1. Introduction

Spatiotemporal optical solitons are among the most intriguing and challenging entities in solitons science [1,2]. While temporal [3] and [4] 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]. Experimentally, however, only 2D spatiotemporal solitons have been demonstrated [6,7]. The experimental realization of 3D spatiotemporal solitons is still considered a ‘grand challenge’ in nonlinear optics [1,2].

Numerous approaches for obtaining light bullets have been considered theoretically, e.g. by using saturation nonlinearity [8,9], nonlocal nonlinearity [10], tandem structures that are composed of linear and nonlinear materials [11], and through manipulating the diffraction and/or dispersion by periodic structures [12–15]. 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]. 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], self-focusing via electrostatic interaction between elongated molecules in liquid crystals [21,22], and thermal nonlinearity [23–25]. Furthermore, slow self-focusing was utilized for discovering new phenomena in solitons science, including incoherent solitons [18–20], incoherent modulation instability [20,26], and random-phase lattice solitons [27]. 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]. 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]. 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]. 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] and dark [31,32] 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 Ψ_train = ∑^∞ _q=-∞ Ψ_q (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:

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

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

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. Ψ_q+1 = Ψ_q , hence we solve Eq. (3) and get Δn_slow = ∫^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 [21,22] and thermal nonlinearity in glass [23–25]. We model the highly nonlocal effect by a spatial convolution between the averaged intensity and a Gaussian response function [34]

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],

Δn^slow = Δn ₀ Pexp [-(x ²+ y ²)/σ ²] ≈ Δn ₀ P[1-(x ²+ y ²)/σ ²], where the averaged power is $P=\frac{1}{T}\underset{-T/2}{\overset{T/2}{\int }}\underset{-\infty }{\overset{\infty }{\int }}{\mid \Psi \mid }^2\mathrm{dxdydt}$ 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]. 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]

where Ψ_B corresponds to temporally-bright spatiotemporal pulse-train solitons that exist in a medium with anomalous group velocity dispersion (β₂<0) and Ψ_D 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 Δn^slow [28], leads to

The condition for self-trapping in the temporal domain, i.e. that the temporal components of Ψ_B and Ψ_D 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]

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

In the previous section, a model for spatiotemporal pulse-train solitons was developed and solved for nonlinear media such as thermal glass [23–25], and liquid crystals [21,22] that exhibit both slow and highly-nonlocal self-focusing nonlinearity as well as optical Kerr nonlinearity. In this section, we demonstrate numerically such solitons and verify their stability. As a concrete and realistic example, we assume that the nonlinear medium is a lead glass sample which was used in Refs. 24, 25 36, and 37. Lead glass exhibits slow and highly nonlocal thermal self-focusing [24,25], as well as the universal optical Kerr nonlinearity. We use the following laser parameters: λ=488 nm, w=21.5 μm, I₀=1.24 GW/cm², τ=164.5 fs, P=29 W, and Δt=9.9 ps. The following parameters correspond to lead glass SF11 from Schott at λ=488 nm: β₂=48.11×l0^-26 sec²/m, n₀=1.806 and n₂=2.23×l0^-19 m²/W. The nonlocal parameter in our simulations is σ=215 μm. The index change parameter is Δn₀=5×l0^-5W^-1 as measured in Ref. 24. Under these conditions, the diffraction length is Z₀=5.36 mm and the dispersion length is L_D=T₀ ²/∣β₂∣=56.3 mm.

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 Ψ_B (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 [37,38]. 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.

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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 Ψ_D (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.

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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.