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, 2]. 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 .
From Silberberg’s work,  much attention has been paid to different mechanisms that can compensate for spatiotemporal self-focusing of ultrashort pulses due to Kerr nonlinearity, resulting in the spontaneous formation of LBs, and to ascertain which of these mechanisms are at work in actual experiments, or could be reliably practiced in future experiments. Until recently, research in this connection has been focused on “phase”, or conservative, mechanisms as saturation of the Kerr nonlinearity,  or chromatic dispersion,  and on the associated LBs of the solitary type, or spatiotemporal solitons. In filamentation experiments, the phase mechanism is widely accepted to be the defocusing effect of the lowering of the refractive index due to the light-induced plasma [7, 8]. This interpretation holds even if self-compression is not directly halted by this defocusing effect, but by the nonlinear wave losses required for ionization, [2,7–10] since pure loss of energy is not considered as a candidate to balance nonlinear compression, and thus to form LBs. In fact, losses eventually destroy any LB, when understood as of solitary type.
Nonlinear losses became an active player in the alternate description of light filaments as conical waves [11, 12]. Motivated by the observed self-healing and self-reconstruction properties of light filaments, [13, 14] conical LBs that are resistant to nonlinear losses [15, 16] 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 . Supporting its relevant role in light self-channelling, conical LBs (in two spatial dimensions) have been described  and observed  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, 18].
The agreement of our simple model with experiments  and “exact” numerical simulations  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, 10].
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,  or with the dissipative light-bullets in three dimensions described recently in Refs.  and , 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 . 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 .
The simplest model accounting for self-focusing with nonlinear losses is
or nonlinear Schrödinger equation (NLSE) for the envelope A of a wave packet E = A exp(-iω 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/dωn∣ω0), 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 0∣k ″ 0∣)1/2, will retain this property on propagation, and can self-focus symmetrically in space and time . 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 . 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 2/λ0 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, 10]. 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 ) 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.
To understand this behavior we consider stationary solutions of the NLSE Eq. (1) of the form A(r,z) = a(r)exp[iφ(r)]exp(-iδz), where the real amplitude a(r) > 0 and phase φ(r) must satisfy
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 0∣k ″ 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 . Their amplitude decays as 1/r, and then they carry infinite energy. In their two-dimensional versions (nonlinear Bessel beams ) an inward energy flux along cones of half-apex angle supplies the energy absorbed in any disk of radius r during propagation. In three dimensions, the energy flows along hyper-cones of angle , 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, 16].
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
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
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)/(3-2Kα)r 1-2α(K-1), and Eq. (2) with δ = 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 - 2Kα)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.
There is remarkable coincidence between the intensity of the long-lived quasi-LBs, or filament segments, in self-focusing experiments with anomalous dispersion, and the intensity I 0,m, of the DLB with maximum losses. This coincidence holds irrespective of the self-focusing pulse, e. g., of its initial power [Figs. 1 from (a) to (c)] and of its width (Figs. 1 and 3). Also, the intensity remains nearly constant in spite that the energy decreases drastically along the segment [Fig. 3(a), dashed blue curve] due to an energy loss per unit length (dotted green curve) comparable to the total energy. The formation of these quasi-LBs upon spatiotemporal self-focusing can be explained from a spontaneous balance of Kerr compression and nonlinear losses to form a DLB. As suggested by previous works,  this equilibrium is more stably reached in the DLB of maximum losses allowed by the medium, which acts as an effective attractor in the self-focusing dynamics. In fact, in the self-focusing stage [Fig. 3(b)], the radial profile of the pulse (solid curves) approaches the radial profile of the DLB of maximum losses (dashed red curve). Once the intensity is stabilized about I 0,m [Fig. 3(c)], the inner part of the radial profile (solid curves) matches that of the attractive DLB (dashed red curve). However, the finite energy of the pulse prevents the pulse from reaching completely the DLB attractor with infinite loses. As z increases, the slowly evolving radial profile of the pulse fits up to a larger radial distance r to the profiles of DLBs with slightly changing intensities extremely close to I 0,m [Fig. 3(c), open circles] and finite losses. At longer propagation distances, the increasingly lack of energy forces the pulse to decay into less lossy conical LBs. The pulse radial profiles [Fig. 3(d) solid curves] at different values of z are seen to match now the radial profiles of conical LBs having same peak intensity and total losses as the pulse (open circles). Contrary to what is stated in Ref.  for the two-dimensional case, the cone angles in the relaxation stage are small but not completely negligible.
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.
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24. This peak power is slightly higher than the power Pcr = 2λ02/[4πn(ω0)n2] for spatial self-focusing.