Supplementary Figure 1: The simulated feedback-defined evolution of the intra-cavity

Transcription

1 Supplementary Figure 1: The simulated feedback-defined evolution of the intra-cavity pulses. The pulse structure is shown for the scheme in Fig. 1a (point B) versus the round- trip number. The zero time offset is bound to the position of the main dissipative soliton. Supplementary Figure : Power profiles and energy fluxes. Power profiles (black) and the corresponding energy flux (red) for the solitons of Supplementary Fig.1 (and Fig. 4b of the paper).

2 Supplementary Figure 3: Spectral and temporal pulse variations along the cavity. Two top panels: variations of the pulse duration and spectral width (at -10 db level) through six cavity

3 points of the cavity for DS (blue) and RDS (red). Six low panels: the corresponding spectral and temporal shapes (insets) at the cavity points 1-6. Supplementary Figure 4: Spectral characteristics of the cavity couplers. Transmission spectra of the WDM1 and WDM couplers of the experimental scheme (Fig.1 of the paper).

4 1 Supplementary Figure 5: Soliton spectra for different coupler parameters. Simulation results for the DS (blue) and RDS (red) spectra at different WDM1 coupler parameters: 1010/1055 nm (solid) and 1015/1060 nm (dashed) at the fixed WDM parameters (the cut-off wavelength at 1005 nm), for 70 ps Raman delay and 105 (1030) nm gain maximum.

5 Supplementary Figure 6: Three-color bound solitons. Results of simulation for the generated spectra in an 80-m-long PM-fiber cavity oscillator with the Raman feedbacks for the first-order Stokes (R 1 = 10-4 ) and the second-order Stokes (R = 10 - ) waves, in the presence of a band-stop filter at 105 nm (10 nm width) and the following parameters of the i-th feedback loop: delay time 140 ps (i=1) and 80 ps (i=), stepwise WDM1 (i) couplers with cut-off wavelengths at 1035 nm (i=1) and 1085 nm (i=).

6 Supplementary Note 1. Feedback-induced evolution from noise to Raman dissipative soliton. Analogies to the known regimes Without feedback, a Raman pre-pulse develops in just a single pass and is characterized by strong phase noise, whereas the chirped Raman dissipative soliton (RDS) reported in the Article is noise-free, being generated in the dissipative soliton (DS) laser cavity with spectrally selective feedback (see Fig.1 of the paper). The feedback loop extracts the Raman signal from the laser cavity and, after suppressing it down to R~10-4 and delaying to 70 ps, sends it back. We have performed simulation of the transition from noise (the pre-pulse) to the steady-state DS (RDS) in the scheme of Fig.1 with feedback, and the corresponding results are shown in Supplementary Fig. 1. In the initial state of the laser, amplification of noise starts at both fundamental and Raman wavelengths. Then, two chirped pulses (DS and RDS) are formed almost simultaneously. They become stable after approximately 30 roundtrips. Note that high losses of the RDS are compensated at each roundtrip by the very high Raman gain (~40 db) induced by a long (~40 ps) high-peak-power (00 W) DS acting as a pump. The resulting complex of two bound dissipative solitons (the main and Raman ones) circulates in the cavity in an extremely stable manner. The maximum intra-cavity energy of the dissipative soliton depends on the feedback value R=P f /P= logarithmically according to the equation for the SRS threshold energy, E th ln(1/r)δv -1 /g R, where δv -1 = v -1 DS -v -1 R is the inverse group velocity difference of the DS and RDS, and g R is the Raman gain coefficient. The threshold energy for the experimental parameters is estimated as E th ~10 nj. At the value R<10-6, the re-injected power P f approaches the energy of the spontaneous emission at the Stokes-shifted wavelength (P sp /P~10-7 ), and the feedback becomes insufficient to overcome noise. At R>10 -, the single DS+RDS complex splits into two

7 similar pulse complexes separated in time. Stable pulses in 40-m cavity are realized for delays in the range ps (with optimum at ps). The RDS shifts towards the DS with the decreasing delay. This regime has to be distinguished from the well-known ones. First of all, the RDS differs from the conventional Raman soliton (see, for example, [1]). The latter develops from noise due to the action of the pump pulse in the low-dissipation system with anomalous dispersion. After formation, such a soliton propagates independently as a Schrödinger one. The RDS propagation is strongly affected by the dissipative factors, and it develops in the all-normaldispersion regime. Continuous interaction with the main DS is required in this case, resulting in transition from a noisy Raman pulse to a coherent steady-state RDS (see Supplementary Fig. 1). This method can be considered as a realization of the chaos control concept [], when a weak (initially chaotic) actuating signal leads to i) stable mode-locking dynamics and ii) mutual coherence of the DS and RDS. One may draw an analogy with the synchronous pumping technique, taking into account that the DS-induced amplification mechanism for the RDS is intrinsically nonlinear and also that the interacting pulses are spectrally separated. In other words, conventional synchronous pumping technique is single-sided: the pumping pulse is not affected by the generated one. The spectral conversion involved in the RDS formation can invoke the analogy with an optical parametric oscillator. However, unlike the latter, the SRS does not require any phase matching. The closest (but also not complete) analogy to the known techniques is a soliton Stokes self-frequency shift in a mode-locked Raman-active laser operating in the anomalous dispersion regime, see, for example, [3]. In the latter case, the overlap of the femtosecond pulse spectrum with the Raman lines of the active crystal causes the Stokes shift of the pulse as a whole in both spectral and temporal domains. The regime realized in the Article

8 may thus be considered as a dissipative realization of the conventional Raman soliton. But contrary to the regime of [3], the RDS is separated from the main DS both spectrally and temporally and, as a result of strong chirp, the DS+RDS complex is energy-scalable. Supplementary Note. Why is the Raman pulse a dissipative soliton? The dissipative nature of the solitons considered can be reasoned on the basis of the approach developed in [4]. From the continuity relation it follows for a soliton that the energy flux ( ) (A is the field amplitude for the soliton envelope) is constant for non-dissipative systems. Supplementary Fig. shows the energy flux for the stable DS+RDS complex corresponding to Supplementary Fig.1 (and Fig.4b of the paper, accordingly). One can clearly see the existence of pronounced (and opposite) energy fluxes inside both DS and RDS, which demonstrates a strong energy exchange between the solitons as well. The central part/leading edges of the soliton exhibit the pronounced energy in/out-flows, which are caused by the gain, SRS and spectral dissipation. This effect can be seen as a kind of nonlinear spectral filtering. It is known [4] that stability of a dissipative soliton is defined by setting a balance between nonlinearity and dispersion, as well as between gain and losses at each round trip. It has been checked that the first balance, γ<p> βδ / (γ is the Kerr coefficient, <P> is the average power, β is the dispersion coefficient, Δ is the spectral half-width), is fulfilled in our case not only for the main DS, but also for the generated RDS. The second balance is clearly seen from

9 the conservation of energy for the RDS over round trips (just as for the DS), see Supplementary Figures 1 and. However, the pulses have different values (and reasons) for the gain and loss. The main DS is amplified in the active fiber due to diode pumping, whereas the RDS amplification is defined by the energy transfer from the main DS during their propagation in the passive fiber, see Supplementary Fig.. This energy transfer defines losses for the main DS together with its outcoupling at each round trip, whereas for the RDS very strong outcoupling is the main factor. High artificial loss of RDS is compensated by very high Raman gain (~40 db per round trip). The Raman pulse is chirped, which means an internal re-distribution of its energy as the consequence of the energy interchange with the environment. The main manifestation of high chirp for the Raman pulse in the experiment is its typical M-shaped ( Batman -type) spectrum just like for DS [5, 6]. Experimental demonstration of dechirping is shown in Fig.5 of the paper. In order to give more evidence of the DS nature of the chirped Raman pulse, we performed a detailed comparison of its evolution with the DS evolution as a template, see Supplementary Fig. 3. The DS enters PM fiber with relatively narrow spectrum (at point 1). It experiences then i) gradual spectral and temporal broadening due to nonlinearity and dispersion (from 1 to ) followed by ii) strong narrowing due to SRS effect ( to 3) converting long-wavelength part of the DS to the Stokes component iii) continuous broadening during interaction-free propagation due to the balanced action of nonlinearity and dispersion (3 to 4), iv) spectral filtering by WDM with a stepwise transmission function which cuts off a short-wavelength part of the DS spectrum accompanied by pulse shortening (point 5), v) amplification by several times in a short piece of Yb-doped fiber (point 6). After spatial separation of the orthogonal polarization components by

10 PBS providing amplitude modulation, the main component is sent to point 1 and a new round trip starts. Evolution of the DS pulse is self-consistent due to the balanced action of nonlinearity and dispersion, loss and gain. The pulse reshaping (spectral filtering) is provided via losses induced by WDM at the short wavelength (linear filtering) and by the Raman process at the long wavelength side (nonlinear filtering), with an additional input of the gain fiber. The amplitude modulation (driven by nonlinear polarization evolution) is provided by PBS. Note that at low pulse energies (below the SRS threshold) a DS is also formed in a short cavity. In this case the combined filtering provided by WDM and PBS accompanied by amplitude modulation is enough for realizing similar pulse evolution like in a classical scheme of ANDi laser with separate spectral filter [5]. Strong nonlinear filtering due to the Raman effect allows to dispense with additional filters at high pulse energies. Both low- and high-energy DSs (in the presence of SRS) have a typical M- ( Batman ) spectral shape like in [5, 6], with the temporal shape close to hyperbolic secant with exponential wings. This equilibrium shape is reached during the interaction-free propagation in a long piece of PM-fiber (from 3 to 4) with relatively low variation in other parts, where spectral reshaping and amplitude recovery occur. The RDS starts from a weak signal which nearly coincides temporally with the DS (point 1). It is then amplified via DS-induced Raman gain which compensates losses and also provides both amplitude modulation (resulting in the pulse shortening) and nonlinear spectral filtering according to the parabolic spectral function of the Raman gain coefficient near its maximum (1 to ), resulting in the Gaussian shape of the spectrum. When the RDS and DS energies become comparable ( to 3), the RDS experiences sufficient shape variation. Continuous broadening occurs during the interaction-free propagation due to the balanced action of nonlinearity and

11 dispersion (3 to 4). In this cavity part, RDS and DS become separated in the time domain showing though quite similar spectral and temporal shapes. After the delay line containing soft WDM1 filter and 1:R coupler, the strongly attenuated part of the RDS becomes coinciding in time with the DS (point 5). After slight amplification by Yb-doped fiber (point 6) and reshaping by PBS, a new round trip starts (point 1). Summarizing the comparison of the DS and RDS evolutions, one can conclude that they are similar in the following aspects: - Variations of the spectral width and pulse duration along the cavity are small relative to their average values. - Below the SRS threshold, the spectral width and pulse duration vary similarly during the propagation along PM fiber and at all point-action cavity elements. - Above the SRS threshold, the DS experiences additional filtering leading to its spectral narrowing in a short part of PM fiber near point, where the RDS also demonstrates narrowing due to amplification. Since the pulses become equal in energy, they vary similarly in the rest part of the cavity. - A huge difference in the gain coefficients (~10 for DS and for RDS) is not important because the Raman gain occurs at the distance of only several meters, much less than the total cavity length (40 m). Therefore, the amplification fiber parts can be considered as point-action elements in both cases. - The gain-to-loss compensation occurs in a short part of the cavity and, in both cases, does not influence the nearly steady-state pulse evolution in the main part of the cavity. - The equilibrium shapes of the pulses in the spectral and time domains are similar ( Batman -type and hyperbolic secant, correspondingly) corresponding to average

12 dissipative solitons generated inside the laser cavity with short filtering/reshaping and gain sections as compared with the long passive part. - Variations of the shapes and their widths in the spectral and time domains around the equilibrium state in the intra-cavity evolution are small. Based on the detailed comparison above, we can conclude that the RDS realised in our manuscript has the same equilibrium characteristics and demonstrates similar evolution dynamics as the chirped pulses generated in the ANDi laser configuration agreed to be dissipative solitons [5, 6]. Supplementary Note 3. Variation of the soliton spectra with filter parameters For quantitative comparison of the calculated and experimental spectra we need to check how the optical spectra of both solitons are sensitive to linear and nonlinear (SRS-induced) filtering. Linear filtering is provided by two WDM couplers, whose spectra are sensitive to temperature and can vary in the range of several nm. The first filter (marked as WDM1 in Fig.1 of the paper) is a self-made PM-fiber coupler based on the internal Lyot filter consisting of a properly oriented short PM-fiber piece and a polarization beam splitter (PBS). Such a WDM coupler/splitter has a typical sinusoidal spectral function (see Supplementary Fig. 4) providing the wavelength division 1015/1060 nm for two output coupler arms. The second coupler (WDM in Fig.1 of the paper) is a commercial one having a stepwise transmission spectrum function with the cut-off at 1005 nm.

13 The influence of the filtering effect of WDM1 on the DS is small, but nonlinear filtering becomes important due to strong Raman conversion, just as in the case of a noisy Raman pulse observed without WDM1 and delay loop in [7]. As follows from our simulations, the SRS effect eats the long-wavelength wing of the DS spectrum thus pushing the soliton spectrum out of the gain maximum ( nm) towards the WDM-induced cut-off at 1005 nm. This effect defines the short-wavelength edge of the generated spectrum. By numerically varying the exact position of the spectral maximum of the WDM1 filter (Supplementary Fig. 4), we found that the RDS spectral width is very sensitive to these variations. As one can see in Supplementary Fig. 5, a shift of only 5 nm (the minimum value we can control in the experiment) leads to a significant change of both the RDS position and width. Because the RDS spectrum obtained with the 1015/1060 nm filter fits better to the transmission function of the WDM1 coupler measured under the experimental conditions (Supplementary Fig. 4), we used this data for comparing the experimental and calculated spectra (Fig. 3 of the paper). Supplementary Note 4. Comparison of the NLSE and CGLE solutions for the Raman dissipative soliton It is worth checking out whether the RP evolution approaches a solution of a modified cubicquintic Ginzburg-Landau equation (CGLE) that is usually used for DS simulations. The CGLE is a master equation that incorporates the main physical ingredients in a distributed way instead of considering point-action components used in a real laser system (see [6, 8]). It is known how

14 to relate, albeit in an approximate way, all the coefficients of the CGLE to the physical parameters of the laser and, in particular, for ANDi fiber lasers [5, 6]. A similar procedure can also be performed for the case of the Raman DS, though it needs sufficient efforts to rigorously derive new master equation based on the pulsed DS-induced Raman gain instead of the normal gain. Additionally, specific mechanisms of amplitude modulation and spectral filtering as well as correct averaging procedure should be involved. However, we can start using CGLE in a phenomenological way. We performed such an analysis based on CGLE in its normalized form [6]: iψ x + Dψ tt / + ψ ψ = iδψ + iβψ tt +iε ψ ψ + iμ ψ 4 ψ, (1) where ψ is the normalized optical field envelope, t is the retarded time in the frame moving with the pulse, x=z/l, the propagation distance normalized on the cavity length L. The left-hand side of the equation contains conservative terms: the dispersion term with the coefficient D (negative for the normal dispersion), and the Kerr nonlinearity term. The quintic effect is neglected. The right-hand side of the equation includes all dissipative terms with the corresponding coefficients δ, β, ε, and μ for the linear loss (negative), spectral filtering, nonlinear gain (positive) and saturation of the nonlinear gain (negative), respectively. The last cubic-quintic terms provide amplitude modulation. For the sake of comparison of the CGLE in the form (1) with the NLSE used in our manuscript for simulation of the SRS effects, we modified the NLSE (see Methods section) in the following way: A i z β A t +γ A A= i point action terms. ()

15 We neglect the third-order dispersion term and consider that the Raman response function h R (t) can be attributed to point-action terms, since the interaction of DS and RDS takes place only in a short part of the cavity (see Supplementary Fig. 3). The point-action terms include also the effects of spectral filtering, gain, loss and amplitude modulation. After replacing z by the normalized propagation distance x = z / L (varied in interval 0 < x < 1), we can set up the correspondence between the terms of equations (1) and (). We take the nonlinearity γl 0.4 W -1, the relation between the amplitudes ψ = Γ A z,t, the net cavity dispersion = β L ps. We can then distribute the point-action terms along the cavity. Based D on the evolution map presented in Supplementary Fig. 3 and the corresponding analysis above, it can be done for the Raman pulse. Its evolution consists of a long distributed part between points 3 and 4 (like for the main DS) and short point-action parts, the most important of which is a path between points 6, 1 and. This path can be described by the following three physical processes: 1) Amplitude modulation ΔA Г = εψ + μψ ψ which results in reshaping and shortening of the pulse between points 6 and (where coefficients and μ can be related to the critical power P cr in a way similar to the DS, see Methods section); 4 ) Amplification (with the net Raman gain coefficient G): A =, where G R 1 GA 1 means loss compensation between points 4 and 1, A = ; 1 RA 4 3) Filtering: ΔA Г = β ψ. The parabolic (-βω ) profile of the RDS spectrum at point t is defined by the Gaussian gain ( Ge ) profile near the gain maximum.

16 After making such a distribution of the point-action terms, we solved the CGLE using numerical methods for simulation of chirped DS [9]. We then compared the CGLE solutions with the local solutions of the NLSE at points 3 and 4 and made the arithmetic mean of them supposing that the solution of the CGLE based on average coefficients should be close to the mean solution of the NLSE covering most of the pulse evolution in the cavity. The results of comparison are summarized in Fig.5 of the paper. The parameters extracted after optimization are δ=-0.3, β=0.016 ps and =0.0813, μ= (and corresponding to the critical power P cr 600 W of the effective saturable absorber, see Methods section) appeared to be close to those for the DS. Supplementary Note 5. Generation of a second RDS: a three-color chirped dissipative complex The scheme shown in Fig.1 of the main text, with additional feedback loops can be used for generating next-order Stokes pulses of similar spectral and temporal shapes. We numerically realized a scheme with two feedback loops, with the corresponding results shown in Supplementary Fig. 6. Figs.e,d of the paper illustrates evolution of the chirped pulse in the stable 3-color regime. To realize it, we made the following modifications of the scheme: (i) long (L 80 m) PM fiber, (ii) two feedback loops, each of which comprise WDM1 (i) splitter and 1:R (i) coupler (similar to the feedback loop in Fig.1) optimized for the 1 st and nd Stokes pulses (i=1,): R (1) = 10-4, R () = The regime becomes stable when we use, instead of sinusoidal WDM1 (see Supplementary Fig. 4), two special WDM1 (1) and WDM1 () splitters of a

17 step-wise spectral function at cut-off wavelengths at 1035 nm and 1085 nm, correspondingly. An implementation of such hard filters eliminates the remaining small-amplitude parts of the Raman pulse which can t be fully suppressed in the case of a soft (sinusoidal) WDM1 splitter (see Fig.c). Since each feedback loop operates independently, the scheme offers a way to generate next Stokes orders by adding corresponding feedback loops in a routine manner. A number of optimization steps to be made in order to realize this regime experimentally, including filter optimization. The open question is whether one can realize the equidistant spectral components shown in Supplementary Fig. 6. Supplementary References 1. Agrawal, G. P. Nonlinear Fiber Optics (Oxford, UK, 007).. González-Miranda, J.M. Synchronization and Control of Chaos: An introduction for Scientists and Engineers (WSP, London, 004) 3. Kalashnikov, V.L., Sorokin, E., Sorokina, I.T. Mechanisms of spectral shift in ultrashortpulse laser oscillators, J. Opt. Soc. Am. B 18, (001). 4. N. Akhmediev, A. Ankiewicz, Dissipative solitons in the complex Ginzburg-Landau and Swift-Hohenberg equations, In: Dissipative Solitons, N. Akhmediev, A. Ankiewicz, Eds., Springer-Verlag, Berlin, 005, pp Chong, A., Renninger, W. H., Wise, F. W. Properties of normal-dispersion femtosecond fiber lasers, J. Opt. Soc. Am. B 5, (008); Renninger, W. H., Chong A., Wise,

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