Stabilization of three-wave vortex beams in the waveguide
We consider two-dimensional (2D) localized vortical modes in the three-wave system with the quadratic () nonlinearity, alias nondegenerate second-harmonic-generating system, guided by the isotropic harmonic-oscillator (HO) (alias parabolic) confining potential. In addition to the straightforward realization in optics, the system models mixed atomic-molecular Bose-Einstein condensates (BECs). The main issue is stability of the vortex modes, which is investigated through computation of instability growth rates for eigenmodes of small perturbations, and by means of direct simulations. The threshold of parametric instability for single-color beams, represented solely by the second harmonic (SH) with zero vorticity, is found in an analytical form with the help of the variational approximation (VA). Trapped states with vorticities in the two fundamental-frequency (FF) components and the SH one [the so-called hidden-vorticity (HV) modes] are completely unstable. Also unstable are semi-vortices (SVs), with component vorticities . However, full vortices, with charges , have a well-defined stability region. Unstable full vortices feature regions of robust dynamical behavior, where they periodically split and recombine, keeping their vortical content.
Keywords: vortex, second-harmonic-generation, parametric instability, azimutal instability,
PACS numbers: 05.45.Yv, 42.65.Tg, 42.65.Lm
The fundamental significance of the quadratic () nonlinearity in optics, including its use for the creation of solitons, is well known -, . In particular, the nonlinearity opens the way to the making of two- and three-dimensional (2D and 3D) solitons, because, unlike the Kerr (cubic) terms, the quadratic ones, which couple the fundamental-frequency (FF) and second-harmonic (SH) components of the optical fields, do not give rise to the collapse in the two- and three-dimensional (2D and 3D) space , which is a severe problem for multidimensional solitons in Kerr media . Stable (2+1)D beams propagating in quadratically nonlinear optical media have been demonstrated in the experiment , and stable spatiotemporal “light bullets” have been predicted in these media too . In the experiment, fully self-trapped “bullets” have not been reported yet, the closest result being a spatiotemporal soliton self-trapped in the longitudinal and one transverse directions, while the confinement in the additional transverse direction was imposed by the guiding structure [11, 12].
A challenging issue for the (2+1)D setting is the search for conditions securing the stability of vortical solitons. In contrast to their fundamental counterparts, vortex solitons supported by the quadratic instability in the free space are always unstable against azimuthal perturbations, which tend to split the vortex into a set of separating fragments. For the degenerate quadratic system, with a single FF component, the splitting instability was predicted theoretically - and demonstrated in the experiment . The same instability occurs in the framework of the full three-wave system, which includes two distinct FF components, which represent mutually orthogonal polarizations of light [19, 20].
Vortex solitons are also known as solutions of the 2D nonlinear Schrödinger (NLS) equation with the self-focusing cubic term . They too are subject to the azimuthal instability, which is actually stronger than the collapse instability driven by the cubic nonlinearity . The 2D NLS equation models not only light beams in bulk media with the Kerr nonlinearity, but also the mean-field dynamics of atomic Bose-Einstein condensates (BECs), trapped in the form of “pancakes” by the confining potential ; in the latter case, the NLS equation is usually called the Gross-Pitaevskii (GP) equation. In the context of the NLS/GP equation, a practically relevant solution of the instability problem was elaborated: both fundamental and vortical solitons, with topological charge and , respectively, can be stabilized by 2D harmonic-oscillator (HO) confining potentials. It is well known - that the HO potential renders the fundamental solitons stable, against small perturbations, in the entire region of their existence. The vortices with , trapped by the same potential, are stabilized in of their existence region (in terms of their norm). Additionally, an adjacent region of width supports a robust dynamical regime featuring periodic splitting and recombination of the vortices, which keep their topological charge . As concerns the BEC realization, stabilization of self-trapped semi-vortex modes in the free 2D space was recently demonstrated in a two-component system with the Kerr nonlinearity and linear spin-orbit coupling between the components .
The effective 2D trapping potential can be also realized in optical waveguides, in the form of the respective profile of the transverse modulation of the local refractive index . This circumstance suggests a natural possibility for the stabilization of (2+1)D vortex solitons in the medium by means of the radial HO potential. For the degenerate (two-wave) system, alias the Type-I second-harmonic-generating interaction -, this possibility was demonstrated in Ref. , and then extended for 3D spatiotemporal vortex solitons in Ref. . The subject of the present work is to investigate this stabilization mechanism for the nondegenerate three-wave system, with the Type-II quadratic interaction, where the results demonstrate essentially new features in comparison with the degenerate model.
A feasible approach to the making of the optical medium combining a nearly-parabolic profile of the refractive index and nonlinearity is the use of a 2D photonic crystal, which can be readily designed to emulate the required index profile, while the quadratic nonlinearity may be provided by poled material filling voids of the photonic-crystal matrix , . In fact, the effective radial potential provides for sufficiently strong localization of the trapped modes, therefore the exact parabolic shape of the radial profile is not crucially important .
Essentially the same system of GP equations for atomic and molecular wave functions models the BEC in atomic-molecular mixtures -. In that case, two different components of the atomic wave function pertain to two different atomic states of the same species, while the quadratic nonlinearity accounts for reactions of the merger of two atoms into a molecule, and splitting of the molecules due to collisions with free atoms. Accordingly, the predicted mechanism of the stabilization of three-component vortex solitons trapped in the HO potential can also be realized in the BEC mixture.
The paper is organized as follows. The model, based on the system of three propagation equation coupled by the terms, is introduced in Section II. In Section III, we introduce four types of three-wave modes considered in this work: the SH-only (single-color) one, with zero FF components; modes with the hidden vorticity (HV, which means opposite vorticities, , in the two FF components, and in the SH component); semi-vortices, with in one FF component and in the other one and in the SH wave (cf. the above-mentioned semi-vortices in the two-component system with the cubic nonlinearity and linear spin-orbit coupling between the components ); and, finally, full vortices, with in both FF components, and in the SH. In Section IV we report simple but nontrivial analytical results, which predict the threshold of the parametric instability for the fundamental and vortical single-color states by means of the variational approximation (VA). In Section V we formulate the eigenvalue problem for small perturbations around stationary solutions, which determines their stability. The numerical results for modes of all the types are reported in Section VI. First, we identify the stability threshold for the fundamental and vortical single-color trapped modes, and compare these results with the above-mentioned analytical predictions provided by the VA. Next, the results are summarized for the HV and SV modes, which are found to be unstable. Finally, the stability region is presented for the most general three-wave states, with vorticities in both FF components and in the SH. In addition, unstable vortices may exist in a quasi-stable dynamical regime, in the form of periodic splitting and recombination of the semi-vortex. The paper is concluded by Section VII.
2 The model
The copropagation of two components of the FF wave, with amplitudes and , and the SH wave, with amplitude , in the bulk waveguide obeys the system of three coupled equations written here in the normalized form [2, 3]:
where is the transverse diffraction operator, the asterisk stands for the complex conjugate, is the birefringence coefficient, is the FF-SH phase mismatch, and the strength of the HO trapping potential. In the most general case, stationary vortex solutions of this system, with two independent FF propagation constants and , can be looked for as
where and are integer vorticities of the two FF components, and , , and are real radial functions with asymptotic forms
The exact equality of the propagation constant and vorticity of the SH component in Eqs. (4) and (5) to sums of the same constants of the FF components is imposed by the coherent coupling between the FF and SH components in Eqs. (1), (2), and (3).
In this work, the analysis is carried out, chiefly, for , in which case the symmetry between the two FF components suggests to consider three-wave states with . For , rescaling of Eqs. (1)-(2) makes it possible to fix . Families of three-wave vortices with are considered too, at the end of the paper.
3 Stationary solutions
3.1 Single-color (second-harmonic) states
where replaces , see Eq. (5). This equation is tantamount to the radial Schrödinger equation for 2D quantum-mechanical states with azimuthal quantum number in the HO potential. The respective solutions to Eq. (11) are
where is an arbitrary constant. The norm (9) of this state is
3.2 Hidden-vorticity (HV) states
where real functions and satisfy the following radial equations (where the birefringence terms are kept, for the time being):
Previously, the HV concept was realized for the three-wave system in the free space , as well as for various systems of coupled continuous - and discrete [42, 43] NLS/GP equations with the cubic nonlinearity, including the system supported by the HO trapping potential . In work , it was demonstrated that the HV essentially suppresses the modulational instability of 2D ring solitons in a two-component system with saturable nonlinearity, in comparison with their counterparts carrying explicit vorticity. However, trapped vortical modes with HV were not previously studied in three-wave systems.
In the limit of small amplitudes, solutions of the linearized version of Eqs. (22) can be found in the form of
where the propagation constant must satisfy, respectively, the following relations for the components :
From Eqs. (24) it follows that, in the small-amplitude limit, the semi-vortex mode exists at the single value of the mismatch, . Below, it is shown that solutions for nonlinear semi-vortex modes lift this constraint.
3.4 Full vortices
In fact, the HV modes and semi-vortices are shown below to be completely unstable (modes of the latter type may be replaced by robust dynamical states which keep the same vortical structure). A stationary topological state which features a well-defined stability region is the full vortex, with the set of in Eqs. (4) and (5). The respective system of stationary equations is [cf. Eqs. (17)-(19) and (22)]:
4 The variational approximation (VA) for the onset of the parametric instability of the single-color modes
The VA is a natural approach for predicting the shape of nonlinear modes, including vortical ones supported by the nonlinearity [31, 30]. In the present context, the VA can be used for the prediction of the instability threshold of the trapped single-color (SH-only) states (the VA may be developed for other situations too, but it then turns out to be rather cumbersome).
For this purpose, it is necessary to consider a perturbed version of the single-color stationary state, which includes the FF components too. In particular, in the case of zero vorticity of the SH component and vorticities of the FF perturbations, the Lagrangian of the respective stationary equations (17)-(19) can be reduced to its radial part:
To develop the VA, the simplest Gaussian ansatz may be adopted, assuming equal amplitudes () and widths of fields and :
( is defined here). The symmetry between the FF components implies that we should set , otherwise the symmetry will be broken by the birefringence. Setting limits the consideration to a particular case of the generic three-wave system, but even this particular case is a nontrivial one, as it has no counterpart in the usually considered degenerate two-wave system, in which the VA was applied before to the description of trapped vortices . The widths of all the components in ansatz (27) are not treated as variational parameters, but are rather taken as per wave functions of states trapped in the 2D harmonic-oscillator potential, cf. Eq. (12).
The objective of the use of the VA in the present context is to predict a point at which a solution with an infinitely small appears, against the background of the fundamental single-color (SH-only) state, given by Eqs. (12), (13) with and arbitrary amplitude . The appearance of this mode signals the onset of the parametric instability of the latter state . The substitution of ansatz (27) into radial Lagrangian (26) yields the following reduced Lagrangian, in which we drop terms that produce no contribution in the subsequent analysis:
This Lagrangian gives rise to the Euler-Lagrange equation, . The parametric-instability threshold, corresponding to the emergence of the solution with an infinitesimal , corresponds to setting in the resulting equation:
Thus, the fundamental single-color state is stable at , and unstable at .
Note that, as it follows from Eq. (30), the parametric instability of the zero-vorticity single-color state is dominated by the vortical FF perturbations with (i.e., give rise to lower than its counterpart produced by the zero-vorticity perturbations, with ) in an interval of negative values of the mismatch, .
It is also relevant to develop a similar analysis for the onset of the parametric instability of the vortical single-color trapped state with even vorticity , for which , cf. Eq. (12), and is replaced by in the FF components of ansatz (27). In this case, the reduced radial Lagrangian (28) is replaced by
Then, the Euler-Lagrange equation, , yields the value of amplitude , the substitution of which into expression (14) gives the instability threshold for the single-color vortex mode:
5 Linearized equations for small perturbations
The stability of the modes under the consideration is the central issue of the present work. It is addressed via computation of eigenvalues for small perturbations. In the general case, perturbed solutions are introduced as
where is the propagation constant of the stationary solution (assuming here that is the same for that both FF components), which is represented by (complex) functions , infinitely small is an amplitude of the perturbation, is the instability growth rate sought for, which is, generally, complex too [the instability takes place if there exists, as usual, at least single with ], the asterisk stands for the complex conjugate, and are components of the perturbation eigenmode, which obey the system of linearized equations:
where is an arbitrary integer (azimuthal index of the perturbation). The substitution of expressions (35) into Eq. (34) leads to the following eigenvalue system which should determine for each integer :
6 The stability of the single-color modes
The objective is to find, for given , a critical (minimum) value of amplitude in solution (12), and, accordingly, the minimum value of the integral power (14), at which Eq. (6) starts to produce eigenvalues with , i.e., the single-color mode becomes unstable against the FF perturbations. Then, we aim to explore the evolution of unstable modes by means of direct simulations. To address these problems, we employed numerical techniques and grids similar to those used in Ref.  for finding the eigenvalues and running direct simulations.
The results for trapped single-color (SH-only) modes are collected in the left panel of Fig. 1, while the right panel illustrates the evolution of an unstable mode with , in terms of the power exchange between the three components (the total power is conserved, as it should be). Here and below, numerical results are displayed for the zero birefringence (, unless it is specified otherwise) and trapping-potential strength .
The left panel of Fig. 1 demonstrates that the VA-predicted instability thresholds (30) and (30) approximate their numerically found counterparts well enough. In particular, the VA exactly predicts that (for and ), the instability threshold vanishes () at .
It is interesting to consider the (in)stability of the single-color trapped vortices with and other odd values of the vorticity, as the respective perturbations in the two FF components cannot be arranged symmetrically, unlike those considered above for even values of . The threshold value of the norm for is shown in the left panel of Fig. 1, and the development of the respective instability is illustrated by Figs. (2) and (3). It can be concluded that the instability does not destroy the vorticity (topological charge) of the SH component, while the SH and FF waves exchange their angular momenta, keeping virtually equal integral powers.
7 Numerical results for three-wave vortices
7.1 Hidden-vorticity (HV) modes with
Families of HV states, defined as per Eqs. (15) and (16), are characterized by dependences between the propagation constant and integral powers of the two FF components and the SH one, see Eq. (9). Typical examples of such dependences, obtained from a numerical solution of Eqs. (17)-(19), are shown in the left panels of Figs. 4 and 5, respectively, for positive () and negative () values of the phase mismatch.
The computation of the stability eigenvalues by means of Eq. (36) demonstrates, in the right panels of Figs. 4 and 5, that all the HV modes are subject to instability against small perturbations with various values of azimuthal index . Direct simulations reveal two different scenarios of the instability development, which are displayed in Figs. 6 and 7. In the former case, the central core is spontaneously expelled from the vortex ring, which tends to transform itself into a fundamental mode. In the latter case, the vortex ring splits into two fragments, which is a typical outcome of the instability development of vortices in models with the cubic nonlinearity -.
7.2 Semi-vortices with
For families of semi-vortex solutions, a numerical solution of Eq. (22) produces dependences between the propagation constant and integral powers of the three components, a typical example of which is displayed in Fig. 8. It is concluded that these families are completely unstable too.
Simulations of the instability development of the semi-vortices reveal two basic outcomes which initially look similar to those shown above for the HV modes, cf. Figs. 6 and 7, i.e., expulsion of the central core from the vortex ring, as shown in Fig. 9, or splitting of the vortex into a set of two fragments, see Fig. 10.
7.3 Full three-wave vortices
The most general three-wave vortex state is built with . Families of these solutions were found for the symmetric system, with and [see Eqs. (4) and (5)] from a numerical solution of Eq. (25). Their stability eigenvalues were then computed using Eq. (36), and the predicted stability or instability was verified by direct simulations of Eqs. (1)-(3).
A crucial difference of the three-wave vortices from the single-color (SH-only), HV and SV states, which were considered above, is that the full vortices have a well-defined stability area. A typical vortex family and its stability are presented in Fig. 11. This figure explicitly displays both the bifurcation, which generates the full vortex from the corresponding single-color state, with (as shown above, the bifurcation simultaneously destabilizes the single-color state), and the point of the destabilization of the three-wave vortices.
The results of the stability analysis are summarized, in the plane of the phase-mismatch parameter, , and total power, , in Fig. 12. The shape of the stability diagram is qualitatively similar to the one which was recently reported, for the degenerate two-wave system, in Ref.  (for the sake of the comparison, note that was defined with the opposite sign in Ref. ). The bottom-right area in the diagram is populated by the single-color (SH-only) vortices with , the bifurcation destabilizing the single-color vortex and replacing it by the three-wave one occurring along the upper boundary of this area.
In the “splitting” area labeled in Fig. 12, the three-wave vortices are subject to an instability which splits them into a set of fragments, the number of which is equal to the dominant (or single) azimuthal index, , of unstable perturbation modes, which is indicated in the right panel of the figure. Further, in the region labeled “ (only)”, the stable static three-wave vortices are replaced by a robust dynamical regime, in the form of periodic splittings of the vortex into two segments and their recombinations, as shown in Fig. 13. In the course of this periodic evolution, the vortical structure of the mode is conserved. A similar dynamical regime, in the form of periodic splittings and recombinations, is known in the 2D GP equation with the cubic self-attractive nonlinearity and HO trapping potential . On the other hand, the instability-induced splitting the vortex into a set of three fragments is irreversible, as shown in Fig. 14.
Finally, the effect of the birefringence of the stability of the three-wave vortices, represented by in Eqs. (1)-(2), was briefly considered too. As shown in Fig. 15, the birefringence terms make the stability area somewhat larger. This result can be explained by the fact that the birefringence renders the system less coherent, while the splitting instability of the vortices is a result of highly coherent interactions.
This work aimed to explore the possibility of the stabilization of various 2D three-wave modes supported by the interactions in the combination with the isotropic HO (harmonic-oscillator) trapping potential. The existence and stability of the modes is determined by powers and vorticities of the three components and the mismatch of the system (). First, using both numerical computations and the VA (variational approximation), stability boundaries were identified for the fundamental (zero-vorticity) and vortical single-color states, in which only the SH (second-harmonic) component is present. On the contrary to the usual assumption that the single-color SH modes are subject to the parametric instability against perturbations in the FF (fundamental-frequency) fields, we have found that they are stable below the respective critical values of the total power. Next, HV (hidden-vorticity) and SV (semi-vortex) states, with vorticities, respectively, or and in the two FF components, were found to be always unstable. The stability region has been identified for the full three-wave vortices. Furthermore, adjacent to it is the region which features the robust dynamical regime of periodic splitting into two fragments and their recombination into the original vortex.
The analysis reported in the present work can be extended in other directions. One possibility is to analyze asymmetric three-wave modes, with unequal propagation constants in the two components of the FF field. It may also be interesting to construct self-trapped three-wave modes supported by a periodic (lattice) potential, instead of the HO, cf. Ref. . Lastly, a challenging possibility is to construct three-dimensional three-wave “light bullets” supported by the HO trapping potential, cf. Ref.  where this was done for the degenerate two-wave system.
A.G. would like to thank the Brazilian funding agencies Fundação de Amparo à Pesquisa do Estado de São Paulo and Conselho Nacional de Desenvolvimento Científico e Tecnológico. B.A.M. acknowledges a visitor’s grant provided by South American Institute for Fundamental Research (Sao Paulo).
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