Mode-locked fiber lasers based on doped fiber arrays Xiao Zhang and Yanrong Song* Institute of Information Photonics Technology and College of Applied Sciences, Beijing University of Technology, Beijing 100124, China *Corresponding author: [email protected] Received 18 December 2013; revised 2 April 2014; accepted 4 April 2014; posted 7 April 2014 (Doc. ID 202894); published 5 May 2014
We designed a new kind of mode-locked fiber laser based on fiber arrays, where the central core is doped. A theoretical model is given for an all-fiber self-starting mode-locked laser based on this kind of doped fiber array. Two different kinds of fiber lasers with negative dispersion and positive dispersion are simulated and discussed. The stable mode-locked pulses are generated from initial noise conditions by the realistic parameters. The process of self-starting mode-locking multipulse transition and the relationship between the energy of the central core and the propagation distance of the pulses are discussed. Finally, we analyze the difference between the averaged mode-locked laser and the discrete mode-locked laser. © 2014 Optical Society of America OCIS codes: (140.4050) Mode-locked lasers; (060.5530) Pulse propagation and temporal solitons; (320.7090) Ultrafast lasers. http://dx.doi.org/10.1364/AO.53.002998
1. Introduction
Since mode-locked lasers have a wide range of applications in military, industry, and medicine, many passive mode-locking elements have been studied in depth in the last decade, e.g., semiconductor saturable absorber mirror (SESAM), carbon nanotubes, and graphene. For the characteristics of nonlinear modecoupling (NLMC), waveguide arrays and fiber arrays could work as novel passive mode-locking mechanisms. Waveguide arrays have been investigated in some laser systems. Numerical studies show that robust and stable mode-locking has been achieved in fiber lasers with waveguide arrays [1–3]. The multipulsing transition [4] and generation of spatiotemporal X-waves [5] were studied in a fully comprehensive physical model of the laser cavity that uses waveguide arrays as a saturable absorber. Nonlinear pulse reshaping of femtosecond pulses was observed in a waveguide array and the power-dependent pulse reshaping agrees with the theory, including the shortening of the pulse in the central waveguide [6]. A fixed-point attractor for chirp in AlGaAs waveguide arrays was demonstrated, and the simulations reproduced the 1559-128X/14/142998-06$15.00/0 © 2014 Optical Society of America 2998
APPLIED OPTICS / Vol. 53, No. 14 / 10 May 2014
experimental results [7]. An AlGaAs waveguide array is used as a saturable absorber to achieve passive mode-locking in an erbium-doped fiber laser, autocorrelations showed that the pulses consist of noise bursts due to incomplete mode-locking. However, unlike a typical Kerr-lens cavity, the waveguide array laser is self-starting [8]. An experimental study of fiber arrays shows that, in the linear case, light either spreads in a diffusive manner or localizes at a few sites. For high excitation power, diffusive spreading is arrested by the focusing nonlinearity [9]. Thus, a fiber array could be used as a saturable absorber like a waveguide array. A mode-locked fiber laser with a fiber array saturable absorber and a gain fiber has been demonstrated theoretically [3]. For its advantages of flexibility, good heat dissipation, all-fiber system, compact configuration, and a better shaping function [3], fiber arrays could be a potential broadband passive modelocking mechanism. In this paper, we designed a novel mode-locked fiber laser which is called a doped fiber array laser (DFAL). In the laser, the central core of the fiber array is doped so that this doped fiber array can be used as both the gain material and mode-locking element at the same time. We also call this kind of laser an averaged mode-locked laser when we compare it with the
discrete mode-locked fiber laser later. Two kinds of DFALs use different doped materials, Yb-doped fiber (central wavelength at 1064 nm) and Er-doped fiber (central wavelength at 1550 nm), which correspond to the positive dispersion and negative dispersion systems. For the negative dispersion system, the output pulse width could reach several tens of femtoseconds, but the pulses would split when the pulse energy increased with increased pump power. If high output energy is required, a DFAL with positive dispersion system, such as Yb-DFAL, is a good choice because the pulses will not split when the pump power is increased and the self-similar system is one solution. The simulation results of the DFAL systems with positive or negative dispersions show that the stable Sech-shaped mode-locked pulses could be reached by realistic parameters. In order to show the advantages of the DFALs model, we compare its output pulse shape with that of a discrete mode-locked fiber laser [3] in the last part of the paper. 2. Doped Fiber Arrays and Mode-Locked Fiber Lasers A.
Doped Fiber Arrays
Figure 1 shows the cross section picture of a homogeneous hexagonal lattice of a fiber array. We set 37 cores in the fiber array. The central core is rareearth-doped, while others are not. Every internal core is surrounded by six cores, and the light in one core could couple to the six nearest neighboring cores and vice versa. In a word, the light of the neighboring cores are coupled with each other. For instance, the mode-coupling between the core No. 5 and its neighboring cores (labeled in Fig. 1) is described as [9] dA5 ∕dz iC25 A2 iC35 A3 iC45 A4 iC65 A6 iC85 A8 iC95 A9 , where An is the electric field envelope in nth core. Cmn is the coupling coefficient between the mth core and the nth core. As a homogeneous hexagonal lattice, C25 C35 C45 C65 C85 C95 C0 , where C0 is a constant. The properties of fiber arrays are determined by designed parameters (refractive indices in the core and
Fig. 1. Cross section of a hexagonal fiber array.
cladding, core spacing, and core diameter) rather than just material properties [8]. There is a significant difference between a photonic crystal fiber (PCF) and a fiber array: their size. In a PCF, to create a band gap, Φcore and the core spacing are close to the light wavelength. However, in a fiber array, the Φcore and the core spacing are far bigger than that. In the fiber array we designed here, the Φcore and core spacing are roughly 5–10 μm and 10–30 μm, respectively. The material parameters of typical silica fiber are used here. For the light in the fiber array, the low intensity portions spread in a diffusive manner and coupled into neighboring cores and then were lost. Meanwhile, the high intensity portions almost remain in the central core for a self-focusing effect (it is also called Kerr lens effect). After traveling a certain distance, the high intensity portion could remain in the central core for the gain and the balance of the nonlinear effect and the dispersion effect, but the low intensity portion decays for the loss only. This characteristic is equivalent to a saturable absorber, such as SESAM and carbon nanotubes in normal mode-locked lasers. B. Model of DFALs
Figure 2 shows the configuration of an all-fiber DFAL. Two couplers are for the input pump light and the output laser. To prevent unwanted feedback, an isolator is placed in the cavity. Doped fiber arrays are used as both a gain material and a saturable absorber at the same time. For Er-doped and Yb-DFAL systems, the wavelengths of the lasers are λ0 1550 nm and 1064 nm, respectively. For the 37 cores in the fiber array, each core should correspond to one equation, but here only ten equations are considered when we simulate the system (labeled by numbers in Fig. 1) because they are both central symmetric and axial symmetric. The evolution of the normalized electric field amplitudes in the nth core An are given by [3,10,11] ∂A1 i ∂ 2 A1 ¯ 2 A3 −sgnβ00 ijA1 j2 A1 3iCA ∂Z 2 ∂T 2 ∂2 (1) gZ 1 τ 2 A1 − r1 A1 ; ∂T
Fig. 2. Configuration of an all-fiber DFAL. 10 May 2014 / Vol. 53, No. 14 / APPLIED OPTICS
2999
gZ
2g0 ; 1 jA1 j2 ∕e0
(2)
∂A2 ¯ 1 A4 2A3 2A5 − r2 A2 ; iCA ∂Z
(3)
∂A3 ¯ 1 A6 2A2 2A5 − r2 A3 ; iCA ∂Z
(4)
∂A4 ¯ 2 A7 2A5 2A8 − r2 A4 ; iCA ∂Z
(5)
∂A5 ¯ 4 A6 A3 A2 A8 A9 − r2 A5 ; (6) iCA ∂Z ∂A6 ¯ 3 A10 2A5 2A9 − r2 A6 ; iCA ∂Z
(7)
∂A7 ¯ 4 2A8 − r2 A7 ; iCA ∂Z
(8)
∂A8 ¯ 7 A 9 A5 A4 − r 2 A8 ; iCA ∂Z
(9)
∂A9 ¯ 8 A10 A6 A5 − r2 A9 ; iCA ∂Z
(10)
∂A10 ¯ 6 2A9 − r2 A10 : iCA ∂Z
(11)
An is normalized by power jA0 j2. Physical time T is normalized by T 0 ∕1.76, where T 0 200 fs is the full width at half-maximum (FWHM) of the pulse. Position Z is scaled on dispersion length Z0 2πc∕ 2 ¯ λ20 jDjT 0 ∕1.76 , which corresponds to the average ¯ [for wavelength 1550 nm, D ¯ ≈ cavity dispersion D ¯ 12 ps∕km · nm, for 1064 nm, D ≈ −30 ps∕km · nm], n2 2.6 × this gives jA0 j2 λ0 Aeff ∕4πn2 Z0 . −16 2 cm ∕W is the nonlinear coefficient in the fiber, 10 and Aeff 60 μm2 is the effective cross-sectional area. c is the light speed. g represents the gain in the central core and bandwidth τ 1∕Ω2 T 0 ∕1.762 , for a gain bandwidth that varies from Δλ 20 to 40 nm, Ω 2πc∕λ20 Δλ, so that τ ≈ 0.08–0.32. Here, we take τ 0.1. r1 Γ1 Z0 and r2 Γ2 Z0 are the loss coefficients in the central core and noncentral core. To let the power of the light in the noncentral core decade quickly, take r2 ≫ r1 . Here, Γ1 0.2 dB∕ km and Γ2 104 dB∕km, so r1 3.88 × 10−5 , r2 2:sgnβ00 represents the sign of β00. gZ is the gain in the central core. g0 and e0 are the gain parameter and saturated gain. The modelocking is highly robust and insensitive to changes in the gain model [3]. For convenience, we use a fixed gain gZ g0 instead of the saturated gain model in Eq. (2). ¯ CZ0 is used for The same coupling coefficient C all cores, which could be easily changed by Φcore, core spacing, the refractive indices in the core and clad¯ 1.17. ding. Take C to be 1.39 m−1 , so that the C 3000
APPLIED OPTICS / Vol. 53, No. 14 / 10 May 2014
The dispersion and nonlinear effects are neglected in noncentral cores because the light intensity is very low and loss is high. The total cavity length is 1 m. Here, we discuss two different kinds of fiber laser systems with negative dispersion (Er-DFAL) and positive dispersion (Yb-DFAL) in the simulations. But this simulation could be extended to different wavelength mode-locked laser systems, such as Nd-, Tm-, Ho-DFALs. 3. Results A. Er-Doped Fiber Array Laser (Negative Dispersion)
Figure 3 demonstrates self-starting mode-locking evolution dynamics in the Er-DFAL. Figure 3(a) shows that the mode-locked pulses start from initial white noise conditions in the central core (g0 1.15). Figures 3(b)–3(d) show the intensity versus time and distance of the noncentral cores jA2 j2, jA4 j2 , and jA7 j2 . Before the laser starts to mode-lock, most of the power of the weak light couples to core 2, core 4, and core 7 (labeled in Fig. 1), then loss away. When stable, about 1% of the power of the mode-locked pulses in the central core couples to core 2, and nearly none of them reach core 4 and core 7. The threshold of gain in the mode-locked laser is around 1.15. When g0 is lower than 1.15, no modelocking occurs. Once it reaches 1.15, single pulse mode-locking occurs. As g0 continues to increase, double and triple pulses mode-locking occur. Figures 4(a), 4(c), and 4(d) show the mode-locking evolution dynamics of the stable single pulse, double pulses, and triple pulses when gain g0 equals 1.15, 1.22, and 1.27, respectively. We show the mode-locked pulses traveling 500 round trips. Figure 4(b) shows the relationship between the energy of the central coreR ‖A1 ‖2 and ∞ the traveling distance, where ‖A1 ‖2 −∞ jA1 j2 dt. The energy of the initial white noise condition is three times bigger than that of the stable mode-locked pulses. ‖A1 ‖2 will stabilize after about 2 m. Figure 5 shows the distribution of the peak amplitude on the cross section of the DFAL at the different
Fig. 3. Self-starting mode-locking evolution dynamics in the Er-DFAL. (a)–(d) Intensity versus time and distance of the central core jA1 j2 , the noncentral core jA2 j2 , jA4 j2 , and jA7 j2 , respectively. g0 1.15.
Fig. 4. Mode-locking with single, double, and triple pulses in the Er-DFAL. (a) and (b) Mode-locking evolution dynamics and the energy of the central core as a function of distance corresponding to g0 as 1.15. (c) and (d) Mode-locking evolution dynamics as a function of distance corresponding to g0 as 1.22 and 1.27.
Fig. 6. Self-starting mode-locking evolution dynamics in the Yb-DFAL. (a), (b), (c), and (d) Intensity versus time and distance of the central core jA1 j2 , the noncentral core jA2 j2 , jA4 j2 , and jA7 j2 , respectively. g0 1.43.
distances at the initiation of self-starting modelocking. At the position of 0 m, there is only a noise initial condition in the central core. Then the low intensity portions couple to noncentral cores and the high intensity portions become stronger for the gain in the central core. After the light in the noncentral cores is almost lost, the distribution of the peak amplitude in the fiber array reaches stability. B.
Yb-Doped Fiber Array Laser (Positive Dispersion)
Similar to the previous section, we get the stable selfstarting mode-locking of the Yb-DFAL under differ¯ and λ0 . Figures 6(a)–6(d) show the evolution ent D dynamics in the central core 1, noncentral core 2, core 4, and core 7, when g0 1.43. Multipulsing transition is also obtained in the Yb-DFAL. Figures 7(a), 7(c), and 7(d) show the modelocking evolution dynamics of the stable single pulse, double pulses, and triple pulses, when g0 equals to 1.43, 1.75, and 2.1, respectively. Figure 7(b) shows the relationship between the energy of the central core ‖A1 ‖2 and the distance. After a huge fluctuation, it gets stable at about 8 m.
Fig. 5. Distribution of peak amplitude on cross section of the Er-DFAL from the initiation of self-starting mode-locking. The figures correspond to the fiber positions at 0, 0.4, 0.8, 1.2, 1.4, 1.6, 2, and 10 m, respectively.
Fig. 7. Mode-locking evolution dynamics with single, double, and triple pulses in the Yb-DFAL. (a) and (b) Mode-locking evolution dynamics and the energy of the central core as a function of distance corresponding to g0 as 1.43. (c) and (d) Mode-locking evolution dynamics when g0 are 1.75 and 2.1.
Figure 8 shows the distribution of the peak amplitude on cross section of the Yb-DFAL from the initiation of self-starting mode-locking. The figures correspond to the fiber positions at 0, 0.4, 0.8, 1.2, 2, 3, 4, and 10 m, respectively.
Fig. 8. Distribution of the peak amplitude on cross section of the Yb-DFAL from the initiation of self-starting mode-locking. The figures correspond to the fiber positions at 0, 0.4, 0.8, 1.2, 2, 3, 4, and 10, respectively. 10 May 2014 / Vol. 53, No. 14 / APPLIED OPTICS
3001
Similar to Fig. 5, about 1% of the energy of the mode-locked pulses in the central core couples to neighboring cores and becomes stable, so neighboring cores keep a little light energy when the laser reaches stable mode-locking. The simulated pulse shapes of the DFALs with negative dispersion and positive dispersion systems are fitted by Sech functions, shown in Fig. 9(a). The phases are also shown in Fig. 9(a). The pulse widths (FWHM) are about 37 fs for the Er-DFAL and 390 fs for the Yb-DFAL. The root mean square errors (RMSE) of fitting are 0.04732 and 0.03565, respectively. For the Yb-DFAL, because the total dispersion of the cavity is positive, the output pulse is chirped. When we compensate the positive dispersion with negative material out of the laser cavity, such as photonics crystal fiber, the pulse width could be compressed. This was calculated and simulation results show that the pulses from the Yb-DFAL can be compressed to 65 fs with dispersion compensation. Figure 9(b) demonstrates the spectra width (FWHM) of 67 nm for the Er-DFAL and 58 nm for the YbDFAL. Mode-locking with such wide spectra could
Fig. 9. (a) Simulated pulse shapes and the phases of the DFALs with negative dispersion and positive dispersion systems and the fitted results. (b) The corresponding spectra. 3002
APPLIED OPTICS / Vol. 53, No. 14 / 10 May 2014
Fig. 10. Discrete mode-locked fiber laser based on a gain fiber and a fiber array.
support a several tens femtoseconds pulse width and has been verified by mode-locked Er/Yb-doped fiber lasers in experiments [12–15]. C. Comparison between the Averaged Mode-Locked Fiber Laser and the Discrete Mode-Locked Fiber Laser
Figure 10 provides a discrete mode-locked fiber laser, with gain material separated from the saturable absorber in the mode-locked fiber laser formed by fiber arrays and doped fiber. It consists of a 2 m long Er-doped fiber and 8 cm long fiber arrays. This modelocked laser is described by the discrete model [3]. For a DFAL (Fig. 2), as the physical effects are averaged inside the system, pulse shape changes rarely in the cavity after gain, dispersion, coupling, and nonlinear effects reach balance. Hence, we also call the DFAL an averaged mode-locked laser. With DFAL, it is easier to generate robust mode-locking rather than the discrete mode-locked laser because the distribution of physical effects is averaged in the whole cavity. By changing the designed parameters (refractive index, size, loss, etc.), one is able to obtain modelocking under different conditions. To compare the difference between the averaged mode-locked laser and the discrete mode-locked laser, the pulse shapes at different positions in the fiber lasers are monitored in both situations.
Fig. 11. (a) Mode-locked pulses of the averaged mode-locked fiber array laser. (b) and (c) Pulses at the end of the fiber array and the end of the gain fiber in the discrete mode-locked laser, respectively. (d) The whole evolution dynamics of pulses in different positions of the discrete mode-locked laser.
the pulse width (FWHM) of mode-locked pulses with negative dispersion. The energies of the central core and noncentral cores as a function of distance and the process of self-starting mode-locking have been studied. To compare the difference between the averaged mode-locked laser and the discrete mode-locked laser, the pulse shapes at different positions in the fiber lasers are monitored in both situations. The authors acknowledge J. Nathan Kutz (University of Washington) for simulation discussions. We acknowledge support from the 973 program (No. 2013CB922404) and the National Natural Science Foundation of China (No. 61177047), the key project of the National Natural Science Foundation of China (No. 61235010). Fig. 12. Pulse shape and corresponding spectrum in the middle of the gain fiber for the discrete model.
We take the Er-DFAL as the example. Figure 11(a) shows the mode-locked pulses of the averaged mode-locked fiber array laser with g0 1.15. The pulse shapes in different positions rarely change. Figures 11(b) and 11(c) show the pulse evolution dynamics in the different positions [labeled (b) and (c) in Fig. 10] in the discrete mode-locked laser with a coupling loss of 20% when the light coupled into the fiber array and g0 2.1. Figure 11(d) shows the whole evolution dynamics of the discrete modelocked laser. The edges of the pulses become steep after they pass through the fiber array [Fig. 11(b)], while the pulses are amplified by the gain fiber and become the shape with pedestal wings [Fig. 11(c)]. Figure 11(d) shows that when the pulses propagate through the whole discrete mode-locked laser, they become wider and higher after the gain fiber and get narrower after the fiber array. Figure 12 shows the pulse shape and corresponding spectrum in the middle position of the gain fiber for the discrete model. The pulse width and spectrum width (FWHM) are 220 fs and 19 nm, respectively. For the strong nonlinear effect in fiber arrays [1], the phase is not in a parabolic shape, which means the nonlinear chirp is pretty high. 4. Conclusion
We designed a new kind of mode-locked fiber laser based on fiber arrays, where the central core is doped, so the fiber array can act as both a saturable absorber and a gain material at the same time. We discuss the all-fiber self-starting mode-locked fiber lasers based on doped fiber arrays with a theoretical model. With the negative dispersion and positive dispersion, the stable multipulsing mode-locking (single pulse, double pulses, and triple pulses) are achieved by the realistic parameters in these novel lasers. Pulse width (FWHM) of mode-locked pulses with positive dispersion is nearly ten times that of
References 1. J. L. Proctor and J. N. Kutz, “Passive mode-locking by use of waveguide arrays,” Opt. Lett. 30, 2013–2015 (2005). 2. J. N. Kutz and B. Sandstede, “Theory of passive harmonic mode-locking using waveguide arrays,” Opt. Express 16, 636–650 (2008). 3. J. Proctor and J. N. Kutz, “Nonlinear mode-coupling for passive mode-locking: application of waveguide arrays, dual-core fibers, and/or fiber arrays,” Opt. Express 13, 8933–8950 (2005). 4. M. O. Williams, E. Shlizerman, and J. N. Kutz, “The multipulsing transition in mode-locked lasers: a low-dimensional approach using waveguide arrays,” J. Opt. Soc. Am. B 27, 2471–2481 (2010). 5. J. N. Kutz, C. Conti, and S. Trillo, “Mode-locked X-wave lasers,” Opt. Express 15, 16022–16028 (2007). 6. D. D. Hudson, K. Shish, T. R. Schibli, J. N. Kutz, D. N. Christodoulides, R. Morandotti, and S. T. Cundiff, “Nonlinear femtosecond pulse reshaping in waveguide arrays,” Opt. Lett. 33, 1440–1442 (2008). 7. D. D. Hudson, J. N. Kutz, T. R. Schibli, Q. Chao, D. N. Christodoulides, R. Morandotti, and S. T. Cundiff, “Fixedpoint attractor for chirp in nonlinear waveguide arrays,” Phys. Rev. A 85, 031806 (2012). 8. Q. Chao, D. D. Hudson, J. N. Kutz, and S. T. Cundiff, “Waveguide array fiber laser,” IEEE Photon. J. 4, 1438–1442 (2012). 9. T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tunnermann, and F. Lederer, “Nonlinearity and disorder in fiber arrays,” Phys. Rev. Lett. 93, 053901 (2004). 10. B. G. Bale, E. Farnum, and J. N. Kutz, “Theory and simulation of passive multifrequency mode-locking with waveguide arrays,” IEEE J. Quantum Electron. 44, 976–983 (2008). 11. J. Proctor and J. N. Kutz, “Averaged models for passive mode-locking using nonlinear mode-coupling,” Math. Comput. Simul. 74, 333–342 (2007). 12. A. Winter, F. O. Ilday, and B. Steffen, “Femtosecond Yb-doped fiber laser system at 1 μm of wavelength with 100-nm bandwidth and variable pulse structure for accelerator diagnostics,” in Proceedings of the 8th European Workshop on Beam Diagnostics and Instrumentation for Particle Accelerators (DIPAC 2007), Venice, Italy (2007), paper WEPB03. 13. J. Hamazaki, N. Sekine, and I. Hosako, “Mode-locked Yb-doped fiber ring-laser for use as a pump pulse source of THz-TDS,” in Proceedings of the Progress in Electromagnetics Research Symposium (PIERS 2012) (2012), pp. 482–485. 14. D. Ma, Y. Cai, C. Zhou, W. J. Zong, L. L. Chen, and Z. G. Zhang, “37.4 fs pulse generation in an Er:fiber laser at a 225 MHz repetition rate,” Opt. Lett. 35, 2858–2860 (2010). 15. Z. H. Yu, Y. G. Wang, X. Zhang, X. Z. Dong, J. R. Tian, and Y. R. Song, “A 66 fs highly stable single wall carbon nanotube mode locked fiber laser,” Laser Phys. 24, 015105 (2014).
10 May 2014 / Vol. 53, No. 14 / APPLIED OPTICS
3003
We designed a new kind of mode-locked fiber laser based on fiber arrays, where the central core is doped. A theoretical model is given for an all-fiber self-starting mode-locked laser based on this kind of doped fiber array. Two different kinds of fiber lasers with negative dispersion and positive dispersion are simulated and discussed. The stable mode-locked pulses are generated from initial noise conditions by the realistic parameters. The process of self-starting mode-locking multipulse transition and the relationship between the energy of the central core and the propagation distance of the pulses are discussed. Finally, we analyze the difference between the averaged mode-locked laser and the discrete mode-locked laser. © 2014 Optical Society of America OCIS codes: (140.4050) Mode-locked lasers; (060.5530) Pulse propagation and temporal solitons; (320.7090) Ultrafast lasers. http://dx.doi.org/10.1364/AO.53.002998
1. Introduction
Since mode-locked lasers have a wide range of applications in military, industry, and medicine, many passive mode-locking elements have been studied in depth in the last decade, e.g., semiconductor saturable absorber mirror (SESAM), carbon nanotubes, and graphene. For the characteristics of nonlinear modecoupling (NLMC), waveguide arrays and fiber arrays could work as novel passive mode-locking mechanisms. Waveguide arrays have been investigated in some laser systems. Numerical studies show that robust and stable mode-locking has been achieved in fiber lasers with waveguide arrays [1–3]. The multipulsing transition [4] and generation of spatiotemporal X-waves [5] were studied in a fully comprehensive physical model of the laser cavity that uses waveguide arrays as a saturable absorber. Nonlinear pulse reshaping of femtosecond pulses was observed in a waveguide array and the power-dependent pulse reshaping agrees with the theory, including the shortening of the pulse in the central waveguide [6]. A fixed-point attractor for chirp in AlGaAs waveguide arrays was demonstrated, and the simulations reproduced the 1559-128X/14/142998-06$15.00/0 © 2014 Optical Society of America 2998
APPLIED OPTICS / Vol. 53, No. 14 / 10 May 2014
experimental results [7]. An AlGaAs waveguide array is used as a saturable absorber to achieve passive mode-locking in an erbium-doped fiber laser, autocorrelations showed that the pulses consist of noise bursts due to incomplete mode-locking. However, unlike a typical Kerr-lens cavity, the waveguide array laser is self-starting [8]. An experimental study of fiber arrays shows that, in the linear case, light either spreads in a diffusive manner or localizes at a few sites. For high excitation power, diffusive spreading is arrested by the focusing nonlinearity [9]. Thus, a fiber array could be used as a saturable absorber like a waveguide array. A mode-locked fiber laser with a fiber array saturable absorber and a gain fiber has been demonstrated theoretically [3]. For its advantages of flexibility, good heat dissipation, all-fiber system, compact configuration, and a better shaping function [3], fiber arrays could be a potential broadband passive modelocking mechanism. In this paper, we designed a novel mode-locked fiber laser which is called a doped fiber array laser (DFAL). In the laser, the central core of the fiber array is doped so that this doped fiber array can be used as both the gain material and mode-locking element at the same time. We also call this kind of laser an averaged mode-locked laser when we compare it with the
discrete mode-locked fiber laser later. Two kinds of DFALs use different doped materials, Yb-doped fiber (central wavelength at 1064 nm) and Er-doped fiber (central wavelength at 1550 nm), which correspond to the positive dispersion and negative dispersion systems. For the negative dispersion system, the output pulse width could reach several tens of femtoseconds, but the pulses would split when the pulse energy increased with increased pump power. If high output energy is required, a DFAL with positive dispersion system, such as Yb-DFAL, is a good choice because the pulses will not split when the pump power is increased and the self-similar system is one solution. The simulation results of the DFAL systems with positive or negative dispersions show that the stable Sech-shaped mode-locked pulses could be reached by realistic parameters. In order to show the advantages of the DFALs model, we compare its output pulse shape with that of a discrete mode-locked fiber laser [3] in the last part of the paper. 2. Doped Fiber Arrays and Mode-Locked Fiber Lasers A.
Doped Fiber Arrays
Figure 1 shows the cross section picture of a homogeneous hexagonal lattice of a fiber array. We set 37 cores in the fiber array. The central core is rareearth-doped, while others are not. Every internal core is surrounded by six cores, and the light in one core could couple to the six nearest neighboring cores and vice versa. In a word, the light of the neighboring cores are coupled with each other. For instance, the mode-coupling between the core No. 5 and its neighboring cores (labeled in Fig. 1) is described as [9] dA5 ∕dz iC25 A2 iC35 A3 iC45 A4 iC65 A6 iC85 A8 iC95 A9 , where An is the electric field envelope in nth core. Cmn is the coupling coefficient between the mth core and the nth core. As a homogeneous hexagonal lattice, C25 C35 C45 C65 C85 C95 C0 , where C0 is a constant. The properties of fiber arrays are determined by designed parameters (refractive indices in the core and
Fig. 1. Cross section of a hexagonal fiber array.
cladding, core spacing, and core diameter) rather than just material properties [8]. There is a significant difference between a photonic crystal fiber (PCF) and a fiber array: their size. In a PCF, to create a band gap, Φcore and the core spacing are close to the light wavelength. However, in a fiber array, the Φcore and the core spacing are far bigger than that. In the fiber array we designed here, the Φcore and core spacing are roughly 5–10 μm and 10–30 μm, respectively. The material parameters of typical silica fiber are used here. For the light in the fiber array, the low intensity portions spread in a diffusive manner and coupled into neighboring cores and then were lost. Meanwhile, the high intensity portions almost remain in the central core for a self-focusing effect (it is also called Kerr lens effect). After traveling a certain distance, the high intensity portion could remain in the central core for the gain and the balance of the nonlinear effect and the dispersion effect, but the low intensity portion decays for the loss only. This characteristic is equivalent to a saturable absorber, such as SESAM and carbon nanotubes in normal mode-locked lasers. B. Model of DFALs
Figure 2 shows the configuration of an all-fiber DFAL. Two couplers are for the input pump light and the output laser. To prevent unwanted feedback, an isolator is placed in the cavity. Doped fiber arrays are used as both a gain material and a saturable absorber at the same time. For Er-doped and Yb-DFAL systems, the wavelengths of the lasers are λ0 1550 nm and 1064 nm, respectively. For the 37 cores in the fiber array, each core should correspond to one equation, but here only ten equations are considered when we simulate the system (labeled by numbers in Fig. 1) because they are both central symmetric and axial symmetric. The evolution of the normalized electric field amplitudes in the nth core An are given by [3,10,11] ∂A1 i ∂ 2 A1 ¯ 2 A3 −sgnβ00 ijA1 j2 A1 3iCA ∂Z 2 ∂T 2 ∂2 (1) gZ 1 τ 2 A1 − r1 A1 ; ∂T
Fig. 2. Configuration of an all-fiber DFAL. 10 May 2014 / Vol. 53, No. 14 / APPLIED OPTICS
2999
gZ
2g0 ; 1 jA1 j2 ∕e0
(2)
∂A2 ¯ 1 A4 2A3 2A5 − r2 A2 ; iCA ∂Z
(3)
∂A3 ¯ 1 A6 2A2 2A5 − r2 A3 ; iCA ∂Z
(4)
∂A4 ¯ 2 A7 2A5 2A8 − r2 A4 ; iCA ∂Z
(5)
∂A5 ¯ 4 A6 A3 A2 A8 A9 − r2 A5 ; (6) iCA ∂Z ∂A6 ¯ 3 A10 2A5 2A9 − r2 A6 ; iCA ∂Z
(7)
∂A7 ¯ 4 2A8 − r2 A7 ; iCA ∂Z
(8)
∂A8 ¯ 7 A 9 A5 A4 − r 2 A8 ; iCA ∂Z
(9)
∂A9 ¯ 8 A10 A6 A5 − r2 A9 ; iCA ∂Z
(10)
∂A10 ¯ 6 2A9 − r2 A10 : iCA ∂Z
(11)
An is normalized by power jA0 j2. Physical time T is normalized by T 0 ∕1.76, where T 0 200 fs is the full width at half-maximum (FWHM) of the pulse. Position Z is scaled on dispersion length Z0 2πc∕ 2 ¯ λ20 jDjT 0 ∕1.76 , which corresponds to the average ¯ [for wavelength 1550 nm, D ¯ ≈ cavity dispersion D ¯ 12 ps∕km · nm, for 1064 nm, D ≈ −30 ps∕km · nm], n2 2.6 × this gives jA0 j2 λ0 Aeff ∕4πn2 Z0 . −16 2 cm ∕W is the nonlinear coefficient in the fiber, 10 and Aeff 60 μm2 is the effective cross-sectional area. c is the light speed. g represents the gain in the central core and bandwidth τ 1∕Ω2 T 0 ∕1.762 , for a gain bandwidth that varies from Δλ 20 to 40 nm, Ω 2πc∕λ20 Δλ, so that τ ≈ 0.08–0.32. Here, we take τ 0.1. r1 Γ1 Z0 and r2 Γ2 Z0 are the loss coefficients in the central core and noncentral core. To let the power of the light in the noncentral core decade quickly, take r2 ≫ r1 . Here, Γ1 0.2 dB∕ km and Γ2 104 dB∕km, so r1 3.88 × 10−5 , r2 2:sgnβ00 represents the sign of β00. gZ is the gain in the central core. g0 and e0 are the gain parameter and saturated gain. The modelocking is highly robust and insensitive to changes in the gain model [3]. For convenience, we use a fixed gain gZ g0 instead of the saturated gain model in Eq. (2). ¯ CZ0 is used for The same coupling coefficient C all cores, which could be easily changed by Φcore, core spacing, the refractive indices in the core and clad¯ 1.17. ding. Take C to be 1.39 m−1 , so that the C 3000
APPLIED OPTICS / Vol. 53, No. 14 / 10 May 2014
The dispersion and nonlinear effects are neglected in noncentral cores because the light intensity is very low and loss is high. The total cavity length is 1 m. Here, we discuss two different kinds of fiber laser systems with negative dispersion (Er-DFAL) and positive dispersion (Yb-DFAL) in the simulations. But this simulation could be extended to different wavelength mode-locked laser systems, such as Nd-, Tm-, Ho-DFALs. 3. Results A. Er-Doped Fiber Array Laser (Negative Dispersion)
Figure 3 demonstrates self-starting mode-locking evolution dynamics in the Er-DFAL. Figure 3(a) shows that the mode-locked pulses start from initial white noise conditions in the central core (g0 1.15). Figures 3(b)–3(d) show the intensity versus time and distance of the noncentral cores jA2 j2, jA4 j2 , and jA7 j2 . Before the laser starts to mode-lock, most of the power of the weak light couples to core 2, core 4, and core 7 (labeled in Fig. 1), then loss away. When stable, about 1% of the power of the mode-locked pulses in the central core couples to core 2, and nearly none of them reach core 4 and core 7. The threshold of gain in the mode-locked laser is around 1.15. When g0 is lower than 1.15, no modelocking occurs. Once it reaches 1.15, single pulse mode-locking occurs. As g0 continues to increase, double and triple pulses mode-locking occur. Figures 4(a), 4(c), and 4(d) show the mode-locking evolution dynamics of the stable single pulse, double pulses, and triple pulses when gain g0 equals 1.15, 1.22, and 1.27, respectively. We show the mode-locked pulses traveling 500 round trips. Figure 4(b) shows the relationship between the energy of the central coreR ‖A1 ‖2 and ∞ the traveling distance, where ‖A1 ‖2 −∞ jA1 j2 dt. The energy of the initial white noise condition is three times bigger than that of the stable mode-locked pulses. ‖A1 ‖2 will stabilize after about 2 m. Figure 5 shows the distribution of the peak amplitude on the cross section of the DFAL at the different
Fig. 3. Self-starting mode-locking evolution dynamics in the Er-DFAL. (a)–(d) Intensity versus time and distance of the central core jA1 j2 , the noncentral core jA2 j2 , jA4 j2 , and jA7 j2 , respectively. g0 1.15.
Fig. 4. Mode-locking with single, double, and triple pulses in the Er-DFAL. (a) and (b) Mode-locking evolution dynamics and the energy of the central core as a function of distance corresponding to g0 as 1.15. (c) and (d) Mode-locking evolution dynamics as a function of distance corresponding to g0 as 1.22 and 1.27.
Fig. 6. Self-starting mode-locking evolution dynamics in the Yb-DFAL. (a), (b), (c), and (d) Intensity versus time and distance of the central core jA1 j2 , the noncentral core jA2 j2 , jA4 j2 , and jA7 j2 , respectively. g0 1.43.
distances at the initiation of self-starting modelocking. At the position of 0 m, there is only a noise initial condition in the central core. Then the low intensity portions couple to noncentral cores and the high intensity portions become stronger for the gain in the central core. After the light in the noncentral cores is almost lost, the distribution of the peak amplitude in the fiber array reaches stability. B.
Yb-Doped Fiber Array Laser (Positive Dispersion)
Similar to the previous section, we get the stable selfstarting mode-locking of the Yb-DFAL under differ¯ and λ0 . Figures 6(a)–6(d) show the evolution ent D dynamics in the central core 1, noncentral core 2, core 4, and core 7, when g0 1.43. Multipulsing transition is also obtained in the Yb-DFAL. Figures 7(a), 7(c), and 7(d) show the modelocking evolution dynamics of the stable single pulse, double pulses, and triple pulses, when g0 equals to 1.43, 1.75, and 2.1, respectively. Figure 7(b) shows the relationship between the energy of the central core ‖A1 ‖2 and the distance. After a huge fluctuation, it gets stable at about 8 m.
Fig. 5. Distribution of peak amplitude on cross section of the Er-DFAL from the initiation of self-starting mode-locking. The figures correspond to the fiber positions at 0, 0.4, 0.8, 1.2, 1.4, 1.6, 2, and 10 m, respectively.
Fig. 7. Mode-locking evolution dynamics with single, double, and triple pulses in the Yb-DFAL. (a) and (b) Mode-locking evolution dynamics and the energy of the central core as a function of distance corresponding to g0 as 1.43. (c) and (d) Mode-locking evolution dynamics when g0 are 1.75 and 2.1.
Figure 8 shows the distribution of the peak amplitude on cross section of the Yb-DFAL from the initiation of self-starting mode-locking. The figures correspond to the fiber positions at 0, 0.4, 0.8, 1.2, 2, 3, 4, and 10 m, respectively.
Fig. 8. Distribution of the peak amplitude on cross section of the Yb-DFAL from the initiation of self-starting mode-locking. The figures correspond to the fiber positions at 0, 0.4, 0.8, 1.2, 2, 3, 4, and 10, respectively. 10 May 2014 / Vol. 53, No. 14 / APPLIED OPTICS
3001
Similar to Fig. 5, about 1% of the energy of the mode-locked pulses in the central core couples to neighboring cores and becomes stable, so neighboring cores keep a little light energy when the laser reaches stable mode-locking. The simulated pulse shapes of the DFALs with negative dispersion and positive dispersion systems are fitted by Sech functions, shown in Fig. 9(a). The phases are also shown in Fig. 9(a). The pulse widths (FWHM) are about 37 fs for the Er-DFAL and 390 fs for the Yb-DFAL. The root mean square errors (RMSE) of fitting are 0.04732 and 0.03565, respectively. For the Yb-DFAL, because the total dispersion of the cavity is positive, the output pulse is chirped. When we compensate the positive dispersion with negative material out of the laser cavity, such as photonics crystal fiber, the pulse width could be compressed. This was calculated and simulation results show that the pulses from the Yb-DFAL can be compressed to 65 fs with dispersion compensation. Figure 9(b) demonstrates the spectra width (FWHM) of 67 nm for the Er-DFAL and 58 nm for the YbDFAL. Mode-locking with such wide spectra could
Fig. 9. (a) Simulated pulse shapes and the phases of the DFALs with negative dispersion and positive dispersion systems and the fitted results. (b) The corresponding spectra. 3002
APPLIED OPTICS / Vol. 53, No. 14 / 10 May 2014
Fig. 10. Discrete mode-locked fiber laser based on a gain fiber and a fiber array.
support a several tens femtoseconds pulse width and has been verified by mode-locked Er/Yb-doped fiber lasers in experiments [12–15]. C. Comparison between the Averaged Mode-Locked Fiber Laser and the Discrete Mode-Locked Fiber Laser
Figure 10 provides a discrete mode-locked fiber laser, with gain material separated from the saturable absorber in the mode-locked fiber laser formed by fiber arrays and doped fiber. It consists of a 2 m long Er-doped fiber and 8 cm long fiber arrays. This modelocked laser is described by the discrete model [3]. For a DFAL (Fig. 2), as the physical effects are averaged inside the system, pulse shape changes rarely in the cavity after gain, dispersion, coupling, and nonlinear effects reach balance. Hence, we also call the DFAL an averaged mode-locked laser. With DFAL, it is easier to generate robust mode-locking rather than the discrete mode-locked laser because the distribution of physical effects is averaged in the whole cavity. By changing the designed parameters (refractive index, size, loss, etc.), one is able to obtain modelocking under different conditions. To compare the difference between the averaged mode-locked laser and the discrete mode-locked laser, the pulse shapes at different positions in the fiber lasers are monitored in both situations.
Fig. 11. (a) Mode-locked pulses of the averaged mode-locked fiber array laser. (b) and (c) Pulses at the end of the fiber array and the end of the gain fiber in the discrete mode-locked laser, respectively. (d) The whole evolution dynamics of pulses in different positions of the discrete mode-locked laser.
the pulse width (FWHM) of mode-locked pulses with negative dispersion. The energies of the central core and noncentral cores as a function of distance and the process of self-starting mode-locking have been studied. To compare the difference between the averaged mode-locked laser and the discrete mode-locked laser, the pulse shapes at different positions in the fiber lasers are monitored in both situations. The authors acknowledge J. Nathan Kutz (University of Washington) for simulation discussions. We acknowledge support from the 973 program (No. 2013CB922404) and the National Natural Science Foundation of China (No. 61177047), the key project of the National Natural Science Foundation of China (No. 61235010). Fig. 12. Pulse shape and corresponding spectrum in the middle of the gain fiber for the discrete model.
We take the Er-DFAL as the example. Figure 11(a) shows the mode-locked pulses of the averaged mode-locked fiber array laser with g0 1.15. The pulse shapes in different positions rarely change. Figures 11(b) and 11(c) show the pulse evolution dynamics in the different positions [labeled (b) and (c) in Fig. 10] in the discrete mode-locked laser with a coupling loss of 20% when the light coupled into the fiber array and g0 2.1. Figure 11(d) shows the whole evolution dynamics of the discrete modelocked laser. The edges of the pulses become steep after they pass through the fiber array [Fig. 11(b)], while the pulses are amplified by the gain fiber and become the shape with pedestal wings [Fig. 11(c)]. Figure 11(d) shows that when the pulses propagate through the whole discrete mode-locked laser, they become wider and higher after the gain fiber and get narrower after the fiber array. Figure 12 shows the pulse shape and corresponding spectrum in the middle position of the gain fiber for the discrete model. The pulse width and spectrum width (FWHM) are 220 fs and 19 nm, respectively. For the strong nonlinear effect in fiber arrays [1], the phase is not in a parabolic shape, which means the nonlinear chirp is pretty high. 4. Conclusion
We designed a new kind of mode-locked fiber laser based on fiber arrays, where the central core is doped, so the fiber array can act as both a saturable absorber and a gain material at the same time. We discuss the all-fiber self-starting mode-locked fiber lasers based on doped fiber arrays with a theoretical model. With the negative dispersion and positive dispersion, the stable multipulsing mode-locking (single pulse, double pulses, and triple pulses) are achieved by the realistic parameters in these novel lasers. Pulse width (FWHM) of mode-locked pulses with positive dispersion is nearly ten times that of
References 1. J. L. Proctor and J. N. Kutz, “Passive mode-locking by use of waveguide arrays,” Opt. Lett. 30, 2013–2015 (2005). 2. J. N. Kutz and B. Sandstede, “Theory of passive harmonic mode-locking using waveguide arrays,” Opt. Express 16, 636–650 (2008). 3. J. Proctor and J. N. Kutz, “Nonlinear mode-coupling for passive mode-locking: application of waveguide arrays, dual-core fibers, and/or fiber arrays,” Opt. Express 13, 8933–8950 (2005). 4. M. O. Williams, E. Shlizerman, and J. N. Kutz, “The multipulsing transition in mode-locked lasers: a low-dimensional approach using waveguide arrays,” J. Opt. Soc. Am. B 27, 2471–2481 (2010). 5. J. N. Kutz, C. Conti, and S. Trillo, “Mode-locked X-wave lasers,” Opt. Express 15, 16022–16028 (2007). 6. D. D. Hudson, K. Shish, T. R. Schibli, J. N. Kutz, D. N. Christodoulides, R. Morandotti, and S. T. Cundiff, “Nonlinear femtosecond pulse reshaping in waveguide arrays,” Opt. Lett. 33, 1440–1442 (2008). 7. D. D. Hudson, J. N. Kutz, T. R. Schibli, Q. Chao, D. N. Christodoulides, R. Morandotti, and S. T. Cundiff, “Fixedpoint attractor for chirp in nonlinear waveguide arrays,” Phys. Rev. A 85, 031806 (2012). 8. Q. Chao, D. D. Hudson, J. N. Kutz, and S. T. Cundiff, “Waveguide array fiber laser,” IEEE Photon. J. 4, 1438–1442 (2012). 9. T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tunnermann, and F. Lederer, “Nonlinearity and disorder in fiber arrays,” Phys. Rev. Lett. 93, 053901 (2004). 10. B. G. Bale, E. Farnum, and J. N. Kutz, “Theory and simulation of passive multifrequency mode-locking with waveguide arrays,” IEEE J. Quantum Electron. 44, 976–983 (2008). 11. J. Proctor and J. N. Kutz, “Averaged models for passive mode-locking using nonlinear mode-coupling,” Math. Comput. Simul. 74, 333–342 (2007). 12. A. Winter, F. O. Ilday, and B. Steffen, “Femtosecond Yb-doped fiber laser system at 1 μm of wavelength with 100-nm bandwidth and variable pulse structure for accelerator diagnostics,” in Proceedings of the 8th European Workshop on Beam Diagnostics and Instrumentation for Particle Accelerators (DIPAC 2007), Venice, Italy (2007), paper WEPB03. 13. J. Hamazaki, N. Sekine, and I. Hosako, “Mode-locked Yb-doped fiber ring-laser for use as a pump pulse source of THz-TDS,” in Proceedings of the Progress in Electromagnetics Research Symposium (PIERS 2012) (2012), pp. 482–485. 14. D. Ma, Y. Cai, C. Zhou, W. J. Zong, L. L. Chen, and Z. G. Zhang, “37.4 fs pulse generation in an Er:fiber laser at a 225 MHz repetition rate,” Opt. Lett. 35, 2858–2860 (2010). 15. Z. H. Yu, Y. G. Wang, X. Zhang, X. Z. Dong, J. R. Tian, and Y. R. Song, “A 66 fs highly stable single wall carbon nanotube mode locked fiber laser,” Laser Phys. 24, 015105 (2014).
10 May 2014 / Vol. 53, No. 14 / APPLIED OPTICS
3003
