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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 7 — Mar. 29, 2010
  • pp: 7376–7383
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A dissipative attractor in the spatiotemporal collapse of ultrashort light pulses

Miguel A. Porras  »View Author Affiliations


Optics Express, Vol. 18, Issue 7, pp. 7376-7383 (2010)
http://dx.doi.org/10.1364/OE.18.007376


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Abstract

Spatiotemporal self-focusing in nonlinear lossy media pushes ultrashort pulses towards a universal, non-solitary and non-conical light-bullet wave state defined by the medium solely, and characterized by maximum energy losses. Its stationary propagation relies on a balance between nonlinear losses and the refuelling effect of self-focusing. No balancing gain is required for stationarity. These purely lossy dissipative light-bullets can explain many aspects of the filamentary dynamics in nonlinear media with anomalous dispersion.

© 2010 Optical Society of America

One of the main goals of modern nonlinear optics is the achievement of the so-called light-bullet (LB), a wave state that remains localized in all dimensions in a stable way during propagation [1

1. F. Wise and P. Di Trapani, “The hunt for light bullets—Spatiotemporal solitons,” Opt. Photon. News , 29–32 (2002).

, 2

2. H. S. Eisenberg, R. Morandotti, and Y. Silberberg, “Kerr spatiotemporal self-Focusing in a planar glass waveguide,” Phys. Rev. Lett. 82, 043902 (2001). [CrossRef]

]. In addition to its intrinsic interest as particle-like waves, LBs would find application in long and short-distance communications, all-optical switching, or digital computing, for instance [3

3. R. McLeod, K. Wagner, and S. Blair, “(3+1)-dimensional soliton dragging logic,” Phys. Rev. A 52, 3254–3278 (1995). [CrossRef] [PubMed]

].

Nonlinear losses became an active player in the alternate description of light filaments as conical waves [11

11. P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91, 093904 (2003). [CrossRef] [PubMed]

, 12

12. M. Kolesik, E. M. Wright, and J. V. Moloney, “Dynamic nonlinear X waves for femtosecond pulse propagation in water,” Phys. Rev. Lett. 92, 253901 (2004). [CrossRef] [PubMed]

]. Motivated by the observed self-healing and self-reconstruction properties of light filaments, [13

13. A. Dubietis, E. Gaizauskas, G. Tamosauskas, and P. Di Trapani, “Light Filaments without Self-Channeling,” Phys. Rev. Lett. 92, 253903 (2004). [CrossRef] [PubMed]

, 14

14. M. Kolesik and J. V. Moloney, “Self-healing femtosecond light filaments,” Opt. Lett. 29, 590–592 (2004). [CrossRef] [PubMed]

] conical LBs that are resistant to nonlinear losses [15

15. M. A. Porras, A. Parola, D. Faccio, A. Dubietis, and P. Di Trapani, “Nonlinear unbalanced Bessel beams: Stationary conical waves supported by nonlinear losses,” Phys. Rev. Lett. 93, 153902 (2004). [CrossRef] [PubMed]

, 16

16. M. A. Porras, A. Parola, and P. Di Trapani, “Nonlinear unbalanced O waves: nonsolitary, conical light bullets in nonlinear dissipative media,” J. Opt. Soc. Am. B 22, 1406–1413 (2005). [CrossRef]

] have been discovered recently. Conical LBs are half linear, half nonlinear waves, whose stationarity in lossy media relies on a conical energy flux from the linear wave periphery towards the nonlinear center, where energy is continuously absorbed. Noticeably, conical LBs are more stable as they loss more energy [15

15. M. A. Porras, A. Parola, D. Faccio, A. Dubietis, and P. Di Trapani, “Nonlinear unbalanced Bessel beams: Stationary conical waves supported by nonlinear losses,” Phys. Rev. Lett. 93, 153902 (2004). [CrossRef] [PubMed]

]. Supporting its relevant role in light self-channelling, conical LBs (in two spatial dimensions) have been described [17

17. M. A. Porras and A. Parola, “Nonlinear unbalanced Bessel beams in the collapse of Gaussian beams arrested by nonlinear losses,” Opt. Lett. 33, 1738–1740 (2008). [CrossRef] [PubMed]

] and observed [18

18. D. Faccio, M. Clerici, A. Averchi, O. Jedrkiewicz, S. Tzortzakis, D. Papazoglou, F. Bragheri, L. Tartara, A. Trita, S. Henin, I. Cristiani, A. Couairon, and P. Di Trapani, “Kerr-induced spontaneous Bessel beam formation in the regime of strong two-photon absorption,” Opt. Express 16, 8213–8218 (2008). [CrossRef] [PubMed]

] to be formed spontaneously in the arrest of spatial collapse by nonlinear losses, and the same is expected in the three dimensions.

In this Letter we show, first, that nonlinear losses can balance nonlinear compression in a similar way as phase effects do, and that this balance results in a novel type of purely nonlinear LBs of dissipative type, or dissipative LBs (DLBs), with the peculiarity of featuring only losses (no gain). Stationarity in purely absorbing media is possible because these DLBs carry, as conical LBs, infinite energy. Further, stationarity does not require the conical inward energy flux characteristic of conical LBs, but nonlinear compression alone provides the needed inward flux to sustain the stationarity.

Second, numerical simulations show that the equilibrium between nonlinear losses and nonlinear compression tends to be spontaneously reached in the spatiotemporal self-focusing of ultrashort pulses. The DLB that tends to be formed is that maximizing energy transfer into the medium, which is defined by the medium properties solely, i. e., is independent of the launched pulse. Reaching completely the attractive LB would require, however, the dissipation of an unbounded amount of energy, and hence an input pulse with infinite energy. Paradoxically, and in contrast with solitary LBs, the dissipative LBs formed upon self-focusing would eventually decay because they cannot dissipate enough energy. The DLB decay proceeds through a sequence of less dissipative conical LBs, as described previously [17

17. M. A. Porras and A. Parola, “Nonlinear unbalanced Bessel beams in the collapse of Gaussian beams arrested by nonlinear losses,” Opt. Lett. 33, 1738–1740 (2008). [CrossRef] [PubMed]

, 18

18. D. Faccio, M. Clerici, A. Averchi, O. Jedrkiewicz, S. Tzortzakis, D. Papazoglou, F. Bragheri, L. Tartara, A. Trita, S. Henin, I. Cristiani, A. Couairon, and P. Di Trapani, “Kerr-induced spontaneous Bessel beam formation in the regime of strong two-photon absorption,” Opt. Express 16, 8213–8218 (2008). [CrossRef] [PubMed]

].

The agreement of our simple model with experiments [9

9. D. Moll and A. L. Gaeta, “Role of dispersion in multiple-collapse dynamics,” Opt. Lett. 29, 995–997 (2004). [CrossRef] [PubMed]

] and “exact” numerical simulations [10

10. L. Bergé and S. Skupin, “Self-channeling of ultrashort laser pulses in materials with anomalous dispersion,” Phys. Rev. E 71, 065601 (2005). [CrossRef]

] supports that this DLB attractor determines the main features of the self-focusing and filamentation dynamics in media with anomalous dispersion, including the particularly long length of light filaments [9

9. D. Moll and A. L. Gaeta, “Role of dispersion in multiple-collapse dynamics,” Opt. Lett. 29, 995–997 (2004). [CrossRef] [PubMed]

, 10

10. L. Bergé and S. Skupin, “Self-channeling of ultrashort laser pulses in materials with anomalous dispersion,” Phys. Rev. E 71, 065601 (2005). [CrossRef]

].

These weakly localized DLBs with only losses and infinite energy should not be confused with the well-known dissipative solitons in one or two dimensions, [19

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

] or with the dissipative light-bullets in three dimensions described recently in Refs. [20

20. P. Grelu, J. M. Soto-Crespo, and N. Akhmediev, “Light bullets and dynamic pattern formation in nonlinear dissipative systems,” Opt. Express 13, 9352–9360 (2005). [CrossRef] [PubMed]

] and [21

21. J. M. Soto-Crespo, P. Grelu, and N. Akhmediev, “Optical bullets and “rockets” in nonlinear dissipative systems and their transformations and interactions,” Opt. Express 14, 4013–4025 (2006). [CrossRef] [PubMed]

], which are strongly localized, carry finite energy, and in which a loss/gain balance between different parts of the soliton, and also between different spectral components, is essential for their stationary propagation. For instance, losses and gain (linear or nonlinear), in addition to spectral filtering, are critical in the formation of the dissipative temporal solitons observed in mode-locked fiber lasers [22

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

]. Similarly, self-localized transversal structures, or cavity solitons, usually in the form of an array of dissipative spatial solitons, are formed in the dissipative environment of the cavity and require gain or external driving [23

23. Y. Tanguy, T. Ackermann, W. J. Firth, and R. Jäger, “Realization of a semiconductor-based cavity soliton laser,” Phys. Rev. Lett. 100, 013907 (2008). [CrossRef] [PubMed]

].

The simplest model accounting for self-focusing with nonlinear losses is

zA=i2k0ΔAik02t2A+ik0n2n0A2Aβ(K)2A2K2A,
(1)

or nonlinear Schrödinger equation (NLSE) for the envelope A of a wave packet E = A exp(- 0 t + ik 0 z) of carrier frequency ω 0, or wave length λ0 = 2πc/ω 0, propagating along z. In Eq. (1), ∆ = x 2 + y 2 t = t - k 0 z is the local time, k = n(ω)ω/c is the propagation constant in the medium, n(ω) the refractive index, c the speed of light in vacuum, k (n) 0 = dnk/nω0), and n 2 > 0 the nonlinear refractive index. In media with anomalous dispersion (k 0 < 0), the diffraction and dispersion terms have equal signs, and a pulse whose envelope A depends on the transversal and temporal coordinates through r = (x 2 + y 2 + t 2/k 0k 0∣)1/2, will retain this property on propagation, and can self-focus symmetrically in space and time [4

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

]. The term with β (K) > 0 in Eq. (1) describes nonlinear losses due K-photon absorption.

For the representative example of fused silica at λ 0 = 1550 nm, Fig. 1 shows the change along z of the peak intensity and width of the pulse, calculated from Eq. (1) for spatiotemporal symmetric Gaussian pulses A = (2P/πσ 2)1/2exp(-r 2/σ 2) launched in the medium with different peak powers P above the critical peak power P cr = 2.157λ 0 2/4πn(ω 0)n 2 for spatiotemporal self-focusing [24

24. This peak power is slightly higher than the power Pcr = 2λ02/[4πn(ω0)n2] for spatial self-focusing.

]. Increasing the energy of the input pulse, the light “segments” of nearly constant high intensity and narrow width become longer, and a number of light “bursts” beyond the segment are formed. These segments may extend beyond the Rayleigh distance zR = πσ 20 of the input pulse, and survive by several hundred times the Rayleigh distance expected from their width. These facts have been observed in self-channelling experiments and numerical simulations [9

9. D. Moll and A. L. Gaeta, “Role of dispersion in multiple-collapse dynamics,” Opt. Lett. 29, 995–997 (2004). [CrossRef] [PubMed]

, 10

10. L. Bergé and S. Skupin, “Self-channeling of ultrashort laser pulses in materials with anomalous dispersion,” Phys. Rev. E 71, 065601 (2005). [CrossRef]

]. The location and intensity of the segments and bursts in Fig. 1 are even in quantitative agreement with accurate simulations under the same conditions (Fig. 2 in [10

10. L. Bergé and S. Skupin, “Self-channeling of ultrashort laser pulses in materials with anomalous dispersion,” Phys. Rev. E 71, 065601 (2005). [CrossRef]

]) that include in the NLSE all relevant higher-order effects in propagation (space-time focusing, self-steepening, higher-order dispersion and plasma defocusing), and with input pulses that are not completely symmetric. We note also that the intensity of segments and bursts stabilize at a value that does not depend on the launched pulse, but only on the medium and carrier wave length.

Fig. 1. Peak intensity (solid curves) and FWHM width (dashed blue curves) along z for input Gaussian pulses of carrier frequency ω 0 = 1.21525 fs-1, Gaussian width σ = 0.00716 cm [duration σ(k 0k 0∣)1/2 = 29 fs], and increasing power above the critical power P cr = 13 MW, calculated from (1) with the parameters of fused silica (k 0 = 5.854 × 104 cm-1, k 0 = -279.4 cm-1fs2, n 2 = 2.2 × 10-16 cm2/W, β (K) = 5.11 × 10-116 cm17/W9, with K = 10[10]).

To understand this behavior we consider stationary solutions of the NLSE Eq. (1) of the form A(r,z) = a(r)exp[(r)]exp(-iδz), where the real amplitude a(r) > 0 and phase φ(r) must satisfy

a+2ar+2k0δaφ2a+2k02n2n0a3=0,
(2)
β(K)4π0rdrr2a2K=4πr2k0a2φ,
(3)

and where prime signs stand for d/dr. The second equation, written as N(r) = -F(r) for short, establishes that in stationary solutions the energy losses N(r) within any sphere or radius r must be refuelled by an inward radial energy flux -F(r) from outside [the actual energy losses and flux per unit propagation length in, e.g., J/cm, are (k 0k 0∣)1/2 times N(r) and F(r)]. Localized solutions (i.e., approaching zero at large r) with δ < 0 are solitary LBs. They exist only in transparent media (β (K) = 0), require a stabilizing phase mechanism other than the Kerr nonlinearity [an additional term in (2)], and a precise relation between the peak intensity I 0 = a 2(0) and the wave vector shift δ. Solutions with δ > 0 are conical LBs, or nonlinear O waves [16

16. M. A. Porras, A. Parola, and P. Di Trapani, “Nonlinear unbalanced O waves: nonsolitary, conical light bullets in nonlinear dissipative media,” J. Opt. Soc. Am. B 22, 1406–1413 (2005). [CrossRef]

]. Their amplitude decays as 1/r, and then they carry infinite energy. In their two-dimensional versions (nonlinear Bessel beams [15

15. M. A. Porras, A. Parola, D. Faccio, A. Dubietis, and P. Di Trapani, “Nonlinear unbalanced Bessel beams: Stationary conical waves supported by nonlinear losses,” Phys. Rev. Lett. 93, 153902 (2004). [CrossRef] [PubMed]

]) an inward energy flux along cones of half-apex angle 2δ/k0 supplies the energy absorbed in any disk of radius r during propagation. In three dimensions, the energy flows along hyper-cones of angle 2δ/k0, supplying the energy absorbed in any sphere of radius r. Given δ, or a cone angle, the conical flux can sustain the stationarity for any I 0 up to a maximum value that depends on the cone angle and the medium properties [15

15. M. A. Porras, A. Parola, D. Faccio, A. Dubietis, and P. Di Trapani, “Nonlinear unbalanced Bessel beams: Stationary conical waves supported by nonlinear losses,” Phys. Rev. Lett. 93, 153902 (2004). [CrossRef] [PubMed]

, 16

16. M. A. Porras, A. Parola, and P. Di Trapani, “Nonlinear unbalanced O waves: nonsolitary, conical light bullets in nonlinear dissipative media,” J. Opt. Soc. Am. B 22, 1406–1413 (2005). [CrossRef]

].

In between these two types of LBs, there exist LB solutions of Eqs. (2) and (3) with δ = 0 whose stationarity with losses does not rely on a specific wave geometry, but only on the nonlinear properties of the medium. Seeking for a balance between the Kerr nonlinearity and nonlinear losses, we try solutions of the form a(r) ~ b/rα at large r, where b > 0 is a constant, and localization requires α > 0. Assuming for the moment that losses occur only in a region about the pulse center, and that the total losses are finite, i.e., that N(r) grows up to a maximum value NT < ∞ with increasing r, Eq. (3) yields ϕ ~ -k 0 NT r 2 (α-1)/4πb 2 at large r, and Eq. (2) yields

α(1α)br2+2k0δbk02NT216π2b3r4(α1)+2k02n2b3n0r2α~0.

The possible decay scalings α must be consistent with this equation. For instance, for δ > 0, the first and the Kerr terms are subleading with respect to the conical term 2k 0 δb, which is constant, and must be balanced by the loss term. This implies the decay with α = 1 of conical LBs. In absence of a cone angle (δ = 0), the decay with α = 1 is no longer consistent. Instead, a balance between the loss and Kerr terms is possible for 4(α - 1) = -2α, which yields α = 2/3, the first term being then subleading. Equating the coefficients of the loss and Kerr terms, we also obtain b 6 = (N T 2 n 0/32π 2 n 2).

These localized solutions are verified to exist by solving numerically Eqs. (2) and (3) with δ = 0 [Fig. 2(a), solid curves], and to present the decay b/r 2/3 for nonlinear losses compensation by Kerr nonlinearity [Fig. 2(a), dotted blue curves]. Losses occur only in the pulse center since the N(r) stabilizes in a constant equal to the total losses NT [Fig. 2(b), solid curves]. The amplitude and phase of these LBs are such that an inward energy flux -F(r) equals to the losses N(r). In absence of a cone angle, this flux is an effect of Kerr compression only. As their peak intensity I 0 = a 2(0) is higher, they become wider and dissipate more total energy NT. The total losses NT with increasing peak intensity are compiled in Fig. 2(c, solid curve). For comparison, the total losses of conical LBs (δ > 0) with increasing peak intensity are also shown in Fig. 2(c, dashed curves), and are seen to be lower.

When the intensity approaches the limiting value

I0,m=(4γKn2k0n0β(K))1/(K2),
(4)

where γK is a number of the order of unity that depends only on the order K [see Fig. 2(d) and its caption], DLBs approach a limiting DLB that also tends to zero but so slowly that its total losses NT are unbounded [dashed red curves in Figs. 2(a) and 2(b), and vertical asymptota in (c)]. No DLBs with intensity above I 0,m exist. This value is fixed by the medium as the maximum intensity at which nonlinear compression and losses can reach a balance. To characterize this limiting DLB, we seek again for localized solutions of Eqs. (2) and (3) of the form b/rα at large r but without assuming that the total losses are finite. In this case (3) yields ϕ ~ -[k 0 b 2K-2 β (K)/(3-2)r 1-2α(K-1), and Eq. (2) with δ = 0,

α(1α)br2[k0b2K2β(K)32]2r24α(K1)+2k02n2b3n0r2α~0.

The loss and Kerr terms are the leading terms and balance each other, remaining the first term subleading, for the choice α = 1/(2K - 3) and b 4K-6 = 2(3 - 2)2 n 2/[β (K)2 n 0]. These formulas give indeed the behavior of the DLB of maximum losses [Fig. 2(d), dotted blue curves], which is numerically calculated for different values of K [Fig.2(d), dashed red curves]. Differently from DLBs of lower intensity, this LB dissipates energy not only in the center, but everywhere along its radial profile, and is unique given a material and a carrier wave length.

Fig. 2. (a) Intensity profile a 2(r) (solid curves) and their asymptotic form (dotted blue curves), and (b) nonlinear losses profile (k 0k 0∣)1/2 N(r) (equal to the inward energy flux profile -(k 0k 0∣)1/2 F(r)) of DLBs in fused silica at λ0 = 1550 nm with increasing peak intensities I 0 = a 2(0). The intensity and loss profiles of the LB with maximum intensity I 0,m = 14.716 TW/cm2 [Eq. (4) with γK = 3.15294 for K = 10] are represented with dashed red curves. (c) Total losses (k 0k 0∣)1/2 NT of DLBs in fused silica at 1550 nm as a function of their peak intensity (solid curves). The total losses of conical LBs with δ = 2 and 7 cm-1 are shown with dashed curves. (d) Shape of the DLBs of maximum intensity and losses (dashed red curves), and their asymptotic forms b 2/ρ 2/(2K-3) (dotted blue curves). The radial profiles are obtained by solving (2) and (3) in the dimensionless form f + 2f /ρ + ϕ 2 f + = 0, 8πγ0 ρ f 2K ρ 2 = 4πρ 2 φ f 2, with f = a/I 0 1/2, ρ = (2k 0 2 n 2 I 0/n 0)1/2 r, and γ = n 0 β (K) IK-2/(4n 2 k 0), and using the maximum values of γ leading to localized solutions, which are γK ≃ 0.88283,1.43838,1.83801,2.16571,2.45008,2.70474,2.93740, 3.15294 for K = 3,4,… 10, respectively.
Fig. 3. (a) Peak intensity (solid curve), energy (dashed blue curve) and losses (dotted green curve) along z in the same conditions as in Fig. 1 except the Gaussian width σ = 0.011 cm [Gaussian duration σ(k 0k 0∣)1/2 =44.5 fs] and the peak power P = 50P cr. (b-d) Radial intensity profile of the pulse with increasing propagation distance z (solid curves), of the DLB of maximum losses (dashed red curve) and of the DLB fitting the pulse (open circles). A vertical off-set is introduced for clarity.

Fig. 4. Left: For the same example as in Fig. 3, animation of the evolution along z of the radial intensity profile (solid curve) as it approaches and relaxes from the DLB of maximum losses (dashed red curve) (Media 1). Right: for the same example as in the left panel, visualization of the evolution as a trajectory of the pulse in the parameter space of DLBs and conical LBs in fused silica at 1550 nm. The inset shows a tiny region close to I 0,m.

Figure 4 offers an overall view of the pulse dynamics. The animation in the left panel (Media 1) shows how the pulse approaches the DLB attractor in the self-focusing stage, and decays from the attractor along the segment. In the right panel, the pulse dynamics is represented in the space of parameters of conical LB (I 0,δ) and of DLBs (I 0,δ= 0). The dashed area indicates the region of parameters where conical and DLBs do not exist. Self-focusing carries the pulse directly to the point (I 0,m,0) representing the DLB with maximum losses in the medium, or to points so close to it that they cannot be discerned at the scale of the figure or of the inset, remaining in this vicinity for about the first half of the collapse segment. Relaxation follows the indicated trajectory, where it is seen that the cone angle grows initially, but tends to zero at the end of the segment. The same dynamics explains also the “bursts” after the segments, if any, but since the remaining energy is considerably smaller, the DLB attractor is less approached and relaxation is faster.

We have shown, to conclude, that nonlinear compression and nonlinear absorption can balance each other in purely nonlinear and lossy stationary wave states localized in all dimensions, which are not solitary, conical or standard dissipative light bullets. There is a preferential lossy wave state defined by the nonlinearities of the medium and characterized by stationarity along with maximum losses, whose attractive property can explain the most relevant features of spatiotemporal collapse in nonlinear media arrested by nonlinear losses.

The author is indebted to Alberto Parola for comments, advices and computational support. This paper is dedicated to the memory of my father, J. J. Porras Écija.

References and links

1.

F. Wise and P. Di Trapani, “The hunt for light bullets—Spatiotemporal solitons,” Opt. Photon. News , 29–32 (2002).

2.

H. S. Eisenberg, R. Morandotti, and Y. Silberberg, “Kerr spatiotemporal self-Focusing in a planar glass waveguide,” Phys. Rev. Lett. 82, 043902 (2001). [CrossRef]

3.

R. McLeod, K. Wagner, and S. Blair, “(3+1)-dimensional soliton dragging logic,” Phys. Rev. A 52, 3254–3278 (1995). [CrossRef] [PubMed]

4.

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

5.

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

6.

G. Fibich and B. Ilan, “Optical light bullets in a pure Kerr medium,” Opt. Lett. 29, 887–889 (2004). [CrossRef] [PubMed]

7.

A. Braun, G. Korn, X. Liu, D. Du, J. Squier, and G. Mourou, “Self-channeling of high-peak-power femtosecond laser pulses in air,” Opt. Lett. 20, 73–75 (1995). [CrossRef] [PubMed]

8.

S. Henz and J. Herrmann, “Two-dimensional spatial optical solitons in bulk Kerr media stabilized by self-induced multiphoton ionization: Variational approach,” Phys. Rev. E 53, 4092–4097 (1996). [CrossRef]

9.

D. Moll and A. L. Gaeta, “Role of dispersion in multiple-collapse dynamics,” Opt. Lett. 29, 995–997 (2004). [CrossRef] [PubMed]

10.

L. Bergé and S. Skupin, “Self-channeling of ultrashort laser pulses in materials with anomalous dispersion,” Phys. Rev. E 71, 065601 (2005). [CrossRef]

11.

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91, 093904 (2003). [CrossRef] [PubMed]

12.

M. Kolesik, E. M. Wright, and J. V. Moloney, “Dynamic nonlinear X waves for femtosecond pulse propagation in water,” Phys. Rev. Lett. 92, 253901 (2004). [CrossRef] [PubMed]

13.

A. Dubietis, E. Gaizauskas, G. Tamosauskas, and P. Di Trapani, “Light Filaments without Self-Channeling,” Phys. Rev. Lett. 92, 253903 (2004). [CrossRef] [PubMed]

14.

M. Kolesik and J. V. Moloney, “Self-healing femtosecond light filaments,” Opt. Lett. 29, 590–592 (2004). [CrossRef] [PubMed]

15.

M. A. Porras, A. Parola, D. Faccio, A. Dubietis, and P. Di Trapani, “Nonlinear unbalanced Bessel beams: Stationary conical waves supported by nonlinear losses,” Phys. Rev. Lett. 93, 153902 (2004). [CrossRef] [PubMed]

16.

M. A. Porras, A. Parola, and P. Di Trapani, “Nonlinear unbalanced O waves: nonsolitary, conical light bullets in nonlinear dissipative media,” J. Opt. Soc. Am. B 22, 1406–1413 (2005). [CrossRef]

17.

M. A. Porras and A. Parola, “Nonlinear unbalanced Bessel beams in the collapse of Gaussian beams arrested by nonlinear losses,” Opt. Lett. 33, 1738–1740 (2008). [CrossRef] [PubMed]

18.

D. Faccio, M. Clerici, A. Averchi, O. Jedrkiewicz, S. Tzortzakis, D. Papazoglou, F. Bragheri, L. Tartara, A. Trita, S. Henin, I. Cristiani, A. Couairon, and P. Di Trapani, “Kerr-induced spontaneous Bessel beam formation in the regime of strong two-photon absorption,” Opt. Express 16, 8213–8218 (2008). [CrossRef] [PubMed]

19.

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

20.

P. Grelu, J. M. Soto-Crespo, and N. Akhmediev, “Light bullets and dynamic pattern formation in nonlinear dissipative systems,” Opt. Express 13, 9352–9360 (2005). [CrossRef] [PubMed]

21.

J. M. Soto-Crespo, P. Grelu, and N. Akhmediev, “Optical bullets and “rockets” in nonlinear dissipative systems and their transformations and interactions,” Opt. Express 14, 4013–4025 (2006). [CrossRef] [PubMed]

22.

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

23.

Y. Tanguy, T. Ackermann, W. J. Firth, and R. Jäger, “Realization of a semiconductor-based cavity soliton laser,” Phys. Rev. Lett. 100, 013907 (2008). [CrossRef] [PubMed]

24.

This peak power is slightly higher than the power Pcr = 2λ02/[4πn(ω0)n2] for spatial self-focusing.

OCIS Codes
(190.4180) Nonlinear optics : Multiphoton processes
(190.5940) Nonlinear optics : Self-action effects
(190.7110) Nonlinear optics : Ultrafast nonlinear optics
(260.5950) Physical optics : Self-focusing

ToC Category:
Nonlinear Optics

History
Original Manuscript: February 11, 2010
Revised Manuscript: March 18, 2010
Manuscript Accepted: March 18, 2010
Published: March 24, 2010

Citation
Miguel A. Porras, "A dissipative attractor in the spatiotemporal collapse of ultrashort light pulses," Opt. Express 18, 7376-7383 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-7-7376


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References

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  24. This peak power is slightly higher than the power Pcr = 2λ20/[4πn(ω0)n2] for spatial self-focusing.

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