We examine the existence and stability of discrete spatial solitons in
coupled nonlinear lasing cavities (waveguide resonators), addressing the case
of active defocusing media, where the gain exceeds damping in the low-amplitude
limit. A new family of stable localized structures is found: these are bright
and grey cavity solitons representing the connections between homogeneous and
inhomogeneous states. Solitons of this type can be controlled by the discrete
diffraction and are stable when the bistability of homogenous states is absent.Comment: 3 pages, 3 figures, accepted to Optics Letters (October 2012

Alexey V. Yulin

Egorov

Jaroslaw E. Prilepsky

Kiselev

Lederer

Magnus Johansson

Peschel

Staliunas

Stanislav A. Derevyanko

Yulin

English

arXiv

We examine the existence and stability of discrete spatial solitons in coupled nonlinear lasing cavities (waveguide resonators), addressing the case of active defocusing media, where the gain exceeds damping in the low-amplitude limit. A new family of stable localized structures is found: these are bright and gray cavity solitons representing the connections between homogeneous and inhomogeneous states. Solitons of this type can be controlled by discrete diffraction and are stable when the bistability of homogenous states is absent

Prilepsky, Jaroslaw E.

Yulin, Alexey V.

Johansson, Magnus

Derevyanko, Stanislav A.

Aston Publications Explorer

arXiv:1202.4660v2  [physics.optics]  3 Oct 2012Discrete solitons in coupled active lasing cavitiesJaroslaw E. Prilepsky,1,∗ Alexey V. Yulin2, Magnus Johansson3, and Stanislav A. Derevyanko11Nonlinearity and Complexity Research Group, Aston University, Aston Triangle, B4 7ET, Birmingham, UK2Centro de Fisica Teorica e Computacional, Universidade de Lisboa, Lisboa 1649-003, Portugal3Department of Physics, Chemistry and Biology (IFM), Linko¨ping University, SE-581 83 Linko¨ping, Sweden∗Corresponding author: y.prylepskiy1@aston.ac.ukCompiled June 23, 2018We examine the existence and stability of discrete spatial solitons in coupled nonlinear lasing cavities(waveguide resonators), addressing the case of active defocusing media, where the gain exceeds damping inthe low-amplitude limit. A new family of stable localized structures is found: these are bright and grey cavitysolitons representing the connections between homogeneous and inhomogeneous states. Solitons of this typecan be controlled by the discrete diffraction and are stable when the bistability of homogenous states is absent.c© 2018 Optical Society of AmericaOCIS codes: 190.1450, 190.4390, 190.4420Due to the huge progress in semiconductor-based pho-tonic devices, a multitude of experimentally achievableconcepts have arisen, aimed at performing optical pro-cessing and light reconfiguration [1, 2]. One of theseconcepts relates to light manipulation in semiconduc-tor microcavities closed by high-reflectivity Bragg re-flectors (microresonators). The cavity medium consiststypically of quantum well structures which can be ab-sorbing or (with electrical or optical pumping) providegain [1]. The soliton solutions in driven optical cavities(cavity solitons, CS) have attracted much attention be-cause of their potential applications in information pro-cessing and optical memory schemes: The ability to con-trol their switch-on/off process by an address beam, andtheir location by introducing gradients in the holdingbeam makes them interesting as mobile pixels for recon-figurable arrays of all-optical processing units [1]. TheCS are a particular realization of the dissipative solitonnotion – localized structures existing due to the balancebetween dissipation, nonlinearity and diffraction [2].A relatively new direction in the light processing inoptical cavities refers to coupled systems of microres-onators [3–9] where the excitation level of each cavity(or of an ordered pattern) can serve as an elementary’pixel’. These ’pixels’ are based on yet another type ofdissipative solitary structures – the discrete cavity soli-tons (DCS), which are localized excitations in coupledarrays of nonlinear cavities. Achieving controlled manip-ulations with ’pixels’ requires a detailed study of theproperties of these objects: nucleation thresholds, stabil-ity, mobility etc [3]. It has to be noted that, unlike thecontinuum case [1], the studies of DCS have so far onlydealt with the case of a passive cavity, i.e. the losses wereassumed to dominate over the lasing gain [4–9]. At thesame time there is an interesting idea of self-sustainedlight spots in cavities with active gain [1], where no ex-ternal driving is needed for supporting the stability ofthe CS (pixels). Therefore additional efforts are neededto address specifically the case of DCS in active media.In this Letter we, for the first time to our knowledge,consider the properties of DCS in coupled active cavitiesand demonstrate that their structure and stability canbe drastically different from their counterparts in passivesystems [3–9]. In addition, we show that by adjusting thecoupling (which is an example of a discrete diffractionmanagement) one can control the stability of the DCS(a similar idea for passive planar resonators was usedin [10]) which can be implemented as a useful tool forall-optical processing in discrete systems.We consider a regular array of evanescently coupledidentical nonlinear cavities with high-reflectivity mirrorsplaced at their facets, driven by a homogeneous holdingbeam having a normal incidence. The mean-field modelfor the light distribution within the array can be derivedfrom the coupled-mode equations [4] assuming that i) thefrequency detuning from the cavity resonance is small(high-finesse cavities) and ii) the temporal dynamics isslow compared to the round-trip time, i.e. the struc-ture is short compared to the coupling and nonlinear-ity lengths (an effective cavity length is on the order of1-2µm, though much longer structures can also be fab-ricated [1]). Then the system of (normalized) equationsgoverning the dynamics of the averaged beam amplitudeinside the n-th cavity, An, reads as:(iddt+∆+ α|An|2)An ++ C(An+1 +An−1 − 2An)− iF (An) = P, (1)where all coefficients are real, C > 0 characterizes thecoupling strength, ∆˜ = ∆ − 2C is the detuning of thepump frequency from the resonant one, α characterizesthe strength of the nonlinear (Kerr) polarization, P > 0is the amplitude of the holding beam and F (An) de-scribes dissipative effects. The field An is normalized tothe lasing saturation intensity I0: An/√I0 → An, andthe time to the cavity round trip time (see also [4]).The existence and stability properties of DCS in model(1) can be essentially different from those of the continu-1ous counterparts [3,4]. In this Letter we will specificallyconcentrate on highly discrete solutions occurring in thecase of defocusing nonlinearity, α < 0, existing up to alimiting value of C and having no continuum limit. Weopt for a simple but physically important and generalform of dissipative function with gain saturation [3,7,11]:F (An) = An[γ(1 + |An|2)−1 − δ], and the active regioncorresponds to amplitudes satisfying F (|A|) > 0 (high-lighted part of the horizontal axis in Figs.1, 3). Here pa-rameters δ, γ > 0 describe linear losses and gain respec-tively. We will assume that in the low-power limit thesystem is active, so that γ > δ, distinguishing the caseconsidered here from other studies of DCS [3–9,11]. Thisimplies that in the absence of a holding beam (P = 0)the trivial solution An ≡ 0 of Eq.(1) is unstable.To specify the order of coefficients entering Eq.(1)we took parameters for GaAs semiconductor lasers [2]with the linear refractive index n0 = 3.55, saturationintensity I0 = 108W/cm2, the Kerr coefficient n2 =−2 × 10−13 cm2/W, the linear gain 1.5cm−1 and theloss 0.5 cm−1. Assuming the cross-section of resonatorsbeing 5µm2, we get that the unit of the normalized am-plitude corresponds to the power 0.5mW. The round tripfrequency is Ω ∼ 10 10s−1. With this in mind we set alldimensionless coefficients ∆, δ, α and γ in order of unity,which appears to be achievable in experiments. For thevisualization of our results we opt for the following par-ticular values: ∆ = 3.5, δ = 0.4, γ = 1.7, α = −0.2, yield-ing |A| < 1.8 as active region. For distances ∼ 1÷ 10µmthe coupling strength varies in a large interval, so we canuse a relatively weak realistic coupling C = 0.15, a valuewhich for the chosen set of parameters is slightly belowthe upper existence limit for the solutions found below.The analysis of DCS in the system (1) starts fromthe identification of stable homogeneous (H) states thatserve as a background for localized solutions [3–7]: set-ting An = A (or C = 0) in Eq.(1) one finds the so-calledresponse curve P (|A|), plotted in Fig.1(a) for the pa-rameters chosen. The static state is stable when Eq.(1)linearized above it has no time-growing solutions. Whentwo stable H-states coexist (thick solid parts marked H1and H2 in Fig.1(a)), solitons for nonzero C can gener-ally be found as connections between these states [7]:the H1-H2-H1 connection for bright solitons, where thecore intensity exceeds the background, or the H2-H1-H2 connection for grey solitons with a ’dent’ in the H2‘substrate’. In our case, as for passive systems [3–7], suchstable DCS exist in a certain interval within the bistabil-ity region of H-states caused by the Kerr nonlinearity;an example of a stable bright soliton is given in Fig.2(upper right pane). The major difference of H-states inthe active case is the instability of the low-amplitudepart of the H1-branch, so that, in contrast to passivesystems [3–7], DCS formed by such H-states connectionscannot be stable at low P values.Now we turn to another possible background for theformation of DCS – inhomogeneous periodic static state(we label it as an I-state). In general, infinitely manyFig. 1. (Color online.) Response curves for (a) homo-geneous (H) and (b) 2-site periodic (I) solutions, withstable regions highlighted by thick solid lines. (a) P (|A|)for H-solutions, with stable lower (H1) and higher (H2)states. (b) P (|Amax|) and P (|Amin|) for the I-solution;the direction of ‘evolution’ from the curve coalescencepoints (marked by ||) is identified with arrows. Regionsof H-H and H-I bistability are marked on the P -axes; theactive region (F (|A|) > 0) is highlighted on the |A|-axes.extended solutions with inhomogeneous patterning canbe found but here, to demonstrate the main features in-troduced by coupling to the I-states, we deal with thesimplest family having a 2-site period. Setting in Eq.(1)An, An+2, . . . = Amax, and An+1, An+3, . . . = Amin,Amax 6= Amin, we solve the system for these two complexfields. The results are given in Fig.1(b) in the form of two(equivalent) response curves: P (|Amax|) and P (|Amin|)(stable regions are marked with a thick solid line). TheI-background is discrete and does not exist in the contin-uum limit C → ∞. One can observe that the regions ofstability for H-states and I-state partially overlap. Thismeans that one can compose new families of localized so-lutions in the form of a fragment of the I-state embeddedin a stable H-background (or vice versa). In addition, theI-state can be stable below the region of H-bistability, i.e.for parameters where one cannot have stable DCS in theform of H-states connections. Thus, in this region theonly stable DCS represent connections of type H2-I.In Figs. 2 and 3 we show examples of the new typeof DCS corresponding to H-I connections found in ourmodel. We present them on snaking diagrams where eachpoint corresponds to a particular DCS. The bright/greyDCS are characterized by the maximal/minimal ampli-tude of the soliton so the snakes corresponding to eachsoliton type are plotted on the curves P (|Amax|), (Fig.2,corresponding to H1-I-H1 connection) and P (|Amin|),(Fig.3, H2-I-H2 connection). These DCS are discrete, i.e.2Fig. 2. (Color online.) Left pane: Snaking diagram forbright DCS corresponding to H1-I-H1 connection (sta-ble regions are highlighted). The inset shows a magni-fied part of the snake. Right panes show particular fielddistributions for DCS: the top pane shows a stable DCScorresponding to the H1-H2-H1 connection (P = 4.3),the two lower panes show the distributions for the stableDCS corresponding to H1-I-H1. The latter correspond tothe points marked • ’1’, ’2’ on the snake.Fig. 3. (Color online.) Snaking diagram P (|Amin|) forthe grey DCS corresponding to the H2-I-H2 connection.The inset shows the particular distribution of stable DCSreferring to the point marked as  ’3’.they can exist only up to a finite value of coupling. Theyhave a nonuniform central excited part correspondingto a fragment of the I-state embedded into either H1or H2 background. Importantly, the grey DCS of thistype can be stable in the region where no solitons rep-resenting the connections of H-states can be stable (seeFig.3), so this stability is achieved solely by the discretediffraction. The solitons outside the stable regions typi-cally display nonoscillatory exponential growth of smallperturbations.We note that, to the best of our knowledge, the DCSwith H-I connections of the type found above have notbeen studied in works on (passive) coupled cavity sys-tems, although a solution involving connections with amore complicated I-state was mentioned in [7]. Solutionsinvolving oscillatory patterns at the tails were discussedin [8, 9]: in [8] DCS with oscillatory decaying tails werefound, while in [9] the staggering low-amplitude plane-wave served as a background I-state for a grey DCS.Notably, both these types of DCS could be stable outof the H-bistability region. Our solutions have the stag-gering in the middle and come from the anticontinuouslimit with generally large staggering oscillations. By de-creasing γ we checked that stable analogues of the H-Itype DCS with a 2-periodic I-state in fact do exist alsofor passive systems.Our findings have several important physical conse-quences. First, we have demonstrated that in active cav-ities, aside from ’traditional’ DCS representing connec-tions between H-states, there can be stable solitary solu-tions with a lower power corresponding to the H-I con-nections, Figs.2, 3. Thus exerting an address beam forthe nucleation of a ’pixel’ one can end up with an H-Irather than an H-H soliton. In turn, the mobility charac-teristics [9] of H-I solitons are likely to be different fromthose of the well-studied H1-H2 solutions, and the con-ditions for optical processing can significantly deviatefrom those of the ’conventional’ H-H DCS. Second, byadjusting the discrete diffraction (i.e. coupling) one cangain stable solitary solutions in the region of parameters,where no stable H1-H2 DCS can exist. Such stability con-trol via a simple discrete diffraction management can beused as an additional tool for light manipulation: onecan control not only the stability, but also the geometryof DCS providing more versatility for optical processing.Finally, the only stable type of DCS suitable for ’pixels’manipulations in active cavities at low P values (or fora relatively high gain) corresponds to the H-I solitons.A.Y. and M.J. thank NCRG, Aston University, forkind hospitality. M.J. also acknowledges support fromthe Swedish Research Council. A.Y. was supported bythe FCT (Portugal), grant PTDC/FIS/112624/2009.References1. T. Ackemann, W.J. Firth and G.-L. Oppo, in “Ad-vances In Atomic Molecular and Optical Physics,” E.Arimondo, P.R. Berman and C.C. Lin (Eds.), vol. 57,323 (2009).2. “Dissipative Solitons,” N. Akhmediev and A. Ankiewicz(Eds.), Series: Lecture Notes in Physics, Vol. 661,(Springer: Berlin, Heidenberg), 2005.3. F. Lederer, G.I. Stegeman, D.N. Christodoulides, G. As-santo, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1(2008).4. U. Peschel, O. Egorov, and F. Lederer, Opt. Lett. 29,1909 (2004).5. O. Egorov, F. Lederer, and K. Staliunas, Opt. Lett. 32,2106 (2007).6. A.V. Yulin, A.R. Champneys, and D.V. Skryabin, Phys.Rev. A 78, 011804(R) (2008).7. A.V. Yulin and A.R. Champneys, SIAM J. Appl. Dy-nam. Syst. 9, 391 (2010).8. O. Egorov, U. Peschel, and F. Lederer, Phys. Rev. E 71,056612 (2005).9. O.A. Egorov, F. Lederer and Y.S. Kivshar, Opt. Express15, 4149 (2007).10. K. Staliunas, O. Egorov, Y.S. Kivshar, and F. Lederer,Phys. Rev. Lett. 101, 153903 (2008).11. Al. S. Kiselev, An. S. Kiselev, and N. N. Rozanov, Opt.Spectrosc. 105, 547 (2008).3Informational Fourth PageIn this section, please provide full versions of citations toassist reviewers and editors (OL publishes a short formof citations) or any other information that would aid thepeer-review process.References1. T. Ackemann, W.J. Firth and G.-L. Oppo, in “Ad-vances In Atomic Molecular and Optical Physics,” E.Arimondo, P.R. Berman and C.C. Lin (Eds.), vol. 57,323-421, 2009.2. “Dissipative Solitons,” N. Akhmediev and A. Ankiewicz(Eds.), Series: Lecture Notes in Physics, Vol. 661,(Springer: Berlin, Heidenberg), 448 p., 2005.3. F. Lederer, G.I. Stegeman, D.N. Christodoulides, G. As-santo, M. Segev, and Y. Silberberg, “Discrete solitons inoptics”, Phys. Rep. 463, 1-126 (2008).4. U. Peschel, O. Egorov, and F. Lederer, “Discrete cavitysolitons”, Opt. Lett. 29, 1909-1912 (2004).5. O. Egorov, F. Lederer, and K. Staliunas, “Subdiffrac-tive discrete cavity solitons”, Opt. Lett. 32, 2106-2109(2007).6. A.V. Yulin, A.R. Champneys, and D.V. Skryabin, “Dis-crete cavity solitons due to saturable nonlinearity,”Phys. Rev. A 78, 011804(R) (2008).7. A.V. Yulin and A.R. Champneys, “Discrete Snaking:Multiple Cavity Solitons in Saturable Media,” SIAM J.Appl. Dynam. Syst. 9 391-431 (2010).8. O. Egorov, U. Peschel, and F. Lederer, “Discretequadratic cavity solitons”, Phys. Rev. E 71, art.no056612 (2005).9. O.A. Egorov, F. Lederer and Y.S. Kivshar, “How doesan inclined holding beam affect discrete modulationalinstability and solitons in nonlinear cavities?”, Opt. Ex-press 15, 4149-4158 (2007).10. K. Staliunas, O. Egorov, Y.S. Kivshar, and F. Lederer,“Bloch Cavity Solitons in Nonlinear Resonators withIntracavity Photonic Crystals,” Phys. Rev. Lett. 101,art.no 153903 (2008).11. Al. S. Kiselev, An. S. Kiselev, and N. N. Rozanov, “Dis-sipative Discrete Spatial Optical Solitons in a System ofCoupled Optical Fibers with the Kerr and ResonanceNonlinearities”, Opt. Spectrosc. 105, 547-556 (2008).4

Discrete solitons in coupled active lasing cavities

We examine the existence and stability of discrete spatial solitons in coupled nonlinear lasing cavities (waveguide resonators), addressing the case of active defocusing media, where the gain exceeds damping in the low-amplitude limit. A new family of stable localized structures is found: these are bright and gray cavity solitons representing the connections between homogeneous and inhomogeneous states. Solitons of this type can be controlled by discrete diffraction and are stable when the bistability of homogenous states is absent.</p

Discrete solitons in coupled active lasing cavities

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