OSA's Digital Library

Optics Express

Optics Express

  • Editor: Martijn de Sterke
  • Vol. 16, Iss. 20 — Sep. 29, 2008
  • pp: 15388–15401
« Show journal navigation

Transformations of continuously self-focusing and continuously self-defocusing dissipative solitons

J. M. Soto-Crespo, N. Akhmediev, N. Devine, and C. Mejía-Cortés  »View Author Affiliations


Optics Express, Vol. 16, Issue 20, pp. 15388-15401 (2008)
http://dx.doi.org/10.1364/OE.16.015388


View Full Text Article

Acrobat PDF (934 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

Dissipative media admit the existence of two types of stationary self-organized beams: continuously self-focused and continuously self-defocused. Each beam is stable inside of a certain region of its existence. Beyond these two regions, beams loose their stability, and new dynamical behaviors appear. We present several types of instabilities related to each beam configuration and give examples of beam dynamics in the areas adjacent to the two regions. We observed that, in one case beams loose the radial symmetry while in the other one the radial symmetry is conserved during complicated beam transformations.

© 2008 Optical Society of America

1. Introduction

The notion of dissipative solitons [1

1. (Eds.)N. Akhmediev and A. Ankiewicz, Dissipative solitons Lecture Notes in Physics, V. 661 (Springer, Heidelberg, 2005). [CrossRef]

] is a useful concept that allows us to describe, in general terms, a variety of phenomena in physics, chemistry, biology and medicine [2

2. Dissipative Solitons: From optics to biology and medicine Lecture Notes in Physics, V 751, (Eds.) N. Akhmediev and A. Ankiewicz, (Springer, Berlin-Heidelberg, 2008).

, 3

3. M. Tlidi, M. Taki, and T. Kolokolnikov, “Dissipative Localized Structures in Extended Systems,” Chaos 17, 037101, 2007. [CrossRef] [PubMed]

, 4

4. M. Tlidi, A. Vladimirov, and P. Mandel, “Curvature instability in passive diffractive resonators,” Phys. Rev. Lett. 98, 233901, (2002). [CrossRef]

]. Some specific features of these formations are common for all of them, independent of the problem that we are solving and the model we are using. Many problems in optics involving gain and loss can be formulated in terms of a single “master equation” that is, in one or another form, of the complex cubic-quintic Ginzburg-Landau equation (CGLE) type [5

5. I. S. Aranson and L. Kramer, “The world of the complex Ginzburg-Landau equation,” Rev. Mod. Phys. 74, 99 (2002). [CrossRef]

, 6

6. O. Descalzi, G. Düring, and E. Tirapegui, “On the stable hole solutions in the complex Ginzburg-Landau equation,” Physica A 356, 66–71 (2005). [CrossRef]

]. This class of problems includes among others, passively mode-locked laser systems [7

7. H. A. Haus, “Theory of mode locking with a slow saturable absorber,” IEEE Journ. of Quantum Electron. , QE-11 (9), 736 (1975). [CrossRef]

, 8

8. W. H. Renninger, A. Chong, and F. W. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A 77, 023814 (2008). [CrossRef]

, 9

9. A. Komarov, H. Leblond, and F. Sanchez, “Quintic complex Ginzburg-Landau model for ring fiber lasers,” Phys. Rev. E 72, 025604(R) (2005). [CrossRef]

], optical parametric amplifiers [10

10. M. Taki, N. Ouarzazi, H. Ward, and P. Glorieux, “Nonlinear front propagation in optical parametric oscillators,” J. Opt. Soc. Am. B 17, 997 (2000). [CrossRef]

], wide aperture lasers [11

11. N. N. Rosanov, Solitons in laser systems with saturable absorption, Dissipative solitons Lecture Notes in Physics, V. 661, (Eds.) N. Akhmediev and A. Ankiewicz, (Springer, Heidelberg, 2005).

], spatial dissipative solitons [12

12. E. A. Ultanir, G. I. Stegeman, D. Michaelis, C. H. Lange, and F. Lederer, “Stable dissipative solitons in semiconductor optical amplifiers,” Phys. Rev. Lett. 90, 253903 (2003). [CrossRef] [PubMed]

] and three-wave mixing in optical fibers [13

13. C. Montes, A. Mikhailov, A. Picozzi, and F. Ginovart, “Dissipative three-wave structures in stimulated backscattering. I. A subluminous solitary attractor,” Phys. Rev. E 55, 1086 (1997). [CrossRef]

]. Many of these problems can be (1+1) dimensional and they deal with the evolution in time of spatially self localized one-dimensional fields [14

14. R. J. Deissler and H. R. Brand, “Periodic, quasiperiodic, and chaotic localized solutions of the quintic complex Ginzburg-Landau equation,” Phys. Rev. Lett. 72, 478–481 (1994). [CrossRef] [PubMed]

]. More complicated are the (2+1) dimensional problems, when the optical field evolving in time is two-dimensional. Examples of two-dimensional dissipative solitons in optics are cavity solitons (CS) [15

15. X. Hachair, F. Pedaci, E. Caboche, S. Barland, M. Giudici, J. R. Tredicce, F. Prati, G. Tissoni, R. Kheradmand, L. A. Lugiato, I. Protsenko, and M. Brambilla, “Cavity solitons in a driven VCSEL above threshold,” J. of Sel. Topics on Quant. Electr. 12, 339–351 (2006). [CrossRef]

] and solitons in wide aperture lasers [11

11. N. N. Rosanov, Solitons in laser systems with saturable absorption, Dissipative solitons Lecture Notes in Physics, V. 661, (Eds.) N. Akhmediev and A. Ankiewicz, (Springer, Heidelberg, 2005).

]. The variety of transverse profiles of two dimensional solitons can be enormous. These profiles can evolve in time as well as can be stationary and stable. An interesting observation is that the majority of these stable stationary profiles are not radially symmetric.

As in conservative homogeneous 2-D systems, radially symmetric node-less beams should be considered as the ground-state modes of its corresponding 2-D nonlinear dynamical system. Ground state modes are stable in certain regions of the parameter space. At the edges of the stability regions, we can expect the appearance of either pulsating radially symmetric solutions or stable radially asymmetric solutions. Transformations happen at the points of bifurcation which in a multi-parameter space can be generalized to surfaces of bifurcations. Other types of solutions can also appear as a result of such bifurcations. Thus, it is natural to find, as a first step, the set of radially symmetric solutions and then extend it to more complicated cases. However, it is occurred that dissipative systems in contrast to conservative ones admit more than one class of radially symmetric solutions.

In particular, we found that in (1+1)-D configuration, dissipative solitons beyond their region of stability can be transformed into pulsating solutions and these into exploding solitons [17

17. J. M. Soto-Crespo, N. Akhmediev, and y A. Ankiewicz. “Pulsating, creeping and erupting solitons in dissipative systems,” hys. Rev. Lett. 85, 2937 (2000). [CrossRef]

]. The latter stay as stationary solitons for certain distance of propagation and intermittently explode into many pieces [18

18. Akhmediev. J. M. Soto-Crespo and G. Town “Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locked lasers: Complex Ginzburg - Landau equation,” Phys. Rev. E. 63, 056602 (2001). [CrossRef]

]. In the (2+1)-D case, this is the only instability we were able to find around continuously self-defocusing beams (dissipative antisolitons). On the other hand, continuously self-focusing beams have a rich structure of bifurcations around the boundaries of the region of their stable existence. Radially symmetric beams can be transformed into beams of elliptic shape, beams of completely asymmetric shape, pulsating beams etc., when we change the parameters of the system. The study of these transformations is one of the interesting aspects of the theory of dissipative solitons. Thus, in this work we give (2+1)-D solutions with distinctive features which appear close to the boundaries of the regions of existence of continuously self-focusing and continuously self-defocusing solitons.

2. Statement of the problem

Our present study is based on an extended complex Ginzburg-Landau equation [16

16. A. Ankiewicz, N. Devine, N. Akhmediev, and J. M. Soto-Crespo “Continuously self-focusing and continuously self-defocusing 2-D beams in dissipative media,” Phys. Rev. A 77, 033840 (2008). [CrossRef]

], that includes cubic and quintic nonlinear terms. In normalized form, this propagation equation reads:

iψz+D22ψ+ψ2ψ+vψ4ψ=iδψ+iεψ2ψ+iβ2ψ+iμψ4ψ.
(1)

where ψ=ψ(x,y, z) is the normalized envelope of the field,

2=2x2+2y2

is the transverse Laplacian, z is the propagation distance and (x,y) are the transversal coordinates, D is the diffraction coefficient that without loss of generality can be set to 1, ν is the saturation coefficient of the Kerr nonlinearity, δ represents linear losses, ε is the nonlinear gain coefficient, β stands for angular spectral filtering of the cavity, and µ characterizes the saturation of the nonlinear gain.

Stationary solutions of Eq. (1) can be found in wide regions of the space of the equation parameters [16

16. A. Ankiewicz, N. Devine, N. Akhmediev, and J. M. Soto-Crespo “Continuously self-focusing and continuously self-defocusing 2-D beams in dissipative media,” Phys. Rev. A 77, 033840 (2008). [CrossRef]

]. They are usually radially symmetric. In this case, the regions of their existence can be found semi-analytically using approximate methods such as in Ref. [16

16. A. Ankiewicz, N. Devine, N. Akhmediev, and J. M. Soto-Crespo “Continuously self-focusing and continuously self-defocusing 2-D beams in dissipative media,” Phys. Rev. A 77, 033840 (2008). [CrossRef]

], with good agreement with the numerically obtained ones by solving directly Eq. (1). Around the boundaries of existence of these radially symmetric stationary solutions, more complicated objects can be observed. Then numerics appears to be the main tool to study them. They can be detected when we start from the region of stationary solutions and continuously change one or two of the equation parameters fixing all the others. At some point, the radially symmetric solution ceases to be stable, and a new solution appears.

The main parameter of the solution that we monitor in simulations is the beam power, Q:

Q(z)=ψ(x,y,z)2dxdy,
(2)

The value of Q for a localized solution is finite and changes smoothly while the solution stays within the region of existence of a certain type of solitons. The value of Q changes abruptly when there is a bifurcation and the solution jumps from a branch of solitons that become unstable to another branch of stable solitons. Thus, monitoring Q allows us to find bifurcations in an easy way. Observing a finite Q also reveals the stability of a solution. As soon as the solution becomes unstable, it diverges and the value of Q either converges to another fixed value, vanishes or goes to infinity. In the case that the resulting solution is pulsating, instead of a fixed value of Q, we obtain a band of Q values that correspond to the changing power of the soliton. As we are interested in bifurcations from radially symmetric solitons, we start simulations from those found in Ref.[16

16. A. Ankiewicz, N. Devine, N. Akhmediev, and J. M. Soto-Crespo “Continuously self-focusing and continuously self-defocusing 2-D beams in dissipative media,” Phys. Rev. A 77, 033840 (2008). [CrossRef]

].

Fig. 1. Regions (in blue) in the parameter space with radially symmetric stable stationary beams. The plot on the left (a) shows the region for continuously self-focusing beams (region I) and on the right (b) the one for continuously self-defocusing beams (region II). The location of these two regions in the five-dimensional parameter space is such that they cannot be represented in the same plane. Soliton solutions beyond these regions are either non-stationary or loose the radial symmetry. Below, we give examples located at the points indicated here by green, yellow and red lines. The change in color corresponds to a bifurcation. The arrows indicate the direction in which the parameters have been changed while obtaining a particular bifurcation diagram.

As the blue region I in Fig. 1(a) represents the “ground state” solutions, there are no solitons below it because the power pumped into the system is smaller for lower values of ε. Solutions with higher power Q exist above the blue region I. The specific cases presented below are obtained on the two vertical lines in Fig.1(a) and on the upper left horizontal line in Fig.1(b). Namely, Fig.2 corresponds to the right hand side vertical line with µ=-0.03 at (a). Figs. 3, 4 and 5 are obtained for µ=-0.08, i.e. on the left hand side vertical line in (a). The horizontal line in (b) for ε=5 corresponds to the case shown in Fig. 12. Each color interval of the lines (green, yellow or red) corresponds to a particular type of solution. The arrows show the direction in which the parameters ε or β were changed when obtaining a bifurcation diagram. When there is no arrow, the bifurcation diagram does not depend on the direction of changing the parameter. The red solid circle in (a) represents the location of the case shown in Fig. 8.

3. Bifurcation diagrams

Fig. 2. Bifurcation of an elliptic soliton from the self-focusing radially symmetric one.

One of the simplest bifurcations that we have found is the transition from a radially symmetric solution to a soliton of elliptic shape that is also stable and stationary. This transformation happens close to the blue region in Fig. 1(a) on the short green vertical line with µ=-0.03. The bifurcation diagram for such transformation is shown in Fig. 2. The beam with radial symmetry loses its stability after the bifurcation (blue dashed curve in the diagram). This can be considered as an example of symmetry breaking bifurcation in the case of 2-D beams. Due to the symmetry of the equation 1 the long axis of the elliptic soliton can have any direction in the (x,y)-plane, what causes the solution to rotate. This elliptic soliton is the only stationary, although rotating, solution different from the radially symmetric one that we were able to find for µ=-0.03. It exists in a stripe (marked in red) above the upper boundary of the blue region (region I). Depending on the initial conditions, the beam may oscillate around the configuration with radial symmetry before finally converging to the stationary beam of elliptic shape. An example of such oscillations for different set of the equation parameters is given in section 5. Although in that case the final stable solution is different.

Fig. 3. (a) Q versus ε diagram as obtained when increasing (red dotted line) or decreasing ε (blue dashed line). (b) magnification of a portion of the upper blue curve around the hysteresis cycle between the two black arrows in (a). The red dots in this case correspond to stationary beams of elliptic shape while the blue dashed line corresponds to stationary beams without any radial symmetry. Typical examples of these two types of solutions are plotted in Figs.4a and 4b respectively for the ε values indicated here with the red and blue arrows. Black vertical lines with continuous values of Q in (b) correspond to pulsating localized solutions.

Figure 3(a) shows, in red dotted line, the beam power of the solutions obtained when we increase ε, starting from a radially symmetric solution at ε=0.2. The power Q increases continuously until an abrupt transition occurs at ε≈0.52. The latter is shown by the black arrow pointing up. It indicates the transformation of the single soliton solution into a stable rotating double-beam complex, which exists in the range of ε from 0.47 to 0.6. Thus, from here, we can either increase or decrease ε and stay at the same branch of solitons. Reaching further ε=0.6, we observe a second bifurcation where Q takes a continuous interval of values. It corresponds to pulsating and simultaneously rotating double beams with an oscillating value of Q.

Alternatively, when we decrease the value of ε, Q follows the blue line. Two solutions can be observed in a certain interval of ε values, manifesting a hysteresis phenomenon. Moreover, we can observe additional bifurcations in this region which can be clearly seen if we magnify properly this region of ε (see Fig. 3(b)). When we move from right to left, three main transitions can be clearly seen: i) transition from a double-beam complex (black solid line) to an asymmetric comma-shaped solution (blue dashed line), both being stationary, ii) transition to pulsating solutions (vertical lines indicating that Q takes a continuous set of values) and iii) transition to elliptically shaped beams along the red dotted line. The red and blue vertical arrows indicate the values of ε for which the solution is plotted in Figs. 4(a) and 4(b) respectively. These 3D-plots illustrate the type of solutions that we obtain in the ranges between the bifurcations. Fig. 4(a) shows the solution with an elliptic profile while Fig. 4(b) shows the solution that looks as composed of two unequal beams closely attached to each other. The profile would look more like a comma-shaped on a contour plot.

Fig. 4. Solitons (a) of an elliptic shape and (b) without any radial symmetry (comma shaped) for the values of ε indicated in Fig.3(b) by the red and blue arrows.
Fig. 5. A portion of the diagram in Fig.3b with a magnified scale. In this case only the maxima (QM) and minima (Qm) of the beam power are shown. The two branches for stationary solutions corresponding to solutions of elliptic shape (left) or highly asymmetric solutions (comma-shaped) with constant Q are shown in green. The red points represent the maxima of the curves Q(z) and the blue points the minima. The data for this plot are obtained when decreasing ε. The period-1 solutions bifurcate from the asymmetric solutions at ε=0.45. Period-doubling bifurcations appear at ε=0.446. A much more complicated type of pulsations appear at ε=0.445. These look like a beating of the two types of stationary solutions (elliptic and highly asymmetric).

Figure 3 shows that soliton solutions are highly sensitive to the change of ε. Indeed, if we further magnify the scale of ε in Fig. 3(b) we will be able to see more bifurcations. These are shown in Fig. 5. For the sake of clarity, we just show here the maximal and minimal values of Q(z) denoted as QM and Qm respectively. The green line at the left hand side of this plot corresponds to rotating solitons with fixed elliptic profile. Their fixed, although rotating, shape results in their fixed values of Q. The green line at the right hand side of the plot corresponds to highly asymmetric solitons with the fixed shape of a comma that are also rotating. Typical examples of the two shapes are shown in Fig. 4. Several bifurcations appear in between these two regions. The first one, when reducing ε appears as a transition from stationary solution to pulsating soliton with a single period of pulsations. The resulting Q oscillates between the red and blue curves that correspond to the maximal and minimal values of Q of the pulsations. The oscillations of Q are very close to being harmonic for pulsating solutions with a single period. The pulsations turn into period doubled ones at the second bifurcation in Fig. 5. As a result, the two curves split into four at this point. Further reduction in ε results in the soliton evolution that looks very much like chaotic before finally turning into an elliptically shaped stationary solitons. Generally speaking, a myriad of other bifurcations may occur at this short interval.

4. Period-1 pulsating solitons

Pulsating solutions appear in dissipative systems as naturally as stationary ones as they represent limit cycles of the infinite-dimensional dynamical system [18

18. Akhmediev. J. M. Soto-Crespo and G. Town “Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locked lasers: Complex Ginzburg - Landau equation,” Phys. Rev. E. 63, 056602 (2001). [CrossRef]

]. For 1-D systems they are usually located near the stationary solutions in the parameter space. The transformation of stationary solutions into pulsating ones occurs in the form of an Andronof-Hopf bifurcation. Due to the relative simplicity of the soliton profile in the 1-D case, this bifurcation can be studied using a trial function approximation in combination with the method of moments [22

22. E. Tsoy and N. Akhmediev, “Bifurcations from stationary to pulsating solitons in the cubic quintic complex Ginzburg Landau equation,” Phys. Lett. A 343, 417–422 (2005). [CrossRef]

]. In the 2-D case, the pulsating soliton profile can be much more complicated. In fact, in most of the cases it lacks the radial symmetry, thus making the approximation with a trial functions difficult. Radially symmetric pulsating solutions exist only in the vicinity of the region of continuously self-defocusing beams (region II). Near the region I, the shape of pulsating solitons look similar to the examples shown in Fig. 6. The soliton profile changes continuously upon propagation. The profiles given in Fig. 6 are taken when the value of Q takes its maximal or minimal value inside each period of pulsations. As we can see, in each case, the beam lacks the radial symmetry.

Fig. 6. Pulsating 2-D soliton profiles when the oscillating power Q takes its (a) maximal and minimal values (dashed and dotted vertical lines in Fig. 7 respectively).

Fig. 7. (a) Periodic evolution of Q versus z for a pulsating soliton with a single period. The red and blue vertical lines show a maximum and a minimum of Q. i The corresponding profiles are shown in Fig. 6(a) and Fig. 6(b) respectively. The actual evolution during a period (marked in green in (a)) is shown in (Media 1) (b).

5. Elliptic beam oscillations

ψ(x,y,0)=5exp[(x1.2)2(y1.1)2]

As the parameters are chosen at the margin of stability of stationary solutions, the beam cannot converge to the radially symmetric solution of Eq. (1). Instead, the small initial asymmetry increases and after an initial transition, the beam converges to a pulsating state without radial symmetry. The beam still has radially symmetric profile when Q takes its maximum value but elongates alternatively either in x or in y directions when Q takes its minimal value. Three consecutive color coded contour plots for the intensity profiles at the extremal points of Q(z) are shown in the lower left inset panels of the figure. The orientation of the elongated profile changes from one consecutive minimum to another one, as shown in the inset. Oscillations of Q are harmonic with high accuracy as shown in the upper left panel of the figure which is a magnified version of the Q(z)-plot between z=100 and z=110. These oscillations are persistent and last from z≈60 till z≈260. However, the pulsating solution at the chosen set of parameters is not completely stable and it is transformed into another stable solution at the end of this interval. Namely, it is transformed into another elongated structure which can be considered as a two-soliton complex. The color contour plot for the intensity profile is shown in the upper right panel of the figure. This beam is also rotating around the central point of its symmetry. Thus, we tend to think that pulsations that keep the radial symmetry at any z do not exist around the region I. Additional studies are needed to give a more definite answer to the question posed in the beginning of this section.

Fig. 8. Dissipative soliton oscillations in two transverse dimensions. The solution converges to these oscillations at around z≈120. The power oscillations are shown in greater resolution in the upper left inset; they are almost harmonic. Color plots in the lower left insets show the intensity profiles when Q reaches two consecutive minima (left and right frames) and any maximum (central frame). The oscillations are weakly unstable and are transformed into a stable configuration at z≈260, consisting of a twin soliton profile shown at the upper right inset that rotates around its center of symmetry

6. More complicated pulsations

Fig. 9. (a) Periodic evolution of Q versus z for three values of epsilon. As epsilon decreases the maxima of Q increases as well as the separation between “bursts”. The red curve is horizontal for large interval of values of z during which the solution is of elliptic shape. (b) (Media 2) showing the evolution of the beam profile.

Figure 10(a) shows the trajectory of the main maximum of the optical field in one full period of evolution. The curve is plotted for the value of ε=0.4446 that corresponds to the blue curve in Fig.9a. Fig.10b reproduces more clearly one period of evolution of Q for ε=0.4446. The “quiet” stage of evolution with almost constant Q is shown in Fig. 10(a) in blue line while the “turbulent” stage of evolution, when Q changes substantially, is shown in green line. The latter corresponds to the interval in Fig. 10(b) enclosed inside the green rectangle. The beam moves intermittently from one position to another one inside this green interval. The beam rotates but stays almost at the same position outside of the green box where Q hardly changes. The blue trajectory in Fig. 10(a) is a circle rather than a point because the maximum of the field is shifted relative to the center of mass of the beam. To complete the description we should add that the evolution of Q is strictly periodic rather than intermittent.

Fig. 10. Periodic evolution of the solution for ε=0.4446 (blue curve in Fig.9). The green trajectory in (a) corresponds to the green zone in (b).

Similar behavior can be observed in a small band of ε values from 0.4446 to 0.4451. Six consecutive periods of evolution of the beam for ε=0.4449 are shown in Fig. 11. Each of the six periods is depicted in a different color. The evolution of the power Q is identical in each of the periods. The trajectory of the maximum of the optical field in each period is similar to the one shown above. Each time, it converges to a circle whose center shifts to another position from one period to the next. The shifts are regular and all circles are located around a fixed point in space. So, in average, the beam stays at the same place in space. The fast part of the trajectory is sensitive to numerical errors. Thus, the boundaries of the used numerical grids are taken to be far away from the beams and calculations have been repeated with several grid sizes and step lengths to be sure that no numerical artifact are presented here.

Fig. 11. a) Trajectory of the peak intensity of the solutions in the (X,Y) plane and b) evolution of Q versus z. Six periods are plotted, each with a different color. The position of the solution behaves somehow chaotic.

7. Beam evolution around the region of continuously self-defocusing beams

A typical example of bifurcation diagram when moving along the tri-color line out of the region of stationary beams is shown in Fig. 12. The power Q is finite and stays constant in z at values of β higher than 0.55. At β≈0.55, we observe an Andronov-Hopf bifurcation where the beam starts to pulsate. These pulsations are revealed with the splitting of the Q value into a maximum, QM, and a minimum Qm values in the bifurcation diagram. This occurs at β≈0.55. An example of oscillating Q(z) is shown in the inset of Fig. 12 by the magenta dashed curve. It is taken at β=0.42. This value of β is indicated by the vertical dashed line in the bifurcation diagram itself. As mentioned, the beam stays radially symmetric in these pulsations. Any perturbation with radial asymmetry vanishes on propagation.

Fig. 12. (a) Bifurcation diagram at the boundary of the region II. Parameters of the system are given inside the plot. An Andronov-Hopf bifurcation of a stationary self-defocusing beam into a pulsating one occurs at β=0.55. The inset shows the power evolution with z for two cases: β=0.42 (magenta) and β=0.4 (amber). In the interval 0.41 <β < 055 the beam is pulsating. The Q(z) curve is harmonic. The evolution is chaotic at the values of β<0.41. The Q(z)-curve (amber) reveals the beam explosions. (Media 3) in (b) shows one period of evolution of the pulsating beam at β=0.42.

8. Conclusion

In conclusion, we studied a variety of two-dimensional beams and their evolution in dissipative media beyond the two main regions of existence of stable stationary beams with radial symmetry (regions of ground state solitons). We have found a multiplicity of transverse shapes of solitons and a myriad of bifurcations between them. We have also found that the beam shapes and their evolution depend strongly on which of the two regions in the parameter space we are close to.

Near the region of stable stationary continuously self-focusing solitons (region I of ground state solitons) we have found several types of bifurcations. (1) Bifurcation into a stable stationary beam of elliptic shape. (2) Bifurcation into pulsating beams with harmonic pulsations or more complicated pulsations with highly involved shape evolution. Beams can be pulsating and rotating simultaneously. Some of the evolution scenarios are caused by the attempts of soliton of splitting into two separate beams. Generally speaking, any bifurcation around this region destroys the radial symmetry of the solution.

On the other hand, near the region of stable stationary continuously self-defocusing beams (region II of ground state solitons), the soliton keeps its radial symmetry after the bifurcation. Beams in this case are pulsating but stay radially symmetric. Pulsating beams can further be transformed into exploding ones, that loose the radial symmetry at the moment of the explosion but recover the original profile between the explosions.

Acknowledgments

J. M. S. C. and C. M. C. acknowledge support from the M.E.y C. under contract FIS2006-03376. The work of NA and ND is supported by the Australian Research Council (Discovery Project scheme DP0663216).

References and links

1.

(Eds.)N. Akhmediev and A. Ankiewicz, Dissipative solitons Lecture Notes in Physics, V. 661 (Springer, Heidelberg, 2005). [CrossRef]

2.

Dissipative Solitons: From optics to biology and medicine Lecture Notes in Physics, V 751, (Eds.) N. Akhmediev and A. Ankiewicz, (Springer, Berlin-Heidelberg, 2008).

3.

M. Tlidi, M. Taki, and T. Kolokolnikov, “Dissipative Localized Structures in Extended Systems,” Chaos 17, 037101, 2007. [CrossRef] [PubMed]

4.

M. Tlidi, A. Vladimirov, and P. Mandel, “Curvature instability in passive diffractive resonators,” Phys. Rev. Lett. 98, 233901, (2002). [CrossRef]

5.

I. S. Aranson and L. Kramer, “The world of the complex Ginzburg-Landau equation,” Rev. Mod. Phys. 74, 99 (2002). [CrossRef]

6.

O. Descalzi, G. Düring, and E. Tirapegui, “On the stable hole solutions in the complex Ginzburg-Landau equation,” Physica A 356, 66–71 (2005). [CrossRef]

7.

H. A. Haus, “Theory of mode locking with a slow saturable absorber,” IEEE Journ. of Quantum Electron. , QE-11 (9), 736 (1975). [CrossRef]

8.

W. H. Renninger, A. Chong, and F. W. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A 77, 023814 (2008). [CrossRef]

9.

A. Komarov, H. Leblond, and F. Sanchez, “Quintic complex Ginzburg-Landau model for ring fiber lasers,” Phys. Rev. E 72, 025604(R) (2005). [CrossRef]

10.

M. Taki, N. Ouarzazi, H. Ward, and P. Glorieux, “Nonlinear front propagation in optical parametric oscillators,” J. Opt. Soc. Am. B 17, 997 (2000). [CrossRef]

11.

N. N. Rosanov, Solitons in laser systems with saturable absorption, Dissipative solitons Lecture Notes in Physics, V. 661, (Eds.) N. Akhmediev and A. Ankiewicz, (Springer, Heidelberg, 2005).

12.

E. A. Ultanir, G. I. Stegeman, D. Michaelis, C. H. Lange, and F. Lederer, “Stable dissipative solitons in semiconductor optical amplifiers,” Phys. Rev. Lett. 90, 253903 (2003). [CrossRef] [PubMed]

13.

C. Montes, A. Mikhailov, A. Picozzi, and F. Ginovart, “Dissipative three-wave structures in stimulated backscattering. I. A subluminous solitary attractor,” Phys. Rev. E 55, 1086 (1997). [CrossRef]

14.

R. J. Deissler and H. R. Brand, “Periodic, quasiperiodic, and chaotic localized solutions of the quintic complex Ginzburg-Landau equation,” Phys. Rev. Lett. 72, 478–481 (1994). [CrossRef] [PubMed]

15.

X. Hachair, F. Pedaci, E. Caboche, S. Barland, M. Giudici, J. R. Tredicce, F. Prati, G. Tissoni, R. Kheradmand, L. A. Lugiato, I. Protsenko, and M. Brambilla, “Cavity solitons in a driven VCSEL above threshold,” J. of Sel. Topics on Quant. Electr. 12, 339–351 (2006). [CrossRef]

16.

A. Ankiewicz, N. Devine, N. Akhmediev, and J. M. Soto-Crespo “Continuously self-focusing and continuously self-defocusing 2-D beams in dissipative media,” Phys. Rev. A 77, 033840 (2008). [CrossRef]

17.

J. M. Soto-Crespo, N. Akhmediev, and y A. Ankiewicz. “Pulsating, creeping and erupting solitons in dissipative systems,” hys. Rev. Lett. 85, 2937 (2000). [CrossRef]

18.

Akhmediev. J. M. Soto-Crespo and G. Town “Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locked lasers: Complex Ginzburg - Landau equation,” Phys. Rev. E. 63, 056602 (2001). [CrossRef]

19.

J. M. Soto-Crespo, Ph. Grelu, and N. Akhmediev “Optical bullets and “rockets” in nonlinear dissipative systems and their transformations and interactions,” Opt. Express 14, 4013 (2006). [CrossRef] [PubMed]

20.

J. M. Soto-Crespo, N. Akhmediev, and Ph. Grelu “Optical bullets and double bullet complexes in dissipative systems,” Phys. Rev. E 74, 046612 (2006). [CrossRef]

21.

N. Akhmediev, J.M. Soto-Crespo, and Ph. Grelu, “Vibrating and shaking soliton pairs in dissipative systems.” Phys. Lett. A 364, 413 (2007). [CrossRef]

22.

E. Tsoy and N. Akhmediev, “Bifurcations from stationary to pulsating solitons in the cubic quintic complex Ginzburg Landau equation,” Phys. Lett. A 343, 417–422 (2005). [CrossRef]

OCIS Codes
(190.3100) Nonlinear optics : Instabilities and chaos
(190.6135) Nonlinear optics : Spatial solitons

ToC Category:
Nonlinear Optics

History
Original Manuscript: July 18, 2008
Manuscript Accepted: August 27, 2008
Published: September 15, 2008

Citation
J. M. Soto-Crespo, N. Akhmediev, N. Devine, and C. Mejía-Cortés, "Transformations of continuously self-focusing and continuously self-defocusing dissipative solitons," Opt. Express 16, 15388-15401 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-20-15388


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. (Eds.) N. Akhmediev and A. Ankiewicz, Dissipative solitons Lecture Notes in Physics, V. 661 (Springer, Heidelberg, 2005). [CrossRef]
  2. Dissipative Solitons: From optics to biology and medicine Lecture Notes in Physics, V 751, (Eds.) N. Akhmediev and A. Ankiewicz, (Springer, Berlin-Heidelberg, 2008).
  3. M. Tlidi, M. Taki, and T. Kolokolnikov, "Dissipative Localized Structures in Extended Systems," Chaos 17, 037101, 2007. [CrossRef] [PubMed]
  4. M. Tlidi, A. Vladimirov and P. Mandel, "Curvature instability in passive diffractive resonators," Phys. Rev. Lett. 98, 233901, (2002). [CrossRef]
  5. I. S. Aranson and L. Kramer, "The world of the complex Ginzburg-Landau equation," Rev. Mod. Phys. 74, 99 (2002). [CrossRef]
  6. O. Descalzi, G. During and E. Tirapegui, "On the stable hole solutions in the complex Ginzburg-Landau equation," Physica A 356, 66-71 (2005). [CrossRef]
  7. H. A. Haus, "Theory of mode locking with a slow saturable absorber," IEEE Journ. of Quantum Electron.,  QE-11 (9), 736 (1975). [CrossRef]
  8. W. H. Renninger, A. Chong, and F. W. Wise, "Dissipative solitons in normal-dispersion fiber lasers," Phys. Rev. A 77, 023814 (2008). [CrossRef]
  9. A. Komarov, H. Leblond, and F. Sanchez, "Quintic complex Ginzburg-Landau model for ring fiber lasers," Phys. Rev. E 72, 025604(R) (2005). [CrossRef]
  10. M. Taki, N. Ouarzazi, H. Ward and P. Glorieux, "Nonlinear front propagation in optical parametric oscillators," J. Opt. Soc. Am. B 17, 997 (2000). [CrossRef]
  11. N. N. Rosanov, Solitons in laser systems with saturable absorption, Dissipative solitons Lecture Notes in Physics, V. 661, (Eds.) N. Akhmediev and A. Ankiewicz, (Springer, Heidelberg, 2005).
  12. E. A. Ultanir and G. I. Stegeman, D. Michaelis, C. H. Lange, and F. Lederer, "Stable dissipative solitons in semiconductor optical amplifiers," Phys. Rev. Lett. 90, 253903 (2003). [CrossRef] [PubMed]
  13. C. Montes, A. Mikhailov, A. Picozzi, and F. Ginovart, "Dissipative three-wave structures in stimulated backscattering. I. A subluminous solitary attractor," Phys. Rev. E 55, 1086 (1997). [CrossRef]
  14. R. J. Deissler and H. R. Brand, "Periodic, quasiperiodic, and chaotic localized solutions of the quintic complex Ginzburg-Landau equation," Phys. Rev. Lett. 72, 478 - 481 (1994). [CrossRef] [PubMed]
  15. X. Hachair, F. Pedaci, E. Caboche, S. Barland, M. Giudici, J. R. Tredicce, F. Prati, G. Tissoni, R. Kheradmand, L. A. Lugiato, I. Protsenko, and M. Brambilla, "Cavity solitons in a driven VCSEL above threshold," J. of Sel. Topics on Quant.Electr. 12, 339-351 (2006). [CrossRef]
  16. A. Ankiewicz, N. Devine, N. Akhmediev and J. M. Soto-Crespo "Continuously self-focusing and continuously self-defocusing 2-D beams in dissipative media," Phys. Rev. A 77, 033840 (2008). [CrossRef]
  17. J. M. Soto-Crespo, N. Akhmediev y A. Ankiewicz. "Pulsating, creeping and erupting solitons in dissipative systems," Phys.Rev. Lett. 85, 2937 (2000). [CrossRef]
  18. Akhmediev. J. M. Soto-Crespo, and G. Town "Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locked lasers: Complex Ginzburg - Landau equation,"Phys. Rev. E. 63, 056602 (2001). [CrossRef]
  19. J. M. Soto-Crespo, Ph. Grelu, and N. Akhmediev"Optical bullets and "rockets" in nonlinear dissipative systems and their transformations and interactions," Opt. Express 14, 4013 (2006). [CrossRef] [PubMed]
  20. J. M. Soto-Crespo, N. Akhmediev and Ph. Grelu "Optical bullets and double bullet complexes in dissipative systems," Phys. Rev. E 74, 046612 (2006). [CrossRef]
  21. N. AkhmedievJ.M. Soto-Crespo and Ph. Grelu, "Vibrating and shaking soliton pairs in dissipative systems." Phys. Lett. A 364, 413 (2007). [CrossRef]
  22. E. Tsoy and N. Akhmediev, "Bifurcations from stationary to pulsating solitons in the cubic quintic complex Ginzburg Landau equation," Phys. Lett. A 343, 417-422 (2005). [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Supplementary Material


» Media 1: MPG (568 KB)     
» Media 2: MPG (1242 KB)     
» Media 3: MPG (636 KB)     

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited