Models of few optical cycle solitons beyond the slowly varying envelope approximation

TL;DR: In this article, the authors used the powerful reductive expansion method (alias multiscale analysis) to derive simple integrable and nonintegrable evolution models describing both nonlinear wave propagation and interaction of ultrashort (femtosecond) pulses.

About: This article is published in Physics Reports.The article was published on 2013-02-01 and is currently open access. It has received 335 citations till now. The article focuses on the topics: Soliton & Korteweg–de Vries equation.

Summary

1. Introduction

• Following a series of challenging works on experimental generation and characterization of two-cycle and even subtwo-cycle pulses from Kerr-lens mode-locked Ti: sapphire lasers [1–4], interest in intense ultrashort light pulses containing only a few optical cycles has grown steadily in recent years since their first experimental realization more than one decade ago.
• Notably, the ultrashort pulses possess extensive applications to the field of light–matter interactions, high-order harmonic generation, extreme [5] and singlecycle [6] nonlinear optics, and attosecond physics [7,8]; see Ref. [9] for a review of earlier works in this area.
• This second revolutionary amplification technique is currently used for broadband, few-optical-cycle pulse amplification.
• Non-SVEA models were proposed within the framework of the unidirectional approximation; see, e.g., Refs. [18] and [56].
• In Section 4 the authors derive a general model for few-cycle optical soliton propagation, namely a modified Korteweg–de Vries–sine–Gordon equation, which describes two-cycle optical pulses propagating in two-component nonlinearmedia.

2.1. Basic notions of the method and its applications

• The reductive expansion method or multiscale analysis is a very powerful way of deriving simplified models describing both nonlinear wave propagation and interaction; see e.g., a tutorial review of this topic [93].
• Here the authors only describe the basic notions of this mathematical method which is widespread in the modern soliton theory.
• From the physical nature of the phenomenon itself, the authors again have multiple scales.
• Then, the complete integrability may still be ensured if and only if some relationships between the coefficients of the nonlinear evolution equation are fulfilled.
• Later on, Taniuti and Wei [95] provided a general method of the derivation of the Korteweg–de Vries equation by using the reductive perturbation technique.

2.1.1. Length scales for envelope solitons

• The nonlinear evolution of dispersivewave packets, e.g., those propagating through optical fibers, involves several length scales, the phase velocity of the wave being a constant in this specific case [85].
• A third scale length is the propagation distance D; see Fig.
• The authors are briefly discussing here the formation of envelope solitons, that is, an equilibrium between the dispersion effect and the nonlinear effect should occur.
• Here k2 = d2ω/dk2 measures the dispersion, and ∆t is the initial pulse duration (in the picosecond range for envelope solitons in fibers).
• In this way the authors have identified the small perturbation parameter ε for this physical situation.

2.1.2. Length scales for long-wave and short-wave expansions

• The authors briefly consider here some typical examples of long-wave expansions in the study of solitons.
• One of the first works devoted to this issuewas that of Su andGardner [96].
• The authors note that the Burgers’ equation describes an equilibriumbetween nonlinearity and dissipation, and the formation of smooth shock profiles, whereas the KdV equation describes an equilibrium between nonlinearity and dispersion and the formation of localized waveforms propagating and interacting with each other without shape deformation; such robust localized waves were called solitons.
• Therefore the nonlinear and dispersion terms should have the same order of magnitude, that is, ε3g = εg2.
• Note that the generic KdV equation describing half-cycle optical solitons in quadratic nonlinear media will be derived by using the quantum formalism in Section 3.3.

2.2. Direct derivation of a macroscopic nonlinear Schrödinger equation from Maxwell–Bloch equations

• It is possible to derive in a rigorousway theNLS equation,which is amacroscopic nonlinear evolution equation for a light pulse in a nonlinear medium; see Ref. [103] for a comprehensive study of this issue.the authors.
• In the following the authors only consider the simple case of a monochromatic plane wave, interacting with independent two-level atoms.
• The authors will show that for the linear part of this particular, though relevant case, the obtained results agree with those of the linear dispersion theory.
• Instead, in these two fields the standard framework is themultiscale analysiswhich allows us to get in a systematic and rigorous way the wave-packets evolution equations directly from the basic equations, such as the Navier–Stokes in hydrodynamics, or the Maxwell–Landau ones in ferromagnetism.

2.2.1. The Maxwell–Bloch equations and the multiscale analysis.

• Which allow us a rigorous derivation of the macroscopic NLS equation from the microscopic quantum theory; see Ref. [103] for a detailed first study of this issue.the authors.
• Notice that the set of Eqs. (2.3)–(2.6) is sometimes called the Maxwell–Bloch equations, although this name usually denotes a reduction of this set of equations; see, e.g., [104].
• When using phenomenological response functions, it is rather difficult to give a reference point for this smallness.
• From the energetic point of view, the authors must compare the energy h̄ω of the incident photon to the energy difference between the atomic levels h̄Ω .
• It is a well-known fact that only the transitions corresponding to frequencies of the same order of magnitude as the wave frequency ω affect appreciably the wave propagation.

2.2.2. The resolution of the perturbative expansion and the obtaining of a NLS equation.

• Wenext give only themain line of the argument; for a detailed account of the derivation of themacroscopic NLS equation from the quantum theory, by using a suitable multiscale expansion; see Ref. [103].
• Notice first that the dispersion relation (2.15) exactly coincides with that found from the linear susceptibility ↔ χ (1) (ω) computed, e.g., in Ref. [104], in the case of density matrix description of a two-level model, as was discussed here.
• As concerning the nonlinear coefficient γ the authors restrict ourselves to a specific physical situation by assuming that the polarization operator µ⃗, in the two-level model, describes oscillations of the molecular dipole along some direction, making an angle α with the propagation direction z (Eq. (2.18)).
• Therefore as a final conclusion of this section the authors state that the multiple scale expansion theory agrees only qualitatively with previous calculations performed by using the susceptibility series expansion, even in the simpler case of a linearly polarized transverse wave.

3.1.1. The multiple scales method and the derivation of the modified KdV equation from quantum equations

• Without the use of the slowly varying envelope approximation.the authors.
• The authors assume that the resonance frequency of the two-level atoms is well above the inverse of the characteristic duration of the pulse (long-wave regime).
• The authors will see that the two-soliton solution of the mKdV equation is very close to the experimentally observed two-cycle pulses as reported, e.g., in Ref. [111].

3.1.2. The breather soliton of the modified KdV equation: a prototype of few-cycle solitons

• The mKdV equation (3.14) is completely integrable by means of the inverse scattering transform; see e.g., [113].
• This occurs in general if χ (3) and n′′0 have opposite sign, which happens most frequently for χ (3) > 0 and anomalous dispersion.
• One notices that both pulse profile and spectrum are comparable to the corresponding results reported in experiments on FCPs; see e.g., [111].
• Thus when the resonance frequency is well above the inverse of the typical pulse width of about few femtoseconds, a long-wave approximation leads to a mKdV equation which adequately describes two-cycle optical solitons.

3.2.1. The multiple scales method and the derivation of the sine–Gordon evolution equation from quantum equations

• I.e., the situation in which the resonance frequencyΩ of the atoms is well below the optical frequencies ω (Ω ≪ ω), in the infrared if the FCP belongs to the visible range.the authors.
• The sGmodel was originally derived in [73] using an alternative scaling, which is fully equivalent to the present one and in which the Hamiltonian H0 of the atom was replaced in the Schrödinger–von Neumann equation (3.3) by εĤ0.
• The authors use here a high-amplitude assumption, in contrast with the low amplitude assumption used in the long-wave approximation.
• The quantities E andw describe here the electric field and population difference themselves, and not amplitudes modulating a carrier with frequency Ω .
• At this point a brief comment should be made concerning the validity of the model equations (3.30)–(3.32).

3.2.2. The breather soliton of the sine–Gordon equation: a few-cycle soliton

• It is a well known fact that the sine–Gordon equation (3.40) is completely integrable [86].
• One notices that when these arbitrary parameters take real values, Eqs. (3.41)–(3.43) describe the interaction of two sine–Gordon solitons.
• As in the case of the long-wave approximation regime discussed in the preceding section, the two-soliton solution (3.41)–(3.43) is also able to describe soliton-type propagation of an ultrashort optical pulse in the two-cycle regime; the obtained analytical solution is very close in shape and spectrum to the ultrashort femtosecond-pulses of this type currently produced in experiments; see e.g., Ref. [111].
• L̂ is inversely proportional to the population difference parameter wi, a relatively small value of the population difference increases the propagation distance at which nonlinear effects occur.
• When the resonance frequency is well below the optical field frequency, a short-wave approximation leads to amodel similar to that describing self-induced transparency, but in very different validity conditions.

3.3. Ultrashort pulses in quadratically nonlinear media: half-cycle optical solitons

• Which exhibit a single hump, with no oscillating tails; for a detailed study of this issue, see Ref. [120].the authors.
• The authors also mention here the important earlier works by Kazantseva et al. [121–123] on propagation and interaction of extremely short electromagnetic pulses in quadratic nonlinearmedia.
• An important issue is the phase invariance of a FCP; in some sense the FCP loses the phase invariance, and the importance of the so-called carrier-envelope phase has been emphasized in a series of works; see e.g., Ref. [7].
• One notices that for long pulse durations, when the slowly varying envelope approximation is valid, the study of optical solitons in quadratic nonlinear media has shown several unique features with respect to the cubic (Kerr) case [125,126].
• These equations present cubic nonlinearities, while the completely integrable KdV equation itself is quadratic, and the authors show in the following, that FCP soliton propagation in a quadratic medium can be adequately described by the latter [120].

3.3.1. Derivation of a KdV equation

• The evolution of the electric field E (for the sake of simplicity the authors restrict to only one field component) is described by the Maxwell wave equations.
• The density matrix obeys the Schrödinger–von Neumann evolution equation ih̄∂tρ = [H, ρ] + R, where as in the case of cubic optical nonlinearity [73], the phenomenological relaxation term R can be neglected.
• It is fully equivalent to assume that the excited state ‘b’ has some permanent dipolar momentum.
• The authors consider here the latter case, i.e. they assume that ωwis much smaller thanΩ , i.e., they consider the long-wave propagation regime.
• A brief comment is necessary at this point: here the authors have a pure quadratic nonlinearity for a single wave, and no phase matching is required,whichmakes a sharp contrastwith the nonlinear propagation of quadratic solitonswithin the SVEA where phase matching is needed.

3.3.2. Half-cycle optical solitons

• At this point let us remind the reader a couple of well known properties of such localized solutions of completely integrable models.
• By further increasing the value of carrier-envelope phase the number of emitted solitons remains equal to two; see Fig.

4.1. Two-cycle optical pulses propagating in two-component nonlinear media: derivation of a governing nonlinear evolution equation

• The authors analyze the response of a two-component medium of two-level atoms driven by a two-cycle optical pulse beyond the traditional approach of slowly varying amplitudes and phases; see Ref. [76] for a detailed study of this problem.
• The FCP soliton depends strongly on the dopant’s matrix elements as well as on the relative population difference of the two components.
• In the general non-integrable case, the existence of two-cycle solitons and the stabilization of the carrier-envelope phase have been demonstrated by adequate numerical techniques; see Ref. [76].
• Thus the authors consider the time-dependent propagation of a femtosecond pulse through a two-component medium.

4.2. The integrable modified Korteweg–de Vries–sine–Gordon equation: envelope, phase, and group velocities for the two-cycle pulses

• Notice that the above exact twosoliton solution Q remains valid for any complex values of the quantities η1,2, c10, and c20.
• An important fact is that the above breather solution can be decomposed into a carrier wave and an envelope, not only in an approximate way in the SVEA limit, but also exactly, for a FCP solution, hence generalizing the notions of carrier and envelope beyond the SVEA regime.
• Indeed, the evolution of the spectral data in the frame of the IST is given by the linear dispersion relation, which ensures that the latter is valid on the imaginary axis.

4.3. The non-integrable mKdV–sG equation: robust two-cycle optical solitons

• In Ref. [76] it was proved by numerical techniques the existence and robustness of two-cycle dispersion-free pulses in the two-cycle regime, in the general, nonintegrable case, i.e., when the coefficients a ≠ 0 and b ≠ 1/2 in the evolution equation (4.16).
• In order to perform this analysis the exact breather solution Utwo of the mKdV equation was used as the input field; see Eq. (3.20).
• For the numerical calculations a fixed value b ≠ 1/2 was considered, and a negative value of the strength a of the resonance term was chosen.
• The exponential time differencing method (see Ref. [115]) along with absorbing boundary conditions in order to avoid numerical instability of the background was used in the computations.
• Notice that for positive values of the parameter a, that is, when both of the initial population differences W1,2(−∞) are of the same sign, the situation is somehow different, in the sense that a robust FCP still exists in this case, but appreciably differs from the mKdV breather; see Ref. [76] for more details.

4.4. Few-optical-cycle solitons: the modified Korteweg–de Vries–sine–Gordon equation versus other non SVEA models

• And so demonstrating its remarkable mathematical capabilities in describing the physics of few-cycle-pulse optical solitons; for a detailed study see Ref. [78].the authors.
• Other non-SVEA models [141–143], especially the so-called short-pulse equation (SPE) [144], have been proposed in the literature.
• The authors then show that both the SPE, and another model put forward in Refs. [141–143,81] can be considered as approximate versions of a generic mKdV–sG equation.
• However the FCP solitons obtained in [81] differ from the breather solutions of mKdV–sG equation considered in Ref. [76].
• Numerical computations confirm the qualitative conclusions, and also that the FCP propagation strongly differs from that one predicted by the SVEA.

4.4.1. A comparison of mKdV, sG, and mKdV–sG models for describing few-cycle solitons

• A temporal soliton, ormore properly a temporal solitarywave, is a pulsewhich propagates in a highly dispersivemedium in such a way that a certain nonlinear effect exactly compensates dispersion (in the sense of the natural tendency of spreading of the pulse), and the pulse shape remains unchanged during propagation.
• The approximation used in deriving the above mKdV–sG equation is quite realistic in the general setting.
• Hence, all optical transitions of the medium can be separated into two distinct groups, some transitions well below ω, and the other ones well above ω.
• If each of these two sets of resonance frequencies is approximated by a single transition, the authors exactly get the assumptions underwhich themKdV–sGmodel has been derived.
• For other values of its coefficients, it has been shown by numerical simulations that FCP solitons (or breathers) still exist, and their robustness has been investigated too; see Ref. [76].

4.4.2. The short-pulse equation: a special case of the mKdV–sG equation

• Also, the reduction of the bi-directional Maxwell equations to a uni-directional one was performed by means of a short wave approximation.
• The mKdV–sG model is able to predict pulse compression, as shown in Fig. 12, which is similar to the result presented in Ref. [81].
• It is worthy to mention that the mKdV–sG equation supports FCP solitons, but only the case of focusing mKdV equation was considered in Ref. [76].
• It is close to, but still differs from the value of group velocity dω/dk ≃ 0.229 predicted by the SVEA by using the corresponding linear dispersion relation k = −(c1/ω)− c3ω3.

5. Few-optical-cycle solitons: their interactions

• In this section the authors consider the problem of few-optical-cycle soliton interactions.
• Following Ref. [77], by using the exact four-soliton solutions of themodified Korteweg–de Vries–sine–Gordon equation describing the propagation of few-opticalcycle pulses in transparent media with instantaneous cubic nonlinearity, the authors study the interaction of two such initially wellseparated pulses.
• One notices that in Ref. [77] the shapes of soliton envelopes, the shifts in the location of envelopesmaxima, and the corresponding phase shifts were explicitly calculated.
• Though the mKdV and sG equations are completely integrable by means of the inverse scattering transform method [85], the mKdV–sG equation is completely integrable only if some condition between its coefficients is satisfied [87].
• Note that the two-breather solution describes the interaction in a Kerr medium of two few-optical-cycle solitons initiallywell separated, in any physical settingwhere one of the three abovementioned integrable models (KdV, sG, mKdV–sG) is a realistic one.

5.1. Exact four-soliton and two-breather solutions of the integrable mKdV–sG equation

• Here the authors consider the propagation of optical FCPs in a one dimensional self-focusing Kerr medium, such as a highly nonlinear optical fiber.
• Therefore they are expected to interact and to cross each other.
• Here the authors briefly discuss what happens during the interaction of such ultrashort pulses; see Ref. [77].
• The evolution of the electric field is governed by the mKdV–sG equation, which is in general a nonintegrable nonlinear evolution equation.
• These solutions are properly speaking solitons only for real values of the parameters kj, but the solution holds for any complex values of kj.

5.2. Interaction of two few-cycle solitons

• The shift in the location of the breathers appears clearly in Fig. 15.
• In Ref. [77], it was seen that the envelopes uj+e, corresponding to the largest values of τ , did not have the correct amplitudes.
• This feature brought forward the existence of the arising of a phase shift during the interaction.
• Concluding this section the authors stress that the shapes of both input and output soliton envelopes as well as the phase and location shifts have been computed in Ref. [77] by using the exact expression for the four-soliton (two-breather) solution of the mKdV–sG equation.

6.1.1. Basic equations for an amorphous optical medium

• The atomsmay present some induced dipolar electricmomentum µ⃗, which is oriented randomly in space.
• The physical values of the relaxation times are indeed in the picosecond range, or even slower, thus very large with regard to the pulse duration, which allows us to neglect them.
• The typical frequencyωw of the wavemust be far away from the resonance frequencyΩ because the transparency of the medium is required for soliton propagation.
• Thus the authors are working in the long wave approximation regime, as defined in the framework of the reductive perturbation method [95,93].
• As in the scalar model [73], the dispersion coefficients A has the same expression as derivedwithin the scalar model (Eq. (3.12)), if the authors consider that only one half of the dipoles are contributing.

6.1.2. Basic equations for a crystal-like optical medium

• This model involved some induced dipolar electric momentum µ⃗, oriented randomly in the transverse plane (x, y), the polarization density P⃗ being averaged over all directions in this plane.
• Wewill show below that the same governing equations (6.6)–(6.7) can be derived from another model, which would rather correspond to a crystalline structure [161].
• This alternative approach involves a two-levelmedium, inwhich the excited level is twice degenerated, with the induced dipole oriented either in the x or in the y direction.
• The retarded time variable τ describes the pulse shape, propagating at speed V in a first approximation, and the ζ variable describes long-distance propagation.
• Regarding the value (6.32) of the coefficient B, the ratio between the corresponding nonlinear coefficients is a bit smaller, 3/8, which is nothing else but the average value cos4 θ of cos4 θ , which is involved in the averaging of the nonlinear polarization density over all orientations θ of µ⃗ in the glass model (Eq. (3.12) and [160]).

6.1.3. The complex mKdV equation

• All linearly polarized FCP solitons are retrieved in this way.
• (6.42) However no exact, even numerical, steady state solution of this type do exist.
• They are studied in detail in the next subsection.

6.1.4. Robust circularly polarized few-optical-cycle solitons

• The authors will next compute an approximate analytic solution to the cmKdV equation (6.40), valid for long pulses, i.e. in the SVEA.
• The numerical resolution of the cmKdV equation is performed using the exponential time differencing second order Runge–Kutta (ETD–RK2) method [115].
• The authors will next show by numerical simulations that they get robust circularly polarized FCP solitons.
• However the mean value of the field is conserved.
• Numerical resolution shows that the pulse, apart from small apparently chaotic oscillations, keeps its shape and characteristics during the propagation.

6.2. Circularly polarized few-optical-cycle solitons: short wave approximation

• In media in which the dynamics of constituent atoms is described by a two-level Hamiltonian, by taking into account the wave polarization and in the short-wave approximation regime.the authors.
• By using the reductive perturbation method (multiscale analysis) the authors then derive from the Maxwell–Bloch–Schrödinger equations the governing evolution equations for the two polarization components of the electric field in the first order of the perturbation approach.
• It is worth mentioning that other vectorial nonSVEA models have been also proposed [158,142], however they were only built from a direct analogy with common SVEA models.
• It is alsoworthmentioning that circularly polarized short pulse propagation in a systemof two-level atoms has been studied more than two decades ago in the framework of the self-induced transparency [172] and the existence of localized solutions ofMaxwell–Bloch type systems beyond the SVEA has been considered too [173,174].
• Not all the coupling mechanisms between the polarization components were taken into account in these earlier studies.

6.2.1. Basic equations and the short-wave approximation

• The short-wave approximation is performed according to the general theory developed in Refs. [73,175,176].
• In the followingthe authors seek for an approximate solution of circularly polarized pulses in the limit of largeω, i.e., in the SVEA limit, bymeans of amultiscale expansion very similar to the standard one for deriving a NLS equationmodel in the SVEA limit [93].

6.2.2. Lifetime of circularly polarized few-cycle pulses and transition to single-humped ones

• One notices that the existence and stability of the circularly polarized FCP does not ensure either its stability or even its existence beyond SVEA [173].
• The Z evolution of u and v is computed by means of a standard fourth-order Runge–Kutta algorithm, at each step and substep of the scheme, the eight other components are computed using the same algorithm but relative to the T variable.
• For the shortest sub-cycle pulses, the instability occurs very fast, the amplitudes of the two single humped pulses strongly differ, and the angle between their polarization directions is not close to π/2.
• By direct numerical simulations the authors calculated the lifetime of circularly polarized few-optical-cycle solitons and they studied their decay into two orthogonally polarized single-humped pulses as a generic route of their instability.
• A challenging extension suggested by these studies is to consider the case of two transitions, one below and one above the range of propagated wavelengths.

7. Few-optical-cycle dissipative solitons

• A generalized modified Korteweg–de Vries partial differential equation is derived,which describes the physics of few-optical-cycle dissipative solitons beyond the slowly-varying envelope approximation; see Ref. [38] for a detailed study of this problem.
• The authors also briefly discuss the output of numerical simulations showing the formation of stable dissipative solitons from arbitrary breather-like few-cycle pulses.
• One of the most important of such models is Haus’ master equation [177], which is in fact the stationary version of the complex Ginzburg–Landau (CGL) equation [178–181]; see, e.g., the comprehensive reviews [182,183] on the CGL equation and its various applications.
• Both the cubic and the cubicquintic CGL models have been derived from a detailed description of the laser cavity, in the case of fiber lasers that are mode-locked by means of nonlinear polarization rotation, or figure-eight ones [184–186].
• KLM is not a mere effective amplitude-dependent gain/loss effect, but a complex phenomenon which involves the spatiotemporal intra-cavity pulse dynamics.

7.1. Maxwell–Bloch equations and their multiscale analysis

• In an important work published more than three decades ago, Haken [196] used a single-mode unidirectional ring laser model (with a homogeneously broadened line) described by the Maxwell–Bloch equations, and after some approximations showed its mathematical equivalence with an appropriate model of the Lorenz oscillator [195].
• Hence not only losses due to reflection at the cavity boundaries, which will be evaluated below, but also the ones of the medium must be taken into account.
• It is due to the correction termw1 to the population difference, and fixes themagnitude of the gain term in the evolution equation governing the dynamics of few-optical-cycle dissipative solitons, as in SVEAmodels of Ginzburg–Landau type [186].
• These terms and the corresponding coefficients A and B have the same expressions as in the conservative counterpart of Eq. (7.21), which is themKdV equation derived in [73] by using a reductive perturbation approach.
• Thewidth of the gain spectrummust be finite, which is ensured by the term−G∂4τ E1.

7.2. Robust ultrashort dissipative optical solitons

• The generalizedmodified KdV dynamical equation (7.21) was solved numerically in Ref. [38] bymeans of an exponential timedifferencing scheme, of secondorder Runge–Kutta type [115].
• The above FCP propagates thus in a stable way, that is, a robust few-optical-cycle dissipative soliton was put forward.
• Using periodic boundary conditions, it was found in Ref. [38] that FCPs form spontaneously, however their number in the numerical box increases during propagation; see Fig. 26.
• It also takes into account gain saturation, through the expression of g(t), and the effect of some instantaneous saturable absorber, through the term ∂x βu3 .
• The multiscale perturbative approach up to the third-order in a certain small perturbation parameter was used.

8. Spatiotemporal few-optical-cycle solitons

• First, the authors will consider ultrashort spatiotemporal optical solitons in quadratic nonlinear media and the generation of both line and lump solitons from few-cycle input pulses [31].
• Third, the authors consider the problem of ultrashort light bullets described by the two-dimensional sine–Gordon equation in the short-wave approximation regime [33].
• By using a reductive perturbation technique applied to a two-level model, it was obtained in Ref. [33] a generic two-dimensional sine–Gordon evolution equation governing the propagation of femtosecond spatiotemporal optical solitons in Kerr media beyond the SVEA.
• Also, in contrast to the case of quadratic nonlinearity, the light bullets oscillate in both space and time, and are therefore not steady-state lumps [33].

8.1. Ultrashort light bullets in quadratic nonlinear media: the long-wave approximation regime

• It was proved in Ref. [120] that the FCP soliton propagation in quadratic nonlinear media can be adequately described by a KdV equation and not by a mKdV equation as in the case of cubic (Kerr) nonlinear media.
• The extensive numerical simulations have shown that in certain conditions, a femtosecond pulse can evolve into a stable few-optical-cycle spatiotemporal soliton [72,74,75].
• The authors then briefly discuss the known analytical solutions of these KP equations, such as line and lump solitons.
• The evolution of the input FCP into a lump is shown in Fig. 32; the authors clearly see the accompanying dispersive waves, which remain captured in the computation box due to the periodic boundary conditions used in the T -direction.
• Note also that this transversely perturbed line soliton, in contrast with the case of KP II equation for normal dispersion shown in Fig. 31, does not recover its initial straight line shape and transverse coherence, but breaks up into localized lumps.

8.2. Collapse of ultrashort spatiotemporal optical pulses

• Following Ref. [32] a cubic generalized Kadomtsev–Petviashvili (CGKP) equation for describing ultrashort spatiotemporal optical pulse propagation in cubic (Kerr-like) media, without the use of the slowly varying envelope approximation, is derived by using a reductive perturbation method.
• The threshold Ath ≃ 3.6 found by numerical methods is about half of the value Ãth, found using the assumptions of the virial theorem.
• A . 7.6 ≃ Ãth), at the beginning of the process, the amplitude is not properly speaking sufficient to initiate collapse but, due to the shape of the pulse, a nonlinear lens effect induces a transverse self-focusing of the pulse, which increases the maximal pulse amplitude.
• Concluding this subsectionthe authors stress that they have introduced amodel beyond the SVEA of the commonly used nonlinear Schrödinger-type evolution equations, for describing the propagation of (2 + 1)-dimensional spatiotemporal ultrashort optical solitons in Kerr nonlinear media.

8.3. Ultrashort light bullets in cubic nonlinear media: the short-wave approximation regime

• In this subsection the authors consider the problem of existence and robustness of ultrashort light bullets described by the twodimensional sine–Gordon equation.
• In Ref. [33] the authors clearly demonstrated that the short wave approximation of the Maxwell–Bloch equations is indeed the two-dimensional sG equation, and not one of the nonlinear systemof equations found in the study of nonlinear electromagneticwaves in ferromagnets [219,221].
• The authors have shown in Section 3.1 that in the framework of a two-level model, a mKdV equation is obtained if the frequency of the transition Ω is far above the characteristic wave frequency ωw (the long-wave approximation regime), while a sG model is valid ifΩ is much smaller than ωw (the short-wave approximation regime); see Section 3.2.
• By using a multiscale analysis, a generic KP evolution equation governing the propagation of femtosecond spatiotemporal optical solitons in quadratic nonlinear media beyond the SVEA was put forward [31].
• Thus in the long wave approximation regime, in a medium with a quadratic nonlinearity, half-cycle light bullets in the form of single lumpsmay exist, while in a mediumwith a cubic nonlinearity collapse occurs.

9. Conclusions

• The above theory is relevant for all phenomena involving ultra-short pulses or very broad spectrum, for which the SVEA fails to be valid.
• It is also suited to the description of supercontinuum generation from femtosecond pulses, which have a broad spectrum from the beginning of the process.
• Within the long-wave approximation, the effects of the various transitions combine themselves so that the general model keeps the same form, the coefficients merely involve the general linear and nonlinear susceptibilities.
• The question, how the various transitions will combine in the general case is still open.
• The authors conclude with the hope that this overview on recent developments in the area of few-optical-cycle solitons will inspire further theoretical and experimental investigations.

Figures

Fig. 31. (Color online) Shape and profile of the line soliton. (a) input, (b) after propagation (Z = 14.95). The parameters are λ = 3, u0 = 10.7, wt = 1.4, and wy = 3.87. After Ref. [31].

Fig. 33. (Color online) Generation of lumps from a perturbed unstable line soliton. (a) input, (b) after propagation Z = 1.45. Parameters are λ = 3, u0 = 10.7, wt = 1.4, and L = 14.14. After Ref. [31].

Fig. 32. (Color online) Shape and profile of the lump. (a) input, (b) after propagation (Z = 14.95). Parameters are λ = 3, u0 = 6.12, wt = 1.2, and wy = 6.36. After Ref. [31].

Fig. 21. Evolution of the polarization of the unstable circularly polarized FCPwith both angular frequency and pulse length 1. Blue (dotted): initial circularly polarized pulse (at Z = 100), red (solid): final linearly polarized pulse (at Z = 1300). After Ref. [161].

Fig. 27. The FCP soliton formed from noise in the presence of velocity filtering yielded by absorbing boundary conditions (thick blue line). The thin red line shows the transverse variations of the linear absorption coefficient α producing the absorbing boundary conditions (more exactly α/15). After Ref. [38].

Fig. 26. Evolution of the electric field from noise. (a) The input (z = 0). (b) Three dissipative FCP solitons are formed (z = 31.9). (c) The interaction of many FCP solitons appears as chaotic (z = 75). After Ref. [38].

Fig. 23. (Color online) The circularly polarized FCP and its decay into two orthogonally polarized single-humped pulses. The shape of the FCP at Z = 280 (left) and Z = 360 (right). Light blue (gray): the amplitude |U| = √ u2 + v2 (and −|U|), red (black): u = Re(U). After Ref. [162].

Fig. 22. (Color online) The circularly polarized FCP and its decay into orthogonally polarized single-humped pulses. Parameters: ω = 5 and b = 2. Here Zdisp = 37. After Ref. [162].

Fig. 24. (Color online) The circularly polarized FCP and its decay into two orthogonally polarized single-humped pulses. The trajectories of the tip of the normalized electric field vector (u, v) in the transverse plane, for two values of Z , showing the polarization. Light blue (gray): at Z = 280, red (black): at Z = 360. After Ref. [162].

Fig. 11. (Color online) Stabilization of the carrier-envelope phase of the two-cycle pulse. The parameters are a = −292.55, b = 2, and p1 = 1 + 4i. After Ref. [76].

Fig. 37. (Color online) Evolution of a perturbed input FCP plane wave into two-dimensional FCP solitons. (a) Input (Z = 0), (b) output (Z = 52.2). After Ref. [33].

Fig. 38. (Color online) The evolution of a two-dimensional soliton, from an input roughly reproducing it. (a) Input (Z = 0), (b) output (Z = 316.8). After Ref. [33].

Fig. 10. (Color online) Robust, dispersion-free propagation of a two-cycle optical pulse for a large resonant term. The parameters are a = −500, b = 2, and p1 = 1 + 4i. After Ref. [76].

Fig. 14. (Color online) Interaction of two FCPs described by the two-breather solution of the mKdV–sG equation. The two envelopes are also shown. Here the parameters of the exact two-breather solution are p1 = 2, p2 = 2.5, and ω1 = ω2 = 8. After Ref. [77].

Fig. 28. The spectral profile of the gain involved by Eq. (7.21) (dashed blue line), compared to the one used in the ‘master mode-locking equation’ for FCPs of Ref. [39] (solid red line).

Fig. 20. Normalized profiles of an unstable circularly polarized FCP. Initial data is defined by the breather soliton with p1 = 1+ i, i.e. both pulse width and angular frequency equal to 1, for the polarization component U , and the same with a π/2 dephasing for V . Light blue (dotted): |f | and −|f |, pink (solid): U = Re(f ), green (dashed): V = Im(f ). After Ref. [161].

Fig. 16. (Color online). Interaction of two FCPs in the non-integrable case. The parameters are c1 = −1, c2 = 0.5, and c3 = 1. The input is a linear superposition of two breather solutions utwo (given by Eq. (3.20)) of the mKdV equation obtained by setting c1 = 0, with p1 = 1 + 2i and p2 = 2 + 2i.

Fig. 15. (Color online) The FCPs and their initial and final envelopes before (15(a)) and after (15(b)) the interaction. The parameters are c1 = −1, c2 = 1/2, p1 = 2, p2 = 2.5, and ω1 = ω2 = 4. After Ref. [77].

Fig. 5. (Color online) Top: the intensity |u|2 of the input FCP [light blue (gray) curve] and its evolution at z = 0.2 according to the focusing mKdV equation (3.21) [dark blue (black) curve]. Bottom: the linear dispersion (the nonlinear term was omitted) of the same input at the same propagation distance [light blue (gray)], and its nonlinear dispersion according to the defocusingmKdV equation (3.22) [dark blue (black) curve]. The input is a breather of the focusing mKdV equation, with angular frequency ω = 4 and inverse pulse duration 1/τ0 = 2. After Ref. [78].

Fig. 36. (Color online) Evolution of the spectrum integrated over Y during collapse. The input data correspond to the case (b) in Fig. 35. After Ref. [32].

Fig. 9. (Color online) (a) The analytical two-cycle solution to Eq. (4.16) in the integrable case, and its envelope. (b) The electric field and its envelope. (c) The ‘resonant’ population difference and its envelope. (d) The optical spectrum (dashed line) compared to the spectrum of the field envelope, shifted to the carrier frequency (solid line). After Ref. [76].

Frequently asked questions

Q1. What are the contributions mentioned in the paper "Models of few optical cycle solitons beyond the slowly varying envelope approximation" ?

This review concentrates on theoretical studies performed in the past decade concerning the description of few optical cycle solitons beyond the slowly varying envelope approximation ( SVEA ). To this aim the authors perform the multiple scale analysis on the Maxwell–Bloch equations and the corresponding Schrödinger–von Neumann equation for the density matrix of two-level atoms. The authors analyze in detail both long-wave and short-wave propagationmodels. Moreover, the authors consider the propagation of few-cycle optical solitons in both ( 1 + 1 ) and ( 2 + 1 ) -dimensional physical settings. The authors investigate in detail the existence and robustness of both linearly polarized and circularly polarized few-cycle solitons, that is, they also take into account the effect of the vectorial nature of the electric field. 

Q2. What is the common formalism used in the field of nonlinear optics?

Notice that in the field of nonlinear optics the so-called ‘nonlinear susceptibilities’ expansion formalism is commonly used; it is an expansion of a phenomenological response function in a power series of the electric field. 

Q3. What is the relevant approach for the infrared transitions?

Regarding the infrared transitions, or more generally the transitions with frequencies below the wave one, the relevant approach is the short wave approximation. 

Q4. What is the nonlinear evolution of dispersivewave packets?

The nonlinear evolution of dispersivewave packets, e.g., those propagating through optical fibers, involves several length scales, the phase velocity of the wave being a constant in this specific case [85]. 

Q5. What is the phenomenological relaxation term for the density matrix?

The density matrix obeys the Schrödinger–von Neumann evolution equation ih̄∂tρ = [H, ρ] + R, where as in the case of cubic optical nonlinearity [73], the phenomenological relaxation term R can be neglected. 

Q6. How is the Z evolution of u and v computed?

The Z evolution of u and v is computed by means of a standard fourth-order Runge–Kutta algorithm, at each step and substep of the scheme, the eight other components are computed using the same algorithm but relative to the T variable. 

Q7. How many cycles of pulses can be produced from a CEP-stabilized system?

As concerning the creation of CEP-stabilized intense pulses the authors also mention the generation of 1.5 cycle pulses at 1.75 µm [28] and the generation of multi-µJ, CEP-stabilized, two-cycle pulses from an optical parametric chirped pulse amplification system with up to 500 kHz repetition rate [29]. 

Q8. What is the amplitude required for the quadratic nonlinearity?

That is why the amplitude required for the quadratic nonlinearity is of order ε2, which is small with respect to the amplitude of order ε required in the present case of the cubic nonlinearity. 

Q9. What is the collapse threshold for the propagation of few-cycle spatiotemporal pulses?

The collapse threshold for the propagation of few-cycle spatiotemporal pulses is calculated by a direct numerical method, and compared to the analytic results based on a rigorous virial theorem. 

Q10. How can the authors think of the two-cycle pulses produced in a series of essential experimental?

Thus it can be thought that the two-cycle pulses produced in a series of essential experimental works on FCPs performedmore than one decades ago [1–4,111] could propagate as true solitons in certain nonlinear media, according to the generic mKdVmodel. 

Q11. What is the numerical value of the amplitude at the boundary between the two distinct domains?

The numerical value of the amplitude at the boundary between the two distinct domains is rather close to the threshold value for collapse (Ãth ≃ 7.6) found from a rigorous mathematical condition; see Ref. [32] for more details of this issue. 

Q12. What other non-SVEA models have been proposed in the literature?

other non-SVEA models [141–143], especially the so-called short-pulse equation (SPE) [144], have been proposed in the literature.