## Manipulating coherence resonance in a quantum dot semiconductor laser via electrical pumping |

Optics Express, Vol. 22, Issue 11, pp. 13288-13307 (2014)

http://dx.doi.org/10.1364/OE.22.013288

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### Abstract

Excitability and coherence resonance are studied in a semiconductor quantum dot laser under short optical self-feedback. For low pump levels, these are observed close to a homoclinic bifurcation, which is in correspondence with earlier observations in quantum well lasers. However, for high pump levels, we find excitability close to a boundary crisis of a chaotic attractor. We demonstrate that in contrast to the homoclinic bifurcation the crisis and thus the excitable regime is highly sensitive to the pump current. The excitability threshold increases with the pump current, which permits to adjust the sensitivity of the excitable unit to noise as well as to shift the optimal noise strength, at which maximum coherence is observed. The shift adds up to more than one order of magnitude, which strongly facilitates experimental realizations.

© 2014 Optical Society of America

## 1. Introduction

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49. A. Vüllings, E. Schöll, and B. Lindner, “Spectra of delay-coupled heterogeneous noisy nonlinear oscillators,” Eur. Phys. J. B **87**, 31 (2014). [CrossRef]

## 2. Quantum dot laser model

51. C. Otto, B. Globisch, K. Lüdge, E. Schöll, and T. Erneux, “Complex dynamics of semiconductor quantum dot lasers subject to delayed optical feedback,” Int. J. Bif. Chaos **22**, 1250246 (2012). [CrossRef]

52. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. **16**, 347–355 (1980). [CrossRef]

*ℰ*of the electric field. Taking into account only one roundtrip of the light in the external cavity, the field amplitude

*ℰ*(

*t*−

*τ*) delayed by the external cavity roundtrip time

*τ*is coupled back into the laser with feedback strength

*k*and rotated by the external cavity phase

*C*.

*J*. Within the QDs formed by pyramidal structures of InGaAs, localized, discrete electron and hole ground states are considered that lead to a wavelength of the optical transition of

*λ*

_{opt}= 1.3

*μ*m. The occupation probabilities of electrons and holes in these states are denoted by

*ρ*and

_{e}*ρ*, respectively.

_{h}53. R. Wetzler, A. Wacker, and E. Schöll, “Non-local Auger effect in quantum dot devices,” Semicond. Sci. Technol. **19**, S43 (2004). [CrossRef]

54. T. R. Nielsen, P. Gartner, and F. Jahnke, “Many-body theory of carrier capture and relaxation in semiconductor quantum-dot lasers,” Phys. Rev. B **69**, 235314 (2004). [CrossRef]

56. N. Majer, S. Dommers-Völkel, J. Gomis-Bresco, U. Woggon, K. Lüdge, and E. Schöll, “Impact of carrier-carrier scattering and carrier heating on pulse train dynamics of quantum dot semiconductor optical amplifiers,” Appl. Phys. Lett. **99**, 131102 (2011). [CrossRef]

*W*) and holes (

_{e}*W*) in the carrier reservoir, and therefore depend on

_{h}*J*. Note, that we use only dimensionless quantities for the rates (details of the non-dimensionalization can be found in [7

7. K. Lüdge, E. Schöll, E. A. Viktorov, and T. Erneux, “Analytic approach to modulation properties of quantum dot lasers,” J. Appl. Phys. **109**, 103112 (2011). [CrossRef]

51. C. Otto, B. Globisch, K. Lüdge, E. Schöll, and T. Erneux, “Complex dynamics of semiconductor quantum dot lasers subject to delayed optical feedback,” Int. J. Bif. Chaos **22**, 1250246 (2012). [CrossRef]

*E*= 210meV and Δ

_{e}*E*= 50meV for electrons and holes, respectively. The latter strongly depend on the size of the QDs and also on their material composition. In comparison to conventional QW lasers the carrier lifetimes

_{h}*τ*(in their dimensionless form

_{e/h}*W*= 0.7

*ns*

^{−1}being the Einstein coefficient of spontaneous emission) constitute additional timescales, which are responsible for the strong suppression of the relaxation oscillations (ROs) of QD lasers [57

57. B. Lingnau, K. Lüdge, W. W. Chow, and E. Schöll, “Influencing modulation properties of quantum-dot semiconductor lasers by electron lifetime engineering,” Appl. Phys. Lett. **101**, 131107 (2012). [CrossRef]

*τ*and

_{e}*τ*can be tuned by the pump current

_{h}*J*, which permits to tune the turn-on damping of the laser.

*ρ*and

_{e}*ρ*as well as for

_{h}*W*and

_{e}*W*. Therefore, the system of coupled delay differential equations reads Here, time

_{h}*t′*≡

*t/τ*

_{ph}is rescaled with respect to the photon lifetime

*τ*

_{ph}, where

*t*denotes the physical time, and (·)

^{′}denotes the derivative respect to

*t′*. The amplitude-phase coupling is modeled by a constant linewidth enhancement factor

*α*to admit analytical insight. Note however, that the validity of this approach depends upon the band structure under consideration. In general, the

*α*-factor is not a reliable parameter in QD lasers as shown recently in [4

4. B. Lingnau, K. Lüdge, W. W. Chow, and E. Schöll, “Failure of the α-factor in describing dynamical instabilities and chaos in quantum-dot lasers,” Phys. Rev. E **86**, 065201(R) (2012). [CrossRef]

5. B. Lingnau, W. W. Chow, E. Schöll, and K. Lüdge, “Feedback and injection locking instabilities in quantum-dot lasers: a microscopically based bifurcation analysis,” New J. Phys. **15**, 093031 (2013). [CrossRef]

*α*-factor is defined as the variation of the real refractive index, which is proportional to the real part of the complex susceptibility with the carrier density divided by the variation of the gain, which is proportional to the imaginary part of the complex susceptibility, with the carrier density. However, each charge carrier transition in the band structure under consideration contributes differently to the complex susceptibility. While the resonant transitions of the QD carriers mainly affect the gain, the main contribution to the change of the refractive index is given by the off resonant carriers of the surrounding carrier reservoir (QW) (see [4

4. B. Lingnau, K. Lüdge, W. W. Chow, and E. Schöll, “Failure of the α-factor in describing dynamical instabilities and chaos in quantum-dot lasers,” Phys. Rev. E **86**, 065201(R) (2012). [CrossRef]

*τ*and

_{e}*τ*). The carrier exchange is then fast enough to synchronize QD and QW carriers, and the approximation of a constant

_{h}*α*-factor holds. But, for very deep dots the carrier exchange between QD and QW can become so slow that it takes place on a similar timescale than the intensity pulsations. As a result, QDs and QW carriers can become desynchronized in feedback regimes, in which intensity pulsations are observed. In this regimes, the approximation of a constant

*α*-factor then fails. Nevertheless, for the band structure discussed in this paper, the approximation of a constant

*α*-factor yields reliable results.

*ρ′*+

_{e}*W′*=

_{e}*ρ′*+

_{h}*W′*holds. Thus,

_{h}*ρ*−

_{e}*ρ*+

_{h}*W*−

_{e}*W*is a constant, which is zero in the intrinsic case [59

_{h}59. K. Lüdge and E. Schöll, “Nonlinear dynamics of doped semiconductor quantum dot lasers,” Eur. Phys. J. D **58**, 167–174 (2010). [CrossRef]

### 2.1. External cavity modes–stationary solutions

*β*= 0) are discussed. These external cavity modes (ECMs) organize the phase space of the system and provide a “backbone” for more complex, e.g., chaotic, dynamics observed in these systems [61

61. V. Rottschäfer and B. Krauskopf, “The ECM-backbone of the Lang-Kobayashi equations: A geometric picture,” Int. J. Bif. Chaos **17**, 1575–1588 (2007). [CrossRef]

*ϕ*≡

*δω*

^{s}

*t*of the electric field amplitude

*s*, and

*δω*

^{s}≡

*τ*

_{ph}(

*ω*−

*ω*

_{th}) is the deviation of the frequency

*ω*of the ECM from the threshold frequency of the solitary laser

*ω*

_{th}. Inserting the ECM-ansatz (3) into Eqs. (1), we find the following expressions for the non-zero intensity solutions (

*δω*

^{s}in terms of

*α*,

*τ*, and

*C*where

*k*

_{eff}< 1 only one solution exists, and at

*k*

_{eff}= 1 a pair of ECMs is created in a saddle-node bifurcation. Increasing

*k*,

*α*, and

*τ*additional pairs of solutions are created in saddle-node bifurcations. The saddle solutions (anti-modes) are always unstable and the stability of the node solutions (modes) has to be determined by a linear stability analysis [51

51. C. Otto, B. Globisch, K. Lüdge, E. Schöll, and T. Erneux, “Complex dynamics of semiconductor quantum dot lasers subject to delayed optical feedback,” Int. J. Bif. Chaos **22**, 1250246 (2012). [CrossRef]

*W*and

_{e}*W*.

_{h}## 3. Bifurcation structure and excitable dynamics

**22**, 1250246 (2012). [CrossRef]

60. B. Globisch, C. Otto, E. Schöll, and K. Lüdge, “Influence of carrier lifetimes on the dynamical behavior of quantum-dot lasers subject to optical feedback,” Phys. Rev. E **86**, 046201 (2012). [CrossRef]

*N*

_{ph}versus feedback strength

*K*for low

*J*= 2

*J*

_{th}(red dots) and for higher

*J*= 3

*J*

_{th}(black dots) and

*J*= 4

*J*

_{th}(gray dots), respectively. Note that for the subsequent discussion we use the feedback strength

*K*, which is more intuitive because it ranges from zero to one. The bifurcation diagrams have been obtained by increasing

*K*stepwise using in each step the last

*τ*-interval of the time series of the previous run as initial condition. Figure 2(b) depicts the frequency deviation

*δω*

^{s}of the ECMs. Solid and dashed lines indicate stable and unstable solutions, respectively. For low

*K*, only one ECM (blue line) exists, which initially is stable. For

*J*= 2

*J*

_{th}this ECM is destabilized in a supercritical Hopf bifurcation at

*K*(

_{H}*J*= 2

*J*

_{th}) = 0.085 (red dot in Fig. 2(b)), which results in a stable solution with a periodically modulated

*N*

_{ph}(see leftmost inset in Fig. 2(a)). Increasing

*K*further, this periodic orbit undergoes a cascade of period doubling bifurcations. After a large period-2 window, the system becomes chaotic at

*K*= 0.21 (see middle inset in Fig. 2(a) for a time series of

*N*

_{ph}). At the end of the region with complex dynamics, the chaotic attractor collapses onto a limit cycle, and periodic pulse packages are observed in the time series of

*N*

_{ph}, which will be discussed in detail below (see rightmost inset in Fig. 2(a)). At

*K*

_{sn}= 0.2290 a new pair of ECMs is created at in a saddle-node bifurcation (limit point) indicated by an open black circle in Fig. 2(b). The position of the limit point is determined by Eq. (6) only, and is thus independent of

*J*. In Fig. 2(b), the stable 2

^{nd}ECM is depicted by a green and the unstable anti-mode by a black dashed line. The photon number

^{nd}ECMs and of the unstable anti-mode are plotted in Figs. 2(a) and 2(c), by thick blue, thick green, and black dashed lines, respectively. For

*J*= 2

*J*

_{th}, bi-stability between the periodic orbit and the 2

^{nd}ECM is observed upon its creation at K

_{sn}, until eventually at

*K*

_{hom}= 0.22920, the periodic orbit is annihilated in a homoclinic bifurcation (brown vertical arrow) with the saddle (anti-mode) of the 2

^{nd}ECM-pair. For

*K*>

*K*

_{hom}, the laser emits in stable cw operation on the 2

^{nd}ECM.

7. K. Lüdge, E. Schöll, E. A. Viktorov, and T. Erneux, “Analytic approach to modulation properties of quantum dot lasers,” J. Appl. Phys. **109**, 103112 (2011). [CrossRef]

*J*= 3

*J*

_{th}and

*J*= 4

*J*

_{th}, the Hopf-bifurcation points

*K*shift toward higher

_{H}*K*-values [9] (red dots and blue arrows in Fig. 2(b). Further, for pump currents larger than

*J*> 2.8

*J*

_{th}the end of the bifurcation cascade is not marked by a homoclinic bifurcation, but by a boundary crisis [62

62. C. Grebogi, E. Ott, and J. A. Yorke, “Crises, sudden changes in chaotic attractors, and transient chaos,” Physica D **7**, 181–200 (1983). [CrossRef]

*K*

_{cris}with the saddle (anti-mode) of the 2

^{nd}ECM pair. Bi-stability is now observed in the interval [K

_{sn},

*K*

_{cris}]. The feedback strengths

*K*

_{hom}and

*K*

_{cris}, at which homoclinic bifurcation and boundary crisis occur, have been found by up- and down-sweeping

*K*with a very small stepsize of Δ

*K*= 1 · 10

^{−5}. For up-sweeping

*K*, the system remains on the periodic orbit (chaotic attractor), up to

*K*

_{hom}(

*K*

_{cris}), while for down-sweeping

*K*the laser emits on the 2

^{nd}ECM down to

*K*

_{sn}. Therefore,

*K*

_{hom}and

*K*

_{cris}are determined by the upper limit of the bi-stability region. In contrast to the homoclinic bifurcation that is independent of the pump current for

*J*∈ [

*J*

_{th}, 2.8

*J*

_{th}], the feedback strength

*K*

_{cris}, at which the boundary crisis occurs, increases with the pump current (see dark and light arrow in Fig. 2(c)).

*K*-values little larger than

*K*

_{hom}for

*J*< 2.8

*J*

_{th}and analogously for

*K*-values little above

*K*

_{cris}for

*J*> 2.8

*J*

_{th}. In both cases, the short unstable manifold of the anti-mode acts as perturbation threshold. For

*J*≤ 2.8

*J*

_{th}, the response of the system to a super-threshold perturbation is a large excursion of the trajectory in phase space close to the “ghost” of the limit-cycle that is destroyed in the homoclinic bifurcation. For

*J*> 2.8

*J*

_{th}, the excursion in phase space is guided by the ruin of the chaotic attractor that collapses at

*K*

_{cris}. In Figs. 2(a) and 2(c) this situation is elucidated, the threshold is given by the difference of the photon numbers

^{nd}ECM (thick green line) and of the anti-mode (black dashed line). The threshold is very low for

*K*=

*K*

_{hom}and increases with

*K*. This implies that for

*J*> 2.8

*J*

_{th}, when the system re-stabilizes in a boundary crisis, the threshold can be tuned by varying the pump current and with it the critical feedback strength

*K*

_{cris}=

*K*

_{cris}(

*J*).

*N*

_{ph},

*W*)-plane for

_{e}*K*= 0.2290, i.e., just below

*K*

_{hom}, where the periodic orbit still exists. A subthreshold perturbation of the system from the stable 2

^{nd}ECM, i.e, the lasing fixed point, (green line) decays rapidly back to this steady state, while a super-threshold perturbation (blue line) yields strictly periodic pulse package, i.e., a motion along the periodic orbit. The green and the blue triangles in the closeup of Fig. 3(a) denote the starting point of the trajectories and the photon number of the anti-mode is plotted as a black line. In Ref. [50] we showed that the inter-pulse interval time

*T*

_{ISI}scales logarithmically with the distance from the bifurcation point, i.e.,

*T*

_{ISI}∼ ln|

*K*−

*K*

_{hom}|, as it is expected close to a homoclinic bifurcation [63

63. Y. A. Kuznetsov, *Elements of Applied Bifurcation Theory* (Springer, New York, 1995). [CrossRef]

^{nd}pair of ECMs has been created at the nearby saddle-node bifurcation (the green dot in Fig. 3(b) indicates the stable 2

^{nd}ECM). Afterwards, it is re-injected into the high gain region during the power dropout.

*K*= 0.23, which is a little below

*K*

_{cris}(

*J*= 3). A super-threshold perturbation (blue line in Fig. 3(d)) yields rather regular pulse packages, although they are not strictly periodic as the ones observed close to the homoclinic bifurcation. Furthermore, the inter-spike interval time does not obey a specific scaling law as the pulse packages described before. From the phase space projection in Fig. 3(b), we see that the trajectory has essentially the same shape observed close to the homoclinic bifurcation, but does not close up, which yields a certain width of the chaotic attractor in phase space. Note that these regular pulse packages are similar to those observed by Heil et al. in a QW laser with short optical feedback [64

64. T. Heil, I. Fischer, W. Elsäßer, and A. Gavrielides, “Dynamics of semiconductor lasers subject to delayed optical feedback: The short cavity regime,” Phys. Rev. Lett. **87**, 243901 (2001). [CrossRef] [PubMed]

65. T. Heil, I. Fischer, W. Elsäßer, B. Krauskopf, K. Green, and A. Gavrielides, “Delay dynamics of semiconductor lasers with short external cavities: Bifurcation scenarios and mechanisms,” Phys. Rev. E **67**, 066214 (2003). [CrossRef]

^{nd}ECM (green line) and

## 4. Coherence resonance

*t*

_{cor}is used. For a stationary stochastic process

*y*, it was introduced by Stratonovich [66] as where

*denotes the normalized autocorrelation function of*

_{y}*y*, 〈·〉 denotes the ensemble average, and the variance is given by

*from the ensemble averaged power spectral density. Here, we take the photon number as stochastic process, i.e.,*

_{y}*y*=

*N*

_{ph}. Another measure for the regularity of the pulse packages is the normalized standard deviation of the inter-spike interval time

*T*

_{ISI}[68

68. N. B. Janson, A. G. Balanov, and E. Schöll, “Delayed feedback as a means of control of noise-induced motion,” Phys. Rev. Lett. **93**, 010601 (2004). [CrossRef] [PubMed]

35. A. S. Pikovsky and J. Kurths, “Coherence resonance in a noise-driven excitable system,” Phys. Rev. Lett. **78**, 775 (1997). [CrossRef]

*T*

_{ISI}not directly from the timeseries of

*N*

_{ph}but from the carrier inversion yields more robust results, because the latter are only indirectly effected by the noise. The dropout of

*N*

_{ph}before the first intensity spike of each pulse package (cf. Fig. 3(a)) corresponds to a spike of the inversion

*ρ*

_{inv}defined in Eq. (5), which is followed by a damped oscillation towards its steady state value

*T*

_{ISI}, a threshold value

*n*-th pulse package, we define a probability density by where

*t′*denotes the time when the leading edge of the

_{n,b}*n*-th pulse first exceeds the threshold

*t′*denotes the time when the trailing edge of the pulse first falls below the threshold value, i.e.,

_{n,e}*n*-th pulse package is then determined by the first moment (mean) of the distribution function

*ρ*(

_{n}*t′*) Eventually, the sequence of inter-spike intervals

*T*

_{ISI}, from which

*R*is calculated, is defined by the difference of the timing positions of the first spikes of subsequent pulse packages. For the chaotic system, we take advantage of the special shape of the chaotic attractor. The trajectory is nearly periodic, meaning that the height of the first spikes of the pulse packages varies little compared to the height difference of the first and the second spike of each pulse package (cf. Figs. 3(c) and 3(d)). Therefore, for the deterministic system it is always possible to find an appropriate threshold value

_{T}^{nd}ECM just behind the bifurcation cascade, where the deterministic system is not bistable anymore. For instance, this implies that the deterministic system would respond to a super-threshold excitation by an excursion in the phase space along the “ghosts” of the attractors destroyed in the homoclinic bifurcation (

*J*≤ 2.8

*J*

_{th}) and the boundary crisis (2.8

*J*

_{th}<

*J*), respectively, and would then return to the stable 2

^{nd}ECM. Meaning that in contrast to the timeseries shown in Figs. 3(a) and 3(c) each super-threshold excitation is followed only by one pulse package. Subject to noise, the system can be excited if the perturbation introduced by the noise is large enough to overcome the excitability threshold. Figure 4(d) depicts

*t*

_{cor}(red triangles, right

*y*-axis) and

*R*(blue dots, left

_{T}*y*-axis) as functions of the noise strength

*β*for

*K*= 0.22921 and

*J*= 2

*J*

_{th}, i.e., for a

*K*-value closely above the homoclinic bifurcation at

*K*

_{hom}= 0.22920. Furthermore,

*t*

_{cor}is shown for

*K*= 0.2314 and

*J*= 3

*J*

_{th}(black stars, right

*y*-axis) as well as for

*K*= 0.24515 and

*J*= 4

*J*

_{th}(gray hexagons, right

*y*-axis), i.e., for

*K*-values closely above the crisis of the chaotic attractor at

*K*

_{cris}(

*J*= 3

*J*

_{th}) = 0.23324 and

*K*

_{cris}(

*J*= 4

*J*

_{th}) = 0.24514, respectively (cf. Fig. 2). A clear maximum of

*t*

_{cor}can be observed in all three cases indicating coherence resonance. Figures 4(a)–4(c) visualize the respective dynamics for values of

*β*below (

*β*= 1 · 10

^{−10}, Fig. 4(a)), at (

*β*

_{opt}= 5 · 10

^{−9}, Fig. 4(b)), and above (

*β*= 6.5 · 10

^{−8}, Fig. 4(c)) the noise strength

*β*

_{opt}, at which the maximum of

*t*

_{cor}is observed for

*J*= 2

*J*

_{th}. Analogously, Figs. 4(e)–4(g) depict time series below (

*β*= 0.02, Fig. 4(e)), at (

*β*

_{opt}= 0.038, Fig. 4(f)), and above (

*β*= 0.08, Fig. 4(g)) the noise strengths

*β*

_{opt}= 0.23325 of the coherence maximum for

*J*= 3

*J*

_{th}. The

*β*-values, at which the time series are taken, are indicated by gray dashed vertical lines in Fig. 4(d).

*T*

_{ISI}can be decomposed into the time needed to activate the system

*t*and the refractory time

_{a}*t*, which the system needs to settle back to the rest state. In our system, the rest state is the stable 2

_{r}^{nd}ECM, and the refractory time is given by the time the system needs to spiral back to the 2

^{nd}ECM after one excitation. This means that

*t*is fixed by the internal dynamics of the system, while

_{r}*t*depends on the noise strength

_{a}*β*. For low values of

*β*, the activation time

*t*is long compared to

_{a}*t*(see Fig. 4(a)). Increasing

_{r}*β*, it becomes easier for the system to overcome the excitation threshold and the pulse packages arise more regularly (see Fig. 4(b)). This is indicated by an increase of

*t*

_{cor}and a decrease of

*R*. Increasing

_{T}*β*further, pulse packages are excited more often, but the regularity of their appearance decreases and they are additionally deformed by the noise (see Fig. 4(c)). This leads to a decrease of

*t*

_{cor}and an increase of

*R*. The maximum of

_{T}*t*

_{cor}does not coincide exactly with the minimum of

*R*. This is expected, because

_{T}*t*

_{cor}accounts for coherence in periodicity of the pulse packages as well as coherence in amplitude fluctuations, while

*R*only measures the periodicity of the pulse packages.

_{T}*J*= 3

*J*

_{th}and

*J*= 4

*J*

_{th}lead to higher excitability thresholds (see Fig. 2(c)). Thus, a maximum of the correlation is therefore expected at a higher level of the noise. This is the reason why the maximum of

*t*

_{cor}shifts to higher values of the noise strength

*β*with increasing

*J*(see black stars and gray hexagons in Fig. 4(d) for

*J*= 3

*J*

_{th}and

*J*= 4

*J*

_{th}, respectively). By comparing the time traces taken at the maxima of

*t*

_{cor}for

*J*= 2

*J*

_{th}and

*J*= 3

*J*

_{th}, which are depicted in Figs. 4(b) and 4(f), respectively, two effects are prominent. On the one hand, the higher noise level in Fig. 4(f) becomes obvious, and, on the other hand, we see that the peak heights of the pulse package are varying more strongly in Fig. 4(f) than in Fig. 4(b), i.e., the amplitude jitter of the pulse packages is larger. However, the measure

*R*fails at higher values of the noise strength, because there is an ambiguity in distinguishing the peak position from positions of extreme noise events. The trajectory is just distorted so much by the noise that the first dropout in

_{T}*ρ*

_{inv}crossing

*R*has not been depicted for

_{T}*J*= 3

*J*

_{th}and

*J*= 4

*J*

_{th}. That for

*J*= 3

*J*

_{th}and

*J*= 4

*J*

_{th}the dynamics beyond the coherence maximum is dominated by the noise can be seen in Fig. 4(g) depicting for

*J*= 3

*J*

_{th}a time trace right to the maximum of

*t*

_{cor}.

*K*

_{cris}, at which the boundary crisis occurs, is depicted as a function of the pump current. It reveals that

*K*

_{cris}increases linearly with the pump current

*J*. As mentioned in Section 3, it was shown in previous works [7

**109**, 103112 (2011). [CrossRef]

*J*. Further, the feedback strengths

*K*of the first Hopf bifurcation marking the beginning of the first bifurcation cascade also reveals a linear dependence on

_{H}*J*as discussed in [9, 51

**22**, 1250246 (2012). [CrossRef]

*K*

_{cris}on

*J*shown in Fig 5(a) now suggest that the linear increase of

*K*

_{cris}with

*J*is also due to the pump dependence of the RO damping. From Fig. 2(c), it can be presumed that this linear dependence of

*K*

_{cris}on

*J*results in a square-root like increase of the projection of the excitability threshold onto the photon number

*J*. This can be seen as follows:

^{nd}ECM (thick green lines in Fig. 2(c)) and its anti-mode (dashed black lines in Fig. 2(c)). The former increases square-root like with

*J*, while the latter decreases with

*J*in the same way, which causes the square-root like increase of

*K*

_{cris}on

*J*is relatively small in the current range plotted in Fig. 5(b), the increase of

*K*

_{cris}, more precisely at

*K*=

*K*

_{cris}+ 1 · 10

^{−5}. Eventually, the dependence of the noise strength at the coherence maximum

*β*

_{opt}on

*J*is depicted in Fig. 5(c). The optimal noise level

*β*

_{opt}increases with

*J*as expected from the increase of the threshold. To our knowledge, this is the first time coherence resonance has been observed close to a boundary crisis. Further, in contrast to coherence resonance close to a homoclinic bifurcation studied in [25

25. H. J. Wünsche, O. Brox, M. Radziunas, and F. Henneberger, “Excitability of a semiconductor laser by a two-mode homoclinic bifurcation,” Phys. Rev. Lett. **88**, 023901 (2001). [CrossRef]

32. B. Krauskopf, K. Schneider, J. Sieber, S. Wieczorek, and M. Wolfrum, “Excitability and self-pulsations near homoclinic bifurcations in semiconductor laser systems,” Opt. Commun. **215**, 367 (2003). [CrossRef]

17. D. Goulding, S. P. Hegarty, O. Rasskazov, S. Melnik, M. Hartnett, G. Greene, J. G. McInerney, D. Rachinskii, and G. Huyet, “Excitability in a quantum dot semiconductor laser with optical injection,” Phys. Rev. Lett. **98**, 153903 (2007). [CrossRef] [PubMed]

19. B. Kelleher, C. Bonatto, G. Huyet, and S. P. Hegarty, “Excitability in optically injected semiconductor lasers: Contrasting quantum-well- and quantum-dot-based devices,” Phys. Rev. E **83**, 026207 (2011). [CrossRef]

39. G. Giacomelli, M. Giudici, S. Balle, and J. R. Tredicce, “Experimental evidence of coherence resonance in an optical system,” Phys. Rev. Lett. **84**, 3298 (2000). [CrossRef] [PubMed]

43. O. V. Ushakov, H. J. Wünsche, F. Henneberger, I. A. Khovanov, L. Schimansky-Geier, and M. A. Zaks, “Coherence resonance near a Hopf bifurcation,” Phys. Rev. Lett. **95**, 123903 (2005). [CrossRef] [PubMed]

25. H. J. Wünsche, O. Brox, M. Radziunas, and F. Henneberger, “Excitability of a semiconductor laser by a two-mode homoclinic bifurcation,” Phys. Rev. Lett. **88**, 023901 (2001). [CrossRef]

26. O. Ushakov, N. Korneyev, M. Radziunas, H. J. Wünsche, and F. Henneberger, “Excitability of chaotic transients in a semiconductor laser,” Europhys. Lett. **79**, 30004 (2007). [CrossRef]

*β*, to excite the system. Since this is also a perturbation of the trajectory in the phase space, we expect that our results can be verified experimentally by the two methods mentioned above.

## 5. Conclusion

## A. Derivation of the dimensionless model

*ℰ*(

*t*) of the electric field

*ν*

_{th}is the optical frequency at lasing threshold, and c.c denotes the complex conjugate. Since different dynamics is taken into account for electrons and holes the carrier subsystem consists of four coupled differential equations for the occupation probabilities

*ρ*and

_{e}*ρ*of electrons and holes in the discrete QD ground states, and the carrier densities for electrons,

_{h}*w*, and holes,

_{e}*w*, in the surrounding QW acting as a carrier reservoir Here, the phase amplitude coupling is described by the linewidth enhancement factor

_{h}*α*. Further, the optical intensity losses for the laser device of length

*L*are given by the inverse of the photon lifetime

*τ*

_{ph}. They are balanced by the linear gain term

*ρ*+

_{e}*ρ*− 1), where

_{h}*W̄*that measures the coherent interaction between the two-level system and the laser mode, and second to the number

*a*is the number of self-organized QD layers,

_{L}*A*is the in-plane area of the QW, and

*N*

^{QD}) matches the mode energies for lasing. Taking into account only one roundtrip of the light in the external cavity, the field amplitude

*ℰ*(

*t*−

*τ*

_{ec}) delayed by the external cavity roundtrip time

*τ*

_{ec}is coupled back into the laser with feedback strength

*K*and rotated by the external cavity phase

*C*≡ 2

*πν*

_{th}

*τ*

_{ec}. The roundtrip time of the light in the laser of length

*L*is denoted by

*ε*

_{bg}, and the speed of light

*c̃*. Although being completely determined by

*ν*

_{th}and

*τ*

_{ec}, the feedback phase

*C*is usually treated as an independent parameter since small variations of the external cavity length cause a variation of the phase

*C*over its full range [0, 2

*π*], while the external roundtrip time

*τ*

_{ec}is hardly affected by these fluctuations [65

65. T. Heil, I. Fischer, W. Elsäßer, B. Krauskopf, K. Green, and A. Gavrielides, “Delay dynamics of semiconductor lasers with short external cavities: Bifurcation scenarios and mechanisms,” Phys. Rev. E **67**, 066214 (2003). [CrossRef]

74. B. Haegeman, K. Engelborghs, D. Roose, D. Pieroux, and T. Erneux, “Stability and rupture of bifurcation bridges in semiconductor lasers subject to optical feedback,” Phys. Rev. E **66**, 046216 (2002). [CrossRef]

75. K. Green, “Stability near threshold in a semiconductor laser subject to optical feedback: A bifurcation analysis of the Lang-Kobayashi equations,” Phys. Rev. E **79**, 036210 (2009). [CrossRef]

*β*is the spontaneous emission factor measuring the probability that a spontaneously emitted photon is emitted into the lasing mode. The Einstein coefficient for spontaneous emission is denoted by

*W*. It can be determined by calculating the coherent interaction of a two-level system, i.e., a single QD, with all resonator modes in the framework of the second quantization [58

58. W. W. Chow and S. W. Koch, *Semiconductor-Laser Fundamentals* (Springer, Berlin, 1999). [CrossRef]

*W̄*and

*W*differ by three orders of magnitude (See Refs. [8

8. J. Pausch, C. Otto, E. Tylaite, N. Majer, E. Schöll, and K. Lüdge, “Optically injected quantum dot lasers - impact of nonlinear carrier lifetimes on frequency locking dynamics,” New J. Phys. **14**, 053018 (2012). [CrossRef]

60. B. Globisch, C. Otto, E. Schöll, and K. Lüdge, “Influence of carrier lifetimes on the dynamical behavior of quantum-dot lasers subject to optical feedback,” Phys. Rev. E **86**, 046201 (2012). [CrossRef]

*ξ*=

*ξ*(

*t*), i.e., Here, subscripts

*a*and

*b*stand for real and imaginary parts, respectively.

54. T. R. Nielsen, P. Gartner, and F. Jahnke, “Many-body theory of carrier capture and relaxation in semiconductor quantum-dot lasers,” Phys. Rev. B **69**, 235314 (2004). [CrossRef]

56. N. Majer, S. Dommers-Völkel, J. Gomis-Bresco, U. Woggon, K. Lüdge, and E. Schöll, “Impact of carrier-carrier scattering and carrier heating on pulse train dynamics of quantum dot semiconductor optical amplifiers,” Appl. Phys. Lett. **99**, 131102 (2011). [CrossRef]

*w*) and holes (

_{e}*w*) in the carrier reservoir, and therefore depend on the pump current density

_{h}*j*. The latter is normalized by the elementary charge

*e*

_{0}in the equations for the reservoir densities (Eqs. (11d) and (11e)). The inverse of the sum of the scattering rates yields the carrier lifetimes

76. E. Schöll, *Nonequilibrium Phase Transitions in Semiconductors* (Springer, Berlin, 1987). [CrossRef]

59. K. Lüdge and E. Schöll, “Nonlinear dynamics of doped semiconductor quantum dot lasers,” Eur. Phys. J. D **58**, 167–174 (2010). [CrossRef]

*e*) and holes (

*h*), respectively. The carrier degeneracy concentrations are given by

*D*

_{e/h}k_{bo}

*𝒯*, where

*D*≡

_{e/h}*m*/(

_{e/h}*πh̄*

^{2}) are the 2D densities of state in the carrier reservoir with the effective masses

*m*. The temperature is denoted by

_{e/h}*𝒯*and

*k*

_{bo}is Boltzmann’s constant (see [60

60. B. Globisch, C. Otto, E. Schöll, and K. Lüdge, “Influence of carrier lifetimes on the dynamical behavior of quantum-dot lasers subject to optical feedback,” Phys. Rev. E **86**, 046201 (2012). [CrossRef]

*B*, where

^{S}w_{e}w_{h}*B*is the band-band recombination coefficient (see Eqs. (11d) and (11e)). All physical parameters used are summarized in Table 2.

^{S}*τ*

_{ph}[77

77. T. Erneux and P. Glorieux, *Laser Dynamics* (Cambridge University Press, UK, 2010). [CrossRef]

*t′*≡

*t/τ*

_{ph}as well as the dimensionless reservoir populations

*W*≡

_{e}*w*/(2

_{e}*N*

^{QD}) and

*W*≡

_{h}*w*/(2

_{h}*N*

^{QD}), the set of dimensionless dynamical equations (1) discussed in the main text can be derived. Where the dimensionless linear gain coefficient

*g*, the rescaled feedback strength

*k*, the dimensionless delay time

*τ*, the dimensionless coefficient of the spontaneous emission

*r*

_{sp}, the ratio of photon and carrier lifetimes

*γ*(Here the lifetime of the carrier subsystem is represented by

*W*

^{−1}.), the ratio of the Einstein-factors of induced and spontaneous emission

*r*, the dimensionless pump rate

_{w}*J*, the coefficient of spontaneous and non-radiative losses in the carrier reservoir

*c*, and the dimensionless scattering rates

*γ*multiplying the right hand sides of Eqs. (1b)–(1e) expresses the timescale separation between the fast field equation and the slow subsystem of the carriers, i.e., the QD laser is a slow-fast system if the scattering rates

## Acknowledgments

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69. | K. Lüdge, “Modeling quantum dot based laser
devices,” in “ |

70. | M. Virte, A. Karsaklian Dal Bosco, D. Wolfersberger, and M. Sciamanna, “Chaos crisis and bistability of self-pulsing dynamics in a laser diode with phase-conjugate feedback,” Phys. Rev. A |

71. | G. Huyet, D. O’Brien, S. P. Hegarty, J. G. McInerney, A. V. Uskov, D. Bimberg, C. Ribbat, V. M. Ustinov, A. E. Zhukov, S. S. Mikhrin, A. R. Kovsh, J. K. White, K. Hinzer, and A. J. SpringThorpe, “Quantum dot semiconductor lasers with optical feedback,” phys. stat. sol. (b) |

72. | M. Radziunas, A. Glitzky, U. Bandelow, M. Wolfrum, U. Troppenz, J. Kreissl, and W. Rehbein, “Improving the Modulation Bandwidth in Semiconductor Lasers by Passive Feedback,” IEEE J. Sel. Top. Quantum Electron. |

73. | A. Argyris, M. Hamacher, K. E. Chlouverakis, A. Bogris, and D. Syvridis, “Photonic integrated device for chaos applications in communications,” Phys. Rev. Lett. |

74. | B. Haegeman, K. Engelborghs, D. Roose, D. Pieroux, and T. Erneux, “Stability and rupture of bifurcation bridges in semiconductor lasers subject to optical feedback,” Phys. Rev. E |

75. | K. Green, “Stability near threshold in a semiconductor laser subject to optical feedback: A bifurcation analysis of the Lang-Kobayashi equations,” Phys. Rev. E |

76. | E. Schöll, |

77. | T. Erneux and P. Glorieux, |

**OCIS Codes**

(140.5960) Lasers and laser optics : Semiconductor lasers

(190.1450) Nonlinear optics : Bistability

(230.5590) Optical devices : Quantum-well, -wire and -dot devices

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: February 20, 2014

Revised Manuscript: April 11, 2014

Manuscript Accepted: May 6, 2014

Published: May 27, 2014

**Citation**

Christian Otto, Benjamin Lingnau, Eckehard Schöll, and Kathy Lüdge, "Manipulating coherence resonance in a quantum dot semiconductor laser via electrical pumping," Opt. Express **22**, 13288-13307 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-11-13288

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