Abstract

In this paper we study the conditions for achieving almost perfect phase locking in large arrays of semiconductor diodes. We show that decayed non-local coupling of diode lasers can provide the necessary conditions for robust phase synchronization of an entire diode laser array. Perfect global coupling is known to allow for robust synchronization, however it is often physically impossible or impractical to achieve. We show that when diodes are coupled via the decayed non-local coupling layout, the dominant transverse mode of the laser array has a uniform phase across the lasers and can be stable. This state is robust to noise and frequency disorder and can be realized under periodic (fixed-intensity limit cycle) continuous-wave and chaotic behavior of lasers.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Semiconductor laser diodes are employed for a wide variety of applications. Such lasers can emit light in wide range of wavelengths, exhibit very high (in the range of 60-70%) electro-optical efficiency, are compact, and are low cost. However, a single diode’s emission power is in the range of Watts or lower. Consequently, beam combining of many diodes is required in order to provide high emission radiance from an array [1,2]. The master oscillator power amplifier (MOPA) designs have been shown to allow for almost perfect semiconductor diode phase locking with arrays as large as 900 lasers [3–5]. For external cavity designs, while excellent beam quality from single mode and broad-area diode arrays has been demonstrated [6–11], the scalability to very large arrays and stacked-arrays still remains an open matter. It is important to elucidate whether phase locking of large semiconductor diode arrays is possible and, if possible, what type of external cavity designs allow for phase synchrony. We would like to note that phase synchrony should not be very sensitive to disorder diode and cavity parameters given the natural heterogeneity of commercial laser diodes used in the experiments [6,11,12].

Spatial mode selection in laser arrays has been widely studied as a plausible mechanism of passive phasing of laser arrays. Modal analysis has also been applied to single resonators [13] as well as compound-resonators [8,10,14–18]. In order to study the stability of spatial modes, we applied a modified version of Master Stability Function (MSF) theory [19–23] to an array of weakly coupled semiconductor lasers described by the Lang-Kobayashi equations [24,25]. MSF theory is essentially a type of modal analysis for coupled nonlinear oscillators.

Using the Lang-Kobayashi equations [24] it is possible to describe a semiconductor laser array as a network of coupled nonlinear oscillators [25]. Lang-Kobayashi equations have been extensively tested both theoretically and experimentally [12,25–28]. It is known that small arrays of semiconductor lasers can be phase synchronized when nearest neighbor coupled [25,26], and even chaotically synchronized for small numbers of lasers [25, and references therein]. For large arrays, as the number of lasers increase the in-phase solutions destabilize in favor of anti-phase, traveling-wave, and chaotic solutions. The coupling strength at which destabilization occurs seems to decrease with array size [26]. This destabilization occurs because the number of fixed-frequency solutions increases as coupling strength increases and the coupled lasers begin to chaotically hop between these solutions [25,27]. In the case of perfect global coupling (also referred to as all-to-all or mean-field coupling) the common perception is that most large systems including semiconductor lasers [28] will synchronize with appropriate parameters and sufficiently low disorder [29]. However, perfect global coupling for diode arrays cannot be experimentally implemented.

In this paper we propose and study an external cavity design where coupling strength between any two lasers decays as the distance between them increases. This type of nonlocal coupling can be implemented, for example, in a V-shape Talbot [6,7], self-Fourier and intra-Talbot second harmonic generation [8,9] cavities. The geometry of the external cavity of the coupled laser array determines the form of the coupling matrix [17], however for Lang-Kobayashi type systems, the coupling matrix can only be specified phenomenologically, as there is no rigorous way to derive the matrix from first principles in the weak-coupling limit under which the Lang-Kobayashi equations are specified. However, if the lasers are identical, then the modes of the external cavity (described by the eigenvectors of the coupling matrix) should be identical to the modes of the gain-free (’cold’) compound resonator system (the lasers coupled with the external cavity) [13,17]. Employing a V-shape cavity, an almost perfect diffraction limited beam (meaning almost perfect phase synchrony) has been experimentally realized [7].

2. Analysis

We start by presenting the dynamical equations for an array of non-locally coupled semiconductor diode lasers. An array of M semiconductor lasers can be modeled using an equation of the general form [24,30, 31]:

X.i(t)=F(Xi(t))+η(Xi(t),t)+κfMj=1MKijC(Xj(tτ),Xi(t))
where κf is the feedback strength. Here F(Xi(t)) represents the noiseless laser dynamics, η(Xi,t) represents phase and carrier noise, and the term proportional to κf represents feedback from the array. Diode laser equations can be viewed as a representation of general system of oscillators coupled through a network with delayed feedback. The state of a laser Xi=(ri,ϕi,Ni)T can be described by a set of the following equations [24,30–34]:
r˙i(t)=12(gNi(t)N01+sri2(t)γ)ri(t)+Rsp(ηEeiϕi(t))+κfMj=1MKijrj(tτ)cos(ϕj(tτ)ϕi(t))
ϕ˙i(t)=α2(gNi(t)N01+sri2(t)γ)+ωiRspri(t)(ηEeiϕi(t))+κfMj=1MKijrj(tτ)ri(t)sin(ϕj(tτ)ϕi(t))
N˙i(t)=J0γnNi(t)gNi(t)N01+sri2(t)ri2(t)+γnNi(t)ηN(t)
where ri is the field magnitude, ϕi is the phase, and Ni is the number of carriers in the gain medium. The electric field in the ith laser isEi=ri(t)eiϕi(t). The parameters we choose are typical for Lang-Kobayashi type models of single-mode semiconductor lasers [35–37]. Here N0=1.5*108 is the number of carriers at transparency, g=1.5*108ps1 is the differential gain coefficient, s=2*107 is the gain saturation coefficient, γ=0.5ps1 is the loss, γn=.5ns1 is the loss term for the carriers, and α=5 is the linewidth enhancement factor [30]. The delay time for feedback isτ=3ns. The large delay-time and line-width enhancement factors are selected so that feedback-induced instabilities occur at lower feedback-strength values [38]. These feedback-induced instabilities seem to be part of the reason semiconductor laser arrays are difficult to synchronize [39], so studying this system in a parameter range where they occur is useful. We use the saturable gain model in this system because we are modeling lasers pumped well above the threshold current where gain saturation becomes apparent in the dynamics [31]. J0=aγn(N0γg) is the pump current, where a=4 is a scalar multiplier denoting the ratio between J0 and the (lasing) threshold current. We choose the large value of pump current to simulate behavior far above threshold for higher power applications. This also implies that the low-frequency-fluctuation region of parameter space is avoided [40]. ωi is the frequency detuning for the ith laser and is distributed according to a Gaussian distribution with mean zero and standard deviation 2πσ. The phase noise and carrier noise are modeled as in [32,33]. The phase noise terms depend on the spontaneous emission rate Rsp=10ns1 and the carrier noise is proportional to the carrier lifetime γn and the number of carriers. ηEis complex uncorrelated Gaussian white noise and ηN is real uncorrelated Gaussian white noise. The sum terms represent feedback from the other lasers in the array to the ith laser. The decayed non-local coupling scheme can be described using a matrix K whose ij element is:
Kij=dx|ij|
wheredx[0,1]. We use the scaling of 1/M in Eq. (1) to ensure that κfjKij<γ for the laser cavities. The reason for such scaling is that the fields in the gain-free system (neglecting delay) E˙i=γEi+κfj=1MKijEj should decay to Ei=0 for all i to guarantee that energy conservation is not violated. This is also the basis of the weak coupling approximation; κf is small enough that the feedback from the external cavity to an individual laser can be treated as a linear addition. Note that K is a symmetric positive definite matrix. So its eigenvalue decomposition isK=i=1MλiViViT, where λ1>λ2>...>λM are the eigenvalues and V1,...,VM are the corresponding eigenvectors.

We can linearize this system (Eq. (1)) and go into the modal basis using a procedure similar to finding the Master Stability Function for a system of coupled oscillators (i.e. linearizing about the synchronous solution and then left-multiplying by a matrix whose row-vectors are the eigenvectors of the coupling matrix) [19]. Our linearization procedure is not identical to the original procedure in [19] and is described in detail in [41]. We present here a condensed version of this derivation. We begin by linearizing to first order about the perfectly synchronous solution X* whereXi*=Xj*. We letξ=XX*. Then Eq. (1) becomes:

ξ˙(t)=[IMF(Xi(t))Xi(t)|Xi(t)=X*(t)+κfMΓC(Xi(t),Xj(tτ))Xi(t)|Xi(t)=X*(t)]ξ(t)+[κfMKC(Xi(t),Xj(tτ))Xj(tτ)|Xi(t)=X*(t)]ξ(tτ)
where Γ is a diagonal matrix whose iith entry is the ith row-sum of K. This system has a 3*M dimensional space. This space is spanned by a set of vectors Vin=Vien (where en is the nth unit vector in a 3-dimensional space). We can then decompose ξ asξ=i=1Mn=13δinVin.Here, δi=(δi1,δi2,δi3) represents the linearized amplitude of the array in the direction of the first mode. If we then left-multiply Eq. (6) by UTI3 where U is a matrix whose ith column isVi. This yields the equation for the modal amplitude:

δ˙i(t)=[Xi(t)F+κfMλ1Xi(t)C]δi(t)+[κfMλiXj(tτ)C]δi(tτ)

Recall that λi is the i th eigenvalue for the eigenvector Vi of the matrixK. We approximate row-sum byλ1. This approximation is valid because the off-diagonal terms of the matrix UTΓU are small. For additional details about the approximation we refer to the reference [41].

For decayed non-local coupling matrix, the first eigenvector (corresponding to the largest eigenvalue) V1 is the principal Gaussian type mode. Lettingκ=κfMλ1, we can write the modal amplitude equation for the first mode as:

δ˙1(t)=[Xi(t)F+κXi(t)C]δi(t)+[κXj(tτ)C]δi(tτ)

Equation (8) is identical to the linearized equation for a single laser with only self-feedback, where ξ(t) is the first-order deviation of the laser dynamics from the solution about which it is linearized (this is verified by linearizing Eq. (1) for a system with a single laser,M=1, and a coupling matrix K=1). Consequently, the linearized equation for the first order deviation of a single diode laser ξ(t) can be written as:

ξ˙(t)=[Xi(t)F+κfXi(t)C]ξ(t)+[κfXj(tτ)C]ξ(tτ)

The implication of this scaling of stability relation is that the solution set and corresponding analysis for the single laser are valid for the first mode of the array. The solutions for the single laser can be found by solving Eq. (1) with K=1 and M=1 for a set of fixed-frequency, fixed-intensity solutions with frequency Ω such that ϕ(t)=Ωt:

Ω=κfαcos(Ωτ)κfsin(Ωτ)r=gJ0γnN0(sγn+g)(γ2κfcos(Ωτ))γnsγn+gN=J0+gN0r21+sr2γn+gr21+sr2

Solutions to this system have been studied in detail for similar Lang-Kobayashi type systems (with various nonlinearities and gain terms but similar dynamics nonetheless) [25,40,42–44]. As κf is increased, the number of solutions (known as external cavity modes) to the above system increases. Each solution has a unique frequency Ω and the solutions have overlapping regions of stability, making the system highly multistable. When κf is low, the single laser solution stays on the continuous-wave solution with a fixed frequencyΩ. However, once κf increases, a series of period-doubling bifurcations leads to quasiperiodic and then chaotic dynamics around the fixed-point solution. Further increase of κf leads to a chaotic state where the system hops between the chaotic attractors formed around the solutions. This is known as the ‘coherence collapse’ region [40,43]. Since this is a delay system, the stability of these solutions cannot easily be found (the characteristic equation is transcendental). However, since the equations used to determine stability of the scaled M-laser system are the same as the equations to determine stability of a single laser (Eq. (8) and Eq. (9)), consequently these solutions should be the same. We show bifurcation diagrams that illustrate this behavior in Fig. 1 for a single laser and a 10-laser array. It is clear from this diagram that the dynamics of the 10-laser array very well resembles the dynamics of the single laser. The yellow line in the figure corresponds to the central solution Ω computed from Eq. (10) for the varying value of κf (or κf=κ for Fig. 1(b)). The bifurcation structure in this system is similar to that in [43].

 figure: Fig. 1

Fig. 1 Poincare sections of (ϕ(t)ϕ(tτ))/τ are plotted for a single laser and a 10-laser array as a function of feedback strength. Each trajectory was generated from a continuation simulation started with random initial conditions andκf=0.01ns1. Every 1000ns, κf was increased by0.01ns1. The yellow line in each plot is the central solution frequency Ω from Eq. (11) for a single laser withκf=κ.It is clear from these diagrams that the dynamics are almost identical except for the scaling.

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3. Results

We find that the effective coupling constant κ can be utilized to predict whether the first mode decays (so that there is perfect synchronization rather than first-mode-selection) or is neutrally stable (so that the first mode is selected). Numerical experiments show that this prediction holds even for very large arrays of lasers.

In Fig. 2 we summarize how phase synchronization S scales with array size when coupled through nonlocal decayed coupling. We use S=<|i=1MEi|2/Mi=1M|Ei|2> as a measure of phase synchrony of the array. As coupling constant κf and size of the array M vary, we observe different levels of phase synchrony including CW, quasi-periodic, and chaotic synchrony. We have also observed spatiotemporal chaos leading to poor phase synchrony. Following the figure, levels of synchronization are sectioned by the effective coupling κ value. When the value of the effective coupling constant κ is below 6ns−1, the array exhibits close-to perfect synchronization. At around κ=6ns−1 the dynamics become quasiperiodic, as shown in Fig. 1. When κ is between 6ns−1 and 16ns−1 there are areas of both chaotic and quasiperiodic synchronization. The bifurcation diagram shown in Fig. 1 corroborates this behavior. Because the equation even for the single laser is highly multi-stable in this area, it is rather tricky to find exact regions of quasiperiodic and chaotic behaviors. Once κ exceeds 16ns−1 phase synchrony destabilizes. We further corroborate this behavior in Fig. 3 where we present the values of cosϕi plotted for 10-laser arrays of identical lasers with varying feedback strength in the presence of carrier and phase noise.

 figure: Fig. 2

Fig. 2 Average phase synchrony of large arrays of lasers. Each point corresponds to a simulation of an array with randomized initial conditions (we started averaging process after 800ns simulation for convergence to occur). Lines denoting regions of effective coupling are related to the degree of synchronization in the system. For very large arrays achieving phase synchronization requires either weak or very strong (though possibly unrealistic) coupling κf between the diodes in the array.

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 figure: Fig. 3

Fig. 3 cosϕi are plotted for 10-laser arrays of identical lasers with varying feedback strength in the presence of carrier and phase noise. Whenκf=5ns1, the effective coupling κ=3.65ns1 and the behavior is CW as predicted in Fig. 1. Whenκf=20ns1, the effective couplingκ=14.6ns1, and the behavior is chaotic with a high degree of synchrony. Whenκf=30ns1, the effective couplingκ=21.9ns1, the behavior is chaotic, and the lasers begin to de-synchronize.

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Figure 3 presents clear evidence that as the coupling strength increases, the behavior of a synchronized array becomes chaotic. For a single laser, the onset of chaos is related to the linear increase of the number of fixed-point solutions of the equations in the laser [25]. Since the equation for the stability of the leading transverse mode of the array is the same as that of the single laser with κf=κ, it is likely that the onset of chaos for the array of lasers can be attributed to the same reason as the onset of chaos for a single laser. The fact that the dynamics of the laser array can be predicted using the effective coupling scaling corroborates this relationship and shows that the synchrony in the array follows from the transverse mode selection.

To show that modal discrimination takes place in the array of lasers, we consider the modal inner productsViTE. This product represents the magnitude of the array field parallel to the ith transverse mode. When V1TE(t)>VjTE(t) for all j (for a given time interval), it is reasonable to state that for that given time interval the array dynamics have converged to the first spatial mode.

In Figs. 4(a)-4(c), we show modal inner products ViTE(t) for a 30-laser array with various coupling strengths in the presence of noise and disorder. This inner product represents the level of mode discrimination in the array. In Figs. 4(b) and 4(c) we observe that even though modal amplitudes are oscillating, the discrimination is still high (the first mode still is fully separated in amplitude from the rest). We observe that in spite of significant noise, disorder and even chaotic behavior, mode selection still takes place and phase synchrony is high. We also plot the far-field behavior of an array of M=100 lasers with various levels of disorder in Fig. 3(d). We observe that forσ=0.1/τ, there is not much of a difference in the central peak intensity compared to the case ofσ=0. However, as the disorder increases, the peak intensity degrades.

 figure: Fig. 4

Fig. 4 (a-c) The inner products of the first four eigenvectors ViTE(t) (i=1,2,3,4) are plotted for an array of 30 lasers with dx=.8 at various values of coupling strength. The simulations have realistic amounts of noise and a frequency disorder ofστ=0.3. The synchronization level S is also given, showing that there is indeed synchronization in the quasiperiodic (b) and chaotic (c) regimes. (d) The central far-field lobes of an array of 100 lasers subject to noise and disorder are plotted. Forσ=0, the synchrony level isS=1.0. Forσ=0.1/τ, S=.97. Forσ=0.5/τ, S=.63. Forσ=1.0/τ, S=.39.

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4. Conclusions

In summary, in this manuscript we have studied and presented conditions for almost perfect phase synchronization of large arrays of semiconductor diode lasers. This phase synchronization can be achieved by employing nonlocal decayed diode coupling structure and choosing array parameters that allow diode laser array to settle at the first spatial mode. We believe examining almost-perfect phase synchronization in the form of mode selection rather than studying perfect in-phase synchronization (as common for global coupling designs) can be of fundamental advantage in order to predict, experimentally achieve and understand the causes of large array phase locking and especially chaotic phase locking.

Our approach may also shed light on how to design a scalable external cavity to phase lock large arrays of semiconductor diode lasers. Indeed we have numerically shown that almost perfect phase synchrony can be achieved for one-dimensional arrays consisting of O (100) coupled diode lasers (see Figs. 2 and 3(d)). Projecting this result to two-dimensional arrays implies O (10,000) diodes (or perhaps more) could be, in principle, coherently phase synchronized provided external cavity (described by the coupling matrix) is properly designed. We would like to note though that engineering challenges in designing such diode array phase locking experiments may be significant. However, it has been demonstrated that using diffractive coupling [45] or optical fiber networks [12] one can design reconfigurable semiconductor laser networks that can show a wide variety of array behaviors so it should be possible to test specific network structures experimentally.

Funding

This research was supported in part by the Office of Naval Research and the Laboratory Directed Research and Development Program of Oak Ridge National Laboratory. Oak Ridge National Laboratory is managed by UT- Battelle, LLC for the U.S. Department of Energy under Contract DE-AC05-00OR22725.

Acknowledgment

The authors would like to thank Alejandro Aceves of Southern Methodist University, Department of Mathematics and Brendan Neschke of Raytheon for valuable discussions that were important for the outcome of this work. Opinions, interpretations, and conclusions, and recommendations are those of the authors and are not necessarily endorsed by the U.S. government.

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References

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  1. T. Y. Fan, “Laser beam combining for high-power, high-radiance sources,” IEEE J. Sel. Top. Quantum Electron. 11(3), 567–577 (2005).
    [Crossref]
  2. A. M. Jones and J. T. Gopinath, “Fast-to-slow axis mode imaging for brightness enhancement of a broad-area laser diode array,” Opt. Express 21(15), 17912–17919 (2013).
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  3. R. K. Huang, L. J. Missaggia, J. P. Donnelly, C. T. Harris, and G. W. Turner, “High-brightness slab-coupled optical waveguide laser arrays,” IEEE Photonics Technol. Lett. 17(5), 959–961 (2005).
    [Crossref]
  4. K. J. Creedon, S. M. Redmond, G. M. Smith, L. J. Missaggia, M. K. Connors, J. E. Kansky, T. Y. Fan, G. W. Turner, and A. Sanchez-Rubio, “High efficiency coherent beam combining of semiconductor optical amplifiers,” Opt. Lett. 37(23), 5006–5008 (2012).
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  6. B. Liu, Y. Liu, and Y. Braiman, “Coherent beam combining of high power broad-area laser diode array with a closed-V-shape external Talbot cavity,” Opt. Express 18(7), 7361–7368 (2010).
    [Crossref] [PubMed]
  7. B. Liu and Y. Braiman, “Coherent beam combining of high power broad-area laser diode array with near diffraction limited beam quality and high power conversion efficiency,” Opt. Express 21(25), 31218–31228 (2013).
    [Crossref] [PubMed]
  8. C. J. Corcoran and F. Durville, “Passive coherent combination of a diode laser array with 35 elements,” Opt. Express 22(7), 8420–8425 (2014).
    [Crossref] [PubMed]
  9. K. Hirosawa, F. Shohda, T. Yanagisawa, and F. Kannari, “In-phase synchronization of array laser using intra-Talbot-cavity second harmonic generation,” Proc. SPIE 9342, 934216 (2015).
  10. D. Mehuys, K. Mitsunaga, L. Eng, W. K. Marshall, and A. Yariv, “Supermode control in diffraction-coupled semiconductor laser arrays,” Appl. Phys. Lett. 53(13), 1165–1167 (1988).
    [Crossref]
  11. J. R. Leger and G. Mowry, “External diode-laser-array cavity with mode-selecting mirror,” Appl. Phys. Lett. 63(21), 2884–2886 (1993).
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  12. A. Argyris, M. Bourmpos, and D. Syvridis, “Experimental synchrony of semiconductor lasers in coupled networks,” Opt. Express 24(5), 5600–5614 (2016).
    [Crossref] [PubMed]
  13. A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40(2), 453–488 (1961).
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  14. E. Kapon, J. Katz, and A. Yariv, “Supermode analysis of phase-locked arrays of semiconductor lasers,” Opt. Lett. 9(4), 125–127 (1984).
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  15. P. K. Jakobsen, H. G. Winful, L. Raman, R. A. Indik, J. V. Moloney, and A. C. Newell, “Diode-laser array modes: discrete and continuous models and their stability,” J. Opt. Soc. Am. B 8(8), 1674 (1991).
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  16. M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, 1974).
  17. N. Nair, E. J. Bochove, A. B. Aceves, M. R. Zunoubi, and Y. Braiman, “Resonator modes and mode dynamics for an external cavity-coupled laser array,” in Proceedings of SPIE-The International Society for Optical Engineering (2015), 9343.
  18. D. Pabœuf, D. Vijayakumar, O. B. Jensen, B. Thestrup, J. Lim, S. Sujecki, E. Larkins, G. Lucas-Leclin, and P. Georges, “Volume Bragg grating external cavities for the passive phase locking of high-brightness diode laser arrays: theoretical and experimental study,” J. Opt. Soc. Am. B 28(5), 1289 (2011).
    [Crossref]
  19. L. M. Pecora and T. L. Carroll, “Master Stability Functions for synchronized coupled systems,” Phys. Rev. Lett. 80(10), 2109–2112 (1998).
    [Crossref]
  20. V. Flunkert, S. Yanchuk, T. Dahms, and E. Schöll, “Synchronizability of networks with strongly delayed links: a universal classification,” J. Math. Sci. 202(6), 809–824 (2014).
    [Crossref]
  21. M. Zigzag, M. Butkovski, A. Englert, W. Kinzel, and I. Kanter, “Zero-lag synchronization of chaotic units with time-delayed couplings,” EPL 85(6), 60005 (2009).
    [Crossref]
  22. F. Sorrentino, L. M. Pecora, A. M. Hagerstrom, T. E. Murphy, and R. Roy, “Complete characterization of the stability of cluster synchronization in complex dynamical networks,” Sci. Adv. 2(4), e1501737 (2016).
    [Crossref] [PubMed]
  23. T. Dahms, J. Lehnert, and E. Schöll, “Cluster and group synchronization in delay-coupled networks,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 86(1), 016202 (2012).
    [Crossref] [PubMed]
  24. R. Lang and K. Kobayashi, “External optical feedback effects on the semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
    [Crossref]
  25. M. C. Soriano, J. García-Ojalvo, C. R. Mirasso, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, and I. Fischer, “Complex photonics: Dynamics and applications of delay-coupled semiconductors lasers,” Rev. Mod. Phys. 85(1), 421–470 (2013).
    [Crossref]
  26. H. G. Winful, “Instability threshold for an array of coupled semiconductor lasers,” Phys. Rev. A 46(9), 6093–6094 (1992).
    [Crossref] [PubMed]
  27. B. Kim, N. Li, A. Locquet, and D. S. Citrin, “Experimental bifurcation-cascade diagram of an external-cavity semiconductor laser,” Opt. Express 22(3), 2348–2357 (2014).
    [Crossref] [PubMed]
  28. G. Kozyreff, A. G. Vladimirov, and P. Mandel, “Global coupling with time delay in an array of semiconductor lasers,” Phys. Rev. Lett. 85(18), 3809–3812 (2000).
    [Crossref] [PubMed]
  29. J. A. Acebrón, L. L. Bonilla, C. J. P. Vicente, F. Ritort, and R. Spigler, “The Kuramoto Model: A simple paradigm for synchronization phenomena,” Rev. Mod. Phys. 77(1), 137–182 (2005).
    [Crossref]
  30. C. Masoller, “Implications of how the linewidth enhancement factor is introduced on the Lang and Kobayashi model,” IEEE J. Quantum Electron. 33(5), 795–803 (1997).
    [Crossref]
  31. C. Masoller, “Comparison of the effects of nonlinear gain and weak optical feedback on the dynamics of semiconductor lasers,” IEEE J. Quantum Electron. 33(5), 804–814 (1997).
    [Crossref]
  32. M. Soriano, T. Berkvens, G. Van der Sande, G. Verschaffelt, J. Danckaert, and I. Fischer, “Interplay of current noise and delayed optical feedback on the dynamics of semiconductor lasers,” IEEE J. Quantum Electron. 47(3), 368–374 (2011).
    [Crossref]
  33. G. V. M. Yousefi, D. Lenstra, M. Yousefi, D. Lenstra, and G. Vemuri, “Carrier inversion noise has important influence on the dynamics of a semiconductor laser,” IEEE J. Sel. Top. Quantum Electron. 10(5), 955–960 (2004).
    [Crossref]
  34. J. Wang and K. Petermann, “Noise analysis of semiconductor lasers within the coherence collapse regime,” IEEE J. Quantum Electron. 27(1), 3–9 (1991).
    [Crossref]
  35. A. Murakami, K. Kawashima, and K. Atsuki, “Cavity resonance shift and bandwidth enhancement in semiconductor lasers with strong light injection,” IEEE J. Quantum Electron. 39(10), 1196–1204 (2003).
    [Crossref]
  36. T. Heil, I. Fischer, W. Elsässer, J. Mulet, and C. R. Mirasso, “Chaos Synchronization and Spontaneous Symmetry-Breaking in Symmetrically Delay-Coupled Semiconductor Lasers,” Phys. Rev. Lett. 86(5), 795–798 (2001).
    [Crossref] [PubMed]
  37. B. Liu, Y. Braiman, N. Nair, Y. Lu, Y. Guo, P. Colet, and M. Wardlaw, “Nonlinear dynamics and synchronization of an array of single mode laser diodes in external cavity subject to current modulation,” Opt. Commun. 324(0), 301–310 (2014).
    [Crossref]
  38. T. Heil, I. Fischer, and W. Elsässer, “Stabilization of feedback-induced instabilities in semiconductor lasers,” J. Opt. B Quantum Semiclass. Opt. 2(3), 413 (2000).
  39. H. Winful and S. Wang, “Stability of phase locking in coupled semiconductor laser arrays,” Appl. Phys. Lett. 53(20), 1894–1896 (1988).
    [Crossref]
  40. T. Erneux and P. Glorieux, Laser Dynamics (Cambridge University, 2010).
  41. N. Nair, E. Bochove, and Y. Braiman, “Transverse modes of coupled nonlinear oscillator arrays,” in Proceedings of the 4th International Conference on Applications in Nonlinear Dynamics (ICAND 2016), (Springer International Publishing, 2017), pp. 277–288.
    [Crossref]
  42. V. Rottschafer and B. Krauskopf, “The ECM-backbone of the Lang-Kobayashi equations: a geometric picture,” Int. J. Bifurcat. Chaos 17(5), 1575–1588 (2007).
    [Crossref]
  43. C. Masoller and N. B. Abraham, “Stability and dynamical properties of the coexisting attractors of an external-cavity semiconductor laser,” Phys. Rev. A 57(2), 1313–1322 (1998).
    [Crossref]
  44. J. Ohtsubo, “Feedback induced instability and chaos in semiconductor lasers and their applications,” Opt. Rev. 6(1), 1–15 (1999).
    [Crossref]
  45. D. Brunner and I. Fischer, “Reconfigurable semiconductor laser networks based on diffractive coupling,” Opt. Lett. 40(16), 3854–3857 (2015).
    [Crossref] [PubMed]

2016 (2)

A. Argyris, M. Bourmpos, and D. Syvridis, “Experimental synchrony of semiconductor lasers in coupled networks,” Opt. Express 24(5), 5600–5614 (2016).
[Crossref] [PubMed]

F. Sorrentino, L. M. Pecora, A. M. Hagerstrom, T. E. Murphy, and R. Roy, “Complete characterization of the stability of cluster synchronization in complex dynamical networks,” Sci. Adv. 2(4), e1501737 (2016).
[Crossref] [PubMed]

2015 (2)

K. Hirosawa, F. Shohda, T. Yanagisawa, and F. Kannari, “In-phase synchronization of array laser using intra-Talbot-cavity second harmonic generation,” Proc. SPIE 9342, 934216 (2015).

D. Brunner and I. Fischer, “Reconfigurable semiconductor laser networks based on diffractive coupling,” Opt. Lett. 40(16), 3854–3857 (2015).
[Crossref] [PubMed]

2014 (4)

C. J. Corcoran and F. Durville, “Passive coherent combination of a diode laser array with 35 elements,” Opt. Express 22(7), 8420–8425 (2014).
[Crossref] [PubMed]

V. Flunkert, S. Yanchuk, T. Dahms, and E. Schöll, “Synchronizability of networks with strongly delayed links: a universal classification,” J. Math. Sci. 202(6), 809–824 (2014).
[Crossref]

B. Kim, N. Li, A. Locquet, and D. S. Citrin, “Experimental bifurcation-cascade diagram of an external-cavity semiconductor laser,” Opt. Express 22(3), 2348–2357 (2014).
[Crossref] [PubMed]

B. Liu, Y. Braiman, N. Nair, Y. Lu, Y. Guo, P. Colet, and M. Wardlaw, “Nonlinear dynamics and synchronization of an array of single mode laser diodes in external cavity subject to current modulation,” Opt. Commun. 324(0), 301–310 (2014).
[Crossref]

2013 (3)

2012 (2)

2011 (2)

M. Soriano, T. Berkvens, G. Van der Sande, G. Verschaffelt, J. Danckaert, and I. Fischer, “Interplay of current noise and delayed optical feedback on the dynamics of semiconductor lasers,” IEEE J. Quantum Electron. 47(3), 368–374 (2011).
[Crossref]

D. Pabœuf, D. Vijayakumar, O. B. Jensen, B. Thestrup, J. Lim, S. Sujecki, E. Larkins, G. Lucas-Leclin, and P. Georges, “Volume Bragg grating external cavities for the passive phase locking of high-brightness diode laser arrays: theoretical and experimental study,” J. Opt. Soc. Am. B 28(5), 1289 (2011).
[Crossref]

2010 (1)

2009 (1)

M. Zigzag, M. Butkovski, A. Englert, W. Kinzel, and I. Kanter, “Zero-lag synchronization of chaotic units with time-delayed couplings,” EPL 85(6), 60005 (2009).
[Crossref]

2007 (1)

V. Rottschafer and B. Krauskopf, “The ECM-backbone of the Lang-Kobayashi equations: a geometric picture,” Int. J. Bifurcat. Chaos 17(5), 1575–1588 (2007).
[Crossref]

2005 (3)

J. A. Acebrón, L. L. Bonilla, C. J. P. Vicente, F. Ritort, and R. Spigler, “The Kuramoto Model: A simple paradigm for synchronization phenomena,” Rev. Mod. Phys. 77(1), 137–182 (2005).
[Crossref]

R. K. Huang, L. J. Missaggia, J. P. Donnelly, C. T. Harris, and G. W. Turner, “High-brightness slab-coupled optical waveguide laser arrays,” IEEE Photonics Technol. Lett. 17(5), 959–961 (2005).
[Crossref]

T. Y. Fan, “Laser beam combining for high-power, high-radiance sources,” IEEE J. Sel. Top. Quantum Electron. 11(3), 567–577 (2005).
[Crossref]

2004 (1)

G. V. M. Yousefi, D. Lenstra, M. Yousefi, D. Lenstra, and G. Vemuri, “Carrier inversion noise has important influence on the dynamics of a semiconductor laser,” IEEE J. Sel. Top. Quantum Electron. 10(5), 955–960 (2004).
[Crossref]

2003 (1)

A. Murakami, K. Kawashima, and K. Atsuki, “Cavity resonance shift and bandwidth enhancement in semiconductor lasers with strong light injection,” IEEE J. Quantum Electron. 39(10), 1196–1204 (2003).
[Crossref]

2001 (1)

T. Heil, I. Fischer, W. Elsässer, J. Mulet, and C. R. Mirasso, “Chaos Synchronization and Spontaneous Symmetry-Breaking in Symmetrically Delay-Coupled Semiconductor Lasers,” Phys. Rev. Lett. 86(5), 795–798 (2001).
[Crossref] [PubMed]

2000 (2)

T. Heil, I. Fischer, and W. Elsässer, “Stabilization of feedback-induced instabilities in semiconductor lasers,” J. Opt. B Quantum Semiclass. Opt. 2(3), 413 (2000).

G. Kozyreff, A. G. Vladimirov, and P. Mandel, “Global coupling with time delay in an array of semiconductor lasers,” Phys. Rev. Lett. 85(18), 3809–3812 (2000).
[Crossref] [PubMed]

1999 (1)

J. Ohtsubo, “Feedback induced instability and chaos in semiconductor lasers and their applications,” Opt. Rev. 6(1), 1–15 (1999).
[Crossref]

1998 (2)

C. Masoller and N. B. Abraham, “Stability and dynamical properties of the coexisting attractors of an external-cavity semiconductor laser,” Phys. Rev. A 57(2), 1313–1322 (1998).
[Crossref]

L. M. Pecora and T. L. Carroll, “Master Stability Functions for synchronized coupled systems,” Phys. Rev. Lett. 80(10), 2109–2112 (1998).
[Crossref]

1997 (2)

C. Masoller, “Implications of how the linewidth enhancement factor is introduced on the Lang and Kobayashi model,” IEEE J. Quantum Electron. 33(5), 795–803 (1997).
[Crossref]

C. Masoller, “Comparison of the effects of nonlinear gain and weak optical feedback on the dynamics of semiconductor lasers,” IEEE J. Quantum Electron. 33(5), 804–814 (1997).
[Crossref]

1993 (1)

J. R. Leger and G. Mowry, “External diode-laser-array cavity with mode-selecting mirror,” Appl. Phys. Lett. 63(21), 2884–2886 (1993).
[Crossref]

1992 (1)

H. G. Winful, “Instability threshold for an array of coupled semiconductor lasers,” Phys. Rev. A 46(9), 6093–6094 (1992).
[Crossref] [PubMed]

1991 (2)

J. Wang and K. Petermann, “Noise analysis of semiconductor lasers within the coherence collapse regime,” IEEE J. Quantum Electron. 27(1), 3–9 (1991).
[Crossref]

P. K. Jakobsen, H. G. Winful, L. Raman, R. A. Indik, J. V. Moloney, and A. C. Newell, “Diode-laser array modes: discrete and continuous models and their stability,” J. Opt. Soc. Am. B 8(8), 1674 (1991).
[Crossref]

1988 (2)

D. Mehuys, K. Mitsunaga, L. Eng, W. K. Marshall, and A. Yariv, “Supermode control in diffraction-coupled semiconductor laser arrays,” Appl. Phys. Lett. 53(13), 1165–1167 (1988).
[Crossref]

H. Winful and S. Wang, “Stability of phase locking in coupled semiconductor laser arrays,” Appl. Phys. Lett. 53(20), 1894–1896 (1988).
[Crossref]

1984 (1)

1980 (1)

R. Lang and K. Kobayashi, “External optical feedback effects on the semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[Crossref]

1961 (1)

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40(2), 453–488 (1961).
[Crossref]

Abraham, N. B.

C. Masoller and N. B. Abraham, “Stability and dynamical properties of the coexisting attractors of an external-cavity semiconductor laser,” Phys. Rev. A 57(2), 1313–1322 (1998).
[Crossref]

Acebrón, J. A.

J. A. Acebrón, L. L. Bonilla, C. J. P. Vicente, F. Ritort, and R. Spigler, “The Kuramoto Model: A simple paradigm for synchronization phenomena,” Rev. Mod. Phys. 77(1), 137–182 (2005).
[Crossref]

Aceves, A. B.

N. Nair, E. J. Bochove, A. B. Aceves, M. R. Zunoubi, and Y. Braiman, “Resonator modes and mode dynamics for an external cavity-coupled laser array,” in Proceedings of SPIE-The International Society for Optical Engineering (2015), 9343.

Argyris, A.

Atsuki, K.

A. Murakami, K. Kawashima, and K. Atsuki, “Cavity resonance shift and bandwidth enhancement in semiconductor lasers with strong light injection,” IEEE J. Quantum Electron. 39(10), 1196–1204 (2003).
[Crossref]

Berkvens, T.

M. Soriano, T. Berkvens, G. Van der Sande, G. Verschaffelt, J. Danckaert, and I. Fischer, “Interplay of current noise and delayed optical feedback on the dynamics of semiconductor lasers,” IEEE J. Quantum Electron. 47(3), 368–374 (2011).
[Crossref]

Bochove, E.

N. Nair, E. Bochove, and Y. Braiman, “Transverse modes of coupled nonlinear oscillator arrays,” in Proceedings of the 4th International Conference on Applications in Nonlinear Dynamics (ICAND 2016), (Springer International Publishing, 2017), pp. 277–288.
[Crossref]

Bochove, E. J.

N. Nair, E. J. Bochove, A. B. Aceves, M. R. Zunoubi, and Y. Braiman, “Resonator modes and mode dynamics for an external cavity-coupled laser array,” in Proceedings of SPIE-The International Society for Optical Engineering (2015), 9343.

Bonilla, L. L.

J. A. Acebrón, L. L. Bonilla, C. J. P. Vicente, F. Ritort, and R. Spigler, “The Kuramoto Model: A simple paradigm for synchronization phenomena,” Rev. Mod. Phys. 77(1), 137–182 (2005).
[Crossref]

Bourmpos, M.

Braiman, Y.

B. Liu, Y. Braiman, N. Nair, Y. Lu, Y. Guo, P. Colet, and M. Wardlaw, “Nonlinear dynamics and synchronization of an array of single mode laser diodes in external cavity subject to current modulation,” Opt. Commun. 324(0), 301–310 (2014).
[Crossref]

B. Liu and Y. Braiman, “Coherent beam combining of high power broad-area laser diode array with near diffraction limited beam quality and high power conversion efficiency,” Opt. Express 21(25), 31218–31228 (2013).
[Crossref] [PubMed]

B. Liu, Y. Liu, and Y. Braiman, “Coherent beam combining of high power broad-area laser diode array with a closed-V-shape external Talbot cavity,” Opt. Express 18(7), 7361–7368 (2010).
[Crossref] [PubMed]

N. Nair, E. J. Bochove, A. B. Aceves, M. R. Zunoubi, and Y. Braiman, “Resonator modes and mode dynamics for an external cavity-coupled laser array,” in Proceedings of SPIE-The International Society for Optical Engineering (2015), 9343.

N. Nair, E. Bochove, and Y. Braiman, “Transverse modes of coupled nonlinear oscillator arrays,” in Proceedings of the 4th International Conference on Applications in Nonlinear Dynamics (ICAND 2016), (Springer International Publishing, 2017), pp. 277–288.
[Crossref]

Brunner, D.

Butkovski, M.

M. Zigzag, M. Butkovski, A. Englert, W. Kinzel, and I. Kanter, “Zero-lag synchronization of chaotic units with time-delayed couplings,” EPL 85(6), 60005 (2009).
[Crossref]

Carroll, T. L.

L. M. Pecora and T. L. Carroll, “Master Stability Functions for synchronized coupled systems,” Phys. Rev. Lett. 80(10), 2109–2112 (1998).
[Crossref]

Citrin, D. S.

Colet, P.

B. Liu, Y. Braiman, N. Nair, Y. Lu, Y. Guo, P. Colet, and M. Wardlaw, “Nonlinear dynamics and synchronization of an array of single mode laser diodes in external cavity subject to current modulation,” Opt. Commun. 324(0), 301–310 (2014).
[Crossref]

Connors, M. K.

Corcoran, C. J.

Creedon, K. J.

Dahms, T.

V. Flunkert, S. Yanchuk, T. Dahms, and E. Schöll, “Synchronizability of networks with strongly delayed links: a universal classification,” J. Math. Sci. 202(6), 809–824 (2014).
[Crossref]

T. Dahms, J. Lehnert, and E. Schöll, “Cluster and group synchronization in delay-coupled networks,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 86(1), 016202 (2012).
[Crossref] [PubMed]

Danckaert, J.

M. Soriano, T. Berkvens, G. Van der Sande, G. Verschaffelt, J. Danckaert, and I. Fischer, “Interplay of current noise and delayed optical feedback on the dynamics of semiconductor lasers,” IEEE J. Quantum Electron. 47(3), 368–374 (2011).
[Crossref]

Donnelly, J. P.

R. K. Huang, L. J. Missaggia, J. P. Donnelly, C. T. Harris, and G. W. Turner, “High-brightness slab-coupled optical waveguide laser arrays,” IEEE Photonics Technol. Lett. 17(5), 959–961 (2005).
[Crossref]

Durville, F.

Elsässer, W.

T. Heil, I. Fischer, W. Elsässer, J. Mulet, and C. R. Mirasso, “Chaos Synchronization and Spontaneous Symmetry-Breaking in Symmetrically Delay-Coupled Semiconductor Lasers,” Phys. Rev. Lett. 86(5), 795–798 (2001).
[Crossref] [PubMed]

T. Heil, I. Fischer, and W. Elsässer, “Stabilization of feedback-induced instabilities in semiconductor lasers,” J. Opt. B Quantum Semiclass. Opt. 2(3), 413 (2000).

Eng, L.

D. Mehuys, K. Mitsunaga, L. Eng, W. K. Marshall, and A. Yariv, “Supermode control in diffraction-coupled semiconductor laser arrays,” Appl. Phys. Lett. 53(13), 1165–1167 (1988).
[Crossref]

Englert, A.

M. Zigzag, M. Butkovski, A. Englert, W. Kinzel, and I. Kanter, “Zero-lag synchronization of chaotic units with time-delayed couplings,” EPL 85(6), 60005 (2009).
[Crossref]

Fan, T. Y.

Fischer, I.

D. Brunner and I. Fischer, “Reconfigurable semiconductor laser networks based on diffractive coupling,” Opt. Lett. 40(16), 3854–3857 (2015).
[Crossref] [PubMed]

M. C. Soriano, J. García-Ojalvo, C. R. Mirasso, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, and I. Fischer, “Complex photonics: Dynamics and applications of delay-coupled semiconductors lasers,” Rev. Mod. Phys. 85(1), 421–470 (2013).
[Crossref]

M. C. Soriano, J. García-Ojalvo, C. R. Mirasso, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, and I. Fischer, “Complex photonics: Dynamics and applications of delay-coupled semiconductors lasers,” Rev. Mod. Phys. 85(1), 421–470 (2013).
[Crossref]

M. Soriano, T. Berkvens, G. Van der Sande, G. Verschaffelt, J. Danckaert, and I. Fischer, “Interplay of current noise and delayed optical feedback on the dynamics of semiconductor lasers,” IEEE J. Quantum Electron. 47(3), 368–374 (2011).
[Crossref]

T. Heil, I. Fischer, W. Elsässer, J. Mulet, and C. R. Mirasso, “Chaos Synchronization and Spontaneous Symmetry-Breaking in Symmetrically Delay-Coupled Semiconductor Lasers,” Phys. Rev. Lett. 86(5), 795–798 (2001).
[Crossref] [PubMed]

T. Heil, I. Fischer, and W. Elsässer, “Stabilization of feedback-induced instabilities in semiconductor lasers,” J. Opt. B Quantum Semiclass. Opt. 2(3), 413 (2000).

Flunkert, V.

V. Flunkert, S. Yanchuk, T. Dahms, and E. Schöll, “Synchronizability of networks with strongly delayed links: a universal classification,” J. Math. Sci. 202(6), 809–824 (2014).
[Crossref]

Fox, A. G.

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40(2), 453–488 (1961).
[Crossref]

Garcia-Ojalvo, J.

M. C. Soriano, J. García-Ojalvo, C. R. Mirasso, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, and I. Fischer, “Complex photonics: Dynamics and applications of delay-coupled semiconductors lasers,” Rev. Mod. Phys. 85(1), 421–470 (2013).
[Crossref]

García-Ojalvo, J.

M. C. Soriano, J. García-Ojalvo, C. R. Mirasso, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, and I. Fischer, “Complex photonics: Dynamics and applications of delay-coupled semiconductors lasers,” Rev. Mod. Phys. 85(1), 421–470 (2013).
[Crossref]

Georges, P.

Gopinath, J. T.

Guo, Y.

B. Liu, Y. Braiman, N. Nair, Y. Lu, Y. Guo, P. Colet, and M. Wardlaw, “Nonlinear dynamics and synchronization of an array of single mode laser diodes in external cavity subject to current modulation,” Opt. Commun. 324(0), 301–310 (2014).
[Crossref]

Hagerstrom, A. M.

F. Sorrentino, L. M. Pecora, A. M. Hagerstrom, T. E. Murphy, and R. Roy, “Complete characterization of the stability of cluster synchronization in complex dynamical networks,” Sci. Adv. 2(4), e1501737 (2016).
[Crossref] [PubMed]

Harris, C. T.

R. K. Huang, L. J. Missaggia, J. P. Donnelly, C. T. Harris, and G. W. Turner, “High-brightness slab-coupled optical waveguide laser arrays,” IEEE Photonics Technol. Lett. 17(5), 959–961 (2005).
[Crossref]

Heil, T.

T. Heil, I. Fischer, W. Elsässer, J. Mulet, and C. R. Mirasso, “Chaos Synchronization and Spontaneous Symmetry-Breaking in Symmetrically Delay-Coupled Semiconductor Lasers,” Phys. Rev. Lett. 86(5), 795–798 (2001).
[Crossref] [PubMed]

T. Heil, I. Fischer, and W. Elsässer, “Stabilization of feedback-induced instabilities in semiconductor lasers,” J. Opt. B Quantum Semiclass. Opt. 2(3), 413 (2000).

Hirosawa, K.

K. Hirosawa, F. Shohda, T. Yanagisawa, and F. Kannari, “In-phase synchronization of array laser using intra-Talbot-cavity second harmonic generation,” Proc. SPIE 9342, 934216 (2015).

Huang, R. K.

R. K. Huang, L. J. Missaggia, J. P. Donnelly, C. T. Harris, and G. W. Turner, “High-brightness slab-coupled optical waveguide laser arrays,” IEEE Photonics Technol. Lett. 17(5), 959–961 (2005).
[Crossref]

Indik, R. A.

Jakobsen, P. K.

Jensen, O. B.

Jones, A. M.

Kannari, F.

K. Hirosawa, F. Shohda, T. Yanagisawa, and F. Kannari, “In-phase synchronization of array laser using intra-Talbot-cavity second harmonic generation,” Proc. SPIE 9342, 934216 (2015).

Kansky, J. E.

Kanter, I.

M. Zigzag, M. Butkovski, A. Englert, W. Kinzel, and I. Kanter, “Zero-lag synchronization of chaotic units with time-delayed couplings,” EPL 85(6), 60005 (2009).
[Crossref]

Kapon, E.

Katz, J.

Kawashima, K.

A. Murakami, K. Kawashima, and K. Atsuki, “Cavity resonance shift and bandwidth enhancement in semiconductor lasers with strong light injection,” IEEE J. Quantum Electron. 39(10), 1196–1204 (2003).
[Crossref]

Kim, B.

Kinzel, W.

M. Zigzag, M. Butkovski, A. Englert, W. Kinzel, and I. Kanter, “Zero-lag synchronization of chaotic units with time-delayed couplings,” EPL 85(6), 60005 (2009).
[Crossref]

Kobayashi, K.

R. Lang and K. Kobayashi, “External optical feedback effects on the semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[Crossref]

Kozyreff, G.

G. Kozyreff, A. G. Vladimirov, and P. Mandel, “Global coupling with time delay in an array of semiconductor lasers,” Phys. Rev. Lett. 85(18), 3809–3812 (2000).
[Crossref] [PubMed]

Krauskopf, B.

V. Rottschafer and B. Krauskopf, “The ECM-backbone of the Lang-Kobayashi equations: a geometric picture,” Int. J. Bifurcat. Chaos 17(5), 1575–1588 (2007).
[Crossref]

Lang, R.

R. Lang and K. Kobayashi, “External optical feedback effects on the semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[Crossref]

Larkins, E.

Leger, J. R.

J. R. Leger and G. Mowry, “External diode-laser-array cavity with mode-selecting mirror,” Appl. Phys. Lett. 63(21), 2884–2886 (1993).
[Crossref]

Lehnert, J.

T. Dahms, J. Lehnert, and E. Schöll, “Cluster and group synchronization in delay-coupled networks,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 86(1), 016202 (2012).
[Crossref] [PubMed]

Lenstra, D.

G. V. M. Yousefi, D. Lenstra, M. Yousefi, D. Lenstra, and G. Vemuri, “Carrier inversion noise has important influence on the dynamics of a semiconductor laser,” IEEE J. Sel. Top. Quantum Electron. 10(5), 955–960 (2004).
[Crossref]

G. V. M. Yousefi, D. Lenstra, M. Yousefi, D. Lenstra, and G. Vemuri, “Carrier inversion noise has important influence on the dynamics of a semiconductor laser,” IEEE J. Sel. Top. Quantum Electron. 10(5), 955–960 (2004).
[Crossref]

Li, N.

Li, T.

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40(2), 453–488 (1961).
[Crossref]

Lim, J.

Liu, B.

Liu, Y.

Locquet, A.

Lu, Y.

B. Liu, Y. Braiman, N. Nair, Y. Lu, Y. Guo, P. Colet, and M. Wardlaw, “Nonlinear dynamics and synchronization of an array of single mode laser diodes in external cavity subject to current modulation,” Opt. Commun. 324(0), 301–310 (2014).
[Crossref]

Lucas-Leclin, G.

Mandel, P.

G. Kozyreff, A. G. Vladimirov, and P. Mandel, “Global coupling with time delay in an array of semiconductor lasers,” Phys. Rev. Lett. 85(18), 3809–3812 (2000).
[Crossref] [PubMed]

Marshall, W. K.

D. Mehuys, K. Mitsunaga, L. Eng, W. K. Marshall, and A. Yariv, “Supermode control in diffraction-coupled semiconductor laser arrays,” Appl. Phys. Lett. 53(13), 1165–1167 (1988).
[Crossref]

Masoller, C.

C. Masoller and N. B. Abraham, “Stability and dynamical properties of the coexisting attractors of an external-cavity semiconductor laser,” Phys. Rev. A 57(2), 1313–1322 (1998).
[Crossref]

C. Masoller, “Implications of how the linewidth enhancement factor is introduced on the Lang and Kobayashi model,” IEEE J. Quantum Electron. 33(5), 795–803 (1997).
[Crossref]

C. Masoller, “Comparison of the effects of nonlinear gain and weak optical feedback on the dynamics of semiconductor lasers,” IEEE J. Quantum Electron. 33(5), 804–814 (1997).
[Crossref]

Mehuys, D.

D. Mehuys, K. Mitsunaga, L. Eng, W. K. Marshall, and A. Yariv, “Supermode control in diffraction-coupled semiconductor laser arrays,” Appl. Phys. Lett. 53(13), 1165–1167 (1988).
[Crossref]

Mirasso, C. R.

M. C. Soriano, J. García-Ojalvo, C. R. Mirasso, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, and I. Fischer, “Complex photonics: Dynamics and applications of delay-coupled semiconductors lasers,” Rev. Mod. Phys. 85(1), 421–470 (2013).
[Crossref]

M. C. Soriano, J. García-Ojalvo, C. R. Mirasso, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, and I. Fischer, “Complex photonics: Dynamics and applications of delay-coupled semiconductors lasers,” Rev. Mod. Phys. 85(1), 421–470 (2013).
[Crossref]

T. Heil, I. Fischer, W. Elsässer, J. Mulet, and C. R. Mirasso, “Chaos Synchronization and Spontaneous Symmetry-Breaking in Symmetrically Delay-Coupled Semiconductor Lasers,” Phys. Rev. Lett. 86(5), 795–798 (2001).
[Crossref] [PubMed]

Missaggia, L. J.

K. J. Creedon, S. M. Redmond, G. M. Smith, L. J. Missaggia, M. K. Connors, J. E. Kansky, T. Y. Fan, G. W. Turner, and A. Sanchez-Rubio, “High efficiency coherent beam combining of semiconductor optical amplifiers,” Opt. Lett. 37(23), 5006–5008 (2012).
[Crossref] [PubMed]

R. K. Huang, L. J. Missaggia, J. P. Donnelly, C. T. Harris, and G. W. Turner, “High-brightness slab-coupled optical waveguide laser arrays,” IEEE Photonics Technol. Lett. 17(5), 959–961 (2005).
[Crossref]

Mitsunaga, K.

D. Mehuys, K. Mitsunaga, L. Eng, W. K. Marshall, and A. Yariv, “Supermode control in diffraction-coupled semiconductor laser arrays,” Appl. Phys. Lett. 53(13), 1165–1167 (1988).
[Crossref]

Moloney, J. V.

Mowry, G.

J. R. Leger and G. Mowry, “External diode-laser-array cavity with mode-selecting mirror,” Appl. Phys. Lett. 63(21), 2884–2886 (1993).
[Crossref]

Mulet, J.

T. Heil, I. Fischer, W. Elsässer, J. Mulet, and C. R. Mirasso, “Chaos Synchronization and Spontaneous Symmetry-Breaking in Symmetrically Delay-Coupled Semiconductor Lasers,” Phys. Rev. Lett. 86(5), 795–798 (2001).
[Crossref] [PubMed]

Murakami, A.

A. Murakami, K. Kawashima, and K. Atsuki, “Cavity resonance shift and bandwidth enhancement in semiconductor lasers with strong light injection,” IEEE J. Quantum Electron. 39(10), 1196–1204 (2003).
[Crossref]

Murphy, T. E.

F. Sorrentino, L. M. Pecora, A. M. Hagerstrom, T. E. Murphy, and R. Roy, “Complete characterization of the stability of cluster synchronization in complex dynamical networks,” Sci. Adv. 2(4), e1501737 (2016).
[Crossref] [PubMed]

Nair, N.

B. Liu, Y. Braiman, N. Nair, Y. Lu, Y. Guo, P. Colet, and M. Wardlaw, “Nonlinear dynamics and synchronization of an array of single mode laser diodes in external cavity subject to current modulation,” Opt. Commun. 324(0), 301–310 (2014).
[Crossref]

N. Nair, E. Bochove, and Y. Braiman, “Transverse modes of coupled nonlinear oscillator arrays,” in Proceedings of the 4th International Conference on Applications in Nonlinear Dynamics (ICAND 2016), (Springer International Publishing, 2017), pp. 277–288.
[Crossref]

N. Nair, E. J. Bochove, A. B. Aceves, M. R. Zunoubi, and Y. Braiman, “Resonator modes and mode dynamics for an external cavity-coupled laser array,” in Proceedings of SPIE-The International Society for Optical Engineering (2015), 9343.

Newell, A. C.

Ohtsubo, J.

J. Ohtsubo, “Feedback induced instability and chaos in semiconductor lasers and their applications,” Opt. Rev. 6(1), 1–15 (1999).
[Crossref]

Pabœuf, D.

Pecora, L. M.

F. Sorrentino, L. M. Pecora, A. M. Hagerstrom, T. E. Murphy, and R. Roy, “Complete characterization of the stability of cluster synchronization in complex dynamical networks,” Sci. Adv. 2(4), e1501737 (2016).
[Crossref] [PubMed]

L. M. Pecora and T. L. Carroll, “Master Stability Functions for synchronized coupled systems,” Phys. Rev. Lett. 80(10), 2109–2112 (1998).
[Crossref]

Petermann, K.

J. Wang and K. Petermann, “Noise analysis of semiconductor lasers within the coherence collapse regime,” IEEE J. Quantum Electron. 27(1), 3–9 (1991).
[Crossref]

Raman, L.

Redmond, S. M.

Ritort, F.

J. A. Acebrón, L. L. Bonilla, C. J. P. Vicente, F. Ritort, and R. Spigler, “The Kuramoto Model: A simple paradigm for synchronization phenomena,” Rev. Mod. Phys. 77(1), 137–182 (2005).
[Crossref]

Rottschafer, V.

V. Rottschafer and B. Krauskopf, “The ECM-backbone of the Lang-Kobayashi equations: a geometric picture,” Int. J. Bifurcat. Chaos 17(5), 1575–1588 (2007).
[Crossref]

Roy, R.

F. Sorrentino, L. M. Pecora, A. M. Hagerstrom, T. E. Murphy, and R. Roy, “Complete characterization of the stability of cluster synchronization in complex dynamical networks,” Sci. Adv. 2(4), e1501737 (2016).
[Crossref] [PubMed]

Sanchez-Rubio, A.

Schöll, E.

V. Flunkert, S. Yanchuk, T. Dahms, and E. Schöll, “Synchronizability of networks with strongly delayed links: a universal classification,” J. Math. Sci. 202(6), 809–824 (2014).
[Crossref]

T. Dahms, J. Lehnert, and E. Schöll, “Cluster and group synchronization in delay-coupled networks,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 86(1), 016202 (2012).
[Crossref] [PubMed]

Shohda, F.

K. Hirosawa, F. Shohda, T. Yanagisawa, and F. Kannari, “In-phase synchronization of array laser using intra-Talbot-cavity second harmonic generation,” Proc. SPIE 9342, 934216 (2015).

Smith, G. M.

Soriano, M.

M. Soriano, T. Berkvens, G. Van der Sande, G. Verschaffelt, J. Danckaert, and I. Fischer, “Interplay of current noise and delayed optical feedback on the dynamics of semiconductor lasers,” IEEE J. Quantum Electron. 47(3), 368–374 (2011).
[Crossref]

Soriano, M. C.

M. C. Soriano, J. García-Ojalvo, C. R. Mirasso, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, and I. Fischer, “Complex photonics: Dynamics and applications of delay-coupled semiconductors lasers,” Rev. Mod. Phys. 85(1), 421–470 (2013).
[Crossref]

Sorrentino, F.

F. Sorrentino, L. M. Pecora, A. M. Hagerstrom, T. E. Murphy, and R. Roy, “Complete characterization of the stability of cluster synchronization in complex dynamical networks,” Sci. Adv. 2(4), e1501737 (2016).
[Crossref] [PubMed]

Spigler, R.

J. A. Acebrón, L. L. Bonilla, C. J. P. Vicente, F. Ritort, and R. Spigler, “The Kuramoto Model: A simple paradigm for synchronization phenomena,” Rev. Mod. Phys. 77(1), 137–182 (2005).
[Crossref]

Sujecki, S.

Syvridis, D.

Thestrup, B.

Turner, G. W.

K. J. Creedon, S. M. Redmond, G. M. Smith, L. J. Missaggia, M. K. Connors, J. E. Kansky, T. Y. Fan, G. W. Turner, and A. Sanchez-Rubio, “High efficiency coherent beam combining of semiconductor optical amplifiers,” Opt. Lett. 37(23), 5006–5008 (2012).
[Crossref] [PubMed]

R. K. Huang, L. J. Missaggia, J. P. Donnelly, C. T. Harris, and G. W. Turner, “High-brightness slab-coupled optical waveguide laser arrays,” IEEE Photonics Technol. Lett. 17(5), 959–961 (2005).
[Crossref]

Van der Sande, G.

M. Soriano, T. Berkvens, G. Van der Sande, G. Verschaffelt, J. Danckaert, and I. Fischer, “Interplay of current noise and delayed optical feedback on the dynamics of semiconductor lasers,” IEEE J. Quantum Electron. 47(3), 368–374 (2011).
[Crossref]

Vemuri, G.

G. V. M. Yousefi, D. Lenstra, M. Yousefi, D. Lenstra, and G. Vemuri, “Carrier inversion noise has important influence on the dynamics of a semiconductor laser,” IEEE J. Sel. Top. Quantum Electron. 10(5), 955–960 (2004).
[Crossref]

Verschaffelt, G.

M. Soriano, T. Berkvens, G. Van der Sande, G. Verschaffelt, J. Danckaert, and I. Fischer, “Interplay of current noise and delayed optical feedback on the dynamics of semiconductor lasers,” IEEE J. Quantum Electron. 47(3), 368–374 (2011).
[Crossref]

Vicente, C. J. P.

J. A. Acebrón, L. L. Bonilla, C. J. P. Vicente, F. Ritort, and R. Spigler, “The Kuramoto Model: A simple paradigm for synchronization phenomena,” Rev. Mod. Phys. 77(1), 137–182 (2005).
[Crossref]

Vijayakumar, D.

Vladimirov, A. G.

G. Kozyreff, A. G. Vladimirov, and P. Mandel, “Global coupling with time delay in an array of semiconductor lasers,” Phys. Rev. Lett. 85(18), 3809–3812 (2000).
[Crossref] [PubMed]

Wang, J.

J. Wang and K. Petermann, “Noise analysis of semiconductor lasers within the coherence collapse regime,” IEEE J. Quantum Electron. 27(1), 3–9 (1991).
[Crossref]

Wang, S.

H. Winful and S. Wang, “Stability of phase locking in coupled semiconductor laser arrays,” Appl. Phys. Lett. 53(20), 1894–1896 (1988).
[Crossref]

Wardlaw, M.

B. Liu, Y. Braiman, N. Nair, Y. Lu, Y. Guo, P. Colet, and M. Wardlaw, “Nonlinear dynamics and synchronization of an array of single mode laser diodes in external cavity subject to current modulation,” Opt. Commun. 324(0), 301–310 (2014).
[Crossref]

Winful, H.

H. Winful and S. Wang, “Stability of phase locking in coupled semiconductor laser arrays,” Appl. Phys. Lett. 53(20), 1894–1896 (1988).
[Crossref]

Winful, H. G.

Yanagisawa, T.

K. Hirosawa, F. Shohda, T. Yanagisawa, and F. Kannari, “In-phase synchronization of array laser using intra-Talbot-cavity second harmonic generation,” Proc. SPIE 9342, 934216 (2015).

Yanchuk, S.

V. Flunkert, S. Yanchuk, T. Dahms, and E. Schöll, “Synchronizability of networks with strongly delayed links: a universal classification,” J. Math. Sci. 202(6), 809–824 (2014).
[Crossref]

Yariv, A.

D. Mehuys, K. Mitsunaga, L. Eng, W. K. Marshall, and A. Yariv, “Supermode control in diffraction-coupled semiconductor laser arrays,” Appl. Phys. Lett. 53(13), 1165–1167 (1988).
[Crossref]

E. Kapon, J. Katz, and A. Yariv, “Supermode analysis of phase-locked arrays of semiconductor lasers,” Opt. Lett. 9(4), 125–127 (1984).
[Crossref] [PubMed]

Yousefi, G. V. M.

G. V. M. Yousefi, D. Lenstra, M. Yousefi, D. Lenstra, and G. Vemuri, “Carrier inversion noise has important influence on the dynamics of a semiconductor laser,” IEEE J. Sel. Top. Quantum Electron. 10(5), 955–960 (2004).
[Crossref]

Yousefi, M.

G. V. M. Yousefi, D. Lenstra, M. Yousefi, D. Lenstra, and G. Vemuri, “Carrier inversion noise has important influence on the dynamics of a semiconductor laser,” IEEE J. Sel. Top. Quantum Electron. 10(5), 955–960 (2004).
[Crossref]

Zigzag, M.

M. Zigzag, M. Butkovski, A. Englert, W. Kinzel, and I. Kanter, “Zero-lag synchronization of chaotic units with time-delayed couplings,” EPL 85(6), 60005 (2009).
[Crossref]

Zunoubi, M. R.

N. Nair, E. J. Bochove, A. B. Aceves, M. R. Zunoubi, and Y. Braiman, “Resonator modes and mode dynamics for an external cavity-coupled laser array,” in Proceedings of SPIE-The International Society for Optical Engineering (2015), 9343.

Appl. Phys. Lett. (3)

D. Mehuys, K. Mitsunaga, L. Eng, W. K. Marshall, and A. Yariv, “Supermode control in diffraction-coupled semiconductor laser arrays,” Appl. Phys. Lett. 53(13), 1165–1167 (1988).
[Crossref]

J. R. Leger and G. Mowry, “External diode-laser-array cavity with mode-selecting mirror,” Appl. Phys. Lett. 63(21), 2884–2886 (1993).
[Crossref]

H. Winful and S. Wang, “Stability of phase locking in coupled semiconductor laser arrays,” Appl. Phys. Lett. 53(20), 1894–1896 (1988).
[Crossref]

Bell Syst. Tech. J. (1)

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40(2), 453–488 (1961).
[Crossref]

EPL (1)

M. Zigzag, M. Butkovski, A. Englert, W. Kinzel, and I. Kanter, “Zero-lag synchronization of chaotic units with time-delayed couplings,” EPL 85(6), 60005 (2009).
[Crossref]

IEEE J. Quantum Electron. (6)

R. Lang and K. Kobayashi, “External optical feedback effects on the semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[Crossref]

C. Masoller, “Implications of how the linewidth enhancement factor is introduced on the Lang and Kobayashi model,” IEEE J. Quantum Electron. 33(5), 795–803 (1997).
[Crossref]

C. Masoller, “Comparison of the effects of nonlinear gain and weak optical feedback on the dynamics of semiconductor lasers,” IEEE J. Quantum Electron. 33(5), 804–814 (1997).
[Crossref]

M. Soriano, T. Berkvens, G. Van der Sande, G. Verschaffelt, J. Danckaert, and I. Fischer, “Interplay of current noise and delayed optical feedback on the dynamics of semiconductor lasers,” IEEE J. Quantum Electron. 47(3), 368–374 (2011).
[Crossref]

J. Wang and K. Petermann, “Noise analysis of semiconductor lasers within the coherence collapse regime,” IEEE J. Quantum Electron. 27(1), 3–9 (1991).
[Crossref]

A. Murakami, K. Kawashima, and K. Atsuki, “Cavity resonance shift and bandwidth enhancement in semiconductor lasers with strong light injection,” IEEE J. Quantum Electron. 39(10), 1196–1204 (2003).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (2)

G. V. M. Yousefi, D. Lenstra, M. Yousefi, D. Lenstra, and G. Vemuri, “Carrier inversion noise has important influence on the dynamics of a semiconductor laser,” IEEE J. Sel. Top. Quantum Electron. 10(5), 955–960 (2004).
[Crossref]

T. Y. Fan, “Laser beam combining for high-power, high-radiance sources,” IEEE J. Sel. Top. Quantum Electron. 11(3), 567–577 (2005).
[Crossref]

IEEE Photonics Technol. Lett. (1)

R. K. Huang, L. J. Missaggia, J. P. Donnelly, C. T. Harris, and G. W. Turner, “High-brightness slab-coupled optical waveguide laser arrays,” IEEE Photonics Technol. Lett. 17(5), 959–961 (2005).
[Crossref]

Int. J. Bifurcat. Chaos (1)

V. Rottschafer and B. Krauskopf, “The ECM-backbone of the Lang-Kobayashi equations: a geometric picture,” Int. J. Bifurcat. Chaos 17(5), 1575–1588 (2007).
[Crossref]

J. Math. Sci. (1)

V. Flunkert, S. Yanchuk, T. Dahms, and E. Schöll, “Synchronizability of networks with strongly delayed links: a universal classification,” J. Math. Sci. 202(6), 809–824 (2014).
[Crossref]

J. Opt. B Quantum Semiclass. Opt. (1)

T. Heil, I. Fischer, and W. Elsässer, “Stabilization of feedback-induced instabilities in semiconductor lasers,” J. Opt. B Quantum Semiclass. Opt. 2(3), 413 (2000).

J. Opt. Soc. Am. B (2)

Opt. Commun. (1)

B. Liu, Y. Braiman, N. Nair, Y. Lu, Y. Guo, P. Colet, and M. Wardlaw, “Nonlinear dynamics and synchronization of an array of single mode laser diodes in external cavity subject to current modulation,” Opt. Commun. 324(0), 301–310 (2014).
[Crossref]

Opt. Express (6)

Opt. Lett. (3)

Opt. Rev. (1)

J. Ohtsubo, “Feedback induced instability and chaos in semiconductor lasers and their applications,” Opt. Rev. 6(1), 1–15 (1999).
[Crossref]

Phys. Rev. A (2)

C. Masoller and N. B. Abraham, “Stability and dynamical properties of the coexisting attractors of an external-cavity semiconductor laser,” Phys. Rev. A 57(2), 1313–1322 (1998).
[Crossref]

H. G. Winful, “Instability threshold for an array of coupled semiconductor lasers,” Phys. Rev. A 46(9), 6093–6094 (1992).
[Crossref] [PubMed]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

T. Dahms, J. Lehnert, and E. Schöll, “Cluster and group synchronization in delay-coupled networks,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 86(1), 016202 (2012).
[Crossref] [PubMed]

Phys. Rev. Lett. (3)

G. Kozyreff, A. G. Vladimirov, and P. Mandel, “Global coupling with time delay in an array of semiconductor lasers,” Phys. Rev. Lett. 85(18), 3809–3812 (2000).
[Crossref] [PubMed]

T. Heil, I. Fischer, W. Elsässer, J. Mulet, and C. R. Mirasso, “Chaos Synchronization and Spontaneous Symmetry-Breaking in Symmetrically Delay-Coupled Semiconductor Lasers,” Phys. Rev. Lett. 86(5), 795–798 (2001).
[Crossref] [PubMed]

L. M. Pecora and T. L. Carroll, “Master Stability Functions for synchronized coupled systems,” Phys. Rev. Lett. 80(10), 2109–2112 (1998).
[Crossref]

Proc. SPIE (1)

K. Hirosawa, F. Shohda, T. Yanagisawa, and F. Kannari, “In-phase synchronization of array laser using intra-Talbot-cavity second harmonic generation,” Proc. SPIE 9342, 934216 (2015).

Rev. Mod. Phys. (2)

J. A. Acebrón, L. L. Bonilla, C. J. P. Vicente, F. Ritort, and R. Spigler, “The Kuramoto Model: A simple paradigm for synchronization phenomena,” Rev. Mod. Phys. 77(1), 137–182 (2005).
[Crossref]

M. C. Soriano, J. García-Ojalvo, C. R. Mirasso, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, and I. Fischer, “Complex photonics: Dynamics and applications of delay-coupled semiconductors lasers,” Rev. Mod. Phys. 85(1), 421–470 (2013).
[Crossref]

Sci. Adv. (1)

F. Sorrentino, L. M. Pecora, A. M. Hagerstrom, T. E. Murphy, and R. Roy, “Complete characterization of the stability of cluster synchronization in complex dynamical networks,” Sci. Adv. 2(4), e1501737 (2016).
[Crossref] [PubMed]

Other (5)

J. L. Levy and K. Roh, “Coherent array of 900 semiconductor laser amplifiers,” in Photonics West ’95, 2382, pp. 58–69. (1995)

M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, 1974).

N. Nair, E. J. Bochove, A. B. Aceves, M. R. Zunoubi, and Y. Braiman, “Resonator modes and mode dynamics for an external cavity-coupled laser array,” in Proceedings of SPIE-The International Society for Optical Engineering (2015), 9343.

T. Erneux and P. Glorieux, Laser Dynamics (Cambridge University, 2010).

N. Nair, E. Bochove, and Y. Braiman, “Transverse modes of coupled nonlinear oscillator arrays,” in Proceedings of the 4th International Conference on Applications in Nonlinear Dynamics (ICAND 2016), (Springer International Publishing, 2017), pp. 277–288.
[Crossref]

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Figures (4)

Fig. 1
Fig. 1 Poincare sections of (ϕ(t)ϕ(tτ))/τ are plotted for a single laser and a 10-laser array as a function of feedback strength. Each trajectory was generated from a continuation simulation started with random initial conditions and κ f =0.01 ns 1 . Every 1000ns, κ f was increased by 0.01 ns 1 . The yellow line in each plot is the central solution frequency Ω from Eq. (11) for a single laser with κ f = κ .It is clear from these diagrams that the dynamics are almost identical except for the scaling.
Fig. 2
Fig. 2 Average phase synchrony of large arrays of lasers. Each point corresponds to a simulation of an array with randomized initial conditions (we started averaging process after 800ns simulation for convergence to occur). Lines denoting regions of effective coupling are related to the degree of synchronization in the system. For very large arrays achieving phase synchronization requires either weak or very strong (though possibly unrealistic) coupling κ f between the diodes in the array.
Fig. 3
Fig. 3 cos ϕ i are plotted for 10-laser arrays of identical lasers with varying feedback strength in the presence of carrier and phase noise. When κ f =5 ns 1 , the effective coupling κ =3.65 ns 1 and the behavior is CW as predicted in Fig. 1. When κ f =20 ns 1 , the effective coupling κ =14.6 ns 1 , and the behavior is chaotic with a high degree of synchrony. When κ f =30 ns 1 , the effective coupling κ =21.9 ns 1 , the behavior is chaotic, and the lasers begin to de-synchronize.
Fig. 4
Fig. 4 (a-c) The inner products of the first four eigenvectors V i T E (t) ( i=1,2,3,4) are plotted for an array of 30 lasers with d x =.8 at various values of coupling strength. The simulations have realistic amounts of noise and a frequency disorder of στ=0.3. The synchronization level S is also given, showing that there is indeed synchronization in the quasiperiodic (b) and chaotic (c) regimes. (d) The central far-field lobes of an array of 100 lasers subject to noise and disorder are plotted. For σ=0, the synchrony level is S=1.0. For σ=0.1/τ, S=.97. For σ=0.5/τ, S=.63. For σ=1.0/τ, S=.39.

Equations (10)

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X . i (t)=F( X i (t))+η( X i (t),t)+ κ f M j=1 M K ij C( X j (tτ), X i (t))
r ˙ i (t)= 1 2 (g N i (t) N 0 1+s r i 2 (t) γ) r i (t)+ R sp ( η E e i ϕ i (t) ) + κ f M j=1 M K ij r j (tτ)cos( ϕ j (tτ) ϕ i (t))
ϕ ˙ i (t)= α 2 (g N i (t) N 0 1+s r i 2 (t) γ)+ ω i R sp r i (t) ( η E e i ϕ i (t) ) + κ f M j=1 M K ij r j (tτ) r i (t) sin( ϕ j (tτ) ϕ i (t))
N ˙ i (t)= J 0 γ n N i (t)g N i (t) N 0 1+s r i 2 (t) r i 2 (t)+ γ n N i (t) η N (t)
K ij = d x |ij|
ξ ˙ (t)=[ I M F( X i (t)) X i (t) | X i (t)= X * (t) + κ f M Γ C( X i (t), X j (tτ)) X i (t) | X i (t)= X * (t) ] ξ (t) +[ κ f M K C( X i (t), X j (tτ)) X j (tτ) | X i (t)= X * (t) ] ξ (tτ)
δ ˙ i (t)=[ X i (t) F+ κ f M λ 1 X i (t) C] δ i (t) +[ κ f M λ i X j (tτ) C] δ i (tτ)
δ ˙ 1 (t)=[ X i (t) F+ κ X i (t) C] δ i (t) +[ κ X j (tτ) C] δ i (tτ)
ξ ˙ (t)=[ X i (t) F+ κ f X i (t) C]ξ(t) +[ κ f X j (tτ) C]ξ(tτ)
Ω= κ f αcos(Ωτ) κ f sin(Ωτ) r= g J 0 γ n N 0 (s γ n +g)(γ2 κ f cos(Ωτ)) γ n s γ n +g N= J 0 + g N 0 r 2 1+s r 2 γ n + g r 2 1+s r 2

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