#### Abstract

We demonstrate that ultrashort optical pulses propagating in a nonlinear dispersive medium are naturally described through incorporation of analytic signal for the electric field. To this end a second-order nonlinear wave equation is first simplified using a unidirectional approximation. Then the analytic signal is introduced, and all nonresonant nonlinear terms are eliminated. The derived propagation equation accounts for arbitrary dispersion, resonant four-wave mixing processes, weak absorption, and arbitrary pulse duration. The model applies to the complex electric field and is independent of the slowly varying envelope approximation. Still the derived propagation equation posses universal structure of the generalized nonlinear Schrödinger equation (NSE). In particular, it can be solved numerically with only small changes of the standard split-step solver or more complicated spectral algorithms for NSE. We present exemplary numerical solutions describing supercontinuum generation with an ultrashort optical pulse.

#### 1. Introduction

Complex envelope adequately describes linear and nonlinear propagation of a wave packet with many field cycles [1]. A slowly varying envelope approximation (SVEA) reduces the full set of Maxwell equations for the pulse field to a much more simple first-order nonlinear Schrödinger equation (NSE) for the complex envelope [2–4]. On the other hand, SVEA lacks precision when the relevant time scales are comparable to a single cycle period. Nonenvelope pulse propagation regimes include self-focusing [5, 6], optical shocks [7, 8], supercontinuum (SC) generation [9], and dynamics of ultrashort pulses [10–15]. In such situations NSE should be replaced by a more general propagation model.

Several simplified unidirectional propagation equations have been derived for special dispersion profiles. Such models do not use the pulse envelope and apply directly to the pulse field (see [16–20] and a review paper [21]). For a general dispersion profile, pulse propagation is commonly described by a generalized NSE [2, 4] in which a polynomial approximation of dispersion in the frequency domain is used. An arbitrary dispersion is then accounted for by a local dispersion operator in the time domain. To resolve convergence problems [22] also rational approximations and nonlocal dispersion operators may be considered [18, 23]. The nonlinear term in the generalized NSE is also modified to capture an arbitrary pulse duration [8, 24], Raman scattering [25, 26], and diffraction effect [24, 27, 28].

Being an envelope model, the generalized NSE was successfully applied to many propagation problems where the pulse envelope evolves as fast as the pulse field and SVEA cannot be used. This is a paradoxical situation especially because the SVEA is implicitly used in a common definition of the envelope. Indeed, an envelope of the field with a central angular frequency is usually introduced through the ansatz (see, e.g., [8, 27, 29–33]) where for the sake of simplicity we (for the moment) consider only time dependence. Now, the imaginary part of remains unspecified but evidently affects, for example, the term in the either standard or generalized NSE. Relation (1) is meaningful if SVEA applies. One can then define as a sliding time average where because is slow on a time scale . Evidently, the sliding time average cannot be used for a few-cycle pulse for which we recall that is as fast as itself (see [34] for a critical review of several possible definitions of the envelope).

This contradiction is addressed in the present paper. Specifically we demonstrate that propagation of an ultrashort pulse can naturally be described in terms of analytic signal instead of the envelope. In other words, the real-valued electric field is replaced with a complex-valued one containing only positive harmonics [1]. The propagation equation for the analytic signal is structurally similar to NSE and can be solved with a small adaptation of the existing NSE solvers. Still the proposed propagation model applies directly to the electric field and avoids questionable use of definition (1). Useful applications of the analytic signal concept to the theory of nonlinear oscillations can be found in [34].

The present paper is organized as follows. Following [31, 35–37] we first derive a set of simplified equations for the field harmonics . Then we introduce an analytic signal and demonstrate how to remove all nonresonant nonlinear terms. The remaining resonant nonlinear terms have a simple “envelope” structure without use of the SVEA or the envelope as such. Finally the analytic signal approach is illustrated by exemplary numerical solutions.

#### 2. Basic Equations

We consider a periodic sequence of linearly polarized electromagnetic pulses propagating along the -axis in a one-dimensional dispersive nonlinear medium. The pulse field is governed by the following nonlinear wave equation: in which is a nonlocal linear susceptibility operator and a constant factor represents an instant nonlinear susceptibility of the third order. An inverse symmetry is assumed such that the quadratic nonlinear term is neglected. To quantify we write in the frequency domain: where is the period of the pulse sequence and . In what follows we assume that , that is, the time-averaged electric field vanishes. Now, is given by a convolution The linear susceptibility yields the dielectric constant and the propagation parameter where and are odd and even functions, respectively. In the following we consider a small absorption limit such that in a transparency window to which an essential part of the pulse spectrum must belong.

To proceed we write the nonlinear wave equation (3) in the frequency domain: where the term is neglected. The summation on the right hand side is performed only over the suitable triads . In what follows such summations will be abbreviated as .

Equation (7) is the starting point of our considerations. It will be simplified in a weakly nonlinear limit. Specifically we introduce a smallness parameter and assume the following scaling: for all fields and frequencies of interest. In particular, both and terms neglected in (7) are estimated as , the latter quantity defines which terms should further be neglected when reducing (7). In the next sections (7) is simplified using an unidirectional approximation, introducing a proper complex electric field and eliminating the nonresonant terms.

#### 3. Unidirectional Approximation

As explained above, both the nonlinear and the absorption terms in (7) are taken small. In a first step, neglecting the small terms, we write (7) as The unidirectional approximation deals with two classes of special solutions of (7), namely, the forward and the backward solutions, where respectively. We further consider the forward wave solution for which the term describes small contributions of the backward waves permanently generated by the nonlinear term in (7). To calculate this contribution we apply an exact identity: where the last term should be neglected because such terms were neglected when deriving (7). The latter is then transformed to the unidirectional form where both the left- and the right-hand sides are scaled as cf. (8).

The propagation model (12) was first suggested in [35] and then generalized in [31, 38] for an arbitrary polarization. As we will see below, it can be significantly simplified by elimination of the non-resonant nonlinear terms.

#### 4. Resonances

A natural approach to (12) is to simplify it by a suitable change of variables. In the spirit of the perturbation theory, we use a power expansion and formally define We apply to both sides of (13). Equations (8) and (12) yield We see that Using the last equation together with definition (13), (12) can be transformed to the form Comparing the latter equation with (12) we see that the cubic nonlinear term is formally eliminated. Strictly speaking, the contribution of the cubic term is “shifted” to the higher order in the spirit of the canonical perturbation theory [39, 40].

It is a good point to stress that such an elimination is possible only for non-resonant triads . The resonant frequencies are defined by the conditions Exemplary solutions of the above resonance conditions in the three-dimensional frequency space are shown by thick solid lines in Figure 1 for fluoride glass. In the vicinity of the resonant lines the transformation (13) is singular and contributions of the nonlinear terms in (12) cannot be eliminated. Far from the resonant curves, the nonlinear terms in (12) lead only to small forced oscillations of and can be neglected.

In what follows we assume that for all solutions of (17) contain one negative and two positive frequencies. In other words, only four-wave mixing (FWM) processes are taken into account. The contribution of FWM processes, for example, when all frequencies in the resonance conditions (17) are positive, is neglected. In such a situation (12) cannot be linearized completely, but can be considerably simplified as explained in the next section.

#### 5. Analytic Signal

To simplify (12) we first write it in such a way that contributions of the positive and negative frequencies are explicitly distinguished. To this end we introduce a complex-valued analytic signal for the electric field [1, 34]. In contrast to the real-valued , the analytic signal contains only positive harmonics such that In other words, because by construction. Inserting (20) into the product in (12) and performing multiplication, we see that all FWM processes are now separated. The reason for such a simplification is that the analytic signal is related to classical creation and annihilation operators as explained in [41] for a more general bidirectional nonlinear wave equation. After neglecting all but FWM processes, the unidirectional (12) finally takes the form where denotes a spectral component, , and denotes a positive frequency part of .

Equation (21) is our main result. It compromises properties of both the spectral propagation models and the envelope models. For instance, an arbitrary is captured like in the spectral model (12); on the other hand a familiar invariance with respect to the phase shifts ( with ) is retained like in the standard NSE.

For the vanishing dissipation term , the analytic signal equation was derived in [41]. We want to emphasize that dissipation enters (21) in a nontrivial way: the nonlinear term is not affected. If one carelessly replaces with in the cubic term, the resulting model will show an unphysical nonlinear gain which will finally spoil pulse propagation.

The nonlinear term in (21) is just a positive frequency part of the standard NSE-type nonlinearity. Still the analytic signal directly represents the electric field and is independent of SVEA. The negative frequency part of corresponds to backscattered light that is generated in the course of pulse propagation. The feedback effect of the backscattered light is neglected in the unidirectional approximation. Finally we note that the analytic signal equation (21) can be solved numerically with only small changes of the standard NSE solvers. Exemplary numerical solutions are given in the next section.

#### 6. Numerical Solutions

The numerical solutions of (21) are described in this section.We use a dealiased pseudospectral method, which originates from the computational fluid dynamics [42]. This method calculates all linear operators and derivatives in the frequency domain and performs the nonlinear multiplications in the time domain, with the transformations between the domains achieved by the fast Fourier transform. The integration for the linear and nonlinear part is performed in the frequency domain by a precise Runge-Kutta integration scheme of order eight with adaptive stepsize control depending on the accuracy as described in [43].

Considering ultrashort optical pulses with the carrier frequencies of several hundreds THz, we use the time step of at least fs. Depending on the initial pulse width, we use a resolution of 2^{14} and 2^{15} harmonics for a periodic time window ps and ps, respectively. Several test calculations were performed for a better resolution, 2^{17}. The increase of the harmonics number does not affect the results.

In the following we study the nonlinear propagation of femtosecond pulses in the anomalous dispersion regime of a microstructured fiber, where complex and comprehensive behavior can be observed. Depending on the input pulse power and width, the interplay of linear and nonlinear effects such as self-phase modulation (SPM), FWM, and soliton dynamics can lead to the generation of octave-spanning spectra. It is well known that the physical mechanism of the dramatic spectral broadening process is related to the break-up of higher-order solitons [35]. The soliton fission is caused by the formation of fundamental soliton pulses and the generation of a nonsolitonic dispersive wave into the phase-matched wavelength, leading to a spectrum broadened over an octave, even if the injected pulse energy is less than a few nanojoules. Besides soliton propagation, the modulation instability (MI) is another general feature in the anomalous dispersion regime, which affects the propagation of an optical pulse. The MI is a well understood instability phenomenon of the NSE, which results from the interplay between SPM and group velocity dispersion. In [44] the ability of the MI to generate SC and the dominance of the MI for short pulses have been demonstrated. In [45] it has been shown that soliton fission dominates for low input power and short pulses (100 fs) and the modulation instability has a strong impact for high input powers at arbitrary pulse widths.

For our simulations the dispersion profile of the highly nonlinear microstructured fiber is taken from [46]. As an initial condition we consider an input pulse electric field having a central angular frequency , and a hyperbolic-secant shape for the initial envelope with the dimensionless amplitude and temporal width fs. The value of corresponds to a pump wavelength nm; this wavelength is in the vicinity of the zero dispersion wavelength in the anomalous dispersion regime.

Figure 2 shows the density plots in the -plane of the spectral evolution for different input pulse widths. The spectra are shown on a logarithmic scale to illustrate the fine structure of the spectrum generated. For a 100 fs pulse spectral broadening in the range mm is mainly dominated by SPM. However, the significant features of the MI can be observed. The underlying MI acts in the initial stage on the pulse and leads to the generation of a Stokes and an anti-Stokes component. Figure 2(a) shows the appearance of two sidebands after mm. This demonstrates that also for short pulses with durations of 100 fs the modulation instability is present and can have an impact on the propagation dynamics.

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The simulations in Figures 2(b) and 2(c) illustrate the typical scenario of spectral broadening by soliton-related dynamics, for input pulses with fs and fs. Three different stages are clearly observed. The initial stage of propagation is dominated by symmetrical spectral broadening induced by SPM. An extreme spectral broadening is then caused by pulse contraction due to the first step of soliton propagation (Figures 3(b) and 3(d) at mm for 50 fs and mm for 10 fs). In the second stage the spectral broadening becomes asymmetric and energy is shifted to the blue side of the spectrum, due to soliton fission accompanied by the excitation of dispersive waves. This is associated with the development of distinct temporal peaks that sit upon a broader low-amplitude background (Figures 3(e) and 3(f) at mm). The extension of the spectrum to the blue side is related to the dispersion profile of the optical fiber and to the input pulse power. The spectral broadening is limited by the broadening of the temporal waveform of the pulses. In the third stage the spectral width is already saturated, but FWM generates complicated substructures. The appearance of the fine structure is an essential phenomenon and is extremely sensitive with respect to the initial pulse energy.

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Equation (21) clearly reproduces all essential features of the SC evolution seen in a number of experiments [9, 47] and in simulations with the generalized NSE, namely, SPM and FWM, MI, soliton fission, and generation of dispersive waves by solitons. Moreover, it goes beyond the envelope approximation and allows for an arbitrary pulse duration.

#### 7. Conclusions

Let us summarize our results. Propagation of spectrally broad ultrashort optical pulses is considered. We first derived the so-called forward Maxwell (12) following [35] but taking into account a small dissipation effect. Then we simplified the forward Maxwell equation by eliminating all non-resonant nonlinear terms. To this end we incorporated an analytic signal representation. The latter is useful as it is intrinsically related to classical creation and annihilation operators [41, 48, 49]. The resulting non-envelope model (21) applies to the analytic signal for the pulse electric field in the frequency domain. The model (21) combines advantages of both envelope and non-envelope approaches; it accounts for arbitrary dispersion, resonant four-wave mixing processes, weak absorption, and arbitrary pulse duration. It is of interest that the linear absorption does not affect the nonlinear term in (21). This is an important issue because careless use of the complex refractive index, for example, in (21), leads to an unphysical nonlinear gain. Finally, the existing numerical approaches to the envelope propagation equations can be easily adopted to solve the non-envelope (21).

#### Acknowledgments

The work of Sh. Amiranashvili was supported by the DFG Research Center MATHEON under project D 14. Helpful discussions with U. Bandelow, U. Leonhardt, M. Lichtner, and A. Mielke are gratefully acknowledged.