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Optical and electrical tuning of the propagation time of 170 fs pulses in a quantum dot semiconductor amplifier at room temperature is demonstrated. Both pulse slowdown and advancement is possible and we achieve fractional delays (delay divided with pulse duration) of up to 40%. The results are explained by a simple gain saturation model.
©2005 Optical Society of America
1. Introduction
Optical pulse propagation in a medium of dipole oscillators has received renewed attention in the optoelectronics and telecommunications community since the recent realization of tunable pulse propagation delay in solid-state and semiconductor optical media [1,2]. This interest is motivated by the fact that a component with tunable optical delay is long sought for as it, among other applications, enables a whole new class of functionalities in optical communications systems and in signal processing [3]. The reported results, however, achieve large pulse delays (group velocities down to 57 m/s in [1]) at the cost of a small bandwidth (37 Hz in the mentioned case). In this paper, we report measurements of slowing down and speeding up the propagation of 170 fs pulses, having a very large bandwidth of 2.6 THz, through a quantum-dot (QD) semiconductor amplifier (SOA) at room temperature. This extremely large bandwidth, on the other hand, is associated with a rather small group index change of ∆ng=4*10-3. The observed delay is demonstrated to be tunable both electrically and optically and is seen to be limited to a fractional delay (ratio of pulse delay to FWHM pulse width) of not more than 0.5. The main features of the experiment are shown to be explained by a simple theoretical model, attributing the pulse transit time modification to absorption and gain saturation of the medium. The explanation is consistent with the mechanism of coherent population oscillations (CPO) phenomenon previously reported.
2. Eksperiment and sample details
Intensity and bias dependent changes in propagation time and shape of 170 fs (FWHM) pulses were measured using a cross-correlation technique (Fig. 1), where the pulse transmitted through the SOA beats with a reference pulse in a balanced, heterodyne detector. The pulse source in the experiment was a Ti:Sapphire, regenerative amplifier and optical parametric amplifier cascade delivering near-transform limited Gaussian pulses at the central wavelength of 1260 nm corresponding to the dot ground-state (GS) transition. From this source, the pump and reference beams were derived using an acousto-optical modulator (AOM) operating at 40 MHz. By varying the amplitude of the AOM drive signal, the intensity of the pump beam could be changed over two orders of magnitude without changing its arrival time at the SOA.
Fig. 2. (a) Measured (dots) and fitted (lines) pulse gain saturation curves for the QD SOA under investigation. See text for explanation. (b) Small signal gain and pulse saturation energy values extracted from the fitted pulse gain saturation curves.
The SOA under investigation had an active layer of 5 sheets of InAs/InGaAs/GaAs self-assembled QDs embedded in a PIN diode structure [4]. Light guiding was accomplished in a 2 mm long and 7 μm wide ridge-waveguide with 7 deg. angled facets to avoid back-reflection. The sample was mounted on a Cu heat sink kept at a constant temperature of 15 degrees C. Light coupling of free-space laser beams into and out of the component was done using high-NA microscope objectives. The guided mode was very tightly confined around the active layer in the vertical direction. This resulted at the same time in a high confinement factor and a large vertical divergence of the guided mode outside the waveguide. As a result of the large divergence, large coupling losses resulting in a net transmission of -20 dB was found when the component was biased at transparency. The dot ensemble featured a GS emission wavelength of 1260 nm and an inhomogeneous linewidth broadening of 96 nm inferred from the amplified spontaneous emission spectrum. Single-pulse gain saturation curves (Fig. 2(a)) as function of bias current were used to characterize the sample’s small-signal gain and saturation energy (Fig. 2(b)). The small-signal gain and saturation energy was extracted from the gain saturation curve by fitting to the gain model detailed in the model section below. The reported pulse energies here and in the following were measured in free-space before incoupling into the waveguide.
3. Experimental results
Fig. 3. Self-slowdown and – advancement vs. power and bias of a 170 ps pulse in a QD SOA. For each current, the fractional delay is defined to relative to the propagation time at small pulse energy.
The fractional delay of a pulse through the SOA (propagation time minus propagation time in the small-signal regime divided by pulse FWHM duration) for increasing pulse energy is shown in Fig. 3 for different bias currents. The pulse position is determined by a Guassian fit to the pulse envelope. In the small signal regime, before the absorption/gain shows any saturation (see Fig. 2(a)) the pulse travel time through the SOA remains constant. As the pulse energy increases into the saturation regime, we see a gradual pulse slowdown/advancement at bias currents in the absorption/gain regime. Close to the transparency current (12 mA) the travel time remains constant throughout the pulse energy range investigated. The absolute value of the relative delay increases monotonously with pulse energy reaching its highest value of 70 fs at zero bias (absorption region) and 30 fs when the SOA is biased to gain.
Fig. 4. Pulse advancement in the gain regime (upper panel) and pulse slowdown and distortion at zero bias (lower panel) by comparison of pulses in the small signal regime and close to saturation. The shown pulses have all been normalized to the same height.
The absolute pulse delay change observed in the experiment is larger in the absorption regime than in the gain regime. The large pulse slowdown is accompanied by a considerable pulse distortion (fig. 4), where the trailing edge of the pulse is elongated to give a slower fall time than rise time. In comparison, the pulse advancement seen in the region of positive gain, preserves the pulse shape. In general, only small pulse distortion was found at any current except zero bias
4. Model
The propagation of a coherent optical pulse in a medium consisting of an ensemble of dipole oscillators display widely different behavior depending on key dipole parameters such as dephasing time T2, lifetime T1, inhomogeneous linewidth ∆E and input pulse parameters such as duration τ and energy E0. We tentatively apply a simple model of the pulse propagation in the adiabatic regime (T2 ≫ τ) knowing that the typical dephasing time of deeply confined QDs at room temperature is comparable to our pulse length of 170 fs [5].
The material gain g of an electromagnetic wave in a resonant optical medium with density N 1 (N 2) of the upper (lower) state is proportional to the population inversion (N 2-N 1)/(N 2+N 1) of the resonance in question. In an SOA, the inversion in absence of an input field is dependent on the injected bias current. As an input pulse of temporal power envelope P in(t) propagates through the medium it is gradually driven toward transparency. At time t the integrated energy E of the pulse is given by
Assuming a slow gain response compared to the pulse length, so that no gain recovery takes place we can evaluate the Einstein A and B coefficients for a given integrated pulse energy to find the steady-state excited state to ground state population ratio N 2/ N 1, which gives us the instantaneous material gain:
In this expression, g 0 is the small-signal material gain, and E S the pulse saturation energy. The pulse exiting the optical medium will display a temporal envelope
where L is the length of the amplifying medium. The model can be derived from the rate and propagation equations of [6] by neglecting internal loss and assuming pulse duration much smaller than the carrier lifetime (but longer than the carrier-carrier scattering time, within which quasi-Fermi equilibrium is reached in the bands [7]). In order to determine the bias dependent g 0 and E S, the only two parameters of this simple model, we fitted Eq. (2) with an additional two-photon absorption term of the form C TPA E 2 to the measured single pulse gain saturation curves of Fig. 2. The fitted curves as well as the fit parameters are also shown in the figure.
In the absorption regime, the leading edge of the pulse will experience less attenuation than the trailing edge of the pulse. This reshaping of the pulse envelope results in a shift of the pulse peak corresponding to a pulse slowdown (Fig. 5). In the gain regime, a pulse reshaping with the opposite sign occurs, leading to pulse advancement.
The fractional pulse delay defined as in the experimental section is shown in Fig. 6 as function of pulse input energy for different bias currents. Pulse position is defined by a Gaussian fit to the calculated pulses to make the results comparable to the experiment. Three distinct pulse energy regimes can be identified. In the small-signal regime when the pulse energy is low relative to the saturation energy, the number of carriers excited or depleted by the pulse is small so that the pulse shape is preserved and the peak shift is linear in energy. In this limit one finds the following expression for the fractional delay ∆t / τ
In the intermediate spectral-hole burning regime, the pulse delay/advancement reaches its maximum value. At very high input powers, the very front of the pulse saturates the gain leading to a pulse shift decreasing toward zero.
Fig. 6. Modelled self-slowdown and -advancement times of a pulse under conditions corresponding to those of Fig. 4.
5. Discussion
A qualitative agreement between the model and the experiment is seen regarding the observed shift in pulse travel time as function of pulse energy and bias current in the pulse energy range over which a comparison was possible. In general the model predicts smaller delays than those observed in the experiment. The calculated extremum in pulse time shift occurs at pulse powers higher than those accessible to the experiment, so this part of the prediction could not be tested. At currents from 5 mA and up, the experiment shows little pulse distortion, in good agreement with the calculation. At zero bias, however, the serious pulse distortion seen in Fig. 4 is not reproduced well by the calculation.
The self-slowdown- and advancement presented here are similar in nature to those in [1] except we use pulses of ca. 11 orders of magnitude larger bandwidth and achieve delays which are smaller by the same amount. The observed pulse delay in [1] was interpreted to be a result of the Coherent Population Oscillations (CPO) mechanism. In [8], we have shown that the group delay induced on an intensity modulated signal can be equally interpreted in terms of absorption/gain saturation or the effect of wave mixing between carrier and sideband, expressing the effect of CPO.
Our model suggests that a fractional delay of not more than ca. 0.5 is possible. This limitation is due to the nature of the gain saturation, which cannot reshape the pulse so as to correspond to a time shift larger than the pulse duration.
A more detailed model of the pulse propagation would need to take into account a number of physical effects of a real-world QD ensemble. For instance, the instantaneous Kerr effect is not taken into account. This effect will lead to self-phase modulation and may possibly explain part of the pulse broadening seen in absorption. Also, the finite QD T 2 time does affect pulse propagation. In an early study, Hopf, Rhodes, and Szöke [9] analyzed experimentally and theoretically the delay of intense pulses in the coherent regime in a strongly inhomogeneously broadened medium (ħ / ∆E ≫ T 2 ≫ τ). They found an intensity dependent delay at absorption qualitatively similar to us. An important conclusion in their case was, however, that the pulse reshaping in the coherent regime not simply stems from a gain saturation effect, but rather from a coherent redistribution of energy from the leading to the trailing pulse edge. Such an effect may lead to larger fractional delays than predicted by our gain saturation model. We also note that the assumption of long pulse duration compared to intra-dot relaxation times can be criticized, as the QDs display fast, phonon-mediated carrier relaxation leading to a gain recovery on the order of a few hundred fs [10].
6. Conclusion
We have investigated Gaussian pulse propagation of 170 fs (FWHM) pulses in a QD SOA. The results show optical and electrical tunability of pulse the propagation leading to fractional delays of up to 40%. The findings were shown to be consistent with a simple model of gain saturation. A severe pulse distortion seen at zero bias current could not be explained by our simple model.
Acknowledgments
Alexey Kovsh of the Ioffe Institute (now at NL semiconductor) is acknowledged for growing the QD sample. Matthias Laemmlin and Dieter Bimberg of Technische Universität Berlin are acknowledged for wafer processing. This project was supported by the EU-IST project DOTCOM (project no. IST-2000–28713).
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