IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 51, NO. 8, AUGUST 2015 1600209 Dynamics of Passively Phased Ring Oscillator Fiber Laser Arrays Sudarshan Sivaramakrishnan, Wei-Zung Chang, Almantas Galvanauskas, and Herbert G. Winful, Fellow, IEEE Abstract—In this paper, we study ﬁber laser arrays coupled in a spatially ﬁltered ring oscillator geometry that produces a tiled-array output. In particular, we examine the passive phasing dynamics, the efﬁciency of coherent combination, and the dependence of the system’s behavior on nonlinearities in the ﬁbers. For an array containing a small number of elements, we ﬁnd that the ﬁbers achieve a co-phased state within two round trips after a perturbation. Steady-state results agree with the previous work. We also ﬁnd that the Kerr nonlinearity decreases the combining efﬁciency as determined from the on-axis intensity in the far-ﬁeld output. Index Terms— Beam combining, ﬁber lasers, ﬁber laser arrays. I. I NTRODUCTION T HERE is currently a great deal of interest in the prospect of achieving a robust, scalable approach to passive coherent beam combining of ﬁber lasers. A number of coupling architectures have been proposed and further studied in the scientiﬁc community, but much of the current literature has focused on steady-state analyses or dynamic but single-mode models. Although these provide valuable insight into the practicality and underlying physics of the proposed approaches, a dynamic multi-mode model can be used to obtain a more complete picture, since the evolution of the system and the stability of steady-state solutions are dictated by the dynamics. An understanding of the dynamics can lead to a more comprehensive demonstration of coherent phasing or lack thereof for the ﬁber laser array. This can be used to compare the different beam combining approaches, understand their limitations, and seek methods to overcome them. Lastly, there is a need for a general metric to characterize combin- ing efﬁciency in order to help compare different combining architectures. To date there have been few dynamic studies of the externally-coupled ﬁber laser array. The initial models required a ﬁxed phase difference as an input and did not yield spectral information [1], [2]. A more recent model focused on a Q-switching instability found from a linear stability analysis and presented only a few preliminary results from the numerical solution of the propagation equations [3], [4]. In the case of internally-coupled arrays [5], Wu, et al. have Manuscript received March 5, 2015; revised May 25, 2015; accepted June 11, 2015. Date of publication June 25, 2015; date of current version July 20, 2015. The authors are with the Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI 48109 USA (e-mail: sivas@umich.edu; wzchang@umich.edu; almantas@umich.edu; arrays@umich.edu). Color versions of one or more of the ﬁgures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identiﬁer 10.1109/JQE.2015.2449310 Fig. 1. Diagram of an array of ﬁber lasers combined via the spatially-ﬁltered ring-geometry architecture. presented a dynamic model based on the amplifying Nonlinear Schrödinger Equation (NLSE) that incorporates the multiple longitudinal modes of a ﬁber laser [6] and allows for the natural selection of the resonant array modes that experience the minimum loss [7], [8]. That model has been used to explain the experimentally observed scaling of combining efﬁciency with the number of laser ampliﬁers [9]. It has also recently been supplemented with saturable absorp- tion to describe experiments on the coherent combining of mode locked ﬁber lasers [10], [11]. In this paper we apply the dynamic model of Wu et al. [7], [8] to the externally-coupled ring oscillator ﬁber array. Direct numerical simulations of the NLSE for each individual laser, along with the external coupling imposed by boundary conditions, facilitate investigation of the growth of coherence in the system for two cases: (i) evolution from an unlocked initial state to the ﬁnal phase-locked state, and (ii) recovery from perturbations applied after reaching steady state. We assess the degree of phase locking by considering the dynamics of the temporal and spectral phase proﬁles, the far-ﬁeld intensity pattern, and an order parameter. The results agree with published steady state and experimental analyses. We also consider the combining efﬁciency, quantifying it by a metric that relies on the far-ﬁeld intensity pattern of the output. We vary the Kerr nonlinearity and investigate its effect on spectral power distribution, spatial distribution of power in the far-ﬁeld, and combining efﬁciency. We conclude with a brief comparison with internally-coupled arrays. II. METHODS Fig. 1 illustrates an array of ﬁber ampliﬁers arranged in a spatially-ﬁltered ring geometry architecture. The individual 0018-9197 © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.