Resonant radiation from oscillating higher order solitons

R. Driben; A. V. Yulin; A. Efimov

doi:10.1364/OE.23.019112

1. Introduction

Nonlinear dynamics of solitary waves [1] is a broad and vibrant research area. In optical fibers while the balanced action of second order dispersion and nonlinearity can lead to formation of fundamental solitons, higher order dispersion causes perturbations to the solitons [2, 3] resulting in radiation of dispersive waves. Studies of interaction between solitons and dispersive waves (DW) led to better understanding of the fundamental physics of solitons and to important advances in applications such as supercontinuum generation, four-waves-mixing, frequency conversion, etc. [4, 5]. A comprehensive theory was developed describing interactions of solitons with weak DWs, which do not drastically alter the properties of solitons [3, 4, 6, 7]. Studies of strong interaction of DWs, significantly modifying temporal and spectral properties of solitons were presented as well [8–14].

Among the various aspects of interaction between DWs and solitons, emission of resonant (Cherenkov) radiation [3] by solitons is of primary importance. It was demonstrated that the spectral recoil produced by the Cherenkov emission [15] resulted in the cancellation of the Raman self-frequency shift [16]. In the process of supercontinuum generation the effect of the resonant radiation plays an important role at the initial stage of spectral broadening [4] as well as at the advanced stage when the radiation gets trapped between a pair of solitons [13]. Cherenkov radiation in quadratic media was demonstrated [17] and trapping of DWs in a tapered fiber by a Raman soliton was also reported [18]. Recently it was shown that in fibers with spatially varying dispersion characteristics the resonant radiation can be polychromatic or, in other words, multiple emissions of DWs at different frequencies may occur [19–26].

The broad studies mentioned above provided deep understanding of the DWs radiated by the fundamental soliton, while the radiation process from second- and higher-order solitons remained obscured. Surprisingly the emission by the second-order soliton reveals a substantially different radiation mechanism with considerably different spectral properties of the radiated DWs. Revealing this mechanism is the aim of the present work.

To describe propagation of high-order solitons we adopt a standard model [5, 27] based on Generalized Nonlinear Schrodinger Equation:

u_{z} = \sum_{m \geq 2} \frac{i^{m + 1} β_{m}}{m!} \frac{\partial^{m} u}{\partial τ^{m}} + i γ (1 + \frac{i \partial}{ω_{0} \partial τ}) [u (z, τ) \int_{- \infty}^{τ} d τ^{'} R (τ - τ^{'}) {| u (z, τ^{'}) |}^{2}] .

The electric field with amplitude

u (z, τ)

propagates along the fiber with longitudinal coordinate,

τ

is the time in a reference frame travelling with the light and

β_{m}

are the m^th-order dispersion coefficients at the central frequency

ω_{0}

. The nonlinear coefficient is given by

γ = 2

W⁻¹km⁻¹. The response function

R (τ) = (1 - f_{R}) δ (τ) + f_{R} h_{R} (τ)

contains both instantaneous and delayed Raman contributions, where

f_{R} = 0.18

is the fraction of Raman contribution to the nonlinear polarization, and

h_{R} (τ)

is the Raman response function of silica fiber, which can be approximated by the expression [5, 25]:

h_{R} (τ) = (τ_{1}^{2} + τ_{2}^{2}) / (τ_{1} τ_{2}^{2}) \exp (\frac{- τ}{τ_{2}}) \sin τ / τ_{1}

, with

τ_{1} = 12.2

fs and

τ_{2} = 32

fs.

2. Resonant radiation of 2-soliton driven only by the third order dispersion (TOD)

First we consider the dynamics of the pulses in a conservative system disregarding the Raman and the shock terms, which can be relevant to hollow core or crystalline fibers. The influence of these effects will be considered in section 4. For the sake of clarity we write the equation for the evolution of a pulse in the vicinity of zero dispersion wavelength (ZDW) in normalized units neglecting fourth and higher order dispersion terms:

i u_{z} + 0.5 u_{t t} + {| u |}^{2} u = i \bar{β_{3}} u_{t t t} .

As usual the time is normalized to the characteristic duration of the initial pulse

T_{0}

, the propagation length is normalized to dispersion length

L_{G V D} = T_{0}^{2} / | β_{2} |

, the intensity is normalized to

γ L_{G V D}

,and

{\bar{β}}_{3} = β_{3} / (6 β_{2} T_{0})

is the normalized third order dispersion. For

{\bar{β}}_{3} = 0

this equation has a family of periodically oscillating soliton solutions with a soliton period

Z_{0} = π T_{0}^{2} / 2 | β_{2} |

[27, 28]. In the presence of higher-order dispersion, however, the perfect periodicity of the soliton evolution is broken by the effect of TOD and the 2-soliton would eventually undergo splitting or fission [28, 29]. Still if the central wavelength of the soliton spectrum is far enough from the ZDW, the effect of TOD will not manifest itself so rapidly, allowing the 2-soliton to propagate robustly for a considerable number of periods [2].

We have simulated light evolution in a standard telecom fiber with Eq. (1) injecting 2-soliton at the input given by $2 P_{0} sech(t / T_{0})$ with $T_{0} = 62.5$ fs, peak power $P_{0} = 1.817$ kW and central wavelength of 1470 nm. The relevant dispersion parameters are $β_{2} = - 14.2$ ps²/km, $β_{3} = 0.087$ ps³/km with zero dispersion length (ZDW) located at 1311 nm. The perturbation caused by TOD is not strong enough to lead to fast splitting [2, 30]. As we can see from Fig. 1 the 2-soliton propagation exhibits typical periodic oscillations while emitting polychromatic radiation during each temporal compression cycle. Polychromatic DWs are clearly seen at the inset of Fig. 1(a) where ${| u |}^{0.25}$ is plotted for better visibility of low intensity waves. DWs manifest interference and starting at $z \approx 0.6$ m we can observe that the radiation band exhibits periodic comb-like structure (Fig. 1(b) and inset). Note that the third- and higher-order solitons would emit DW radiation with the same inter-harmonic frequency spacing because the soliton period is the same for all higher-order solitons. However, for a detectable effect the amount of TOD should be metered more carefully since higher-order solitons are much more efficiently disintegrated by TOD than 2-solitons. As a result, too much TOD will lead to splitting within one soliton period and too little TOD will render DW emission unobservable.

Fig. 1 Evolution of 2-soliton under the action of weak TOD in (a) temporal and (b) spectral domains. Inset to (a) demonstrates the evolution of ${| u |}^{0.25}$ instead of the intensity ${| u |}^{2}$ shown in main panel for better visibility of low intensity waves. Inset to (b) displays the zoomed-in region of radiation from 1030 to 1080nm.

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Similar type of radiation band was also observed in more complex optical settings such as fiber lasers with periodically modulated gain [22, 31], periodically oscillating solitons in fibers with dispersion management [23, 25] and dissipative solitons in optical fiber cavities [24].

3. Resonance condition for the second-order soliton radiation in the presence of TOD

We now consider the phase matching condition for the soliton of the second order. An oscillating soliton can be represented as a sum of functions localized in space and harmonically oscillating in time. We can find a phase matching condition for each of the temporal harmonics and the propagating eigenmodes of the medium. The resonance with the zero temporal harmonic can be referred to as Cherenkov radiation while the other temporal harmonics will generate additional radiation lines. Using an analogy with vacuum electronics one can refer to these radiation lines located in the vicinity of the Cherenkov resonance as synchrotron radiation. Of course NLSE is not invariant in respect to Lorentz transform and so all the properties of synchrotron radiation related to its relativistic nature are not applicable to the case of oscillating solitons but the synchronism conditions are the same.

To derive the resonant terms we consider the third-order dispersion as a perturbation and write a linear equation for the perturbed field. Then, a driving force proportional to the third-order dispersion appears on the right-hand side of the equation. The resonant radiation emerges when the right hand side of the equation contains harmonics moving with velocities equal to the phase velocity of the eigenmode with the same wave vector [22, 24–26].For the case of an oscillating soliton and third order dispersion the resonance condition takes the form:

\frac{β_{2}}{2} δ^{2} + \frac{β_{3}}{6} δ^{3} - τ_{g} δ = \frac{P γ}{8} + \frac{2 π N}{Z_{0}} .

Here we use physical units:

δ

is detuning of the frequency of the resonant wave from the soliton frequency, P is the soliton’s peak power,

Z_{0}

is the soliton period, and N is an integer number. The soliton group delay

τ_{g}

is defined as a rate of change in soliton’s center of mass position (in the moving frame of reference) with propagation distance z.

Graphical solutions of Eq. (3) are presented in Fig. 2. Figure 2(b) is the zoomed version of Fig. 2(a) focused on the spectral domain of the radiation band. The solutions shown in the lower panel of Fig. 2(b) (intersections of the dispersion blue curve with colored lines corresponding to different values of N in Eq. (3)) are compared to the numerically obtained spectrum for $z = 2$ m, shown in the upper panel. We can clearly see a very good agreement when predicting each peak’s spectral location with accuracy up to 2 nm. This small inaccuracy can be attributed to small variations in 2-soliton’s velocity in a course of its propagation.

Fig. 2 (a) Graphical solution of Eq. (3) for obtaining spectral positions of resonances. The vertical dashed line designates the position of ZDW. (b) Spectrum of radiation (upper panel, blue curve) and zoomed graphical solutions (the intersections of the dispersion blue curve with lines of other colors, lower panel).

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It is relevant to notice here that, strictly speaking, the resonance condition is just a necessary condition for the radiation. It is known that the coupling coefficient between the soliton and the resonant wave can be exactly equal to zero and thus the emission can be completely suppressed giving rise to so called embedded solitons [32, 33].

We have also studied border of existence of the observed mechanism and conducted a set of numerical simulations to study the radiation frequency comb’s characteristics. If we launch our pulse too close to ZDW the relative strength of TOD ${\bar{β}}_{3} = β_{3} / (6 β_{2} T_{0})$ is high and the 2-soliton undergoes fast splitting. On the other hand, working too far from ZDW will lead to very weak radiation and quite slow accumulation of light in the radiation band.

Figure 3 demonstrates the dependence of the comb’s central peak location and inter-peak spectral separation as a function of input wavelength of the 2-soliton. We can conclude from Fig. 3(b) that the phenomenon is observable over a wide range of physical parameters.

Fig. 3 (a) Spectral position of the central radiation peak (solid blue curve) and (b) separation between the peaks in the radiation band vs. input wavelength. All units are nanometers.

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4. The influence of Raman effect on the synchrotron radiation of second order solitons

We have also simulated the radiation of 2-soliton in presence of high order nonlinear effects such as Raman and self-steepening using the full generalized nonlinear Schrodinger equation, Eq. (1). Clearly, for short pulses the action of Raman term will immediately lead to the splitting of a high-order soliton. Thus, in order to observe the phenomenon described above we have to work with longer pulses. Moreover the effect of TOD must be strong, therefore we need to launch the pulse closer to ZDW.

Figure 4 demonstrates the evolution of a 1 ps pulse in a standard telecom fiber. We can see that 2-soliton undergoes strong Raman shift (Fig. 4(a)), but still robustly oscillates twice and emits polychromatic DWs during each temporal contraction that interfere and form a comb-like radiation band, see Fig. 4(b) and the inset.

Fig. 4 Temporal, (a), and spectral (b) evolution of 2-soliton in standard telecom fiber with all higher order terms included. Inset in (b) displays the zoomed region of radiation from 1270 to 1300 nm.

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4. Conclusions

By a synthesis of direct numerical simulations with analytical approach we have studied the emission of polychromatic radiation by a periodically breathing second-order soliton. The radiation band appears as a sequence of peaks in spectral domain resulting from a coherent interference of periodically emitted dispersive waves. The spectral location and inter-peak separation of this band can be effectively controlled by the strength of the TOD.

The observed mechanism appears to be robust in a wide range of physical parameters in Raman-free fibers. The influence of the Raman term partially washes out the radiation band structure, although the comb-like structure of the radiation band is still observable under appropriate conditions.

Acknowledgment

R.D. and A.V.Y gratefully acknowledges the support by the Russian Federation Grant 074-U01 through ITMO Early Career Fellowship scheme.

This work was performed, in part, at the Center for Integrated Nanotechnologies, a U.S. Department of Energy, Office of Basic Energy Sciences user facility. Los Alamos National Laboratory, an affirmative action equal opportunity employer, is operated by Los Alamos National Security, LLC, for the National Nuclear Security Administration of the U.S. Department of Energy under contract DE-AC52-06NA25396.

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Resonant radiation from oscillating higher order solitons

Abstract

1. Introduction

2. Resonant radiation of 2-soliton driven only by the third order dispersion (TOD)

3. Resonance condition for the second-order soliton radiation in the presence of TOD

4. The influence of Raman effect on the synchrotron radiation of second order solitons

4. Conclusions

Acknowledgment

References and links

Cited By

Figures (4)

Equations (3)

Optics Express