Topological three-dimensional dissipative optical solitons

Abstract

This article presents a review of recent investigations of topological three-dimensional (3D) dissipative optical solitons in homogeneous laser media with fast nonlinearity of amplification and absorption. The solitons are found numerically, with their formation, by embedding two-dimensional laser solitons or their complexes in 3D space after their rotation around a vortex straight line with their simultaneous twist. After a transient, the ‘hula-hoop’ solitons can form with a number of closed and unclosed infinite vortex lines, i.e. the solitons are tangles in topological notation. They are attractors and are characterized by extreme stability. The solitons presented here can be realized in lasers with fast saturable absorption and are promising for information applications. The tangle solitons of the type described present an example of self-organization that can be found not only in optics but also in various distributed dissipative systems of different types.

This article is part of the theme issue ‘Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 1)’.

1. Introduction

Prigogine's concept of dissipative structures arising in open nonlinear systems, as a result of the dynamic balance between energy importation and dissipation [1], has proved its fundamental importance not only in chemistry but also in a wide range of scientific directions in physics, biology and industry. Since the invention of lasers, with distinct mechanisms of energy input (pump) and output (losses inside lasers and due to laser radiation escape from the cavity) and with the principal role of optical nonlinearity, laser physics and nonlinear optics provide an almost ideal testing area for the realization and investigation of the physics of dissipative systems. Because the typical size of optical schemes essentially exceeds the usual optical wavelength, these systems are spatially distributed, allowing the controllable formation of various complex structures.

Solitons—particle-like, localized structures of fields in distributed nonlinear media and schemes—are one of the most exciting objects in nonlinear science. Although they exist both in conservative (without dissipation) and in dissipative optical systems [2–4], their features are very distinct: conservative optical solitons form families with continuously varying characteristics, including their shape and energy, whereas dissipative optical solitons are ‘calibrated’ and have a discrete spectrum of their main parameters for fixed parameters of the scheme. This difference results in much higher stability of dissipative optical solitons and in the possibility to generate almost arbitrary complex spatio-temporal localized structures.

An intriguing class of topological, knotted solitons was proposed by Faddeev [5] for conservative systems, but this has not yet been studied in sufficient detail in dissipative optical systems. The goal of this paper is to present the first, as far as we know, review of the theory of such topological tangle dissipative optical solitons based mainly on our recent publications [6–8]. Some related papers on the subject can also be found in [6–8].

2. Model and initial equations

We consider the propagation of quasi-monochromatic, nearly linearly polarized radiation with angular divergence close to the diffractive limit through a weakly nonlinear active (with laser amplification) medium (e.g. [3,4]). The electric field is represented in the following form:

2.1

Here, e is the unit vector along the direction of the field polarization, t is the time, is the radius vector, x, y and z are Cartesian coordinates, the z-axis coincides with the main direction of radiation propagation, is the carrier frequency of the field and is the wavenumber of a plane wave with frequency propagating along the z-axis through the corresponding linear medium. The medium consists of a matrix doped with active and passive centres. The matrix is a linear medium with frequency dispersion of the refractive index and weak dichroism–angular dependence of absorption (with its minimum for waves propagating along the z-axis). The centres provide saturable (decreasing with the increase in radiation intensity ) amplification and absorption. The interaction of radiation with the doped centres is resonant, and the centres of the corresponding spectral lines are close to the carrier frequency . We assume that the radiation pulse duration exceeds the relaxation times of the centres (their response to the radiation is fast). Then, Maxwell's equations result, in the framework of the slowly varying envelope approximation, in the following governing equation [3,4,6–8]:

2.2

The above equation is presented here in the dimensionless form. In equation (2.2), is the transverse Laplacian, x and y are transverse coordinates and is the time in the co-propagating system of coordinates moving along the z-axis with group velocity . The coefficient describes the frequency dispersion of the medium gain and loss in the second order of the dispersion theory. The medium matrix dichroism is represented by the coefficient . Both ‘diffusion’ coefficients, and , are positive; otherwise, the propagation of the nearly axial plane monochromatic waves would be unstable even in linear media. We suppose that the medium is only weakly dispersive and dichroic, i.e. the diffusion coefficients are small, . For the equal diffusion coefficients, , the three-dimensional Laplacian in the isotropic space presented in equation (2.2) becomes

2.3

Finally, the nonlinear function of the radiation intensity I corresponds to the intensity-dependent medium amplification and absorption. For a two-level scheme of active (saturable laser gain) and passive (saturable absorber) centres, this function, for the case of exact frequency tuning, is as follows [9]:

2.4

Here, —the small-signal gain coefficient—and —the small-signal absorption coefficient—are determined by parameters of the centres, the term (−1) describes the non-resonant linear absorption in the matrix (after the normalization of the longitudinal coordinate z) and is the ratio of the intensities of saturation for gain and absorption; the intensity I is normalized on the intensity of absorption saturation. Bright localized structures can exist only if the linear absorption overcomes the small-signal gain,

2.5

Under condition (2.5), the trivial solution of equations (2.2) and (2.3), , is stable; otherwise, any bright localized solution of these equations would be unstable because of the field growth at the structure's periphery. Another restriction on the coefficients comes from the condition of bistability: the equation should have a non-trivial solution with positive intensity corresponding to the homogeneous balance of gain and loss [4]. For function given by equation (2.4),

2.6

where and .

Equations (2.2) and (2.3) have translational symmetry, i.e. symmetry with respect to the shifts of coordinates x, y and z, and with respect to the shift of time τ; they also have symmetry of rotation around the axis τ and are invariant to the inversions , and . Correspondingly, the medium is homogeneous and weakly anisotropic, like a uniaxial crystal. However, equations (2.2) and (2.3) have no Galilean symmetry for non-zero diffusion coefficients. Additionally, equation (2.3) has spherical symmetry in the space .

3. Initial and boundary conditions

An evolution variable in equations (2.2) and (2.3) is the longitudinal coordinate z. Correspondingly, the initial condition is the specification of the envelope E at the medium input:

3.1

Specifying various ‘initial conditions’ , one can obtain a large variety of different field structures formed with increasing z; therefore, the choice of these conditions is very important. Let us describe the approach to this choice accepted below for the case of equation (2.3) with spherical symmetry; for corresponding to equation (2.2), the approach is similar [7].

The choice is illustrated in figure 1. We use well-known structures of the field of two-dimensional (2D) laser solitons [3,4] or their complexes as the three-dimensional (3D) initial field distribution in the plane . The 2D field can have an N-fold axis of symmetry and include a number of vortices—points with vanishing field , around which the phase varies by with an integer topological charge of the ith vortex . The total topological charge of the 2D structure is the sum of the individual ones: . We set such a 2D distribution in the cross-section of the 3D initial distribution as with constant shifts of the distribution centre . A natural choice of these constants is one-half of the equilibrium distance between 2D solitons; however, small variations in these parameters result in the same stable final 3D structures. Next, we rotate this 2D structure around the axis τ by polar angle φ by 2π; the trace of the 2D structure centre forms a circle in the 3D space r. Simultaneously, we twist the structure around its centre in the poloidal plane by twist angle with integer s (N is also integer). Then, we introduce, for the 3D field , a phase multiplier with an integer . As a result, we obtain a continuous toroidal initial field 3D distribution with one or many infinite unclosed vortex lines characterized by the total topological charge and one or many closed vortex lines originating from the vortices of the generating 2D structures. Hence, for 3D structures, we get two new, when compared with the generating 2D structures, topological indices: charge M and fractional twist index . Such preparation of the initial conditions [7] differs from that in [10,11] by the introduction of an unclosed vortex line with charge M₀ and by the choice of generating 2D structures as single 2D dissipative solitons or their complex.

Figure 1.

Preparation of the field initial distribution. (a) Generating 2D structure (here, a strongly coupled pair of 2D vortex solitons). (b) Initial 3D distribution results using rotation of the 2D structure around axis τ with its simultaneous twist.

As we are interested here in bright, localized structures with finite energy, the field should vanish at the structures' periphery, at . There, according to equation (2.5), the medium absorption dominates and the field decays exponentially. Correspondingly, the exact form of the boundary condition is not essential: very close results are obtained both for zero, , where B is some bounding surface sufficiently far from the structure centre, and for periodic conditions.

4. Structure characterization and energy flows

The structures can be characterized by the following ‘mechanical’ integral quantities: electromagnetic energy

4.1

vector radius of the structure's energetic centre and the centre's velocity

4.2

torque

4.3

and inertia tensor with matrix elements

4.4

where is the Kronecker symbol. M and are calculated with respect to the structure centre .

Eigenvectors of the inertia tensor, i.e. the three mutually orthogonal principal axes of tensor , form a trihendron characterizing the intensity distribution orientation. After determination of such an orientation, one can introduce a z-dependent vector of angular velocity Ω.

The energy flow, or normalized, averaged over the optical temporal period Poynting vector, is

4.5

where is the phase of the quasi-monochromatic radiation. A point r corresponds to the energy source if the energy flow divergence and to the energy sink if . Evidently, these values are local. The separating surfaces where serve as simple, but important characteristics of the internal structure of dissipative solitons.

Similar to the case of 2D dissipative structures [3,4], the type of the structures' motion as a whole depends on the symmetry of the distributions of radiation energy and energy flows (Poynting vector). For example, it follows from equation (2.2) or (2.3) that if these distributions have in space r stable symmetry with respect to a plane parallel to, for example, the axis τ, then the normal to the plane component of the structure velocity turns to zero. Similarly, if there is an N-fold axis, that is, symmetry to rotation with axis τ through angle with integer N, then the velocity has no components orthogonal to the axis τ. And, the centre of the structures with inversion symmetry should be motionless, .

A much more detailed description of a soliton's internal structure at fixed z is given by the lines of energy flow; the lines are determined by the condition that the tangent to them at any point r has the same direction as . They can be found, for a known field of the Poynting vector , by solution of the autonomous (without explicit dependence of the right parts of the equations on l) system of three ordinary differential equations of the first order:

4.6

Here, l is the line length from some (arbitrary) initial point to a current point r, which is positive if l increases in the direction of energy flow and negative otherwise, . For axially symmetric structures such as the axially symmetric ‘apple solitons’ (see §5b), it is convenient to use cylindrical coordinates with and ; axis τ is made coincident with the symmetry axis. Such a structure field has the following form:

4.7

with integer m, the topological charge of the vortex line being the symmetry axis. In this case, the partial separation of variables is possible in equation (4.6),

4.8

and

4.9

One can solve equations (4.8) independent of equation (4.9). Analysis of the corresponding phase plane includes the determination of singular, or fixed, points, both isolated and uninsulated, and their local characteristics, separatrices of saddle singular points and limit cycles [12]. The uninsulated fixed points, separatrices and limit cycles divide the phase plane of energy flows into a number of cells with quantitatively different types of energy flow lines. In 3D space, involving equation (4.9), the separating lines are transformed into cylindrical separating surfaces, also characterizing the 3D soliton internal structure. Examples are given below, see §5b, where the structure of the energy flow lines near the uninsulated fixed points (unclosed vortex lines) is also considered.

5. Stable tangle solitons

The dynamics of the structures was studied numerically using the finite difference, Crank–Nicolson method. The number of nodes was up to 320 × 320 × 320, depending on the structure type; the steps were decreased until appropriate precision was obtained. The soliton's stability was typically checked by the structure transient after a sudden change in the scheme parameters. One can show that the stabilization of the solitons for new parameters is fairly fast (a typical propagation length of approx. 1000), and the stabilization is very precise, with relative errors of less than 10⁻⁸.

In all the simulations, we fix the values of two parameters: and , and vary the other parameters, such as laser gain and diffusion coefficients and . Additionally, we use various 2D solitons for construction of the 3D initial distributions and various values of topological charge M₀ of the initial unclosed vortex line. Let us remind ourselves also that inversion of the coordinate(s) gives additional new soliton structures.

The 3D stable localized structures found using the initial distributions as described above after a sufficiently long transient are presented in figure 2 as their ‘skeletons’—arrays of unclosed, ending at infinity and finite closed vortex lines. The vortex lines have some positive or negative topological charge m defined as the phase variation during path-tracing around the line in its small vicinity in the plane orthogonal to it divided by . Next, the vortex line is oriented according to the sign of its topological index.

Figure 2.

‘Skeletons’—arrays of unclosed and closed vortex lines—of various laser solitons: (a) a ‘precesson’, (b) an axially symmetric ‘apple’, (c) an ‘apple’ with sixfold symmetry axis, (d) an ‘apple’ with the fifth and sixth angular harmonics in the shape of a closed vortex line, (e) a ‘corkscrew apple’, (f) a soliton with two closed vortex lines, (g) a soliton with an unknotted closed vortex line, (h) a soliton with a trefoil closed vortex line, and (i) a soliton with Solomon's knot (two-linked) closed vortex lines. Wide inner tubes in (f–i) consist of a number of isolated vortex lines (isosurfaces at level ). (a,b,g), 0.035 (c,d), 0.05 (e,h), 0.04 (f) and 0.068 (i); (a), 2.11 (b), 2.127 (c–e), 2.115 (f,g), 2.114 (h) and 2.118 (i).

The number of closed vortex lines varies from 0 (a) to 2 (f,i). In (a)–(e), there is only one unclosed vortex line with a unit topological charge. In (f)–(i), there are two or three, all with a unit topological charge. Note that, in (f)–(i), the unclosed vortex lines are represented by common wide tubes because they are relatively close.

Various types of solitons, presented in figure 2, exist in different domains of parameters. As shown in figure 3, these domains overlap, and this is an indication of the various hysteretic phenomena when one or many control parameters slowly vary.

Figure 3.

Domains of existence and stability of topological solitons in the plane of parameters for precessons (I) and solitons with closed vortex line(s): axially symmetric ring (II), trefoil (III), unknot (IV) and Solomon's knot (V). Double-ring solitons exist in the domain more narrow than V.

Because the skeletons of solitons presented in figure 2b–i include both unclosed vortex lines and closed ones encircling them, they can be called ‘hula-hoop’ structures [7] or, in topological notation, tangles [13]. Now, we move to a more detailed characterization of these topological solitons.

(a). Precessons

The generating 2D structure here is a fundamental, axially symmetric soliton () without vortices [3,4]. The unclosed vortex line coincides, for example, for the case , with axis τ and has a unit topological charge. The distance of the 2D soliton centre from the axis of the 3D rotation can be chosen as one-half of the equilibrium distance between two weakly coupled 2D solitons [4]; however, the choice is not critical, because, during the next evolution, this distance varies considerably. After 2D soliton rotation, we get the axially symmetric vortex toroidal intensity and phase distribution. However, the next evolution demonstrates its instability. Finally, an asymmetric soliton arises with solid-like intensity distribution. The main features of these precessons are illustrated in figure 4.

Figure 4.

(a) Intensity isosurface demonstrating the dimensions of the precesson. (b) Divergence of the Poynting vector. A single hollow toroid corresponds to the domain of energy sources (red domains here and below). (c) Energy flows near the unclosed vortex line 1 with closed limit lines and a spiral on the surface separating the domain of attraction of the vortex line and the domain of flow lines moving away outside the surface [8]. Arrows show the direction of the tangential component of the energy flow near line 1. The parameters are the same as in figure 2a.

Owing to inversion symmetry, the precesson's centre is motionless. However, the solid-like intensity distribution rotates as a whole with constant angular velocity; for , precesson with a weak modulation of the intensity distribution can take place [7].

(b). ‘Apple’ solitons

For the next structure, we use a single 2D vortex soliton with axially symmetric distributions of intensity and the Poynting vector. The unclosed vortex line is also rectilinear and has a unit topological charge. The trace of the 2D vortex during rotation around the axis forms a circle. Depending on the parameters, we get a number of solitons with different shapes of skeletons (figure 2b–e) and the same topology. They all have solid-like intensity distribution (for ).

Figure 5 shows the main features of the axially symmetric ‘apple’ solitons. Owing to symmetry, their centre moves with constant velocity along the symmetry axis. The energy sources are distributed inside two hollow, nearly concentric toroids (figure 5b, red). The axial symmetry allows one to reduce the problem of 3D space of the energy flows to that for the 2D phase plane [8]. The analysis reveals the existence of three degenerate fixed points on the unclosed vortex line (as well as for precessons; figure 4c). Near them, the tangential component of the energy flow has opposite directions. The rotation of the 2D phase plane (figure 5c) around axis gives the 3D space of flows with separating surfaces as indicated in figure 5d.

Figure 5.

(a) Intensity isosurface; (b) divergence of the Poynting vector; (c) singular points, curves and surfaces in the distribution of energy flows for an axially symmetric apple soliton. Singular curves and surfaces in the 3D distribution (d) are obtained from singular points and curves in the 2D distribution (c) by rotation around the symmetry axis . Lines 1 and 2 are of a general position; they illustrate the strong coupling of localized and moving to infinity lines of energy flow. The surfaces in (d) separate the domains with quantitatively different behaviour of lines of energy flow. The parameters are the same as in figure 2b.

(i). Deformed apple solitons

With changes in the scheme parameters, the axially symmetric apple solitons can transform to less symmetric ones (figure 6). Asymmetry is more pronounced for closed vortex lines (figure 2). In figures 2c and 6a,b, this line has sixfold symmetry; for figures 2d and 6c,d, the closed vortex line is asymmetric, with domination of the sixth and fifth angular harmonics in its form. In addition to translational motion (rectilinear for solitons with sixfold symmetry), the intensity distribution for these structures rotates as a whole.

Figure 6.

Deformed apple solitons. Isointensity surfaces at level (a,c,e), divergence of the Poynting vector (b,d,f) and skeletons at propagation distance (g), 57 (h) and 114 (i) for solitons with a sixfold axis of symmetry (a,b), solitons with sixth and fifth angular harmonics of closed line modulation (c,d), and for a ‘corkscrew’ soliton (e–i). The parameters are the same as in figure 2.

A different type of apple soliton is shown in figure 6e–i. The unclosed vortex line moves continuously with a corkscrew-like rotation simultaneously with propagation. Figure 6g–i shows the positions of the unclosed vortex line at three progressive values of propagation distance z during one period.

(c). Double-ring soliton

For the next initial condition for formation of all topological solitons presented below, we use a generating 2D structure as a pair of strongly coupled vortex solitons with an equal unit topological charge (for this pair ) [4] and introduce an initial vortex line with charge .

Without twist (), after a sufficiently long propagation, a double-ring soliton forms with three infinite unclosed and two closed unlinked vortex lines periodically, with the increase of z, passing one through the other; see figures 2f and 7 [7]. All vortex lines have a unit topological charge. The structure moves along the axis τ with small variations in its shape and velocity.

Figure 7.

Intensity isosurface (a), energy flow divergence (b) and a skeleton with an initial unclosed vortex line with topological charge 3 split into three unclosed vortex lines with a unit topological charge (c) for a double-ring soliton. The parameters are the same as in figure 2f.

(d). Soliton with an unknotted closed vortex line

With the same generating 2D structure, we introduce a twist with index . Then we get, after the transient, a soliton with a single two-branched closed vortex line and three unclosed vortex lines (figures 2g and 8). Once again, the intensity distribution slowly moves and rotates with weak oscillations. The closed vortex line itself is an unknot: without unclosed vortex lines, it could be transformed to a circle by smooth deformations, without intersections of vortex lines. However, the presence of infinite vortex lines prevents the realization of such an operation.

Figure 8.

Intensity isosurface (a), energy flow divergence (b) and splitting of the unclosed vortex line (c) for an unknot soliton. The parameters are the same as in figure 2g.

(e). Soliton with a trefoil closed vortex line

Similar to the previous case, but choosing a twist with index , we finally obtain a soliton with a single closed vortex line that is a trefoil and again three curved infinite unclosed vortex lines (figures 2h and 9). The 3D soliton is chiral, i.e. its mirror image (replacement ) cannot be superposed with the initial soliton by its rotation. The intensity distribution is nearly solid-like and symmetric, as in the previous case. The motion of the trefoil soliton is also similar to the previous case.

Figure 9.

Intensity isosurface (a), energy flow divergence (b) and splitting of the unclosed vortex line (c) for a trefoil soliton. The parameters are the same as in figure 2h.

(f). Soliton with a Solomon's knot closed vortex line

Similar to the previous procedure, but with a twist with index and with an initial vortex line with charge , we found a soliton with two closed vortex lines that are linked and only two infinite unclosed vortex lines (figures 2i and 10). Closed vortex lines present a link known as a Solomon's knot.

Figure 10.

Intensity isosurface (a), energy flow divergence (b) and splitting of the unclosed vortex line (c) for a Solomon's knot soliton. The parameters are the same as in figure 2i.

6. Conclusion

We have presented here a new wide class of topological 3D dissipative solitons—‘hula-hoop solitons’—in homogeneous, one-component, active (with gain) nonlinear media. Because they include both closed and unclosed vortex lines, they are tangles [13]. Such solitons can be realized experimentally in laser media or in lasers of sufficiently large size (larger than approx. 10 µm) and fast nonlinearity. They are robust and extremely stable because they are attractors. This stability in combination with conservation of the topological characteristics even for large perturbations make this class of solitons promising for information applications.

Although the scheme used is purely optical, it is natural to expect that the approach developed can be applied to much a wider range of spatially distributed dissipative systems—with essential input and output of energy and/or matter, including various physical, chemical, biological and cosmological objects. For example, it is likely that similar phenomena of self-organization in reaction–diffusion systems will be found [14,15].

Data accessibility

This article has no additional data.

Competing interests

We declare we have no competing interests.

Funding

The research was performed in accordance with the plan of grant no. 18-12-00075 of the Russian Science Foundation.