Solitons are wave packets that can propagate without changing shape by balancing nonlinear effects with the effects of dispersion. In photonics, they have underpinned numerous applications, ranging from telecommunications and spectroscopy to ultrashort pulse generation. Although traditionally the dominant dispersion type has been quadratic dispersion, experimental and theoretical research in recent years has shown that high-order, even dispersion enriches the phenomenon and may lead to novel applications. In this Tutorial, which is aimed both at soliton novices and at experienced researchers, we review the exciting developments in this burgeoning area, which includes pure-quartic solitons and their generalizations. We include theory, numerics, and experimental results, covering both fundamental aspects and applications. The theory covers the relevant equations and the intuition to make sense of the results. We discuss experiments in silicon photonic crystal waveguides and in a fiber laser and assess the promises in additional platforms. We hope that this Tutorial will encourage our colleagues to join in the investigation of this exciting and promising field.

## I. GENERAL INTRODUCTION

### A. Brief history and context

While studying water waves in the Glasgow–Edinburgh union canal, Russell observed a boat being pulled by a pair of horses. When the horses abruptly stopped, it gave rise to a large surface wave at the front of the boat that stretched across the width of the canal. Russell followed the traveling wave and realized that it was propagating on the canal undisturbed and that it propagated for a few of miles before eventually decaying. Following this observation, he described this special phenomenon as the *great solitary wave*.^{1}

Since this original report in 1834, these solitary waves, more commonly known as *solitons*, have intrigued generations of physicists and mathematicians, and they have become one of the most widely studied forms of excitation in nonlinear systems.^{2–6} They are localized structures that can be seen as self-reinforcing waves that propagate while maintaining their shape, a remarkable property enabled by balancing spreading and focusing effects. Solitons are ubiquitous in nonlinear systems and have provided a framework to explain physical phenomena in topics as varied as oceanography,^{7,8} atmospheric science,^{9} plasma physics,^{10} Bose–Einstein condensates,^{11} and even astronomy.^{12,13}

In optics, solitons result from the interplay between the linear and the nonlinear interaction of the light and matter. The linear response of a medium leads to dispersion, the frequency dependence of the refractive index *n*(*ω*).^{14} In contrast, the nonlinear light–matter interaction means that the response medium is no longer proportional to the electric field. The nonlinear effect of interest here, the Kerr effect, is the dependence of the refractive index on the square modulus of the electric field or, equivalently, on the local light intensity *I*. We take this dependence to lowest order to be linear, so the nonlinear change in the refractive index Δ*n* is written as Δ*n* = *n*^{(2)}*I*, where *n*^{(2)} is the nonlinear refractive index, the value of which depends on the medium.^{15} The value of *n*^{(2)} is usually very small, and therefore, nonlinear effects remained unobserved until the invention of the laser, which allowed for the generation of high optical intensities.^{16,17}

There are two types of optical solitons, *spatial* and *temporal* solitons. Temporal solitons, such as the one observed by Russell, are as described above: they balance the nonlinear effect with dispersion. In contrast, spatial solitons balance the nonlinear effect with diffraction–spatial solitons maintain their transverse shape while propagating. This is because the nonlinear effect causes the refractive index to be larger at the center of the beam, where the optical intensity is higher, than at the edges, thus acting as a lens. Building on earlier theoretical work of Chiao *et al.*,^{18} Bjorkholm and Ashkin reported the experimental observation of self-trapping of a laser beam in 1974^{19} in a cell of sodium vapor, which was the nonlinear medium. By adjusting the optical power (i.e., the nonlinearity), the induced self-focusing effect exactly canceled the self-defocusing of the laser beam.^{19} However, spatial solitons were reported first by Barthelemy *et al.*^{20} and Aitchison *et al.*^{21} In their experiments, the light propagated in a planar waveguide, so the light can diffract freely in a single transverse direction. In this geometry, the self-trapping occurs under very general conditions. Many generalizations have been investigated, but the discussion of these phenomena is beyond the scope of this Tutorial. There exist many excellent books and review articles on optical spatial solitons, and the interested reader is encouraged to consult them (see, for example, Refs. 22–25).

Here, we focus on their temporal equivalent. Optical *temporal* solitons were suggested theoretically in 1973,^{26} before the development of low-loss optical fibers,^{27} and mode-locked lasers emitting short optical pulses with high peak power^{28} allowed for their experimental confirmation in an optical fiber, a few years later.^{29} In this case, the dispersion, i.e., the different frequencies in a pulse propagate at different speeds, is compensated by self-phase modulation (SPM), which is a consequence of the fact that the effective value of the refractive index is higher at the peak of the pulse than on the edges,^{30} so that they maintain their temporal and spectral shape during propagation.^{29} While in these first experimental observations, the generation of temporal solitons relied on pulsed sources, it has been shown that they can also be excited by a continuous wave that breaks up into a periodic pulse train of solitons through a nonlinear process known as modulation instability (MI).^{31–34}

Since their discovery, solitons have impacted a wide range of photonic applications. In the early 1990s, temporal solitons were considered for telecommunications. In this approach, each soliton represents a bit 1 in a binary coded-based message. These bits can propagate in optical fibers without dispersing, and therefore avoiding temporal overlap, to be finally recovered at the output.^{35} Despite promising early demonstrations, this method became rapidly impractical due to the careful gain/loss and nonlinearity management required.^{36} Soliton effects have also greatly enhanced mode-locked laser performances and allowed for the direct generation of optical pulses with ultrashort duration well below 100 fs.^{37,38} Tremendous engineering and technological effort has been invested in the development of solid-state, fiber, and semiconductor lasers emitting solitons, and these sources have become a key tool in many areas of science.^{39} Over the last decade, temporal solitons have experienced renewed interest through *dissipative solitons*^{40} and have become the central element in the state-of-the-art generation of frequency combs^{41}—they enable these optical combs to be generated in miniature sized ring resonators, opening the way for extending precision metrology applications from the laboratory to everyday devices.^{42,43}

In addition to the direct use of solitons as ultrashort optical pulses, their high-order dynamics have also offered a framework to understand more complex nonlinear phenomena. Coherent supercontinuum generation, for example, is dominated by soliton-related dynamics.^{44} Soliton fission and self-soliton frequency shift allow for the extension of the spectral broadening toward longer wavelengths and can be exploited to reach spectral regions that are otherwise difficult to access,^{45,46} while dispersive wave emission allows for spectral broadening toward shorter wavelengths.^{47,48} Higher-order soliton dynamics can explain exotic nonlinear phenomena, such as optical rogue waves and soliton explosions.^{49–51} It is now well established that optical temporal solitons provide an ideal platform for the study of complex nonlinear dynamics because of their one-dimensional character. This allows us to perform a wide range of experiments, with some analogies in oceanography, where the study of these phenomena is substantially more complicated.^{8,52} Finally, it is worth noting that almost four decades after the first observation of optical solitons, these pulses continue to support the development of emerging applications, such as topological photonics,^{53,54} machine learning,^{55} and nonlinear interaction of light with biological soft-matter.^{56}

Historically, stable temporal solitons have been based almost exclusively on the balance of Kerr nonlinearity and negative quadratic (second order) dispersion. Dispersion is discussed in detail in Sec. II A. Briefly though, the order of dispersion describes how the inverse group velocity depends on frequency. Quadratic dispersion corresponds to a linear dependence, at least approximately—when it is positive, the inverse group velocity increases linearly, whereas when it is negative, it decreases linearly. Generically, one would expect the dispersion to be quadratic. Quartic dispersion refers to the non-generic situation in which the inverse group depends cubically on frequency. Higher-order dispersion was historically seen as a detrimental factor, limiting the pulse duration achievable^{57,58} or leading to soliton instabilities and energy loss.^{59,60} This thinking changed in 2016 when some of us reported the experimental demonstration of optical pulses with soliton characteristics but arising from the interplay between SPM and negative quartic dispersion.^{61} To highlight this fundamental difference and to distinguish them from previous theoretical studies^{60,62–65} in which the quartic dispersion is considered but is not the dominant dispersion effect, these novel soliton pulses were named *pure-quartic solitons* (PQSs). As we will discuss in this Tutorial, conventional solitons and PQSs share several features, and some of their differences can appear to be subtle at first sight. However, studies following the discovery of PQSs showed that one particular property, the *energy-width scaling*, is significantly different. Concretely, it implies that the energy of PQSs can be a few orders of magnitude higher than conventional solitons for the same pulse duration.^{66,67} This could open a whole new range of applications for soliton fiber lasers. While these compact, efficient, low-cost systems allow for the direct generation of ultrashort pulses, these properties are often eclipsed by the strongly limited peak power achievable, as illustrated in Fig. 1.

Finally, it has been recently experimentally demonstrated and theoretically explained that solitons and PQSs are just the two lowest-order members of a infinite hierarchy of soliton pulses arising from the interaction of Kerr nonlinearity and negative pure high, even-order dispersion.^{68} Manipulating the linear dispersion offers, in effect, a new way to access an infinite family of nonlinear pulses.

### B. Scope of this Tutorial

In the context of these recent works on temporal solitons arising from Kerr nonlinearity and high-order dispersion, we offer in this Tutorial a detailed perspective on PQSs and the opportunities provided by tailoring high-order dispersion for the generation of nonlinear optical pulses. We discuss the theoretical, experimental, and technological outcomes of these interactions. While we focus here on the optical physics aspect, this problem opens the way for future investigation in the context of applied mathematics and engineering. We organized this Tutorial as follows: in Sec. II, we present the underlying fundamental concepts describing the formation of nonlinear optical pulses, namely, dispersion and nonlinearity. Building upon these concepts, in Sec. III, we give a brief introduction to optical temporal solitons. In Sec. IV, we extend and generalize these concepts and their implications for the formation of PQSs and their high-order counterparts. These results are illustrated by the detailed review of the original discovery of PQSs. We then discuss the implication of solitons and PQSs in the development of laser systems in Sec. V, followed by experimental results in this area. The prospect of PQSs in other photonic platforms is discussed in Sec. VI. In Sec. VII, we expand the PQS concept to pure-high, even-order dispersion solitons (PHEODS) and to optical solitons arising from combinations of different even orders of dispersion. Finally, general remarks and future prospects for this field are outlined in Sec. VIII.

## II. INTRODUCTION TO NONLINEAR WAVEGUIDE OPTICS

There exist many excellent books and review articles on nonlinear guided-wave optics and on optical solitons (see, for example, the textbooks by Agrawal^{69} or by Kivshar and Agrawal^{70}), and it is not our intention to replicate these. However, we aim for this Tutorial to be self-contained, at least to some degree, and therefore, in this section, we review the main theoretical concepts that will be required, namely, dispersion (Sec. II A) and the Kerr nonlinearity (Sec. II B). The propagation of light in the presence of these effects can be described by the nonlinear Schrödinger equation and its generalizations, which is discussed in Sec. II C.

### A. Dispersion

Dispersion is the general phenomenon that describes that the (effective) refractive index *n* of a medium depends on wavelength or, equivalently, on frequency *ω* so that *n* = *n*(*ω*). In a bulk medium, dispersion is entirely caused by the frequency-dependent response of the medium. This chromatic dispersion is responsible, for example, for the wavelength separation of white light through a prism.

In this Tutorial, we consider light propagating in a guided-wave structure. Such waveguides can be described by a core with refractive index *n*_{core} surrounded by a cladding with lower refractive index *n*_{cladding}. When light propagates in such a structure, its propagation constant *β* can be used to define an effective refractive index *n*_{eff} ≡ *β*/*k*_{0}, with *k*_{0} = *ω*/*c*, where *c* is the speed of light in vacuum and *ω* is the free space wave number. Intuitively, *n*_{eff} describes the average refractive index seen by the field, and thus, in general, *n*_{clad} ≤ *n*_{eff} ≤ *n*_{core}.

At short wavelengths (high frequencies), the light is almost entirely confined to the core and, therefore, *n*_{eff} ≈ *n*_{core}, whereas at long wavelengths, close to the modal cutoff, the light is poorly confined and thus *n*_{eff} ≈ *n*_{cladding}. Therefore, when light propagates in a waveguide, it experiences waveguide dispersion, even in the absence of material dispersion. The total dispersion is approximately the sum of these contributions. For this reason, it is possible to tailor the dispersion of a waveguide, even though the material dispersion is fixed.

The propagation constant of a waveguide can, in general, be written as *β* = *β*(*ω*). However, for the experiments in which we are interested, the frequency spectrum has a limited width. Typically, we will be discussing pulses with a pulse length Δ*t* such that Δ*t* > 250 fs and at a center wavelength of *λ* = 1550 nm, corresponding to *ω*_{0} = 1.2 × 10^{15} s^{−1} and an optical period of ∼5 fs. Thus, these pulses contain 100 s of optical periods, and the fractional bandwidth of the pulses is thus approximately Δ*ω*/*ω* < 1/100. Under this condition, we only require *β*(*ω*) over a relatively narrow bandwidth centered at *ω*_{0}, which allows us to write it as a Taylor series around *ω*_{0},

where Ω = *ω* − *ω*_{0} and *β*_{m} = *∂*^{m}*β*/*∂ω*^{m}, evaluated at *ω* = *ω*_{0}. *β*_{m} is referred to as the *m*th dispersion order coefficient. Of course, the number of terms that is needed depends on the frequency interval of interest and whether or not some of the low-order terms vanish.

The dispersion discussed up until now arises upon propagation of the light. In addition to this, dispersion can be applied at a particular position, without the light needing to propagate. This “localized” approach, which is essential for this Tutorial, can be applied using prisms, gratings, or a spectral pulse-shaper,^{71} as discussed in Sec. V B.

The dispersion is important because of its relation to the group velocity *v*_{g} of the light through

Combining this with Eq. (1), we find that, in general, the dispersion order coefficients *β*_{m} are defined as

and so we can expand the inverse group velocity around *ω*_{0} as

This Taylor expansion shows that *β*_{2} provides the lowest order description of dispersion. Generically, therefore, the effect of *β*_{2} would be expected to dominate over higher-order effects.

We now discuss the effect of dispersion on the propagation of an optical pulse, starting with *β*_{2}. Equation (4) shows that the effect of *β*_{2} is a linear variation in the inverse group velocity with frequency—thus, when *β*_{2} > 0, the inverse group velocity increases monotonically and, in fact, linearly, with increasing frequency, and when *β*_{2} < 0, it decreases with frequency. Thus, a pulse propagating through a medium with quadratic dispersion broadens with, for *β*_{2} > 0, the red frequencies on the leading edge and the blue frequencies on the trailing edge (and vice versa when *β*_{2} < 0); this is referred to as chirping. We note that the effect of all high even-orders of dispersion is qualitatively the same, as illustrated in Fig. 2. For *β*_{4} < 0 (red) and *β*_{6} < 0 (blue), the inverse group velocity also monotonically decreases with increasing frequency. In contrast, the effect of odd orders dispersion is very different; although these also lead to pulse broadening, a positive *β*_{3}, for example, causes low and high frequency to travel more slowly than intermediate frequencies. The group velocity dependence associated with odd orders of dispersion is not monotonic. For this reason, we only consider even order orders of dispersion for the remainder of this Tutorial. We return to this in Sec. VII B.

### B. Nonlinearity

The description of optical nonlinearities starts with the generalization of the well-known linear response of a medium to an electric field **E** described by

where **P** is the polarization. It describes the formation of a macroscopic dipole element under the influence of an applied electric field in which the force on the electrons and on the nuclei have opposite signs. When the strength of the applied electric field increases, this linear relationship has to be generalized to^{15}

where *χ*^{(2)} and *χ*^{(3)} describe the anharmonic response. For *χ*^{(2)} to be nonzero, it is necessary for the medium to lack inversion symmetry.^{15} Thus, it vanishes for silicon, which has inversion symmetry, and also for silica glass, which, as an amorphous material, has inversion symmetry when averaged over volumes of approximately a cubic wavelength. For the materials of interest to us, *χ*^{(3)} is the lowest order nonlinearity. One of the consequences of a nonzero *χ*^{(3)} is that the material response can be written as a refractive index that depends on the optical intensity *I* as

where *n*_{0} is the refractive index at low intensities, and the nonlinear refractive index *n*^{(2)} is proportional to *χ*^{(3)}. This is known as the optical Kerr effect.^{15} Although *n*^{(2)} can be positive or negative, we will here take it to be positive consistent with the materials used in our experiments.

The nonlinear refractive index is a property of a bulk medium. The equivalent expression for a waveguide is

where *β*_{0} is the low-intensity propagation constant, *γ* is the nonlinear parameter, and *P* is the power. Just like the dispersion parameter, *γ* is a property of the waveguide and its constituent materials and can, in general, be expressed as an integral over the waveguide’s modal field and the linear and nonlinear refractive indices. For a standard optical fiber, *γ* takes the particularly simple form

which defines the effective area *A*_{eff} and where the integrals are over the entire fiber cross section. For the standard single-mode fiber (SMF), *γ* = 1.2 W^{−1} km^{−1}.^{72} We note that the expression for *γ* depends on the fact that the waveguide mode is normalized such that the square modulus of its amplitude corresponds to the power in the mode.

Tapering a standard SMF reduces the core diameter, leading to a transverse confinement of the field, which reduces the effective area *A*_{eff} and thus, according to Eq. (9), enhances the nonlinearity.^{73} The effective nonlinearity can also be enhanced by confining the field in the longitudinal direction. This can be achieved by slow light, light that propagates at a (group) velocity that is well below $\beta 1\u22121$ of the constituent materials. Optical pulses that enter a slow-light medium are spatially compressed by the slow-down factor *S*, the ratio of the group velocities in the two media. Slow light can be achieved by engineering the dispersion and is discussed further in Sec. IV C.

Now that we have discussed how nonlinear effects may be described, we discuss how they affect light propagation. Consider first a harmonically varying plane wave. The effect of the nonlinearity then is to change the propagation constant via Eqs. (7) or (8), the effect of which is that the phase of the light differs from that in the linear case. Although this can be measured using an interferometer, in itself it is of limited interest. On the other hand, the intensity of an optical pulse depends, by definition, on time. Then, by the argument above, the phase depends on time as well, and a phase that depends on time corresponds to a shift in frequency *δω* via

Since *ϕ* ∝ *I*, we see that the largest frequency shifts occur at the times with the largest intensity gradient, i.e., at the leading and the trailing edges of the pulse. At the leading edge, where the intensity increases with time, the frequency shift is negative, i.e., toward the red, whereas, by the same argument, on the trailing edge, the frequency shifts to the blue. Thus, the Kerr nonlinearity causes the leading edge of the pulse to shift to the red and the trailing edge to the blue. As a consequence, the pulse undergoes self-phase modulation (SPM): the nonlinearity broadens the pulse spectrum, even though in the time domain the pulse is unchanged. This is illustrated in Fig. 3.

The presence of *χ*^{(3)} in Eq. (6) not only leads to SPM but also to other nonlinear effects. Examples are cross-phase modulation, by which the presence of one pulse affects the phase of another, and four-wave mixing, by which new frequencies can be generated. Other nonlinear effects that may occur in optical fibers are two-photon absorption, Brillouin and Raman scattering, and nonlinear effects associated with even higher order terms in the expansion in Eq. (6).^{69} These effects, though very interesting, do not play a major role in this Tutorial.

### C. The nonlinear Schrödinger equation and generalizations

The propagation of light in a waveguide is ultimately described by the Maxwell equations. However, solving these equations to describe the nonlinear propagation of an optical pulse is very cumbersome. Instead, the nonlinear Schrödinger equation (NLSE) and its generalization are usually used. There are many excellent derivations of this equation,^{69,70} so we here limit ourselves to a heuristic derivation and a general discussion. Taylor series [Eq. (1)], for the dispersion, which is a linear effect, is consistent with the differential equation that describes the evolution of the electric field envelope *ψ* (apart from a constant that does not matter),

which can be seen by taking *ψ* = *e*^{i(βz−Ωt)}. The full NLSE is obtained by realizing that the nonlinearity is cubic and contributes nonlinearly to the propagation constant. The NLSE can be written in its final form by changing to the time *τ* = *t* − *z*/*v*_{g} (*ω*_{0}) propagating with the pulse. With this definition, we obtain the generalized NLSE

where we truncated the Taylor expansion after the quartic term, and we have used that a real *γ* only affects the phase of the light. It is common that Eq. (12) is truncated after the term representing the quadratic dispersion, in which case the NLSE results. We discuss it in more detail in Sec. III.

The function *ψ* in NLSE only describes the electric field envelope and not the electric field itself. This simplification is commonly used as the envelope contains almost all the relevant physical information. The actual electric field can be found by multiplying *ψ* by exp [*i* (*β*_{0}*z* − *ω*_{0}*τ*)] and by the shape of the modal fields and by then taking the real part.

## III. INTRODUCTION TO SOLITONS

Recall from Sec. II B that a Kerr nonlinearity leads to SPM, which causes the leading edge of a pulse to shift to longer wavelengths and the trailing edge to shorter wavelengths. We now combine this with the effect of quadratic dispersion. According to Eq. (3), when *β*_{2} > 0 is positive, this means that $vg\u22121$ increases with frequency and thus that redshifted frequencies travel faster than blueshifted frequencies. Thus, the combined effects of the nonlinearity and the dispersion are for the pulse to broaden faster than it would in the absence of the nonlinearity. In contrast, when *β*_{2} < 0, blueshifted frequencies travel faster than redshifted frequencies. As a consequence, the newly generated frequencies on the leading and trailing edges of the pulse tend toward the center of the pulse by the effect of the dispersion. This leads to a stable pulse, a *soliton*, that balances dispersion and the nonlinearity (see Fig. 3). The argument presented above implies that the shape of soliton pulses results from the physical processes (dispersion and the Kerr effect) and cannot be imposed (Fig. 4).

In fact, solitons that are solutions to the NLSE with only quadratic dispersion have a hyperbolic secant shape and so has their spectrum. This can be seen by rewriting the generalized NLSE [Eq. (12)] and dropping dispersion order higher than 2. The NLSE thus becomes

This equation is so widely used that it is easy to forget the approximations needed to derive it.^{69} Apart from those already mentioned, the most important of these is that the propagation is unidirectional and that the nonlinear effect changes the propagation constant of the mode but not its shape. In its present form, it also requires the pulse length to be much longer than the period of the light and to have no other nonlinear effects besides the Kerr effect (see Sec. II B).

We look for stationary solutions—solutions with an unchanged intensity as they evolve—of the form *ψ*(*z*, *τ*) = *u*(*τ*)*e*^{iμz}, where *μ* is the change in propagation constant due to the presence of the nonlinearity. Function *u* thus satisfies

where we now explicitly took |*β*_{2}| to be negative. Note also that for a solution to be stationary, there cannot be energy flow within the pulse—this means that *u* needs to be real or at least can be made real, and so |*u*|^{2}*u* can be replaced by *u*^{3}. Equation (14) has the fundamental soliton solution

where *μ* can be chosen freely and *T*_{0} is a measure of the soliton’s width. It is related to the full-width at half-maximum (FWHM) Δ*t* through $\Delta t=2T0arcosh(2)$. This means that there is a family of solutions including narrow solutions with high peak power (*μ* large) and wide solutions with low peak power (*μ* small). Note, however, that the soliton area $\u222bud\tau =\pi |\beta 2|/\gamma $ and is thus independent of *μ* and thus independent of *T*_{0} as well. Since solutions of Eq. (15) are real, they have a uniform phase and are therefore unchirped. This illustrates that the dispersion, which acting on its own leads to strong chirping, is truly balanced by the nonlinear effects.

These solitons, to which we will refer as conventional solitons, have profound mathematical properties that derive from the fact that the governing equation (13) is integrable.^{74} At an operational level, this means that upon propagation, these solitons retain their properties (amplitude and speed) and they even retain these after colliding with another such soliton.^{70} In fact, according to their more restrictive definition, solitons are pulse-like solutions of integrable systems; non-integrable systems cannot have soliton solutions according to this definition. However, here we are less restrictive and consider solitons to be pulse-like solution to any nonlinear, dispersive wave-equation.

A consequence of the fact that solitons maintain both their temporal and spectral shapes is that the time-bandwidth product (TBP), the product of their full-width at half maximum in time (Δ*t*) and in frequency (Δ*ν*), remains constant upon propagation. In addition, since the solitons are unchirped, the TBP takes a minimum value and is said to be transform limited. For a soliton with a hyperbolic secant squared intensity, TBP = Δ*ν*Δ*t* ≈ 0.315. The TBP can be seen as an indication of how efficiently the pulse is using the spectral bandwidth. Typically, high quality pulses have a TBP well below 0.5.

The peak power and (temporal) width of conventional solitons can also be found from an argument introduced by Agrawal.^{69} We introduce it here as it will be useful in Sec. IV A. We can define a nonlinear length

where *P*_{0} is the square modulus of the amplitude as the length scale over which nonlinear effects become important; strictly speaking, *L*_{NL} is the length over which the peak of the pulse experiences a nonlinear phase shift of 1 rad. Similarly, the length over which dispersion becomes important, i.e., the (quadratic) dispersion length, is defined as

This dispersion length is often referred to as *L*_{D} in other reference books and review articles.^{44,69} Here, we use this notation as it is more convenient to differentiate between the dispersion lengths arising from different dispersion orders as discussed in Sec. IV.

A soliton, a consequence of the balancing of the dispersion and the nonlinearity, then exist when *L*_{2} ≈ *L*_{NL}. In fact, conventional solitons satisfy *L*_{2} = *L*_{NL}. From this, we see that for conventional solitons, $P0=|\beta 2|/(\gamma T02)$, which can straightforwardly be shown to be consistent with Eq. (15). By integrating over time, which in essence corresponds to a multiplication by Δ*t* and a constant of order unity that depends on the pulse shape, it is found that the energy of a soliton *U*_{sol} is equal to

The physical argument of soliton formation discussed earlier in this section implies that even when *L*_{NL} and *L*_{2} are initially not exactly the same, a soliton forms upon propagation, and this is indeed correct. The pulse oscillates and may shed energy until it settles on a solution of the type in Eq. (15). This begs the question how the field evolves when the difference between *L*_{NL} and *L*_{2} is more significant. To discuss this issue, it is useful to define the soliton number *N*^{2} = *L*_{2}/*L*_{NL}. Satsuma and Yajima showed that for the specific case of hyperbolic secant-shaped initial conditions, a soliton of the form Eq. (15) eventually forms when $12<N<32$.^{75} When $N<12$, no soliton forms, whereas for $N>32$, a higher order soliton forms. Higher-order solitons can be considered to be superpositions of solitons of the type in Eq. (15). The beating of these leads to a propagation that varies periodically with position. For initial conditions other than the hyperbolic secant, the behavior qualitatively similar, although the boundaries for *N* may differ. We discuss the observation of solitons in optical fibers in Sec. V A.

The qualitative description early in this section and Fig. 2 imply that solitons may also occur through the balance of nonlinearity and quartic dispersion or even higher even orders, and indeed, this is true. These quartic and higher order solitons are the main subject of this Tutorial. We turn to these solitons next.

## IV. PURE-QUARTIC SOLITONS

### A. Background

The name *pure-quartic soliton* was introduced in the seminal 2016 paper to comprise a class of solitary waves arising, exclusively, from the interaction of negative quartic dispersion and Kerr nonlinearity.^{61} The word *pure* was used to distinguish this class from previously studied *quartic solitons*,^{76} occurring in scenarios with anomalous second order dispersion and quartic dispersion. Although recently we demonstrated that PQSs, quartic solitons, and conventional solitons are special members of the same family of *generalized-dispersion Kerr solitons*,^{77} in this section, we aim to give a brief historical perspective of decades of research that culminated in the discovery of PQSs.

The impact of higher orders of dispersion on the propagation of conventional solitons in fiber became an active field of study in the early 1990s. It was established that the presence of third order dispersion, of any sign, leads to radiation^{78} and that positive fourth order dispersion (*β*_{4} > 0) also led to soliton decay via dispersive radiation.^{59} Interestingly, a series of theoretical studies demonstrated that solitons could be stable in the presence of negative fourth order dispersion (*β*_{4} < 0).^{62,63,65,76,79,80} In 1994, Karlsson and Höök presented an analytic, stable, pulse-like hyperbolic secant squared solution to the generalized NLSE in the presence of negative quadratic and quartic dispersion.^{63} Shortly after, Piché *et al.* re-derived this solution including also the effect of third order dispersion.^{65} Akhmediev *et al.* showed that these pulse-like solutions can have oscillations in their exponentially decaying tails.^{64} In all these studies, however, the presence of negative quadratic dispersion was crucial, and indeed, when *β*_{2} = 0, these analytic solutions cease to exist.^{63,65}

In parallel, the ultrafast laser community at that time was pushing the limits of titanium:sapphire soliton lasers toward shorter pulse duration.^{58,81–83} The realization that dispersion was a limiting factor toward shorter pulses led to the exploration of new regimes of operation close to zero-quadratic and cubic dispersions. It was in this regime that Christov *et al.* theoretically suggested that the pulse shaping behavior in the laser cavity for pulses below 10 fs may arise from the balance of SPM and quartic dispersion.^{58}

We note that the spatial soliton community has previously investigated the effects of higher-order diffraction as well.^{84,85}

### B. Theory

According to the argument in Sec. III, the requirements for the formation of a soliton is the presence of a Kerr nonlinearity and, assuming this nonlinearity to be positive, negative dispersion, which means light with low frequencies travel slower than high frequencies. In Sec. III, this dispersion was taken to be quadratic as is common in this field because it is lowest order of dispersion and it therefore would generically be expected to the leading order. We now consider a dispersion relation such that *β*_{2} = 0 and *β*_{3} = 0 but *β*_{4} < 0. For this type of dispersion, the group velocity increases monotonically with frequency, as seen in Fig. 2, just as for conventional solitons, and thus, based on the argument from Sec. III, one might expect the existence of solitons—even though the soliton shape differs for different dispersion orders, the underlying physical phenomena are the same (see Fig. 5 for illustration). The existence of these PQSs has been confirmed experimentally and numerically. We now briefly review some of their key properties as following from theory and numerical simulations;^{77} we will discuss experimental results in Secs. IV C and V B.

PQSs are stationary solutions of Eq. (12) with *β*_{4} as the only nonzero dispersion term or

where we assumed that *β*_{4} < 0. Similar to the conventional soliton case (see Sec. III), we look for stationary solutions *ψ*(*z*, *τ*) = *u*(*z*)*e*^{iμz} of the same form. Thus, *u* satisfies

where we again used that *u* is real, indicating that there is no internal energy flow in the soliton. Analytic solutions to Eq. (20) are not known, but it can be solved using a numerical method such as the Newton conjugate gradient method developed by Yang.^{86}

The results obtained for *β*_{4} = −2.2 ps^{4} mm^{−1}, *γ* = 4.072 W^{−1} mm^{−1}, and *μ* = 1.76 mm^{−1} are shown in Fig. 6. These correspond to the experimental parameters of the original observation of PQS (see Sec. IV C).^{61} Figures 6(a) and 6(b) show the temporal profile on linear and logarithmic axis, respectively. The latter shows an important difference with conventional solitons, i.e., oscillations in the PQS tails, which occur at low intensities and cannot be seen on a linear scale. These oscillations can be found analytically from Eq. (20) by dropping the nonlinear term that is negligible in the tails,^{64,66,77} which gives

This equation has solitons of the type $u=\u2211jaje\lambda j\tau $, where the *λ*_{j} are solutions of the characteristic equation

The four solutions form a square in the complex *λ* plane of the form $\lambda =(24\mu /|\beta 4|)14e\xb1i\pi /4,e\xb13i\pi /4$. Two of these roots correspond to exponential growth and are associated with the soliton’s leading edge, whereas the other two are associated with the soliton’s trailing edge. Since the soliton can be made real, the solutions in the tails must be of the form cos (*ητ*) *e*^{±ητ}, where $\eta =(6\mu /|\beta 4|)1/4$, and indeed, the oscillations seen in Fig. 6(b) are consistent with this result. This argument is a very general one and applies to all solutions considered in this Tutorial. It also applies to the solutions of the NLSE [Eq. (13)], although it is unnecessary there since the exact solution (15) is known.

Figures 6(c) and 6(d) show the associated normalized spectra again, respectively, on linear and logarithmic scales (blue solid curves). Also shown for comparison are the associated curves for a conventional soliton with the same pulse width at FWHM Δ*t*, for which the spectrum has a hyperbolic secant shape (red dashed). Note that the PQS spectrum is more blunt near the peak, which is a consequence of the temporal oscillations.^{66} The temporal oscillations also cause the TBP of PQSs to be somewhat larger than that of conventional solitons as TBP_{PQS} = 0.53.^{66} Despite these differences, on a linear scale, the distinction between the shape of conventional solitons and PQSs is quite subtle.

A key difference between PQSs and conventional solitons is their energy scaling. As discussed in Sec. III, for a given quadratic dispersion (*β*_{2}) and nonlinearity (*γ*), the energy of a conventional soliton varies linearly with the inverse of the pulse-width [see Eq. (18)]. To find the equivalent relation for PQSs, we define the characteristic quartic dispersion length *L*_{4} = Δ*t*^{4}/|*β*_{4}| in analogy to Eq. (17). We would expect a quartic soliton when *L*_{4} ≈ *L*_{NL}, from which we find that^{87}

The constant *M*_{4} cannot be found by this argument, but numerical solutions show that *M*_{4} ≈ 2.87.^{66} This different scaling behavior of PQSs compared to conventional solitons is potentially very significant as it implies that if a PQS becomes sufficiently short, then its energy exceeds that of a conventional solitons. This has applications in solitons lasers^{88} that are currently limited by the energy of the output pulses. This is illustrated in Fig. 7 where we show Eqs. (18) and (23) vs the pulse width Δ*t*. The inset compares the conventional soliton (red) and PQS (blue) for the same FWHM pulse width Δ*t* = 0.25 ps. The PQS peak power exceeds the conventional soliton by almost fourfold.

As a final point in this section, we note that Tam *et al.* performed a rigorous stability analysis of PQSs.^{66,89} They found that these solitons are stable in that perturbations do not grow exponentially with propagation.

### C. Experiments

Pure-quartic solitons were first observed in the slow-light regime of a silicon photonic crystal waveguide (PhC-wg).^{61} In this section, we analyze the features of this platform and the conditions of the experiment and use this analysis to distill the general requirements to build PQS-supporting platforms.

Photonic crystals are periodic arrangements of dielectric materials.^{90} Analogously to atomic or molecular crystals—where the periodic potential caused by the crystalline structure can give rise to regions of forbidden electron energies—photonic crystals can exhibit photonic bandgaps, frequency regions in which light cannot propagate through the structure and in which any incident light is reflected.^{90} The introduction of linear defects in an otherwise perfectly periodic structure can lead to the appearance of guided modes within a photonic bandgap, forming a photonic crystal waveguide (PhC-wg).^{91} One way to see this is that the defect is surrounded by media that reflect any incoming light so that the light is confined to the defect. PhC-wg’s offer unparalleled possibilities for dispersion engineering, thanks to the generally large index contrast between the different elements of the structure and to the large number of degrees of freedom to be tuned. Furthermore, the combination of sub-wavelength confinement and the possibility of slow-light propagation in PhC-wg’s render an excellent platform to explore enhanced nonlinear effects.^{92–94} The combination of these superb capabilities for dispersion and nonlinearity control makes PhC-wg’s an extraordinary medium to study novel soliton effects.^{61,95–97}

The PhC-wg used in the original PQS experiment,^{61} shown in Fig. 8(a), was a silicon air-suspended slab with a hexagonal lattice of air holes (p6m symmetry group) with lattice constant *a* = 404 nm, hole radius *r* = 116 nm, and thickness *t* = 220 nm. A 396 *μ*m-long dispersion engineered PhC-wg was created by removing a row of holes and shifting the two innermost adjacent rows 50 nm away from the center of the line defect. These parameters lead to the dispersion characteristics shown in Figs. 8(b)–8(e). The group index *n*_{g} ≡ *cβ*_{1} dependence on wavelength [Fig. 8(b)] reveals a relatively flat slow-light region around 1550 nm. At this wavelength, the group-velocity dispersion is positive and small, *β*_{2} = +1 ps^{2}/mm [Fig. 8(c)], and the third order dispersion is virtually zero, *β*_{3} = +0.02 ps^{3}/mm [Fig. 8(d)]. The fourth order dispersion is, however, strong and negative, *β*_{4} = −2.2 ps^{4}/mm [Fig. 8(e)].

The experiment was performed by injecting Gaussian-shaped pulses from a mode-locked laser with a pulse duration Δ*t* = 1.3 ps and a center wavelength of 1550 nm into the PhC-wg. At this pulse duration, the physical length scales for the different orders of dispersion are^{69} $L2=T02/|\beta 2|=0.615mm$, $L3=T03/|\beta 3|=22.6mm$, and $L4=T04/|\beta 4|=0.168mm$, highlighting the dominance of the quartic dispersion, as *L*_{4} is almost four times smaller than *L*_{2} and more than a hundred times smaller than *L*_{3}. The pulses at the output of the PhC-wg were characterized using an in-house built frequency resolved electrical gating (FREG)^{98} apparatus, which allows for ultrasensitive characterization of phase-resolve measurements. The specific implementation of this FREG apparatus is described in the supplementary material of Ref. 94.

It is instructive to discuss the measured outputs at different input powers, as shown in Fig. 9, which match well with simulations of the NLSE, including linear loss, high-order dispersion, and two-photon absorption and its associated free-carrier effects (see Ref. 61 for details). At *P*_{0} = 0.07 W, corresponding to the linear regime, the pulse shows moderate dispersive temporal broadening from 1.3 to 1.4 ps and no remarkable spectral effect. At *P*_{0} = 0.7 W, the nonlinearity balances the quartic dispersion and a fundamental PQS is observed, where the measured (red dashed) and numerical (solid blue) outputs match well with the measured input (solid green). This shape-preserving propagation, both in time and frequency, together with the flat temporal phase (black dashed) is the key feature of solitons. At higher input powers *P*_{0} = 2.5 and *P*_{0} = 4.5 W, the output pulse exhibits temporal compression with respect to the input. Unfortunately, at these powers, the competing nonlinear effects typical of silicon, i.e., two-photon absorption and its associated free-carrier dispersion and free-carrier absorption,^{94,99} become important, leading to strong asymmetries in the pulse, both in time and frequency.

From this successful observation of PQSs, we can extract the following lessons. First, the quartic dispersion must be dominant over the other orders of dispersion. This requires a platform with both negative quartic dispersion at certain wavelengths and wide-enough bandwidth to support short pulses. Note that for arbitrarily short pulses, the quartic dispersion always dominates, according to the relation $L4/L2=T02|\beta 2/\beta 4|$. The impact of a moderate presence of either negative or positive second order dispersion on the propagation of PQSs was discussed by Tam *et al.*^{77} Second, the third order dispersion must be as close to zero as possible at the center wavelength of the pulse to guarantee a symmetric non-radiating output pulse.^{78} Finally, the platform must be sufficiently (Kerr) nonlinear so that the nonlinear length (*L*_{NL} = 1/*γP*_{0}) is commensurate with the quartic dispersion length at reasonable input powers. What constitutes a “reasonable power” is determined by the available sources, the damage threshold of the material, and, very importantly, the point at which higher-order nonlinearities become dominant over the Kerr nonlinearity.

In silicon, as we saw in the high-power measurements in Fig. 9, the most deleterious competing nonlinear effects are two-photon absorption and its associated free-carrier effects.^{61} By using a slow-light PhC-wg, we were able to observe a fundamental PQS at powers in which the Kerr effect was still dominant. This is because the Kerr nonlinearity scales as *S*^{2}, where *S* is the slowdown factor (see Sec. II B), whereas the free carrier dispersion and absorption effects simply scale with *S*.^{94} The realization of the different scaling of the Kerr effect and the free-carrier effects with *S* earlier enabled the observation of conventional solitons in silicon.^{96}

## V. PURE-QUARTIC SOLITON LASERS

### A. Introduction to soliton lasers and dispersion management

Conventional soliton lasers have their roots in early work that predicted and demonstrated soliton propagation in optical fibers. In this section, we explore the trajectory of the conventional soliton laser as it emerged from the basic demonstration of soliton propagation to laser systems with ever increasing peak power. These advances led to the development of new types of pulses that significantly diverge from soliton propagation. This story spans nearly 50 years and has ultimately enabled a wide range of applications and led to a proliferation of commercial ultrashort pulse lasers. While conventional soliton lasers were eventually left behind in the rapid progress of subsequent laser systems, the arrival of the PQS laser—discussed in detail in Sec. V B—and its new energy scaling law warrants a discussion on its potential for performance improvements and an examination of how far the performance can be pushed.

In 1973, Hasegawa and Tappert^{26} presented theoretical and numerical work that predicted solitons could be realized using light pulses propagating in optical fiber, which was rapidly developing at the time with ever lower propagation loss. Their paper focused on the ability of the self-phase modulation (see Sec. II B) to balance the spreading effect of the fiber’s group-velocity dispersion. Above a certain power threshold, they observed stable “stationary pulses” that exhibited striking robustness to large perturbations. Their work presented an exciting story of non-spreading pulses that could have major implications for data transmission, but the experiments would have to wait for the technology to catch up.

It was not until 1980 that Mollenauer *et al.*^{29} were able to demonstrate optical solitons in fiber waveguides experimentally. This demonstration was possible due to the newly available Rayleigh scattering-limited optical fiber (loss ≈0.2 dB/km), with anomalous dispersion at wavelengths *λ* > 1.3 *μ*m. Also key to this work was the development of mode-locked color center lasers operating in the 1.35–1.75 *μ*m range, which allowed the team to position their pulses at the minimum loss wavelength of their fiber (i.e., 1.55 *μ*m). By launching 7 ps pulses into the anomalous dispersion fiber, they demonstrated both fundamental and higher-order soliton propagation in 700 m of silica fiber.

In 1984, Mollenauer and Stolen^{88} took the soliton from purely propagation-based experiments and brought it into the world of laser physics. By incorporating a small section of anomalous optical fiber into the feedback loop of a mode-locked color center laser, the team was able to shape pulses via soliton effects producing pulse widths down to 200 fs. Furthermore, they experimentally demonstrated the inverse relationship between the pulse energy and pulse duration in a soliton laser system (see Sec. III). This relationship, the “soliton area theorem,” sets the achievable pulse energy for a given pulse width. Increased gain in the cavity eventually results in a nonlinear phase accumulation (*ϕ*_{NL} ≈ 1) that induces wave-breaking and multiple-pulsing, thereby limiting the maximum pulse energy. Subsequent work by Duling^{100} in 1991 demonstrated solitons in a purely fiber laser-based system with sub-picosecond performance.

To overcome this pulse energy limitation, in 1993 Tamura *et al.*^{101} added a normal dispersion fiber section to a soliton laser that allowed the pulse to stretch by an order of magnitude within the cavity, thereby mitigating the effects of the nonlinear phase accrued as the pulse circulated in the cavity. By designing the laser with an appropriately placed output coupler, they were able to achieve 77 fs pulses with kW level peak power, a full order of magnitude higher than conventional soliton lasers. This stretched-pulse or dispersion managed (DM) soliton laser represents a breathing solution of the NLSE.

While the few nJ pulse energy of the DM soliton laser was enough to enable new applications such as fiber-based frequency combs,^{102} it was not enough to compete with more established laser systems for high peak power applications such as laser machining. This was achieved through parabolic pulse amplification, which had been proposed theoretically by Anderson *et al.*^{103} in 1993 and demonstrated experimentally by Fermann *et al.*^{104} in 2000. An optical pulse with a parabolic shape propagating in a gain region is amplified in a self-similar manner: both the pulse duration and amplitude scale up exponentially.^{105} Mathematically, these self-similar solutions or similaritons are an asymptotic solution of the NLSE and exhibit a linear chirp that is readily compressible via a diffraction grating compressor. This idea was implemented in a fiber laser cavity by Ilday *et al.*^{106} in 2004, resulting in 15 nJ pulses with peak powers up to 5× that of the stretched-pulse laser. With its improved performance, the similariton laser demonstrated a route to ∼100 kW peak powers and made impact in fields such as high-harmonic generation.^{107}

In 2006, yet another scheme to increase pulse energy was introduced in the form of an all-normal dispersion (ANDi) cavity by Chong *et al.*^{108} The ANDi laser gets rid entirely of anomalous dispersion in the cavity and introduces a spectral filter that resets the pulse shape on each round trip. With the large normal cavity dispersion completely unbalanced by nonlinearity, the pulse circulating in the cavity can vary from 10 to 20 times its transform-limited duration, thus allowing for extremely high pulse energies. Furthermore, the long duration pulse has a linear chirp (similar to the similariton), which means the pulse can be compressed close to its transform-limit after exiting the cavity. ANDi lasers have progressed to pulse energies in the 100 s of nJ range^{109} with >MW peak powers and have been employed in commercial laser machining systems.

Where do PQSs fit in this picture? From a practical perspective, it is unlikely that a PQS laser could match an ANDi system in terms of performance. While the energy scaling of the PQS area theorem is advantageous for short pulses relative to the conventional soliton laser (i.e., 1/Δ*t*^{3} vs 1/Δ*t*), the nonlinear phase shift is still likely to pose significant limits at high pulse energies since the PQS pulses remain reasonably short throughout the cavity. However, PQSs bring back the simplicity and high efficiency of soliton lasers. Given the novelty of PQS lasers, it is reasonable to expect significant performance improvements soon, which should elucidate the point where the 1/Δ*t*^{3} scaling runs into these nonlinear phase limitations. An important note, however, is that the advantageous soliton area theorem for PQSs relative to conventional solitons raises interesting prospects for normal dispersion systems based on quartic dispersion.

### B. The PQS laser

As discussed in Sec. IV B, one of the key properties of PQSs is their energy-width scaling relationship, which allows them to potentially carry significantly more energy than conventional solitons for the same pulse duration (see Fig. 7). Recalling Sec. V A, we saw that the development of soliton-based laser systems followed two approaches: (i) soliton lasers made of elements with negative quadratic (*β*_{2} < 0) dispersion,^{29} in which the optical pulses propagate as a conventional soliton (Sec. III); and (ii) dispersion-managed cavities,^{101} in which the pulse “breaths” as it propagates through sections with negative and positive quadratic dispersion.

This latter approach can be achieved by different techniques, such as adding sections of fiber with opposite dispersion or using an intracavity arrangement of prisms, gratings, or chirped mirrors.^{101,110,111} The effect of dispersion, in the absence of nonlinear effects, is equivalent to applying a phase transformation to the electric field in the spectral domain as

where $\psi \u0303$ is the Fourier transform of the pulse envelope *ψ*, *ϕ*(*ω*) = *β*(*ω*)*L*, with *β*(*ω*) defined as in Eq. (1), and *L* is the propagation length. However, adjusting the overall net-cavity dispersion *βL*, using these techniques requires modification to the cavity (splicing fibers) or careful realignment of the laser cavity (prisms, gratings). Moreover, these approaches are usually limited to second (*β*_{2}) and third (*β*_{3}) order dispersion.

A more flexible technique to adjust the net-cavity dispersion is to include a *spectral pulse-shaper* in the laser cavity.^{71,112} These devices, which are available commercially (see, e.g., WaveShaper® from II–VI), can apply a frequency-dependent phase to a light pulse and hence dispersion. Spectral pulse-shapers operate by splitting the constituent wavelengths of a beam into separate spatial channels. Each individual channel then undergoes a phase and/or amplitude modulation, before being recombined. This procedure allows for shaping of ultrashort pulses that would be far too fast for direct temporal shaping via a modulator. A standard implementation of such a device is to employ a diffraction grating for wavelength separation. A spatial-light modulator (SLM) is then used either in reflection or transmission to apply a phase and amplitude to each individual wavelength channel before a second diffraction grating (or the same grating if the system is operated in reflection) is used to recombine the beam. In this way, any type of dispersion, subject to the resolution of the SLM device, can be applied.^{71}

Spectral pulse-shapers can be straightforwardly implemented in a fiber laser cavity. By applying the appropriate spectral phase profile *ϕ*(*ω*), the net-cavity dispersion can be adjusted.^{113} Using this approach, mode-locked fiber lasers operating in net anomalous (*β*_{2}*L* < 0), dispersion managed (*β*_{2}*L* ≈ 0), or net normal (*β*_{2}*L* > 0) dispersion regimes, without modifying the cavity configuration, have been reported.^{114–116} While these earlier works focused on adjusting the group velocity dispersion (*β*_{2}), we can use a similar approach to tailor multiple orders of dispersion.

We used this approach to build the first PQS laser.^{67} A schematic of the cavity is shown in Fig. 10(a). The laser is an erbium-doped, passively mode-locked fiber laser, which uses nonlinear polarization evolution (NPE) for mode-locking.^{117–119} In our configuration, this is achieved by a set of polarization controllers and fiber polarizer that are adjusted so that the maximum transmission at the polarizer occurs for the highest possible optical intensity. This technique acts as a fast saturable absorber (SA). The spectral pulse-shaper (i) compensates for the second (*β*_{2} = −21.4 ps^{2} km^{−1}) and third (*β*_{3} = 0.12 ps^{3} km^{−1}) order dispersion of the single-mode fiber (SMF) segments and (ii) introduces the desired quartic dispersion value (*β*_{4}).^{72} We use a phase profile of the form

where *L* = 18.2 m is the cavity length. Figure 10(b) illustrates the effects of the quadratic (top), cubic (middle), and quartic (bottom) dispersive phase imparted by the cavity (blue) and pulse-shaper (black). The red-dashed curves correspond to the net dispersion.

#### 1. PQS laser cavity output

This laser cavity can operate in two different regimes. When no phase mask is applied by the spectral pulse-shaper [*ϕ*(*ω*) = 0], the laser operates in the conventional soliton regime, with the intrinsic anomalous quadratic dispersion of the SMF segments balancing the SPM. The corresponding laser output is shown in Figs. 11(a) and 11(c). The pulses display the well-known hyperbolic secant spectrum (solid blue curve). The output spectrum is slightly truncated at long wavelengths due to the finite bandwidth of the pulse-shaper. The temporal intensity of the output pulse, shown on a linear scale in Fig. 11(c) (solid blue curve), is also in good agreement with the temporal shape of the conventional soliton from Eq. (15).

We then program the pulse-shaper to induce a spectral phase profile following Eq. (25). For the results discussed below, *β*_{2} = +21.4 ps^{2} km^{−1}, *β*_{3} = −0.12 ps^{3} km^{−1}, and we initially take *β*_{4} = −80 ps^{4} km^{−1}. This value of applied quartic dispersion is several order of magnitude larger than the intrinsic quartic dispersion of SMF, which is approximately *β*_{4} = −2.2 × 10^{−3} ps^{4} km^{−1}.^{120}

The laser output is shown in Figs. 11(b) and 11(d). The spectrum of the laser operating in this regime is shown in Fig. 11(b). Its shape differs significantly from that of conventional solitons but is in very good agreement with the theoretically predicted spectral profile of the PQS for the same spectral bandwidth (red dashed line),^{66} particularly the distinct flatness of the spectral maximum [cf. Fig. 6(d)]. Note also several strong, narrowly spaced spectral sidebands that we discuss in more detail in Sec. V B 2. The retrieved temporal pulse shape, shown in Fig. 11(d), is again in good agreement with the theoretically predicted PQS shape for the same pulse duration (red dashed line).^{66} The predicted periodic oscillations in the tails [see Fig. 6(b)] are not observed in the measured temporal profile. This is because the first lobe is expected to appear ∼28 dB below the pulse’s central maximum, which is below our experiment’s detection background.

#### 2. Spectral sidebands

One important difference between the laser output spectrum (solid blue) and the predicted spectral shape (red dashed-line) in Fig. 11(b) is the presence of spectral sidebands. These *Kelly sidebands* are specific to soliton lasers.^{121–124} This phenomenon can be understood by recalling that solitons arise from the balance of dispersion and nonlinearity. However, when propagating in the laser cavity, a soliton pulse undergoes periodic perturbations and reshapes to maintain its shape. During this reshaping, the soliton sheds energy through dispersive radiation over its entire spectrum. These linear waves are generated every round-trip, but only certain frequencies interfere constructively, leading to a resonant enhancement of the dispersive wave to form a strong, narrow sideband in the soliton spectrum. This occurs when

where *β*_{sol} and *β*_{lin} are the propagation constants of the soliton and the linear waves, respectively, and *m* is an integer. If we consider a linear wave propagating in a quartic dispersion cavity, $\beta lin=\u2212|\beta 4|(\omega \u2212\omega 0)4/24$, whereas the PQS has constant dispersion (i.e., it is unchirped) across its entire bandwidth of *β*_{PQS} = *K*|*β*_{4}|/*τ*^{4}, where *K* ≈ 0.62 is a constant that is found numerically.^{66} By substituting *β*_{sol} and *β*_{lin} in Eq. (26), it is straightforward to show that the *m*th-order spectral resonances *ω*_{m} satisfy

Thus, the fourth power of the sideband frequency offsets Δ*ω*^{4} are equally spaced by 48*π*/(|*β*_{4}|*L*).

Figure 12(a) shows the output optical spectrum of the PQS for an applied quartic dispersion *β*_{4} = −100 ps^{4} km^{−1} centered around *ω*_{0}. The white circles and diamonds mark the measured spectral positions of the low and high frequency sidebands, respectively. Figure 12(b) shows the fourth power of these sideband positions Δ*ω*^{4} vs the sideband order *m*, confirming a linear relation, consistent with Eq. (27). The spacing calculated from the measured spectral position of the sidebands is fixed at 0.824 and 0.829 ps^{−4} for the low (circles) and high (diamonds) frequencies, respectively. These are in excellent agreement with the theoretical value of 0.83 ps^{−4} following from Eq. (27). The small discrepancies can be attributed to the limited resolution of the sideband measurements or residual, uncompensated quadratic and cubic dispersion. This technique provides a simple way to confirm the nature and the magnitude of the net-cavity dispersion.^{123}

#### 3. Energy-width scaling

The laser can be used to confirm the energy-width scaling of PQSs predicted from Eq. (23). As the cavity is made of standard silica SMF, pulses do not suffer from nonlinear absorption such as free-carrier absorption (FCA) or two-photon absorption as in Si PhC-wg’s discussed in Sec. IV.

To assess the energy-width scaling of the cavity operating in the PQS regime, we adjust the pulse energy by tuning the pump power. For each value of pump power, we measure the average output power *P*_{av} and calculated the pulse energy *U*_{p} = *P*_{av}/*τ*_{rep}, where *τ*_{rep} is the repetition rate of the cavity. We also deduct the portion of the energy in the spectral sidebands, and we take into account the constant output coupling and the pulse-shaper insertion losses (∼5.6 dB). We measure the pulse width using the FREG setup.

The results for different values of applied quartic dispersion *β*_{4} are summarized in Fig. 13. The triangles show the pulse energies vs the pulse duration Δ*t*, each color corresponding to a different value of quartic dispersion. The solid color curves correspond to fits following Eq. (23). The consistency between the measured data and the prediction from Eq. (23) demonstrates that when operating in the PQS regime, the laser pulses do follow the new Δ*t*^{−3} energy-width scaling.

#### 4. Numerical modeling

Numerical modeling of the laser dynamics is an important tool that serves two objectives. First, it allows important insight into the physics and operational dynamics of the laser. Second, simulations provide a tool for simple and fast exploration of various operating regimes. Due to the number of variable parameters and the complexity of the laser, experimental exploration of the full parameter space is rarely possible and instead must be carried out numerically. Successful calculations require that the model adequately captures the complex dynamics of realistic experiments with minor approximations. We used the generalized nonlinear envelope equation to obtain a minimum-approximation description of light propagation through each fiber segment. This approach successfully models stable and highly nonlinear transient dynamics in advanced mode-locked lasers.^{125–128}

In a frame of reference moving at the group velocity of a frequency *ω*_{0}, the evolution of the complex electric field *ψ* in the fiber sections of the laser cavity is modeled by the modified generalized NLSE [Eq. (12)] that includes the effects of gain *g*,

where $D\u0302$ is the dispersion operator, which includes the intrinsic dispersion effects of the SMF section of the cavity but not that of the pulse pulse-shaper,

where *β*_{k} is the *k*th order of dispersion. For the examples below, we used standard SMF-28 with the following parameters: mode-field diameter (MFD) of 10.4 *μ*m, numerical aperture (NA) of 0.14, and *γ* = 1.3 W^{−1} km^{−1} at *λ*_{0} = 1560 nm. The dispersion coefficients are *β*_{2} = −21.4 ps^{2} km^{−1}, *β*_{3} = 0.12 ps^{3} km^{−1} and *β*_{4} = −0.0022 ps^{4} km^{−1} (see Refs. 72 and 120).

The first term on the right-hand side of Eq. (28) represents the gain in the doped fiber section and is calculated using

where *g*_{0} is the small-signal gain (corresponding to 25 dB in power) and is taken to be non-zero only in the doped-fiber section of the cavity. The other parameters in Eq. (30) are the pulse energy *E*(*z*) = *∫*|*A* (*z*, *T*)|^{2}*dT*, where the integration is over the entire time domain, and the saturation energy *E*_{sat}. The latter can be adjusted to simulate changing the pump power. To account for the spectral dependence of the gain, we multiply *g*(*z*) with a Lorentzian profile of 50 nm width to form the finite gain bandwidth *g* (*z*, *ω*_{0}). More advanced models have been suggested to include specific characteristic of active dopants and frequency-dependent saturation of ultrashort pulses.^{129–131} However, Eq. (30) is a reasonable, widely used approximation (see, e.g., Refs. 126 and 128). Equation (28) is solved with a standard symmetric split-step Fourier method algorithm.^{69} The dispersion and gain are calculated in the frequency domain, while the nonlinear term is calculated in the time domain.

The other elements of the laser cavity are modeled as lumped transmission elements. The mode-locking element is modeled by a transfer function that describes its transmittance

where *q*_{0} is the unsaturated loss of the saturable absorber, *P*(*τ*) = |*A* (*z*, *τ*)|^{2} is the instantaneous pulse power, and *P*_{0} is the saturation power. Finally, the spectral pulse-shaper is modeled by multiplying the electric field by a phase following the expression in Eq. (25) in the spectral domain as

The insertion losses (≈5.6 dB) of the spectral pulse-shaper are also included. Having separately simulated the light propagating in each component of the laser, it is possible to concatenate the model to produce a single model of the cavity using an iterative loop. Finally, we used an initial field composed of Gaussian random noise multiplied by a hyperbolic secant shape in the time domain and we ascertained that the same stable solutions are reached for different initial noise in a reasonable time. A detailed discussion of the numerical requirements for the split-step Fourier method can be found in Refs. 44 and 69.

Simulated steady-state PQS temporal and associated spectral profiles at the output of the laser for applied quartic dispersion (*β*_{4} = −80 ps^{4} km^{−1}), and corresponding to similar spectral bandwidth and output energy as in Fig. 11(b), are shown in Figs. 14(a) and 14(b), respectively. The other parameters values are *q*_{0} = 0.7 and *P*_{0} = 200 W. The saturation energy was set at *E*_{sat} = 105 pJ. The simulated laser output is in excellent agreement with our experimental results and is consistent with the change in the pulse-shaping mechanism. In particular, we observe similar spectral sidebands as in the experimental output spectrum. We also note that while these parameters and models are reasonable approximations, they correspond to physical quantities that are not always directly measurable experimentally.

The corresponding simulated evolution of pulse durations at FWHM (Δ*t*) and spectral bandwidths at −3 dB (Δ*ν*) for each saved step along the cavity are shown in Figs. 14(c) and 14(d), respectively. The pulse duration decreases in the SMF section following the gain fiber. In the spectral pulse-shaper, the pulse duration greatly increases due to the large applied negative quartic phase. In contrast, the pulse bandwidth changes only slightly over a round-trip. This highlights the lumped dynamics of the PQS pulse in a similar way that conventional solitons can arise in dispersion-managed lasers.

## VI. PROSPECTS IN OTHER PLATFORMS

To date, PQSs have been experimentally observed in two platforms, as discussed in Secs. IV C and V, both resulting in invaluable insight into underlying new physics. However, these two platforms suffer from technical limitations, which limit the range of experiments that can be performed. These could be overcome by using other photonic platforms, which could also greatly benefit from the advantageous properties of PQS. Below, we discuss the photonic platforms that are currently investigated to extend the range of applications of PQSs.

### A. Photonic crystal fibers

Photonic crystal fibers (PCFs) are the fiber equivalent of the PhC-wg’s discussed in Sec. IV C. These optical fibers guide light through a modified form of total internal reflection in a solid core, usually silica, surrounded by air holes.^{132} Due to the high contrast between core and cladding, PCFs present stronger optical confinement than conventional optical fibers, facilitating nonlinearities, and also superior possibilities for dispersion engineering.^{133,134} In contrast to silicon-based PhC-wg’s, PCFs can be fabricated using materials that are free from undesirable two-photon absorption and free-carrier effects and can support much higher pulse energies.

In this section, we discuss the design of PCF that can support PQSs. The fiber is not meant to be included in a laser cavity, and indeed, the modeling that we used for this does not involve gain or loss. Rather, the fiber is meant to be used for free propagation experiments in which a pulse with approximately the right width and energy is coupled into the fiber. It then evolves into a PQS upon propagation, similar to the original soliton experiments of Mollenauer *et al.*^{29}

Recall from Sec. IV C that a PQS supporting platform must have *β*_{2}, *β*_{3} ≈ 0 and *β*_{4} < 0. Silica, however, has a monotonically decreasing *β*_{2} and a large *β*_{3} around 1550 nm. To achieve the requirements for PQS propagation, we can find inspiration in the literature of PCFs with three zero-dispersion wavelengths and in ultraflattened PCFs, which can be based on two different approaches. The first of these rely on arrays of uniform hole size,^{135–137} which is simple to fabricate but usually suffer from high confinement loss due to a small hole radius/pitch ratio. The second relies on using a ring-to-ring gradient in the hole radius,^{138,139} which leads to generally lower confinement loss but increased complexity of fabrication, since the applied pressure to each of the rings during drawing must be different.

Some of us reported realistic designs for PQS-supporting PCFs^{140} based on a compromise between the two approaches above: a PCF comprising rings of only two different hole radii. The two innermost rings comprise holes with a small radius, which have a strong impact on shaping the dispersion. The three outermost rings consist of holes of larger radius, which form a barrier and thus reduce confinement loss. The exact parameters small hole diameter (*d*), big hole diameter (*D*), and pitch (*a*) were determined through a careful simulation and optimization process,^{140} which can be summarized as follows. First, we chose a pitch value range according to our desired operational wavelengths, since the pitch determines the frequency at which *β*_{3} = 0.^{138} Second, we performed dispersion and loss calculations within that range by sweeping *d* and *D* and looking for combination of these three parameters that yield dominant quartic dispersion at the desired wavelengths. Then, we fixed a maximum acceptable loss, depending on the application, and choose the design that simplifies fabrication from those with low enough loss. This usually entails minimizing the contrast between *d* and *D*.

The inset of Fig. 15(d) shows a concrete example of a PQS-supporting PCF design with *a* = 1.92 *μ*m, *d* = 578 nm, and *D* = 1.027 *μ*m.^{140} The dispersion parameters shown in Figs. 15(a)–15(c) reveal that *β*_{2} is an approximately quadratic function of frequency with a maximum at 1550 nm, that *β*_{3} = 0, and that *β*_{4} < 0 at these wavelengths. The different curves in Fig. 15(d) show the pulse duration (*T*_{0}) under which the quartic dispersion is increasingly dominant over the second and third order dispersion: *L*_{2}, *L*_{3} > *L*_{4} (cyan), *L*_{2}, *L*_{3} > 2*L*_{4} (yellow), and *L*_{2}, *L*_{3} > 4*L*_{4} (green) at each wavelength. Using these dispersion parameters, the effective area of the fiber *A*_{eff} = 14.5 *μ*m^{2} and the well-known nonlinear coefficient of silica *n*_{2} = 2.6 × 10^{−20} m^{2}/W, it was shown using NLSE simulations that fundamental PQS propagation would be supported for pulses of duration *T*_{0} = 40 fs and peak power *P*_{0} = 2 W. There are, obviously, many other possible combinations of pulse width and peak power that would result in PQS propagation but always bearing in mind that the pulse duration must be kept under the curves in Fig. 15(d) to guarantee quartic dispersion dominance.

PCFs could become a promising platform for the development of PQS functionality. Specially designed PCFs could become central pieces in PQS laser fiber cavities, potentially substituting the role of the pulse-shaper and even of the conventional Erbium doped fibers in Ref. 67, as PCFs can also be doped and provide gain.^{141} However, we should not underestimate the complexity of fabrication of PCFs with more than one hole radius. Assessing the real potential of PCFs for supporting PQSs and their generalizations will become possible only once realistic designs, such as those proposed in Ref. 140, are successfully fabricated.

### B. Microring resonators (ABRs)

Microring resonators (MRRs) are ring-shaped micrometer-scale waveguides forming optical cavities supporting a series of optical standing wave modes or resonances.^{43} MRRs have recently become a blooming field of study given their compact footprint, dispersion engineering possibilities, and most importantly, their role in shrinking frequency comb technology, and its applications.^{43,142} MRRs can operate in a regime in which a continuous-wave pump is converted through nonlinear effects into a pulse train in time, equivalent to a frequency comb in the spectral domain.^{42} These pulses are referred to as *cavity solitons* because, again, they arise from the balance nonlinearity and dispersion.^{143}

Cavity solitons in MRR can be described by a generalized NLSE.^{144} Additionally though, the MRR configuration imposes a boundary condition on the circulating field at the beginning of the (*m* + 1)th roundtrip that depends on the field at the end of the *m*th roundtrip. Applying this boundary condition and assuming that the field varies slightly between consecutive roundtrips, the generalized NLSE can be averaged into an externally driven NLSE,^{145} which corresponds to the generalized Lugiato–Lefever equation (LLE).^{146} For anomalous group-velocity dispersion, the intracavity field solution of the LLE corresponds to a cavity soliton with a hyperbolic secant shape.^{42,142}

While quadratic dispersion usually dominates in MRRs, as in a standard optical fiber, early work made clear that higher-order dispersion, in particular, quartic, plays an important role in extending the spectral bandwidth of octave-spanning frequency combs.^{147} Through simulations of the LLE, Bao *et al.* highlighted the importance of including all orders of dispersion when studying Kerr combs in MRRs, as high-order dispersion affects crucial performance parameters of the comb, such as its bandwidth, conversion efficiency, output pulse peak power and temporal shape, and the position of the dispersive waves.^{148} Moreover, they showed that Kerr comb generation should be possible in a MRR with pure-quartic dispersion and that, in the more realistic case of a hybrid quadratic-quartic dispersion, increasing the relative strength of the quartic dispersion leads to increased pulse energy and pump-comb conversion efficiency.

In 2019, Taheri and Matsko showed theoretically the existence of *cavity pure-quartic solitons* in MRRs with dominant quartic dispersion.^{149} They solved the LLE with *β*_{4} < 0 as the only dispersion term and showed that *cavity PQSs* can arise from the interaction of quartic dispersion and Kerr nonlinearity in MRR cavities. Analytical expressions for the pulse amplitude, width, and energy in a MRR with pure-quartic dispersion were provided in this study. This was done using the Lagrangian variational method with a Gaussian ansatz, which is justified given that PQSs have an approximately Gaussian shape near the center of the pulse.^{61} These expressions showed that the pulse peak power of PQSs in MRRs scales linearly with the detuning of the pump frequency from the closest cavity resonance. They also predicted a linear relation between the fourth power of the width and the fourth order dispersion coefficient and with the inverse fourth power of the detuning. Taheri and Matsko’s analysis concluded with an area theorem for PQSs in MRRs that confirmed that the energy of PQSs in MRRs also scales up as 1/Δ*t*^{3}.^{149} This is evidence that in this platform as well, the energy of PQS increases more strongly with decreasing pulse width than that of their conventional dissipative Kerr soliton counterpart.

These theoretical studies were recently complemented by a numerical investigation of the effect of stimulated Raman scattering on PQSs in micro-cavities.^{150} The soliton self-frequency shift induced by stimulated Raman scattering in microresonators is non-negligible in materials with high Raman gain.^{151} In their recent study, Liu *et al.*^{150} showed that stimulated Raman scattering leads to redshift and temporal and spectral asymmetry of dissipative PQSs in MRRs. It also obstructs the linear scaling between pump detuning and peak power, much like in the case of dissipative Kerr solitons with quadratic dispersion. Interestingly, Liu *et al.* found an intermediate stable regime between the region of existence of breathers and chaos when increasing the pump power,^{150} which has not been known to occur for any soliton regime other than PQSs.

The aforementioned works simply assumed dominant negative quartic dispersion without considering the physical requirements (material, geometry) of the MRR to achieve such requirement. MRRs’ dispersion can be easily tailored by modifying the ring waveguide geometry.^{147,152,153} Recently, Yao *et al.* reported a design for a MRR made of aluminum nitride (AlN) with realistic structural dimensions, which exhibits dominant negative quartic dispersion.^{154} This approach is promising and could be further supported by novel fabrication techniques that offer greater engineering flexibility, including atomic layer deposition, and by the design notion of concentric racetrack-shaped geometries.^{155,156}

These set of results confirm that PQSs can be generated in the MRR platform, which could have an impact on integrated photonics applications by enabling for the direct generation of ultrashort high-energy pulses on-chip.^{157,158}

## VII. OTHER GENERALIZATIONS

Until now, we focused our analysis around the case where *β*_{4} is the only nonzero term in the Taylor expansion of the propagation constant *β*(*ω*) [see Eq. (19)]. Here, we briefly discuss the generalization to other dispersion and combinations thereof, which also lead to soliton formation.

### A. Pure high, even-order dispersion solitons

As discussed in Secs. III and IV B, both conventional and pure-quartic solitons arise from the balance between SPM and a group velocity *v*_{g} monotonically increasing with frequency. This second condition is satisfied for negative quadratic or quartic dispersion and, in fact, for all higher, even-order types of negative dispersion, as illustrated in Fig. 2. Thus, perhaps the most obvious generalization is the case of solitons arising from pure negative dispersion orders higher than four.

Such optical pulses can be described by generalizing Eq. (19) to

where *β*_{k} is the dispersion coefficient, which is taken to be negative. For *k* = 2 and *k* = 4, we see that Eq. (33) describes the conventional solitons [Eq. (13)] and PQSs [Eq. (19)], respectively. Using a similar ansatz as for the PQS case, we solved Eq. (33) and found stationary solutions corresponding to pure high, even-order dispersion (PHEOD) solitons for orders up to *k* = 16, limited only by the ability to evaluate high derivatives numerically.^{68} As for conventional and the pure-quartic solitons, each of these solitons follow a different energy-width scaling relation that takes the form

where *M*_{k} are constants that can be found numerically.^{68,86} These numerical results imply the existence of an infinite family of solitons for any negative even order of dispersion.

By using a similar experimental setup as the one schematized in Fig. 10 and by simply reprogramming the spectral pulse-shaper, we experimentally generated pure-sextic (*β*_{6}), pure-octic (*β*_{8}), and pure-decic (*β*_{10}) solitons. Examples of the measured spectral (left column) and temporal (right column) intensity shapes are shown in Fig. 16 (solid blue curves). For all cases, the experimental and numerically calculated (red dashed lines) shapes agree well. We note, in particular, spectral fluctuations away from the pulses. This is because the pulse-shaper has a limited spectral resolution. As a consequence, for rapidly varying functions (i.e., high order dispersion), the pulse-shaper undersamples the phase profile, leading to aliasing in the applied phase mask. However, this limitation does not affect the pulse dynamics significantly as the spectral fluctuations appear far from the central frequency and at least 10 dB below the peak of the spectrum.

### B. Hybrid dispersion

So far in this Tutorial, we considered optical solitons arising from the interplay between Kerr nonlinearity and a single order of dispersion, while all other dispersion coefficients were assumed to be zero or were negligible.^{61,68} However, some recent studies considered *hybrid* dispersion that includes several non-zero even orders of dispersion.

According to the heuristic argument in Sec. III, a soliton should also arise for any combination of negative quadratic and negative quartic dispersion. This was confirmed numerically by Tam *et al.*^{77} A tail analysis similar to that in Sec. IV B indicates the boundary between the parameter regions where the soliton tails oscillate while they decay, as for PQSs, and where they decay monotonically, as for conventional solitons.

The case with *β*_{4} < 0 and *β*_{2} > 0 is more complicated and more interesting, as the dispersion relation now has two equivalent maxima. Tam *et al.*^{77} showed that a novel type of solitons can exist in the presence of negative quartic and positive quadratic dispersion. These solitons can be thought of as a sinusoidal function modulated by an envelope that satisfies an NLSE. This novel type of optical solitons was independently discovered by Melchert *et al.* in numerical studies in a different context.^{159,160}

In this Tutorial, we have only considered even orders of dispersion. However, we note that exact soliton solutions in the presence of odd orders of dispersion are known in systems in which the dispersion is generated by a grating.^{161,162} However, in these solutions, the quadratic dispersion dominates, and the odd dispersion can be considered to be a weak effect. Finally, we mention that early studies considered the effects of higher-order dispersion on MI, another nonlinear process that can lead to the generation of a train of ultrashort pulses.^{163–165}

## VIII. DISCUSSION AND CONCLUSION

Since the earliest demonstration of solitons in optical fibers,^{29} these pulses have always been at the forefront of the development of nonlinear optics, and soliton-based systems have enhanced numerous photonic applications, such as telecommunications,^{36,166} ultrafast lasers,^{37,38,88} and frequency comb generation.^{42,43} These works have been based on pulses arising from balancing the Kerr nonlinearity and the quadratic dispersion. This is because second order dispersion is the dominant dispersion contribution in conventional waveguides. Moreover, a series of studies in the early 1990s concluded that higher-order dispersion was detrimental for the stability of soliton.^{59,60}

In this Tutorial, we have shown that changing the linear properties of the medium can change the properties of solitons in unexpected, interesting, and potentially useful ways. This represents a paradigm shift, highlighted by the experimental discovery of PQSs in 2016 by some of us, in a dispersion engineered PhC-wg.^{61} Such structures offer unparalleled dispersion engineering possibilities.

While quartic dispersion changes the shape of the soliton, the most important of which are oscillating tails in time and a flattened shape in frequency, this novel type of linear/nonlinearity interaction has also deeper consequences on the properties of this type of nonlinear pulses. The most important probably being the energy-width scaling relation of PQSs (*U*_{PQS} ∝ Δ*t*^{−3}), which could have significant implications for soliton-based laser systems or MRR frequency combs.^{66,149}

Ongoing efforts focus on generating PQSs in other platforms such as PCFs and MRRs, but none are without challenges.^{140,154} In parallel, we developed a fiber laser with a programmable spectral pulse-shaper. This allows for the flexible, precise tailoring of the net-cavity dispersion. We can cancel the native dispersion of the cavity and to impose any other type of dispersion (subject to the specification of the device), including negative pure-quartic dispersion. The drawback is that this is now an active system, and the solitons that we observe are thus strictly speaking dissipative solitons, solitons that balance not only dispersion and nonlinearity but also gain and loss.^{167} However, our experimental results are in excellent agreement with theory and with numerical simulations for lossless systems. We note that the incorporation of the spectral pulse-shaper in optical systems offers considerable flexibility to access new regimes of operations and generate novel nonlinear pulses. This approach is currently a growing trend in photonics.^{114,168}

Subsequent studies investigated the existence of sub-families of PQSs such as *vector*^{169} or *dissipative*^{149} PQSs. In both cases, the formation of the localized structure still primarily relies on the interplay between quartic dispersion and nonlinearity, but other physical effects also play an important role. Vector solitons have two components usually corresponding to the two orthogonal modes in a birefringent waveguide. For dissipative solitons, the pulse generation and propagation requires a second balance between two competing effects, parametric gain and loss.^{40}

While we focused here on systems with negative dispersion, as required for soliton generation, higher-order dispersion also expands the family of nonlinear waves with unique shapes and properties, such as self-similarity and Riemann waves.^{170,171} Positive quartic dispersion might also enable the existence of dark pure-quartic solitons, a region with vanishing or low intensity in a constant, higher-intensity background. Although these have not been reported, recent theoretical studies of dark solitons in related systems^{172} suggest the likely existence of dark solitons in the presence of pure-quartic and combinations of quartic and quadratic dispersion.

The novel PQS concept has also triggered several studies in applied mathematics. Some of these studies investigate the dynamical properties and multiple higher-order dispersion solitons. For example, the interaction between temporally adjacent solitons depends on the relative phase of the tails. Therefore, the oscillations in the tails of PQSs affect the way these solitons interact. Some recent research has investigated the properties of multipeak solutions including their stability.^{173,174} The discovery of PQSs has also encouraged investigation of exact solutions of the generalized NLSE with different orders of dispersion and different orders of nonlinearity. A typical example is the paper by Biswas *et al.*^{175}

In conclusion, considering higher-order dispersion allows for revisiting the role of linear effects in NLSE-governed systems. Instead of acting as a fixed, limiting parameter of the soliton problem, dominant higher-order dispersion opens a novel route to an infinite family of nonlinear pulses with soliton properties. This has both technological and fundamental consequences as it allows for the generation of high-energy optical solitons in lasers or integrated photonic systems.

## ACKNOWLEDGMENTS

The authors thank their colleagues and collaborators who contributed to the cited work, particularly Kevin Tam, Joshua Lourdesamy, Long Qiang, Tristram Alexander, Mohammad Rafat, and Chad Husko. They also thank many of their colleagues for soliton-related discussions over many years, including Nail Akhmediev, Neil Broderick, John Dudley, Ben Eggleton, Irina Kabakova, Yuri Kivshar, John Sipe, Mike Steel, and Eduard Tsoy. This publication was supported by the following research funds: Australian Research Council (ARC) Discovery Project (Grant No. DP180102234) and the Asian Office of Aerospace R&D (AOARD) (Grant No. FA2386-19-1-4067).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## REFERENCES

_{2}:Eu

^{2+}

We have used Δt here since there is no equivalent of *T*_{0} for PQSs. This does not matter as it does not affect the shape of the scaling relation but only the prefactor in Eq. (23), which we find numerically anyway.

^{14}W/cm

^{2}peak intensity at 136 MHz

^{3+}-doped photonic crystal fibers

_{3}N

_{4}resonators

^{4}model with quadratic and quartic dispersion