# Pattern Dynamics in Nonlinear Optical Cavities

For each poster contribution there will be one poster wall (width: 97 cm, height: 250 cm) available. Please do not feel obliged to fill the whole space. Posters can be put up for the full duration of the event.

### Flaming 2$\pi$-kinks in parametrically driven systems

Berríos, Ernesto

In the present work we study dissipative sine-Gordon 2$\pi$-kinks under parametric oscillatory forcing. These kinks solutions are characterized by the emission of evanescent waves traveling from the front position in opposite directions ("flaming kinks"). We show how these waves are modified in the space of parameters, i.e., amplitude and frequency of the forcing. Also we study the interaction between these kinks, mediated by the waves, resulting in a family of localized structures. We apply these results to a ferromagnetic magnetic wire in presence of a oscillatory external magnetic field.

### Mode-Switching induced super-thermal bunching in quantum-dot microlasers

Kreinberg, Sören

Vertical cavity surface emitting lasers (VCSEL) have proven in many experiments to exhibit fascinating nonlinear dynamical behaviour including polarization mode instabilities and polarization mode switching. Advancing this research towards the domain of cavity quantum electrodynamics, we present here a comprehensive study investigating mode switching characteristics in a high-\beta quantum dot micropillar laser. Due to a slightly elliptical cross-section the fundamental of the micropillar is split into two orthogonal mode components. In good agreement with a stochastic multi-mode rate equation model, autocorrelation measurements and acquired time traces reveal a distinctive above-threshold region in which mode instability and mode switching leads to super-thermal bunching of one of the orthogonal modes [1]. As it turns out, both the difference in intensity of the modes and the correlation time of the weaker mode rise with increasing injection current until a stable operation regime is reached. Calculations show further that a higher \beta-factor reduces the consolidating effect of strong injection currents and hence shifts the onset of the stable regime to higher injection currents. [1] C. Redlich et al., New J. Phys., accepted for publication

Marconi, Mathias

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### Robustness and symmetry breaking of nonlinear Bessel-vortex transverse laser patterns supported by dissipation without gain

Porras, Miguel A.

Authors: Miguel A. Porras, Marcio Carvalho, Hervé Leblond, and Boris Malomed Abstract: Usual dissipative vortex solitons in the complex Ginzburg-Landau (CGL) model are supported by a balance between diffraction, self-focusing, losses and gain. Lifting the restriction of strong transversal localization, we have found stationary Bessel-vortex states in the nonlinear Schrödinger model with added dissipation supported by a balance between dissipation and an inward power flux from their intrinsic reservoir. With low dissipation and slow inward flux, these transverse patterns suffer from azimuthal instability, but high dissipation and fast restoring influx stabilize these weakly localized vortex-carrying structures. Similar results can readily obtained in the CGL model, where a three-fold balance between gain, losses and influx power can be given.

### Nonlinear dynamics in quantum dot micropillar lasers subject to on-chip optoelectronic feedback

Porte, Javier

Quantum dot micropillar lasers are mature and compact devices that exhibit many interesting properties like ultra-low threshold or high-beta lasing. However, in contrast to conventional semiconductor lasers, research on nonlinear properties emerging from external feedback coupling and optical injection of nanophotonic systems has been elusive. In this contribution, the effects of on-chip optoelectronic feedback on the emission of micropillar lasers are studied. Our system consists of an integrated high-beta laser and photodetector pair based on quantum dot micropillars. The laser emits laterally (in whispering gallery modes), feeding the negatively-biased micropillar photodetector. The generated photocurrent is amplified and fed back into the microlaser, constituting an ultra-compact on-chip optoelectronic feedback experiment. The effects of delayed optoelectronic feedback on the optical spectrum, the input-output characteristics and the spatio-temporal dynamics are characterized for different experimental parameters. Characteristic feedback phenomenology is observed for this system, like the reduction of the lasing threshold and the presence of chaotic instabilities in the laser emission. This work pioneers the characterization of the nonlinear emission of a quantum dot microlaser subject to optoelectronic feedback, opening the possibility to use such devices in nonlinear photonic applications like ultra-fast random number generation or chaos encrypted communications.

### Analysis of the Effects of Periodic Forcing in the Spike Rate and Spike Correlation's in Semiconductor Lasers with Optical Feedback.

Quintero-Quiroz, Carlos Alberto

We study the dynamics of semiconductor lasers with optical feedback and direct current modulation, operating in the regime of low frequency fluctuations (LFFs). In the LFF regime the laser intensity displays abrupt spikes: the intensity drops to zero and then gradually recovers. We focus on the inter-spike-intervals (ISIs) and use a method of symbolic time-series analysis, which is based on computing the probabilities of symbolic patterns. We show that the variation of the probabilities of the symbols with the modulation frequency and with the intrinsic spike rate of the laser allows to identify different regimes of noisy locking. Simulations of the Lang-Kobayashi model are in good qualitative agreement with experimental observations.

### The skeleton of chaotic eigenfunctions

Revuelta, Fabio

We demonstrate the feasibility of using basis sets formed by phase space localized wave functions (the so called 'tube and scar wavefunctions') to accuratelly compute the eigenenergies and the corresponding eigenfunctions of classically chaotic systems. The efficiency of the method is demonstrated by application to a highly chaotic system, a coupled quartic oscillator, and the LiNC/LiCN molecule, a system with a mixed dynamics that combines regular and irregular motion at the same energy.

### Towards AC-induced optimum control of dynamical localization

Revuelta, Fabio

It is shown that dynamical localization (quantum suppression of classical diffusion) in the context of ultracold atoms in periodically shaken optical lattices subjected to time-periodic modulations having equidistant zeros depends on the impulse transmitted by the external modulation over half-period rather than on the modulation amplitude. This result provides a useful principle for optimally controlling dynamical localization in general periodic systems, which is capable of experimental realization.

### Injection locking of quantum dot microlasers operating in the few photon regime

Schlottmann, Elisabeth

Injection locking of standard semiconductor lasers, where the slave adapts to the master laser’s frequency is well known and is widely applied e.g. for laser stabilization. Here we go beyond the classical injection locking by exploring the quantum limit of injection locking by using a microscopic quantum dot laser as a slave. This device with a high quality factor of Q ? 70000 and a low mode volume exhibits high spontaneous emission enhancement due to the Purcell effect (high $\beta$-factor), enabling stable lasing at intra-cavity photon numbers as low as a few tens. In contrast to predictions of classical deterministic rate equations, we find the laser in a superposition of oscillating synchronized to the external signal and at its solitary frequency. With semi-classical rate equations based on a quantum Langevin approach we can show that our experimental results on "partial injection locking" are specific to non-linear cQED systems where cavity enhanced spontaneous emission noise plays an important role. Measurements of the second-order autocorrelation function $g^{(2)}(\tau)$ prove this simultaneous presence of both master and slave-like emission, where the former has coherent character with $g^{(2)}(0)=1$ while the latter one has thermal character with $g^{(2)}(0)=2$ [1]. These results approach the limits of external quantum control of nanophotonic systems subject to spontaneous emission. [1] arXiv:1604.02817

### Mode-locking via dissipative parametric instability

Tarasov, Nikita

The dissipative parametric, or Faraday, instability can be induced by spatially periodic zig-zag modulation of a dissipative parameter of the system, i.g. spectrally dependent loss. In this work we experimentally demonstrate the dissipative parametric instability as a novel mode-locking mechanism in a simple fibre laser setup. Shifting in the spectral domain the reflection profiles of the linear cavity mirrors we were able to induce the parametric instability and achieve high order harmonic mode-lock, without any additional mode-locking mechanisms.

### Multistability in a semiconductor laser with intracavity saturable absorber and delayed optical feedback

Terrien, Soizic

As compact sources of pulses, passively Q-switched semiconductor lasers are suitable for high bit-rate telecommunications. However, such devices are very sensitive to perturbations, and a small amount of noise is enough to induce timing-jitter of the pulses. We consider here a micropillar semiconductor laser with an intracavity saturable absorber, subject to delayed optical feedback. In the absence of feedback, this device exhibits a wealth of dynamics, including excitability and self-pulsations. By introducing a delayed optical feedback, we aim at achieving a better control of the pulses properties, including their repetition rate. To investigate the effect of the delayed optical feedback on the laser behaviour, we focus on the theoretical investigation of the Yamada rate equations, with a delay term. This system of three delay differential equations for the gain G, the absorption Q and the intensity I can be considered as the simplest mathematical model for such a laser. We perform a numerical bifurcation analysis of this model, where both the feedback strength $\kappa$ and the feedback delay $\tau$ are considered as bifurcation parameters. The resulting bifurcation diagram divides the ($\kappa,\tau$)-plane in regions with different qualitative dynamics, thus providing a deep understanding of the system dynamics. This analysis highlights that, despite its simplicity, the Yamada model with delay displays complex feedback-induced dynamics. This includes multistability between several periodic solutions: when the delay is increased, different pulse-like periodic solutions are created, coexist, and are really close to each other when the intensity is close to zero along the orbits. Using time-domain simulations, we show that the basins of attraction of these different stable pulsing solutions have a complex Cantor set-like structure. Consequently, the laser is very sensitive to small perturbations while in multistable configuration. Noise-induced jumping between stable periodic orbits then emerges as an interpretation of complex dynamics observed experimentally, such as multipulses and coexistence between pulses with different amplitudes.

### Determining the photon number distribution of a quantum dot microlaser via a transition edge sensor

von Helversen, Martin

Microlasers operating in the regime of cavity quantum electrodynamics (cQED) exhibit enhanced coupling of spontaneous emission in the cavity mode (high $\beta$-factor) which greatly reduce their threshold powers if compared to standard semiconductor lasers. High $\beta$-factors lead to a smooth transition from thermal to coherent emission and a smeared out threshold characteristics which disappears completely in the limiting case of a thresholdless laser with beta = 1. In this respect it is crucial to determine the second order autocorrelation function $g^{(2)}(\tau)$ as a unambiguous proof of laser action. Here a power dependent study of $g^{(2)}(\tau)$ shows a transition from value of 2 at zero time delay ($g^{(2)}(0)$) for thermal light towards a value of 1 in the coherent regime. However, the standard characterization of $g^{(2)}(\tau)$ suffers from the limited temporal resolution of the applied detectors and the underlying photon number distribution is also not accessible. Here we report on a comprehensive study of a high-beta quantum dot microlaser, which includes for the first time a detailed analysis of the photon number distribution in the transition region between spontaneous emission to stimulated emission. The micropillar laser under study possesses two orthogonal modes which share the same gain medium. To determine the excitation power dependent photon number distribution we apply a state-of-the-art photon number resolving transition edge sensor (TES). In this way, we obtain important and direct insight into the relative contribution spontaneous emission and stimulated emission in the two modes of the microlaser which complements the usual analysis of the photon statistics via a Hanbury-Brown and Twiss configuration. As such TESs prove to be powerful and valuable tool to characterize and deeply understand microlasers in the few photon regime. Their sensitivity to small changes in the laser statistics makes them particularly attractive to measure the influence of external coupling on microlasers.

### Spiny Solitons

Vouzas, Peter

Various effects in nature or in a physical system may be described by evolution equations, which include nonlinear and non-conservative phenomena. One example of a dissipative dynamical system is the Ginzburg-Landau equation, which characterizes the physical properties of a passive mode locked laser system. This is an extension of the nonlinear Schrödinger equation (NLSE). The complex cubic-quintic Ginzburg- Landau equation (CCQGLE) has several parameter coefficients; it is expressed as $i\psi_z$ + $\frac{D}{2}$ $\psi_{tt}$ + $|\psi|^2$ $\psi$ + $\nu$ $|\psi|^4\psi$ = (i$\beta$ $\psi_{tt}$ + $\delta$ $\psi$ + $\epsilon$ $|\psi|^2$ $\psi$ + $\mu$ $|\psi|^4$ $\psi$) . (1) Active, or dissipative, terms are on the right-hand-side of the equation while all reactive terms are on the left. The coefficients $\delta$, $\epsilon$, and $\mu$ are mainly determined by the gain in the system, cavity losses and transmission characteristics of the mode-locking device, while roughly describes the main part of the spectral response of the cavity. On the left-hand-side, D is responsible for the net dispersion in the cavity; it is anomalous when D > 0 and normal if D < 0. Then $\nu$ describes the active part of the reactive non-linearity. The cubic part is assumed to be positive; it is normalized and thus taken to be equal to one. Numerical investigation of this master equation[1] for laser systems has revealed new phenomena such as dissipative solitons[2], and dissipative soliton resonances[3] which have guided experiments and also have been verified by them[4]. More recently, a new phenomenon has been revealed for this equation. We have called this feature 'spiny solitons'. It has the formation of a spike signal occurring in a chaotic matter, resembling a rogue wave, arising from a base soliton signal[5,6]. We anticipate that this will be experimentally verified in a mode-locked laser system. In our example, we have (D = -2.7, $\nu$ = -0.002, $\delta$ = -0.08, $\beta$ = 0.18, $\epsilon$ = 0.04 and $\mu$ = - 0.000025). These values may relate to super-continuum generation. Dissipative soliton processes are known to exist also in water[7,8] matter[9,10,11]. They offer potential insights into biological life itself[12,13]. Existing light sources may possibly be utilized to emit spiny light signal pulsations from within a base soliton, with the base soliton acting as if it were a viable secondary medium of propagation itself. Spiny solitons require the presence of the high-order differential terms in the CCQGLE. Hamiltonian soliton/rogue structures, based on the NLSE, have been known to resemble basic atomic structures[14]. There is scope to examine such classical dissipative subtleties further, perhaps even at atomic scales. 'Spiny solitons' show how interesting and bizarre classical non-linear evolution equations can be, where mode-locked lasers and the CCQGLE are key to understanding this new phenomenon. Perhaps there can be applications elsewhere. [1] A. Zaviyalov, R. Iliew, O. Egorov, and F. Lederer, “Lumped versus distributed description of mode-locked fiber lasers”, J. Opt. Soc. Am. B 27, 2313-2321 (2010). [2] N. Akhmediev and A. Ankiewicz, “Dissipative Solitons, Lecture Notes in Physics” (Springer Berlin Heidelberg, 2005). [3] W. Chang, A. Ankiewicz, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative soliton resonances”, Phys. Rev. A 78, 023830 (2008). [4] Z. Cheng, H. Li, H. Shi, J. Ren, Q.-H. Yang, and P. Wang, “Dissipative soliton resonance and reverse saturable absorption in graphene oxide mode-locked all-normal-dispersion yb-doped fiber laser," Opt. Express 23, 7000-7006 (2015). [5] W. Chang, J. M. Soto-Crespo, P. Vouzas, and N. Akhmediev, “Spiny solitons and noise-like pulses”, J. Opt. Soc. Am. B 32,1377-1383 (2015). [6] N. Akhmediev, B. Kibler, F. Baronio, M. Beli, W.-P. Zhong,Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser,G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R.Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bor-tolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events”, Journal of Optics 18, 063001 (2016). [7] A. Chabchoub, N. Hoffmann, M. Onorato, and N. Akhmediev, “Super rogue waves: Observation of a higher-order breather in water waves”, Phys. Rev. X 2, 011015 (2012). [8] A. Chabchoub, O. Kimmoun, H. Branger, N. Homann, D. Pro-ment, M. Onorato, and N. Akhmediev, “Experimental observation of dark solitons on the surface of water”, Phys. Rev. Lett. 110, 124101 (2013). [9] N. Akhmediev, M. P. Das, and A. V. Vagov, “Bose-einstein condensation of atoms with attractive interaction in a harmonic trap”, Aust. J. Phys. 53, 157-165 (2000). [10] A. D. Martin, C. S. Adams, and S. A. Gardiner, “Bright matter-wave soliton collisions in a harmonic trap: Regular and chaotic dynamics”, Phys. Rev. Lett. 98, 020402 (2007). [11] N. G. Parker, A. M. Martin, S. L. Cornish, and C. S. Adams, “Collisions of bright solitary matter waves”, Journal of Physics B: Atomic, Molecular and Optical Physics 41, 045303 (2008). [12] N. Akhmediev and A. Ankiewicz, “Dissipative Solitons: From Op-tics to Biology and Medicine, Lecture Notes in Physics” (Springer Berlin Heidelberg, 2008). [13] N. Akhmediev, J. Soto-Crespo, and H. Brand, “Dissipative solitons with energy and matter flows: Fundamental building blocks for the world of living organisms”, Physics Letters A 377, 968-974 (2013). [14] D. J. Kedziora, A. Ankiewicz, and N. N. Akhmediev, “Rogue wave clusters with atom-like structures”, in Advanced Photonics Congress (Optical Society of America, 2012) p. NW3D.6.

### Amplitude and timing pattern quantification of passively mode-locked semiconductor laser

Weber, Christoph

Semiconductor based ultra-short pulse laser sources for non-linear imaging or time-critical telecommunication applications require high pulse repetition rates and a precise knowledge of the stability of the emitted pulse train. Potential instabilities can manifest in amplitude modulations patterns or influence the timing jitter and they can originate from a net gain window outside the optical pulse or an imbalance of gain and absorption. By radio-frequency spectra analysis of a mode-locked semiconductor quantum dot laser we quantify the amplitude and timing pattern stability based on amplitude jitter and timing jitter. We thus can experimentally identify different mode-locking operation regimes depending on the laser operating conditions.

### Four-wave Mixing in Quantum Dot Semiconductor Optical Amplifiers

Zajnulina, Marina

Quantum Dot Semiconductor Optical Amplifiers (QD SOAs) possess a high gain bandwidth which makes them attractive for deployment in optical networks either to compensate the optical losses of a broadband signal or for the wavelength conversion for wavelength division multiplexing. [1] For the latter, the response of the medium needs to be nonlinear. In the case of QD SOAs, the medium nonlinearity is driven by the charge carrier dynamics within the optically active medium. The wavelength conversion in QD SOAs is achieved via the nonlinear parametric process called four-wave mixing (FWM). It relies on the third-order susceptibility of the QD SOA medium and is induced if two injected optical signals are mixed due to effects such as the carrier-density pulsation, the spectral hole-burning, and the carrier heating. So far, the theoretical treatment of the FWM in QD SOAs has mainly consisted of the analytical evaluation of the optical field at the amplifier's end facet. This usually involves a spatially averaged gain and a constant linewidth-enhancement factor for the analytic calculation of the third-order susceptibility. [2], [3] Our microscopically motivated model for studying of FWM in QD SOAs is based on a travelling-wave equation of the first order in time and space implemented as a delayed differential equation for the spatially resolved nonlinear light propagation through the quantum dot amplifier. Within this model, we assume the light-matter interaction for all quantum dots within the inhomogeneous ensemble. The charge-carrier dynamics within the quantum dot excited state as well as the carrier dynamics of the surrounding quantum well are modelled by means of balance equations using microscopically calculated Auger scattering rates. [4] This model features the evolution of the optical gain along the amplifying medium and does not need to introduce a constant linewidth-enhancement factor. Using this model that has proven to be quantitatively successful in description of experimental results presented in Ref. 5, we numerically study the FWM wavelength conversion efficiency and the corresponding maximum spectral bandwidth as a function of the pump current and the device temperature using the characteristics of a 3mm long amplifier device embedding InAs quantum dots into a InGaAs quantum well. An increase of the efficiency with the pump current and the device temperature is observed which is in agreement with the experimental results presented in Ref. 1. Compared to the model presented in Ref. 2 and Ref. 3, our model provides a more accurate description of the material third-order susceptibility. The results we obtained show an increase of the FWM efficiency due to the spectral hole-burning observed for low values of the detuning which has not been reported yet. Further, we can relate different charge-carrier scattering processes to their effect on the susceptibility of the device and, thus, control the wavelength conversion efficiency by optimising the parameters of the quantum dot amplifier. References: [1] C. Meuer, H. Schmeckebier, G. Fiol, D. Arsenijevi?, J. Kim, G. Eisenstein, D. Bimberg: Cross-Gain Modulation and Four-Wave Mixing for Wavelength Conversion in Undoped and p-Doped 1.3-?m Quantum Dot Semiconductor Optical Amplifiers, IEEE Photonics Journal Vol. 2, No. 2 (2010) [2] D. Nielsen, S. Lien Chuang: Four-wave mixing and wavelength conversion in quantum dots. Physical Review B 81 (2010) 035305 [3] A. H. Flayyih, A. H. Al-Khursan: Theory of pulse propagation and four-wave mixing in a quantum dot semiconductor optical amplifier. Current Applied Physics 14 (2014), pp. 946-953 [4] B. Lingnau, W. W. Chow, E. Schöll, K. Lüdge: Feedback and injection locking instabilities in quantum-dot lasers: a microscopically based bifurcation analysis. New Journal of Physics 15 (2013) 093031 [5] B. Lingnau: Nonlinear and Nonequilibrium Dynamics of Quantum-Dot Optoelectronic Devices. Springer International Publishing (2015)