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Period-doubling of vector solitons in a ring fiber laser
Optics Communications 281 (2008) 5614–5617
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Period-doubling of vector solitons in a ring fiber laser L.M. Zhao a, D.Y. Tang a,*, H. Zhang a, X. Wu a, C. Lu b, H.Y. Tam c a b c
School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore Department of Electronic and Information Engineering, Hong Kong Polytechnic University, Hung Hom, Hong Kong, China Department of Electrical Engineering, Hong Kong Polytechnic University, Hung Hom, Hong Kong, China
a r t i c l e
i n f o
Article history: Received 27 May 2008 Received in revised form 5 July 2008 Accepted 2 August 2008
a b s t r a c t Period-doubling of vector solitons in a ring fiber laser was experimentally observed for the first time. Apart from period-doubling of single vector soliton, period-doubling of multiple vector solitons was observed as well. Numerical simulation verified the experimental observation. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Soliton dynamics of fiber lasers passively mode-locked by the nonlinear polarization rotation (NPR) technique has been extensively investigated [1–8]. It was shown that soliton circulation in the fiber cavity could accumulate a large amount of nonlinear phase shift, which, combined with the cavity feedback, could lead to a number of nonlinear soliton dynamic effects. Tamura et al. firstly documented a state of the soliton period-doubling and tripling of the fiber laser [1]. Akhmediev et al. theoretically studied the parameter regime of soliton period-doubling in the modelocked lasers whose dynamics is governed by the complex Ginzburg–Landau equation [2], Soto-Crespo et al. further observed both numerically and experimentally the generation of single and double periodic pulsation of solitons in a passively mode-locked fiber laser [3], and Ilday et al. reported a period-doubling route to stable multiple-pulse in a fiber laser [4]. Recently we have studied both experimentally and numerically the soliton period-doubling and route to chaos in a dispersion-managed fiber ring laser mode locked by the NPR technique [5,6]. The phenomenon of period bifurcation based on soliton pairs was also experimentally reported [7] and theoretically presented [8]. We also demonstrated experimentally and numerically the period-doubling of multiple solitons in a passively mode-locked fiber laser [9]. A characteristic of the NPR mode-locking is that a polarizer must be inserted in the laser cavity to take advantage of the NPR of the optical pulses for the generation of an artificial saturable absorption effect [10]. As the intracavity polarizer fixes light polarization at the cavity position, solitons formed in the lasers are scalar solitons. Therefore, all the aforementioned soliton period-doubling studies have been focused on the case of scalar solitons. Due to the technical limitation in fabricating a perfect-circular core cross-section fiber and/or the random mechanical stresses, a * Corresponding author. Tel.: +65 790 4337; fax: +65 792 0415. E-mail address: [email protected] (D.Y. Tang). 0030-4018/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.08.014
single-mode fiber always supports two polarization eigenmodes, or in another word, possesses weak birefringence. It was shown that without a polarizer in cavity, soliton circulation in a fiber laser cavity could exhibit complicate polarization dynamics. Cundiff et al. have demonstrated the formation of vector solitons in a fiber laser mode-locked with a semiconductor saturable absorber mirror (SESAM) [11,12]. Both the polarization locked vector solitons (PLVSs) and the group velocity locked vector solitons (GVLVSs) were observed. A vector soliton formed in the fiber lasers differs from a scalar soliton by that it consists of two orthogonal polarization components. Moreover, the two polarization components are nonlinearly coupled, which leads to the formation of the various types of vector solitons, such as the PLVS, GVLVS, and polarization rotating vector soliton. Except their novel polarization dynamics, experimental studies revealed that the formed vector solitons exhibit similar features as those of the scalar solitons formed in a fiber laser, like the multiple vector soliton formation, vector soliton energy quantization, and vector soliton harmonic mode-locking [13]. In view of these observed vector soliton features, one would like to ask whether the period-doubling observed on the scalar solitons could occur on the vector solitons. In this paper, we report the first experimental observation and numerical demonstration of period-doubling of vector soliton in fiber lasers. Apart from period-doubling of the single vector soliton, period-doubling of multiple vector solitons is also observed. The vector soliton fiber laser shown in Fig. 1 has a similar configuration as reported in Ref. [14]. The ring cavity of the laser has a length of about 9.40 m, which consists of 2.63 m erbium-doped fiber (StockerYale EDF-1480-T6) and all other fibers used are the standard single mode fiber (SMF). With the help of a 3-port polarization independent circulator, a SESAM was introduced in the ring cavity of the laser. A 1480 nm Raman fiber laser with maximum output of 220 mW was used to pump the laser. Backward pump scheme was adopted to avoid the CW overdriving of the SESAM by the residual pump strength. The laser output was through a
L.M. Zhao et al. / Optics Communications 281 (2008) 5614–5617
Circulator
Fiber PC Output Coupler
SESAM
WDM
EDF Fig. 1. Schematic of the fiber laser. SESAM: semiconductor saturable absorber mirror; PC: polarization controller; WDM: wavelength-division multiplexer; EDF: erbium-doped fiber.
10% fiber coupler. A fiber-based polarization controller was inserted in the cavity to control the cavity birefringence. The SESAM used has a saturable absorption of 8%, and a recovery time of 2 ps. As no explicit polarization discrimination components were used in the cavity, and all the fibers used have weak birefringence, vector solitons were easily obtained by simply increasing the pump power above the laser mode-locking threshold. Determined
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by the detailed laser operation conditions, various types of vector solitons, such as the PLVS [11,12], incoherently coupled vector soliton [15], and polarization rotating vector solitons were observed. In particular, we found that the polarization rotation of the formed polarization rotating vector solitons could be locked to the laser cavity roundtrip time or multiple of it [14]. In the parameter regime of a polarization rotation locked vector soliton, we have further experimentally observed a state of the period-doubling of the vector solitons, as shown in Fig. 2a. The vector soliton output of the fiber laser is directly monitored by a 2 GHz photodetector. Although the pulse intensity difference between two adjacent cavity roundtrips is weak, intensity period-doubling of the soliton pulse train is evident. We also simultaneously measured the RF spectrum of Fig. 2a as shown in Fig. 2b. A weak but clearly visible frequency component appeared at the position of the half of the cavity fundamental repetition frequency. We note that when obtaining the Fig. 2, no polarizer was inserted before the photodetector. Measured after passing through an external polarizer, a polarization rotation state with polarization rotation locked to twice of the cavity roundtrip time would have the same result as shown in Fig. 2. However, such a state is not a period-doubled state as the pulse intensity alternation observed on the oscilloscope trace is due to the polarization rotation of the vector
Fig. 2. The (a) oscilloscope trace and (b) corresponding RF spectrum of direct intensity period-doubling.
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L.M. Zhao et al. / Optics Communications 281 (2008) 5614–5617
Fig. 3. Period-doubling of multiple vector solitons. (a) Two vector solitons; (b) eight vector solitons (the numbers and the supplemental marks identify the corresponding pulses experiencing period-doubling).
soliton. The period-doubling shown in Fig. 2 is formed due to the intrinsic dynamic feature of the laser. After the period-doubling state was achieved, keeping all other laser operation parameters fixed but increasing the pump power the number of vector solitons circulating in the cavity increased. The new vector solitons generated exhibited the same period-doubling feature. Fig. 3a and b shows for example the cases where two vector solitons and eight vector solitons coexist in the cavity, respectively. Period-doubling of the vector solitons can be clearly identified by the RF spectrum monitoring. Decreasing pump power reduced the number of vector solitons in cavity. Varying pump power could also change the soliton pulse intensity within a small range. However, no period-one state could be obtained by simply decreasing the pump power. Period-one state could be obtained only if the cavity birefringence was changed. Similarly, after a period-one state was obtained, the period-doubling state could not be obtained by simply increasing the pump strength, which is different from the case for scalar solitons. Polarization rotation of the vector solitons can be identified with an external polarizer [11,12,14]. Experimentally we found that in a period-doubling state the polarization rotation of the vector solitons is locked to twice of the cavity roundtrip time. That is, both the intensity and the polarization ellipse of the vector soliton recurred every two cavity-lengths. To verify the experimental observations, we numerically simulated the laser operation based on the coupled Ginzburg–Landau equations (GLEs) and the pulse tracing technique described in [15]. Briefly speaking, the pulse propagation in the fibers is de-
Fig. 4. Numerically simulated results of period-doubling/period one of (a/b) the vector soliton, (c/d) the vector soliton along the horizontal birefringent axis, and (e/f) the vector soliton along the vertical birefringent axis with the cavity roundtrips (all simulation parameters are same but the cavity birefringence: Lb = L/0.09 for period-doubling; Lb = L/0.4 for period-one).
L.M. Zhao et al. / Optics Communications 281 (2008) 5614–5617
scribed by the coupled GLEs; whenever the pulse propagates through an individual component, the Jone’s matrix of the component is multiplied. We began from an arbitrary weak pulse, after one roundtrip, the obtained pulse is used as a new initial pulse and the calculation is repeated until a steady state is obtained. In our simulations we used the following parameters: c = 3 W1 km1, Xg = 24 nm, Psat = 100 pJ, kSMF00 = 23 ps2/km, kEDF00 = 13 ps2/km, k000 = 0.13 ps3/km, Esat = 1 pJ, l0 = 0.15, Trec = 6 ps, and G = 75. Fig. 4 shows the results of simulations obtained under different net cavity birefringence but the same other laser parameters. Numerically it was found that when the cavity birefringence was selected as Lb = L/0.4, where L is the cavity length, a period-one vector soliton state was obtained. When the cavity birefringence was selected as Lb = L/0.09, a period-doubled vector soliton was then obtained. Fig. 4a and b shows the calculated vector soliton evolution with the cavity roundtrips. The soliton period-doubling of the state shown in Fig. 4a in comparison with that shown in Fig. 4b is evident. Fig. 4c/e and d/f shows further the evolution of the vector soliton components, of the period-doubled vector soliton and the periodone vector soliton, respectively. Each component of the period-doubled vector soliton also exhibited period-doubling. However, the pulse intensity variation between the two orthogonal polarization components is anti-phase. It is because of this anti-phase pulse intensity evolution of the period-doubled vector soliton, only a weak period-doubling could be observed on the vector soliton intensity evolution, as shown in Fig. 4a. The result agrees with our experimental observations.
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In conclusion, period-doubling of the vector solitons formed in a passively mode-locked fiber ring laser was first experimentally observed and numerically confirmed. Not only period-doubling of single vector soliton but also multiple vector solitons was demonstrated. Our experimental results and numerical simulations suggest that the phenomenon of soliton period-doubling is a feature of fiber laser which is independent of the type of the formed soliton. References [1] K. Tamura, C.R. Doerr, H.A. Haus, E.P. Ippen, IEEE Phot. Tech. Lett. 6 (1994) 697. [2] N. Akhmediev, J.M. Soto-Crespo, G. Town, Phys. Rev. E 63 (2001) 056602. [3] J.M. Soto-Crespo, M. Grapinet, P. Grelu, N. Akhmediev, Phys. Rev. E 70 (2004) 066612. [4] F. Ilday, J. Buckley, F. Wise, Proceedings of the Nonlinear Guided Waves Conference, Toronto, OSA, Washington, DC, 2004, Paper MD9. [5] L.M. Zhao, D.Y. Tang, F. Lin, B. Zhao, Opt. Express 12 (2004) 4573. [6] D.Y. Tang, L.M. Zhao, F. Lin, Europhys. Lett. 71 (2005) 56. [7] L.M. Zhao, D.Y. Tang, B. Zhao, Opt. Commun. 252 (2005) 167. [8] N. Akhmediev, J.M. Soto-Crespo, Ph. Grelu, Phys. Lett. A 364 (2007) 413. [9] L.M. Zhao, D.Y. Tang, T.H. Cheng, C. Lu, Opt. Commun. 273 (2007) 554. [10] Hermann A. Haus, James G. Fujimoto, Erich P. Ippen, IEEE J. Quantum Electron 28 (1992) 2086. [11] S.T. Cundiff, B.C. Collings, W.H. Knox, Opt. Express 1 (1997) 12. [12] S.T. Cundiff, B.C. Collings, N.N. Akhmediev, J.M. Soto-Crespo, K. Bergman, W.H. Knox, Phys. Rev. Lett. 82 (1999) 3988. [13] B.C. Collings, K. Bergman, S. Tsuda, W.H. Knox, Lasers and Electro-Optics CLEO97, vol. 11, 1997, p. 343. [14] L.M. Zhao, D.Y. Tang, H. Zhang, X. Wu, Opt. Express 16 (2008) 10053. [15] L.M. Zhao, D.Y. Tang, H. Zhang, X. Wu, N. Xiang, Opt. Express 16 (2008) 9528.
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Period-doubling of vector solitons in a ring fiber laser L.M. Zhao a, D.Y. Tang a,*, H. Zhang a, X. Wu a, C. Lu b, H.Y. Tam c a b c
School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore Department of Electronic and Information Engineering, Hong Kong Polytechnic University, Hung Hom, Hong Kong, China Department of Electrical Engineering, Hong Kong Polytechnic University, Hung Hom, Hong Kong, China
a r t i c l e
i n f o
Article history: Received 27 May 2008 Received in revised form 5 July 2008 Accepted 2 August 2008
a b s t r a c t Period-doubling of vector solitons in a ring fiber laser was experimentally observed for the first time. Apart from period-doubling of single vector soliton, period-doubling of multiple vector solitons was observed as well. Numerical simulation verified the experimental observation. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Soliton dynamics of fiber lasers passively mode-locked by the nonlinear polarization rotation (NPR) technique has been extensively investigated [1–8]. It was shown that soliton circulation in the fiber cavity could accumulate a large amount of nonlinear phase shift, which, combined with the cavity feedback, could lead to a number of nonlinear soliton dynamic effects. Tamura et al. firstly documented a state of the soliton period-doubling and tripling of the fiber laser [1]. Akhmediev et al. theoretically studied the parameter regime of soliton period-doubling in the modelocked lasers whose dynamics is governed by the complex Ginzburg–Landau equation [2], Soto-Crespo et al. further observed both numerically and experimentally the generation of single and double periodic pulsation of solitons in a passively mode-locked fiber laser [3], and Ilday et al. reported a period-doubling route to stable multiple-pulse in a fiber laser [4]. Recently we have studied both experimentally and numerically the soliton period-doubling and route to chaos in a dispersion-managed fiber ring laser mode locked by the NPR technique [5,6]. The phenomenon of period bifurcation based on soliton pairs was also experimentally reported [7] and theoretically presented [8]. We also demonstrated experimentally and numerically the period-doubling of multiple solitons in a passively mode-locked fiber laser [9]. A characteristic of the NPR mode-locking is that a polarizer must be inserted in the laser cavity to take advantage of the NPR of the optical pulses for the generation of an artificial saturable absorption effect [10]. As the intracavity polarizer fixes light polarization at the cavity position, solitons formed in the lasers are scalar solitons. Therefore, all the aforementioned soliton period-doubling studies have been focused on the case of scalar solitons. Due to the technical limitation in fabricating a perfect-circular core cross-section fiber and/or the random mechanical stresses, a * Corresponding author. Tel.: +65 790 4337; fax: +65 792 0415. E-mail address: [email protected] (D.Y. Tang). 0030-4018/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.08.014
single-mode fiber always supports two polarization eigenmodes, or in another word, possesses weak birefringence. It was shown that without a polarizer in cavity, soliton circulation in a fiber laser cavity could exhibit complicate polarization dynamics. Cundiff et al. have demonstrated the formation of vector solitons in a fiber laser mode-locked with a semiconductor saturable absorber mirror (SESAM) [11,12]. Both the polarization locked vector solitons (PLVSs) and the group velocity locked vector solitons (GVLVSs) were observed. A vector soliton formed in the fiber lasers differs from a scalar soliton by that it consists of two orthogonal polarization components. Moreover, the two polarization components are nonlinearly coupled, which leads to the formation of the various types of vector solitons, such as the PLVS, GVLVS, and polarization rotating vector soliton. Except their novel polarization dynamics, experimental studies revealed that the formed vector solitons exhibit similar features as those of the scalar solitons formed in a fiber laser, like the multiple vector soliton formation, vector soliton energy quantization, and vector soliton harmonic mode-locking [13]. In view of these observed vector soliton features, one would like to ask whether the period-doubling observed on the scalar solitons could occur on the vector solitons. In this paper, we report the first experimental observation and numerical demonstration of period-doubling of vector soliton in fiber lasers. Apart from period-doubling of the single vector soliton, period-doubling of multiple vector solitons is also observed. The vector soliton fiber laser shown in Fig. 1 has a similar configuration as reported in Ref. [14]. The ring cavity of the laser has a length of about 9.40 m, which consists of 2.63 m erbium-doped fiber (StockerYale EDF-1480-T6) and all other fibers used are the standard single mode fiber (SMF). With the help of a 3-port polarization independent circulator, a SESAM was introduced in the ring cavity of the laser. A 1480 nm Raman fiber laser with maximum output of 220 mW was used to pump the laser. Backward pump scheme was adopted to avoid the CW overdriving of the SESAM by the residual pump strength. The laser output was through a
L.M. Zhao et al. / Optics Communications 281 (2008) 5614–5617
Circulator
Fiber PC Output Coupler
SESAM
WDM
EDF Fig. 1. Schematic of the fiber laser. SESAM: semiconductor saturable absorber mirror; PC: polarization controller; WDM: wavelength-division multiplexer; EDF: erbium-doped fiber.
10% fiber coupler. A fiber-based polarization controller was inserted in the cavity to control the cavity birefringence. The SESAM used has a saturable absorption of 8%, and a recovery time of 2 ps. As no explicit polarization discrimination components were used in the cavity, and all the fibers used have weak birefringence, vector solitons were easily obtained by simply increasing the pump power above the laser mode-locking threshold. Determined
5615
by the detailed laser operation conditions, various types of vector solitons, such as the PLVS [11,12], incoherently coupled vector soliton [15], and polarization rotating vector solitons were observed. In particular, we found that the polarization rotation of the formed polarization rotating vector solitons could be locked to the laser cavity roundtrip time or multiple of it [14]. In the parameter regime of a polarization rotation locked vector soliton, we have further experimentally observed a state of the period-doubling of the vector solitons, as shown in Fig. 2a. The vector soliton output of the fiber laser is directly monitored by a 2 GHz photodetector. Although the pulse intensity difference between two adjacent cavity roundtrips is weak, intensity period-doubling of the soliton pulse train is evident. We also simultaneously measured the RF spectrum of Fig. 2a as shown in Fig. 2b. A weak but clearly visible frequency component appeared at the position of the half of the cavity fundamental repetition frequency. We note that when obtaining the Fig. 2, no polarizer was inserted before the photodetector. Measured after passing through an external polarizer, a polarization rotation state with polarization rotation locked to twice of the cavity roundtrip time would have the same result as shown in Fig. 2. However, such a state is not a period-doubled state as the pulse intensity alternation observed on the oscilloscope trace is due to the polarization rotation of the vector
Fig. 2. The (a) oscilloscope trace and (b) corresponding RF spectrum of direct intensity period-doubling.
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L.M. Zhao et al. / Optics Communications 281 (2008) 5614–5617
Fig. 3. Period-doubling of multiple vector solitons. (a) Two vector solitons; (b) eight vector solitons (the numbers and the supplemental marks identify the corresponding pulses experiencing period-doubling).
soliton. The period-doubling shown in Fig. 2 is formed due to the intrinsic dynamic feature of the laser. After the period-doubling state was achieved, keeping all other laser operation parameters fixed but increasing the pump power the number of vector solitons circulating in the cavity increased. The new vector solitons generated exhibited the same period-doubling feature. Fig. 3a and b shows for example the cases where two vector solitons and eight vector solitons coexist in the cavity, respectively. Period-doubling of the vector solitons can be clearly identified by the RF spectrum monitoring. Decreasing pump power reduced the number of vector solitons in cavity. Varying pump power could also change the soliton pulse intensity within a small range. However, no period-one state could be obtained by simply decreasing the pump power. Period-one state could be obtained only if the cavity birefringence was changed. Similarly, after a period-one state was obtained, the period-doubling state could not be obtained by simply increasing the pump strength, which is different from the case for scalar solitons. Polarization rotation of the vector solitons can be identified with an external polarizer [11,12,14]. Experimentally we found that in a period-doubling state the polarization rotation of the vector solitons is locked to twice of the cavity roundtrip time. That is, both the intensity and the polarization ellipse of the vector soliton recurred every two cavity-lengths. To verify the experimental observations, we numerically simulated the laser operation based on the coupled Ginzburg–Landau equations (GLEs) and the pulse tracing technique described in [15]. Briefly speaking, the pulse propagation in the fibers is de-
Fig. 4. Numerically simulated results of period-doubling/period one of (a/b) the vector soliton, (c/d) the vector soliton along the horizontal birefringent axis, and (e/f) the vector soliton along the vertical birefringent axis with the cavity roundtrips (all simulation parameters are same but the cavity birefringence: Lb = L/0.09 for period-doubling; Lb = L/0.4 for period-one).
L.M. Zhao et al. / Optics Communications 281 (2008) 5614–5617
scribed by the coupled GLEs; whenever the pulse propagates through an individual component, the Jone’s matrix of the component is multiplied. We began from an arbitrary weak pulse, after one roundtrip, the obtained pulse is used as a new initial pulse and the calculation is repeated until a steady state is obtained. In our simulations we used the following parameters: c = 3 W1 km1, Xg = 24 nm, Psat = 100 pJ, kSMF00 = 23 ps2/km, kEDF00 = 13 ps2/km, k000 = 0.13 ps3/km, Esat = 1 pJ, l0 = 0.15, Trec = 6 ps, and G = 75. Fig. 4 shows the results of simulations obtained under different net cavity birefringence but the same other laser parameters. Numerically it was found that when the cavity birefringence was selected as Lb = L/0.4, where L is the cavity length, a period-one vector soliton state was obtained. When the cavity birefringence was selected as Lb = L/0.09, a period-doubled vector soliton was then obtained. Fig. 4a and b shows the calculated vector soliton evolution with the cavity roundtrips. The soliton period-doubling of the state shown in Fig. 4a in comparison with that shown in Fig. 4b is evident. Fig. 4c/e and d/f shows further the evolution of the vector soliton components, of the period-doubled vector soliton and the periodone vector soliton, respectively. Each component of the period-doubled vector soliton also exhibited period-doubling. However, the pulse intensity variation between the two orthogonal polarization components is anti-phase. It is because of this anti-phase pulse intensity evolution of the period-doubled vector soliton, only a weak period-doubling could be observed on the vector soliton intensity evolution, as shown in Fig. 4a. The result agrees with our experimental observations.
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In conclusion, period-doubling of the vector solitons formed in a passively mode-locked fiber ring laser was first experimentally observed and numerically confirmed. Not only period-doubling of single vector soliton but also multiple vector solitons was demonstrated. Our experimental results and numerical simulations suggest that the phenomenon of soliton period-doubling is a feature of fiber laser which is independent of the type of the formed soliton. References [1] K. Tamura, C.R. Doerr, H.A. Haus, E.P. Ippen, IEEE Phot. Tech. Lett. 6 (1994) 697. [2] N. Akhmediev, J.M. Soto-Crespo, G. Town, Phys. Rev. E 63 (2001) 056602. [3] J.M. Soto-Crespo, M. Grapinet, P. Grelu, N. Akhmediev, Phys. Rev. E 70 (2004) 066612. [4] F. Ilday, J. Buckley, F. Wise, Proceedings of the Nonlinear Guided Waves Conference, Toronto, OSA, Washington, DC, 2004, Paper MD9. [5] L.M. Zhao, D.Y. Tang, F. Lin, B. Zhao, Opt. Express 12 (2004) 4573. [6] D.Y. Tang, L.M. Zhao, F. Lin, Europhys. Lett. 71 (2005) 56. [7] L.M. Zhao, D.Y. Tang, B. Zhao, Opt. Commun. 252 (2005) 167. [8] N. Akhmediev, J.M. Soto-Crespo, Ph. Grelu, Phys. Lett. A 364 (2007) 413. [9] L.M. Zhao, D.Y. Tang, T.H. Cheng, C. Lu, Opt. Commun. 273 (2007) 554. [10] Hermann A. Haus, James G. Fujimoto, Erich P. Ippen, IEEE J. Quantum Electron 28 (1992) 2086. [11] S.T. Cundiff, B.C. Collings, W.H. Knox, Opt. Express 1 (1997) 12. [12] S.T. Cundiff, B.C. Collings, N.N. Akhmediev, J.M. Soto-Crespo, K. Bergman, W.H. Knox, Phys. Rev. Lett. 82 (1999) 3988. [13] B.C. Collings, K. Bergman, S. Tsuda, W.H. Knox, Lasers and Electro-Optics CLEO97, vol. 11, 1997, p. 343. [14] L.M. Zhao, D.Y. Tang, H. Zhang, X. Wu, Opt. Express 16 (2008) 10053. [15] L.M. Zhao, D.Y. Tang, H. Zhang, X. Wu, N. Xiang, Opt. Express 16 (2008) 9528.