coworkers  predict that even the continuum states do retain their coherence for several 10 fs. The transmitted spectra do not only show a very slight broadening on the low-energy side and a small redshift with increasing intensity. Therefore we think that we can rule out self-phase modulation due to Kerr nonlinearities as reason for the pulse breakup, as it would give substantially larger pulse broadening. However, this continuum breakup behavior is different from the on-resonance breakup: The transmitted pulse becomes broader than the initial pulsewidth if breakup occurs. If the breakup is due to coherent Rabi oscillations of the carrier density in the continuum, this would mean that the dipole matrix element is much smaller for continuum transitions because the breakup occurs at much larger intensities. The described behavior is not theoretically understood so far because solving the semiconductor Maxwell-Bloch equations for such large pulse areas and for such long propagation distances in the continuum is presently not a simple computational task.
In conclusion, we have shown coherent self-induced transmission in semiconductors on the free exciton resonance. Tuning the laser into the continuum leads to a pulse shortening and to a reduction of the transmission. Increasing the intensity of the pulses leads again to a pulse breakup.
This work was supported by the DFG through the Schwerpunktprogramm ‘Quantenkohärenz in Halbleitern’, the Sonderforschungsbereich 383, and the Leibniz price. We would like to acknowledge continuous support from W.W. Rühle.
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