Dissipative Raman Solitons Vladimir L. Kalashnikov1,2,∗ and Evgeni Sorokin1 1 Photonics
Institute, Vienna University of Technology, Gusshausstr. 27/387, A-1040, Vienna, Austria 2 Aston University, Institute of Photonic Technologies, School of Engineering & Applied Science, Aston Triangle, Birmingham, B4 7ET, UK ∗ [email protected]
http://info.tuwien.ac.at/kalashnikov
Abstract: A new type of dissipative solitons - dissipative Raman solitons - are revealed on the basis of numerical study of the generalized complex nonlinear Ginzburg-Landau equation. The stimulated Raman scattering significantly affects the energy scalability of the dissipative solitons, causing splitting to multiple pulses. We show, that an appropriate increase of the group-delay dispersion can suppress the multipulsing instability due to formation of the dissipative Raman soliton, which is chirped, has a Stokes-shifted spectrum, and chaotic modulation on its trailing edge. The strong perturbation of a soliton envelope caused by the stimulated Raman scattering confines the energy scalability, preventing the so-called dissipative soliton resonance. We show that in practical implementations, a spectral filter can extend the stability regions of high-energy pulses. © 2014 Optical Society of America OCIS codes: (140.7090) Ultrafast lasers; (190.5530) Pulse propagation and temporal solitons; (190.5650) Raman effect; (270.3100) Instabilities and chaos.
References and links 1. N. N. Akhmediev and A. Ankiewicz (Eds.), Dissipative Solitons (Springer, 2005). 2. E. Podivilov and V.L. Kalashnikov, “Heavily-chirped solitary pulses in the normal dispersion region: new solutions of the cubic-quintic complex Ginzburg-Landau equation,” JETP Lett. 82, 467–471 (2005). 3. W. H. Renninger, A. Chong, and F. W. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A 77, 023814 (2008). 4. V.L. Kalashnikov and A. Apolonski, “Chirped-pulse oscillators: a unified standpoint,” Phys. Rev. A 79, 043829 (2009). 5. Ph. Grelu and N. Akhmediev, “Dissipative solitons for mode-locked lasers,” Nature Photon. 6, 84–92 (2012). 6. V. L. Kalashnikov, E. Podivilov, A. Chernykh, and A. Apolonski, “Chirped-pulse oscillators: theory and experiment,” Applied Physics B 83, 503–510 (2006). 7. W. Chang, A. Ankiewicz, J.M. Soto-Crespo, and N. Akhmediev, “Dissipative soliton resonances,” Phys. Rev. A 78, 023830 (2008). 8. A. Fernandez, T. Fuji, A. Poppe, A. F¨urbach, F. Krausz, and A. Apolonski, “Chirped-pulse oscillators: a route to high-power femtosecond pulses without external amplification,” Opt. Lett. 29, 1366–1368 (2004). 9. S. Naumov, A. Fernandez, R. Graf, P. Dombi, F. Krausz, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators,” New J. Phys. 7, 216 (2005). 10. W. H. Renninger and F. W. Wise, “Fundamental Limits to Mode-Locked Lasers: Toward Terawatt Peak Powers,” IEEE J. Sel. Topics Quantum Electron. 21, 1-8 (2015). 11. T. S¨udmeyer1, S. V. Marchese, S. Hashimoto, C. R. E. Baer, G. Gingras, B. Witzel, and U. Keller, “Femtosecond laser oscillators for high-field science,” Nat. Phot. 2, 599–604 (2008). 12. Y. Liu, S. Tschuch, A. Rudenko, M. D¨urr, M. Siegel, U. Morgner, R. Moshammer, and J. Ullrich, “Strong-field double ionization of Ar below the recollision threshold,” Phys. Rev. Lett. 101, 053001 (2008).
#224304 - $15.00 USD Received 2 Oct 2014; accepted 15 Oct 2014; published 24 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.030118 | OPTICS EXPRESS 30118
13. E. Seres, J. Seres, and Ch. Spielmann, “Extreme ultraviolet light source based on intracavity high harmonic generation in a mode locked Ti:sapphire oscillator with 9.4 MHz repetition rate,” Opt. Express 20, 6185–6190 (2012). 14. G.Sciaini and R. J. D. Miller, “Femtosecond electron diffraction: heralding the era of atomically resolved dynamics,” Rep. Prog. Phys. 74, 096101 (2011). 15. R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nature Photon. 2, 219–225 (2008). 16. A. A. Lanin, V. Fedotov, D. A. Sidorov-Biryukov, L. V. Doronina-Amitonova, O. I. Ivashkina, M. A. Zots, ¨ Ilday, A. B. Fedotov, K. V. Anokhin, and A. M. Zheltikov, “Air-guided photonic-crystalChi-Kuang Sun, F. O. fiber pulse-compression delivery of multimegawatt femtosecond laser output for nonlinear-optical imaging and neurosurgery,” Appl. Phys. Lett. 100, 101104 (2012). 17. V. L. Kalashnikov, “Chirped-Pulse Oscillators: Route to the Energy-Scalable Femtosecond Pulses,” in Solid State Laser, Amin H. Al-Khursan (Ed.), 145–184 (InTech, 2012). 18. S. K. Turitsyn, B. G. Bale, and M. P. Fedoruk, “Dispersion-managed solitons in fibre systems and lasers,” Phys. Rep. 521, 135–204 (2012). 19. E. Sorokin, N. Tolstik, V. L. Kalashnikov, and I. T. Sorokina, “Chaotic chirped-pulse oscillators,” Opt. Express 21, 29567–29577 (2013). 20. V. L. Kalashnikov and A. Apolonski, “Energy scalability of mode-locked oscillators: a completely analytical approach to analysis,” Opt. Express 18, 25757–25770 (2010). 21. E. Ding, Ph. Grelu, and J. N. Kutz, “Dissipative soliton resonance in a passively mode-locked fiber laser,” Opt. Lett. 36, 1146–1148 (2011). 22. V. L. Kalashnikov, “Dissipative solitons in presence of quantum noise,” Chaotic modeling and simulation 1, 29–37 (2014). 23. A. E. Bednyakova, S. A. Babin, D. S. Kharenko, E. V. Podivilov, M. P. Fedoruk, V. L. Kalashnikov, and A. Apolonski, “Evolution of dissipative solitons in a fiber laser oscillator in the presence of strong Raman scattering,” Opt. Express 21, 20556–20564 (2013). 24. G. Agrawal. Nonlinear Fiber Optics (Elsevier, 2013). 25. H. A. Haus. Electromagnetic Noise and Quantum Optical Measurements (Springer, 2000). 26. R. Paschotta, “Noise of mode-locked lasers (Part II): timing jitter and other fluctuations,” Appl. Phys. B 79, 163–173 (2004). 27. C. Headley and G. P. Agrawal. Raman Amplification in Fiber Optical Communication Systems (Elsevier, 2005). 28. A. Ankiewicz and N. Akhmediev, “Moving fronts for complex Ginzburg-Landau equation with Raman term,” Phys. Rev. E 58, 6723–6727 (1998). 29. V. L. Kalashnikov, E. Sorokin, S. Naumov, and I. T. Sorokina, “Spectral properties of the Kerr-lens mode-locked Cr4+ :YAG laser,” J. Opt. Soc. Am. B 20, 2084–2092 (2003). 30. V. L. Kalashnikov, E. Sorokin, and I. T. Sorokina,“ Multipulse operation and limits of the Kerr-lens mode locking stability,” IEEE J. Quantum Electron. 39, 323–336 (2003). 31. L. Zhu, A. J. Verhoef, K. G. Jespersen, V. L. Kalashnikov, L. Gr¨uner-Nielsen, D. Lorenc, A. Baltuska, and A. Fernandez, “Generation of high fidelity 62-fs, 7-nJ pulses at 1035 nm from a net normal-dispersion Yb-fiber laser with anomalous dispersion higher-order-mode fiber,” Opt. Express 21, 16255–16262 (2013). 32. D. S. Kharenko, E. V. Podivilov, A. A. Apolonski, and S. A. Babin, “20 nJ 200 fs all-fiber highly-chirped dissipative soliton oscillator,” Opt. Lett. 19, 4104–4106 (2012). 33. S. A. Babin, E. V. Podivilov, D. S. Kharenko, A. E. Bednyakova, M. P. Fedoruk, V. L. Kalashnikov, and A. Apolonski, “Multicolour nonlinearly bound chirped dissipative solitons,” Nat. Commun. 5, 4653 (2014).
1.
Introduction
The concept of the dissipative soliton (DS) has an extremely wide horizon of application. It is useful in quantum optics and modeling of Bose-Einstein condensation, condensed matter physics and study of non-equilibrium phenomena, nonlinear dynamics and quantum mechanics of self-organizing dissipative systems [1]. Non-equilibrium character of a system, where a DS emerges, requires a well-organized energy exchange between the soliton and the environment. The energy flow forms a non-trivial internal structure of a DS, which provides the energy redistribution inside it [1]. As a result, the DS phase φ (t) becomes inhomogeneous with a nonzero chirp: Q ≡ ∂ 2 φ /∂t 2 6= 0 (here t is the local time). A DS with the nontrivial phase distribution is called the “chirped dissipative soliton” (CDS) [2, 3]. The unique feature of a CDS is its capacity to accumulate energy E without stability loss: E ∝ Q [4, 5]. This results in the energy scalability of CDS [4–6]. Such phenomenon resembles a resonant enhancement of oscillations in environment-coupled systems and was named the “dissipative soliton resonance” (DSR) [7]. #224304 - $15.00 USD Received 2 Oct 2014; accepted 15 Oct 2014; published 24 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.030118 | OPTICS EXPRESS 30119
The ability of a CDS to accumulate energy is of interest for a number of applications. For instance, it allows energy scaling of ultrashort laser pulses so that >10 MW peak powers and, respectively, > 1014 W/cm2 intensities can be obtained directly from a laser operating at few MHz repetition rates [8–10]. Such pulse powers bring the high-field physics on tabletops of the mid-level university labs [11]. In particular, high-energy ultrashort pulse lasers allow nowadays such experiments as direct gas ionization and high-harmonic generation, pump-probe diffraction experiments with electrons and production of nm-scale structures at the surface of transparent materials, characterization and control of the electronic dynamics, a variety of biophotonic and biomedical applications, etc. [12–16]. The MHz pulse repetition rates provide signal rate improvement factor of 103 ÷ 104 as compared to usual chirped-pulse amplifiers, with a corresponding signal-to-noise ratio improvement. Mechanisms of the CDS formation and stabilization can be described qualitatively in the framework of the complex nonlinear Ginzburg-Landau equation [1,17,18]. The contribution of the group-delay dispersion (GDD) to the soliton phase can be expressed as (β /2)∂ 2 a(t)/∂t 2 (here β is the GDD coefficient, a(t) is the complex slowly-varying envelope of DS). For a CDS, such contribution can be compensated by the term −(β /2)a(t) (∂ φ (t)/∂t)2 if p the dimensionless chirp parameter, also called the DS stretching parameter ψ ' QT 2 equals 2 + γP0 T 2 /β (here γ is the coefficient of self-phase modulation (SPM), P0 is the DS peak power, and T is the DS width). It is important that such phase compensation becomes possible in the normal GDD range (β > 0 in our notations), where the classical Schr¨odinger soliton does not exist. Additional insight into the CDS stabilization mechanism can be obtained from the dispersion relations for a DS and linear perturbation waves. In the normal dispersion regime, a DS with the wave number q = γP0 [2, 19] can interact resonantly with a linear perturbation wave if the wave number of the latter is k(ω) ≡ β ω 2 /2 = q (here ω is a frequency deviation from the DS carrier frequency). Hence, a stable DS must have its spectrum truncated at the frequencies ±∆: k(±∆) = q where ∆2 = 2γP0 /β . On the other hand, the DS spectral loss ∼ α∆2 (here α is the squared inverse bandwidth of the spectral filter) has to be compensated by the nonlinear gain ∼ κP0 (here κ is the nonlinear gain or, in another words, the self-amplitude modulation (SAM) coefficient). This condition in combination with the dispersion relation gives the DS stability criterion: 2αγ/β κ ≤ 1. A more precise analysis [17] within the framework of the complex nonlinear cubic-quintic Ginzburg-Landau model gives the conditions of the DS asymptotical stability: ( 2/3, E → ∞ 2αγ ≤ (1) βκ 2, E →0 The asymptotics E → ∞ in Eq. (1) is equivalent to the DSR phenomenon [7] and reveals the two-dimensionality of the CDS parametric space. The latter can be described in the form of the so-called “master-diagram” on a plane of the control parameter C ≡ 2αγ/β κ vs. the dimensionless energy E ∗ ≡ Eκ 3/2 ζ 1/2 /γα 1/2 , where ζ is the SAM saturation parameter in the complex nonlinear cubic-quintic Ginzburg-Landau model (see below) [6]. The DSR regime, i.e. the perfect energy-scalability of the DS requires extensive energy exchange with the environment followed by the energy redistribution inside the soliton. As already mentioned, such redistribution results in the phase inhomogeneity, i.e. a strong chirp, accompanied by the temporal broadening. Since the DS peak power is fixed by the SAM saturation so that P0 ∝ 1/ζ , the law limC→2/3 E ∗ = ∞ promises perfect energy scalability of the DS, for instance, by simply increasing the fiber laser length [20]. However, the recent studies have demonstrated that such scalability can be destroyed by gain saturation, stimulated Raman scattering (SRS), and noise amplification [21–23]. The two latter factors become especially important with the DS energy growth and are subjects of the study presented below. #224304 - $15.00 USD Received 2 Oct 2014; accepted 15 Oct 2014; published 24 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.030118 | OPTICS EXPRESS 30120
2.
Model
The main originality of our study is taking into account the effects of quantum noise and SRS on the DS stability and energy scalability. The presented analysis is based on the numerical solution of the generalized complex nonlinear Ginzburg-Landau equation, which is a testbed for modeling of nonlinear dissipative dynamics far from the thermodynamic equilibrium (for an overview, see [1]). The master equation under consideration is the distributed complex nonlinear cubic-quintic Ginzburg-Landau equation [2, 4]: 2 ∂2 β ∂2 ∂ a (z,t) 2 2 − γ|a| a + −σ + α 2 + κ 1 − ζ |a| |a| a, (2) =i ∂z 2 ∂t 2 ∂t where a(z,t) is the slowly varying complex field amplitude depending on the propagation distance z and the local time t. It is convenient for an oscillator to normalize the propagation length z to the oscillator length [6]. Then all coefficients in Eq. (2) can be treated as those averaged over an oscillator period. Terms in the first square brackets correspond to the so-called nonlinear Schr¨odinger equation [24] which takes into account the GDD (the β coefficient) and the SPM (the γ coefficient). With the usual condition γ > 0, no soliton can exist in the normal GDD region (β > 0 in our notations), if only the first bracketed terms of Eq. (2) contribute. Terms in the second square brackets correspond to dissipative factors and describe the net-loss (the σ coefficient), the spectral dissipation (the squared inverse bandwidth α), and the saturable nonlinear gain (saturable SAM) with the corresponding inverse power-dimensional coefficients κ and ζ . This part is decisive for the further consideration because the dissipative factors are formative for the DS. The unique property of Eq. (2) is that it has a CDS solution, which can be characterized by a two-dimensional parametric space, or “master diagram” [4]. The structure of this diagram is defined by a set of the “isogain” curves σ = const ≥ 0 on the (C, E ∗ )-plane. The zeroisogain σ = 0 (Fig. 1, black solid curve) defines both, the CDS stability threshold and its energy scalability law limC→2/3 E ∗ = ∞. However, the model based on Eq. (2) is not complete. First, it does not account for the quantum noises, which are inherent in any dissipative system [25]. The quantum noise can be introduced by a complex white noise term s(z,t) with the correlation function
0 0 ∗ s z ,t s (z,t) = Γδ z0 − z δ t 0 − t , (3) where Γ is the noise power coefficient [26]. Another important generalization of the Eq. (2) is including the stimulated Raman scattering (SRS), which is a significant factor in fiber lasers. One may assume that the phonon basin acts as a damped harmonic oscillator, which is nonlinearly coupled with the photon basin [24, 27]. The corresponding Raman response function can then be written as h(t) =
T12 + T22 t t exp(− ) sin( ), T2 T1 T1 T22
(4)
where T2 = 32 fs and T1 = 12.2 fs define the effective relaxation time and resonant frequency for phonons in fused silica, respectively [24, 27]. In fiber optics, the SRS contribution is often included as the first moment of the response R function [24, 28]: −iTR γa∂ |a|2 /∂t, where TR ≈ fR 0∞ h(t)tdt and fR is the fractional contribution of the Raman response to the nonlinear polarization, valid for pulses with sufficiently narrow spectra (∆ω 1/T1 ). We use the more adequate and general approximation (4), thus including the Raman self-scattering, which is important for the femtosecond pulse dynamics [29].
#224304 - $15.00 USD Received 2 Oct 2014; accepted 15 Oct 2014; published 24 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.030118 | OPTICS EXPRESS 30121
The resulting generalized stochastic nonlinear Ginzburg-Landau equation has the following form: 2 β ∂2 ∂2 ∂ a (z,t) 2 2 =i − (1 − fR ) γ|a| a(z,t) + −σ + α 2 + κ 1 − ζ |a| |a| a(z,t)− ∂z 2 ∂t 2 ∂t − iγ fR a(z,t)
Zt
(5) 0
0 2
dt h(t − t)|a(z,t )| + s(z,t).
−∞
This equation will be the subject of our analysis. The main difficulty here is the incommensurability of the temporal scales: from the femtosecond scale, corresponding to the Raman response, to over 100 ps duration of the strongly chirped CDS. We used the time mesh with 1 fs step and up to 221 points (2 ns time window), with a propagation distance up to z = 104 roundtrips. The numerical calculations have been realized using the symmetrized split-step Fourier method with the convolution of the SRS term performed in the spectral domain. 3.
Results
The master diagram is a network of “isogains” with σ = const > 0 [4]. The zero-level isogain σ = 0 defines the threshold of CDS stability against the vacuum excitation. Such excitation may cause, e.g., a collapsing DS instability or a multiple pulse generation [30]. The CDS stability border in the absence of noise and SRS is shown by a solid black curve in Fig. 1. A CDS is stable below this curve. A “plateau” in the dependence of the stability threshold on the energy for E ∗ 1 corresponds to the DSR. A typical Wigner function of a stable CDS in the vicinity of DSR is shown in Fig. 2(a). CDSs along the DSR curve in Fig. 1 have a characteristic finger-like spectrum (a truncated Lorenzian) and a flat-top temporal profile [2, 4, 31]. If now we include the complete set of terms in Eq. (5), the DS stability at high energies dramatically changes. If only the noise term in Eq. (5) contributes, then the DSR regime is suppressed. The corresponding stability border is shown by the red dashed curve in Fig. 1. The DSR disappearance means that reaching higher pulse energy requires smaller C-parameter (e.g., by adding more dispersion β ). Even more important, there might exist no single-pulse CDS solution beyond some maximum E ∗ (see e.g. the break of the red dashed curve in Fig. 1). The situation becomes more complicated in the presence of the SRS. The corresponding stability border is shown by the blue dashed-dot curve in Fig. 1. Fig. 3 shows evolution of the CDS spectra with increasing energy, corresponding to the horizontal trajectory 1 in Fig. 1. With increasing E ∗ , the spectrum broadens and becomes Lorentz-profiled, truncated at the deviation frequencies ±∆ (Section 1). When the CDS spectral width reaches the resonant frequency ∼ 1/πT1 = 0.026 fs−1 , the SRS causes self-scattering followed by the self-Stokes-shift. This is a dissipative Raman soliton (DRS). It is chirped and possesses Stokes-shifted spectrum with a truncated Lorentzian shape. The spectrum also has an anti-Stokes component which corresponds to a perturbation of the DS trailing edge and causes a chaotic evolution of the DRS instantaneous power. Finally, the red spectrum shift due to the SRS in the normal dispersion regime results in its acceleration (group velocity increase). When spectral shift of the DRS becomes sufficiently strong, the multiple pulse generation develops. The multipulsing is seen as modulated spectra in Fig. 3, which begin when the trajectory 1 in Fig. 1 intersects the stability border at E ∗ ≈ 30. We suggest, that the spectral shift induced by the SRS acts an additional mechanism of the nonlinear spectral filtering. The additional spectral dissipation initiates multiple pulsing [23, 30]. The multipulsing instability has been experimentally identified as a typical regime of an all-fiber laser with the strong SRS [23, 32] and only stabilization of this regime has allowed generation of the DRS paired #224304 - $15.00 USD Received 2 Oct 2014; accepted 15 Oct 2014; published 24 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.030118 | OPTICS EXPRESS 30122
2
DS is unstable 1 noiseless
DS is stable 2
noise
noise+SRS
1
noise+SRS, narrow filter
0.1
DS is stable DS is unstable
0.01 0.1
1
10
100
∗
1000
Fig. 1. The CDS master diagram. Black solid curve corresponds to the stability threshold in the absence of noise and SRS. The stability thresholds in the presence of the quantum noise only (Γ = 10−10 /γ, fR = 0) and noise with SRS (Γ = 10−10 /γ, fR =0.22) are shown by the red dashed and the blue dashed-dot curves, respectively. Spectral filter parameter α corresponds to ≈40 nm (α =366 fs2 ) at 1 µm central wavelength and to ≈20 nm (α =1460 fs2 ) for the magenta dotted curve. κ = 0.1γ, ζ = 0.05γ.
Fig. 2. Wigner function of an unperturbed CDS (a) and in the presence of the quantum noise and SRS (b). The CDS temporal and spectral profiles are shown as the corresponding axis-projections. The top wavelength shift scale corresponds to an Yb:fiber laser centered around 1070 nm. The parameters along the trajectory 2 in Fig. 1 are C =0.59, E ∗ =111.
#224304 - $15.00 USD Received 2 Oct 2014; accepted 15 Oct 2014; published 24 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.030118 | OPTICS EXPRESS 30123
Fig. 3. Averaged CDS spectra along the trajectory 1 in Fig. 1. Spectral power profiles are averaged over z = 1000 round-trips.
with the ordinary CDS [33]. Hence, the contribution of SRS and noise limits the maximum DS soliton energy (blue dashed-dot curve in Fig. 1). Let us consider the change of the DS stability with the C-parameter variation (vertical trajectory 2 in Fig. 1). This can be achieved e.g. by increasing the dispersion, since C ∝ 1/β . At low energy values, the multipulsing above the stability threshold shown by the solid black curve in Fig. 1 can always be suppressed by an appropriate decrease of the C-parameter, but if the energy E ∗ is too high, the noise contribution can prevent the DS stabilization (red dashed curve in Fig. 1). The situation becomes more complicated in the presence of SRS, which causes pulse spectrum splitting when C-parameter decreases (Fig. 2(b)). Note the perturbation caused by the anti-Stokes component localized on the trailing edge of CDS. This perturbation induces chaotization of the CDS dynamics, in a certain analogy to chaotic behavior of a DS interacting with the dispersive wave [19]. Such chaotization grows with smaller C values provoking earlier pulse splitting (intersection of trajectory 2 with the blue dashed-dot curve in Fig. 1). A Wigner function corresponding to the double-pulse CDS regime is shown in Fig. 4. One should note the “bugles” and “branches” on the Wigner function edges (Fig. 4). They can be considered as a manifestation of the CDS chaotization. Such a chaotization reveals itself in irregular oscillations of the soliton instantaneous power, but may remain unnoticed in conventional experimental conditions when all measurements effectively average over many round-trips [19]. The double-pulse complex, despite the chaotization at the trailing edges, remains very stable and mutually coherent, as confirmed by the interference fringes with 100% contrast in the spectrum (Fig. 4). Further decrease of the C-parameter can suppress the multiple pulse instability (the trajectory 2 intersects again the blue dashed-dot curve in Fig. 1). The reason for this is that the inverse filter bandwidth parameter α enters the C-parameter in the numerator, so that a smaller C can equally be represented by a more relaxed spectral filtering, allowing the soliton to shift away from the gain maximum. The resulting pulse is the chirped DRS with a red-shifted finger-like
#224304 - $15.00 USD Received 2 Oct 2014; accepted 15 Oct 2014; published 24 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.030118 | OPTICS EXPRESS 30124
-12
0
12
nm
Fig. 4. Wigner function of a double CDS in the presence of the quantum noise and SRS. The CDS temporal and spectral profiles are shown as the corresponding axis-projections. The spectrum is modulated by the interference fringes with 100% visibility, which are not resolved. The top wavelength shift scale corresponds to an Yb:fiber laser centered around 1070 nm. The parameters along the trajectory 2 in Fig. 1 are C =0.037, E ∗ =111.
6
12
18
24
nm
Fig. 5. Wigner function of a chirped DRS. The DRS temporal and spectral profiles are shown as the corresponding axis-projections. The top wavelength shift scale corresponds to an Yb:fiber laser centered around 1070 nm. The parameters along the trajectory 2 in Fig. 1 are C =0.021, E ∗ =111.
#224304 - $15.00 USD Received 2 Oct 2014; accepted 15 Oct 2014; published 24 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.030118 | OPTICS EXPRESS 30125
regular spectrum (Fig. 5) that propagates faster than an ordinary DS. Simultaneously, the DRS width can be appreciably less than that of an ordinary CDS with the same C- and E ∗ -parameters. Note that the edges of the chirped DRS Wigner-function (Fig. 5) are bent in the opposite directions relatively to those of ordinary CDS (Fig. 2(a)), and the DRS fidelity (i.e., quality of its compressibility provided by the “flatness” of the spectral chirp frequency dependence [31]) is in fact higher than that of an ordinary CDS. The fingerprints of the DRS reported here are the pronounced perturbations of envelope trailing edge (Fig. 5) and chaotic evolution of its instantaneous power. With even smaller C parameter, the Stokes-shift decreases but the spectrum tends to a “triangular” shape, which corresponds to the breezer-like dynamics of the DRS [28]. The chaotization of the high-energy DRS dynamics as well as the multiple pulse instability become an acute problem for the DS energy scalability. Since the main source of such instability is a perturbation of the anti-Stockes spectral component localized on the CDS trail, one may suggest a stabilization technique based on the manipulation with a spectral filter. The magenta dotted curve in Fig. 1 demonstrates the corresponding stability threshold for a two-fold narrower filter bandwidth. One can see, that while the maximum dimensionless energy E ∗ becomes stringer confined, the stable range protracts towards smaller C values. Note, however, that confinement of E ∗ by increasing √the α parameter does not, in fact, decrease the physical energy E, since it scales as E ∝ E ∗ α, while the extended range of the C-parameter can be exploited e.g. by manipulation with the dispersion only (β parameter). It thus seems that narrowing the spectral filter is one of the ways to counteract the Raman destabilization at high energies in the real-world systems. 4.
Conclusion
We demonstrated the effects of quantum noise and stimulated Raman scattering (SRS) on the chirped dissipative soliton stability. These factors suppress the dissipative soliton resonance and reduce the soliton energy scalability by increasing the multiple pulse instability region. Simultaneously, the SRS induces the spectral Stokes-shift and perturbs the soliton trailing edge, where the anti-Stokes spectral components are localized. Such perturbation causes chaotic evolution of the instantaneous power and can initiate pulse splitting with increasing dispersion. However, if the dispersion becomes sufficiently large, a stable dissipative Raman soliton develops. It has a finger-like and Stokes-shifted spectrum, higher group velocity, and shorter duration in comparison with an ordinary dissipative soliton. Narrowing of the spectral filter can further reduce the tendency to multipulsing caused by the stimulated Raman scattering. Acknowledgments This work was supported by the Austrian Science Fund (FWF), project P24916-N27. VLK acknowledges the support of the FP7-PEOPLE-2012-IAPP (project GRIFFON, no. 324391). The computational results have been achieved using the Vienna Scientific Cluster (VSC).
#224304 - $15.00 USD Received 2 Oct 2014; accepted 15 Oct 2014; published 24 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.030118 | OPTICS EXPRESS 30126
Institute, Vienna University of Technology, Gusshausstr. 27/387, A-1040, Vienna, Austria 2 Aston University, Institute of Photonic Technologies, School of Engineering & Applied Science, Aston Triangle, Birmingham, B4 7ET, UK ∗ [email protected]
http://info.tuwien.ac.at/kalashnikov
Abstract: A new type of dissipative solitons - dissipative Raman solitons - are revealed on the basis of numerical study of the generalized complex nonlinear Ginzburg-Landau equation. The stimulated Raman scattering significantly affects the energy scalability of the dissipative solitons, causing splitting to multiple pulses. We show, that an appropriate increase of the group-delay dispersion can suppress the multipulsing instability due to formation of the dissipative Raman soliton, which is chirped, has a Stokes-shifted spectrum, and chaotic modulation on its trailing edge. The strong perturbation of a soliton envelope caused by the stimulated Raman scattering confines the energy scalability, preventing the so-called dissipative soliton resonance. We show that in practical implementations, a spectral filter can extend the stability regions of high-energy pulses. © 2014 Optical Society of America OCIS codes: (140.7090) Ultrafast lasers; (190.5530) Pulse propagation and temporal solitons; (190.5650) Raman effect; (270.3100) Instabilities and chaos.
References and links 1. N. N. Akhmediev and A. Ankiewicz (Eds.), Dissipative Solitons (Springer, 2005). 2. E. Podivilov and V.L. Kalashnikov, “Heavily-chirped solitary pulses in the normal dispersion region: new solutions of the cubic-quintic complex Ginzburg-Landau equation,” JETP Lett. 82, 467–471 (2005). 3. W. H. Renninger, A. Chong, and F. W. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A 77, 023814 (2008). 4. V.L. Kalashnikov and A. Apolonski, “Chirped-pulse oscillators: a unified standpoint,” Phys. Rev. A 79, 043829 (2009). 5. Ph. Grelu and N. Akhmediev, “Dissipative solitons for mode-locked lasers,” Nature Photon. 6, 84–92 (2012). 6. V. L. Kalashnikov, E. Podivilov, A. Chernykh, and A. Apolonski, “Chirped-pulse oscillators: theory and experiment,” Applied Physics B 83, 503–510 (2006). 7. W. Chang, A. Ankiewicz, J.M. Soto-Crespo, and N. Akhmediev, “Dissipative soliton resonances,” Phys. Rev. A 78, 023830 (2008). 8. A. Fernandez, T. Fuji, A. Poppe, A. F¨urbach, F. Krausz, and A. Apolonski, “Chirped-pulse oscillators: a route to high-power femtosecond pulses without external amplification,” Opt. Lett. 29, 1366–1368 (2004). 9. S. Naumov, A. Fernandez, R. Graf, P. Dombi, F. Krausz, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators,” New J. Phys. 7, 216 (2005). 10. W. H. Renninger and F. W. Wise, “Fundamental Limits to Mode-Locked Lasers: Toward Terawatt Peak Powers,” IEEE J. Sel. Topics Quantum Electron. 21, 1-8 (2015). 11. T. S¨udmeyer1, S. V. Marchese, S. Hashimoto, C. R. E. Baer, G. Gingras, B. Witzel, and U. Keller, “Femtosecond laser oscillators for high-field science,” Nat. Phot. 2, 599–604 (2008). 12. Y. Liu, S. Tschuch, A. Rudenko, M. D¨urr, M. Siegel, U. Morgner, R. Moshammer, and J. Ullrich, “Strong-field double ionization of Ar below the recollision threshold,” Phys. Rev. Lett. 101, 053001 (2008).
#224304 - $15.00 USD Received 2 Oct 2014; accepted 15 Oct 2014; published 24 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.030118 | OPTICS EXPRESS 30118
13. E. Seres, J. Seres, and Ch. Spielmann, “Extreme ultraviolet light source based on intracavity high harmonic generation in a mode locked Ti:sapphire oscillator with 9.4 MHz repetition rate,” Opt. Express 20, 6185–6190 (2012). 14. G.Sciaini and R. J. D. Miller, “Femtosecond electron diffraction: heralding the era of atomically resolved dynamics,” Rep. Prog. Phys. 74, 096101 (2011). 15. R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nature Photon. 2, 219–225 (2008). 16. A. A. Lanin, V. Fedotov, D. A. Sidorov-Biryukov, L. V. Doronina-Amitonova, O. I. Ivashkina, M. A. Zots, ¨ Ilday, A. B. Fedotov, K. V. Anokhin, and A. M. Zheltikov, “Air-guided photonic-crystalChi-Kuang Sun, F. O. fiber pulse-compression delivery of multimegawatt femtosecond laser output for nonlinear-optical imaging and neurosurgery,” Appl. Phys. Lett. 100, 101104 (2012). 17. V. L. Kalashnikov, “Chirped-Pulse Oscillators: Route to the Energy-Scalable Femtosecond Pulses,” in Solid State Laser, Amin H. Al-Khursan (Ed.), 145–184 (InTech, 2012). 18. S. K. Turitsyn, B. G. Bale, and M. P. Fedoruk, “Dispersion-managed solitons in fibre systems and lasers,” Phys. Rep. 521, 135–204 (2012). 19. E. Sorokin, N. Tolstik, V. L. Kalashnikov, and I. T. Sorokina, “Chaotic chirped-pulse oscillators,” Opt. Express 21, 29567–29577 (2013). 20. V. L. Kalashnikov and A. Apolonski, “Energy scalability of mode-locked oscillators: a completely analytical approach to analysis,” Opt. Express 18, 25757–25770 (2010). 21. E. Ding, Ph. Grelu, and J. N. Kutz, “Dissipative soliton resonance in a passively mode-locked fiber laser,” Opt. Lett. 36, 1146–1148 (2011). 22. V. L. Kalashnikov, “Dissipative solitons in presence of quantum noise,” Chaotic modeling and simulation 1, 29–37 (2014). 23. A. E. Bednyakova, S. A. Babin, D. S. Kharenko, E. V. Podivilov, M. P. Fedoruk, V. L. Kalashnikov, and A. Apolonski, “Evolution of dissipative solitons in a fiber laser oscillator in the presence of strong Raman scattering,” Opt. Express 21, 20556–20564 (2013). 24. G. Agrawal. Nonlinear Fiber Optics (Elsevier, 2013). 25. H. A. Haus. Electromagnetic Noise and Quantum Optical Measurements (Springer, 2000). 26. R. Paschotta, “Noise of mode-locked lasers (Part II): timing jitter and other fluctuations,” Appl. Phys. B 79, 163–173 (2004). 27. C. Headley and G. P. Agrawal. Raman Amplification in Fiber Optical Communication Systems (Elsevier, 2005). 28. A. Ankiewicz and N. Akhmediev, “Moving fronts for complex Ginzburg-Landau equation with Raman term,” Phys. Rev. E 58, 6723–6727 (1998). 29. V. L. Kalashnikov, E. Sorokin, S. Naumov, and I. T. Sorokina, “Spectral properties of the Kerr-lens mode-locked Cr4+ :YAG laser,” J. Opt. Soc. Am. B 20, 2084–2092 (2003). 30. V. L. Kalashnikov, E. Sorokin, and I. T. Sorokina,“ Multipulse operation and limits of the Kerr-lens mode locking stability,” IEEE J. Quantum Electron. 39, 323–336 (2003). 31. L. Zhu, A. J. Verhoef, K. G. Jespersen, V. L. Kalashnikov, L. Gr¨uner-Nielsen, D. Lorenc, A. Baltuska, and A. Fernandez, “Generation of high fidelity 62-fs, 7-nJ pulses at 1035 nm from a net normal-dispersion Yb-fiber laser with anomalous dispersion higher-order-mode fiber,” Opt. Express 21, 16255–16262 (2013). 32. D. S. Kharenko, E. V. Podivilov, A. A. Apolonski, and S. A. Babin, “20 nJ 200 fs all-fiber highly-chirped dissipative soliton oscillator,” Opt. Lett. 19, 4104–4106 (2012). 33. S. A. Babin, E. V. Podivilov, D. S. Kharenko, A. E. Bednyakova, M. P. Fedoruk, V. L. Kalashnikov, and A. Apolonski, “Multicolour nonlinearly bound chirped dissipative solitons,” Nat. Commun. 5, 4653 (2014).
1.
Introduction
The concept of the dissipative soliton (DS) has an extremely wide horizon of application. It is useful in quantum optics and modeling of Bose-Einstein condensation, condensed matter physics and study of non-equilibrium phenomena, nonlinear dynamics and quantum mechanics of self-organizing dissipative systems [1]. Non-equilibrium character of a system, where a DS emerges, requires a well-organized energy exchange between the soliton and the environment. The energy flow forms a non-trivial internal structure of a DS, which provides the energy redistribution inside it [1]. As a result, the DS phase φ (t) becomes inhomogeneous with a nonzero chirp: Q ≡ ∂ 2 φ /∂t 2 6= 0 (here t is the local time). A DS with the nontrivial phase distribution is called the “chirped dissipative soliton” (CDS) [2, 3]. The unique feature of a CDS is its capacity to accumulate energy E without stability loss: E ∝ Q [4, 5]. This results in the energy scalability of CDS [4–6]. Such phenomenon resembles a resonant enhancement of oscillations in environment-coupled systems and was named the “dissipative soliton resonance” (DSR) [7]. #224304 - $15.00 USD Received 2 Oct 2014; accepted 15 Oct 2014; published 24 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.030118 | OPTICS EXPRESS 30119
The ability of a CDS to accumulate energy is of interest for a number of applications. For instance, it allows energy scaling of ultrashort laser pulses so that >10 MW peak powers and, respectively, > 1014 W/cm2 intensities can be obtained directly from a laser operating at few MHz repetition rates [8–10]. Such pulse powers bring the high-field physics on tabletops of the mid-level university labs [11]. In particular, high-energy ultrashort pulse lasers allow nowadays such experiments as direct gas ionization and high-harmonic generation, pump-probe diffraction experiments with electrons and production of nm-scale structures at the surface of transparent materials, characterization and control of the electronic dynamics, a variety of biophotonic and biomedical applications, etc. [12–16]. The MHz pulse repetition rates provide signal rate improvement factor of 103 ÷ 104 as compared to usual chirped-pulse amplifiers, with a corresponding signal-to-noise ratio improvement. Mechanisms of the CDS formation and stabilization can be described qualitatively in the framework of the complex nonlinear Ginzburg-Landau equation [1,17,18]. The contribution of the group-delay dispersion (GDD) to the soliton phase can be expressed as (β /2)∂ 2 a(t)/∂t 2 (here β is the GDD coefficient, a(t) is the complex slowly-varying envelope of DS). For a CDS, such contribution can be compensated by the term −(β /2)a(t) (∂ φ (t)/∂t)2 if p the dimensionless chirp parameter, also called the DS stretching parameter ψ ' QT 2 equals 2 + γP0 T 2 /β (here γ is the coefficient of self-phase modulation (SPM), P0 is the DS peak power, and T is the DS width). It is important that such phase compensation becomes possible in the normal GDD range (β > 0 in our notations), where the classical Schr¨odinger soliton does not exist. Additional insight into the CDS stabilization mechanism can be obtained from the dispersion relations for a DS and linear perturbation waves. In the normal dispersion regime, a DS with the wave number q = γP0 [2, 19] can interact resonantly with a linear perturbation wave if the wave number of the latter is k(ω) ≡ β ω 2 /2 = q (here ω is a frequency deviation from the DS carrier frequency). Hence, a stable DS must have its spectrum truncated at the frequencies ±∆: k(±∆) = q where ∆2 = 2γP0 /β . On the other hand, the DS spectral loss ∼ α∆2 (here α is the squared inverse bandwidth of the spectral filter) has to be compensated by the nonlinear gain ∼ κP0 (here κ is the nonlinear gain or, in another words, the self-amplitude modulation (SAM) coefficient). This condition in combination with the dispersion relation gives the DS stability criterion: 2αγ/β κ ≤ 1. A more precise analysis [17] within the framework of the complex nonlinear cubic-quintic Ginzburg-Landau model gives the conditions of the DS asymptotical stability: ( 2/3, E → ∞ 2αγ ≤ (1) βκ 2, E →0 The asymptotics E → ∞ in Eq. (1) is equivalent to the DSR phenomenon [7] and reveals the two-dimensionality of the CDS parametric space. The latter can be described in the form of the so-called “master-diagram” on a plane of the control parameter C ≡ 2αγ/β κ vs. the dimensionless energy E ∗ ≡ Eκ 3/2 ζ 1/2 /γα 1/2 , where ζ is the SAM saturation parameter in the complex nonlinear cubic-quintic Ginzburg-Landau model (see below) [6]. The DSR regime, i.e. the perfect energy-scalability of the DS requires extensive energy exchange with the environment followed by the energy redistribution inside the soliton. As already mentioned, such redistribution results in the phase inhomogeneity, i.e. a strong chirp, accompanied by the temporal broadening. Since the DS peak power is fixed by the SAM saturation so that P0 ∝ 1/ζ , the law limC→2/3 E ∗ = ∞ promises perfect energy scalability of the DS, for instance, by simply increasing the fiber laser length [20]. However, the recent studies have demonstrated that such scalability can be destroyed by gain saturation, stimulated Raman scattering (SRS), and noise amplification [21–23]. The two latter factors become especially important with the DS energy growth and are subjects of the study presented below. #224304 - $15.00 USD Received 2 Oct 2014; accepted 15 Oct 2014; published 24 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.030118 | OPTICS EXPRESS 30120
2.
Model
The main originality of our study is taking into account the effects of quantum noise and SRS on the DS stability and energy scalability. The presented analysis is based on the numerical solution of the generalized complex nonlinear Ginzburg-Landau equation, which is a testbed for modeling of nonlinear dissipative dynamics far from the thermodynamic equilibrium (for an overview, see [1]). The master equation under consideration is the distributed complex nonlinear cubic-quintic Ginzburg-Landau equation [2, 4]: 2 ∂2 β ∂2 ∂ a (z,t) 2 2 − γ|a| a + −σ + α 2 + κ 1 − ζ |a| |a| a, (2) =i ∂z 2 ∂t 2 ∂t where a(z,t) is the slowly varying complex field amplitude depending on the propagation distance z and the local time t. It is convenient for an oscillator to normalize the propagation length z to the oscillator length [6]. Then all coefficients in Eq. (2) can be treated as those averaged over an oscillator period. Terms in the first square brackets correspond to the so-called nonlinear Schr¨odinger equation [24] which takes into account the GDD (the β coefficient) and the SPM (the γ coefficient). With the usual condition γ > 0, no soliton can exist in the normal GDD region (β > 0 in our notations), if only the first bracketed terms of Eq. (2) contribute. Terms in the second square brackets correspond to dissipative factors and describe the net-loss (the σ coefficient), the spectral dissipation (the squared inverse bandwidth α), and the saturable nonlinear gain (saturable SAM) with the corresponding inverse power-dimensional coefficients κ and ζ . This part is decisive for the further consideration because the dissipative factors are formative for the DS. The unique property of Eq. (2) is that it has a CDS solution, which can be characterized by a two-dimensional parametric space, or “master diagram” [4]. The structure of this diagram is defined by a set of the “isogain” curves σ = const ≥ 0 on the (C, E ∗ )-plane. The zeroisogain σ = 0 (Fig. 1, black solid curve) defines both, the CDS stability threshold and its energy scalability law limC→2/3 E ∗ = ∞. However, the model based on Eq. (2) is not complete. First, it does not account for the quantum noises, which are inherent in any dissipative system [25]. The quantum noise can be introduced by a complex white noise term s(z,t) with the correlation function
0 0 ∗ s z ,t s (z,t) = Γδ z0 − z δ t 0 − t , (3) where Γ is the noise power coefficient [26]. Another important generalization of the Eq. (2) is including the stimulated Raman scattering (SRS), which is a significant factor in fiber lasers. One may assume that the phonon basin acts as a damped harmonic oscillator, which is nonlinearly coupled with the photon basin [24, 27]. The corresponding Raman response function can then be written as h(t) =
T12 + T22 t t exp(− ) sin( ), T2 T1 T1 T22
(4)
where T2 = 32 fs and T1 = 12.2 fs define the effective relaxation time and resonant frequency for phonons in fused silica, respectively [24, 27]. In fiber optics, the SRS contribution is often included as the first moment of the response R function [24, 28]: −iTR γa∂ |a|2 /∂t, where TR ≈ fR 0∞ h(t)tdt and fR is the fractional contribution of the Raman response to the nonlinear polarization, valid for pulses with sufficiently narrow spectra (∆ω 1/T1 ). We use the more adequate and general approximation (4), thus including the Raman self-scattering, which is important for the femtosecond pulse dynamics [29].
#224304 - $15.00 USD Received 2 Oct 2014; accepted 15 Oct 2014; published 24 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.030118 | OPTICS EXPRESS 30121
The resulting generalized stochastic nonlinear Ginzburg-Landau equation has the following form: 2 β ∂2 ∂2 ∂ a (z,t) 2 2 =i − (1 − fR ) γ|a| a(z,t) + −σ + α 2 + κ 1 − ζ |a| |a| a(z,t)− ∂z 2 ∂t 2 ∂t − iγ fR a(z,t)
Zt
(5) 0
0 2
dt h(t − t)|a(z,t )| + s(z,t).
−∞
This equation will be the subject of our analysis. The main difficulty here is the incommensurability of the temporal scales: from the femtosecond scale, corresponding to the Raman response, to over 100 ps duration of the strongly chirped CDS. We used the time mesh with 1 fs step and up to 221 points (2 ns time window), with a propagation distance up to z = 104 roundtrips. The numerical calculations have been realized using the symmetrized split-step Fourier method with the convolution of the SRS term performed in the spectral domain. 3.
Results
The master diagram is a network of “isogains” with σ = const > 0 [4]. The zero-level isogain σ = 0 defines the threshold of CDS stability against the vacuum excitation. Such excitation may cause, e.g., a collapsing DS instability or a multiple pulse generation [30]. The CDS stability border in the absence of noise and SRS is shown by a solid black curve in Fig. 1. A CDS is stable below this curve. A “plateau” in the dependence of the stability threshold on the energy for E ∗ 1 corresponds to the DSR. A typical Wigner function of a stable CDS in the vicinity of DSR is shown in Fig. 2(a). CDSs along the DSR curve in Fig. 1 have a characteristic finger-like spectrum (a truncated Lorenzian) and a flat-top temporal profile [2, 4, 31]. If now we include the complete set of terms in Eq. (5), the DS stability at high energies dramatically changes. If only the noise term in Eq. (5) contributes, then the DSR regime is suppressed. The corresponding stability border is shown by the red dashed curve in Fig. 1. The DSR disappearance means that reaching higher pulse energy requires smaller C-parameter (e.g., by adding more dispersion β ). Even more important, there might exist no single-pulse CDS solution beyond some maximum E ∗ (see e.g. the break of the red dashed curve in Fig. 1). The situation becomes more complicated in the presence of the SRS. The corresponding stability border is shown by the blue dashed-dot curve in Fig. 1. Fig. 3 shows evolution of the CDS spectra with increasing energy, corresponding to the horizontal trajectory 1 in Fig. 1. With increasing E ∗ , the spectrum broadens and becomes Lorentz-profiled, truncated at the deviation frequencies ±∆ (Section 1). When the CDS spectral width reaches the resonant frequency ∼ 1/πT1 = 0.026 fs−1 , the SRS causes self-scattering followed by the self-Stokes-shift. This is a dissipative Raman soliton (DRS). It is chirped and possesses Stokes-shifted spectrum with a truncated Lorentzian shape. The spectrum also has an anti-Stokes component which corresponds to a perturbation of the DS trailing edge and causes a chaotic evolution of the DRS instantaneous power. Finally, the red spectrum shift due to the SRS in the normal dispersion regime results in its acceleration (group velocity increase). When spectral shift of the DRS becomes sufficiently strong, the multiple pulse generation develops. The multipulsing is seen as modulated spectra in Fig. 3, which begin when the trajectory 1 in Fig. 1 intersects the stability border at E ∗ ≈ 30. We suggest, that the spectral shift induced by the SRS acts an additional mechanism of the nonlinear spectral filtering. The additional spectral dissipation initiates multiple pulsing [23, 30]. The multipulsing instability has been experimentally identified as a typical regime of an all-fiber laser with the strong SRS [23, 32] and only stabilization of this regime has allowed generation of the DRS paired #224304 - $15.00 USD Received 2 Oct 2014; accepted 15 Oct 2014; published 24 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.030118 | OPTICS EXPRESS 30122
2
DS is unstable 1 noiseless
DS is stable 2
noise
noise+SRS
1
noise+SRS, narrow filter
0.1
DS is stable DS is unstable
0.01 0.1
1
10
100
∗
1000
Fig. 1. The CDS master diagram. Black solid curve corresponds to the stability threshold in the absence of noise and SRS. The stability thresholds in the presence of the quantum noise only (Γ = 10−10 /γ, fR = 0) and noise with SRS (Γ = 10−10 /γ, fR =0.22) are shown by the red dashed and the blue dashed-dot curves, respectively. Spectral filter parameter α corresponds to ≈40 nm (α =366 fs2 ) at 1 µm central wavelength and to ≈20 nm (α =1460 fs2 ) for the magenta dotted curve. κ = 0.1γ, ζ = 0.05γ.
Fig. 2. Wigner function of an unperturbed CDS (a) and in the presence of the quantum noise and SRS (b). The CDS temporal and spectral profiles are shown as the corresponding axis-projections. The top wavelength shift scale corresponds to an Yb:fiber laser centered around 1070 nm. The parameters along the trajectory 2 in Fig. 1 are C =0.59, E ∗ =111.
#224304 - $15.00 USD Received 2 Oct 2014; accepted 15 Oct 2014; published 24 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.030118 | OPTICS EXPRESS 30123
Fig. 3. Averaged CDS spectra along the trajectory 1 in Fig. 1. Spectral power profiles are averaged over z = 1000 round-trips.
with the ordinary CDS [33]. Hence, the contribution of SRS and noise limits the maximum DS soliton energy (blue dashed-dot curve in Fig. 1). Let us consider the change of the DS stability with the C-parameter variation (vertical trajectory 2 in Fig. 1). This can be achieved e.g. by increasing the dispersion, since C ∝ 1/β . At low energy values, the multipulsing above the stability threshold shown by the solid black curve in Fig. 1 can always be suppressed by an appropriate decrease of the C-parameter, but if the energy E ∗ is too high, the noise contribution can prevent the DS stabilization (red dashed curve in Fig. 1). The situation becomes more complicated in the presence of SRS, which causes pulse spectrum splitting when C-parameter decreases (Fig. 2(b)). Note the perturbation caused by the anti-Stokes component localized on the trailing edge of CDS. This perturbation induces chaotization of the CDS dynamics, in a certain analogy to chaotic behavior of a DS interacting with the dispersive wave [19]. Such chaotization grows with smaller C values provoking earlier pulse splitting (intersection of trajectory 2 with the blue dashed-dot curve in Fig. 1). A Wigner function corresponding to the double-pulse CDS regime is shown in Fig. 4. One should note the “bugles” and “branches” on the Wigner function edges (Fig. 4). They can be considered as a manifestation of the CDS chaotization. Such a chaotization reveals itself in irregular oscillations of the soliton instantaneous power, but may remain unnoticed in conventional experimental conditions when all measurements effectively average over many round-trips [19]. The double-pulse complex, despite the chaotization at the trailing edges, remains very stable and mutually coherent, as confirmed by the interference fringes with 100% contrast in the spectrum (Fig. 4). Further decrease of the C-parameter can suppress the multiple pulse instability (the trajectory 2 intersects again the blue dashed-dot curve in Fig. 1). The reason for this is that the inverse filter bandwidth parameter α enters the C-parameter in the numerator, so that a smaller C can equally be represented by a more relaxed spectral filtering, allowing the soliton to shift away from the gain maximum. The resulting pulse is the chirped DRS with a red-shifted finger-like
#224304 - $15.00 USD Received 2 Oct 2014; accepted 15 Oct 2014; published 24 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.030118 | OPTICS EXPRESS 30124
-12
0
12
nm
Fig. 4. Wigner function of a double CDS in the presence of the quantum noise and SRS. The CDS temporal and spectral profiles are shown as the corresponding axis-projections. The spectrum is modulated by the interference fringes with 100% visibility, which are not resolved. The top wavelength shift scale corresponds to an Yb:fiber laser centered around 1070 nm. The parameters along the trajectory 2 in Fig. 1 are C =0.037, E ∗ =111.
6
12
18
24
nm
Fig. 5. Wigner function of a chirped DRS. The DRS temporal and spectral profiles are shown as the corresponding axis-projections. The top wavelength shift scale corresponds to an Yb:fiber laser centered around 1070 nm. The parameters along the trajectory 2 in Fig. 1 are C =0.021, E ∗ =111.
#224304 - $15.00 USD Received 2 Oct 2014; accepted 15 Oct 2014; published 24 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.030118 | OPTICS EXPRESS 30125
regular spectrum (Fig. 5) that propagates faster than an ordinary DS. Simultaneously, the DRS width can be appreciably less than that of an ordinary CDS with the same C- and E ∗ -parameters. Note that the edges of the chirped DRS Wigner-function (Fig. 5) are bent in the opposite directions relatively to those of ordinary CDS (Fig. 2(a)), and the DRS fidelity (i.e., quality of its compressibility provided by the “flatness” of the spectral chirp frequency dependence [31]) is in fact higher than that of an ordinary CDS. The fingerprints of the DRS reported here are the pronounced perturbations of envelope trailing edge (Fig. 5) and chaotic evolution of its instantaneous power. With even smaller C parameter, the Stokes-shift decreases but the spectrum tends to a “triangular” shape, which corresponds to the breezer-like dynamics of the DRS [28]. The chaotization of the high-energy DRS dynamics as well as the multiple pulse instability become an acute problem for the DS energy scalability. Since the main source of such instability is a perturbation of the anti-Stockes spectral component localized on the CDS trail, one may suggest a stabilization technique based on the manipulation with a spectral filter. The magenta dotted curve in Fig. 1 demonstrates the corresponding stability threshold for a two-fold narrower filter bandwidth. One can see, that while the maximum dimensionless energy E ∗ becomes stringer confined, the stable range protracts towards smaller C values. Note, however, that confinement of E ∗ by increasing √the α parameter does not, in fact, decrease the physical energy E, since it scales as E ∝ E ∗ α, while the extended range of the C-parameter can be exploited e.g. by manipulation with the dispersion only (β parameter). It thus seems that narrowing the spectral filter is one of the ways to counteract the Raman destabilization at high energies in the real-world systems. 4.
Conclusion
We demonstrated the effects of quantum noise and stimulated Raman scattering (SRS) on the chirped dissipative soliton stability. These factors suppress the dissipative soliton resonance and reduce the soliton energy scalability by increasing the multiple pulse instability region. Simultaneously, the SRS induces the spectral Stokes-shift and perturbs the soliton trailing edge, where the anti-Stokes spectral components are localized. Such perturbation causes chaotic evolution of the instantaneous power and can initiate pulse splitting with increasing dispersion. However, if the dispersion becomes sufficiently large, a stable dissipative Raman soliton develops. It has a finger-like and Stokes-shifted spectrum, higher group velocity, and shorter duration in comparison with an ordinary dissipative soliton. Narrowing of the spectral filter can further reduce the tendency to multipulsing caused by the stimulated Raman scattering. Acknowledgments This work was supported by the Austrian Science Fund (FWF), project P24916-N27. VLK acknowledges the support of the FP7-PEOPLE-2012-IAPP (project GRIFFON, no. 324391). The computational results have been achieved using the Vienna Scientific Cluster (VSC).
#224304 - $15.00 USD Received 2 Oct 2014; accepted 15 Oct 2014; published 24 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.030118 | OPTICS EXPRESS 30126
