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Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 11 — Jun. 2, 2014
  • pp: 13288–13307
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Manipulating coherence resonance in a quantum dot semiconductor laser via electrical pumping

Christian Otto, Benjamin Lingnau, Eckehard Schöll, and Kathy Lüdge  »View Author Affiliations


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

Semiconductor quantum dot (QD) lasers [1

1. D. Bimberg, M. Grundmann, and N. N. Ledentsov, Quantum Dot Heterostructures (John Wiley & Sons Ltd., New York, 1999).

, 2

2. K. Lüdge, Nonlinear Laser Dynamics-From Quantum Dots to Cryptography (Wiley-VCH, Weinheim, 2012).

] are promising candidates for optical communication applications and high-speed data transmission, since they are singled out by a narrow linewidth [1

1. D. Bimberg, M. Grundmann, and N. N. Ledentsov, Quantum Dot Heterostructures (John Wiley & Sons Ltd., New York, 1999).

] due to small phase-amplitude coupling [3

3. M. Gioannini and I. Montrosset, “Numerical analysis of the frequency chirp in quantum-dot semiconductor lasers,” IEEE J. Quantum Electron. 43, 941–949 (2007). [CrossRef]

5

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]

], and by strongly suppressed relaxation oscillations. The latter can be attributed to the special carrier scattering dynamics of QD lasers [6

6. T. Erneux, E. A. Viktorov, and P. Mandel, “Time scales and relaxation dynamics in quantum-dot lasers,” Phys. Rev. A 76, 023819 (2007). [CrossRef]

, 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]

]. It results in an higher dynamical stability of these lasers with respect to perturbation, e.g., external optical injection [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]

] or optical feedback [9

9. C. Otto, K. Lüdge, E. A. Viktorov, and T. Erneux, “Quantum dot laser tolerance to optical feedback,” in “Nonlinear Laser Dynamics - From Quantum Dots to Cryptography,”, K. Lüdge, ed. (WILEY-VCH, Weinheim, 2012), Chap. 6, pp. 141–162.

]. This results in simpler bifurcation scenarios and therefore in a better observability of nonlinear effects in experiments, which will be crucial for the noise induced dynamics discussed in this paper.

Understanding noise-induced effects, e.g., due to spontaneous emission noise, is indispensable for a variety of semiconductor devices, to name just a few: the performance of QD optical amplifiers [10

10. S. Wilkinson, B. Lingnau, J. Korn, E. Schöll, and K. Lüdge, “Influence of noise on the signal properties of quantum-dot semiconductor optical amplifiers,” IEEE J. Sel. Top. Quantum Electron. 19, 1900106 (2013). [CrossRef]

], the polarization dynamics of surface emitting lasers [11

11. S. Perrone, R. Vilaseca, and C. Masoller, “Stochastic logic gate that exploits noise and polarization bistability in an optically injected vcsel,” Opt. Express 20, 22692–22699 (2012). [CrossRef] [PubMed]

] or the synchronization properties of coupled laser systems [12

12. V. Flunkert, O. D’Huys, J. Danckaert, I. Fischer, and E. Schöll, “Bubbling in delay-coupled lasers,” Phys. Rev. E 79, 065201 (R) (2009). [CrossRef]

, 13

13. M. C. Soriano, J. García-Ojalvo, C. R. Mirasso, and I. Fischer, “Complex photonics: Dynamics and applications of delay-coupled semiconductors lasers,” Rev. Mod. Phys. 85, 421–470 (2013). [CrossRef]

]. A special situation emerges for devices that can be operated as excitable systems, i.e., systems that rest in a stable steady state, but can be excited to emit a spike by a super-threshold perturbation (e.g. noise). Well-known examples for excitable systems are spiking neurons [14

14. A. L. Hodgkin, “The local electric changes associated with repetitive action in a medullated axon,” J. Physiol. 107, 165 (1948). [PubMed]

], cardiac dynamics [15

15. J. D. Murray, Mathematical Biology, 2nd ed. of Biomathematics Texts (Springer, Berlin Heidelberg, 1993), vol. 19.

], and nonlinear chemical reactions [16

16. A. N. Zaikin and A. M. Zhabotinsky, “Concentration wave propagation in two-dimensional liquid-phase self-oscillating system,” Nature 225, 535–537 (1970). 10.1038/225535b0. [CrossRef] [PubMed]

]. Excitability in laser systems received considerable interest in the last years. It was observed experimentally [17

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]

20

20. B. Kelleher, S. P. Hegarty, and G. Huyet, “Modified relaxation oscillation parameters in optically injected semiconductor lasers,” J. Opt. Soc. Am. B 29, 2249–2254 (2012). [CrossRef]

] and studied theoretically [21

21. S. Wieczorek, B. Krauskopf, and D. Lenstra, “Multipulse excitability in a semiconductor laser with optical injection,” Phys. Rev. Lett. 88, 063901 (2002). [CrossRef] [PubMed]

24

24. L. Olejniczak, K. Panajotov, H. Thienpont, and M. Sciamanna, “Self-pulsations and excitability in optically injected quantum-dot lasers: Impact of the excited states and spontaneous emission noise,” Phys. Rev. A 82, 023807 (2010). [CrossRef]

] in lasers with optical injection. Furthermore, excitability was found in lasers with short optical feedback [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]

, 26

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]

] as well as in lasers with a long external cavity [27

27. M. Giudici, C. Green, G. Giacomelli, U. Nespolo, and J. R. Tredicce, “Andronov bifurcation and excitability in semiconductor lasers with optical feedback,” Phys. Rev. E 55, 6414–6418 (1997). [CrossRef]

], and it was investigated theoretically in lasers with saturable absorbers [28

28. J. L. A. Dubbeldam, B. Krauskopf, and D. Lenstra, “Excitability and coherence resonance in lasers with saturable absorber,” Phys. Rev. E 60, 6580 (1999). [CrossRef]

, 29

29. J. L. A. Dubbeldam and B. Krauskopf, “Self-pulsations of lasers with saturable absorber: dynamics and bifurcations,” Opt. Commun. 159, 325–338 (1999). [CrossRef]

].

Recently, it has been demonstrated that an excitable optical unit may be used as an optical tongue wrench permitting to sense single perturbation events [30

30. F. Pedaci, Z. Huang, P. van Hese, S. Barland, and L. Deuker, “Excitable particles in an optical torque wrench,” Nat. Phys. 7, 259–264 (2011). [CrossRef]

]. Data transmission systems based on excitable optical units confer a high degree of robustness due to their inherent signal reshaping capabilities. Therefore, it has been suggested to use an excitable optical unit as optical switch for all-optical-signal processing where it only reacts on sufficiently high optical input signals [31

31. V. Z. Tronciu, “Excitability and coherence resonance of a DFB laser with passive dispersive reflector,” Moldavian Journal of the Physical Sciences 7, 516 (2008).

] or for noise reduction in optical telecommunication applications [32

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]

]: a noisy input pulse triggers a “clean” output pulse. Generation of nanosecond pulses by an excitable semiconductor laser in an integrated optoelectronic circuits was already experimentally demonstrated [33

33. B. Romeira, J. Javaloyes, C. N. Ironside, J. M. L. Figueiredo, S. Balle, and O. Piro, “Excitability and optical pulse generation in semiconductor lasers driven by resonant tunneling diode photo-detectors,” Opt. Express 21, 20931–20940 (2013). [CrossRef] [PubMed]

]. However, in the conventional setup of a quantum well (QW) semiconductor laser under long external optical feedback bifurcation points lie very dense. This makes it hard to experimentally address the small regions of excitability, which occur only close to certain bifurcation points. Instead, the QD laser with short optical feedback studied in this paper is dynamically more stable, and showing a simpler bifurcation scenario, it is thus more promising for this kind of application.

The counter-intuitive effect that an increase of the noise can lead to an increase of correlation, i.e., to an increase of the regularity of the spikes observed in the excitable regime, is known as coherence resonance [34

34. G. Hu, T. Ditzinger, C. Z. Ning, and H. Haken, “Stochastic resonance without external periodic force,” Phys. Rev. Lett. 71, 807 (1993). [CrossRef]

36

36. A. B. Neiman, P. I. Saparin, and L. Stone, “Coherence resonance at noisy precursors of bifurcations in nonlinear dynamical systems,” Phys. Rev. E 56, 270 (1997). [CrossRef]

]. In contrast to stochastic resonance (see [37

37. L. Gammaitoni, P. Hänggi, P. Jung, and F. Marchesoni, “Stochastic resonance,” Rev. Mod. Phys. 70, 223–287 (1998). [CrossRef]

] for a review) the effect occurs without periodic forcing of the system. Coherence resonance is already an intensively studied effect and was shown theoretically in quantum well (QW) lasers with saturable absorber [28

28. J. L. A. Dubbeldam, B. Krauskopf, and D. Lenstra, “Excitability and coherence resonance in lasers with saturable absorber,” Phys. Rev. E 60, 6580 (1999). [CrossRef]

], in QD lasers under optical injection [38

38. D. Ziemann, R. Aust, B. Lingnau, E. Schöll, and K. Lüdge, “Optical injection enables coherence resonance in quantum-dot lasers,” Europhys. Lett. 103, 14002 (2013). [CrossRef]

], in lasers subject to long optical feedback [32

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]

, 39

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]

], in laser systems with polarization instabilities [40

40. F. T. Arecchi and R. Meucci, “Stochastic and coherence resonance in lasers: homoclinic chaos and polarization bistability,” Eur. Phys. J. B 67, 93–100 (2009). [CrossRef]

, 41

41. S. Sergeyev, K. O’Mahoney, S. Popov, and A. T. Friberg, “Coherence and anticoherence resonance in high-concentration erbium-doped fiber laser,” Opt. Lett. 35, 3736 (2010). [CrossRef] [PubMed]

], in semiconductor super-lattices [42

42. J. Hizanidis, A. G. Balanov, A. Amann, and E. Schöll, “Noise-induced front motion: signature of a global bifurcation,” Phys. Rev. Lett. 96, 244104 (2006). [CrossRef] [PubMed]

], as well as in non-excitable systems below a subcritical Hopf-bifurcation [43

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]

45

45. A. Zakharova, A. Feoktistov, T. Vadivasova, and E. Schöll, “Coherence resonance and stochastic synchronization in a nonlinear circuit near a subcritical Hopf bifurcation,” Eur. Phys. J. Spec. Top. 222, 2481–2495 (2013). [CrossRef]

].

In this paper, excitability and coherence resonance close to a boundary crisis bifurcation found in a QD laser subject to short optical feedback are studied. It is known that coherence resonance can be controlled by delayed feedback, e.g. for neural systems in the framework of the FitzHugh-Nagumo model (type-II excitability) [46

46. 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]

48

48. S. A. Brandstetter, M. A. Dahlem, and E. Schöll, “Interplay of time-delayed feedback control and temporally correlated noise in excitable systems,” Phil. Trans. R. Soc. A 368, 391 (2010). [CrossRef]

], for systems close to a saddle-node infinite period bifurcation (type-I excitability), and close to a subcritical Hopf-bifurcation [49

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

], however so far it has not been investigated close to a boundary crises. Using a sophisticated microscopically motivated rate equation approach, we show that the interesting effect of tunable regularity of emitted spikes strongly depends on the operating pump current and is thus easily accessible in experiments.

The paper is structured as follows: At first, in Sec. 2 a dimensionless version of the dynamic equations is introduced and the structure of the basic continuous wave (cw) solutions is discussed. Next, in Sec. 3 the bifurcation structure of the deterministic system is analyzed and its dynamics in the bistable regimes close to the loci of the bifurcation points, which render the system excitable, is studied in detail. Then, in Sec. 4 coherence resonance of the system subject to Gaussian white spontaneous emission noise is discussed in dependence of the pump current, before concluding in Sec. 4. Eventually, in Appendix A the dimensionless version of the model equations is derived.

2. Quantum dot laser model

The microscopically based rate equation model for the QD laser under optical feedback was previously discussed in [9

9. C. Otto, K. Lüdge, E. A. Viktorov, and T. Erneux, “Quantum dot laser tolerance to optical feedback,” in “Nonlinear Laser Dynamics - From Quantum Dots to Cryptography,”, K. Lüdge, ed. (WILEY-VCH, Weinheim, 2012), Chap. 6, pp. 141–162.

, 50

50. C. Otto, K. Lüdge, and E. Schöll, “Modeling quantum dot lasers with optical feedback: sensitivity of bifurcation scenarios,” Phys. Stat. Sol. (B) 247, 829–845 (2010).

, 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]

]. Here a dimensionless form of the dynamical equations is used, which is derived in Appendix A. A sketch of the edge-emitting single-mode laser device is shown in Fig. 1(a). The light in the cavity is modeled by a semiclassical Lang-Kobayashi-type [52

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

] equation for the slowly varying complex amplitude of the electric field. Taking into account only one roundtrip of the light in the external cavity, the field amplitude (tτ) 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.

Figure 1 (a): Sketch of the laser under delayed optical feedback. (b): Sketch of band structure.

The energy-band diagram of the dot-in-a-well structure under consideration is sketched in Fig. 1(b). The carriers are first injected in the InGaAs quantum well (QW), which acts as a carrier reservoir, with the dimensionless pump rate 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 λopt = 1.3μm. The occupation probabilities of electrons and holes in these states are denoted by ρe and ρh, respectively.

Coulomb scattering (nonlocal Auger scattering) is the dominating scattering process for high carrier densities in the lasing regime [53

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

]. Therefore, electron-phonon scattering is neglected for the carrier exchange between QW and QDs, but it is taken into account for the intraband transitions within the carrier reservoir. In the model the carrier exchange between QW and QDs is mediated by non-constant microscopically calculated Coulomb in- ( se/hin) and out-scattering ( se/hout) rates [54

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

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]

], which are nonlinear functions of the dimensionless carrier densities of electrons (We) and holes (Wh) in the carrier reservoir, and therefore depend on 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

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]

]). The scattering rates also strongly depend on the energy spacings between the QW band edges and the discrete QD levels, which are given by ΔEe = 210meV and ΔEh = 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 τe/h (in their dimensionless form te/h=τe/hW(se/hin+se/hout)1 with W = 0.7ns−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]

] mentioned in the introduction. The order of magnitude of τe and τh can be tuned by the pump current J, which permits to tune the turn-on damping of the laser.

In the subsystem of the carriers, different dynamics is taken into account for ρe and ρh as well as for We and Wh. Therefore, the system of coupled delay differential equations reads
(t)=1+iα2[g(ρe+ρh1)1](t)+keiC(tτ)+βrspρeρhξ(t),
(1a)
ρe=γ[Ferw(ρe+ρh1)||2ρeρh],
(1b)
ρh=γ[Fhrw(ρe+ρh1)||2ρeρh],
(1c)
We=γ[JFecWeWh],
(1d)
Wh=γ[JFhcWeWh].
(1e)
Here, time t′t/τph is rescaled with respect to the photon lifetime τph, 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

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]

]. The 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]

] for details). For shallow QDs the timescale of the intensity pulsations, which are, as the ROs, on the timescale of nano-seconds, is large compared to the timescale of the carrier exchange between QW and QDs (given by τe and τh). The carrier exchange is then fast enough to synchronize QD and QW carriers, and the approximation of a constant 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.

Table 1. Parameter values used in the numerical simulations.

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It is crucial to note for the subsequent analysis that the carrier equations (1b)(1e) are not independent but contain carrier conservation, which can be seen by verifying that ρ′e + W′e = ρ′h + W′h holds. Thus, ρeρh + WeWh is a constant, which is zero in the intrinsic case [59

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

]. Due to the carrier conservation, one dynamical variable can be eliminated by expressing one carrier type in terms of the others, i.e.,
Wh=ρe+Weρh.
(2)

2.1. External cavity modes–stationary solutions

In this section, the basic solutions of the dynamical equations (1) without noise (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]

], and therefore it is crucial to understand their bifurcation structure. They are cw solutions with constant photon number Nph=Nphs and carrier densities ρe/hs, We/hs, and a phase ϕδωst of the electric field amplitude Npheiϕ that varies linearly in time
(,ρe/h,We/h)=(Nphseiδωst,ρe/hs,We/hs),
(3)
where the steady states of the dynamic equations (1) with feedback are denoted by the superscript s, and δωsτph(ωωth) is the deviation of the frequency ω of the ECM from the threshold frequency of the solitary laser ωth. Inserting the ECM-ansatz (3) into Eqs. (1), we find the following expressions for the non-zero intensity solutions ( Nphs0)
ρinvs=kcos(δωsτ+C),
(4a)
δωs=αρinvsksin(δωsτ+C),
(4b)
0=γ[Fesrw(ρes+ρhs1)Nphsρesρhs],
(4c)
0=γ[Fhsrw(ρes+ρhs1)Nphsρesρhs],
(4d)
0=JFescWesWhs,
(4e)
0=JFhscWesWhs,
(4f)
where a rescaled inversion
ρinv12[g(ρe+ρh1)1]
(5)
has been introduced. Inserting Eq. (4a) into Eq. (4b), we obtain a transcendental equation for δωs in terms of α, τ, and C
δωs=keffsin(δωsτ+C+arctan(α)),
(6)
where keffkτ1+α2. For keff < 1 only one solution exists, and at keff = 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]

]. Taking advantage of the carrier conservation (Eq. (2)), we can reformulate Eq. (4a) to express ρes and ρhs in terms of Wes and Whs
ρes=12[1+g2kcos(δωs+C)g+WhsWes],
(7a)
ρhs=12[1+g2kcos(δωs+C)g+WesWhs].
(7b)
Further, an expression for Nphs as a function of the carrier populations can be obtained, by inserting the sum of Eqs. (4c) and (4d) into the sum of Eqs (4e) and (4f)
Nphs=grw(12kcos(δωs+C))[JρesρhscWesWhs]=grw(12kcos(δωs+C))[JJth],
(8)
where the pump current at lasing threshold JthρesρhscWesWhs has been introduced in the second line. Eventually, the steady states Wes and Whs may be determined by solving Eqs. (4e) and (4f) self-consistently, which has to be done numerically, because se/hin/out=se/hin/out(We/Wh) are nonlinear functions of We and Wh.

3. Bifurcation structure and excitable dynamics

Figure 2 Deterministic dynamics: (a): Bifurcation diagram of local maxima of photon number Nph vs. feedback strength K for pump current J = 2Jth (brown dots), where Jth ist the threshold current. Thick blue and green lines denote the steady state photon numbers Nphs of the stable parts of the first and the second ECM, respectively, and the black dashed line denotes Nphs of the unstable antimode. Insets show time traces of Nph for fixed K. (b): Frequency deviations δωs of the ECMs vs. K. Solid and dashed lines denote stable and unstable solutions, respectively. Hopf and limit points are denoted by red dots and open black circles, respectively. Blue, red, and black (gray) arrows indicate the feedback strengths of the Hopf points (KH), the homoclinic bifurcation (Khom), and the boundary crisis (Kcris), respectively. (c): Same as (a) but for higher J = 3Jth (black dots) and J = 4Jth (gray dots). Parameters as in Table 1.

With varying current, the bifurcation scenario changes. In Ref. [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]

], it was shown that the RO damping increases linearly with the pump current. This is the reason why for higher J = 3Jth and J = 4Jth, the Hopf-bifurcation points KH shift toward higher K-values [9

9. C. Otto, K. Lüdge, E. A. Viktorov, and T. Erneux, “Quantum dot laser tolerance to optical feedback,” in “Nonlinear Laser Dynamics - From Quantum Dots to Cryptography,”, K. Lüdge, ed. (WILEY-VCH, Weinheim, 2012), Chap. 6, pp. 141–162.

] (red dots and blue arrows in Fig. 2(b). Further, for pump currents larger than J > 2.8Jth 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]

] of the chaotic attractor that collides at Kcris with the saddle (anti-mode) of the 2nd ECM pair. Bi-stability is now observed in the interval [Ksn, Kcris]. The feedback strengths Khom and Kcris, at which homoclinic bifurcation and boundary crisis occur, have been found by up- and down-sweeping K with a very small stepsize of ΔK = 1 · 10−5. For up-sweeping K, the system remains on the periodic orbit (chaotic attractor), up to Khom (Kcris), while for down-sweeping K the laser emits on the 2nd ECM down to Ksn. Therefore, Khom and Kcris 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 ∈ [Jth, 2.8Jth], the feedback strength Kcris, at which the boundary crisis occurs, increases with the pump current (see dark and light arrow in Fig. 2(c)).

The laser is excitable for K-values little larger than Khom for J < 2.8Jth and analogously for K-values little above Kcris for J > 2.8Jth. In both cases, the short unstable manifold of the anti-mode acts as perturbation threshold. For J ≤ 2.8Jth, 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 > 2.8Jth, the excursion in phase space is guided by the ruin of the chaotic attractor that collapses at Kcris. In Figs. 2(a) and 2(c) this situation is elucidated, the threshold is given by the difference of the photon numbers Nphs of the 2nd ECM (thick green line) and of the anti-mode (black dashed line). The threshold is very low for K = Khom and increases with K. This implies that for J > 2.8Jth, 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 Kcris = Kcris(J).

Next, the dynamics in phase space is discussed to gain a better understanding of the difference between the excitable behavior close to the homoclinic bifurcation and close to a boundary crisis. Figures 3(a) and 3(b) depict time series and phase space projections onto the (Nph, We)-plane for K = 0.2290, i.e., just below Khom, where the periodic orbit still exists. A subthreshold perturbation of the system from the stable 2nd 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

50. C. Otto, K. Lüdge, and E. Schöll, “Modeling quantum dot lasers with optical feedback: sensitivity of bifurcation scenarios,” Phys. Stat. Sol. (B) 247, 829–845 (2010).

] we showed that the inter-pulse interval time TISI scales logarithmically with the distance from the bifurcation point, i.e., TISI ∼ ln|KKhom|, as it is expected close to a homoclinic bifurcation [63

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

]. In the phase space projection in Fig. 3(b), it can be seen that after a power dropout at the end of each pulse package (nearly vertical part of the trajectory), the trajectory at first performs pronounced damped oscillations spiraling around the point in phase space, where the 2nd pair of ECMs has been created at the nearby saddle-node bifurcation (the green dot in Fig. 3(b) indicates the stable 2nd ECM). Afterwards, it is re-injected into the high gain region during the power dropout.

Figure 3 Subthreshold (green lines) and super-threshold (blue lines) excitations of deterministic system in the bistable regime. (a) and (b): Close to a homoclinic bifurcation for K = 0.229 and J = 2Jth. (c) and (d): Close to a boundary crisis of chaotic attractor for K = 0.23 and J = 3Jth. Blue and green triangles in the closeups mark the starting points of the perturbed trajectories for super- and subthreshold perturbations, respectively. Black lines denote the steady state photon number of the unstable anti-mode of the 2nd ECM-pair. (a) and (c): Time series of the perturbed trajectories. (b) and (d): Projections of the trajectories onto the (Nph, We)-plane. Green dots indicate the position of the stable 2nd ECM-mode. Parameters as in Table 1.

The lower panel of Fig. 3 depicts the excitability of the laser close to the boundary crisis for K = 0.23, which is a little below Kcris(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

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]

]. Comparing the distance of Nphs of the 2nd ECM (green line) and Nphs of the anti-mode (black line) in Figs. 3(a) and 3(c), we see that the excitation threshold is much larger close to the crisis than close to the homoclinic bifurcation. Thus, close to the crisis larger perturbations (higher noise levels) are needed to excite the system and cause a phase space excursion.

4. Coherence resonance

To study coherence resonance, the QD laser is operated on the stable 2nd 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 ≤ 2.8Jth) and the boundary crisis (2.8Jth < J), respectively, and would then return to the stable 2nd 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 tcor (red triangles, right y-axis) and RT (blue dots, left y-axis) as functions of the noise strength β for K = 0.22921 and J = 2Jth, i.e., for a K-value closely above the homoclinic bifurcation at Khom = 0.22920. Furthermore, tcor is shown for K = 0.2314 and J = 3Jth (black stars, right y-axis) as well as for K = 0.24515 and J = 4Jth (gray hexagons, right y-axis), i.e., for K-values closely above the crisis of the chaotic attractor at Kcris(J = 3Jth) = 0.23324 and Kcris(J = 4Jth) = 0.24514, respectively (cf. Fig. 2). A clear maximum of tcor 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 (βopt = 5 · 10−9, Fig. 4(b)), and above (β = 6.5 · 10−8, Fig. 4(c)) the noise strength βopt, at which the maximum of tcor is observed for J = 2Jth. Analogously, Figs. 4(e)–4(g) depict time series below (β = 0.02, Fig. 4(e)), at (βopt = 0.038, Fig. 4(f)), and above (β = 0.08, Fig. 4(g)) the noise strengths βopt = 0.23325 of the coherence maximum for J = 3Jth. The β -values, at which the time series are taken, are indicated by gray dashed vertical lines in Fig. 4(d).

Figure 4 Stochastic dynamics: (a)(c): Time series for J = 2Jth and K = 0.2292 for different β indicated by gray dashed lines in (d). βopt denotes the noise strength at the maximum of the coherence time. Central panel (d): Normalized standard deviation of inter-spike interval RT (blue dots) for J = 2Jth and coherence time tcor (normalized to its maximum value tcormax) versus noise strength β for J = 2Jth (red triangles), J = 3Jth (black stars), and J = 4Jth (gray hexagons). In physical units the maximal coherence times are τphtcormax(J=2Jth)=2.50ns, τphtcormax(J=3Jth)=2.39ns, and τphtcormax(J=4Jth)=1.93ns. The feedback strength is K = 0.22921 for J = 2Jth, K = 0.23325 for J = 3Jth, and K = 0.24515 for J = 4Jth, respectively. (e)(g): Time series for J = 3Jth and K = 0.233325 for different β indicated by gray dashed lines in (d). Parameters as in Table 1.

Let us first discuss the coherence resonance close to the homoclinic bifurcation. Generally, the time between two excitations TISI can be decomposed into the time needed to activate the system ta and the refractory time tr, which the system needs to settle back to the rest state. In our system, the rest state is the stable 2nd ECM, and the refractory time is given by the time the system needs to spiral back to the 2nd ECM after one excitation. This means that tr is fixed by the internal dynamics of the system, while ta depends on the noise strength β. For low values of β, the activation time ta is long compared to tr (see Fig. 4(a)). Increasing 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 tcor and a decrease of RT. Increasing 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 tcor and an increase of RT. The maximum of tcor does not coincide exactly with the minimum of RT. This is expected, because tcor accounts for coherence in periodicity of the pulse packages as well as coherence in amplitude fluctuations, while RT only measures the periodicity of the pulse packages.

Higher pump currents of J = 3Jth and J = 4Jth 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 tcor shifts to higher values of the noise strength β with increasing J (see black stars and gray hexagons in Fig. 4(d) for J = 3Jth and J = 4Jth, respectively). By comparing the time traces taken at the maxima of tcor for J = 2Jth and J = 3Jth, 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 RT 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 ρinv crossing ρinvthr is not necessarily the beginning of a pulse package. Therefore, RT has not been depicted for J = 3Jth and J = 4Jth. That for J = 3Jth and J = 4Jth the dynamics beyond the coherence maximum is dominated by the noise can be seen in Fig. 4(g) depicting for J = 3Jth a time trace right to the maximum of tcor.

Figure 5 Dependence of (a): the feedback strength Kcris, at which the boundary crisis, occurs (red triangles), (b): the photon number of the threshold ΔNphthr (normalized to the photon number without feedback Nph0) (green hexagones), and (c): the optimal noise strength βopt (blue circles) on pump current J (normalized to its threshold value Jth). The dashed lines are best fits. Parameters as in Table 1.

In the remainder of this section, some practicalities of finding excitability and coherence resonance in laser systems are discussed. Experimentally it has been shown that spontaneous emissions noise is sufficient to excite a semiconductor laser under optical injection operating in a stable locked cw state close to the boundary of the locking tongue [17

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

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]

]. Further, excitability and coherence resonance has been experimentally observed in semiconductor lasers under optical feedback by adding broadband Gaussian white noise to the pump current [39

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

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]

]. Moreover, excitability close to a homoclinic bifurcation [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]

] and close to a crises [26

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]

] has been verified experimentally in an integrated multi-section semiconductor QW integrated feedback laser by perturbing the laser with short external optical pulses. Both methods, adding noise to the pump current and external optical pulses cause well tunable perturbations of the trajectory in the phase space. In our simulations, for simplicity we use the spontaneous emission noise in the field equation, i.e., the coefficient 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.

Further, our simulations have been performed for a fixed amplitude phase coupling α, a fixed band structure, and a fixed feedback phase C, but they are robust under changes of these parameters as discussed in the following. From the transcendental Eq. (6), it can be seen that the number of ECMs and thus the number of bifurcation cascades increases with τ and α (See [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]

] for a detailed discussion.). We have focused on the simplest scenario, of a short external cavity and a small α-factor, where only one instability region is observed. However, excitable regimes and crises can also be found for larger values of α and τ for which several bifurcation cascades occur [50

50. C. Otto, K. Lüdge, and E. Schöll, “Modeling quantum dot lasers with optical feedback: sensitivity of bifurcation scenarios,” Phys. Stat. Sol. (B) 247, 829–845 (2010).

,70

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 84, 043836 (2011). [CrossRef]

]. The band structure mainly impacts on the damping of the ROs, which in terms influences the dynamical stability of the laser. For experimental realizations the enhanced dynamical stability of QD lasers is a big advantage, because the structure of the bifurcation cascade is simpler [50

50. C. Otto, K. Lüdge, and E. Schöll, “Modeling quantum dot lasers with optical feedback: sensitivity of bifurcation scenarios,” Phys. Stat. Sol. (B) 247, 829–845 (2010).

, 71

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) 201, 345–352 (2004). [CrossRef]

], and they are thus less sensitive to perturbations unavoidable in experiments, e.g., small temperature fluctuations. Therefore, we expect that with a QD lasers it should be easier to detect the excitable regime and stay close to a homoclinic bifurcation or a boundary crises. Eventually, our results are robust under changes of the feedback phase C, that mainly shifts the range of K values at which the bifurcations occur. As mentioned above excitability has been demonstrated in multi-section integrated feedback laser, which reveals that the feedback parameters, especially the phase C and the feedback strength K, can be controlled well enough to stay close to the homoclinic bifurcation or the crises of the chaotic attractor. Indeed it has been shown that the bifurcation cascade can be scanned by careful tuning of external cavity phase C and pump current [72

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. 13, 136–142 (2007). [CrossRef]

, 73

73. A. Argyris, M. Hamacher, K. E. Chlouverakis, A. Bogris, and D. Syvridis, “Photonic integrated device for chaos applications in communications,” Phys. Rev. Lett. 100, 194101 (2008). [CrossRef] [PubMed]

].

5. Conclusion

We found that a QD laser subject to optical feedback can be operated in an excitable regime, where the regularity of the emitted spikes is sensitive to the noise strength as well as to the pump current. More precisely, we have shown that coherence resonance exists close to a boundary crisis of a chaotic attractor. In contrast to coherence resonance close to a homoclinic bifurcation, which was theoretically predicted previously, this type of coherence resonance has the advantage to be highly sensitive to variations of the pump current. This permits to shift the excitability threshold and, consequently, the maximum of the coherence found in the emitted spikes. Further, our findings are robust over a large range of pump currents facilitating the experimental accessibility of the excitable regime. Since the operating pump current is easily accessible in an experiment, it opens up the possibility to experimentally observe coherence resonance in semiconductor QD lasers. Further, we connect the pump current induced shift of the boundary crisis to the damping of the turn-on relaxations, as it also increases linearly with the pump strength.

A. Derivation of the dimensionless model

In this section, the dimensionless version of the dynamical equations (1) used in the main text is derived from the physical model. The optical subsystem of the QD laser model with feedback is described by a Lang-Kobayashy type delay differential equation for the normalized slowly varying complex amplitude (t) of the electric field E(t)=12((t)ei2πνtht+c.c), where νth 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 ρe and ρh of electrons and holes in the discrete QD ground states, and the carrier densities for electrons, we, and holes, wh, in the surrounding QW acting as a carrier reservoir
ddt=1+iα2[2W¯ZaQD(ρe+ρh1)τph1]+KτineiC(ttec)+βZaQDWρeρhξ,
(11a)
dρedt=Sein(1ρe)SeoutρeW¯(ρe+ρh1)NphWρeρh,
(11b)
dρhdt=Shin(1ρh)ShoutρhW¯(ρe+ρh1)NphWρeρh,
(11c)
dwedt=je02NQD[Sein(1ρe)Seoutρe]BSwewh,
(11d)
dwhdt=je02NQD[Shin(1ρh)Shoutρh]BSwewh.
(11e)
Here, the phase amplitude coupling is described by the linewidth enhancement factor α. Further, the optical intensity losses for the laser device of length L are given by the inverse of the photon lifetime τph. They are balanced by the linear gain term 2W¯ZaQD (ρe + ρh − 1), where W¯ZaQD is the linear gain coefficient for the processes of induced emission and absorption. The gain coefficient is proportional first to the Einstein coefficient of induced emission that measures the coherent interaction between the two-level system and the laser mode, and second to the number ZaQD of lasing QDs inside the waveguide (the factor 2 is due to spin degeneracy). The number of lasing QDs, ZaQD, is given by ZaQDaLANaQD, where aL is the number of self-organized QD layers, A is the in-plane area of the QW, and NaQD is the density per unit area of the active QDs. As a result of the size distribution and of the material composition fluctuations of the QDs, the gain spectrum is inhomogeneously broadened, and only a subgroup (density NaQD) of all QDs (NQD) matches the mode energies for lasing. Taking into account only one roundtrip of the light in the external cavity, the field amplitude (tτec) delayed by the external cavity roundtrip time τec is coupled back into the laser with feedback strength K and rotated by the external cavity phase C ≡ 2πνthτec. The roundtrip time of the light in the laser of length L is denoted by τin2Lεbg/c˜ with the background permittivity εbg, and the speed of light . Although being completely determined by νth and τec, 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, 2π], while the external roundtrip time τec 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

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

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]

].

The spontaneous emission is modeled by bimolecular recombination βZaQDWρeρh, where 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]

]. Note that the coefficients 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

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]

] for details of their derivation.). In a semiclassical approach, the process of spontaneous emission is modeled by a complex Gaussian white noise term ξ = ξ (t), i.e.,
ξ(t)=ξa(t)+iξb(t),ξi(t)=0,ξa(t)ξb(t˜)=δa,bδ(tt˜),forξi(t),i{a,b}.
Here, subscripts a and b stand for real and imaginary parts, respectively.

In the model, the carrier exchange between QW and QDs is mediated by non-constant microscopically calculated Coulomb in- ( Se/hin) and out-scattering ( Se/hout) rates [54

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

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]

], which are nonlinear functions of the carrier densities of electrons (we) and holes (wh) in the carrier reservoir, and therefore depend on the pump current density j. The latter is normalized by the elementary charge e0 in the equations for the reservoir densities (Eqs. (11d) and (11e)). The inverse of the sum of the scattering rates yields the carrier lifetimes τe/h(Se/hin+Se/hout)1 in the QD levels. Note that in thermodynamic equilibrium, there is a detailed balance between in- and out-scattering rates, which allows one to relate the coefficients of in- and out-scattering even away from the thermodynamic equilibrium [76

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

]. The detailed balance relation for in- and out-scattering rates for the quasi-equilibrium then reads [59

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

]
Se/hout(we,wh)=Se/hin(we,wh)eΔEe/hkbo𝒯[we/heDe/hkbo𝒯1]1.
(12)
It shows that the scattering rates strongly depend on the energy differences ΔEeEeQWEeQD and ΔEhEhQDEhQW between the QD levels, EeQD and EhQD, and the band edges of the QW, EeQW and EhQW, for electrons (e) and holes (h), respectively. The carrier degeneracy concentrations are given by De/hkbo𝒯, where De/hme/h/(πh̄2) are the 2D densities of state in the carrier reservoir with the effective masses me/h. The temperature is denoted by 𝒯 and kbo 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]

] for fit functions for the in-scattering rates Se/hin). Analogously to the spontaneous emission in the field equations, the spontaneous emission in the QW is incorporated by the bimolecular term BSwewh, where BS is the band-band recombination coefficient (see Eqs. (11d) and (11e)). All physical parameters used are summarized in Table 2.

Table 2. Physical parameters used in the simulation of the QD laser model unless stated otherwise.

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In the main text, a dimensionless form of the dynamical equations is used, which emphasizes the different timescales involved. As it is usually done for rate equation models of semiconductor lasers, time is rescaled with respect to the photon lifetime τph [77

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

]. Introducing the dimensionless time t′t/τph as well as the dimensionless reservoir populations Wewe/(2NQD) and Whwh/(2NQD), 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 rsp, 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 rw, the dimensionless pump rate J, the coefficient of spontaneous and non-radiative losses in the carrier reservoir c, and the dimensionless scattering rates se/hin/out have been introduced as
g2aLW¯ANaQDτph,k=Kτphτin,τ=τecτph,rspWZaQDτph,γτphW,rwW¯W,Jj2NQDe0W,cBS2NQDW,andse/hin/out1WSe/hin/out.
The values of the dimensionless parameters corresponding to the physical parameters of Table 2, are listed in Table 1. Note, that the small parameter 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 se/hin/out are not to large.

Acknowledgments

We thank R. Aust and A. Zakharova for fruitful discussions. This work was supported by DFG in the framework of SFB 910 and SFB 787.

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M. Gioannini and I. Montrosset, “Numerical analysis of the frequency chirp in quantum-dot semiconductor lasers,” IEEE J. Quantum Electron. 43, 941–949 (2007). [CrossRef]

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]

6.

T. Erneux, E. A. Viktorov, and P. Mandel, “Time scales and relaxation dynamics in quantum-dot lasers,” Phys. Rev. A 76, 023819 (2007). [CrossRef]

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]

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]

9.

C. Otto, K. Lüdge, E. A. Viktorov, and T. Erneux, “Quantum dot laser tolerance to optical feedback,” in “Nonlinear Laser Dynamics - From Quantum Dots to Cryptography,”, K. Lüdge, ed. (WILEY-VCH, Weinheim, 2012), Chap. 6, pp. 141–162.

10.

S. Wilkinson, B. Lingnau, J. Korn, E. Schöll, and K. Lüdge, “Influence of noise on the signal properties of quantum-dot semiconductor optical amplifiers,” IEEE J. Sel. Top. Quantum Electron. 19, 1900106 (2013). [CrossRef]

11.

S. Perrone, R. Vilaseca, and C. Masoller, “Stochastic logic gate that exploits noise and polarization bistability in an optically injected vcsel,” Opt. Express 20, 22692–22699 (2012). [CrossRef] [PubMed]

12.

V. Flunkert, O. D’Huys, J. Danckaert, I. Fischer, and E. Schöll, “Bubbling in delay-coupled lasers,” Phys. Rev. E 79, 065201 (R) (2009). [CrossRef]

13.

M. C. Soriano, J. García-Ojalvo, C. R. Mirasso, and I. Fischer, “Complex photonics: Dynamics and applications of delay-coupled semiconductors lasers,” Rev. Mod. Phys. 85, 421–470 (2013). [CrossRef]

14.

A. L. Hodgkin, “The local electric changes associated with repetitive action in a medullated axon,” J. Physiol. 107, 165 (1948). [PubMed]

15.

J. D. Murray, Mathematical Biology, 2nd ed. of Biomathematics Texts (Springer, Berlin Heidelberg, 1993), vol. 19.

16.

A. N. Zaikin and A. M. Zhabotinsky, “Concentration wave propagation in two-dimensional liquid-phase self-oscillating system,” Nature 225, 535–537 (1970). 10.1038/225535b0. [CrossRef] [PubMed]

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]

18.

B. Kelleher, D. Goulding, S. P. Hegarty, G. Huyet, D. Y. Cong, A. Martinez, A. Lemaitre, A. Ramdane, M. Fischer, F. Gerschütz, and J. Koeth, “Excitable phase slips in an injection-locked single-mode quantum-dot laser,” Opt. Lett. 34, 440–442 (2009). [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]

20.

B. Kelleher, S. P. Hegarty, and G. Huyet, “Modified relaxation oscillation parameters in optically injected semiconductor lasers,” J. Opt. Soc. Am. B 29, 2249–2254 (2012). [CrossRef]

21.

S. Wieczorek, B. Krauskopf, and D. Lenstra, “Multipulse excitability in a semiconductor laser with optical injection,” Phys. Rev. Lett. 88, 063901 (2002). [CrossRef] [PubMed]

22.

S. Wieczorek and D. Lenstra, “Spontaneously excited pulses in an optically driven semiconductor laser,” Phys. Rev. E 69, 016218 (2004). [CrossRef]

23.

S. Wieczorek, B. Krauskopf, T. Simpson, and D. Lenstra, “The dynamical complexity of optically injected semiconductor lasers,” Phys. Rep. 416, 1–128 (2005). [CrossRef]

24.

L. Olejniczak, K. Panajotov, H. Thienpont, and M. Sciamanna, “Self-pulsations and excitability in optically injected quantum-dot lasers: Impact of the excited states and spontaneous emission noise,” Phys. Rev. A 82, 023807 (2010). [CrossRef]

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]

27.

M. Giudici, C. Green, G. Giacomelli, U. Nespolo, and J. R. Tredicce, “Andronov bifurcation and excitability in semiconductor lasers with optical feedback,” Phys. Rev. E 55, 6414–6418 (1997). [CrossRef]

28.

J. L. A. Dubbeldam, B. Krauskopf, and D. Lenstra, “Excitability and coherence resonance in lasers with saturable absorber,” Phys. Rev. E 60, 6580 (1999). [CrossRef]

29.

J. L. A. Dubbeldam and B. Krauskopf, “Self-pulsations of lasers with saturable absorber: dynamics and bifurcations,” Opt. Commun. 159, 325–338 (1999). [CrossRef]

30.

F. Pedaci, Z. Huang, P. van Hese, S. Barland, and L. Deuker, “Excitable particles in an optical torque wrench,” Nat. Phys. 7, 259–264 (2011). [CrossRef]

31.

V. Z. Tronciu, “Excitability and coherence resonance of a DFB laser with passive dispersive reflector,” Moldavian Journal of the Physical Sciences 7, 516 (2008).

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]

33.

B. Romeira, J. Javaloyes, C. N. Ironside, J. M. L. Figueiredo, S. Balle, and O. Piro, “Excitability and optical pulse generation in semiconductor lasers driven by resonant tunneling diode photo-detectors,” Opt. Express 21, 20931–20940 (2013). [CrossRef] [PubMed]

34.

G. Hu, T. Ditzinger, C. Z. Ning, and H. Haken, “Stochastic resonance without external periodic force,” Phys. Rev. Lett. 71, 807 (1993). [CrossRef]

35.

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

36.

A. B. Neiman, P. I. Saparin, and L. Stone, “Coherence resonance at noisy precursors of bifurcations in nonlinear dynamical systems,” Phys. Rev. E 56, 270 (1997). [CrossRef]

37.

L. Gammaitoni, P. Hänggi, P. Jung, and F. Marchesoni, “Stochastic resonance,” Rev. Mod. Phys. 70, 223–287 (1998). [CrossRef]

38.

D. Ziemann, R. Aust, B. Lingnau, E. Schöll, and K. Lüdge, “Optical injection enables coherence resonance in quantum-dot lasers,” Europhys. Lett. 103, 14002 (2013). [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]

40.

F. T. Arecchi and R. Meucci, “Stochastic and coherence resonance in lasers: homoclinic chaos and polarization bistability,” Eur. Phys. J. B 67, 93–100 (2009). [CrossRef]

41.

S. Sergeyev, K. O’Mahoney, S. Popov, and A. T. Friberg, “Coherence and anticoherence resonance in high-concentration erbium-doped fiber laser,” Opt. Lett. 35, 3736 (2010). [CrossRef] [PubMed]

42.

J. Hizanidis, A. G. Balanov, A. Amann, and E. Schöll, “Noise-induced front motion: signature of a global bifurcation,” Phys. Rev. Lett. 96, 244104 (2006). [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]

44.

A. Zakharova, T. Vadivasova, V. Anishchenko, A. Koseska, and J. Kurths, “Stochastic bifurcations and coherencelike resonance in a self-sustained bistable noisy oscillator,” Phys. Rev. E 81, 011106 (2010). [CrossRef]

45.

A. Zakharova, A. Feoktistov, T. Vadivasova, and E. Schöll, “Coherence resonance and stochastic synchronization in a nonlinear circuit near a subcritical Hopf bifurcation,” Eur. Phys. J. Spec. Top. 222, 2481–2495 (2013). [CrossRef]

46.

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]

47.

A. G. Balanov, N. B. Janson, and E. Schöll, “Control of noise-induced oscillations by delayed feedback,” Physica D 199, 1–12 (2004). [CrossRef]

48.

S. A. Brandstetter, M. A. Dahlem, and E. Schöll, “Interplay of time-delayed feedback control and temporally correlated noise in excitable systems,” Phil. Trans. R. Soc. A 368, 391 (2010). [CrossRef]

49.

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

50.

C. Otto, K. Lüdge, and E. Schöll, “Modeling quantum dot lasers with optical feedback: sensitivity of bifurcation scenarios,” Phys. Stat. Sol. (B) 247, 829–845 (2010).

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]

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]

55.

K. Lüdge and E. Schöll, “Quantum-dot lasers – desynchronized nonlinear dynamics of electrons and holes,” IEEE J. Quantum Electron. 45, 1396–1403 (2009). [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]

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]

58.

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

59.

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

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]

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]

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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]

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R. L. Stratonovich, Topics in the Theory of Random Noise (Gordon and Breach, New York, 1963), vol. 1.

67.

C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences (Springer, Berlin, 2002).

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]

69.

K. Lüdge, “Modeling quantum dot based laser devices,” in “Nonlinear Laser Dynamics - From Quantum Dots to Cryptography,” Volume [2], Chap. 1, pp. 3–34(Wiley, 2012).

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 84, 043836 (2011). [CrossRef]

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) 201, 345–352 (2004). [CrossRef]

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. 13, 136–142 (2007). [CrossRef]

73.

A. Argyris, M. Hamacher, K. E. Chlouverakis, A. Bogris, and D. Syvridis, “Photonic integrated device for chaos applications in communications,” Phys. Rev. Lett. 100, 194101 (2008). [CrossRef] [PubMed]

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]

76.

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77.

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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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References

  1. D. Bimberg, M. Grundmann, N. N. Ledentsov, Quantum Dot Heterostructures (John Wiley & Sons Ltd., New York, 1999).
  2. K. Lüdge, Nonlinear Laser Dynamics-From Quantum Dots to Cryptography (Wiley-VCH, Weinheim, 2012).
  3. M. Gioannini, I. Montrosset, “Numerical analysis of the frequency chirp in quantum-dot semiconductor lasers,” IEEE J. Quantum Electron. 43, 941–949 (2007). [CrossRef]
  4. B. Lingnau, K. Lüdge, W. W. Chow, 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, K. Lüdge, “Feedback and injection locking instabilities in quantum-dot lasers: a microscopically based bifurcation analysis,” New J. Phys. 15, 093031 (2013). [CrossRef]
  6. T. Erneux, E. A. Viktorov, P. Mandel, “Time scales and relaxation dynamics in quantum-dot lasers,” Phys. Rev. A 76, 023819 (2007). [CrossRef]
  7. K. Lüdge, E. Schöll, E. A. Viktorov, T. Erneux, “Analytic approach to modulation properties of quantum dot lasers,” J. Appl. Phys. 109, 103112 (2011). [CrossRef]
  8. J. Pausch, C. Otto, E. Tylaite, N. Majer, E. Schöll, K. Lüdge, “Optically injected quantum dot lasers - impact of nonlinear carrier lifetimes on frequency locking dynamics,” New J. Phys. 14, 053018 (2012). [CrossRef]
  9. C. Otto, K. Lüdge, E. A. Viktorov, T. Erneux, “Quantum dot laser tolerance to optical feedback,” in “Nonlinear Laser Dynamics - From Quantum Dots to Cryptography,”, K. Lüdge, ed. (WILEY-VCH, Weinheim, 2012), Chap. 6, pp. 141–162.
  10. S. Wilkinson, B. Lingnau, J. Korn, E. Schöll, K. Lüdge, “Influence of noise on the signal properties of quantum-dot semiconductor optical amplifiers,” IEEE J. Sel. Top. Quantum Electron. 19, 1900106 (2013). [CrossRef]
  11. S. Perrone, R. Vilaseca, C. Masoller, “Stochastic logic gate that exploits noise and polarization bistability in an optically injected vcsel,” Opt. Express 20, 22692–22699 (2012). [CrossRef] [PubMed]
  12. V. Flunkert, O. D’Huys, J. Danckaert, I. Fischer, E. Schöll, “Bubbling in delay-coupled lasers,” Phys. Rev. E 79, 065201 (R) (2009). [CrossRef]
  13. M. C. Soriano, J. García-Ojalvo, C. R. Mirasso, I. Fischer, “Complex photonics: Dynamics and applications of delay-coupled semiconductors lasers,” Rev. Mod. Phys. 85, 421–470 (2013). [CrossRef]
  14. A. L. Hodgkin, “The local electric changes associated with repetitive action in a medullated axon,” J. Physiol. 107, 165 (1948). [PubMed]
  15. J. D. Murray, Mathematical Biology, 2nd ed. of Biomathematics Texts (Springer, Berlin Heidelberg, 1993), vol. 19.
  16. A. N. Zaikin, A. M. Zhabotinsky, “Concentration wave propagation in two-dimensional liquid-phase self-oscillating system,” Nature 225, 535–537 (1970). 10.1038/225535b0. [CrossRef] [PubMed]
  17. D. Goulding, S. P. Hegarty, O. Rasskazov, S. Melnik, M. Hartnett, G. Greene, J. G. McInerney, D. Rachinskii, G. Huyet, “Excitability in a quantum dot semiconductor laser with optical injection,” Phys. Rev. Lett. 98, 153903 (2007). [CrossRef] [PubMed]
  18. B. Kelleher, D. Goulding, S. P. Hegarty, G. Huyet, D. Y. Cong, A. Martinez, A. Lemaitre, A. Ramdane, M. Fischer, F. Gerschütz, J. Koeth, “Excitable phase slips in an injection-locked single-mode quantum-dot laser,” Opt. Lett. 34, 440–442 (2009). [CrossRef] [PubMed]
  19. B. Kelleher, C. Bonatto, G. Huyet, S. P. Hegarty, “Excitability in optically injected semiconductor lasers: Contrasting quantum-well- and quantum-dot-based devices,” Phys. Rev. E 83, 026207 (2011). [CrossRef]
  20. B. Kelleher, S. P. Hegarty, G. Huyet, “Modified relaxation oscillation parameters in optically injected semiconductor lasers,” J. Opt. Soc. Am. B 29, 2249–2254 (2012). [CrossRef]
  21. S. Wieczorek, B. Krauskopf, D. Lenstra, “Multipulse excitability in a semiconductor laser with optical injection,” Phys. Rev. Lett. 88, 063901 (2002). [CrossRef] [PubMed]
  22. S. Wieczorek, D. Lenstra, “Spontaneously excited pulses in an optically driven semiconductor laser,” Phys. Rev. E 69, 016218 (2004). [CrossRef]
  23. S. Wieczorek, B. Krauskopf, T. Simpson, D. Lenstra, “The dynamical complexity of optically injected semiconductor lasers,” Phys. Rep. 416, 1–128 (2005). [CrossRef]
  24. L. Olejniczak, K. Panajotov, H. Thienpont, M. Sciamanna, “Self-pulsations and excitability in optically injected quantum-dot lasers: Impact of the excited states and spontaneous emission noise,” Phys. Rev. A 82, 023807 (2010). [CrossRef]
  25. H. J. Wünsche, O. Brox, M. Radziunas, 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, F. Henneberger, “Excitability of chaotic transients in a semiconductor laser,” Europhys. Lett. 79, 30004 (2007). [CrossRef]
  27. M. Giudici, C. Green, G. Giacomelli, U. Nespolo, J. R. Tredicce, “Andronov bifurcation and excitability in semiconductor lasers with optical feedback,” Phys. Rev. E 55, 6414–6418 (1997). [CrossRef]
  28. J. L. A. Dubbeldam, B. Krauskopf, D. Lenstra, “Excitability and coherence resonance in lasers with saturable absorber,” Phys. Rev. E 60, 6580 (1999). [CrossRef]
  29. J. L. A. Dubbeldam, B. Krauskopf, “Self-pulsations of lasers with saturable absorber: dynamics and bifurcations,” Opt. Commun. 159, 325–338 (1999). [CrossRef]
  30. F. Pedaci, Z. Huang, P. van Hese, S. Barland, L. Deuker, “Excitable particles in an optical torque wrench,” Nat. Phys. 7, 259–264 (2011). [CrossRef]
  31. V. Z. Tronciu, “Excitability and coherence resonance of a DFB laser with passive dispersive reflector,” Moldavian Journal of the Physical Sciences 7, 516 (2008).
  32. B. Krauskopf, K. Schneider, J. Sieber, S. Wieczorek, M. Wolfrum, “Excitability and self-pulsations near homoclinic bifurcations in semiconductor laser systems,” Opt. Commun. 215, 367 (2003). [CrossRef]
  33. B. Romeira, J. Javaloyes, C. N. Ironside, J. M. L. Figueiredo, S. Balle, O. Piro, “Excitability and optical pulse generation in semiconductor lasers driven by resonant tunneling diode photo-detectors,” Opt. Express 21, 20931–20940 (2013). [CrossRef] [PubMed]
  34. G. Hu, T. Ditzinger, C. Z. Ning, H. Haken, “Stochastic resonance without external periodic force,” Phys. Rev. Lett. 71, 807 (1993). [CrossRef]
  35. A. S. Pikovsky, J. Kurths, “Coherence resonance in a noise-driven excitable system,” Phys. Rev. Lett. 78, 775 (1997). [CrossRef]
  36. A. B. Neiman, P. I. Saparin, L. Stone, “Coherence resonance at noisy precursors of bifurcations in nonlinear dynamical systems,” Phys. Rev. E 56, 270 (1997). [CrossRef]
  37. L. Gammaitoni, P. Hänggi, P. Jung, F. Marchesoni, “Stochastic resonance,” Rev. Mod. Phys. 70, 223–287 (1998). [CrossRef]
  38. D. Ziemann, R. Aust, B. Lingnau, E. Schöll, K. Lüdge, “Optical injection enables coherence resonance in quantum-dot lasers,” Europhys. Lett. 103, 14002 (2013). [CrossRef]
  39. G. Giacomelli, M. Giudici, S. Balle, J. R. Tredicce, “Experimental evidence of coherence resonance in an optical system,” Phys. Rev. Lett. 84, 3298 (2000). [CrossRef] [PubMed]
  40. F. T. Arecchi, R. Meucci, “Stochastic and coherence resonance in lasers: homoclinic chaos and polarization bistability,” Eur. Phys. J. B 67, 93–100 (2009). [CrossRef]
  41. S. Sergeyev, K. O’Mahoney, S. Popov, A. T. Friberg, “Coherence and anticoherence resonance in high-concentration erbium-doped fiber laser,” Opt. Lett. 35, 3736 (2010). [CrossRef] [PubMed]
  42. J. Hizanidis, A. G. Balanov, A. Amann, E. Schöll, “Noise-induced front motion: signature of a global bifurcation,” Phys. Rev. Lett. 96, 244104 (2006). [CrossRef] [PubMed]
  43. O. V. Ushakov, H. J. Wünsche, F. Henneberger, I. A. Khovanov, L. Schimansky-Geier, M. A. Zaks, “Coherence resonance near a Hopf bifurcation,” Phys. Rev. Lett. 95, 123903 (2005). [CrossRef] [PubMed]
  44. A. Zakharova, T. Vadivasova, V. Anishchenko, A. Koseska, J. Kurths, “Stochastic bifurcations and coherencelike resonance in a self-sustained bistable noisy oscillator,” Phys. Rev. E 81, 011106 (2010). [CrossRef]
  45. A. Zakharova, A. Feoktistov, T. Vadivasova, E. Schöll, “Coherence resonance and stochastic synchronization in a nonlinear circuit near a subcritical Hopf bifurcation,” Eur. Phys. J. Spec. Top. 222, 2481–2495 (2013). [CrossRef]
  46. N. B. Janson, A. G. Balanov, E. Schöll, “Delayed feedback as a means of control of noise-induced motion,” Phys. Rev. Lett. 93, 010601 (2004). [CrossRef] [PubMed]
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