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As recently revealed, chirped dissipative solitons (DSs) generated in a long cavity fiber laser are subject to action of stimulated Raman scattering (SRS). Here we present theoretical and experimental study of the DS formation and evolution in the presence of strong SRS. The results demonstrate that the rising noisy Raman pulse (RP) acts not only as an additional channel of the energy dissipation destroying DS, but on the contrary can support it that results in formation of a complex of the bound DS and RP of comparable energy and duration. In the complex, the DS affords amplification of the RP, whereas the RP stabilizes the DS via temporal-spectral filtering. Stable 25 nJ SRS-driven chirped DS pulses are generated in all-fiber ring laser cavities with lengths of up to 120 m. The DS with duration up to 70 ps can be externally dechirped to <300 fs thus demonstrating the record compression factor.
© 2013 Optical Society of America
1. Introduction
Introduced in 1990s [1,2] as an extension of the term “soliton”, dissipative soliton (DS) describes localized structures of an electromagnetic field in non-conservative systems, where an energy exchange with environment becomes important in addition to a balance between nonlinearity and dispersion (or diffraction). A balance between gain and loss is also needed for a stable structure formation (see [3, 4] for a review of the DS basic principles and their emergence in optics, hydrodynamics, biology etc.). Since the gain-to-loss balance is a typical requirement for laser generation, the concept of dissipative solitons has been successfully applied recently (see [5–7] for a review) for description of the dynamics of optical pulses in a laser cavity, especially for a broad class of passively mode-locked lasers [8]. The DS concept is also useful for practical purposes implying for the new cavity designs, in particular those providing generation of high-energy femtosecond pulses. An important step in this direction is a laser cavity with normal net dispersion. It has been shown that the pulses generated in such a cavity are stretched significantly acquiring frequency modulation (so called “chirp”) that offers much higher pulse energy compared to transform-limited soliton pulses generated in cavities with anomalous dispersion. As one of the first examples, a Ti:Sa laser with 14-m normal-dispersion cavity generated 0.2 μJ linearly chirped pulses compressed down to ~30 femtoseconds [9].
The practical achievements of fiber lasers in the last decade have initiated further intensive research and development on generation of high-power femtosecond pulses in fiber lasers. Note that in a fiber waveguide the diffraction effects are eliminated and the system becomes effectively one-dimensional. For the sake of mode-locking, bulk elements are usually inserted in the scheme, such as semiconductor saturable absorber mirror (SESAM) or polarization optics providing intensity modulation based on the nonlinear polarization evolution (NPE) of elliptically polarized light in a single-mode fiber (SMF), see [5–7] and citation therein. There is a tendency to replace the free space bulk elements by their fiber analogues resulted in the so-called all-fiber design, see e.g [10]. In this case, the laser becomes much less sensitive to environment and operates “hands free”, although its output parameters (e.g. pulse energy) do not yet exceed those in fiber lasers comprising bulk optical elements.
Advances in physics of pulse evolution in fiber lasers based on the DS concept resulted in more than one order increase in pulse energy during the past few years, see [5–7] and citation therein. The most powerful fiber source, an Yb-doped fiber laser (YDFL), operates in ~1 μm spectral range, where conventional single-mode fibers and fiber components have normal dispersion. An all-normal dispersion YDFL based on NPE mode-locking mechanism has exhibited several nJ output pulse energy already at first demonstration [11]. Scaling of the pulse energy by increasing length L and core diameter d of the fiber resulted in generation of <200 fs dechirped pulses with ~20 nJ energy at repetition rate f = c/Ln = 12.5 MHz in 6.5-μm core fiber [12, 13] and >100 nJ pulses at rate f = 84 MHz in a large-mode-area (d = 25 μm) photonic crystal fiber [14]. The peak power demonstrated recently approaches [14] or exceeds [15] MW level.
Since the pulse is being continuously stretched and chirped during propagation in an all-normal-dispersion cavity as a result of the balanced action of nonlinear and dispersion phase shifts, a spectral filtering at each round trip is usually used to provide self-consistent evolution [5–7, 11–14]. Efficient filtering becomes more important in longer cavities with larger phase shifts. Another problem at the cavity lengthening is NPE overdriving which results in the loss of mode-locking stability already at L~15 m for high-energy pulses [12]. The reason of the stability break is shown to be an excessive nonlinear rotation of the polarization ellipse in a long SMF fiber, therefore alternative cavity designs are discussed implying non-NPE mode-locking mechanisms and/or PM (polarization maintaining) fibers [16–18] that also improve environmental stability. Recently, implementation of PM fiber with d = 5.5 μm and L~30 m in [17] resulted in generation of highly-chirped DS pulses with energy ~20 nJ that is >5 times more than in the best all-fiber YDFL DS realization [10] and comparable with the best result obtained in a hybrid cavity [12] based on NPE mode-locking in SMF with larger core diameter. As has been found, a new energy limiting factor comes into play for such long (~30 ps) and powerful (~0.5 kW) DS pulses. That is the Raman conversion of the DS energy, when the latter exceeds the SRS threshold. Lower energies have been obtained in the all-PM configuration based on the nonlinear amplifying loop mirror [18] that exhibits chirped pulse destabilization for the lengths >100 m [19] attributed by the authors to the SRS-induced noise influence. Thus, a question arises about the mechanisms of the back influence of the SRS effect on the DS parameters.
In this paper we report on the comprehensive theoretical and experimental study of the new physical effects defining the DS formation and evolution in the presence of strong SRS. We demonstrate that SRS acts not only as an additional channel of energy dissipation destroying DS, but can also support it enabling thus the generation of a stable “DS - Raman pulse” bound complex in all-fiber ring laser cavities of order-of-magnitude larger lengths (>100 m) than in conventional NPE-mode-locked DS fiber lasers [5–7, 12–14]. We also discuss the influence of SRS conversion and noise influence on the DS energy scalability. As a result, we have identified the effects which confine the maximal reachable energy of DS for long cavity lasers.
2. Experimental setup
Figure 1 illustrates the YDFL scheme under study that is similar to the basic ring scheme of NPE mode-locked DS fiber lasers [5, 6, 11–14]. A key difference is the all-fiber design of the cavity divided on two parts: very long (L1 = 30-120 m) passive PM fiber part and very short (L2~1.5 m) SMF part between polarization controller (PC) and polarization beam splitter (PBS). The SMF part comprises 15-cm piece of active Yb3+-doped fiber pumped by 976 nm single-mode laser diode (LD) via 976/1030 nm wavelength division multiplexer (WDM). Small-signal Yb3+ gain at the spectral maximum (~1025 nm) reaches ~25 dB for the available pump of ~400 mW. A coupler with 1% output port is inserted into the cavity to monitor the intracavity parameters of the generated pulses.
Fig. 1 (Left) Schematic of the experimental setup. (Right) The calculated shapes of the generated pulses in corresponding points (A,B,C,D) of the scheme.
The main feature of the scheme is that it provides stable (overdriving-free) mode-locking via NPE in a short SMF part, whereas the cavity length is increased independently by means of PM fiber. The SMF part is also responsible for the pulse amplification balancing the losses on the PBS output and spectral filtering naturally defined by the Yb gain bandwidth and the spectral transmission function of WDM, see Fig. 2. Because the SMF part is much shorter than the PM part, the effects of mode-locking, gain and filtering can be treated as a point-action, while DS evolution defined by nonlinearity, dispersion and the studied SRS effect is distributed along the PM fiber (its dispersion in the spectral domain of the DS and Raman pulses is shown in Fig. 2).
The fiber laser exhibits generation of stable pulses in the all studied range of the cavity lengths (30-120 m) with their duration 30-70 ps nearly proportional to the cavity length. The SRS effect is present already at 30 m like that in [17] and becomes stronger for longer cavities, where a noisy Raman pulse develops with the energy comparable with that for a DS.
3. Numerical simulation
3.1 Basic equations and parameters
To prove a possibility of stable pulse generation in such a long cavity, we performed a numerical simulation using the generalized nonlinear Schrödinger equation (NLSE), see [5–7] and citation therein. Note that in our previous studies of the highly-chirped DSs in fiber lasers [20, 21] we used cubic-quintic Ginzburg-Landau equation (see e.g [22].) that is valid only for small round-trip power variations but directly involves saturable absorber terms providing mode-locking. Though it describes well the experimentally observed SRS-free DS pulses [21], it is hardly applicable for description of the SRS-driven pulse evolution. On the contrary, NLSE enables easy treatment of the SRS effect at pulse propagation in a passive fiber, whereas mode-locking, amplification and filtering effects may be treated independently as point-action ones. To include the effect of the DS spectrum conversion via SRS (the DS and Raman pulses are equally polarized in the PM fiber that was directly checked), we added the corresponding SRS term in NLSE [22]:
where is the electric field envelope, and are the second- and third-order dispersion coefficients at the central frequency , is the Kerr nonlinearity coefficient with the nonlinear refractive index and the effective fiber cross-section areafor a fundamental mode. The response function includes both instantaneous electronic and delayed Raman contributions, where represents the fractional contribution of the delayed Raman response to the instantaneous nonlinear polarization. The Raman response function defines the Raman gain coefficient, whose spectrum is given by . At typical value [22] and specified for the PM fiber used, the calculated peak Raman gain (with the Stokes shift = 45 nm) is nearly equal to the experimental value of [17]. In our simulations, we use the multiple-vibrational-mode model for the Raman response described in [23]. The equation was solved by using the standard split-step Fourier-transform method. The simulations are run until the pulse field reaches the steady state after a certain number of cavity round trips, taking into consideration the contribution of the point-action WDM having a stepwise transmission spectrum (see Fig. 2), amplification in Yb3+-fiber and the NPE-induced intensity modulation described below. The following fiber parameters are used in the simulations: γ = 6 W−1 km−1, = 22 ps2/km and = 0.037 ps3/km (see Fig. 2).3.1 Amplification model
Amplification in an Yb-doped active fiber with the Lorentzian line of a 40 nm bandwidth, a central wavelength of λ0 = 1025 nm, and a small-signal gain coefficient of g0(λ0) ≈170 dB/m was simulated in the spectral domain. The gain saturation is modeled according to
where the total gain g0 equals to 25 dB, E is the signal energy given by , and is the saturation energy, is the cavity round trip time. The amplifier noise is simulated by an additive complex “white” noise with 10−7 W average power, which results in spectral noise level −40 dB relative to the DS power (Fig. 4(b)).3.2 NPE mode-locking
Since the SMF part in our scheme is short (<1.5 m) compared to PMF (30-120 m), the angle of the NPE-induced polarization rotation in SMF is also small. So, we can describe the action of the NPE-based modulator in a scalar form by the qubic-quintic nonlinear term for the pulse amplitude A(t) [24] and rewrite the nonlinear modulator transmission ρ dependent on the incident power P = |A|2 in the following form:
wherecharacterizes the minimal transmission at low powers and is the maximum transmission at the critical powerW, which corresponds to the experimental data for the PBS splitting ratio in the DS regime.4. Results and discussions
Here we present in comparison the experimental and numerical results. First, such a comparison was made for L = 30 m. A stationary pulse evolution along the PM fiber is shown in Fig. 3.
Fig. 3 Evolution of the pulse shape (a) and spectrum (b) along PMF of L = 30 m in the scheme of Fig. 1.
A 500-W DS with a duration of 30 ps started at point A (z = 0), propagates without significant changes of the shape, but its intensity is decreased down to 350 W (z = 30 m) due to generation of a noisy RP (represented by the lower stripe of “<” –shaped trace in Fig. 3(a)). The DS and RP become fully separated due to dispersion at point B (z = 30 m), with spacing between the centers up to 50 ps. Pulse shapes at points A and B corresponding to the entrance and exit points of the PM fiber are also shown in Fig. 1. Then, in a short SMF (not shown in Fig. 3) the pulses are filtered and amplified in an Yb3+ fiber (point C in Fig. 1). The power of the DS pulse reaches ~1 kW while amplification of the Stokes pulse is much smaller as its spectrum is far off the Yb3+gain maximum. The amplified pulse is then equally divided by a) PBS dumping the counter-polarized part (point D in Fig. 1) out of the cavity and b) sending the co-polarized part back into the PM fiber. The DS evolution then starts again at point A (z = 0) of the PMF. Note that the residual part of the RP generated on the previous round trip enters the PMF together with the DS (the weak lower stripe in Fig. 3(a) and the relevant forerunner (left) pulse in Fig. 1 and Fig. 4(a)).
Fig. 4 Calculated and measured pulse shapes (a) and spectra (b) at the output port D in a scheme of Fig. 1.
The corresponding spectra of the DS (centered at 1015 nm) and the RP (centered at 1055 nm) shown in Fig. 3(b) do not vary substantially and demonstrate both significant noise of the RP and appearance of the additional Raman components: the first anti-Stokes (<990 nm) and the second Stokes (>1080 nm) waves at z = 0 (point A in Fig. 1), and at z = 30 m (point B in Fig. 1), correspondingly. The calculated output spectra (point D in Fig. 1) for the PMF length 30 m are compared in Fig. 4 with the experimental ones demonstrating their good agreement. The corresponding pulse shapes in the time domain are also shown here. One can see that the spectral structure realized in the experiment and numerical simulation does not correspond to that of a DS in the absence of SRS (dashed curves in Fig. 4(b), fR = 0). The blue shift of the DS spectrum can be attributed to the SRS, which “eats” a red spectral part of DS. The spectral shift is limited by the WDM loss which is high at <1005 nm (see Fig. 2).
The corresponding temporal pulse profiles calculated with and without the SRS term are shown in Fig. 4(a). The SRS-driven DS has 1.5-times higher power and 3-times shorter duration than the Raman-free DS. This means that the SRS effect leads to strong temporal-spectral filtering of the DS, resulting in its specific shape.
The autocorrelation function (ACF) and cross-correlation function (CCF) shown in Fig. 5(a) were reconstructed from the experimentally measured FROG trace shown in Fig. 5(b) [25]. They agree with the calculated ones: the triangular ACF confirms the rectangular DS shape, and the CCF trace shows 50-ps delay of RP relative to DS. The FROG trace (Fig. 5(b)) reveals partial coherence between DS and RP pulses, which can be explained in the following way. Different temporal and spectral parts of the Stokes-shifted pulse correspond to the different parts of the chirped DS, and the level of coherence is defined by the ratio of the Raman gain and the DS spectral widths. The relevant ACF traces measured before and after the compression are shown in Fig. 6(a). The compression factor of the RP (with an external compressor providing only second-order compensation (Fig. 6(a), inset)) does not exceed 4, whereas the DS is compressed by a factor of 150 down to ~200 fs for L = 30 m.
Fig. 6 (a) Experimental ACF traces at different cavity lengths (30-120 m) for the DS and the RP (inset) with external compression from the broad to the narrow contour marked by arrows; (b) Energy of the DS in the experiment (green triangles) and simulation (green squares connected by the solid line). The calculated total energy of the DS and RP (dashed line). The experimental point obtained for L = 120 m (E≈20 nJ) is not added because the pump power was reduced due to lower stability at this length.
We have also checked the possibility to generate SRS-driven DS pulses in longer cavities. To our surprise, we were able to generate stable chirped pulses at lengths up to maximum available length of PM fiber (L = 120 m) without special measures. The ACF of the DSs remains triangular for cavities up to 120 m, indicating that the duration increases up to 70 ps (Fig. 6(a)). It can be seen that a stable chirped DS can be generated in the whole 30-120 m range, with the dechirped duration down to 200-300 fs, correspondingly. The output DS energy is saturated at 20-25 nJ level in the experiment and simulation for cavities longer than 60 m (Fig. 6(b)). The maximum DS energy is defined by the intra-cavity SRS threshold [17] estimated as nJ. Here P~500 W is the DS power in PMF, is the level of spontaneous emission at the Stokes wavelength, is the group velocity difference between the DS and RP due to dispersion in PMF (see Fig. 2), and is the Raman gain coefficient. Thus, the SRS defines a strict upper limit for the DS energy, and so the RP energy starts to grow with increasing pump power and length. As a result, the RP energy becomes comparable to that for DS, and such a strong SRS effect does not deteriorate the DS stability.
As the calculations demonstrate, the SRS-induced passive negative feedback substantially changes the DS energy scalability. In the absence of white noise and SRS, a DS can be perfectly energy-scalable, which is revealed in the form of a plain asymptotic E→∞ on the so-called DS master diagram [26]. The emergence of such a type of asymptotic corresponds to an appearance of the so-called DS resonance [27].We found, that the noise contribution destroys a perfect energy scalability of DS so that the DS resonance disappears and the maximum reachable energy of DS becomes to be confined by spontaneous noise amplification. The SRS makes this confinement tighter owing to energy transfer into the Stokes-shifted RP, which works as a nonlinear analog of spectral filtering.
5. Discussion and conclusions
We numerically confirmed the experimental results of [17]: so that the upper limit for the maximum DS energy was revealed for a 30-m-long fiber oscillator. This limit is defined by the SRS that converting the excess energy of the DS to the RP. In addition, we proved both experimentally and theoretically that longer cavities do not lead to higher DS energy. The energy of the generated RP becomes comparable to the DS level, but, surprisingly, such efficient conversion does not destroy a stable DS. For cavities up to 90 m, the DS energy limit defined by the excess energy conversion into the RP is lower than the limit defined by destabilization of the DS because of SRS-induced noise. This relation becomes reversed for longer cavities (120 m), but needs additional study. The maximum RP energy is also limited because of its sufficient attenuation and generation of higher-order Stokes pulses. The total energy of the DS-RP complex in the experiment and numerical simulation is also saturated at 30 nJ level (see Fig. 6(b)). The DS duration grows linearly with the cavity length, up to ~70 ps at 120 m. The DS has a rectangular shape with high linear chirp, which is confirmed by external compression down to 200-300 fs, depending on the cavity length. The resulting compression factor of >200 seems to be a record value for the pulses generated in fiber lasers.
Let us now discuss the results obtained in this work in a more general way, with their possible extensions. We found that the DS with variable chirp and the energy defined by the fiber core diameter (~25 nJ at a wavelength of 1 μm for single-mode PM fiber with d = 5.5 μ) can be generated even in the presence of strong SRS. The pulse energy realized in our experiment is higher than that obtained in other all-fiber schemes in the absence of SRS [10, 18, 19]. Note that the chirped RP has partial coherence and can therefore be compressed by only a factor of 4. To push the maximum DS energy, one can try large-mode area fibers [13–15]. The SRS in this case will be suppressed owing to a lower coefficient gR. The main drawback of this approach is the loss of all-fiber concept that has many advantages over hybrid schemes, as we pointed out above.
We have demonstrated that the stable DS pulses can be generated in a fiber oscillator in spite of strong SRS, in contrast to conclusions of [19]. The generated noisy RP does not destroy the DS and can even form a stable bound compleх with it. In this complex the DS provides amplification for the RP, whereas the RP stabilizes the DS energy acting as a temporal and spectral filter for DS. Intuitively, because the Raman process is incoherent and acts in our case as an additional energy loss channel. The DS stability can’t be affected by the RP, at least if we compensate the extra loss by proportional pumping. The pulse energy realized in our experiment is higher than that obtained in other all-fiber schemes in the absence of SRS, though SRS-induced energy conversion defines an upper limit for the DS energy and substantially changes the DS energy scalability.
We recently found that the RP noise seen in Figs. 1 and 4, can be successfully suppressed by the feedback loop. As a result, a stable chirped dissipative Raman soliton has been demonstrated for the first time. These results will be published soon.
Acknowledgments
The authors acknowledge financial support from Russian Ministry of Education and Science (programs 1.5. and 1.9), Russian Foundation for Basic Research and integration project of Siberian Branch of Russian Academy of Sciences. V.L.K. acknowledges the Austrian Science Fund (Project P24916-N27) for financial support.
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