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Ultrahigh-repetition-rate bound-soliton harmonic passive mode-locked fiber lasers Andrey Komarov,1,2,* Adil Haboucha,1 and François Sanchez1 1
Laboratoire des Propriétés Optiques des Matériaux et Applications, FRE CNRS 2988, Université d’Angers, 2 Bd. Lavoisier, 49000 Angers, France 2 Present address: Institute of Automation and Electrometry, Russian Academy of Sciences, Acad. Koptyug Pr. 1, 630090 Novosibirsk, Russia *Corresponding author:
[email protected] Received May 29, 2008; revised July 23, 2008; accepted August 26, 2008; posted September 3, 2008 (Doc. ID 96809); published September 30, 2008 On the basis of numerical simulation results, we put forward a way to realize harmonic passive mode locking of fiber lasers with an ultrahigh-repetition-rate pulse train. The equidistant distribution of ultrashort pulses filling the total laser cavity is due to bound-soliton mechanisms. In the case of large bound energy, such long soliton trains are very stable and have the ideal periodic structure as a soliton crystal. © 2008 Optical Society of America OCIS codes: 140.3510, 140.7090, 190.5530.
Ultrashort optical pulse sources with a high repetition rate are a key element in high-speed optical communications and are also of primary importance in a number of other applications [1–3]. One of the greatest promising ways for the creation of perfect sources of such a type is related to harmonic passive mode locking of fiber lasers with the nonlinear polarization rotation technique [3,4]. These lasers have unique potentialities. They are reliable and compact and can be pumped with commercially available semiconductor lasers. Mode locking in such lasers is realized by fast practically inertia-free nonlinear losses [5]. Parameters of the nonlinear losses are easily controlled through the orientation angles of intracavity phase plates. The multiple pulse passive mode locking is a usual operating regime for these lasers in both anomalous and normal dispersion regimes [6–9]. In the case of soliton repulsion through sufficiently longdistance mechanisms, solitons are uniformly distributed along the cavity; this means that all solitons are equidistant in such a way that passive harmonic mode locking is realized [3,4]. In the case of ultrashort pulse attraction, structures of bound solitons can be formed in the laser cavity. The structures of a few solitons have been predicted theoretically in the frame of the complex Ginzburg–Landau equation [10,11] and have been observed experimentally [12–15]. In this Letter we report the numerical simulation results on a long train of bound solitons in the cavity of passive modelocked fiber lasers with the nonlinear polarization rotation technique. In these systems, through a rotation of intracavity phase plates, one can change the characteristics of nonlinear losses over a wide range and, as a result, cardinally change the properties of dissipative solitons [16]. In such a way one can realize the strong interaction between neighboring solitons with large bound energy. As a consequence, a very long bound-soliton train becomes stable. The periodic structure of such a train is the structure of an ideal soliton crystal [17,18]. When the whole laser cavity is filled by such a train, harmonic passive 0146-9592/08/192254-3/$15.00
mode locking is realized. The rate of repetition of ultrashort pulses in the output of this laser is determined by the distance between neighboring solitons. That is, it can be as high as the inverse ultrashort pulse duration. That means that the pulse-repetition rate of such a bound-soliton harmonic passive mode locked fiber laser can lay in the terahertz frequency range for subpicosecond pulses. The analyzed setup is presented in Fig. 1. It consists of a unidirectional ring cavity in which mode locking is achieved thanks to the combined effects of a nonlinear polarization rotation and an intracavity polarizer. The equations describing this setup have the following form [8]:
E
= 共Dr + iDi兲
2E 2
+ 共G + iq兩E兩2兲E,
共1兲
En+1共兲 = − 关cos共pIn + ␣0兲cos共␣1 − ␣3兲 + i sin共pIn + ␣0兲sin共␣1 + ␣3兲兴En共兲,
共2兲
where E共 , 兲 is the electric field amplitude, is a time coordinate expressed in units ␦t = 冑兩2兩L / 2 (here 2 is the second-order group-velocity dispersion for the fiber and L is the fiber length), is the normalized propagation distance (the number of passes of radiation through the laser cavity), Dr and Di are the frequency dispersions for a gain–loss and for a refractive index, respectively, and q is the Kerr nonlinear-
Fig. 1. Schematic of the investigated laser. The ring laser resonator consists of the fiber gain medium, polarizer, two quarter-wave plates and one half-wave plate; ␣1, ␣2, and ␣3 are the orientation angles of the phase plates. © 2008 Optical Society of America
October 1, 2008 / Vol. 33, No. 19 / OPTICS LETTERS
ity. The term G describes the saturable amplification determined by the total energy of the intracavity radiation G = a / 共1 + b 兰 兩E兩2d兲, where the integration is carried out on the whole round-trip period, a is the pumping parameter, and b is the saturation one. Equation (2) determines the relation between the time distributions of the field before and after the nth pass of radiation through the polarizer ( is the transmission coefficient of the intracavity polarizer). The values ␣1, ␣3, and ␣2 are orientation angles of the quarter-wave plates 1 and 3 and of the half-wave plate 2, respectively. Parameters ␣0, I, and p are determined by relations ␣0 = 2␣2 − ␣1 − ␣3, I = 兩E兩2, and p = sin共2␣3兲 / 3. The amplitude E共兲 is subject to periodic boundary conditions with a period equal to one cavity round trip. A numerical simulation has been performed for typical parameters of an Er-doped fiber laser with the anomalous net dispersion of group velocity 2 [18]. The temporal and spectral profiles of a single soliton are presented in Fig. 2. The soliton has powerful wings. The sidebands in the soliton spectrum are typical for a case of anomalous dispersion. There exists the sequence of the bound stationary states for a pair of such solitons. The bound steady state with a minimal distance between solitons is described by an antisymmetric field function E共兲 = −E共−兲. The distances between solitons for other bound steady states are a multiple of the minimal one. One can introduce the soliton binding energy as the difference between the energy of two single solitons placed at a large distance one from other (the double energy of the single soliton) and the energy of two bound solitons. With increasing the distance between bound solitons, the bounding energy in the steady state decreases monotonously. Each bound steady state is described by even or odd functions E共兲 = ± E共−兲. In the first case, the solitons have the same phase in points with a peak intensity. In the second case, the phase difference equals . The neighboring energy states are described by functions with opposite parity. Figure 3 shows the temporal distributions of intensities for ground steady state (the minimal distance between solitons) and “the first excited steady state” (the distance between solitons equals the double minimal one). The latter is described by even function E共兲 = E共−兲. The binding energies for ground and several first excited steady states in relative units (the binding energy divided by the double energy of the single
Fig. 2. (a) Temporal and (b) spectral distributions of radiation for a single steady-state soliton in the fiber laser with anomalous frequency dispersion of the intracavity medium. The upper right inset in (a) shows the multiplied soliton pedestal. In all figures we use arbitrary units, a = 1.1, q = 2, Di = 0.13, ␣0 = 0.2, ␣1 = −1.64, and ␣3 = 0.2. Dr is determined by the amplification medium Dr = Dr0G, Dr0 = 0.085.
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Fig. 3. Temporal distributions of intensity for (a) ground state, with minimal distance between solitons and for (b) first excited state for a pair of bound solitons. The laser parameters are the same as in the case of Fig. 2.
soliton) are, respectively, equal to 11.8%, 4.9%, 2.9%, and 2.5%. Such large binding energies are due to powerful soliton wings [16] by which solitons interact with each other. In the case of normal dispersion (the other parameters are the same) the relative binding energy is less by approximately a factor of 10. The large binding energies (especially for the ground steady state) result in the formation of long trains of bound solitons in the case of multiple pulse passive mode locking. These trains are very stable. The train with 154 equidistant solitons is shown in Fig. 4. The pulse width equals 0.1 ps. The distance between the next solitons equal to 0.5 ps corresponds to the ground steady state of the pair of bound pulses [see Fig. 3(a)]. The phase difference for neighboring pulses is equal to . We have also obtained the trains in which both the ground bound steady state and some lower excited bound steady states coexist. However, some sufficiently strong perturbation transforms such trains into ones with the lowest energy (only the bound steady states with the greatest binding energy are realized). In our numerical experiment, as a rule, such perturbation was related to a transient process. For the initial conditions, we have taken the chaotic solution obtained for high pumping powers. After that, the pumping has been reduced to observe a progressive self-organization of the pulses. The temporal and spectral distributions (Fig. 4) demonstrate the ideal periodical structure of the soliton train after 2500 round trips. We have used the calculation box equal to 175 ps (0.4% of the resonator
Fig. 4. (a) Temporal and (b) spectral distributions of intensity for the long train of equidistant bound solitons (154 ultrashort pulses) a = 3.8; the other parameters are the same as in the case of Fig. 2.
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length L ⬇ 10 m; in the other parts of the round-trip period the field is assumed to be absent). To break the periodicity of pulses in the train it is necessary to transform the ground bound steady state for some pair of two neighboring solitons into some excited bound steady state. There exist two reasons that prevent such a transformation. The first reason is related to the large energetic difference for ground and first excited bound states. Second, the perturbation energy that was initially localized in the vicinity of some pair of bound solitons is quickly collectivized among all solitons of the train. As a result, such an ideal periodic structure is very stable. In the numerical simulation we have also used the random radiation noise to prove this stability. This noise induces up to 10% fluctuation of peak soliton intensity but does not change the structure of the soliton train. The higher pumping produces the longer soliton train. When the duration of the train with the rigorous periodic structure becomes equal to the cavity round-trip time, harmonic passive mode locking is realized (see Fig. 5). For the economy of computer time we analyzed the 175 ps calculation box (in the other boxes of the round-trip period the field distribution is assumed to be the same). After 50,000 round trips the established regime with 314 ultrashort pulses at this box is realized. We have also used the calculation boxes with other lengths. For the ring train with a periodic soliton structure (Fig. 5), the rigorous periodicity is conserved under an adiabatic change of the resonator length up to 30% (therewith the number of pulses is not changed). Transition into the harmonic passive mode locking regime due to increased pumping and reverting into the original operation due to decreased pumping are described by a hysteresis dependence. Until now, it was obtained experimentally by our team that a long train of bound solitons fills approximately one tenth of the total resonator length [18]. We have demonstrated in this Letter that the realization of a train having a length equal to the resonator length can result in harmonic passive mode locking. Indeed, we have pointed out that an increase of the pumping power together with a suitable choice of the laser parameters are the key elements for the realization of high-repetition-rate fiber lasers.
Fig. 5. Modeling of the regime of harmonic passive mode locking due to the bound-soliton mechanism (314 ultrashort pulses) a = 7; the other parameters are the same as in the case of Fig. 2.
On the basis of numerical simulation we have found that in passive mode-locked fiber lasers with anomalous dispersion the bound states of dissipative solitons with a large relative binding energy (greater than 10%) are realized. As a result, long multiple soliton trains with a high level of stability are formed. The state of such a train, with the largest binding energy and equidistant interval between the next pulses, is more stable. With sufficiently large pumping, such a train fills in the total laser resonator and bound-soliton harmonic passive mode-locking is realized. The expected rate of repetition of ultrashort pulses in the output radiation of this laser is of the order of inverse ultrashort pulse duration and can lay in the terahertz frequency range for femtosecond pulses. A. Komarov thanks the European Community that supports his research through a Marie Curie International Fellowship within the sixth European Community Framework Programme (N 039942-PMLFL). References 1. K. S. Abedin, J. T. Gopinath, L. A. Jiang, M. E. Grein, H. A. Haus, and E. P. Ippen, Opt. Lett. 27, 1758 (2002). 2. C. X. Yu, H. A. Haus, E. P. Ippen, W. S. Wong, and A. Sysoliatin, Opt. Lett. 25, 1418 (2000). 3. B. Ortaç, A. Hideur, G. Martel, and M. Brunel, Appl. Phys. B 81, 507 (2005). 4. A. Komarov, H. Leblond, and F. Sanchez, Opt. Commun. 267, 162 (2006). 5. V. J. Matsas, T. P. Newson, D. J. Richardson, and D. N. Payne, Electron. Lett. 28, 1391 (1992). 6. A. B. Grudinin, D. J. Richardson, and D. N. Payne, Electron. Lett. 28, 67 (1992). 7. D. Y. Tang, W. S. Man, and H. Y. Tam, Opt. Commun. 165, 189 (1999). 8. A. Komarov, H. Leblond, and F. Sanchez, Phys. Rev. A 71, 053809 (2005). 9. A. Haboucha, A. Komarov, H. Leblond, F. Sanchez, and G. Martel, Opt. Fiber Technol. 14, 262 (2008). 10. V. V. Afanasjev, B. A. Malomed, and P. L. Chu, Phys. Rev. E 56, 6020 (1997). 11. N. N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, J. Opt. Soc. Am. B 15, 515 (1998). 12. P. Grelu, F. Belhache, and J. M. Soto-Crespo, Opt. Lett. 27, 966 (2002). 13. D. Y. Tang, W. S. Man, H. Y. Tam, and P. D. Drummond, Phys. Rev. A 64, 033814 (2001). 14. A. Hideur, B. Ortaç, T. Chartier, M. Brunel, H. Leblond, and F. Sanchez, Opt. Commun. 225, 71 (2003). 15. B. Ortaç, A. Hideur, T. Chartier, M. Brunel, Ph. Grelu, H. Leblond, and F. Sanchez, IEEE Photon. Technol. Lett. 16, 1274 (2004). 16. A. Komarov and F. Sanchez, Phys. Rev. E 77, 066201, 2008. 17. S. Rutz and F. Mitschke, J. Opt. B 2, 364 (2000). 18. A. Haboucha, H. Leblond, M. Salhi, A. Komarov, and F. Sanchez, Opt. Lett. 33, 524 (2008).
