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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 9 — Apr. 27, 2009
  • pp: 7052–7058
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Spatiotemporal pulse-train solitons

Hassid C. Gurgov and Oren Cohen  »View Author Affiliations


Optics Express, Vol. 17, Issue 9, pp. 7052-7058 (2009)
http://dx.doi.org/10.1364/OE.17.007052


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Abstract

We propose spatiotemporal solitons that consist of trains of short pulses. The pulses are collectively trapped in the transverse directions by a slow nonlinearity and each pulse is self-trapped in the longitudinal direction by a fast nonlinearity. We demonstrate numerically spatiotemporal bright pulse-train solitons (trains of light bullets) and temporally-dark spatiotemporal pulse-train solitons in an experimentally feasible scheme.

© 2009 Optical Society of America

1. Introduction

Spatiotemporal optical solitons are among the most intriguing and challenging entities in solitons science [1

1. F. Wise and P. Trapani, “The hunt for light bullets - spatiotemporal solitons,” Opt. Photonics News 13, 28–32 (2002).

,2

2. B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B 7, R53–R72 (2005).

]. While temporal [3

3. A. Hasegawa, “Optical solitons in fibers,” Opt. Photonics News 13, 33–37 (2002).

] and [4

4. G. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,” Science 286, 1518–1523 (1999). [PubMed]

] spatial solitons are self-trapped and retain their intensity profile only in the temporal (longitudinal) or only in the spatial (transversal) directions, respectively, spatiotemporal solitons are self-localized in both the temporal and spatial directions. Self-trapping is obtained through a robust balance between diffraction, group velocity dispersion and nonlinearity. Three dimensional (3D) spatiotemporal solitons that are self-localized in two transverse dimensions and one longitudinal dimension were predicted by Silberberg in 1990 and termed light bullets [5

5. Y. Silberberg, “Collapse of optical pulses,” Opt. Lett. 15, 1282–1284 (1990). [PubMed]

]. Experimentally, however, only 2D spatiotemporal solitons have been demonstrated [6

6. X. Liu, L. J. Qian, and F.W. Wise, “Generation of optical spatiotemporal solitons,” Phys. Rev. Lett. 82, 4631–4634 (1999).

,7

7. X. Liu, K. Beckwitt, and F. Wise, “Two-dimensional optical spatiotemporal solitons in quadratic media,” Phys. Rev. E 62, 1328–1340 (2000).

]. The experimental realization of 3D spatiotemporal solitons is still considered a ‘grand challenge’ in nonlinear optics [1

1. F. Wise and P. Trapani, “The hunt for light bullets - spatiotemporal solitons,” Opt. Photonics News 13, 28–32 (2002).

,2

2. B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B 7, R53–R72 (2005).

].

Numerous approaches for obtaining light bullets have been considered theoretically, e.g. by using saturation nonlinearity [8

8. D. Edmundson and R. H. Enns, “Robust bistable light bullets,” Opt. Lett. 17, 586–588 (1992). [PubMed]

,9

9. N. Akhmediev and J. M. Soto-Crespo, “Generation of a train of three-dimensional optical solitons in self-focusing medium,” Phys. Rev. A 47, 1358–1364 (1993). [PubMed]

], nonlocal nonlinearity [10

10. D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. V. Kartashov, L. C. Crasovan, and L. Torner, “Three-dimensional spatiotemporal optical solitons in nonlocal nonlinear media,” Phys. Rev. E 73, 025601R (2006).

], tandem structures that are composed of linear and nonlinear materials [11

11. L. Torner, S. Carrasco, J. P. Torres, L. C. Crasovan, and D. Mihalache, “Tandem light bullets,” Opt. Commun. 199, 277–281 (2001).

], and through manipulating the diffraction and/or dispersion by periodic structures [12–15

12. A. B. Aceves, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, “Multidimensional solitons in fiber arrays,” Opt. Lett. 19, 329–331 (1994). [PubMed]

]. Also, trains of spatiotemporal localized structures have been predicted in a Raman-active medium due to spatio-temporal coupling that is induced by four wave mixing [16

16. A.V. Gorbach and D. V. Skryabin, “Cascaded Generation of Multiply charged Optical Vortices and Spatiotemporal Helical Beams in a Raman Medium,” Phys. Rev. Lett. 98, 243601 (2007). [PubMed]

]. A common feature in all these proposals, however, is that the same nonlinear mechanism(s) counteract both the diffraction and the group velocity dispersion. As a result, the formation of light bullets requires that the characteristic lengths of the diffraction and of the group velocity dispersion coincide with the characteristic length of the nonlinearity. Even when this nontrivial but necessary condition can be satisfied experimentally, it might not be sufficient since the light bullet may not be robust, i.e. stable to inevitable noise. In this paper we propose a new approach for constructing spatiotemporal solitons, which is based on employing slow nonlinearity for removing the condition that the dispersion length must be equal to the diffraction length. Furthermore, the slow nonlinearity facilitates the soliton stability.

Self-focusing nonlinearities that respond slowly to variations in the light intensity have been used for demonstrating and investigating optical spatial solitons (i.e. solitons that are only trapped in the transverse directions). Examples in which the response time is in the range of - or longer than - one second include screening nonlinearity in photorefractives [17–20

17. M. Segev, B. Crosignani, P. DiPorto, G. C. Valley, and A. Yariv, “Steady state spatial screening-solitons in photorefractive media with external applied field,” Phys. Rev. Lett. 73, 3211–3214 (1994). [PubMed]

], self-focusing via electrostatic interaction between elongated molecules in liquid crystals [21

21. M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, “Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells,” Appl. Phys. Lett. 77, 7 (2000).

,22

22. M. Peccianti and G. Assanto, “Nematic liquid crystals: A suitable medium for self-confinement of coherent and incoherent light,” Phys. Rev. E 65, R035603 (2002).

], and thermal nonlinearity [23–25

23. M. D. Iturbe Castillo, J. J. Sánchez-Mondragón, and S. Stepanov, “Formation of steady-state cylindrical thermal lenses in dark stripes,” Opt. Lett. 21, 1622–1624 (1996).

]. Furthermore, slow self-focusing was utilized for discovering new phenomena in solitons science, including incoherent solitons [18–20

18. M. Mitchell, Z. Chen, M. Shih, and M. Segev, “Self-trapping of partially spatially-incoherent light,” Phys. Rev. Lett. 77, 490–493 (1996). [PubMed]

], incoherent modulation instability [20

20. M. Segev and D. N. Christodoulides, “Incoherent solitons: self-trapping of weakly-correlated wave-packets,” Opt. Photonics News 13, 70–76 (2002).

,26

26. D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. N. Christodoulides, “Modulation instability and pattern formation in spatially incoherent light beams,” Science 290, 495–498(2000) [PubMed]

], and random-phase lattice solitons [27

27. O. Cohen, G. Bartal, H. Buljan, T. Carmon, J. W. Fleischer, M. Segev, and D. N. Christodoulides, “Observation of random-phase lattice solitons,” Nature 433, 500–503 (2005). [PubMed]

]. In such incoherent solitons, the characteristic fluctuation time of the light is much shorter than the characteristic response-time of the nonlinearity. Thus, the nonlinearity averages over the fluctuating optical field, leading to induced waveguide that is stationary in time, smooth in space, and is only determined by the averaged intensity. Thus far, this decoupling between the detailed temporal structure of a light and its nonlinear dynamics in the spatial domain has not been utilized for manipulating the spatiotemporal dynamics of pulse-train beams.

Here, we propose a new type of spatiotemporal solitons: spatiotemporal pulse-train solitons, or trains of light bullets. Such solitons consist of sequence of short pulses that are collectively trapped in the transverse directions by a slow nonlinearity and each pulse is self-trapped in the longitudinal (temporal) direction by a fast nonlinearity. In spatiotemporal pulse-train solitons, the characteristic length of the slow nonlinearity corresponds to the diffraction length, while the length of the fast nonlinearity matches the dispersion length. Stability of the soliton is facilitated by the slow nonlinearity and requires that the diffraction length is much shorter than the dispersion length. We formulate a model for describing the spatiotemporal propagation of pulse-train beams in a nonlinear medium that exhibit both slow and fast nonlinearities and then solve the model for spatiotemporal pulse-train solitons. Finally, we demonstrate numerically a train of bright and a train of dark light bullets in lead glass which shows thermo-optical and optical Kerr nonlinearities.

2. Concept

Consider a beam of short pulses that propagates in a medium with self-focusing nonlinearity whose characteristic response time is much longer than the interval between consecutive pulses. In this case, the nonlinear index change is transparent to the fact that the beam consists of ultrashort pulses and, in fact, only depends on the averaged intensity. If the slow nonlinearity is also highly nonlocal, then the nonlinear index change only depends on the averaged power [28

28. A. W. Snyder and D. J. Mitchell, “Accessible solitons,” Science 276, 1538–1541 (1997).

]. When the beam populates the first guided mode of the induced waveguide in a self-consistent fashion, it may form an optical spatial soliton [29

29. A. W. Snyder, D. J. Mitchell, L. Poladian, and F. Ladouceur, “Self-induced optical fibers: spatial solitary waves,” Opt. Lett. 16, 21–23 (1991). [PubMed]

]. Next, consider a single pulse from the pulse-train that forms an optical spatial soliton. The pulse is injected into a waveguide that was induced by pulses which propagated in the medium at earlier times. In fact, since the response time of the nonlinearity is much longer than the temporal width of the pulse, a pulse only influences the propagation of future pulses and not its own propagation. Thus, it is fair to say that each pulse propagates linearly in a “fixed” fiber that was induced by earlier pulses of the train. Next, consider that the intensity of the pulse is large, such that it can form a temporal soliton through the optical Kerr self phase modulation effect in the “fixed” fiber. Bright or dark temporal solitons can form in a medium with anomalous or normal group velocity dispersion, respectively [3

3. A. Hasegawa, “Optical solitons in fibers,” Opt. Photonics News 13, 33–37 (2002).

]. Following the arguments above, we conclude that a proper combination of slow and fast nonlinearities can be used for spatiotemporal self-trapping of a pulse-train beam. Similarly to temporal bright [30

30. L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental Observation of Picosecond Pulse Narrowing and Solitons in Optical Fibers,” Phys. Rev. Lett. 45, 1095–1048 (1980).

] and dark [31

31. D. Krökel, N. J. Halas, G. Giuliani, and D. Grischkowsky, “Dark-Pulse Propagation in Optical Fibers,” Phys. Rev. Lett. 60, 29–32 (1988). [PubMed]

,32

32. A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirschner, D. E. Leaird, and W. J. Tomlinson, “Experimental Observation of the Fundamental Dark Soliton in Optical Fibers,” Phys. Rev. Lett. 61, 2445–2448 (1988). [PubMed]

] Kerr solitons that were experimentally demonstrated in optical fibers, stability of spatiotemporal pulse-train solitons requires that the transverse confinement is more substantial than the longitudinal (temporal) confinement. An equivalent requirement is that the change in the index of refraction due to the Kerr nonlinearity should be significantly smaller than the index change of the waveguide that in our case is induced through the slow nonlinearity.

3. Model and analysis

We first construct a scheme for analyzing the propagation of pulse-train beams in media that exhibit both fast and slow nonlinearities, and then solve it for spatiotemporal pulse-train solitons. The complex slowly varying envelope of the pulse-train beam is given Ψtrain = ∑ q=-∞ Ψq (x,y,z,t-qT), where q is the pulse order, x and y are the transverse coordinates, z is the longitudinal coordinate, t is time in the frame of pulse q=0, and T is the time interval between consecutive pulses which is much larger than their widths. The spatiotemporal dynamics of pulse q is determined by the following nonlinear Schröinger equation:

iΨqzβ222Ψqt2+12kT2Ψq+Δnqkn0Ψq=0
(1)

where β2 is the group velocity dispersion, k=2πn0/λ is the central wavenumber, λ the central wavelength, n0 the linear refractive index at λ, and ∇T 2 = 2 /∂x 2 + 2 /∂y 2 is the transverse Laplacian. The nonlinear index change of pulse q is given by

Δnq=n2Ψq2+Δnqslow
(2)

where n2 is the Kerr coefficient and Δnslowq is the index change that is induced through the slow self-focusing effect. Assuming that the relaxation time of Δnslowq is much larger than the pulse-train periodicity, tNL≫T, the temporal dynamics of the slow index change can be determined by a discrete relaxation equation.

tNLΔnq+1slowΔnqslowT+Δnqslow=T/2T/2f(Ψ2)dt
(3)

where f is the self-focusing function. Equations (1)–(3) model the propagation of a pulse-train beam in nonlinear media with Kerr and slow self-focusing effects. Here, we are interested in spatiotemporal pulse-train solitons where the propagation is stationary, i.e. Ψq+1 = Ψq , hence we solve Eq. (3) and get Δnslow = ∫T/2 -T/2 f (∣Ψ∣2)dt, (we omit the subscript q). We continue by analyzing a concrete example. In order to get simple analytic solutions and conditions of pulse-train solitons, we choose the slow self-focusing to be highly nonlocal, i.e. the refractive index change at a given location is a function of the intensity at some nonlocality range surrounding that location and that range is much larger than the width of the beam. Examples of slow and highly nonlocal self-focusing mechanisms include long-range electrostatic interaction in liquid crystals [21

21. M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, “Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells,” Appl. Phys. Lett. 77, 7 (2000).

,22

22. M. Peccianti and G. Assanto, “Nematic liquid crystals: A suitable medium for self-confinement of coherent and incoherent light,” Phys. Rev. E 65, R035603 (2002).

] and thermal nonlinearity in glass [23–25

23. M. D. Iturbe Castillo, J. J. Sánchez-Mondragón, and S. Stepanov, “Formation of steady-state cylindrical thermal lenses in dark stripes,” Opt. Lett. 21, 1622–1624 (1996).

]. We model the highly nonlocal effect by a spatial convolution between the averaged intensity and a Gaussian response function [34

34. O. Bang, W. Krolikowski, J. Wyller, and J. Rasmussen, ”Collapse arrest and soliton stabilization in nonlocal nonlinear media,” Phys. Rev. E 66, 046619 (2002).

]

Δnslow(x,y)=Δn0exp[(x2+y2)/σ2]1TT/2T/2Ψ2dt
(4)

In the highly nonlocal regime, where the width of the response function, σ, is much larger than the spatial width of the beam, the induced index change due to the slow self-focusing effect is approximately parabolic [28

28. A. W. Snyder and D. J. Mitchell, “Accessible solitons,” Science 276, 1538–1541 (1997).

],

Δnslow = Δn 0 Pexp [-(x 2+ y 2)/σ 2] ≈ Δn 0 P[1-(x 2+ y 2)/σ 2], where the averaged power is P=1TT/2T/2Ψ2dxdydt and Δn0P is a constant such that Δn0P is the peak index change. Fundamental spatial solitons in highly nonlocal self-focusing media are approximately Gaussian [28

28. A. W. Snyder and D. J. Mitchell, “Accessible solitons,” Science 276, 1538–1541 (1997).

]. Since the slow self-focusing effect is much larger than the Kerr nonlinearity, the spatiotemporal pulse-train soliton is also Gaussian. In the temporal domain, spatiotemporal pulse-train solitons resemble solitons in fixed fiber. Hence, we ansatz the following trial functions [35

35. S. Raghavan and G. P. Agrawal, “Spatiotemporal solitons in inhomogeneous nonlinear media,” Opt. Commun. 180, 377–382 (2000).

]

ΨB=I0exp[(x2+y2)/w2]sech(t/τ)exp((z))
(5)
ΨD=I0exp[(x2+y2)/w2]exp[(t/Δt)2N]tanh(t/τ)exp((z))
(6)

where ΨB corresponds to temporally-bright spatiotemporal pulse-train solitons that exist in a medium with anomalous group velocity dispersion (β2<0) and ΨD corresponds to temporally-dark spatiotemporal pulse-train solitons that exist in a medium with normal group velocity dispersion (β2>0). Also, I0 is the peak intensity, w and τ are the transverse and temporal width parameters, respectively, φ is a z-dependent phase, N=5 is the order of the super-Gaussian background pulse and Δt is its width. The averaged power of the temporally-bright pulse-train beam is P=I0πw2τ/T, where the averaged power of the temporally-dark pulse-train beam is calculated numerically.

The self-consistency condition for self-trapping in the spatial domain, which requires exp [-(x 2 + y 2)/w 2] to be a guided mode of the parabolic Δnslow [28

28. A. W. Snyder and D. J. Mitchell, “Accessible solitons,” Science 276, 1538–1541 (1997).

], leads to

P=2n0σ2Δn0k2w4
(7)

The condition for self-trapping in the temporal domain, i.e. that the temporal components of ΨB and ΨD are bright and dark temporal solitons in the “fixed” fiber Δn 0 P[1-(x 2+y 2)/σ 2] , with Kerr self phase modulation respectively, leads to [35

35. S. Raghavan and G. P. Agrawal, “Spatiotemporal solitons in inhomogeneous nonlinear media,” Opt. Commun. 180, 377–382 (2000).

]

I0=2β2n0/n2kτ2
(8)

Interestingly, the pulse-train beam has four free parameters (e.g. P, I0, τ, and w) while there are only two conditions. Thus, 3D spatiotemporal pulse-train solitons contain two free parameters (e.g. τ and w, or τ and P, etc.).

4. Concrete examples

To demonstrate a temporally-bright 3D spatiotemporal pulse-train soliton (train of light bullets), anomalous dispersion is required (β2<0). Thus we changed the sign of the group velocity dispersion while keeping its absolute value (all the other parameters still correspond to SF11 lead glass). We launch the beam ΨB (z = 0) (Eq. (5)) with 5% noise in amplitude and phase into the lead glass sample and calculate its evolution by integrating Eqs. (1), (2) and (4) for propagation distance of 1130 mm which corresponds to 20 dispersion lengths and 210 diffraction lengths. Figures 1(a) and 1(b) present the nonlinear propagation in the spatial and temporal domains, respectively, showing that spatiotemporal soliton is indeed formed. The intensity profiles at the input and output faces of the sample show that the high-frequencies components of the noise radiate away while the low-frequencies components stay trapped within the soliton (Figs. 1(c)–1(e)). This behavior results from the fact that the highly nonlocal nonlinearity induces a multi-mode waveguide [37

37. C. Rotschild, T. Schwartz, O. Cohen, and M. Segev, “Incoherent solitons in effectively-instantaneous nonlocal nonlinear media,” Nat. Photonics 2, 371–376 (2008).

,38

38. O. Cohen, H. Buljan, T. Schwartz, J. W. Fleischer, and M. Segev, “Incoherent solitons in instantaneous nonlocal nonlinear media,” Phys. Rev. E 73, 015601(R) (2006).

]. Finally, the evolution of the intensity widths (FWHM) in x and in t are shown in Figs. 1(f) and 1(g), respectively, showing the stationary evolution for 20 dispersion lengths and 210 diffraction lengths. The tiny fluctuations in the widths (inset of Fig. 1(f)) result from mode beating between multiple modes that are populated by the initial noise and are trapped within the soliton.

Fig. 1. Temporally-bright spatiotemporal pulse-train soliton. Intensity of the soliton in plane x-z (a) and t-z (b) demonstrating the stationary propagation. (c) Intensity in x-y-t at the input and output faces of the sample. Intensity profiles in x-direction (d) and t-direction (e) at the input and output faces of the sample. (f) Width (FWHM) of the intensity in the x-direction (f) and t-direction (g) versus propagation distance for linear and nonlinear propagations.

Next, we demonstrate temporally-dark 3D spatiotemporal pulse-train soliton in SF11 lead glass. The dark notch is inserted within a super-Gaussian background pulse. We launch the beam ΨD (z =0) (Eq. (6)) with 5% noise in amplitude and phase into the lead glass sample and calculate its evolution by integrating Eqs. (1), (2) and (4) for propagation distance of 565 mm. Figure 2 shows that temporally-dark spatiotemporal soliton is indeed formed. Figures 2(a) and 2(b) present the nonlinear propagation in the spatial and temporal domains, respectively. The intensity profiles at the input and output faces of the sample are shown in Figs. 1(c)–1(e). Note that while the background pulse somewhat broadens during propagation, the dark notch stays intact. The continuing broadening of the background pulse eventually also leads, at very large propagation distances, to broadening of the dark notch. The evolution of the intensity widths (FWHM) in x and in t are shown in Figs. 2(f) and 2(g), respectively. The oscillations in the temporal FWHM at the linear propagation case (Fig. 2(g)) results from the non-smooth diffraction pattern of the dark notch. Finally, we emphasize that the existence and stability of the temporally-bright and temporally-dark spatiotemporal solitons are insensitive to small changes in all the parameters of the system.

Fig. 2. Temporally-dark spatiotemporal pulse-train soliton. Intensity of the soliton in plane x-z (a) and t-z (b) demonstrating the stationary propagation of the soliton. (c) Intensity in x-y-t at the input and output faces of the sample. Intensity profiles in x-direction (d) and t-direction (e) at the input and output faces of the sample. (f) Width (FWHM) of the intensity in the x-direction (f) and t-direction (g) versus propagation distance for linear and nonlinear propagations.

5. Summary and conclusions

In conclusion, we have predicted spatiotemporal pulse-train solitons: a train of short pulses that are collectively trapped in the transverse directions by a slow nonlinearity and each pulse is self-trapped in the longitudinal direction by a fast nonlinearity. Pulse-train solitons exhibit a new strategy for realizing experimentally 3D optical solitons. In this approach, diffraction and dispersion are counteracted by different nonlinear mechanisms. In this way, the most challenging requirement for realizing light bullets - finding a setting in which the nonlinearity balances both the diffraction and dispersion - is removed. We believe that spatiotemporal pulse-train solitons are experimentally accessible in various systems that show both fast and slow nonlinearities such as thermal glass, liquid crystals, and photorefractives. As a concrete example, we analyze and demonstrate numerically spatiotemporal pulse-train solitons in lead glass. Spatiotemporal pulse-train solitons open new opportunities in solitons science. For example, the interaction between pulse-train solitons or the coupling between components of vector solitons can be controlled by a new knob: the degree of spatiotemporal overlap between the different fields.

References and links

1.

F. Wise and P. Trapani, “The hunt for light bullets - spatiotemporal solitons,” Opt. Photonics News 13, 28–32 (2002).

2.

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B 7, R53–R72 (2005).

3.

A. Hasegawa, “Optical solitons in fibers,” Opt. Photonics News 13, 33–37 (2002).

4.

G. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,” Science 286, 1518–1523 (1999). [PubMed]

5.

Y. Silberberg, “Collapse of optical pulses,” Opt. Lett. 15, 1282–1284 (1990). [PubMed]

6.

X. Liu, L. J. Qian, and F.W. Wise, “Generation of optical spatiotemporal solitons,” Phys. Rev. Lett. 82, 4631–4634 (1999).

7.

X. Liu, K. Beckwitt, and F. Wise, “Two-dimensional optical spatiotemporal solitons in quadratic media,” Phys. Rev. E 62, 1328–1340 (2000).

8.

D. Edmundson and R. H. Enns, “Robust bistable light bullets,” Opt. Lett. 17, 586–588 (1992). [PubMed]

9.

N. Akhmediev and J. M. Soto-Crespo, “Generation of a train of three-dimensional optical solitons in self-focusing medium,” Phys. Rev. A 47, 1358–1364 (1993). [PubMed]

10.

D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. V. Kartashov, L. C. Crasovan, and L. Torner, “Three-dimensional spatiotemporal optical solitons in nonlocal nonlinear media,” Phys. Rev. E 73, 025601R (2006).

11.

L. Torner, S. Carrasco, J. P. Torres, L. C. Crasovan, and D. Mihalache, “Tandem light bullets,” Opt. Commun. 199, 277–281 (2001).

12.

A. B. Aceves, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, “Multidimensional solitons in fiber arrays,” Opt. Lett. 19, 329–331 (1994). [PubMed]

13.

E. W. Laedke, K. H. Spatschek, and S. K. Turitsyn, “Stability of discrete solitons and quasicollapse to intrinsically localized modes,” Phys. Rev. Lett. 73, 1055–1058 (1994). [PubMed]

14.

Z. Xu, Ya.V. Kartashov, L.C. Crasovan, D. Mihalache, and L. Torner, “Spatiotemporal discrete multicolor solitons,” Phys. Rev. E 70, 066618 (2004).

15.

A. S. Sukhorukov and Y. S. Kivshar, “Slow-Light Optical Bullets in Arrays of Nonlinear Bragg-Grating Waveguides,” Phys. Rev. Lett. 97, 233901 (2006).

16.

A.V. Gorbach and D. V. Skryabin, “Cascaded Generation of Multiply charged Optical Vortices and Spatiotemporal Helical Beams in a Raman Medium,” Phys. Rev. Lett. 98, 243601 (2007). [PubMed]

17.

M. Segev, B. Crosignani, P. DiPorto, G. C. Valley, and A. Yariv, “Steady state spatial screening-solitons in photorefractive media with external applied field,” Phys. Rev. Lett. 73, 3211–3214 (1994). [PubMed]

18.

M. Mitchell, Z. Chen, M. Shih, and M. Segev, “Self-trapping of partially spatially-incoherent light,” Phys. Rev. Lett. 77, 490–493 (1996). [PubMed]

19.

M. Mitchell and M. Segev, “Self-trapping of incoherent white light,” Nature 387, 880–883 (1997).

20.

M. Segev and D. N. Christodoulides, “Incoherent solitons: self-trapping of weakly-correlated wave-packets,” Opt. Photonics News 13, 70–76 (2002).

21.

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, “Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells,” Appl. Phys. Lett. 77, 7 (2000).

22.

M. Peccianti and G. Assanto, “Nematic liquid crystals: A suitable medium for self-confinement of coherent and incoherent light,” Phys. Rev. E 65, R035603 (2002).

23.

M. D. Iturbe Castillo, J. J. Sánchez-Mondragón, and S. Stepanov, “Formation of steady-state cylindrical thermal lenses in dark stripes,” Opt. Lett. 21, 1622–1624 (1996).

24.

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95, 213904 (2005). [PubMed]

25.

A. Barak, C. Rotschild, B. Alfassi, M. Segev, and D. N. Christodoulides, “Random-phase surface wave solitons in nonlocal nonlinear media,” Opt. Lett. 32, 2450–2452 (2007). [PubMed]

26.

D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. N. Christodoulides, “Modulation instability and pattern formation in spatially incoherent light beams,” Science 290, 495–498(2000) [PubMed]

27.

O. Cohen, G. Bartal, H. Buljan, T. Carmon, J. W. Fleischer, M. Segev, and D. N. Christodoulides, “Observation of random-phase lattice solitons,” Nature 433, 500–503 (2005). [PubMed]

28.

A. W. Snyder and D. J. Mitchell, “Accessible solitons,” Science 276, 1538–1541 (1997).

29.

A. W. Snyder, D. J. Mitchell, L. Poladian, and F. Ladouceur, “Self-induced optical fibers: spatial solitary waves,” Opt. Lett. 16, 21–23 (1991). [PubMed]

30.

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental Observation of Picosecond Pulse Narrowing and Solitons in Optical Fibers,” Phys. Rev. Lett. 45, 1095–1048 (1980).

31.

D. Krökel, N. J. Halas, G. Giuliani, and D. Grischkowsky, “Dark-Pulse Propagation in Optical Fibers,” Phys. Rev. Lett. 60, 29–32 (1988). [PubMed]

32.

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirschner, D. E. Leaird, and W. J. Tomlinson, “Experimental Observation of the Fundamental Dark Soliton in Optical Fibers,” Phys. Rev. Lett. 61, 2445–2448 (1988). [PubMed]

33.

S. K. Turitsyn, “Spatial dispersion of nonlinearity and stability of multidimensional solitons,” Theor. Math. Phys. 64, 226–232 (1985).

34.

O. Bang, W. Krolikowski, J. Wyller, and J. Rasmussen, ”Collapse arrest and soliton stabilization in nonlocal nonlinear media,” Phys. Rev. E 66, 046619 (2002).

35.

S. Raghavan and G. P. Agrawal, “Spatiotemporal solitons in inhomogeneous nonlinear media,” Opt. Commun. 180, 377–382 (2000).

36.

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, “Long-range interactions between optical solitons,” Nature Phys. 2, 769–774 (2006).

37.

C. Rotschild, T. Schwartz, O. Cohen, and M. Segev, “Incoherent solitons in effectively-instantaneous nonlocal nonlinear media,” Nat. Photonics 2, 371–376 (2008).

38.

O. Cohen, H. Buljan, T. Schwartz, J. W. Fleischer, and M. Segev, “Incoherent solitons in instantaneous nonlocal nonlinear media,” Phys. Rev. E 73, 015601(R) (2006).

OCIS Codes
(190.5530) Nonlinear optics : Pulse propagation and temporal solitons
(190.6135) Nonlinear optics : Spatial solitons

ToC Category:
Nonlinear Optics

History
Original Manuscript: March 9, 2009
Revised Manuscript: April 10, 2009
Manuscript Accepted: April 10, 2009
Published: April 14, 2009

Citation
Hassid C. Gurgov and Oren Cohen, "Spatiotemporal pulse-train solitons," Opt. Express 17, 7052-7058 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-9-7052


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