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Quantum Dot Laser Under Optical Injection

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Dynamics of Quantum Dot Lasers

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Abstract

In this chapter, the complex dynamics of QD lasers under optical injection is discussed. In the typical injection setup sketched in Fig. 3.1, the light of a laser (the master laser) is injected into a second laser (the slave laser).

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Notes

  1. 1.

    Alternatively, we can directly plug the ansatz \({\mathcal {E}}\equiv \sqrt{N_{\mathrm{ph }}}e^{i(\Psi +\delta \omega t')}\) in the field Eq. (3.6).

  2. 2.

    Typically, \(N_{\mathrm{ph }}=\mathcal {O}(10^{4})\), which implies that the product of \(r_wN_{\mathrm{ph }}^{0}\) is a \(\mathcal {O}(1)\).

  3. 3.

    As a shortcut, we can directly plug the ansatz \({\mathcal {E}}\equiv \sqrt{N_{\mathrm{ph }}^{0}}Re^{i(\Psi +\Delta \omega t)}\) in the field Eq. (3.3).

  4. 4.

    Note that \(\Delta \nu _{\mathrm{inj }}\) and \(K\) are related to the dimensionless quantities \(\delta \omega \) and \(\tilde{k}\) defined in the previous section by \(\Delta \nu _{\mathrm{inj }}=\kappa \delta \omega /\pi \) and \(K=2\kappa \tau _{\mathrm{in }}\tilde{k}\), respectively.

  5. 5.

    Note that for Hopf bifurcations there exists a different definition of super- and subcritical. For the two-dimensional Hopf normal form, a supercritical Hopf bifurcation takes place when a stable limit-cycle is created that destabilized the stable fixed point, while in subcritical Hopf bifurcation, an unstable limit-cycle destabilizes the stable fixed-point. This definition can be generalized to higher dimensional systems by defining that in a supercritical Hopf bifurcation, the dimension of the unstable manifold of the fixed point is increased by two, while the limit-cycle gains a pair of complex conjugate Floquet-multipliers \(\sigma _{\pm }\) with \(\vert \sigma _{\pm }\vert <1\). In a subcritical Hopf bifurcation, the unstable dimension of the fixed point increases by two, and a complex conjugate pair of Floquet-multipliers of the limit-cycle leaves the unit-cycle. In the two dimensional center-manifold of a Hopf bifurcation, the high-dimensional then reduces to the generic two dimensional case of the Hopf normal form [38]. In this work, we call in the high dimensional case only those Hopf bifurcation supercritical, in which a stable limit cycle is created.

  6. 6.

    The following trigonometric relations are needed

    $$\begin{aligned} \cos (\arctan (\alpha ))&=\frac{1}{\sqrt{1+\alpha ^2}},&\qquad \sin (\arctan (\alpha ))&=\frac{\alpha }{\sqrt{1+\alpha ^2}},\end{aligned}$$
    (3.20a)
    $$\begin{aligned} \sin (x\pm y)=\sin (x)\cos (y)&\pm \cos (x)\sin (y),&\cos (x\pm y)=\cos (x)\cos (y)&\mp \sin (x)\sin (y). \end{aligned}$$
    (3.20b)
  7. 7.

    For the special case mentioned above, we obtain \((K^{\mathrm{ZH,2 }},\Delta ^{\mathrm{ZH,2 }})=(1.83, -1.74)\).

  8. 8.

    To see this, note that in Ref. [81] the RO frequency is given by \(\omega ^{\mathrm{RO }}=\sqrt{2\epsilon P}\), where in the notation of Sect. 2.5.1 \(\epsilon =\gamma ^{\mathrm{QW }}\), and \(P=r^{\mathrm{QW }}N_{\mathrm{ph }}^{0}\) is the pump parameter (see Eq. (2.23a)). Further, the \(\alpha \)-factor was denoted by \(b\) in Ref. [81].

  9. 9.

    To compare expression (3.157) with the one obtained in Ref. [81], note that for the QW rate equation model the RO damping is given by \(\Gamma ^{\mathrm{RO }}=\epsilon (1+2P)/2\), where \(\epsilon =\gamma ^{\mathrm{QW }}\) and \(P=r^{\mathrm{QW }}N_{\mathrm{ph }}^{0}\) is the pump parameter (see Eq. (2.23a)). Further, the \(\alpha \)-factor was denoted by \(b\) in Ref. [81].

  10. 10.

    Injection strength \(K\) and frequency detuning \(\Delta \nu _{\mathrm{inj }}\) may be expressed \(\tilde{k}\) and \(\delta \omega \) as \(K=2\kappa \tau _{\mathrm{in }}\tilde{k}\) and \(\Delta \nu _{\mathrm{inj }}=\kappa \delta \omega /\pi \).

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  1. Research Domain II - Climate Impacts and Vulnerabilities, Potsdam Institute for Climate Impact Research, Potsdam, Germany

    Christian Otto

  2. Institute of Theoritical Physics, Berlin Institute of Technology, Berlin, Germany

    Christian Otto

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Otto, C. (2014). Quantum Dot Laser Under Optical Injection. In: Dynamics of Quantum Dot Lasers. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-03786-8_3

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