Solitons are localized waves that propagate without decaying. They occur in a variety of physical systems, including liquids [1,2], optical fibers [3], plasmas [4], and condensed matter [5,6]. They generate significant interest fundamentally as well as for applications such as telecommunications. Initial research focused on the nonlinear localization of wave packets in one dimension, such as temporal solitons in optical fibers or spatial solitons in planar waveguides. One of the major goals in the field of nonlinear science is the generation of waves that are localized in all three dimensions of space as well as time [7]. Spatiotemporal solitons (STSs) have been investigated since the early days of nonlinear optics [8,9]. Aside from their intrinsic scientific interest, STSs have potential utility in ultrafast optical information processing [10].

Although soliton solutions of nonlinear wave equations may exist in higher dimensions, they are often unstable. There are only isolated reports of optical STSs: two-dimensional (2D) STSs have been observed in quadratic nonlinear media [11], and three-dimensional (3D) STSs were generated in an array of waveguides [12]. In each case the solitons were only stable for a few characteristic lengths, and there is still no report of 3D solitons in a homogeneous medium.

An alternative approach to STSs relies on dissipative processes. Systems that incorporate dissipation (in addition to diffractive, dispersive, and nonlinear phase modulations) have been shown theoretically to support stable solitons that are referred to as “dissipative optical bullets” [13–16]. While STSs in conservative systems require anomalous group-velocity dispersion, in dissipative systems stable STSs can also exist with normal dispersion. 2D spatial dissipative solitons in a cavity have been the subject of major interest recently [17]. One potential route to the experimental observation of a dissipative optical bullet would be an extension of this work to include localization in the time domain [17–19].

Lasers based on dielectric media doped with active ions may also be able to stabilize STSs. Temporal dissipative solitons have already been observed in both solid-state [20,21] and fiber [22] lasers. STSs would extend dissipative solitons to the two transverse dimensions. Theoretical work on models that are generally relevant to 3D dissipative systems strongly suggests that stable STSs are possible [13–16,18,19] and motivates further research to identify realistic experiments.

Here, we show through numerical modeling that stable STSs form in a solid-state laser with large self-focusing nonlinearity, significant normal group-velocity dispersion, and an unstable resonator (Fig. 1). To ensure that spatial localization is achieved primarily by optical nonlinearity, linear focusing elements must be removed from the cavity. In order to compensate for the diffraction in a realistic laser resonator, the total nonlinear phase shift must be significant, and this is accumulated in normal-dispersion material. Operation of the laser yields STSs that are stable over numerous propagation lengths. In addition to nearly static solutions, we observe qualitatively new phenomena, such as spontaneous spatiotemporal symmetry breaking and breathing limit cycles. As a secondary point, in the future this type of laser may offer performance advances over current designs [23].

 

Fig. 1. Illustration of an STS laser.

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A mode-locked laser must include a gain medium and a saturable absorber. The gain medium also provides refractive nonlinearity, group-velocity dispersion, diffraction, and filtering by the gain spectrum. We assume flat cavity mirrors. In any real system with discrete optical elements, the majority of diffraction will occur from propagation in air. Intensity-dependent loss from a saturable absorber is needed to start a pulsed laser from noise and to stabilize the steady-state solutions. Initial simulations show that ordinary gain media could provide adequate nonlinear refraction with microjoule pulses, in which case chirped mirrors could provide the needed normal dispersion. The simplicity of this approach is attractive, but it will require a kilowatt-class pump laser. To achieve a practical design based on 100 W pump lasers, we must add a material with high refractive nonlinearity and normal group-velocity dispersion to the cavity. The STS laser therefore consists of a gain medium, air, a nonlinear material, a saturable absorber, and a spectral filter. The filter helps compensate the nonlinear phase in time, as is done in temporal dissipative-soliton lasers [22]. A ring cavity is assumed.

Numerical simulations are employed to find stable solutions to this system. The gain is modeled by a simple transfer function,

The search for stable solutions begins with the choice of roughly equal diffraction, dispersion, and nonlinear lengths, and we have confined our search to physically realizable parameter values. For example, stable solutions are found with a standard Yb:KGW crystal as the gain medium, SF11 glass as the nonlinear material, and a semiconductor saturable absorber mirror (the parameters are listed in the caption of Fig. 2). It is important to emphasize that the laser only reaches threshold if STSs form. The intracavity pulse energy smoothly converges to 1.5 μJ [Fig. 2(a)]. This corresponds to a $3\mathrm{W}/{\mathrm{\mu m}}^2$ peak intensity, 4 ps pulse duration, 9 nm bandwidth, and 250 μm beam waist (Fig. 2). The pulse is symmetric in space and time and exhibits spatial narrowing at the center of the pulse, which illustrates the strong role of nonlinearity in the spatial domain [Fig. 2(b)]. At the center of the beam, the temporal frequency spectrum has steep sides [Fig. 2(c)] and the pulse is highly chirped [Fig. 2(d)], as is the case for dissipative solitons in normal-dispersion fiber lasers [22]. At the center of the pulse, the spatial frequency spectrum has multiple sidebands [Fig. 2(e)] and $>90\%$ of the beam is well fitted by a hyperbolic-secant profile [Fig. 2(f)], as is the case for solitons in a cavity. Nonlinear focusing is primarily responsible for compensating diffraction, with a small ($\sim 1\%$) contribution from gain guiding. To summarize, diffraction and nonlinear phase balance as in solitons of the 1D nonlinear Schrodinger equation, while dispersion, spectral filtering, saturable absorption, and nonlinear phase balance as in 1D dissipative solitons. The pulse (beam) actually breathes by 20% (50%) as the pulse traverses the laser; the solution shown in Fig. 2 is the pulse after the saturable absorber.

 

Fig. 2. Typical 2D simulation result: (a) energy convergence, (b) spatiotemporal color map of the pulse, (c) spectrum at the center of the beam, (d) temporal profile along with ${\mathrm{sech}}^2$ fit (dashed line), (e) spatial-frequency spectrum at the center of the pulse, and (f) spatial profile along with ${\mathrm{sech}}^2$ fit (dashed line). Parameters: the nonlinear material is 40 cm of SF11 glass ($n=1.76$, ${\beta }_2=1520{\mathrm{fs}}^2/\mathrm{cm}$ [Schott glass], and $n_2=41\times {10}^{-8}{\mathrm{\mu m}}^2/\mathrm{W}$ [24]). The gain bandwidth is 40 nm, $E_{\text{sat}}=600\mathrm{nJ}$, $w_p=308\mathrm{\mu m}$ (assuming 400 μm multimode pump fiber), there is 10 cm of air, the filter bandwidth is 12 nm, 2% power is coupled out, $l_o=0.03$, and $P_{\text{sat}}=51\mathrm{kW}/\mathrm{\mu m}$ (saturable absorber and output coupling parameters from Ref. [25]).

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With a slightly higher saturation energy of 650 nJ and all other parameters as above, a distinct change occurs in the solution. The pulse converges to a symmetric solution (middle panel of Fig. 3) similar to that in Fig. 2, but after several hundred round trips, this symmetry breaks and the solution finally converges to one which is asymmetric in space at every point in time. Seeding the simulation with the asymmetric pulse or its spatial mirror image shows that the asymmetric pulses are stable nodes of the system.

 

Fig. 3. Evolution of the pulse energy as a function of round trip around the oscillator and spatiotemporal profiles of the output at the indicated numbers of round trips.

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A further increase in the saturation energy, to 740 nJ, reveals a final solution, which is a long-period oscillation (Fig. 4 and Media 1. The symmetry, peak intensity, and spatiotemporal widths oscillate with a period of $\sim 14$ round trips (Fig. 4, left panel). This solution is a spatiotemporal limit cycle of the governing equation. Previously, oscillatory solutions were identified in a homogeneous model [18] and in the context of double bullet complexes [13,15].

 

Fig. 4. Evolution of the energy as a function of round trip for the spatiotemporal limit cycle, with representative spatiotemporal profiles of the output field at the half-periods of the oscillation (Media 1).

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To verify and refine the results quantitatively, 3D simulations were performed. With the same parameters as above and with a saturation energy of 450 nJ, a stable optical bullet converges (Fig. 5). Similar to the behavior in two dimensions (Fig. 3), symmetry breaking occurs after several hundred round trips [Fig. 5(a)], and the solution converges to a spatiotemporally asymmetric optical bullet [Fig. 5(b)]. The profiles in the $x$ and $y$ dimensions are nearly identical [Fig. 5(c)] and correspond to the symmetry of the spatial effects. The resulting three-dimensionally localized wave packet resembles a bean in an isointensity plot [Fig. 5(d)].

 

Fig. 5. Results of 3D simulations. (a) Evolution of the energy as a function of round trip, (b) spatiotemporal profile at $x=0$ or $y=0$, (c) beam at the center of the pulse, and (d) isointensity surface of the bullet at 5% of the peak intensity.

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As the nonlinear processes must create dissipative-soliton solutions that are stable against perturbations, a primary consideration in the design of an STS laser is the choice of nonlinear material. The refractive nonlinearity must balance the effects of diffraction. However, because the nonlinear and diffractive segments are in separate locations in the cavity, there is an upper limit to the nonlinear phase that can be fully compensated. Guided by the numerical results, we assume that the ideal accumulated nonlinear phase shift (and thus the diffractive phase) will be $\pi$. The nonlinear phase is determined by the intensity, nonlinear index, and material length. The intensity is determined by the pump power, the loss of the material (including effective loss in the cavity due to wavefront distortions), the nonlinear loss of the material, the loss of the cavity, the gain medium, and the length of the cavity. Many of these variables are coupled. The relevant equations can be readily solved. While it is difficult to achieve the needed combination of length and loss with ordinary glasses such as fused silica, the requirements become more reasonable as the nonlinear index increases.

The beam size and the length of the air section also have significant influence on the design. The nonlinear material determines the size of the beam in the cavity (that is, total diffraction is related to the length of propagation through the material and the amount of nonlinear phase available to compensate for this diffraction, as well as the size of the beam). To study STS formation, it will be advantageous to minimize the effect of any aperture in the cavity by reducing the beam size. Therefore, it will be preferable to use the shorter, high-index materials for the smaller beam sizes that result. The air section should be as short as practically possible to minimize diffraction and therefore reduce the requirements on the nonlinear material. With care this could probably be reduced, which in turn would reduce the length of the nonlinear material.

A final practical issue will be the stringent requirements on the homogeneity of materials that are placed inside a low-loss laser cavity. We constructed a femtosecond Yb:KGW laser [25] as a test bed for candidate nonlinear materials. The internal transmittance of 10 cm of SF11 glass is $>99\%$ at the operating wavelength. However, we find that insertion of 50 cm of SF11 in the cavity introduces an effective loss of $\sim 25\%$, which arises from wavefront distortion. It will be important to find the appropriate lengths of laser-quality nonlinear materials. It may be more practical, however, to compensate for imperfect materials by instead relying on higher pump powers. As an example, a short segment of chalcogenide glass with high pump power is both theoretically suitable and seems to be within current capabilities, albeit with the possible challenges of larger material absorption and photodarkening.

Alternative embodiments of the STS laser are worth noting. For example, the narrow gain bandwidth of Yb:YAG will eliminate the need for a filter, and the larger pump beam in a thin-disk geometry will avoid aperture effects. In such a system STSs could be stable with a dense flint material and several hundred watts of pump power. Ultimately, it should be possible to generate femtosecond pulses with microjoule energies, which will be attractive for some applications.

In conclusion, we have demonstrated the existence and stability of STSs in a realistic model of a solid-state laser. Key to the design is recognition of the role of normal dispersion in stabilizing the localized wave packet. Numerical simulations reveal unique features at high pulse energies, such as spontaneous spatiotemporal symmetry breaking and breathing limit cycles. Experimental realization of an STS laser will be challenging but seems to be technologically feasible.