OSA's Digital Library

Optics Express

Optics Express

  • Editor: Micha
  • Vol. 13, Iss. 23 — Nov. 14, 2005
  • pp: 9217–9223
« Show journal navigation

Pulse interactions in the stretched-pulse fiber laser

V. Roy, M. Olivier, and M. Piché  »View Author Affiliations


Optics Express, Vol. 13, Issue 23, pp. 9217-9223 (2005)
http://dx.doi.org/10.1364/OPEX.13.009217


View Full Text Article

Acrobat PDF (359 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

The possible outcomes of a collision between a two-pulse bound state and a single pulse in the stretched-pulse fiber laser are examined experimentally. The changes experienced by the pulse bound state during the collision process account for the various observed scenarios. In particular, it is seen that the relative speed and pulse number need not be conserved in the process. A mechanism that helps explain interactions among the colliding pulses is discussed.

© 2005 Optical Society of America

Pulse collisions in the stretched-pulse fiber laser have recently been reported [1–4

1 . M. Olivier , V. Roy , F. Babin , and M. Piché , “ Pulse collisions in the stretched-pulse fiber laser ,” Opt. Lett. 29 , 1461 – 1463 ( 2004 ). [CrossRef] [PubMed]

]. The collisions take place between two or more bound states of pulses traveling in the laser cavity with different group velocities. In several instances, this process has been observed to repeat itself indefinitely with a relatively long period [1–2

1 . M. Olivier , V. Roy , F. Babin , and M. Piché , “ Pulse collisions in the stretched-pulse fiber laser ,” Opt. Lett. 29 , 1461 – 1463 ( 2004 ). [CrossRef] [PubMed]

]. In that particular case, the colliding pulse bound states appear to leave one another unaltered and move apart with conserved initial momentum. Hence, this collision scenario has been dubbed “elastic” with reference to the laws of classical mechanics [2

2 . Ph. Grelu and N. Akhmediev , “ Group interactions of dissipative solitons in a laser cavity: the case of 2+1 ,” Opt. Express 12 , 3184 – 3189 ( 2004 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-14-3184 . [CrossRef] [PubMed]

]. More recently, a time-resolved measurement of the collision process was reported [3

3 . V. Roy , M. Olivier , F. Babin , and M. Piché , “ Dynamics of periodic pulse collisions in a strongly dissipative-dispersive system ,” Phys. Rev. Lett. 94 , 203903 ( 2005 ). [CrossRef] [PubMed]

]. The measurement showed the collision to proceed through the interaction between the tails of the stretched pulses.

A variety of other collision scenarios may occur in such a system. Indeed, energy and momentum conservation do not hold for pulse propagation and collision in a laser cavity since it is a dissipative system. In ref. [4

4 . N. Akhmediev , J. M. Soto-Crespo , M. Grapinet , and Ph. Grelu , “ Dissipative soliton interactions inside a fiber laser cavity ,” Opt. Fiber Technol. 11 , 209 – 228 ( 2005 ). [CrossRef]

], a range of outcomes for the collision of a two-pulse bound state with a single pulse have been predicted from a series of numerical simulations. In this paper, we report the experimental observation of several of the possible outcomes theoretically described in that paper. The observed collision scenarios result from single events since the colliding pulse structures are modified after the collisions. More specifically, it is shown that the relative speed and pulse number need not be conserved as pulse bound states may rearrange in several ways during the process. As well, a mechanism that may be responsible for the formation and interaction of pulse bound states is discussed.

The stretched-pulse fiber ring laser used for the experiment is illustrated in Fig. 1. The gain is provided by a 2.4-m-long erbium-doped fiber (19.3-dB/m peak absorption at 1530 nm) pumped by a 130-mW laser diode at 980 nm. The laser cavity also includes a WDM coupler for injecting the laser pump, a 50/50 broadband output coupler and a polarization-insensitive Faraday isolator to ensure unidirectional propagation of the signal. Ultrashort-pulse emission is achieved through polarization additive-pulse mode-locking (P-APM) with the help of a polarizer inserted between two polarization controllers. In addition to the gain fiber, the laser cavity is made of 3.4 m of Corning SMF-28 and 1.2 m of Corning Flexcor fiber. The dispersion of the erbium-doped fiber is estimated as 36 ps2/km, whereas the dispersion of the SMF-28 and Flexcor fibers is -22 and -6 ps2/km, respectively. Thus, the average cavity dispersion is slightly positive, with D = 0.004 ps2. The output fiber length is adjusted such as to get chirp-free pulses at the laser output. To characterize the laser signal, we used a fast photodiode (~ 1 GHz) connected to a digital oscilloscope that allows multiple-frame acquisition. As well, the signal was sent to an optical spectrum analyzer and an autocorrelator in non-collinear geometry.

Fig. 1. Configuration of the stretched-pulse fiber ring laser.

Operation of the laser at a pump power around 60 mW yields 80-fs pulses with a period ~ 35 ns between consecutive pulses. The energy per pulse is then ~ 300 pJ, and the time-bandwidth product is close to 0.6. Increasing the pump power to about twice that level (and above) results in the formation of bound states of two or more identical pulses with a pulse spacing of a few ps [5

5 . Ph. Grelu , J. Béal , and J. M. Soto-Crespo , “ Soliton pairs in a fiber laser: from anomalous to normal average dispersion regime ,” Opt. Express 11 , 2238 – 2243 ( 2003 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-18-2238 . [CrossRef] [PubMed]

]. As described in [1–4

1 . M. Olivier , V. Roy , F. Babin , and M. Piché , “ Pulse collisions in the stretched-pulse fiber laser ,” Opt. Lett. 29 , 1461 – 1463 ( 2004 ). [CrossRef] [PubMed]

], it is common to observe in this same regime multiple bound states traveling with different group velocities and undergoing collisions of various nature. This situation arises when one proceeds to perturb a bound state that would normally be stable if not for this slight disturbance. For instance, a perturbation may be created by changing the polarization ellipse at some point along the laser cavity. A bound state of three pulses usually breaks up into a two-pulse bound state and a single pulse, both of them traveling with different group velocities and colliding once their relative delay amounts to a full cavity round trip. Earlier work has focused on the periodicity that sometimes characterizes this phenomenon and the fact that the colliding structures appeared undistorted after the collision in this particular instance. However, it happens that this distinct case is not the only possible outcome [4

4 . N. Akhmediev , J. M. Soto-Crespo , M. Grapinet , and Ph. Grelu , “ Dissipative soliton interactions inside a fiber laser cavity ,” Opt. Fiber Technol. 11 , 209 – 228 ( 2005 ). [CrossRef]

]. In the following, different collision scenarios are reported.

A situation where the relative speed of the initial structures is changed after the collision is illustrated in Fig. 2. The evolution of the laser pulse train recorded from the oscilloscope is illustrated in Fig. 2(a). The oscilloscope trigger is set on the central peak which corresponds to the two-pulse bound state. As well, the pulse spectra and the intensity autocorrelation of the pulse train, before (red) and after (blue) the collision, are shown in Fig. 2(b) and Fig. 2(c), respectively. The existence of the pulse bound state is evidenced by the presence of the interference fringes in the pulse spectra and the cross-correlation peaks in the autocorrelation traces. The change in the relative speed between the single pulse and the bound state is obvious in Fig. 2(a). Indeed, the single pulse delay on successive cavity round trips is larger after than before the collision. The modification of the pulse spacing within the two-pulse bound state (see Fig. 2(c)) as a result of the collision accounts for the relative speed change. The pulses in the bound state are closer to each other after the collision and travel faster, in agreement with the findings reported in [1

1 . M. Olivier , V. Roy , F. Babin , and M. Piché , “ Pulse collisions in the stretched-pulse fiber laser ,” Opt. Lett. 29 , 1461 – 1463 ( 2004 ). [CrossRef] [PubMed]

].

Fig. 2. (a) Collision of a two-pulse bound state and a single pulse with a change in their relative speed; (b) the pulse spectrum before (red) and after (blue) the collision and (c) the intensity autocorrelation of the pulse train before (red) and after (blue) the collision.

In some situations, the collision may have a much greater impact on the structure of the pulse bound state. In these instances, the change in the relative speed between the colliding pulses will usually be more considerable than in the case examined previously. In some circumstances, even the relative motion between the single pulse and the two-pulse bound state may get reversed, i.e. the single pulse is being repelled by the pulse bound state (or vice-versa). Note that a similar scenario was described in [4

4 . N. Akhmediev , J. M. Soto-Crespo , M. Grapinet , and Ph. Grelu , “ Dissipative soliton interactions inside a fiber laser cavity ,” Opt. Fiber Technol. 11 , 209 – 228 ( 2005 ). [CrossRef]

]. This situation was observed experimentally and is depicted in Fig. 3. Again, the oscilloscope trigger is set on the highest peak, which corresponds to the two-pulse bound state. From Fig. 3(a), we see that initially the single pulse is traveling faster than the pulse bound state whereas the reverse is true after the collision. Moreover, the relative speed is significantly larger after than before the collision. One should notice the break mark on the y axis in Fig. 3(a) and the different scales on both sides of the mark. This was made necessary in order to better illustrate the evolution of the pulse train. The intensity autocorrelations confirm that the pulse spacing in the bound state has changed significantly. It has reduced from about 6 ps before the collision to nearly 2 ps after. This fact is consistent with the fringe spacing in the pulse spectra illustrated in Fig. 3(b) and Fig. 4. In another case (not shown here), the pulse bound state was left nearly unchanged after the collision (i.e. the forward and recoil speeds were about the same).

Fig. 3. (a) A single pulse repelled by a two-pulse bound state; (b) the pulse spectrum before (red) and after (blue) the collision and (c) the intensity autocorrelation of the pulse train before (red) and after (blue) the collision.
Fig. 4. Enlarged portion of the pulse spectra depicted in Fig. 3(b).

Even more dramatic changes can occur during the collision between the two-pulse bound state and the single pulse. In some cases, the single pulse and the pulse bound state may be seen to merge together; the result is a new pulse bound state. During this process, the pulse number needs not be conserved. For instance, the single pulse and the two-pulse bound state may proceed to form a three-pulse bound state, thus conserving the pulse number. As well, one of the initial pulses may get annihilated whereas the remaining pulses will form a two-pulse bound state distinct from the original one. Both of these outcomes have been observed experimentally and are illustrated in Fig. 5. Such scenarios have been predicted in [4

4 . N. Akhmediev , J. M. Soto-Crespo , M. Grapinet , and Ph. Grelu , “ Dissipative soliton interactions inside a fiber laser cavity ,” Opt. Fiber Technol. 11 , 209 – 228 ( 2005 ). [CrossRef]

].

Fig. 5. (a) A single pulse merging with a two-pulse bound state; (b) the pulse spectrum before (red) and after the collision, with (blue) or without (green) pulse number conservation, and (c) the intensity autocorrelation of the pulse train before (red) and after the collision, with (blue) or without (green) pulse number conservation.

The single pulse merging with the two-pulse bound state is evident from the pulse train illustrated in Fig. 5(a). Examples where the pulse number is conserved after the collision correspond to the blue curves in Fig. 5(b)–(c) while the converse is depicted by the green curves. (The pulse train dynamics in both cases were almost identical, except for a slight difference in the relative speed between the single pulse and the two-pulse bound state before the collision. Hence, only the pulse train in the case where the pulse number is conserved is shown in Fig. 5). In the case where a three-pulse bound state is formed, the pulse spacing between subsequent pulses is roughly the same as for the initial two-pulse bound state. For instance, this may be seen from the pulse spectra illustrated in Fig. 5(b) and Fig. 6, and also from the pulse train intensity autocorrelation in Fig. 5(c). In the case where a pulse is annihilated, the newly-formed bound state consists of higher-energy pulses located further apart from one another, as can be inferred from Fig. 5(b)–(c) and Fig. 6. In particular, the latter is revealed from the height and the position of the autocorrelation and cross-correlation peaks in Fig. 5(c).

Fig. 6. Enlarged portion of the pulse spectra depicted in Fig. 5(b).

A method for resolving the collision dynamics was reported in a recent work [3

3 . V. Roy , M. Olivier , F. Babin , and M. Piché , “ Dynamics of periodic pulse collisions in a strongly dissipative-dispersive system ,” Phys. Rev. Lett. 94 , 203903 ( 2005 ). [CrossRef] [PubMed]

]. Unfortunately, this method cannot be used to study the dynamics of the collision scenarios described in this paper. The periodicity of the collision process is a prerequisite to perform the time-resolved measurement discussed in the aforementioned work while the observations reported here results from single collision events. Nonetheless, we may still assume the general conclusion of ref. [3

3 . V. Roy , M. Olivier , F. Babin , and M. Piché , “ Dynamics of periodic pulse collisions in a strongly dissipative-dispersive system ,” Phys. Rev. Lett. 94 , 203903 ( 2005 ). [CrossRef] [PubMed]

] to hold here as well, namely that the colliding pulses interact with each other through their tails. How can this interaction account for the various outcomes of the collision between a single pulse and a pulse bound state is discussed next.

The observation of pulse bound states in passively mode-locked fiber lasers has drawn attention in recent years [5–9

5 . Ph. Grelu , J. Béal , and J. M. Soto-Crespo , “ Soliton pairs in a fiber laser: from anomalous to normal average dispersion regime ,” Opt. Express 11 , 2238 – 2243 ( 2003 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-18-2238 . [CrossRef] [PubMed]

]. Multiple pulse emission occurs in additive-pulse mode-locked fiber lasers because of the pulse-limiting response of the mode-locking mechanism at high power. In the soliton regime, the bound states were shown to arise from the nonlinear interaction between solitons and the resonant dispersive waves emitted as a result of the periodic perturbations they undergo along propagation around the laser cavity on successive round trips [8

8 . J. M. Soto-Crespo , N. Akhmediev , Ph. Grelu , and F. Belhache , “ Quantized separations of phase-locked soliton pairs in fiber lasers ,” Opt. Lett. 28 , 1757 – 1759 ( 2003 ). [CrossRef] [PubMed]

]. In the stretched-pulse regime, the resonant sidebands are strongly reduced because of the alternating broadening and compression the pulses experience in a single cavity round trip [10–11

10 . H. A. Haus , K. Tamura , L. E. Nelson , and E. P. Ippen , “ Stretched-pulse additive pulse mode-locking in fiber ring lasers: Theory and Experiment ,” IEEE J. Quantum Electron. 31 , 591 – 598 ( 1995 ). [CrossRef]

]. In fact, direct pulse interaction is more significant since the stretched pulses in a pulse bound state overlap in a sizeable portion of the laser cavity [5

5 . Ph. Grelu , J. Béal , and J. M. Soto-Crespo , “ Soliton pairs in a fiber laser: from anomalous to normal average dispersion regime ,” Opt. Express 11 , 2238 – 2243 ( 2003 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-18-2238 . [CrossRef] [PubMed]

].

The nonlinear transmission that results from P-APM is, to first order, proportional to the nonlinear phase accumulated during a complete cavity round trip [10

10 . H. A. Haus , K. Tamura , L. E. Nelson , and E. P. Ippen , “ Stretched-pulse additive pulse mode-locking in fiber ring lasers: Theory and Experiment ,” IEEE J. Quantum Electron. 31 , 591 – 598 ( 1995 ). [CrossRef]

]. Thus, the nonlinear transmission profile extends way beyond the minimum pulse duration as a result of the broadening and compression sequence the pulses are going through. In turn, the transmission profile around each pulse in a pulse bound state will be modified because of contributions from adjacent pulses. For instance, the transmission peaks of the individual pulses in a two-pulse bound state will be shifted toward or away from each other, depending on the spacing and relative phase between the two pulses. For given pulse spacing and relative phase, the attractive/repulsive force that results from this interaction vanishes and there should result stable bound states of pulses, in a way similar to ref. [12

12 . B. A. Malomed , “ Bound solitons in the nonlinear Schrödinger-Ginzburg-Landau equation ,” Phys. Rev. A 44 , 6954 – 6957 ( 1991 ). [CrossRef] [PubMed]

]. As well, higher-order terms of the P-APM nonlinear transmission should further help stabilize the formation of these bound states against small perturbations.

Following the latter reasoning, a single pulse colliding with a pulse bound state represents a perturbation large enough to induce changes to the initially stable bound state. In the cases discussed above, the difference in group velocity between the single pulse and the two-pulse bound state is of the order of a few meters per second, which is about 10-8 the speed of light. The corresponding delay on a single cavity round trip is shorter than the oscillation cycle of light! Thus, the phase difference between the single pulse and the closest pulse in the pulse bound state should have a decisive influence on the outcome of the collision, i.e. whether the pulses attract or repel each other. However, specific outcomes may depend on the particular shape of the nonlinear transmission curve and possibly on other pulse shaping mechanisms (including dispersive waves emitted as a result of such mechanisms).

In conclusion, collisions between a single pulse and a two-pulse bound state in the stretched-pulse fiber laser may result in various outcomes. Among the many observed scenarios were the formation of a new pulse bound state, annihilation of a pulse and the “recoil” of the colliding pulse relative to the pulse bound state. These results have shown that the relative speed and pulse number need not be conserved in the interaction between a single pulse and a pulse bound state.

Acknowledgements

This work was supported by Exfo Electro-Optical Engineering Inc., the Natural Science and Engineering Research Council of Canada, the Fonds Québécois de la Recherche sur la Nature et les Technologies, the Canadian Institute for Photonic Innovations and the Femtotech Consortium.

References and links

1 .

M. Olivier , V. Roy , F. Babin , and M. Piché , “ Pulse collisions in the stretched-pulse fiber laser ,” Opt. Lett. 29 , 1461 – 1463 ( 2004 ). [CrossRef] [PubMed]

2 .

Ph. Grelu and N. Akhmediev , “ Group interactions of dissipative solitons in a laser cavity: the case of 2+1 ,” Opt. Express 12 , 3184 – 3189 ( 2004 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-14-3184 . [CrossRef] [PubMed]

3 .

V. Roy , M. Olivier , F. Babin , and M. Piché , “ Dynamics of periodic pulse collisions in a strongly dissipative-dispersive system ,” Phys. Rev. Lett. 94 , 203903 ( 2005 ). [CrossRef] [PubMed]

4 .

N. Akhmediev , J. M. Soto-Crespo , M. Grapinet , and Ph. Grelu , “ Dissipative soliton interactions inside a fiber laser cavity ,” Opt. Fiber Technol. 11 , 209 – 228 ( 2005 ). [CrossRef]

5 .

Ph. Grelu , J. Béal , and J. M. Soto-Crespo , “ Soliton pairs in a fiber laser: from anomalous to normal average dispersion regime ,” Opt. Express 11 , 2238 – 2243 ( 2003 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-18-2238 . [CrossRef] [PubMed]

6 .

D. Y. Tang , W. S. Man , H. Y. Tam , and P. D. Drummond , “ Observation of bound states of solitons in a passively mode-locked fiber laser ,” Phys. Rev. A 64 , 033814 ( 2001 ). [CrossRef]

7 .

Ph. Grelu , F. Belhache , F. Gutty , and J. M. Soto-Crespo , “ Phase-locked soliton pairs in a stretched-pulse fiber laser ,” Opt. Lett. 27 , 966 – 968 ( 2002 ). [CrossRef]

8 .

J. M. Soto-Crespo , N. Akhmediev , Ph. Grelu , and F. Belhache , “ Quantized separations of phase-locked soliton pairs in fiber lasers ,” Opt. Lett. 28 , 1757 – 1759 ( 2003 ). [CrossRef] [PubMed]

9 .

D. Y. Tang , B. Zhao , L. M. Zhao , and H. Y. Tam , “ Soliton interaction in a fiber ring laser ,” Phys. Rev. E 72 , 016616 ( 2005 ). [CrossRef]

10 .

H. A. Haus , K. Tamura , L. E. Nelson , and E. P. Ippen , “ Stretched-pulse additive pulse mode-locking in fiber ring lasers: Theory and Experiment ,” IEEE J. Quantum Electron. 31 , 591 – 598 ( 1995 ). [CrossRef]

11 .

D. J. Jones , Y. Chen , H. A. Haus , and E. P. Ippen , “ Resonant sideband generation in stretched-pulse fiber lasers ,” Opt. Lett. 23 , 1535 – 1537 ( 1998 ). [CrossRef]

12 .

B. A. Malomed , “ Bound solitons in the nonlinear Schrödinger-Ginzburg-Landau equation ,” Phys. Rev. A 44 , 6954 – 6957 ( 1991 ). [CrossRef] [PubMed]

OCIS Codes
(140.3510) Lasers and laser optics : Lasers, fiber
(140.4050) Lasers and laser optics : Mode-locked lasers
(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

ToC Category:
Research Papers

History
Original Manuscript: October 18, 2005
Revised Manuscript: October 27, 2005
Published: November 14, 2005

Citation
V. Roy, M. Olivier, and M. Piché, "Pulse interactions in the stretched-pulse fiber laser," Opt. Express 13, 9217-9223 (2005)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-23-9217


Sort:  Journal  |  Reset  

References

  1. M. Olivier, V. Roy, F. Babin, and M. Piché, �??Pulse collisions in the stretched-pulse fiber laser,�??Opt. Lett. 29, 1461-1463 (2004) [CrossRef] [PubMed]
  2. Ph. Grelu and N. Akhmediev, �??Group interactions of dissipative solitons in a laser cavity: the case of 2+1,�?? Opt. Express 12, 3184-3189 (2004). <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-14-3184"http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-14-3184">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-14-3184</a> [CrossRef] [PubMed]
  3. V. Roy, M. Olivier, F. Babin, and M. Piché, �??Dynamics of periodic pulse collisions in a strongly dissipative-dispersive system,�?? Phys. Rev. Lett. 94, 203903 (2005) [CrossRef] [PubMed]
  4. N. Akhmediev, J. M. Soto-Crespo, M. Grapinet, and Ph. Grelu, �??Dissipative soliton interactions inside a fiber laser cavity,�?? Opt. Fiber Technol. 11, 209-228 (2005). [CrossRef]
  5. Ph. Grelu, J. Béal, and J. M. Soto-Crespo, �??Soliton pairs in a fiber laser: from anomalous to normal average dispersion regime,�?? Opt. Express 11, 2238-2243 (2003), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-18-2238">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-18-2238</a>. [CrossRef] [PubMed]
  6. D. Y. Tang, W. S. Man, H. Y. Tam, and P. D. Drummond, �??Observation of bound states of solitons in a passively mode-locked fiber laser,�?? Phys. Rev. A 64, 033814 (2001). [CrossRef]
  7. Ph. Grelu, F. Belhache, F. Gutty, and J. M. Soto-Crespo, �??Phase-locked soliton pairs in a stretched-pulse fiber laser,�?? Opt. Lett. 27, 966-968 (2002). [CrossRef]
  8. J. M. Soto-Crespo, N. Akhmediev, Ph. Grelu, and F. Belhache, �??Quantized separations of phase-locked soliton pairs in fiber lasers,�?? Opt. Lett. 28, 1757-1759 (2003). [CrossRef] [PubMed]
  9. D. Y. Tang, B. Zhao, L. M. Zhao, and H. Y. Tam, �??Soliton interaction in a fiber ring laser,�?? Phys. Rev. E 72, 016616 (2005). [CrossRef]
  10. H. A. Haus, K. Tamura, L. E. Nelson, and E. P. Ippen, �??Stretched-pulse additive pulse mode-locking in fiber ring lasers: Theory and Experiment,�?? IEEE J. Quantum Electron. 31, 591-598 (1995). [CrossRef]
  11. D. J. Jones, Y. Chen, H. A. Haus, and E. P. Ippen, �??Resonant sideband generation in stretched-pulse fiber lasers,�?? Opt. Lett. 23, 1535-1537 (1998) [CrossRef]
  12. B. A. Malomed, �??Bound solitons in the nonlinear Schrödinger�??Ginzburg-Landau equation,�?? Phys. Rev. A 44, 6954-6957 (1991). [CrossRef] [PubMed]

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.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited