## Abstract

Gain and phase dynamics in InAs/GaAs quantum dot semiconductor optical amplifiers are investigated. It is shown that gain recovery is dominated by fast processes, whereas phase recovery is dominated by slow processes. Relative strengths and time constants of the underlying processes are measured. We find that operation at high bias currents optimizes the performance for nonlinear cross-gain signal processing if a low chirp is required.

©2008 Optical Society of America

## 1. Introduction

Self-assembled quantum dot (QD) semiconductor optical amplifiers (SOA) are anticipated to replace more conventional bulk and quantum well devices due to their broad spectral amplification range, low chirp and due to a low temperature sensitivity which holds promise for future uncooled operation [1–3].

Of paramount importance is the dynamic response of an SOA when it becomes saturated. This will ultimately determine the performance of QD SOA for use as broadband amplifiers or as fast nonlinear signal processing elements. Therefore, pump-probe measurements of gain parameters have been performed for QD SOA both in the 1.55 µm and in the 1.3 µm wavelength region [4–8]. Fast gain recovery has been observed, but fast gain dynamics does not necessarily come along with fast phase dynamics. Particularly, long-reach telecommunications require unchirped signals with a fast response both for gain and phase. However, so far the strength and relevance of the slow and fast phase dynamics under various operating conditions have never been clarified.

In this paper, we present gain and phase dynamics of 1.3 µm InAs/GaAs QD SOA extracted from heterodyne pump-probe measurements. Fast and slow processes of both gain and phase are quantified. Also the relative strength of the various processes with respect to the current bias and input power operating conditions is shown. It is found that gain recovery is predominantly fast, while phase recovery is predominantly slow under almost any operation conditions of a QD SOA. These findings have strong implications for potential applications: QD SOA are efficient nonlinear amplitude modulators and thus well suited for cross-gain signal processing applications. If phase effects should be avoided then QD SOA are best operated under high bias current densities with very small or very high input power levels.

## 2. Experimental setup and device structure

For a measurement of the amplitude and phase dynamics we implemented a heterodyne pump-probe technique with sub-picosecond resolution [4,9]. Figure 1 shows the schematic of our setup.

Pulses with a FWHM of 130 fs are generated by an optical parametric oscillator (OPO) and split into pump, probe and reference pulses. Probe and reference pulses are tagged by a frequency shift *f*
_{prb} and *f*
_{ref}, respectively, which is induced by acousto-optic modulators (AOM). The pulse repetition rate is *R*=80 MHz. The pulses are coupled into a PM fiber from where they are launched into the SOA. First, the weak reference pulse is guided through the unperturbed device under test (DUT). Next, a strong pump pulse drives the SOA into its nonlinear regime. A weak pulse probes these nonlinearities in gain and phase. A polarizing beam splitter (PBS) acts as a variable attenuator for the pump and ensures the co-polarization of all pulses. After the device, the pulse train is split in a Michelson interferometer with unbalanced arm lengths. Both copies of the pulse train are then superimposed in such a way that the reference pulse leaving the long arm coincides with the probe pulse leaving the short arm. Because of the respective frequency shifts, the lock-in amplifier detects amplitude and phase of the photodiode current at the difference frequency *f*
_{ref}-*f*
_{prb}. For each data point, a reference measurement without pump pulses is used to correct for phase and amplitude drift. By varying the optical delay τ between -10 ps and 300 ps, the temporal evolution of gain and phase can be measured. To reduce the impact of the mean-zero noise, the data is over-sampled by using step width much smaller than the pulse width.

The investigated samples are QD SOA grown with MBE on a GaAs substrate. The active region consists of a stack of 10 layers of self-assembled InAs quantum dots with a density of approximately 1.3×10^{11}cm^{-2}, each layer covered with InGaAs and a spacer of 33 nm thickness. The active region is surrounded by 1.5 µm thick p- and n-doped Al_{0.35}Ga_{0.65}As cladding layers. The deeply etched ridge waveguide is *W*=4 µm wide and *L*=2 mm long. The facets are AR coated and tilted by 7° to avoid lasing. The SOA chip gain is 18 dB and the coupling losses of the structure are about 3 dB per facet.

Figure 2(a) shows the amplified spontaneous emission (ASE) of the SOA and the spectrum of the pulsed laser source that was tuned to *λ*=1295 nm. Figure 2(b) shows the suppression of the measured gain relative to the small-signal gain under pump probe conditions, where the pump beam is blocked and the power of probe and reference pulses are varied. The obvious outliers are insignificant, as they are outnumbered by more than a factor of 20 by the total number of measurement points (620). For a gain compression of 3 dB, an input saturation power of -11.5 dBm is found, which equals a pump pulse energy of 0.89 pJ. Pump, probe and reference pulses are co-polarized and coupled into the same quasi-TE mode of the waveguide, where the dominant component of the electric field is parallel to the substrate plane.

Figure 3 shows experimental data of a pump-probe measurement of a typical QD SOA for a bias current of 100 mA which corresponds to a current density of *i*=1.25 kA/cm^{2} and a pump power of -11 dBm (dots). The empirical model and procedure leading to the fit (solid line) will be explained in the next section.

## 3. Device model and fitting procedure

We fitted the measured data by a simple empirical model that reproduces all observed features. Starting with a physical model and taking into account the finite temporal width of the laser pulses, the complex amplitude transmission function is calculated and finally used to fit the measured data.

Detailed models predict complex carrier dynamics taking into account several different relaxation processes [7], each associated with a characteristic time constant. However, in pump-probe experiments the number of measurable time constants is limited by temporal resolution and by noise. Therefore, it is only possible to draw conclusions regarding the dominant relaxation processes, each of which is commonly approximated by an exponential time dependence [8].

In this paper, we consider a fast process associated with the quantum dot carrier capture time τ_{1} and a slow process associated with the wetting layer capture time τ_{2}. Both processes influence the material gain Δ*g*(*t*) and the phase dependence Δ*n*(*t*) simultaneously, but with different magnitudes Δ*g*
_{1}, Δ*g*
_{2} and Δ*n*
_{1}, Δ*n*
_{2}. To confirm this assumption, we have fitted the measured data allowing for individual time constants for the gain and phase recovery. These time constants turned out to be equal within the experimental margin of error. In addition, the probe also experiences instantaneous two-photon absorption (TPA) under the influence of a strong pump.

The change of the net gain Δ*g* and the refractive index Δ*n* in response to an ideal impulse δ(*t*) then becomes

$$\Delta n\left(t\right)=u\left(t\right)\left[\Delta {n}_{1}\mathrm{exp}\left\{-\frac{t}{{\tau}_{1}}\right\}+\Delta {n}_{2}\mathrm{exp}\left\{-\frac{t}{{\tau}_{2}}\right\}\right],$$

where *u*(*t*) is the unit step function and Δ*g _{TPA}*, Δ

*g*

_{1}, Δ

*g*

_{2}, Δ

*n*

_{1}, Δ

*n*

_{2}are gain and phase parameters, respectively. To keep the number of free parameters small, the instantaneous influence on the phase due to Kerr-nonlinearities was neglected in this empirical model.

The measurement setup has a limited temporal resolution, caused by the finite width of pump and reference pulses. Therefore, before fitting the experimental data, we need to convolve the ideal impulse response in Eq. (1) with the pump pump-probe cross-correlation *F*
^{(2)} [9].

Fitting of the measured data is then performed to the normalized complex amplitude transmission factor *T*(τ)=|*T*|*e*
^{jφ} of the probe pulse

where the delay τ is the pump-probe-delay, Γ denotes the field confinement factor, *L* is the length of the SOA, and *k*
_{0}=2*π*/*λ* the vacuum wave vector at the center wavelength *λ*. In relation to the single pass gain *G*
_{0}, the gain suppression *G*/*G*
_{0} is then related to the normalized amplitude transmission by *G*/*G*
_{0}=|*T*|^{2}. Figure 3 shows the results of the fitting procedure (solid line) and the separate contributions by fast (dashed line), slow (long-dashed line) and TPA processes (dash-dotted line).

## 4. Dependence of fast and slow processes on pump power and bias current density

The dependence of the process time constants on pump power and bias current density for both the fast and the slow process are plotted in Fig. 4. The time constants depend only weakly on the input pump power. For all measurements we found a fast process with a typical time constant of τ_{1}<10 ps and a slow process with a typical time constant of τ_{2}>100 ps up to 500 ps. The two outliers in Fig. 4(a) are due to the small changes introduced by low input powers, which marks the noise limit.

We attribute the smaller time constant to carrier capture from the wetting layer into the QD states, and the larger one to the recovery of the carrier numbers in the wetting layer. When changing the pump power by more than two orders of magnitude, the time constants change by less than a factor of two. The dependence on the bias current density shown in Fig. 4(b) is much stronger. For the fast process a power-law behaviour is found (see inset),

the exponent of which slightly differs from the value of *β*=0.69 expected for an Auger assisted capturing process [5]. We estimate the bias current density where the fast process becomes nearly instantaneous on the scale of the pump pulse width by extrapolating the fit to a value of 10 kA/cm^{2}, in agreement with theoretical predictions [10].

For all practical applications it is not only important to know the time constants but rather the strengths of the respective processes. Figure 5 shows to what extent the various slow and fast nonlinear processes contribute to the overall nonlinearity, depicted as a function of input pump power and bias current density. Figure 5(a) shows that the gain dynamics is mostly dominated by the fast process while the slow process hardly contributes. This finding has often led to the conclusion that QD dynamics are fast in general. However, our measurements of the phase changes depicted in Fig. 5(b) confirm theoretical predictions, that phase recovery is to a large extent dominated by slow processes [11].

The relevance of the contributions of fast and slow processes are probably best visualized in Fig. 6(a), where the ratios of the parameters Δg_{1}/Δg_{2} and Δ*φ*
_{1}/Δ*φ*
_{2} are plotted. The figure shows that the gain is dominated by the fast process (the ratio is larger than 1). For the phase change both fast and slow processes contribute alike (a ratio around 1).

It is interesting to note, that the bias current density has no measurable influence on the ratio Δ*φ*
_{1}/Δ*φ*
_{2}, while the relative strength of the fast process for the gain rapidly increases, Fig. 6(a). Identical changes of the number of free carriers in the wetting layer and inside the QD lead to virtually identical phase changes, even if the current density increases. For the gain, which is determined by the QD states, the situation is different. The QD carriers deplete strongly and rapidly, and the refilling of the QD states is mostly determined by the capture time of carriers into the quantum dots but also limited by carrier number in the wetting layer. This is the reason why Δg_{1}/Δg_{2} increases with the bias current density.

By comparing the relative amplitudes of the net gain- and phase changes we calculate the alpha-factors *α*
_{1}, *α*
_{2} from Eq. (1) for the slow and fast processes independently,

Alpha-factors calculated from these coefficients are shown in Fig. 6(b). The fast process has a very low alpha-factor *α*
_{1}<1 that is close to zero for small current densities and low input pump powers, which is expected for quantum dot devices [7]. The slow process has an alpha-factor of 1<*α*
_{2}<22 that strongly increases with the bias current density.

To visualize the temporal evolution of the phase response of Eq. (1), it is convenient to introduce an effective time dependent alpha-factor [12],

In the literature, the absolute phase change Δ*φ*(*t*) is often related to the absolute gain change Δ*g*(*t*) by using an alpha-factor definition that integrates over the infinitesimal changes used in our definition of *α _{eff}*(

*t*) in Eq. (5) [6]. This integrated alpha-factor then takes the form

This definition is useful to calculate the absolute phase change at a given time Δ*φ*(*t*) in comparison to the reference phase for an undepleted device. The differential definition of *α _{eff}*(

*t*) in Eq. (5) gives insight to the temporal evolution of the alpha-factor experienced by the pulse.

Figure 7 shows a characteristic plot of the time dependent effective alpha-factors *α _{eff}*(

*t*) and

*α*(

_{int}*t*) based on data from Fig. 3. The plot shows an almost negligible effective alpha-factor during the first 10 ps. The quantum dot processes therefore can be regarded as almost chirp free in this time window. The situation dramatically changes when the slow regime becomes dominant. The alpha-factor becomes large and might contribute significantly to the overall chirp. As predicted, the integrated alpha-factor also shows strong dynamics during the first 20 ps [7]. After this time, both dynamic alpha-factors give the same value, which is identical to the alpha-factor

*α*

_{2}of the slow recovery process.

For practical applications effective 90% to 10% recovery times of gain *G* and phase *φ* might be more useful. Fig. 8 shows the dependence on pump power and bias current density of the effective 90/10 recovery times calculated from the fitted device response. The phase recovery is clearly dominated by the slow process. It shows an effective recovery time of hundreds of picoseconds and cannot be improved considerably by changing the input power or the bias current density. The gain recovery is considerably faster. The associated effective time constant increases with input power, because the gain depletion is much stronger, but rapidly decreases for higher bias current densities, as the capturing process into the QD states becomes more efficient.

## 5. Summary

Pump-probe measurements of QD SOA reveal two characteristic timescales, firstly a small time constant attributed to quantum dot carrier capture times. This fast process that dominates the quantum dot dynamics during the first 10 picoseconds determines mainly the material gain response and shows a very small alpha-factor, i.e. a small chirp. Secondly a large time constant associated with the slow refilling of the wetting layers governs the quantum-dot dynamics in the time frame after 15 ps. It contributes only little to the gain dynamics but since the associated alpha-factor is large (between 1 and 22 depending on the operating conditions) it dominates the phase response of the device and may significantly contribute to chirp.

Because of their fast gain dynamics, QD SOA are well suited as ultra-fast nonlinear elements for switching applications and regeneration schemes based on cross-gain modulation [13]. Distortions from chirp can be avoided either by using high input powers thereby improving the effective phase recovery time, or by using very low input powers where the total phase distortion becomes negligible. For both operating points, the effective gain recovery can be additionally improved by using high bias current densities.

## Acknowledgments

The authors would like to thank Dr. Andrew Ellis, Tyndall National Institute, University College Cork, for discussion. This work was supported by the European project TRIUMPH (grant IST-027638 STP) and by the Center for Functional Nanostructures (CFN) of the Deutsche Forschungsgemeinschaft (DFG) within Project A4.4.

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