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Numerical analysis of stimulated Brillouin scattering in high-power double-clad fiber lasers
ARTICLE IN PRESS
Optik
Optics
Optik 120 (2009) 24–28 www.elsevier.de/ijleo
Numerical analysis of stimulated Brillouin scattering in high-power double-clad fiber lasers Guohua Liu, Deming Liu Department of Optoelectronics, Huazhong University of Science and Technology, Wuhan 430074, PR China Received 1 February 2007; accepted 10 June 2007
Abstract A theoretical analysis of stimulated Brillouin scattering (SBS) in linear cavity Yb3+-doped double-clad fiber lasers is presented by solving the steady-state rate equations with the SBS. The effects of cavity length, fiber core diameter, input mirror reflectivity at Stokes wavelength, Yb3+ concentration and laser linewidth on the SBS are discussed. Numerical results show that the SBS threshold power can be improved by shortening the cavity length, using large mode area fiber, reducing the input mirror reflectivity at Stokes wavelength, lowering the Yb3+ concentration and broadening the laser linewidth, and the influence of the laser linewidth on the SBS threshold power is more noticeable than other system parameters. r 2007 Published by Elsevier GmbH. Keywords: Fiber lasers; Stimulated Brillouin scattering; Threshold power
1. Introduction Yb3+-doped double-clad (YDDC) fiber lasers pumped by low-cost laser diodes have attracted considerable attention recently in commercial and military applications due to their high brightness, eminent efficiency, good compactness, excellent beam quality, efficient heat dissipation, etc., compared to traditional gas and solid-state lasers [1]. With the availability of high-power laser diode bars and cladpumping techniques, the output power of YDDC fiber lasers is able to reach hundreds of watts, even a kilowatt, in the continuous-wave (CW) regime [2–4]. However, the scalability of output powers can be limited by amplified spontaneous emission and nonlinear processes such as stimulated Brillouin scattering (SBS), stimulated Raman scattering and the optical Kerr effect. Although Corresponding author. Tel.: +86 27 87545578.
E-mail address: [email protected] (G. Liu). 0030-4026/$ - see front matter r 2007 Published by Elsevier GmbH. doi:10.1016/j.ijleo.2007.06.007
these nonlinear effects could be of interest for specific applications [5–7], they can also lead to some unexpected instabilities in the laser signal. In particular, the SBS is expected to be the origin of instabilities in highpower fiber lasers [8] or deformation of pulses in fiber amplifiers [9]. The aim of this paper is to investigate theoretically the dependence of the SBS on system parameters in YDDC fiber lasers. By solving a set of laser rate equations with the SBS, the SBS thresholds are obtained under different fiber conditions. The results and analysis presented facilitate the design and optimization of YDDC fiber lasers.
2. Numerical model The configuration of a typical YDDC fiber laser under CW end pump ðPþ p and Pp Þ is schematically shown in Fig. 1. The YDDC fiber has a length L and a
ARTICLE IN PRESS G. Liu, D. Liu / Optik 120 (2009) 24–28
R1
intrinsic SBS gain constant. Aeff is the effective area associated with the laser and Stokes waves. The boundary conditions at the input (Z ¼ 0) and output (Z ¼ L) mirrors are given by
R2 P sout
0
Z
L
P +p
25
P –p
Pþ s ð0Þ ¼ R1s Ps ð0Þ,
Fig. 1. Schematic illustration of end pumped fiber lasers.
Pþ B ð0Þ ¼ R1B PB ð0Þ,
uniform dopant concentration in the fiber core. R1 is the reflectivity of the input mirror and R2 is the output coupler reflectivity. This input mirror can be either a dichroic mirror or a fiber grating, and the output coupler can be either a cleaved fiber facet or a fiber grating. The steady-state rate equations to describe the pump, laser and Stokes waves in CW lasers [10,11] are as follows:
þ P s ðLÞ ¼ R2s Ps ðLÞ,
ð5Þ
þ P B ðLÞ ¼ R2B PB ðLÞ,
ð6Þ
where R1s and R1B are the input mirror reflectivities at the laser and Stokes wavelength, respectively. R2s and R2B are the output mirror reflectivities at the laser and Stokes wavelength, respectively. We must numerically solve the nonlinear Eqs. (1)–(4) with the specified conditions (Eqs. (5) and (6)) at two
h i Pþ p ðzÞ þ Pp ðzÞ sap Gp
þ Ps ðzÞ þ P s ðzÞ sas Gs þ N 2 ðzÞ hup A hus A i ¼h , þ þ N Pp ðzÞ þ Pp ðzÞ ðsap þ sep ÞGp 1 P ðzÞ þ Ps ðzÞ ðsas þ ses ÞGs þ þ s t hup A hus A ¼ Gp ðsap þ sep ÞN 2 ðzÞ sap N
P p ðzÞ
a p P p ðzÞ,
(2) dP s ðzÞ ¼ Gs ½ðsas þ ses ÞN 2 ðzÞ sas N P s ðzÞ dz gB as P P ðzÞP s ðzÞ s ðzÞ, Aeff B dP gB B ðzÞ ¼ aB P P ðzÞP B ðzÞ B ðzÞ, dz Aeff s
points. We apply a shooting method for such a boundary-value problem and choose the magnitudes ðP s ð0Þ; PB ð0ÞÞ of backward powers at Z ¼ 0 as unknown initial parameters. For the backward powers at Z ¼ 0, we calculate the forward powers 500
(a)
ð3Þ
L = 30m
400
(4)
where N is Yb3+ concentration in the core with a crosssection area A and N2(z) is the population density of the upper level. P p ðzÞ is the pump power, Ps ðzÞ is the laser power and PB ðzÞ is the first-order Brillouin Stokes power (7 correspond to forward and backward propagations, respectively). Since the threshold conditions of the first-order Stokes wave are dealt with in this work, the higher-order Stokes waves can hence be neglected. Gp and Gs are power filling factors. sep and sap are the emission and absorption cross sections of the pump light, whereas ses and sas are the emission and absorption cross sections of the laser. up is the pump frequency, us is the laser frequency and t is spontaneous lifetime. h is Planck’s constant. ap and as represent scattering loss coefficients of pump light and laser. Because the SBS is less than 0.1 nm away from the laser wavelength, aBEas. gB is the SBS gain. When Dus, the laser linewidth, is much larger than the SBS gain bandwidth DuB, gB ¼ g0DuB/Dus+DuB, where g0 is the
Output power / W
dz
out
Ps
300
out B0 out P BL
P
200 100 0 0
100
200
300
400
500
600
700
Pump power / W 300
(b) Output power / W
dP p ðzÞ
(1)
L = 50m out
Ps
200
out B0 out P BL
P
100
0 0
100
200 300 Pump power / W
400
500
Fig. 2. Output laser and Stokes power for different cavity lengths (L). dq, the output laser power; Pout B0 , the backward output Stokes power at Z ¼ 0; Pout BL , the forward output Stokes power at Z ¼ L.
ARTICLE IN PRESS 26
G. Liu, D. Liu / Optik 120 (2009) 24–28
þ ðPþ s ð0Þ; PB ð0ÞÞ at Z ¼ 0 through Eq. (5) and determine þ two components ðPþ s ðLÞ; PB ðLÞ; Ps ðLÞ; PB ðLÞÞ of laser and Stokes powers at Z ¼ L by numerically integrating Eqs. (1)–(4). In order to seek solutions that satisfy the boundary conditions (Eq. (6)) at Z ¼ L, we use the Marquardt method, which is an algorithm for least-squares estimation of nonlinear parameters. Once intracavity powers are determined, we can calculate the output power of the laser and Stokes waves.
3. Simulation results and discussion For the purposes of illustration, unless indicated otherwise, the parameters used in the simulation are lp ¼ 975 nm, ls ¼ 1082.8 nm, R1s ¼ 0.99, R2s ¼ 0.04, R1B ¼ 0.04, R2B ¼ 0.04, L ¼ 40 m, t ¼ 0.8 ms, sap ¼ 2.0 1024 m2, sep ¼ 2.0 1024 m2, Gp ¼ 0.0012, 27 2 sas ¼ 3.1 10 m , ses ¼ 4.2 1025 m2, Gs ¼ 0.85, ap ¼ 3.0 103 m1, as ¼ aB ¼ 5.0 103 m1, DuB ¼ 50 MHz, Dus ¼ 10 GHz, N ¼ 6.0 1025 m3, g0 ¼ 5.0 1011 m/W, fiber core diameter Dc ¼ 20 mm and NA ¼ 0.05. Equal forward and backward pump power is assumed in the numerical calculations. Fig. 2 shows the output power versus pump power for L ¼ 30 and 50 m. In each figure, there are three curves. Pout is the output laser power, Pout s B0 is the backward output Stokes power at Z ¼ 0, and Pout BL is the forward output Stokes power at Z ¼ L. SBS begins once the pump power reaches the Brillouin threshold. One finds
that the threshold increases as the cavity length decreases. It is 336 W for a 50-m-long cavity and 514 W for a 30-m-long cavity. Near the SBS threshold, the slope efficiency of the backward output Stokes power with respect to the total pump power is 57% for a 50-m-long cavity and 60% for a 30-m-long cavity. In the presence of the SBS, the laser efficiency decreases, and the considerable forward propagating laser is converted to the backward Stokes wave. Fig. 3 shows the output power versus pump power for Dc ¼ 20 and 30 mm. One finds that the SBS threshold increases as the fiber core diameter increases. It is 402 W for Dc ¼ 20 mm and 644 W for Dc ¼ 30 mm. The slope efficiencies of the backward output Stokes power are 60% and 58% for Fig. 3(a) and (b), respectively. Since the Stokes wave frequency is downshifted by 16 GHz (0.06 nm) from the laser frequency, when the input mirror is a Bragg grating, which reflects 99% of laser over a bandwidth of 0.1 nm near 1082.8 nm, the reflectivity of the input mirror is also 99% for the Stokes wave. Indeed, the spectral bandwidth of the Bragg grating can be low enough to permit laser oscillation on a stable, narrow linewidth can be smaller than the Brillouin shift, so the reflectivity for the Stokes waves can be reduced. Fig. 4 shows the output power versus pump power for R1B ¼ 0.10 and 0.99. The SBS threshold is 346 W for R1B ¼ 0.10 and 214 W for R1B ¼ 0.99, so the SBS threshold increases as the input mirror reflectivity for the Stokes waves decreases. When 300
400
(a) Output power / W
Output power / W
(a) Dc = 20µm 300 out
Ps
200
out B0 out P BL
P
100
out
400
100
out
Ps
out B0 out P BL
P
200
400
500
150 out
Ps
400 600 Pump power / W
800
Fig. 3. Output laser and Stokes power for different core diameters (Dc). The explanation of curves is the same as Fig. 2.
out B0 out P BL
P
100
50
0
200
200 300 Pump power / W
(b) R1B = 0.99
(b) Dc = 30µm
0
100
200
400
0
0
500
Output power / W
Output power / W
600
200 300 Pump power / W
out B0 out P BL
P
0
100
Ps
200
0 0
R1B = 0.10
0
100 200 Pump power / W
300
Fig. 4. Output laser and Stokes power for different input mirror reflectivities at Stokes wavelength (R1B). The explanation of curves is the same as Fig. 2.
ARTICLE IN PRESS G. Liu, D. Liu / Optik 120 (2009) 24–28 2000
SBS threshold / W
the input mirror reflectivity for the Stokes waves is low, the backward output Stokes power is higher than the forward output Stokes power. However, when the input mirror reflectivity for the Stokes waves is high, the forward output Stokes power is higher than the backward output Stokes power. Fig. 5 shows the output power versus pump power for N ¼ 3.0 1025 and 6.94 1025 m3. The SBS threshold increases as the Yb3+ concentration decreases. However, the SBS threshold does not significantly vary with the Yb3+ concentration. The SBS threshold is 680 W for N ¼ 3.0 1025 m3 and 642 W for N ¼ 6.94 1025 m3. For fiber lasers, the spectral linewidth may be 0.01 nm or broader, depending on cavity design. For example, the spectral linewidth with grating reflector may be o0.1 nm, but with end-face polishing it may be higher. The dependence of the SBS threshold on the laser linewidth is shown in Fig. 6. In the calculation, the SBS gain linewidth is 50 MHz. When the laser linewidths are 1, 10, 20, 30 and 50 GHz, the SBS thresholds are 42, 402, 804, 1200 and 2000 W, respectively. It is obvious that the SBS threshold increases in direct ratio with the laser linewidth. The reason for this is that DusbDuB, gBEg0DuB/Dus. Once the laser linewidth becomes broader than the SBS gain linewidth, the SBS gain is reduced significantly. In other words, the SBS threshold increases significantly as the laser linewidth increases. Comparing Fig. 6 to Figs. 2–5, one finds that the SBS threshold depends very much on the laser linewidth.
27
1500 1000 500 0
0
10
20
30
40
50
Laser linewidth / GHz
Fig. 6. SBS threshold as a function of laser linewidth.
4. Conclusion The SBS of linear cavity high-power Yb3+-doped double-clad fiber lasers has been studied theoretically. By solving the steady-state rate equations with SBS, we have investigated the effects of cavity length, fiber core diameter, input mirror reflectivity at Stokes wavelength, Yb3+ concentration and laser linewidth on the SBS threshold power. Numerical results show that the SBS threshold power can be improved significantly by broadening the laser linewidth, and it can also be improved effectively by using large mode area fiber, shortening the cavity length and reducing the input mirror reflectivity at Stokes wavelength.
Acknowledgment This work was supported by the key technology R&D project of Hubei Province under Grant no. 2005AA101B10.
600
Output power / W
(a) N = 3X1025m-3 400
References
out
Ps
out B0 out P BL
P 200
0 500
550
600 650 Pump power / W
700
750
600
Output power / W
(b) N = 6.94X1025m-3 400
out
Ps
out B0 out P BL
P 200
0 500
550
600 Pump power / W
650
700
Fig. 5. Output laser and Stokes power for different Yb3+ concentrations (N). The explanation of curves is the same as Fig. 2.
[1] L. Zenteno, High-power double-clad fiber lasers, J. Lightwave Technol. 11 (1993) 1435–1446. [2] J. Limpert, A. Liem, H. Zellmer, et al., 500 W continuous wave fiber laser with excellent beam quality, Electron. Lett. 39 (2003) 645–647. [3] Y. Jeong, J.K. Sahu, S. Baek, et al., Cladding-pumped ytterbium-doped large-core fiber laser with 610 W of output power, Opt. Commun. 234 (2004) 315–319. [4] Y. Jeong, J. Sahu, D. Payne, et al., Ytterbium-doped large-core fiber laser with 1.36 kW continuous-wave output power, Opt. Express 12 (2004) 6088–6092. [5] A. Hideur, T. Chartier, M. Brunel, et al., Generation of high energy femtosecond pulses from a side-pumped Ybdoped double-clad fiber laser, Appl. Phys. Lett. 79 (2001) 3389–3391. [6] V.I. Karpov, E.M. Dianov, V.M. Paramonov, et al., Laser-diode-pumped phosphosilicate-fiber Raman laser with an output power of 1 W 1.48 mm, Opt. Lett. 24 (1999) 887–889. [7] Z.J. Chen, A.B. Grudinin, J. Porta, et al., Enhanced Q switching in double-clad fiber lasers, Opt. Lett. 23 (1998) 454–456.
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[8] A. Hideur, T. Chartier, C. Ozkul, et al., Dynamics and stabilization of a high power side-pumped Yb doped double-clad fiber laser, Opt. Commun. 186 (2000) 311–317. [9] K. Shimizu, T. Horiguchi, Y. Koyamada, Coherent lightwave amplification and stimulated Brillouin scattering in
an erbium-doped fiber amplifier, IEEE Photon Technol. Lett. 4 (1992) 564–567. [10] A. Liu, Novel SBS suppression scheme for high power fiber amplifiers, SPIE 6102 (2006) 6102R1–6102R9. [11] I. Kelson, A.A. Hardy, Strongly pumped fiber lasers, IEEE J. Quantum Electron. 34 (1998) 1570–1577.
Optik
Optics
Optik 120 (2009) 24–28 www.elsevier.de/ijleo
Numerical analysis of stimulated Brillouin scattering in high-power double-clad fiber lasers Guohua Liu, Deming Liu Department of Optoelectronics, Huazhong University of Science and Technology, Wuhan 430074, PR China Received 1 February 2007; accepted 10 June 2007
Abstract A theoretical analysis of stimulated Brillouin scattering (SBS) in linear cavity Yb3+-doped double-clad fiber lasers is presented by solving the steady-state rate equations with the SBS. The effects of cavity length, fiber core diameter, input mirror reflectivity at Stokes wavelength, Yb3+ concentration and laser linewidth on the SBS are discussed. Numerical results show that the SBS threshold power can be improved by shortening the cavity length, using large mode area fiber, reducing the input mirror reflectivity at Stokes wavelength, lowering the Yb3+ concentration and broadening the laser linewidth, and the influence of the laser linewidth on the SBS threshold power is more noticeable than other system parameters. r 2007 Published by Elsevier GmbH. Keywords: Fiber lasers; Stimulated Brillouin scattering; Threshold power
1. Introduction Yb3+-doped double-clad (YDDC) fiber lasers pumped by low-cost laser diodes have attracted considerable attention recently in commercial and military applications due to their high brightness, eminent efficiency, good compactness, excellent beam quality, efficient heat dissipation, etc., compared to traditional gas and solid-state lasers [1]. With the availability of high-power laser diode bars and cladpumping techniques, the output power of YDDC fiber lasers is able to reach hundreds of watts, even a kilowatt, in the continuous-wave (CW) regime [2–4]. However, the scalability of output powers can be limited by amplified spontaneous emission and nonlinear processes such as stimulated Brillouin scattering (SBS), stimulated Raman scattering and the optical Kerr effect. Although Corresponding author. Tel.: +86 27 87545578.
E-mail address: [email protected] (G. Liu). 0030-4026/$ - see front matter r 2007 Published by Elsevier GmbH. doi:10.1016/j.ijleo.2007.06.007
these nonlinear effects could be of interest for specific applications [5–7], they can also lead to some unexpected instabilities in the laser signal. In particular, the SBS is expected to be the origin of instabilities in highpower fiber lasers [8] or deformation of pulses in fiber amplifiers [9]. The aim of this paper is to investigate theoretically the dependence of the SBS on system parameters in YDDC fiber lasers. By solving a set of laser rate equations with the SBS, the SBS thresholds are obtained under different fiber conditions. The results and analysis presented facilitate the design and optimization of YDDC fiber lasers.
2. Numerical model The configuration of a typical YDDC fiber laser under CW end pump ðPþ p and Pp Þ is schematically shown in Fig. 1. The YDDC fiber has a length L and a
ARTICLE IN PRESS G. Liu, D. Liu / Optik 120 (2009) 24–28
R1
intrinsic SBS gain constant. Aeff is the effective area associated with the laser and Stokes waves. The boundary conditions at the input (Z ¼ 0) and output (Z ¼ L) mirrors are given by
R2 P sout
0
Z
L
P +p
25
P –p
Pþ s ð0Þ ¼ R1s Ps ð0Þ,
Fig. 1. Schematic illustration of end pumped fiber lasers.
Pþ B ð0Þ ¼ R1B PB ð0Þ,
uniform dopant concentration in the fiber core. R1 is the reflectivity of the input mirror and R2 is the output coupler reflectivity. This input mirror can be either a dichroic mirror or a fiber grating, and the output coupler can be either a cleaved fiber facet or a fiber grating. The steady-state rate equations to describe the pump, laser and Stokes waves in CW lasers [10,11] are as follows:
þ P s ðLÞ ¼ R2s Ps ðLÞ,
ð5Þ
þ P B ðLÞ ¼ R2B PB ðLÞ,
ð6Þ
where R1s and R1B are the input mirror reflectivities at the laser and Stokes wavelength, respectively. R2s and R2B are the output mirror reflectivities at the laser and Stokes wavelength, respectively. We must numerically solve the nonlinear Eqs. (1)–(4) with the specified conditions (Eqs. (5) and (6)) at two
h i Pþ p ðzÞ þ Pp ðzÞ sap Gp
þ Ps ðzÞ þ P s ðzÞ sas Gs þ N 2 ðzÞ hup A hus A i ¼h , þ þ N Pp ðzÞ þ Pp ðzÞ ðsap þ sep ÞGp 1 P ðzÞ þ Ps ðzÞ ðsas þ ses ÞGs þ þ s t hup A hus A ¼ Gp ðsap þ sep ÞN 2 ðzÞ sap N
P p ðzÞ
a p P p ðzÞ,
(2) dP s ðzÞ ¼ Gs ½ðsas þ ses ÞN 2 ðzÞ sas N P s ðzÞ dz gB as P P ðzÞP s ðzÞ s ðzÞ, Aeff B dP gB B ðzÞ ¼ aB P P ðzÞP B ðzÞ B ðzÞ, dz Aeff s
points. We apply a shooting method for such a boundary-value problem and choose the magnitudes ðP s ð0Þ; PB ð0ÞÞ of backward powers at Z ¼ 0 as unknown initial parameters. For the backward powers at Z ¼ 0, we calculate the forward powers 500
(a)
ð3Þ
L = 30m
400
(4)
where N is Yb3+ concentration in the core with a crosssection area A and N2(z) is the population density of the upper level. P p ðzÞ is the pump power, Ps ðzÞ is the laser power and PB ðzÞ is the first-order Brillouin Stokes power (7 correspond to forward and backward propagations, respectively). Since the threshold conditions of the first-order Stokes wave are dealt with in this work, the higher-order Stokes waves can hence be neglected. Gp and Gs are power filling factors. sep and sap are the emission and absorption cross sections of the pump light, whereas ses and sas are the emission and absorption cross sections of the laser. up is the pump frequency, us is the laser frequency and t is spontaneous lifetime. h is Planck’s constant. ap and as represent scattering loss coefficients of pump light and laser. Because the SBS is less than 0.1 nm away from the laser wavelength, aBEas. gB is the SBS gain. When Dus, the laser linewidth, is much larger than the SBS gain bandwidth DuB, gB ¼ g0DuB/Dus+DuB, where g0 is the
Output power / W
dz
out
Ps
300
out B0 out P BL
P
200 100 0 0
100
200
300
400
500
600
700
Pump power / W 300
(b) Output power / W
dP p ðzÞ
(1)
L = 50m out
Ps
200
out B0 out P BL
P
100
0 0
100
200 300 Pump power / W
400
500
Fig. 2. Output laser and Stokes power for different cavity lengths (L). dq, the output laser power; Pout B0 , the backward output Stokes power at Z ¼ 0; Pout BL , the forward output Stokes power at Z ¼ L.
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G. Liu, D. Liu / Optik 120 (2009) 24–28
þ ðPþ s ð0Þ; PB ð0ÞÞ at Z ¼ 0 through Eq. (5) and determine þ two components ðPþ s ðLÞ; PB ðLÞ; Ps ðLÞ; PB ðLÞÞ of laser and Stokes powers at Z ¼ L by numerically integrating Eqs. (1)–(4). In order to seek solutions that satisfy the boundary conditions (Eq. (6)) at Z ¼ L, we use the Marquardt method, which is an algorithm for least-squares estimation of nonlinear parameters. Once intracavity powers are determined, we can calculate the output power of the laser and Stokes waves.
3. Simulation results and discussion For the purposes of illustration, unless indicated otherwise, the parameters used in the simulation are lp ¼ 975 nm, ls ¼ 1082.8 nm, R1s ¼ 0.99, R2s ¼ 0.04, R1B ¼ 0.04, R2B ¼ 0.04, L ¼ 40 m, t ¼ 0.8 ms, sap ¼ 2.0 1024 m2, sep ¼ 2.0 1024 m2, Gp ¼ 0.0012, 27 2 sas ¼ 3.1 10 m , ses ¼ 4.2 1025 m2, Gs ¼ 0.85, ap ¼ 3.0 103 m1, as ¼ aB ¼ 5.0 103 m1, DuB ¼ 50 MHz, Dus ¼ 10 GHz, N ¼ 6.0 1025 m3, g0 ¼ 5.0 1011 m/W, fiber core diameter Dc ¼ 20 mm and NA ¼ 0.05. Equal forward and backward pump power is assumed in the numerical calculations. Fig. 2 shows the output power versus pump power for L ¼ 30 and 50 m. In each figure, there are three curves. Pout is the output laser power, Pout s B0 is the backward output Stokes power at Z ¼ 0, and Pout BL is the forward output Stokes power at Z ¼ L. SBS begins once the pump power reaches the Brillouin threshold. One finds
that the threshold increases as the cavity length decreases. It is 336 W for a 50-m-long cavity and 514 W for a 30-m-long cavity. Near the SBS threshold, the slope efficiency of the backward output Stokes power with respect to the total pump power is 57% for a 50-m-long cavity and 60% for a 30-m-long cavity. In the presence of the SBS, the laser efficiency decreases, and the considerable forward propagating laser is converted to the backward Stokes wave. Fig. 3 shows the output power versus pump power for Dc ¼ 20 and 30 mm. One finds that the SBS threshold increases as the fiber core diameter increases. It is 402 W for Dc ¼ 20 mm and 644 W for Dc ¼ 30 mm. The slope efficiencies of the backward output Stokes power are 60% and 58% for Fig. 3(a) and (b), respectively. Since the Stokes wave frequency is downshifted by 16 GHz (0.06 nm) from the laser frequency, when the input mirror is a Bragg grating, which reflects 99% of laser over a bandwidth of 0.1 nm near 1082.8 nm, the reflectivity of the input mirror is also 99% for the Stokes wave. Indeed, the spectral bandwidth of the Bragg grating can be low enough to permit laser oscillation on a stable, narrow linewidth can be smaller than the Brillouin shift, so the reflectivity for the Stokes waves can be reduced. Fig. 4 shows the output power versus pump power for R1B ¼ 0.10 and 0.99. The SBS threshold is 346 W for R1B ¼ 0.10 and 214 W for R1B ¼ 0.99, so the SBS threshold increases as the input mirror reflectivity for the Stokes waves decreases. When 300
400
(a) Output power / W
Output power / W
(a) Dc = 20µm 300 out
Ps
200
out B0 out P BL
P
100
out
400
100
out
Ps
out B0 out P BL
P
200
400
500
150 out
Ps
400 600 Pump power / W
800
Fig. 3. Output laser and Stokes power for different core diameters (Dc). The explanation of curves is the same as Fig. 2.
out B0 out P BL
P
100
50
0
200
200 300 Pump power / W
(b) R1B = 0.99
(b) Dc = 30µm
0
100
200
400
0
0
500
Output power / W
Output power / W
600
200 300 Pump power / W
out B0 out P BL
P
0
100
Ps
200
0 0
R1B = 0.10
0
100 200 Pump power / W
300
Fig. 4. Output laser and Stokes power for different input mirror reflectivities at Stokes wavelength (R1B). The explanation of curves is the same as Fig. 2.
ARTICLE IN PRESS G. Liu, D. Liu / Optik 120 (2009) 24–28 2000
SBS threshold / W
the input mirror reflectivity for the Stokes waves is low, the backward output Stokes power is higher than the forward output Stokes power. However, when the input mirror reflectivity for the Stokes waves is high, the forward output Stokes power is higher than the backward output Stokes power. Fig. 5 shows the output power versus pump power for N ¼ 3.0 1025 and 6.94 1025 m3. The SBS threshold increases as the Yb3+ concentration decreases. However, the SBS threshold does not significantly vary with the Yb3+ concentration. The SBS threshold is 680 W for N ¼ 3.0 1025 m3 and 642 W for N ¼ 6.94 1025 m3. For fiber lasers, the spectral linewidth may be 0.01 nm or broader, depending on cavity design. For example, the spectral linewidth with grating reflector may be o0.1 nm, but with end-face polishing it may be higher. The dependence of the SBS threshold on the laser linewidth is shown in Fig. 6. In the calculation, the SBS gain linewidth is 50 MHz. When the laser linewidths are 1, 10, 20, 30 and 50 GHz, the SBS thresholds are 42, 402, 804, 1200 and 2000 W, respectively. It is obvious that the SBS threshold increases in direct ratio with the laser linewidth. The reason for this is that DusbDuB, gBEg0DuB/Dus. Once the laser linewidth becomes broader than the SBS gain linewidth, the SBS gain is reduced significantly. In other words, the SBS threshold increases significantly as the laser linewidth increases. Comparing Fig. 6 to Figs. 2–5, one finds that the SBS threshold depends very much on the laser linewidth.
27
1500 1000 500 0
0
10
20
30
40
50
Laser linewidth / GHz
Fig. 6. SBS threshold as a function of laser linewidth.
4. Conclusion The SBS of linear cavity high-power Yb3+-doped double-clad fiber lasers has been studied theoretically. By solving the steady-state rate equations with SBS, we have investigated the effects of cavity length, fiber core diameter, input mirror reflectivity at Stokes wavelength, Yb3+ concentration and laser linewidth on the SBS threshold power. Numerical results show that the SBS threshold power can be improved significantly by broadening the laser linewidth, and it can also be improved effectively by using large mode area fiber, shortening the cavity length and reducing the input mirror reflectivity at Stokes wavelength.
Acknowledgment This work was supported by the key technology R&D project of Hubei Province under Grant no. 2005AA101B10.
600
Output power / W
(a) N = 3X1025m-3 400
References
out
Ps
out B0 out P BL
P 200
0 500
550
600 650 Pump power / W
700
750
600
Output power / W
(b) N = 6.94X1025m-3 400
out
Ps
out B0 out P BL
P 200
0 500
550
600 Pump power / W
650
700
Fig. 5. Output laser and Stokes power for different Yb3+ concentrations (N). The explanation of curves is the same as Fig. 2.
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