1. Introduction

Spectral broadening of ultrashort pulses is one of the central effects in nonlinear optics [1]. While in linear optics, spectral broadening is forbidden by the second law of thermodynamics, nonlinear-optical systems and materials offer a unique resource for a spectral transformation of ultrashort optical pulses [2], enabling numerous pulse-compression schemes [3], supercontinuum-seeded optical parametric amplification [4], supercontinuum-assisted spectroscopy [5] and microscopy [6], as well as multioctave supercontinuum generation for ultrafast photonics [7, 8] and frequency-comb technologies [9].

Self-phase modulation (SPM) is known to dominate early stages of spectral broadening, seeding cascades of parametric processes, kicking off soliton dynamics, and providing a spectral content for subsequent pulse self-compression in the regime of anomalous dispersion [7, 8]. The basic theory of SPM [1, 2] neglects dispersion effects to derive an analytical result for the Kerr-effect-induced phase shift that is linear in the peak power, optical nonlinearity, and the propagation distance. This approximation provides a powerful tool for the analysis of both desirable and unwanted nonlinear phase shifts in a vast class of laser systems and parametric converters [3], helping to tailor the short-pulse output of such systems [10].

However, as the pulse propagates further on down an extended nonlinear medium, e.g., an optical fiber, dispersion effects inevitably come into the picture. Acting jointly with nonlinearity, these effects can give rise to a broad diversity of soliton phenomena [1, 11–13], enabling efficient supercontinuum generation [7, 8] and pulse self-compression [2], including compression to single-cycle and even subcycle pulse widths [14]. With dispersion completely out of picture in the canonical basic SPM theory, one faces an uncomfortable gap between the SPM and soliton theory accounts of pulse evolution. Indeed, in basic SPM theory, the nonlinear phase shift and, hence, the pulse bandwidth display an unbounded growth, increasing with the propagation path. In soliton dynamics, on the other hand, the spectral bandwidth of an optical pulse can become a nonmonotonous function of the propagation length, often undergoing signature, breather oscillations [2, 11], in which an explosion-like bandwidth growth is followed by a stage of spectral narrowing, translating into pulse-self-compression − pulse-stretching cycles in the time domain [15].

The main goal of this work is to close this gap by providing a closed-form analytical description of the early decisive stages of spectral broadening of an ultrashort laser pulse that would allow the basic SPM theory to be continuously extended to the soliton-number-based description of nonlinear pulse evolution. This analytical description will be shown to shed light on how dispersion effects enter the picture of pulse evolution, leading to a spatial-scale renormalization, decelerating the spectral broadening in the regime of normal dispersion, and giving rise to an explosion-like bandwidth growth in anomalous-dispersion high-soliton-number pulse evolution scenarios. Based on this formalism, we will provide an analytical derivation for the relation between the maximum soliton self-compression length and the soliton number, which has been previously treated as purely empirical.

2. Self-phase modulation: basic theory

As a starting point, we consider the nonlinear Schrödinger equation (NSE) [1, 2]

With the dispersion term dropped, Eq. (1) is reduced to an elementary SPM-theory equation, with a solution given by [1, 2]

The η-dependent nonlinear phase shift [Eq. (3)] gives rise to a deviation of the instantaneous frequency from the central frequency of the input field, $\delta \omega \left(z,\eta \right)=\partial {\varphi }_{SPM}\left(z,\eta \right)/\partial \eta$, leading to a standard order-of-magnitude estimate for the SPM-induced spectral broadening [1, 2]

3. Soliton dynamics

The full NSE, with both nonlinearity and dispersion terms included in Eq. (1), allows an infinite number of soliton solutions [2, 11–13]. Within a vast parameter space and across a broad diversity of soliton phenomena, the soliton dynamics is controlled by the soliton number N, defined as N = (l_d/l_nl), where $l_d={\tau }_0^2/\left|{\beta }_2\right|$ is the dispersion length and l_nl = 1/(γP₀) is the nonlinear length. However, only a few of these solutions can be written in a closed analytical form. Among those, the fundamental, N = 1, soliton solution [1, 2, 11], $u\left(\xi ,\theta \right)=\sec \text{h}\left(\theta \right)\exp \left(i\xi /2\right)$, with θ = t/τ₀ and ξ = z/l_d, plays a very special role, as such a soliton pulse propagates in a lossless medium without changing its envelope and dominates over other features in spectral and temporal dynamics as long as N < 2 [2, 11].

The N = 2 soliton is a rare example of a closed-form analytical solution for an N > 1 soliton (see, e.g., Refs. 2, 11). This solution exhibits a periodic behavior of |u|² as a function of ξ. This oscillatory behavior, observed both in the spectral (Fig. 1) and temporal (Fig. 2) domain, is one of the key properties of higher order optical solitons. However, except for a few rare examples, the dynamics of N > 1 solitons has to be studied with numerical methods. As an example of such numerical analysis, Figs. 1(a) and 1(b) present the maps of temporal evolution of NSE solitons with N = 5 and N = 7, respectively. At the initial stage of their oscillatory dynamics, the solitons undergo pulse self-compression, reaching a minimum pulse width τ_m at ξ = ξ_m. This self-compression stage is followed by pulse stretching, revealing a multisoliton structure of the waveform [Figs. 1(a), 1(b)]. For ideal NSE solitons, such cycles recur with an ideal periodicity and a period ξ_s = π/2, or, with the physical units of length restored, z_s = πl_d/2. High-order dispersion leads to a fission of this bound high-N soliton solution, giving rise to efficient supercontinuum generation [7, 8, 16]. Individual soliton features emerging from this soliton-fission dynamics can still be coherently combined to yield a high-energy supercontinuum output [17, 18].

 

Fig. 1 Temporal dynamics of high-N NSE solitons found by numerically solving the NSE [Eq. (1)] with N = 5 (a) and 7 (b).

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Fig. 2 Spectral broadening in the regime of (a, b) normal (sgnβ₂ = + 1) and (c - f) anomalous (sgnβ₂ = −1) dispersion: (maps) numerical solution of the NSE [Eq. (1)] and (solid lines) calculation using Eq. (14) (a, b) and Eq. (13) (c - f) with N = 5 (a, c, d) and 7 (b, e, f). The spectral bandwidth is normalized to the spectral bandwidth Δω₀ of the input pulse. The propagation coordinate z is normalized to the dispersion length l_d.

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Numerical analysis of high-N soliton dynamics, supported by a wealth of experimental data, suggests purely empirical, yet practically useful design rules for devices implementing soliton pulse self-compression. As perhaps the most significant design rule, the optimal pulse compression distance z_m = ξ_ml_d, that is, the distance at which the soliton pulse width reaches its minimum τ_m, has been found to obey the following scaling with the soliton number N [2]:

We will show below that this important relation, viewed in the earlier work as purely empirical [2], can, in fact, be derived analytically through an appropriate extension of the basic SPM theory.

4. Dispersion and scale renormalization

To extend the analytical description of the early stages of spectral broadening of ultrashort laser pulses beyond the simplest equations of the SPM theory [Eqs. (2)–(5)], we first resort to the solution of Eq. (1) without the nonlinear term. For an input pulse with an initial chirp α₀, ${\left|u\left(0,\eta \right)\right|}^2=\exp \left[-\left({\tau }_0^{-2}+i{\alpha }_0\right){\eta }^2\right]$, the solution to this equation is written as

Seeking an approximate description of spectral broadening beyond the basic SPM theory in the high-N regime, we note that, since N² >> 1 is equivalent to l_nl < l_d, the nonlinear phase shift in the case of high N is accumulated much faster than the dispersion-induced phase shift. We use this observation as a rationale, which in no way qualifies as a rigorous justification, for the separation of l_nl and l_d scales and plug an ansatz α₀ = α_SPM, with α_SPM as defined by Eq. (5), into Eq. (9) to find

Equation (10) describes pulse self-compression in the case of anomalous dispersion, sgn(β₂) = −1, and pulse stretching when sgn(β₂) = + 1, i.e., dispersion is normal.

We restrict our analysis to propagation distances z < (l_dl_nl)^1/2 = l_b. Notably, this restriction does not prevent us from analyzing soliton dynamics up to the point of maximum pulse compression, as, in accordance with Eq. (6), z_m ≈l_b/2. Then, dropping the small ${\left(z/l_d\right)}^2$ and $z^4/{\left(l_d{l}_{nl}\right)}^2$ terms in Eq. (10), we find from the resulting expression for τ(z) that the minimum pulse width (zero τ in our approximation) is achieved, in the case of anomalous dispersion, at

This is a rewarding result. Indeed, the empirical design rule of Eq. (6) can be rewritten, for a convenience of comparison, as z_m ≈1.0048l_d/(2N), which agrees with Eq. (11) within 0.5%.

We now modify Eq. (4) for the SPM-induced spectral broadening to reflect variations in the pulse width τ and, hence, the peak power P as functions of z. To this end, we replace τ₀ in Eq. (4) by τ(z) as defined by Eq. (10), and express the z-dependent peak power as P(z) = P₀/[τ₀ψ(z)], with ψ(z) as defined by Eq. (8). Then, within the same approximation as in the analysis above, we arrive at

Integration in Eq. (12) leads to

Equations (13) and (14) provide an approximate description of spectral broadening beyond the basic SPM theory in the high-N regime. As one important finding, these equations reveal the significance of the l_c length as the key spatial scale of the problem not only in the regime of soliton self-compression, when l_c defines the length of maximum pulse compression (Figs. 1a, 1b), but also in the case of normal dispersion. Indeed, for propagation paths z << l_c, $\arcsin \left(z/l_c\right)\approx z/l_c$ and $\arctan \left(z/l_c\right)\approx z/l_c$. In this limit, both Eq. (13) and Eq. (14) recover the celebrated result of the basic SPM theory [Eq. (4)], $\Delta \omega \left(z\right)\approx \gamma P_{0}z/{\tau }_0=\Delta {\omega }_{SPM}\left(z\right)$.

5. Discussion

In Figs. 2(a)–2(e), we compare predictions of Eqs. (13) and (14) with the numerical solution of the NSE [Eq. (1)]. To this end, we plot the normalized spectral bandwidth $\left[\Delta {\omega }_0+\Delta \omega \left(z\right)\right]/\Delta {\omega }_0$, Δω₀ being the spectral bandwidth of the input pulse, for two different values of N in the regime of normal [Figs. 2(a), 2(b)] and anomalous [Figs. 2(c)–2(f)] dispersion. For both positive and negative β₂, our approximation provides a reasonably accurate description of SPM-induced spectral broadening within the entire applicability range of Eqs. (13) and (14), i.e., all the way up to z ≈l_c. When the propagation path is no longer much smaller than l_c, Eqs. (13) and (14) are instructive in showing how dispersion effects enter the scene, leading to spatial-scale renormalization. Indeed, for z << l_c, spectral broadening for both positive and negative β₂ is given by Eq. (4), with the nonlinear phase shift accumulating as z/l_nl, articulating the significance of the nonlinear length l_nl as the spatial scale of the process. For longer propagation paths, however, the scenario of spectral broadening changes. As z ceases to be small compared to l_c, l_nl no longer provides a spatial scale that fully defines spectral broadening. Instead, as dispersion effects come into play, l_nl has to share its role as the spatial scale with the dispersion length l_d, through a geometric mean (l_nll_nl)^1/2, defining a new spatial scale l_c. This l_nl-to-l_c scale renormalization is analytically described by $\arcsin \left(z/l_c\right)$ in the anomalous dispersion regime [Eq. (13)] and $\arctan \left(z/l_c\right)$ in the case of normal dispersion [Eq. (14)], both providing a smooth transition from the z << l_c regime, where the l_c scale is completely eliminated, to the z ~l_c regime, where l_nl is suppressed as an isolated factor in Eqs. (13) and (14), entering the problem only through the l_c scale instead.

Unlike the basic SPM theory, whose predictions for the nonlinear phase shift [Eq. (3)] and the related spectral broadening [Eq. (4)] are insensitive to the sign of dispersion, Eqs. (13) and (14) provide a closed-form analytical description of how the difference between spectral broadening in the normal and anomalous dispersion regimes builds up as the pulse propagates along a nonlinear, dispersive medium. As z ceases to be much smaller than l_c, the difference becomes striking. For positive β₂, normal dispersion results in pulse stretching, decelerating spectral broadening. This effect, which is grasped by Eq. (14) and is clearly seen in Figs. 2(a) and 2(b), manifests itself in a broad class of nonlinear fiber-optic experiments [7, 8], including recent studies on the spectral broadening of high-energy short laser pulses in normally-dispersive large-mode-area photonic-crystal fibers (PCFs) [19, 20].

In the case of negative β₂, anomalous dispersion, acting jointly with nonlinearity, gives rise to soliton pulse self-compression [Figs. 1(a), 1(b)]. In the frequency domain, this translates into enhanced spectral broadening [Figs. 2(c)–2(f)], giving rise to an explosion-like growth in Δω near the point of maximum pulse compression, z ~l_c. As can be seen in Figs. 2(c)–2(f), our approximate analytical result of Eq. (13) agrees reasonably well with numerical simulations, thus offering a helpful analytical tool for the design of fiber-optic pulse compressors [7, 8], including PCF-based sources of single-cycle and few-cycle field waveforms [14, 21].

6. Conclusion

To summarize, we presented a closed-form analytical description of the early stages of spectral broadening of ultrashort laser pulses beyond the basic SPM theory. This formalism closes the gap between the basic SPM theory and the soliton-number-based description of nonlinear pulse evolution. Based on this formalism, we have provided an analytical derivation for the relation between the maximum soliton self-compression length and N, which has been previously treated as purely empirical.