1. Introduction

Optical nonlinearity sets fundamental limits on the peak power that an intense light field can accommodate without losing its connectedness in space and time through the exponential growth of modulation instabilities (MIs) across the beam and along the pulse. In the time domain, nonlinearity and dispersion make a high-peak-power light pulse unstable with respect to small modulations of its temporal envelope, which tend to split the pulse into multiple peaks and give rise to spectral sideband generation [1]. As their manifestation in spatial nonlinear dynamics, MIs cause a light beam with a peak power above the critical power of self-focusing to develop a complex transverse mode structure [2], eventually breaking up into multiple filaments [3,4].

In optical fibers, temporal MIs play a central role in parametric frequency conversion and soliton pulse transformation scenarios [1]. Since the peak powers needed for temporal MIs to take off in optical fibers are typically much lower than the critical power of self-focusing, manifested as nonlinearity-induced mode coupling, such MIs can often be treated separately from their spatial counterpart. Spatial MIs, on the other hand, are currently the focus of much interest in the context of multiple filamentation [3,4], which sets limits on the peak powers, and, hence, radiation energies, that can be transmitted or, with properly chosen parameters, compressed in a single filament, that is, without compromising the output beam quality. Although solutions for spatial MIs coupled to temporal MIs have been known for decades [5], spatial MIs of self-focusing beams and laser filaments are often examined separately from temporal MIs. This tendency owes much to the physically transparent picture of spatial MIs and simple, closed-form solutions for the MI gain provided in the seminal work by Bespalov and Talanov [2]. As filamentation-based approaches find growing applications in remote sensing [6] and high-power pulse compression [7], expanding into new wavelength ranges [8], helping generate high-power subcycle pulses [9], and offering new solutions for standoff detection using filamentation-driven lasing [10], a more detailed understanding of multiple filamentation regimes is needed [11] with a special emphasis on effects that go beyond the standard treatment of MIs, based on the slowly varying envelope approximation (SVEA).

Here, we show that useful insights into the properties of spatiotemporal MIs can be gained from the analysis of noncollinear four-wave mixing (FWM) leading to a coupled parametric amplification of MIs in space and time. We demonstrate that, with the MI gain and sideband frequencies expressed through the wave-vector mismatch for such an FWM process, the main results of the canonical SVEA treatment of spatiotemporal MIs can be fully recovered. Moreover, such analysis will be shown to offer important physical insights into non-SVEA effects in spatiotemporal MIs.

2. Linear stability analysis of modulation instabilities

Within the framework of the canonical linear-stability analysis [1,12], the gain and sideband frequencies of spatiotemporal MIs are derived by considering a weak plane-wave perturbation [5]

The perturbed field can then be represented as

Substituting Eq. (3) into Eq. (2) and linearing in $\tilde{\epsilon }$, we find

Linearized equations for small perturbation amplitudes ε₁ and ε₂ lead to the dispersion relation $4k_0^2{K}_z^2=\left(K_{\perp }^2-k_0{k}_2{\Omega }^2\right)\left(K_{\perp }^2-k_0{k}_2{\Omega }^2-4k_0\gamma I_0\right)$, where $K_{\perp }={\left(K_x^2+K_y^2\right)}^{1/2}$ and K_z are the transverse and longitudinal components of K = {K_x, K_y, K_z}. When K_z has a negative imaginary part, the steady-state solution to Eq. (2) becomes unstable with respect to plane-wave perturbations (1). The gain of this instability is found from the imaginary part of K_z as [5]

As can be seen from Eq. (3), in a striking contrast to purely temporal MIs [1], spatiotemporal MIs are not restricted to the region of anomalous dispersion, but exist for group-velocity dispersion k₂ and Kerr nonlinearity n₂ of any sign except when both k₂ and n₂ are negative. When the spatial part of MI is suppressed in Eq. (5), K_⏊ = 0, the properties of purely temporal MIs are recovered. In this regime, in media with n₂ > 0, such as standard materials for optical, MIs can only occur when dispersion is anomalous (k₂ < 0), with the maximum gain achieved with ${\Omega }_0^2=-2\gamma I_0/k_2$ [1, 5, 12].

3. Modulation instabilities as parametric amplification via four-wave mixing

We now examine an alternative perspective of spatiotemporal MIs by considering a noncollinear FWM 2ω₀ = (ω₀ + Ω) + (ω₀ − Ω), which couples the pump field with frequency ω₀ and wave vector k_p to its Stokes and anti-Stokes sidebands with frequencies ω₀ ± Ω and wave vectors k _± that are allowed to be generated in the off-axial direction. With k_p = k₀ + γI₀ and $k_{\pm }\approx k_0\pm \Omega /u+k_2{\Omega }^2/2+2\gamma I_0$, we find that the wave-vector mismatch for this FWM process can be found as $\Delta k=\left|k_1+k_2-2k_p\right|$, where $k_{1,2}=k_{\pm }{\left(1-K_{\perp }^2/k_{\pm }^2\right)}^{1/2}$. With $k_0>>\left|\Omega /u\right|,\left|k_2\right|{\Omega }^2,\left|\gamma \right|I,K_{\perp }$, we find $\Delta k\approx k_2{\Omega }^2+2\gamma I_0-K_{\perp }^2/k_0=$$=\Delta k_0+2\gamma I_0$, where Δk₀ is the intensity-independent part of Δk, determined only by dispersion and wave-mixing geometry.

With K_⏊ = 0 for on-axis MIs, phase matching Δk = 0 is then achieved at ${\Omega }_0^2=-2\gamma I_0/k_2$, which exactly recovers the SVEA expression for the sideband frequencies in purely temporal MI [1]. In this regime phase matching and, hence, sideband generation is possible only when the nonlinearity n₂ and the second-order dispersion coefficient k₂ have opposite signs. In most of the nonlinear materials, including silica, the material most commonly used in optical fibers, n₂ > 0, and the ω₀ ± Ω sidebands are generated only in the region of anomalous dispersion.

With second-order dispersion effects included in Δk, the buildup of the Stokes and anti-Stokes sidebands occurring as a part of FWM parametric amplification can be described [13] with coupled SVEA equations for the sideband amplitudes A _± at ω₀ ± Ω: $d{A}_{-}/dz=i\gamma \left[2I_0{A}_{-}+I_0{A}_{+}^{\ast }\exp \left(-iqz\right)\right]$ and $d{A}_{+}^{\ast }/dz=-i\gamma \left[2I_0{A}_{+}^{\ast }+I_0{A}_{-}\exp \left(-iqz\right)\right]$, where $q=\Delta k_0-3\gamma I_0$. The solution to these equations is written as [13]

Representing the MI gain Γ given by Eq. (5) as $\Gamma \left(\Omega ,K_{\perp }\right)={\left(\gamma I_0-\Delta k/2\right)}^{1/2}{\left(\gamma I_0+\Delta k/2\right)}^{1/2}$, we find that g = Im(Γ). Thus, the gain of ω₀ ± Ω sidebands in FWM parametric amplification is equal to the gain predicted by the field evolution equation [Eq. (2)] for the instabilities of the form of Eq. (1). As a straightforward consequence, all the properties of the ω₀ ± Ω sidebands in FWM parametric amplification, including their central frequencies, gain bands, and directions of maximum growth rates, are identical to the properties of the spatiotemporal MIs [Eq. (1)] of the steady-state solutions of Eq. (2). Specifically, the gain of spatiotemporal MIs peaks at Δk = 0. Physically, this implies that the maximum MI gain is achieved when the nonlinear phase shifts compensates for the material dispersion, Δk₀ = 2γI₀,. For anomalously dispersive self-focusing materials (k₂ < 0, n₂ > 0), spatiotemporal MIs are confined to a region within an ellipse ${\Omega }^2+{\Omega }_s^2<{\Omega }_c^2$ in the $\Omega {\Omega }_s^{}$ plane, ${\Omega }_s={\left(k_0\left|k_2\right|\right)}^{-1/2}{K}_{\perp }$and ${\Omega }_c=2{\left(\left|\gamma \right|I_0/\left|k_2\right|\right)}^{1/2}$, with the maximum MI gain achieved along the ellipse ${\Omega }^2+{\Omega }_s^2={\Omega }_c^2/2$. In the normal dispersion region, spatiotemporal MIs exist when the conditions ${\Omega }_s^2-{\Omega }_c^2\le {\Omega }^2\le {\Omega }_s^2$ and ${\Omega }_s^2\le {\Omega }^2\le {\Omega }_s^2+{\Omega }_c^2$ are satisfied for materials with positive and negative n₂, respectively [5].

The physics behind the equivalence of ω₀ ± Ω sidebands in FWM parametric amplification and spatiotemporal MIs of the form of Eq. (1) is, however, anything but clear. Indeed, there is a clear gap between the canonical treatment of FWM parametric amplification based on coupled equations for A _± and the standard linear stability analysis of MIs, which treats MIs as an amplification of weak, yet complex harmonic perturbations [Eq. (1)] of the steady-state solution of Eq. (2).

4. Physical equivalence between two pictures of modulation instabilities

To close this gap and to shed light on the physics behind the equivalence of ω₀ ± Ω sidebands in FWM parametric amplification and complex spatiotemporal MIs of the field evolution equation, we represent the perturbed solution to the generic nonlinear field evolution equation as a sum of a steady-state, background field A and a small perturbation [cf. Equation (3)]: $\tilde{A}=A+4^{-1}{\pi }^{-2}{\displaystyle \int \epsilon \left(K,z\right)}\exp \left(-iKr\right)d^2{K}_{\perp }$. With the perturbation taken in the form of a complex harmonic field, $\epsilon \left(K,z\right)=u_K\left(z\right)+i{v}_K\left(z\right)$, exactly as prescribed by Eq. (1), the field-evolution equation yields a typical coupled-wave solution [14]: $u_K\left(z\right)=\cos h\left(\kappa z\right)u_K\left(0\right)+k_0/\left(4\kappa \right)\sin h\left(\kappa z\right)v_K\left(0\right)$, $v_K\left(z\right)=4\kappa /k_0\sin h\left(\kappa z\right)u_K\left(0\right)+\cos h\left(\kappa z\right)v_K\left(0\right)$, where $\kappa =\left(K_{\perp }/2\right){\left(2\gamma I_0-K_{\perp }^2/k_0\right)}^{1/2}$. Such a solution is typical of a vast variety of wave-mixing processes [13, 15], where the fields are coupled by the nonlinear polarization induced in a medium by an intense pump field and where properly phase-matched weak fields can experience amplification, gaining energy from the pump.

To connect these solutions to the solutions for parametrically amplified ω₀ ± Ω sidebands in FWM, as described by Eqs. (6) and (7), we represent the evolution of u_K(z) and v_K(z) in the matrix form [14], $w_K\left(z\right)=T_K\left(z\right)w_K\left(0\right)$, where $w_K=\left(u_K,v_K\right)$ is a two-component vector and T(z) is a 2x2 matrix whose four components are defined by the four coefficients in front of u_K(0) and v_K(0) in the coupled-wave solution governing the evolution of u_K(z) and v_K(z). Expanding w(z) in the eigenvectors of T(z), $w_K^{\pm }={\left[1+{\left(4\kappa /k_0\right)}^2\right]}^{-1/2}\left(1,\pm 4\kappa /k_0\right)$, and expressing the result in terms of the respective eigenvalues, ${\theta }^{\pm }=\exp \left(\pm \kappa z\right)$, we find $w_K\left(z\right)=b_K^{+}\left(0\right)\exp \left(\kappa z\right)w_K^{+}+b_K^{-}\left(0\right)\exp \left(-\kappa z\right)w_K^{-}$, where $b_K^{\pm }\left(0\right)$ are defined by the initial conditions (e.g., the input noise).

It is straightforward to see from these equations that a weak complex harmonic perturbation [Eq. (1)] gives rise to two waves, $w_K^{\pm }$, of which one is exponentially growing and the other one is exponentially decreasing. This is exactly what the two parametrically amplified waves $A_1^{\pm }$ and $A_2^{\pm }$ in Eqs. (6) and (7) do. Thus, although the complex harmonic wave of Eq. (1) appears in the standard MI treatment as a purely nominal trial function, which helps isolate MIs in space and time, allowing their dispersion relation to be defined through the pertinent characteristic equation for K_z, such a two-wave field structure is, in fact, more than a purely formal substitution. Rather, it is inherent in secondary waves generated in a broad class of FWM processes as a part of nonlinear spatiotemporal field evolution. Since this two-wave structure of MI-type fields arising as a result of FWM processes reflects the universal properties of nonlinear field evolution, it is manifested in MIs of different nature and is independent of beam geometry.

5. High-order dispersion in spatiotemporal modulation instabilities

We are going to show now that the parametric amplification perspective of spatiotemporal MIs offers a convenient framework to understand the role of high-order dispersion in spatiotemporal MIs and suggests a physically transparent method to calculate corrections to the MI gain due to these effects. Such an extension of MI analysis beyond the standard SVEA procedures would offer valuable physical insights into spatial instabilities of few-cycle and subcycle field waveforms [16] and help understand multiple filamentation of high-power ultrashort laser pulses [3, 4].

Effects related to high-order dispersion can be examined by using Eq. (8) for the parametric gain g with Δk including dispersion terms beyond k₂, $\Delta k\approx 2{\displaystyle {\sum }_{m=2j}\left(k_m/m!\right){\left(\omega -{\omega }_0\right)}^m}+2\gamma I_0-K_{\perp }^2/k_0$, j = 1, 2, …. The MI gain is thus given by

As can be seen from this expression, only even-order dispersion terms contribute to spatiotemporal MIs. In a particular case of purely temporal MIs, Eq. (9) with K_⏊ = 0 recovers the results of standard linear stability analysis of Eq. (2) without the diffraction term using the complex harmonic field ansatz of Eq. (1) [17–19], which shows that the third-order dispersion term does not appear in the gain of purely temporal MI. Equation (9) extends the analysis to MIs with K_⏊ ≠ 0 and allows effects related to the dispersion of any order to be quantified.

It is straightforward to see from Eq. (9) that, with higher order dispersion included, the maximum gain is still g₀ = γI₀, and it is still achieved when phase matching is satisfied, Δk = 0. However, as can also be seen from Eq. (9), high-order dispersion gives rise to new effects in spatiotemporal MIs, never observed when only the k₂ term is included. In Figs. 1(a)–1(l), these new effects are illustrated for the parameters of dispersion and nonlinearity similar to those of YAG (n₀ ≈1.76, n₂ ≈4 10⁻¹⁶ cm²/W, zero group-velocity dispersion (GVD) wavelength is λ_z ≈1605 nm) – material that enables efficient filamentation-assisted compression of high-peak-power laser pulses [20]. Deep in the anomalous dispersion region [λ₀ = 2π/ω₀ = 1.8 μm, Figs. 1(j)–1(l)], the second-order dispersion plays the dominant role, with higher order dispersion only slightly distorting the MI gain bands, giving rise to small corrections to the ${\Omega }^2+{\Omega }_s^2={\Omega }_c^2/2$ phase-matching condition, needed for the maximum MI gain.

 

Fig. 1 The MI gain g(Ω, K_⏊) for dispersion and nonlinearity of YAG for a pump field with I₀ = 0.5 TW/cm² and λ₀ = 1200 nm (a, b, c), 1400 nm (d, e, f), 1600 nm (g, h, i), 1800 nm (j, k, l). Dispersion is included up to the second (a, d, g, j), fourth (b, e, h, k), and twelfth (c, f, i, l) order.

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The scenery, however, changes drastically near λ_z, where |k₂| is small. Here, the k₂-only approximation fails both to the left and to the right of λ_z,, while Eq. (9) predicts smooth and continuous transformation of elliptical MI gain bands, typical of the k₂ < 0 regime, toward more complicated profiles, with the curvature of these bands first decreasing and then reversing its sign around Ω ≈0 [λ₀ = 1.6 μm, Figs. 1(g)–1(i)]. Deeper into the normal dispersion region [1.2 to 1.4 μm, Figs. 1(a)–1(f)], high-order dispersion effects are striking, as Eq. (9) reveals the existence of new MI gain bands, in addition to the signature hyperbolic MI gain bands [5], predicted by the k₂-only approximation. As a result, within a finite range of K_⏊, two pairs of MI sidebands experience an exponential buildup through phase-matched FWM parametric amplification.

To gain deeper insights into new MI features induced by high-order dispersion, we focus now on a practically significant case when all the k_2j effects with j ≥ 3 are negligible, that is, dispersion up to the fourth-order needs to be included. In this case, the Δk = 0 phase-matching condition allows instructive analytical solutions for the MI sideband frequencies. In particular, near the zero-GVD wavelength, where the k₂ term in Δk is small, phase matching, Δk = 0, is achieved at ${{\Omega }^{\prime }}_0^2\approx {\left[12\left(K_{\perp }^2/k_0-2\gamma I_0\right)/k_4\right]}^{1/2}-6k_2/k_4$. As can be seen from Figs. 1(b), 1(e), 1(h), 1(k), this approximation is reasonably accurate in reproducing all the main tendencies in the behavior of the MI gain. Keeping only the leading term in this expression and setting K_⏊ = 0, we recover the results of the earlier studies for purely temporal MIs [17, 19], ${{\Omega }^{\prime }}_0^{}\approx \pm {\left(24\gamma I_0/\left|k_4\right|\right)}^{1/4}$. MIs of this type can only occur when k₄ is negative. With the spatial part of MI included, however, the MIs, due to the K_⏊ term in Δk, are no longer limited to the region of negative k₄ [Figs. 1(b), 1(e), 1(h), 1(k)].

In the general case, when both k₂ and k₄ effects are nonnegligible, the fourth-order dispersion may dramatically change the entire picture of spatiotemporal MI [Figs. 1(b), 1(h)]. As perhaps its most striking MI manifestation, the k₄ term in Eq. (9) can give rise to the second MI gain band [Fig. 1(b)]. Indeed, solving the equation $\Delta k=2\gamma I_0+k_2{\Omega }_{}^2+k_4{\Omega }_{}^4/12-K_{\perp }^2/k_0=0$, we find

When the k₄ term is small, Eq. (10) gives two solutions, ${\Omega }_{-}^2\approx \left(K_{\perp }^2/k_0-2\gamma I_0\right)/k_2$and ${\Omega }_{+}^2\approx -12k_2/k_4-{\Omega }_{-}^2$, corresponding to two pairs of MI sidebands. This result is fully consistent with MI gain calculations with all the dispersion orders included [Figs. 1(c), 1(f)], providing important insights into the physics behind this effect. It is straightforward to see now that the first of these sideband pairs represents the ordinary MI gain band of k₂-only MI analysis. This band has a shape of an ellipse for k₂ < 0 [Figs. 1(j), 1(k)] and a hyperbola for k₂ > 0 [Figs. 1(a), 1(b)].

As long as k₄ is small, the second MI band never shows up, lost at Ω → ± ∞. With k₂ < 0 and n₂ > 0, spatiotemporal MIs in this regime can only exist within the region ${\Omega }^2\left[1-k_4{\Omega }^2/\left(12\left|k_2\right|\right)\right]+{\Omega }_s^2<{\Omega }_c^2$, whose boundary is slightly distorted relative to the ellipse ${\Omega }^2+{\Omega }_s^2={\Omega }_c^2$, which bounds the region of ordinary, k₂-only MIs. However, when the k₄ term in Δk becomes comparable to the k₂ term, it can manifest itself in a very prominent way, dramatically distorting MI gain bands and giving rise to the second pair of MI sidebands [Fig. 1(b)], in agreement with the full MI gain analysis including all the dispersion orders [Fig. 1(c)]. Clearly, the approximation that neglects all the k_2j effects for j ≥ 3 fails within a narrow region near λ_z where k₂ and k₄ have the same sign [Figs. 1(d)–1(f)]. Higher order dispersion effects have to be included for an accurate description of spatiotemporal MIs in this regime.

As shown in the earlier work [21], transverse MIs in counterpropagating optical waves can be understood in terms of FWM processes that couple counterpropagating pump waves and their Kerr-nonlinearity-induced sidebands. The case of transverse MIs in counterpropagating waves is, however, drastically different from single-beam spatiotemporal MIs considered in this work in its physics and mathematical framework, as well as in its place in the context of ultrafast optical science and its current challenges. Physically, transverse MIs in counterpropagating waves are induced through the FWM of sidebands generated by the counterpropagating fields due to the Kerr nonlinearity. The sidebands induced as a part of single-beam spatiotemporal MIs considered in this work are all generated in the direction of the only input laser field involved in the problem. The generation of these sidebands is not possible without group-velocity dispersion of the medium. In a striking contrast, the GVD does not even enter the analysis of transverse MIs in counterpropagating waves. Mathematically, analysis of transverse MIs in counterpropagating waves is based on coupled-wave equations for the slowly varying envelopes of the counterpropagating fields that do not include the higher-order time derivative terms [21]. For single-beam spatiotemporal MIs examined in this work, these terms are, on the contrary, are of crucial significance. As a result, the question of how important high-order dispersion effects are, which is one of the key questions of concern in ultrafast photonics of high-peak-power ultrafast optical waveforms (see, e.g., [3, 4, 10, 11, 20]), as well as in this study, is not even relevant in the context of transverse MIs in counterpropagating waves. Finally, the role that the MIs of these two types play in the context of optical science and technologies is very different. While transverse MIs in counterpropagating waves play an important role in phase-conjugation applications, single-beam spatiotemporal MIs considered in this work is one of the key factors in a predominantly unidirectional evolution of high-peak-power ultrashort laser pulses. Laser-induced filamentation, where single-beam spatiotemporal MIs examined in this work lead to beam breakup into multiple filaments (e.g., [3, 4, 10, 11, 20]), is, perhaps, one of the most prominent examples, explaining the motivation behind this study.

6. Conclusion

To summarize, the canonical linear-stability analysis of MIs, which treats MIs as instabilities of the steady-state solution of the field evolution equation with respect to weak complex harmonic perturbations has been shown to be physically equivalent to a coupled-wave analysis of off-axis parametric amplification of the Stokes and anti-Stokes fields by an intense pump wave. The physics behind this equivalence has been revealed via a space-evolution transfer matrix, which transforms a complex harmonic trial function used in linear-stability analysis of MIs into a pair of coupled off-axis waves, thus fully recovering a two-wave field structure of the waves undergoing parametric amplification as a part of FWM. This perspective of spatiotemporal MIs has been shown to offer a convenient framework to understand the role of high-order dispersion in spatiotemporal MIs and to suggest a physically transparent method to calculate corrections to the MI gain due to high-order dispersion in a broad class of problems, including multiple filamentation of high-power ultrashort laser pulses. Within a vast parameter space, high-order dispersion has been shown to drastically change the entire picture of spatiotemporal MIs, substantially modifying MI gain bands and giving rise to new pairs of exponentially growing MI sidebands.