## Abstract

We investigate the dependence of the amplitude-phase coupling in quantum-dot (QD) lasers on the charge-carrier scattering timescales. The carrier scattering processes influence the relaxation oscillation parameters, as well as the frequency chirp, which are both important parameters when determining the modulation performance of the laser device and its reaction to optical perturbations. We find that the FM/AM response exhibits a strong dependence on the modulation frequency, which leads to a modified optical response of QD lasers when compared to conventional laser devices. Furthermore, the frequency response curve changes with the scattering time scales, which can allow for an optimization of the laser stability towards optical perturbations.

© 2014 Optical Society of America

## Corrections

Benjamin Lingnau, Weng W. Chow, and Kathy Lüdge, "Erratum: Amplitude-phase coupling and chirp in quantum-dot lasers: influence of charge carrier scattering dynamics," Opt. Express**22**, 9413-9413 (2014)

https://www.osapublishing.org/oe/abstract.cfm?uri=oe-22-8-9413

## 1. Introduction

Self-organized quantum-dot (QD) lasers have been the subject of extensive experimental and theoretical studies [1]. Their low threshold currents, high temperature stability, and low sensitivity to unwanted optical perturbations in comparison to conventional lasers make them potential candidates for many types of applications. Of particular interest in recent studies are the dynamical response of lasers to the deliberate injection of an optical signal into the cavity [2–10], their sensitivity to time-delayed optical self-feedback [11–20], as well as their modulation properties [21–27].

The unique dynamical behavior of QD lasers stems from the existence of localized electronic QD states. The charge carriers in these states are coupled to each other and to the surrounding charge-carrier reservoir via nonlinear scattering processes. This scattering can occur on different timescales, depending on, e.g., the material system and temperature of the device. If they are in the order of the relevant dynamical processes, they introduce additional dynamic degrees of freedom. Furthermore, the amplitude-phase coupling in QD lasers, i.e. the dynamic change of the active medium refractive index in relation to changes of the gain, commonly characterized by the linewidth enhancement factor *α*, has been shown to lie within a wide range of possible values [21, 28] and to exhibit strongly nonlinear dependences on the operating conditions [29]. A description of the amplitude-phase coupling that goes beyond the *α*-factor has been previously considered [9, 10, 30, 31], showing that in QD lasers the use of an *α*-factor is not always justified. Nevertheless, the characterization of QD laser devices in terms of an *α*-factor is still widely employed [29].

In this manuscript, we explore the influence of the charge-carrier dynamics on the dynamic response of QD lasers. Specifically, we employ a rate equation model to determine the dependence of the amplitude-phase coupling on the charge-carrier scattering processes. The model is based on the microscopically based balance equation (MBBE) model used in [10] which was further simplified to allow for numerical path continuation of the occurring bifurcations [32]. We evaluate the laser dynamics under injection of an external optical signal and compare our results with a model that treats the amplitude-phase coupling as a fixed parameter. We highlight the modifications to the laser behavior arising due to the complex carrier dynamics in QD lasers and clarify under which circumstances the approximation of the amplitude-phase coupling by an *α*-factor is valid. We then proceed by analyzing the amplitude and frequency response of the laser device under small-signal modulation conditions, and show that it exhibits a strong dependence on the frequency of the modulation as well as on the origin of the modulation.

## 2. Model

We consider a 1.2mm long ridge waveguide edge-emitting dot-in-a-well (DWELL) single-mode laser device, consisting of a number of *a*_{L} stacked InGaAs quantum-well (QW), each embedding a density of *N*^{QD} InAs QDs. The QDs are assumed to have localized bound electron and hole states, with a ground state (GS) centered at a transition frequency corresponding to an emission linewidth of *λ* = 1.3*μ*m, and the twofold degenerate first excited state (ES). Due to the inhomogeneous broadening of the QD states not all QDs contribute equally to the gain. An intuitive approach would be to calculate the dynamics of individual QD groups, each with a common transition energy, separately [10, 33, 34]. However, such an approach greatly increases the dimensionality of the dynamical system. A much simpler approach is to distinguish only between the resonant (active) and off-resonant (inactive) part of the QD distribution, where only the resonant part contributes to stimulated emission, as we have previously used in [23]. We extend the previous approach by explicitly tracking the occupation of inactive QDs. We assume a fraction of *f*^{act} of QDs to be resonant (*f*^{inact} ≡ 1 − *f*^{act}). This approach was found to exhibit excellent quantitative agreement with the predictions of the multi-subgroup model.

The QD laser is described by dynamic equations for the electric field inside the cavity *E*, the occupation probabilities of active and inactive QDs in the GS
${\rho}_{\text{GS},\text{b}}^{(\text{in})\text{act}}$ and the ES *ρ*_{ES,b}, as well as the 2D charge-carrier density in each QW layer *w*_{b} (b ∈ {e, h}):

*κ*is the cavity loss rate,

*K*is the optical injection strength,

*E*

^{0}is the steady-state electric-field amplitude,

*ω*

_{inj}is the injection frequency. The number of active layers is

*a*

_{L},

*N*

^{QD}is the 2D QD density per QW layer, ${R}_{\text{sp},\text{m}}^{\left(\text{in}\right)\text{act}}={W}_{\text{m}}{\rho}_{\text{m},\text{e}}^{\left(\text{in}\right)\text{act}}\cdot {\rho}_{\text{m},\text{h}}^{\left(\text{in}\right)\text{act}}$ describes spontaneous recombination losses in the QD states (m ∈ {GS, ES}), ${r}_{\text{loss}}^{\text{w}}={R}_{\text{loss}}^{\text{w}}\hspace{0.17em}{w}_{\text{e}}\cdot {w}_{\text{h}}$ is an effective QW loss rate, taking into account spontaneous emission losses as well as defect and Auger recombination as one phenomenological rate,

*J*is the pump current density per QW layer. In the description of the stimulated emission, only the active QD GS states are assumed to contribute to the gain, while all off-resonant states only lead to an index change [31]. The magnitude of the gain and index change coefficients were calculated with a model treating the QD subgroups and the QW

*k*-states separately, each contributing to the total gain and index change [9]. Using the complete microscopically based laser equations, the complex gain was found to be very well described by a linear dependence on the respective carrier occupations and densities over the operating parameter range typically encountered in the laser device. We can thus write the amplitude gain

*g*and change of instantaneous frequency

*δω*due to refractive index changes as follows:

The scattering processes between the different charge-carrier states are given by

We approximate the nonlinear in-scattering rates by the following expressions, derived from fits to the full microscopically calculated rates [34]:

The corresponding out-scattering rates are given by the detailed balance relationships [35]:

*ε*

_{m,b}is the mean energy of the corresponding localized QD state,

*k*

_{B}is the Boltzmann constant, and

*T*is the charge-carrier temperature.

The numerical model treats the carrier dynamics within the QD laser on a microscopic level and thus allows for quantitative predictions of the QD laser dynamics if the bandstructure of the device is known. Thus, with the correct energy structure as an input parameter, it can be used for device optimization. In our simulations, we use scattering rates that were microscopically calculated for a given QD structure with confinement energies as given in Tab. 2. Since the scattering rates depend crucially on, e.g., the material system and device temperature, real devices can exhibit scattering rates that may differ from our calculated rates. Thus, in the following investigation, we will vary the rates and present results in dependence on their magnitude, in order to make predictions for a variety of possible QD laser devices.

## 3. Results

#### 3.1. Solitary laser dynamics

We simulate the laser device with parameters as given in Table 1. One of the most important properties of the QD laser defining the behavior under optical perturbations as well as its modulation capabilities are its relaxation oscillation angular frequency *ω*_{RO} and damping Γ_{RO}. We therefore determine the dependence of these quantities on the charge-carrier scattering lifetime. We thus define an effective GS scattering lifetime, quantifying the average time needed for the QD GS to be filled with a carrier:

*I*(

*t*) ≡ |

*E*(

*t*)|

^{2}:

*I*

^{0},

*a*,

*b*,

*γ*are fit parameters, with

*I*

^{0}the steady state intensity and

*γ*≪ Γ

_{RO}describing an underlying slow change of the intensity. The resulting relaxation oscillation parameters are plotted as a function of the inverse effective electron scattering lifetime

*τ*

_{e}

^{−1}(the scattering lifetime is evaluated at steady-state). The QD laser exhibits pronounced relaxation oscillations both for very slow and very fast scattering rates, which become increasingly damped when the rates approach the overdamped region around

*τ*

_{e}

^{−1}≈ 10

^{11}s

^{−1}(gray shaded area) [25, 36, 37]. The relaxation oscillation parameters for the microscopic rates given in Tab. 2 are denoted by the vertical line labeled “rates×1”. Apart from the relaxation oscillation parameters, another important property determining the dynamics is the amplitude-phase coupling, i.e., the change of the instantaneous electric-field frequency under changes of the charge-carrier number due to refractive index variations. This effect becomes important as soon as electric-field interference effects arise, i.e., under optical perturbations, such as optical feedback [16] and optical injection. Since in our model the amplitude-phase coupling arises directly from the carrier dynamics, we will in the following investigate its dependence on the charge-carrier scattering timescales in an optical injection setup.

#### 3.2. Optical injection

In an optical injection setup, the laser dynamics depend on the strength of the optical signal *K*, as well as the frequency detuning
$\mathrm{\Delta}{\nu}_{\text{inj}}\equiv \frac{1}{2\pi}\left({\omega}_{\text{inj}}-{\omega}^{0}\right)$, where *ω*^{0} denotes the optical frequency of the free-running laser. The dynamics of semiconductor lasers under optical injection is already well understood [4, 5, 7, 10, 37, 38], but now our microscopically based MBBE model [10] that is reduced to a 10-variable system allows for a thorough investigation of an optically injected QD laser by path continuation methods and opens possibilities for a deeper analytical understanding of the system.

By direct integration as well as numerical continuation of occurring bifurcations using the program
`AUTO07p` [32, 39], we create bifurcation diagrams of the optically injected QD laser, which is plotted in Fig. 2(a). Around zero detuning there exists a parameter region for which the laser is phase-locked to the injected signal. This phase-locked solution is delimited by either saddle-node on an invariant cycle (SNIC) or Hopf-bifurcations, both of which lead to periodic solutions outside of that region. These periodic solutions can undergo a series of bifurcations [4], which may lead to solutions of higher periodicity and to irregular or chaotic behavior, denoted by the darker color shading in Fig. 2. The specific solutions within the different regions have been discussed elsewhere [4, 7, 10]. Our focus in this paper is to determine how these regions change if different quantum-dots are used and to understand the conditions that allow the description of the QD laser with an *α*-factor.

The dynamics of the QD laser under optical injection depends crucially on the frequency and damping of its relaxation oscillations [7, 40] and the amplitude-phase coupling [4, 7]. A common way to describe this change of instantaneous laser frequency is the *α*-factor, which relates the changes in gain and refractive index by a proportionality factor. Such an approach may, however, lead to inaccurate predictions by neglecting the independent dynamics of gain and index change [9, 10]. Therefore, we compare our predictions using Eq. (7) with those obtained when using an *α*-factor, which we calculate from the changes of gain and frequency under a small injection strength. For the QD structure used here, we obtain *α*_{E} = 1.42. The index E is used to denote the *α*-factor stemming from the variation of the electric field due to the injection [41]. The dynamic frequency change is then given by *δω* = *α _{E}g*, and the calculated bifurcation diagram is shown in Fig. 2(b). A generally good agreement between the two approaches is visible. The period doubling curves span a slightly larger parameter region and are shifted towards higher injection strengths when using an

*α*-factor, while the Hopf and SNIC bifurcations closely resemble those obtained using the full model equations. Compared to our previous results published in [10], the agreement between both descriptions is improved, which is related to the higher relaxation scattering rate in the used QDs, as discussed in the following.

Given the carrier-dependent refractive index change, it is now interesting to see how the charge-carrier dynamics influence the bifurcation structure of the QD laser. We therefore simulate the laser with the scattering rates reduced by a factor of 50. Lower scattering rates can occur, e.g., in devices with stronger electronic confinement or at reduced temperature. The chosen slower carrier scattering leads to a regime where the QD states are only weakly coupled to the carrier reservoir, such that while the output power is decreased compared to the faster rates, the relaxation oscillation frequency and damping remain comparable [25,40] (compare the two cases indicated by the vertical lines in Fig. 1).

The resulting bifurcation diagram is shown in Fig. 3(a) and 3(b). The dynamics surprisingly exhibits a drastically different behavior, with a more symmetric shape of the bifurcations with respect to the frequency detuning. These modifications can be explained by the reduced variation of off-resonant carrier occupation under optical injection due to the weaker coupling to the resonant QD GS carriers, which then leads to a lower refractive index change. This can be expressed by a lower alpha factor, which evaluated for the reduced scattering rates yields *α*_{E} = 0.29. Comparing Fig. 3(a) and 3(b) reveals again a shift of the bifurcation structure of the periodic orbit outside of the phase-locked area when assuming an *α*-factor, which introduces an asymmetry in the bifurcation diagram that is not observed when using the full model. The bifurcations of the locked steady-state solutions, however, again show a very good agreement. The changes in the bifurcation structure introduced by altering the scattering processes reveals a strong dependence of the amplitude-phase coupling on the internal charge-carrier scattering timescales, as evident from the varying asymmetry of the locking regions. Even though two laser devices can have near-identical steady-state gain spectra, their response to optical perturbations can be very different (compare Figs. 2 and 3).

Apart from the carrier scattering processes, also the non-radiative losses in the surrounding reservoir indicated with
${r}_{\text{loss}}^{\text{w}}$ in Eq. (5) play an important role for the carrier dynamics. Lower losses are associated with reduced threshold currents and improved laser efficiency. In addition to the reduced scattering rates, we now calculate the bifurcation diagram for reduced nonradiative losses in the reservoir, which is shown in Fig. 3(c) 3(d). Here, the modifications to the bifurcation diagram arising from reduced scattering rates become even more pronounced, with a strongly asymmetric locking region and symmetric bifurcations of periodic solutions. Clearly, in this case, the discrepancy between the full modeling and the *α*-factor approach is very strong. A simple fit of the shape of the locking area in order to retrieve a value for *α* can therefore not be expected to accurately reproduce the dynamics of periodic solutions and the bifurcations of the phase-locked solution simultaneously.

The above discussion shows a strong dependence of the amplitude-phase coupling on the charge-carrier scattering dynamics. We thus evaluate the dependence of the static *α _{E}*-factor under optical injection, i.e., the alpha-factor that is able to describe the phase locking boundary, on the charge-carrier scattering lifetime. We evaluate

*α*from the gain and index variation under injection of a small resonant optical signal. The resulting

_{E}*α*is plotted in Fig. 4. A strong dependence of the evaluated

_{E}*α*on the scattering timescales can be seen. Its value approaches ≈ 1.7 for very fast scattering and steadily decreases with lower scattering rates. As seen from the optical injection bifurcation diagrams before, a slower scattering of the charge carriers can help to increase the stability of the QD laser by reducing the amplitude-phase coupling. With decreased charge-carrier losses in the reservoir, on the other hand,

_{E}*α*increases over the whole scattering rate interval. Nevertheless, as previously seen, the increased value of

_{E}*α*for reduced losses does not lead to a strong amplitude-phase coupling for periodic solutions in the optical injection setup, as evident from the symmetric shape of the period doubling regions in Fig. 3. Therefore, in order to fully understand the phase response of the QD laser, we perform a frequency resolved analysis of the laser modulation dynamics in the following.

_{E}#### 3.3. Modulation dynamics

The previous results imply a complex dependence of the amplitude-phase coupling on the charge-carrier dynamic timescales. Furthermore, different types of dynamics vary in their corresponding refractive index dynamics, i.e., the amplitude-phase coupling for steady-state and periodic solutions under optical injection behaves differently. We thus calculate the laser response to different types of external modulation (either optical or electrical), in order to reveal the underlying effects of the changes in the amplitude-phase coupling.

The small-signal modulation with frequency *f* of either the pump current or the electric field leads to a variation of the charge-carrier occupations in the system. These will in turn induce a variation of the electric-field gain and refractive index. The instantaneous electric-field frequency is then modulated as

*δω*

^{0}, the frequency chirp Δ

*ω*(

*f*), and a constant phase shift ${\varphi}_{\omega}^{0}$. Similarly, the amplitude gain also undergoes a modulation, with Δ

*g*(

*f*) describing its modulation depth. When assuming the amplitude-phase coupling to be described by an

*α*-factor, the frequency chirp of the laser would be given by Therefore, we define the frequency response as

*α*(

_{X}*f*) ≡ |Δ

*ω*(

*f*)|/|Δ

*g*(

*f*)|, where

*f*is the modulation frequency. The index

*X*distinguishes between a modulation of either the pump current (

*X*≡

*J*), with small Δ

*J*(chosen equal to 0.01 ×

*J*

^{0}), or a modulation of the the electric field (

*X*≡

*E*), modeled by

*k*(chosen equal to 10

^{−4}ns

^{−1}). The modulation amplitudes were chosen such that nonlinearities in the response of the laser are suppressed, i.e., only the small-signal response is evaluated.

Figure 5(a) shows *α _{X}* (

*f*) of the QD laser device for either a direct modulation of the cavity field (solid red line) or a modulation of the pump current (solid blue line). For the pump current modulation, this value corresponds to the evaluation of 2

*β/m*in experimental FM/AM measurements [42], where $\beta \equiv \frac{\mathrm{\Delta}\omega}{{\omega}^{0}}$, $m\equiv 2\frac{\left|\mathrm{\Delta}E\right|}{{E}^{0}}$ are the frequency and amplitude modulation coefficients, respectively. It can be seen that the frequency response curve exhibits a strong frequency dependence. Under pump current modulation

*α*is very high at low modulation frequencies and decreases with increasing

_{J}*f*, until it reaches a minimum around 10GHz before it increases again at high

*f*≵ 50GHz [30]. The decrease at low frequencies is related to nonlinear gain suppression [43], while the increase at fast modulation is a direct result of the finite charge carrier scattering rate, which limits the maximum frequency up to which the gain modulation can follow the pump current modulation. In terms of

*α*(

*f*) this means that the amplitude-phase coupling itself is frequency dependent, and modulations at low or very high frequencies will induce a chirp of higher magnitude than at intermediate frequencies.

A direct modulation of the electric field, as occurs when considering optical perturbations either by feedback or injection of an external optical signal, leads to a different response of the QD laser (red solid curves, *α _{E}*, in Fig. 5). The frequency response under such modulation displays a plateau for low modulation frequencies and a steady decrease for faster modulation. The plateau value of

*α*was used previously in Figs. 2 and 3 to approximate the locking regions in the optical injection setup. It is important to note that up until the cutoff frequency at ≈ 4GHz the gain and frequency modulations are nearly in phase, i.e., gain and frequency variations are proportional. Thus it is possible to accurately characterize the response of the laser under optical perturbations with frequencies below this cutoff value with an

_{E}*α*-factor. This explains the good agreement of the bifurcations of steady-state solutions under optical injection between the two descriptions of the amplitude-phase coupling seen before. Modulations at higher frequency, however, will introduce a more complex phase response.

The decrease of the phase response at high modulation frequencies is due to the finite scattering lifetime. Only the case of instantaneous charge-carrier scattering would lead to a frequency independent amplitude-phase coupling. Furthermore, *α _{E}*(

*f*) would then coincide with the value of

*α*(

_{J}*f*

_{min}) at the frequency response minimum under pump current modulation. It was suggested to use this minimal value as the device’s

*α*-factor [30]. Generally, however, an evaluation of the frequency response under pump current modulation will only give an upper limit for

*α*(

_{E}*f*), and the amplitude-phase coupling under optical perturbations will be weaker than

*α*(

_{J}*f*

_{min}). Thus trying to characterize the laser response with a single

*α*will fail [9, 10, 30, 44].

As before, we want to clarify the impact of the scattering timescales and therefore the frequency dependent analysis is repeated for the reduced scattering rates that we already used in Fig. 3. Figure 5(b) shows the modulation behavior when considering slower scattering rates. The modulation capabilities of the QD laser become worse [25, 45], as the frequency chirp under pump current modulation is increased. This is due to a less efficient modulation of the resonant charge carriers, which increases the effect of the off-resonant carriers on the refractive index. Also, now a very pronounced minimum around *f* ≈ 4GHz is visible, with a sharp increase for higher frequencies. Under a modulation of the electric field, however, a lower frequency chirp is observed, since here the resonant carriers are directly modulated, and a slower scattering thus leads to a decreased modulation of off-resonant carrier occupation. Slow scattering processes can therefore greatly reduce the frequency chirp of QD lasers under optical perturbations and increase their stability.

The modulation dynamics of lasers is also crucially affected by charge-carrier losses. Under modulations of the pump current, a slight reduction of the amplitude-phase coupling can be observed, which is more pronounced for lower modulation frequencies. Under optical modulations, however, the frequency variation is increased for low frequencies, and approaches the value of *α _{E}* in the full loss case for higher

*f*. While reduced non-radiative losses improve the efficiency of the laser device, they can therefore lead to a stronger amplitude-phase coupling and thus a potentially higher sensitivity to optical perturbations. The observed change of the bifurcation diagram under optical injection with decreased carrier losses can be explained from the changes to the frequency response curve. The low frequency

*α*is higher at lower losses, which explains the increased asymmetry of the locking region, while it remains the same at higher frequencies, thus leading to a similar bifurcation structure of the periodic solutions.

_{E}## 4. Summary

We have presented a rate-equation model based on a microscopic approach, that implements the dynamic refractive index changes explicitly from off-resonant charge carriers in a QD laser. This model allows the evaluation of the amplitude-phase coupling without the need to assume a value for the *α*-factor. The model is thus suited for predicting the chirp of QD lasers under different kinds of modulation and for comparisons with experimentally determined *α*-factors.

We have investigated the dynamic response of QD lasers to optical perturbations, specifically external optical injection, and to small-signal modulation of the pump current and cavity optical field, in dependence of the charge-carrier scattering and loss timescales. We calculated the bifurcation structure of the QD laser under optical injection, and compared the results with predictions when using a fixed *α*-factor, and found that in the case of fast carrier scattering the QD laser behavior can be reasonably well characterized by an *α*-factor. For slower scattering timescales, as, e.g., present at low temperatures, the bifurcation structure exhibits differences between the two approaches that become even more pronounced when carrier losses are low. In these cases, the bifurcations of periodic solutions and of the phase-locked solution can then no longer be simultaneously described by a single *α*. Instead, the bifurcation structure outside the locking region is found to be more symmetric, which is due to a reduced amplitude-phase-coupling at modulation frequencies of the periodic oscillations. The stability of the QD laser towards optical perturbations is thus increased for slower intrinsic time scales of the carrier dynamics, i.e., the minimum strength of optical perturbations required to induce instabilities is increased and the extent of chaotic regions in the parameter space is reduced.

Simulations of the small-signal response of the QD laser reveal a strong dependence of the FM/AM response on the modulation frequency. Under pump current modulation, the QD laser reproduces the well-known behavior of QW lasers with a sharp decrease of the FM/AM response at first for increasing modulation frequencies until it reaches a constant plateau. For even higher frequencies, however, an increase is predicted that is especially pronounced for slow scattering rates.

For direct modulations of the electric field, as occurring under optical perturbation, the FM/AM response is always smaller than under a modulation of the pump current. It shows a plateau for low modulation frequencies, and a steady decrease for modulation frequencies above a cutoff-frequency that depends on the charge-carrier scattering rates. Thus, for slower scattering between the QD and reservoir states, the overall chirp of the laser is reduced and cuts off at lower frequency, such that the amplitude-phase coupling for high-frequency modulations is significantly reduced. A reduction of charge-carrier losses, on the other hand, leads to an increase of the amplitude-phase coupling under optical perturbations and thus to a higher chirp, while it leads to a slightly reduced frequency variation under pump current modulation. The laser stability under optical perturbations can thus be improved by slow carrier scattering and sufficiently large carrier losses.

## Acknowledgments

This work was supported by Deutsche Forschungsgemeinschaft within SFB 787, and by Sandia’s Solid-State Lighting Science Center, an Energy Frontier Research Center (EFRC) funded by the U. S. Department of Energy, Office of Science, Office of Basic Energy Sciences. The authors thank E. Schöll for fruitful discussions and careful reading of the manuscript.

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