Numerical studies together with asymptotic and spectral analysis establish
regimes where soliton pairs in degenerate optical parametric oscillators fuse,
repel, or form bound states. A novel bound state stabilized by coupled internal
oscillations is predicted.Comment: 3 page

Brambilla

Dmitry V. Skryabin

Etrich

Gorshkov

Malomed

Oppp

Richy

Rosanov

Staliunas

William J. Firth

English

arXiv

Crossref

Interaction of cavity solitons in degenerate optical parametric oscillators.

Numerical studies, together with asymptotic and spectral analysis, establish regimes in which soliton pairs in degenerate optical parametric oscillators fuse, repel, or form bound states. A novel bound state that is stabilized by coupled internal oscillations is predicted

Skryabin, Dmitry V.

Firth, William J.

Name not available

Interaction of cavity solitons in degenerate optical parametric oscillators

University of Strathclyde Institutional Repository

Strathprints Institutional RepositorySkryabin, Dmitry V. and Firth, William J. (1999) Interaction of cavity solitons in degenerate opticalparametric oscillators. Optics Letters, 24 (15). pp. 1056-1058. ISSN 0146-9592Strathprints is designed to allow users to access the research output of the University of Strathclyde.Copyright c© and Moral Rights for the papers on this site are retained by the individual authorsand/or other copyright owners. You may not engage in further distribution of the material for anyprofitmaking activities or any commercial gain. You may freely distribute both the url (http://strathprints.strath.ac.uk/) and the content of this paper for research or study, educational, ornot-for-profit purposes without prior permission or charge.Any correspondence concerning this service should be sent to Strathprints administrator:mailto:strathprints@strath.ac.ukhttp://strathprints.strath.ac.uk/  Skryabin, Dmitry V.* and Firth, William J.* (1999) Interaction of cavity solitons in degenerate optical parametric oscillators. Optics Letters, 24 (15). pp. 1056-1058. ISSN 0146-9592      http://eprints.cdlr.strath.ac.uk/6472/   This is an author-produced version of a paper published in Optics Letters, 24 (15). pp. 1056-1058. ISSN 0146-9592. This version has been peer-reviewed, but does not include the final publisher proof corrections, published layout, or pagination.  Strathprints is designed to allow users to access the research output of the University of Strathclyde. Copyright © and Moral Rights for the papers on this site are retained by the individual authors and/or other copyright owners. You may not engage in further distribution of the material for any profitmaking activities or any commercial gain. You may freely distribute both the url (http://eprints.cdlr.strath.ac.uk) and the content of this paper for research or study, educational, or not-for-profit purposes without prior permission or charge. You may freely distribute the url (http://eprints.cdlr.strath.ac.uk) of the Strathprints website.  Any correspondence concerning this service should be sent to The Strathprints Administrator: eprints@cis.strath.ac.uk  arXiv:patt-sol/9906004 v1   4 Jun 1999Interaction of cavity solitons in degenerate optical parametric oscillators.Dmitry V. Skryabin and William J. FirthDepartment of Physics and Applied Physics, University of Strathclyde, Glasgow, G4 0NG, Scotland(June 29, 2006)Numerical studies together with asymptotic and spectral analysis establish regimes where solitonpairs in degenerate optical parametric oscillators fuse, repel, or form bound states. A novel boundstate stabilized by coupled internal oscillations is predicted.Many nonlinear media can support soliton-like struc-tures when contained in a driven optical cavity [1–4]. Wewill refer to such structures as cavity solitons (CS). Inquadratic nonlinear media CS have recently been pre-dicted in both optical parametric oscillator (OPO) [3]and second harmonic generation [4] configurations. Al-though experimental observation of χ(2)-CS remains achallenge, impressive bistability results [5] demonstratethe required level of nonlinearity and thus pave the waytowards this goal.The large values of effective χ(2) accessible in artifi-cially phase-matched materials in combination with theirpractically instantaneous response are important advan-tages of using quadratic nonlinearity for implementationof CS for all-optical processing of information. They thusrepresent an interesting alternative to the CS which canbe created in cavities with dispersive-absorptive [1],andresonant electron-hole [2] types of nonlinearities. In allsuch schemes high CS density is desirable and thereforeunderstanding of their interaction is a practically impor-tant problem which is still largely open. In this Letterwe focus on the interactions of CS found in the below-threshold regime of a degenerate doubly resonant OPO,under conditions where the signal field has three coexis-tent plane-wave states [3].Assuming phase-matching, a plane-wave input field,and ignoring walk-off, the mean-field OPO equations canbe presented in the following dimensionless form [3]−i∂tE1 = (α1∂2x + δ1 + iγ1)E1 + (E2 + µ)E∗1 , (1)−i∂tE2 = (α2∂2x + δ2 + iγ2)E2 + E21/2,Here E1 and (E2 + µ) are the signal and pump fields,respectively, at frequencies ω and 2ω (we use µ as a mea-sure of the pump strength). The slow time t is scaled sothat γm (proportional to the cavity damping rates) andδm (to the detunings from its resonances) are of orderunity. Here and below m = 1, 2.This system can describe either diffractive or dispersiveeffects. We consider x a dimensionless transverse coor-dinate, and so set αm = 1/m. For this case, existenceof CS for δm < 0 was numerically demonstrated [3] forµL < µ < µR, where µL = |γ1δ2 + γ2δ1|/√δ22 + γ22 , andµR =√δ21 + γ21 is the OPO threshold. Within this rangetwo different non-trivial homogeneous solutions (Em 6= 0,∂xEm = 0) coexist with the trivial one (Em = 0), and theCS are sech-like localized states on the zero background.We start our analysis by applying a perturbativemethod [6] to the problem of CS interaction. We seeksolutions of Eqs. (1) in the formEm(x, t) = Am(x − xA) +Bm(x− xB) +ǫ(am(x− xA, xB, t) + bm(x− xB, xA, t)) +O(ǫ2), (2)where Am(x − xA) and Bm(x − xB) are CS centred onxA,B. Note that Eqs. (1) are invariant with respect to aπ phase flip of the signal field, so that A and B can be ei-ther in-phase or out-of-phase CS. We assume 0 < ǫ≪ 1,and that the perturbation functions am, bm are negligibleexcept close to xA, xB respectively. We further assumethat xA,B vary on the slow time scale τ = ǫt and thatd = |xA − xB | is large enough that the overlap functionsI1 = A2B∗1 +B2A∗1 and I2 = A1B1 are of order ǫ.Substituting ansatz (2) into Eqs. (1) and truncatingO(ǫ2) terms we obtain two analogous systems of equa-tions for am and bm, the former expressible in the form:(LˆA − ∂t)~a = −(∂τxA)~ξA + ~I/ǫ, (3)Here ~a = (Rea1, Rea2, Ima1, Ima2)T ; operator LˆA is thelinearization of Eqs. (1) around the soliton Am; ~ξA =∂x(ReA1, ReA2, ImA1, ImA2)T is the neutral eigenmodeof LˆA associated with translational symmetry, LˆA~ξA = 0;and ~I = (−ImI1,−ImI2, ReI1, ReI2)T controls the in-teraction of the two CS.The solution of Eq. (3) should in general be expressedas a superposition of the eigenmodes ~ξn of LˆA, LˆA~ξn =λn~ξn, with time dependent coefficients, because the CSinteraction will couple to them all. However, apart fromthe above-mentioned neutral eigenmodes, the only ana-lytic knowledge about the eigensystems of LˆA and LˆBis that they have two bands of continuum modes witheigenvalues λ lying on Reλ = −γm, i.e. that all extendedeigenmodes are damped. We have obtained their fulleigensystems numerically, using finite-differences, overwide ranges of all relevant parameters. We find that forsufficiently large dissipation all cavity solitons are stablethroughout the entire region of their existence. A Hopfbifurcation can occur as photon lifetime is increased, butwe will not consider here any parameter regions whereisolated CS are unstable. With oscillatory eigenmodesabsent or well damped, only the neutral mode is easilyexcited by external perturbations, and so we meantimeneglect all other modes. This enables us to obtain semi-analytic results on CS interactions.1To exclude secularly growing solutions the right-handside of Eq. (3) must be orthogonal to the neutral eigen-mode of Lˆ†A (which we calculated numerically). Thissolvability condition, together with that for the B soli-ton, defines a function f which governs the dynamicalevolution of the distance d between the soliton centers:∂td = f(d). (4)We computed f for both in-phase and out-of-phase inter-acting CS, for many parameter values. Typical examplesare plotted in Fig. 1. Regions where f is negative (pos-itive) correspond to CS attraction (repulsion). Zeros off(d) thus mark stationary bound states of CS pairs, whichare stable if ∂df < 0 where f = 0.We find that this equation gives generally correct pre-dictions of the inter-soliton forces, in particular that in-phase CS attract and out-of-phase CS repel. Both re-pulsion and attraction become stronger as µ increases,presumably because the signal component (E1) of theCS becomes less localized as µ approaches plane-wavethreshold at µR. A similar effect can be envisaged inother CS models. For in-phase CS the function f can de-velop pairs of zeros, see Fig. 1. This predicts birth of newpairs of CS bound states, one stable and one unstable.In Fig. 2 we present simulation results showing differ-ent interaction scenarios for two CS initially separated byabout three soliton widths. First, we consider interactionof in-phase solitons. For small µmutual attraction resultsin fusion of two solitons into one (Fig. 2(a)). Graduallyincreasing µ we first observe formation of a stable oscilla-tory bound state (Fig. 2(b)), then of a stationary boundstate (Fig. 2(c)) which is stable (the radiation visiblein Fig. 2(c) decays, albeit slowly). Note that the equi-librium separation in Fig. 2(c) is predicted quite wellby the appropriate zero of f(d) in Fig. 1, even thoughthese CS are close enough to endanger the assumptionsof our perturbation method. Stationary two-hump soli-tary states have been found previously [3] as solutions ofan approximate equation derived from Eqs. (1), but noanalysis of soliton interactions was performed. Note thatwe have found not only two-hump but also multi-humpsolitary states. However the latter were usually dynam-ically unstable. Further details on this issue will be re-ported elsewhere. Close to the upper boundary of CSexistence the interaction of two solitons excites a globalpattern, see Fig. 2(d), via generation of a switching wavefrom the stable trivial solution up into the modulation-ally unstable nontrivial homogeneous state. As predictedby Eq. (4), out-of-phase CS repel each other throughoutthe entire region of their existence - contrast Fig. 2(e)with Fig. 2(d), which corresponds to the same value ofµ.Now we will describe numerical results of the inter-action of CS where weakly-damped oscillatory modesstrongly influence the soliton interactions. Oscillat-ing solitons generally radiate energy, which can becometrapped between neighbouring solitons, exerting a radia-tion force which may lead to formation of a bound state.An effect of this kind has been reported for solitons inmodels with a weak global dissipation [7]. We investi-gated a quite different situation, where linear waves es-caping from the soliton are strongly damped. Here stronginteraction between the solitons is due, not to radiationmodes, but to proto-Hopf modes, and thus has novel as-pects.The effect is strong providing that two conditions aresatisfied. First, and crucially, the corresponding eigen-modes must have tails with well pronounced and weaklydecaying oscillatory structure, see Fig. 3(a). Second,as might be expected, the oscillatory mode should beweakly damped (see Fig. 3(b)), i.e. the CS is close toa Hopf instability. If both conditions hold, then, even ifthe global damping due to the γm is strong, a CS acts asa guide for waves weakly damped in both space and time.If a second CS is close enough, these guided waves cancouple and reinforce each other. Fig. 3(c, d) illustratesthe dynamics of two interacting CS having eigenmodesshown in Fig. 3(a). Note that the separation of the in-teracting solitons in Fig. 3(c) is much greater than theirwidth. Comparison with Fig. 3(b) clearly indicates thatthe undamped pulsations shown in Fig. 3(d) originatefrom coupling and mutual reinforcement of the oscilla-tory modes of the two solitons. A further interestingpoint is that we find these dynamic bound states also forout-of-phase solitons, balancing the repulsion induced bytheir neutral mode interaction.Quadratic nonlinearity is also known to support soli-tons in free propagation geometry and in particularly in-teraction of these solitons has recently been studied bothexperimentally and theoretically [8,9]. Hamiltonian na-ture of free propagating solitons results in their interac-tion obeying the laws of Newtonian dynamics [9]. An-other important difference is that the relative phase ofthe interacting solitons can take only two discrete valuesin a cavity while it is a continuous free parameter in apropagation scheme. In spite of these differences fusion ofthe in-phase solitons and repulsion of the out-phase arecommon features in both schemes. However, existenceof the stationary and oscillatory bound states coupledeither via translational neutral modes or via internal os-cillatory modes are novel important features arising dueto presence of the external pump and cavity losses. Bal-ance between the pump and losses acts as an additional,equally important with the balance between diffractionand nonlinearity, mechanism of soliton formation insidean optical cavity.In summary, we have presented the analytical and nu-merical study of the interaction of cavity solitons in a de-generate OPO and identified distinct static and dynamicbinding mechanisms.D.V.S. thanks C. Etrich, F. Lederer, D. Michaelis andU. Peschel for warm hospitality and illuminating discus-sions of many relevant questions during his short visit toJena. He also acknowledges financial support from theRoyal Society of Edinburgh and British Petroleum. The2work is partially supported by ESPRIT project PIANOSand EPSRC grant GR/M19727.[1] N.N. Rosanov, Progress in Optics 35, 1 (1996); M. Tlidi,P. Mandel, and R. Lefever, Phys. Rev. Lett. 73, 640(1994); W.J. Firth and A.J. Scroggie, Phys. Rev. Lett.76, 1623. For a recent review of CS, see W.J. Firth andG.K. Harkness, Asian J. Phys. 7, 665 (1998).[2] M. Brambilla, L.A. Lugiato, F. Prati, L. Spinelli, andW.J. Firth, Phys. Rev. Lett. 79, 2042 (1997); D.Michaelis, U. Peschel, and F. Lederer, Phys. Rev. A 56,R3366 (1997).[3] K. Staliunas and V.J. Sa´nchez-Morcillo, Opt. Commun.139, 306 (1997); S. Longhi, Phys. Scripta 56, 611 (1997);S. Trillo and M. Haelterman, Opt. Lett. 23, 1514 (1998).[4] C. Etrich, U. Peschel, and F. Lederer, Phys. Rev. Lett.79, 2454 (1997); S. Longhi, Opt. Lett. 23, 346 (1998).[5] C. Richy, K.I. Petsas, E. Giacobino, C. Fabre, and L. Lu-giato, J. Opt. Soc. Am. B 12, 456 (1995); A.G. White, J.Mlynek, and S. Schiller, Europhys. Lett. 35, 425 (1996).[6] Various versions of this method can be found, e.g., in:K. A. Gorshkov and L. A. Ostrovsky, Physica D 3, 428(1981); S. Longhi, Phys. Rev. E 55, 1060 (1997).[7] B.A. Malomed, Phys. Rev. E 47, 2874 (1993).[8] C. Etrich, U. Peschel, F. Lederer, and B. Malomed, Phys.Rev. A 52, R3444 (1995); C.B. Clausen, P.L. Chris-tiansen, and L. Torner, Opt. Commun. 136, 185 (1997);Y. Baek, R. Schiek, G.I. Stegeman, I. Baumann, andW. Sohler, Opt. Lett. 22, 1550 (1997); D.M. Baboiu,G.I. Stegeman, Opt. Quant. Electr. 30, 849 (1998); A.Barthelemy, B. Bourliaguet, V. Kermene, B. Costantini,C. De Angelis, D. Modotto, and G. Assanto, Opt. Quant.Electr. 30, 923 (1998).[9] V.V. Steblina, Y.S. Kivshar, and A.V. Buryak, Opt. Lett.23, 156 (1998); A.V. Buryak and V.V. Steblina, J. Opt.Soc. Am. B 16, 245 (1999).[10] Only the central third of the computational window in xis shown in Figs. 2(a)-(c),(e), 3(c).2 4 6 8 10 12d-0.10-0.050.000.050.10fFIG. 1. Plots of the CS velocity function f vs d. Full(dashed) lines correspond to in-phase (out-of-phase) solitonsand thin (thick) lines correspond to µ = 1.6(1.9). Other pa-rameters are δ2 = −4, δ1 = −1.8, γ1 = 1, γ2 = 0.8.0 100 200 300 400-6-4-20246x(a)0 100 200 300 400t-6-4-20246x(b)0 100 200 300 400-6-4-20246x(c)0 100 200 300 400t-20-1001020x(d)0 100 200 300 400t-505x(e)FIG. 2. Interaction dynamics of χ(2) cavity solitons [9].At different values of pump parameter µ, in-phase CS: (a)merge, µ = 1.6; (b) form oscillatory bound state, µ = 1.8; (c)form stable stationary bound state, µ = 1.9; (d) generate apattern via a switching wave, µ = 2. Out-of-phase solitonsrepel, e.g. at µ = 2, (e). Other parameters as for Fig. 1.0 20 40 60 80 100t230240250260270Q(b)550 560 570 580t-505x (c)0 10 20 30 40x-0.10.00.10.2Re(u 1,2)(a)550 560 570 580 590t220240260280300Q A,B(d)FIG. 3. Dynamic interaction of CS for µ = 2, δ1 = −3,δ2 = −12, γ1 = 0.3, γ2 = 1, for which the CS has a modewith eigenvalue pair λ ≃ −0.03± i4.14: (a) Spatial structureof the eigenmode, Re(u1) - full lines, Re(u2) -dashed lines;(b) Temporal evolution of signal energy Q =∫dx|E1|2 forslightly perturbed single soliton, showing damped oscillation;(c) Spatio-temporal evolution [9] of |E1| (time window muchlater than in (b)), showing dynamic bound state; (d) Tem-poral evolution of signal energies of the two CS in (c), in thesame time window, showing rapid undamped oscillations andslow energy exchange between the two solitons.3

Interaction of cavity solitons in degenerate optical parametric
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Interaction of cavity solitons in degenerate optical parametric oscillators

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