## Abstract

We show in theory and simulation that the supercontinuum generation from an initial continuous wave field in a highly nonlinear fiber operating near the zero-dispersion point can be significantly enhanced with the aid of dispersion management. We characterize the spectral broadening as a process initiated by modulational instability, but driven by the zero-dispersion dynamics of an *N*-soliton interacting with the asymmetric phase profile generated by the Raman effect, self-steepening effect, and/or higher-order dispersion. Higher *N*-soliton values lead to shorter pulses and a broader spectrum. This insight allows us to use dispersion management in conjunction with modulational instability to effectively increase the *N* value and greatly enhance the supercontiuum generation process.

© 2005 Optical Society of America

## 1. Introduction

Supercontinuum generation in optical fibers is of growing practical and fundamental interest. The ultra-broadband light generated has a variety of potential applications, including in optical coherence tomography, sensing, and metrology. Although supercontinuumgeneration has been achieved with femtosecond pulses [1, 2], surprisingly one can also generate a supercontinuum with a continuous wave (CW) or nearly CW source. Particularly strong supercontinuum generation has been found near the zero-dispersion wavelength of optical fibers [1, 3, 4, 5]. And to enhance the underlying nonlinear effects in the fiber, highly-nonlinear fiber (HNLF) is used in conjunction with advantageous low cost, robust and compact continuous wave (CW) pumping technology in the ≈0.5-4 W regime. Such experiments have achieved spectral bandwidths of several hundred nanometers. Although modulational instability is understood to be of fundamental importance in initiating the supercontinuum from CW pumping [6, 7, 8], it is the subtle interplay of nonlinear effects that generates the ultra-short pulses required for the supercontinuum. Indeed, the underlying theory considered here also applies to the supercontinuum generation in microstructured optical fibers (MOFs) [9].

In this manuscript, we show that the supercontinuum generation from an initial CW excitation can result from the interaction of the Raman effect (and/or nonlinear self-steepening term, higher-order dispersion, or pre-chirping) with the zero-dispersion (semi-classical) behavior of the nonlinear Schrödinger equation. Specifically, these effects generate an asymmetric chirp across an *N*-soliton pulse which in turn creates and ejects ultra-short one-soliton pulses. The one-soliton pulse widths, which are narrower for higher *N* values, can be controlled and shortened substantially through dispersion management, thus allowing for enhanced spectral broadening. Specifically, dispersion management allows us to control the modulational instability frequencies which initially determine the energy contained in each N-soliton pulse created. A similar idea has been used by Hori *et al*. [10] to generate supercontinuumby managing the fiber nonlinearity, which is equivalent in principle to dispersion management.

Modulational instability (MI) is a well understood instability phenomena of the nonlinear Schrödinger equation (NLS) which governs the underlying wave propagation in an optical fiber in the presence of chromatic dispersion and self-phase modulation. In the anomalous dispersion regime, the dominant unstable MI frequencies can be shown to be proportional to the CW amplitude and inversely proportional to the square root of the dispersion [11, 12]. Thus, the modulation which grows on the CW field is of increasing frequency as the dispersion goes to zero. Even in the normal dispersion regime, MI can occur due to third- and fourth-order dispersion corrections [6].MI is responsible for initiating supercontinuumgeneration by destabilizing the launched CW light. Figure 1 illustrates the fundamental process of MI induced instability followed by the creation of an ultra-short (broadband) pulse train responsible for the supercontinuum. In this figure, dispersion management is used to enhance the spectral broadening. This aspect of the dynamics is fully treated and motivated in Sec. 3. It is known from NLS theory that the CW light eventually forms a train of interacting one-solitons. However, the simple NLS model fails to capture the ultra-short pulses and spectral broadening observed in experiment [3, 4, 5]. To capture the observed supercontinuum generation, the NLS equation must be modified to include the Raman effect, self-steepening effect, higher-order dispersion and/or pre-chirping. Acting individually or in concert, these effects will all result in the same basic spectral-broadening behavior. However, for the physical parameters considered here, the Raman effect dominates the self-steepening and higher-order dispersion. Thus we only consider Raman in what follows in order to more clearly exemplify the underlying theory. As will be shown, the Raman response generates a large, asymmetric chirp across the pulses formed by MI. This large phase variation in conjuction with the zero-dispersion limit, also known as the semi-classical limit of NLS, splits the propagating pulse into a series of one-solitons [13, 14]. The largest of the ejected one-solitons is typically an ultra-short pulse whose temporal duration is determined by the energy of the initial pulse and the strength of the phase-variation across it. The underlying theory for this pulse ejection behavior has been considered extensively for the case of symmetric phase variations [13, 14, 15]. It is this nonlinear, semi-classical behavior which ultimately gives the observed supercontinuum generated by the soliton fission process [16, 17, 18, 19].

The paper is composed of two primary sections. Section 2 discusses the fundamental ideas of *N*-soliton fission. Of critical importance in this section is the role that asymmetric phase variations, generated, for instance, by the Raman effect, play in the fission process. Section 3 builds on the insights of Sec. 2 and makes connection between the MI observed with CW light to the *N*-soliton fission dynamics. The key observation is made that the *N*-soliton energy, which ultimately determines the soliton fission, can be controlled via dispersion management. A brief summary and conclusion is provided in Sec. 4.

## 2. N-soliton Fission

This section is concerned with reviewing some of the basic ideas associated with *N*-soliton fission. Special emphasis will be given to the *N*-soliton dynamics under the influence of the Raman effect. However, higher-order dispersion, self-steepening or pre-chirping all have a qualitatively similar effect. Ultimately, it is the fission process that is responsible for the generated supercontiuum.

To illustrate and quantify the zero-dispersion (semi-classical) behavior, we consider the NLS equation with the Raman term:

where *Q* is the normalized electric field envelope, the variable *T* represents the physical time normalized by *T*
_{0}/1.76 where *T*
_{0} is the full-width half-maximum (FWHM) of an intial onesoliton pulse, and the variable *Z* is the physical distance divided by the dispersion length *Z*
_{0}=(2*πc*)/(${\lambda}_{\mathit{0}}^{\mathit{2}}$*D̄*)(*T*
_{0}/1.76)^{2} where *D̄* is the average dispersion of the signal field. This gives a peak field power of a one-soliton solution of the NLS to be |*E*
_{0}|^{2}=(*λ*_{0}*A*
_{eff})/(2*πn2Z*_{0}
) where *n*
_{2}=2.6×10^{-16} cm^{2}/W is the nonlinear coefficient in the fiber, *A*
_{eff}=11 *μ*m^{2} is the cross-sectional area of the HNLF fiber, and *λ*
_{0}=1.554*µ*m and c are the free-space wavelength and speed of light respectively. Finally, *τ*=*T*_{R}*/T*_{0}
where *T*_{R}
=1-5 femtoseconds is the response time of the Raman effect.

The semi-classical limit is achieved by considering an initial *N*-soliton *Q*(0,*T*)=*N*sech*T* where *N*≫1. Thus the nonlinear term is *O*(*N*
^{2}) larger than the dispersive term and dominates the dynamics, i.e. the zero-dispersion limit [12, 13, 14, 15]. The Raman term acts to shift the phase in a nontrivial manner. For a qualitative understanding, consider the following heuristic argument based upon the Raman response on an initial *N*-soliton:

In the absence of dispersion and self-phase modulation, the propagation Eq. (1) can be solved with the asymmetric phase-variation given by the Raman term in Eq. (2). This yields the approximate evolution *i∂Q*/*∂Z*+[2*N*
^{2}τsech2*T*tanh*T*]*Q*=0 which has the solution

If τ*N*
^{2}≫1, the phase profile generated is large and asymmetric. It is known that a large, symmetric phase splits the eigenvalue spectrum associated with the semi-classical NLS like a *zipper* [13], ejecting one-solitons in an ordered fashion [14]. Specifically, the ejected one-solitons are arranged by height with the tallest (narrowest) solitons moving with the largest group velocity. This analysis is based upon the Zakharov-Shabat eigenvalue problem [20] associated with Eq. (1) and has been observed experimentally [15] for symmetric phase profiles. More precisely, the semi-classical eigenvalues of the Zakharov-Shabat problem in general have a real and imaginary part: *λ*_{n}
=*k*_{n}
+*iη*_{n}
. The relationship between the eigenvalues and the fundamental 1-solitons contained in a given *N*-soliton is given by [20, 21]:

Thus the real part of the eigenvalue *k*_{n}
determines the ejection velocity while the imaginary part *η*_{n}
determines the ejected soliton pulse height and width. Figure 2 demonstrates the qualitative distribution of the eigenvalue spectrum for an *N*-soliton with and without phase chirp [13]. Note the splitting of the spectrum into a Y-shaped zipper with the addition of phase chirp. The resulting nonzero real part of the eigenvalue *k*_{n}
begins the ejection process since it determines the group velocity for the fundamental solitons of Eq. (4). Additionally, to achieve the maximal amount of supercontinuum generation, a large value of *η*_{n}
is desired. This is strictly controlled by the *N*-soliton considered, i.e. *η*_{n}
scales with *N* so that the larger the *N*-soliton the shorter (spectrally broader) the ejected solitons. For the Raman case considered here, a large asymmetric chirp is generated for *N*-soliton initial data which is capable of splitting the spectrum like a zipper in an asymmetric fashion. The ejected solitons are now ultra-short femtosecond solitons which are broad in the spectral domain. Soliton ejection due to asymmetric perturbations has also been observed experimentally [21].

Evidence for this soliton ejection process is provided by simulating Eq. (1) with an initial *N*-soliton. A *T*
_{0}=1 ps soliton pulse is considered with cavity dispersion *D̄*=0.1 ps/(km-nm) and Raman response time of *T*_{R}
=3 fs. Figures 3 illustrates the time domain evolution along with the spectral evolution over 1.4 kilometers of HNLF for an initial *N*=5 soliton. The higher the *N* value, the further into the semi-classical regime and the broader the spectrum. Indeed, the resulting pulse width for the narrowest one-soliton ejected is ≈600,250,100 and 40 fs for *N*=2,4,6 and 8 respectively. Thus, the hallmark feature of the dynamics is the asymmetric ejection of ultra-short one-solitons from the inital *N*-soliton data. Further, it shows that the larger the initial *N*-soliton energy (∫^{∞}_{−∞}|*Q*|^{2}
*dT*=2*N*
^{2}) the narrower the ejected pulses. This is critical in using dispersion management to enhance the supercontinuum generation. Further, since the asymmetric phase variation, such as that given by Eq. (3), drives the ultra-short one-soliton ejection process, it shows that strong pre-chirping can also be used to help enhance the spectral broadening. This may be difficult to achieve for CW light, but can be accomplished with pulses [15].

## 3. CWModulational Instability and N-soliton Fission

With the mechanism for ultra-short pulse generation established, we turn to the role of dispersion management and MI in enhancing the spectral broadening. The onset of MI from an initial 800 mW CW wave is determined from simulations of Eq. (1) and illustrated in Fig. 4 for the dispersion values *D̄*=0.1 ps/(km-nm) and *D̄*=1 ps/(km-nm). The resulting modulations are consistent with the theoretical predictions for the MI instability [11]. Of importance is the energy captured in each *modulation cell*. For this specific example, the modulational cells capture energies of ≈0.2 pJ Fig. 4(top) and ≈1.6 pJ Fig. 4(bottom) respectively. In general, the modulation cell energy is simply the product of the dominant modulation period [11] ($(=2\pi \u2044\sqrt{2\gamma {P}_{0}\u2044\mid {\beta}_{2}\mid})$) with the initial CW power level (=*P*
_{0}), i.e. ${E}_{\mathrm{cell}}=\sqrt{2{\pi}^{2}\mid {\beta}_{2}\mid {P}_{0}\u2044\gamma}$. The modulation cells, which are enclosed by the dotted lines, begin the soliton ejection process illustrated, for instance, by Fig. 3. The energy in each modulation cell contains a given *N*-soliton and some dispersive radiation [13, 14]. The critical observation is that the modulation cell generated with higher dispersion contains significantly more energy than its low dispersion counterpart. However, because of its high dispersion, it only generates one-solitons on the order of picoseconds. Dispersion management allows us to first initiate the MI process with high dispersion fiber so that a larger modulation cell is created. We then switch to a fiber operating as close to the zero-dispersion (anomalous) point as possible. The energy captured in the modulation cell corresponds to a large *N*-soliton, leading to ultra-short pulse formation and enhanced supercontinuum generation. The basic supercontinuum dynamics and experimental configuration, which includes a concatenation of high dispersion fiber followed by low dispersion fiber, is depicted in Fig. 1. Note that for pulsed femtosecond supercontinuum generation, the modulation cell is pre-determined by the launched pulse energy. For the CW case, it is the MI which is responsible for generating the pulsed structure which is then spectrally broadened.

To demonstrate the supercontinuum generation, Eq. (1) is numerically integrated over 7 km of HNLF with *T*_{R}
=3 fs. Although many effects are neglected, i.e. linear attenuation, higher-order dispersion, self-steepening, and discrete signal-stokes coupling, the critical issue here is to demonstrate the fundamental phenomena illustrated in Figs. 1, 3 and 4. To observe the supercontinuumgenerated via dispersion management, we propagate again over 7 km. But in this case, the first five kilometers are a dispersion-shifted HNLF with *D̄*=5 ps/(km-nm). Note that the launched CW light is seeded with a small amount of white-noise (≈50 dB lower than the launched signal) in order to more accurately model experiments and induce MI. Further note that dispersion tapering of a fiber may be an equivalently efficient method for performing the necessary dispersion management. Figure 5 demonstrates the evolution of the pulse over 7 km with and without dispersion management. In the case of no dispersion management the resulting FWHM pulse train is on the order of picoseconds. In contrast, the case of dispersion management results in FWHM pulses on the order of femtoseconds, demonstrating the enhanced ultra-short pulse generation. Figure 6 (light line) demonstrates the ≈60 nm supercontinuum generated in the HNLF fiber for *D̄*=0.2 ps/(km-nm). The enhanced supercontinuum is shown in Fig. 6 (bold line). The moderate amount of dispersion-management gives more than four times the spectral width, which is a significant improvement in the supercontinuumgeneration. To further enhance the broadening, higher dispersion fiber can be used in the first segment to increase the modulation cell, further enhancing in the supercontinuum. Note that simulations of Eq. (1) operating closer to the zero-dispersion point or for much stronger dispersion maps require significant computational resources in order to avoid numerical high-frequency instabilities. In the simulations, care must also be taken to ensure that the periodic boundary conditions imposed by the pseudo-spectral split-step technique do not generate any numerical artifacts. Specifically, the computational domain is taken to be large enough so that the observed dynamics is independent of the computational window. Thus as the domain is increased the solution converges to a domain independent solution in both the time- and Fourier-domain.Note that for illustrative purposes only, Figs. 4 and 5 show a magnified view of the time-domain evolution.

To further validate the theoretical arguments presented, Eq. (1) can be modified to include several neglected effects, namely third-order dispersion, self-steepening, and linear attenuation. Thus the governing equation now includes the standard terms

where *β* is the normalized third-order dispersion, G is the normalized attenuation, and σ measures the normalized self-steepening response [11]. Simulations which include the additional terms show only minor qualitative differences from the spectral broadening which occurs from the simulations of Eq. (1) alone, i.e. the qualitative predictions and spectral broadening made previously are unaltered by the inclusion of these additional terms. Indeed, numerous studies which include some or all of these terms [16, 17, 18, 19] show the same basic spectral broadening characteristic of the zero-dispersion limit. We emphasize again here that we have only considered Raman as an illustrative example. Which of these terms might dominate can vary from experiment to experiment depending upon the physical parameters of each specific configuration. It suffices here to point out that the numerous examples considered previously [16, 17, 18, 19] already attest to the underlying theoretical arguments concerning the spectral broadening.

## 4. Conclusions

In conclusion, the ultra-short pulse generation and associated spectral broadening achieved near the zero-dispersion point of optical fibers is a consequence of one-soliton ejection due to the semi-classical NLS dynamics in optical fibers [13, 14, 15, 21]. The ejected solitons result from the semi-classical dynamics interacting with the asymmetric phase variations generated by the Raman effect, higher-order dispersion, self-steepening, and/or pre-chirping. Dispersion management can be used to control the MI induced modulation cell and resulting ejected solitons. Specifically, a high-dispersion fiber which is followed by a HNLF operating near the zero-dispersion point (anomalous side) gives significant enhancement of the supercontinuum generation. Alternatively, dispersion tapered fibers or nonlinearity managed fibers, when used appropriately and in accordancewith the theory, should be equally effective with a concatenated fiber dispersion map.

## Acknowledgments

We would especially like to thank the reviewers for their helpful and insightful comments. The manuscript has been greatly improved due to their remarks.

* permanent address Department of Applied Mathematics, University of Washington, Seattle, WA 98195 and acknowledges support from the National Science Foundation (DMS-0092682). This work was produced with the assistance of the Australian Research Council under the ARCS Centres of Excellence.

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