August 15, 2014 / Vol. 39, No. 16 / OPTICS LETTERS
4655
Prospects for diode-pumped alkali-atom-based hollow-core photonic-crystal fiber lasers Yoav Sintov,1,2,* Dror Malka,1 and Zeev Zalevsky1 1
Faculty of Engineering, Bar Ilan University, Ramat-Gan 52900, Israel 2 Soreq NRC, Applied Physics Division, Yavne 81800, Israel *Corresponding author: [email protected]
Received May 27, 2014; revised June 19, 2014; accepted June 23, 2014; posted June 24, 2014 (Doc. ID 212922); published August 5, 2014 By employing large hollow-core Kagome fiber in a double-clad configuration, the performance of a potentially rubidium vapor-based fiber laser is explored. The absorbed power and laser efficiency versus pump power are calculated utilizing a simple laser model. Our results show that a Kagome-based high-power fiber laser is feasible provided that the value of the collisional fine-structure mixing rate will be elevated by increasing the ambient temperature or by increasing the helium pressure. © 2014 Optical Society of America OCIS codes: (140.3510) Lasers, fiber; (060.4005) Microstructured fibers; (140.1340) Atomic gas lasers; (140.3480) Lasers, diode-pumped. http://dx.doi.org/10.1364/OL.39.004655
In the last 10 years there has been a marked advance in the field of a diode-pumped alkali lasers (DPALs). Since the introduction of the concept of DPAL by Krupke [1] in 2001, there has been an effort to exploit the beneficial characteristics of alkali lasers for generating high power at ∼800 nm. Even though one of the beneficial characteristics of DPALs is low quantum defect, the major limitation in achieving high power in DPALs is the waste heat generated in the gain medium. One of the ways to leverage the heat effect is to flow the gas through the gain media. Recently, a 1 kW DPAL laser was demonstrated with an optical efficiency of 48% [2]. Another way of removing the heat in a sealed fiberlaser system is by embedding the alkali gas in a hollowcore photonic-crystal fiber (HC-PCF), which was first suggested by Payne et al. [3]. By utilizing the beneficial characteristics of high surface to volume ratio of optical fibers [4], one may exploit the low quantum defect of DPAL to generate high power without circulating the gas in a compact fiber-laser design. The possibility of introducing rubidium (Rb) atoms in a sufficient density (∼1013 cm−3 ) into a standard hollowcore fiber was demonstrated by Slepkov et al. [5]. By using light-induced atomic desorption (LIAD) technique [6], Slepkov et al. demonstrated optical depths greater than 1200, which implies sufficient Rb density for laser operation, as noted in previously reported works [7]. In this Letter we report on the possibility of achieving laser operation in hollow-core Rb-vapor doped Kagome HC-PCF [8], with a core diameter of 63 μm, in a doubleclad potential version, with an inner-clad diameter of 125 μm. The proposed design and optimization in this Letter were not reported before, to our knowledge, and they are to be used as part of the experimental realization of high-power alkali fiber lasers. A simple laser model as suggested by Beach et al. [9] is used to demonstrate the laser feasibility. It is assumed that a linear laser-cavity configuration is used in which the output coupler reflectivity at 795 nm is Roc 5%. The cavity transmission per one pass through the fiber is Tl 80% and the back mirror has 100% reflectivity 0146-9592/14/164655-04$15.00/0
at 795 nm and 90% pump reflectivity at 780.2 nm (Rp 0.9). Under these assumptions the double-pass gain required for the laser operation is given by G exp2 gL
T 2l
1 ; · Roc
(1)
where G is the double pass gain, g is the gain per unit length, and L is the active fiber length. Assuming L 150 cm, one can get from Eq. (1) the gain per unit length at the Rb D1 line resonance gν ν0;D1 0.012 cm−1 . Using this assumption, one can derive the effective refractive index of the gain material at ν ν0;D1 to be [10] gνν0;D1 c ; nν ν0;D1 ≅ n0 1 − i 4n0 πν0
(2)
where ν0;D1 is the D1 line frequency, c 3 · 1010 cm∕s is the speed of light, and n0 ≈ 1 is the unexcited core refractive index. The gain is represented by the right imaginary term in Eq. (2). By applying the effective refractive index into a beam propagating method (BPM) simulation for 63 μm core Kagome HC-PCF [8], a numerical calculation for the gain versus the fiber’s d∕Λ structural property (d is a hole diameter and Λ is the spacing between the air holes) is shown in Fig. 1(a) for Λ 18 μm [8]. It can be seen that the maximum gain is achieved for d∕Λ 0.91, at which the core losses are minimal. From the gain optimized transverse mode profile, resulting from the BPM analysis [Fig. 1(b)], the overlap integral between the amplified signal and the doped core is ηmode 0.9. The Rb density (nRb ) in the HC-PCF is given by [11] 4040 133.32 · 10 1.193− T cm−3 ; nRb T (3) kB T where T is the temperature and kB is the Boltzmann constant. As mentioned in previous works [1,2,7,9], the addition of He as a buffer gas plays an important role in the laser © 2014 Optical Society of America
4656
OPTICS LETTERS / Vol. 39, No. 16 / August 15, 2014
performance in terms of D1 and D2 linewidth broadening, as well as in collisional mixing of the Rb 52 P3∕2 and 52 P1∕2 fine structure levels. In addition, it provides good heat conduction from the excited Rb atoms to the fiber’s core walls. The He collisional mixing cross section of Rb fine structure levels is provided in the Beach model [9] with a binary approximation. However, recently it was proven [12] that the three-body collisional mixing rate may be of the same value as the binary rate, especially at high He densities and high operating temperatures. The addition of small hydrocarbon molecules like Et expedites the collisional mixing and compensates for the relatively low He collisional mixing in low He pressure setups. The two-direction collisional mixing rate dependence on temperature and He and Et density are [12,13] γ 2 P3∕2 →2 P1∕2 T σ He;2 P3∕2 →2 P1∕2
T 294
σ Et;2 P3∕2 →2 P1∕2 vEt−rel nEt ; ΔE ; γ 2 P3∕2 ←2 P1∕2 T γ 2 P3∕2 →2 P1∕2 · 2 exp − kB T
(6)
where Acore and Aclad are the core area and the innerclad area of the double-clad Kagome fiber. σ He−broadened D2 is the He broadened atomic D2 cross section. λD2 and ΔλFWHM are the D2 centerline wavelength and the D2 He-broadened full width at half-maximum (FWHM) linewidth, respectively. In addition, the D1 Rb line effective wavelengthdependent cross section includes the calculated mode overlap integral with the Rb-doped core, ηmode 0.9: σ D1;eff λ
σ He−broadened D1 h i · ηmode ; D1 1 2λ−λ FWHM Δλ
(7)
D1
(4)
where nHe and nEt are the He and Et densities, respectively. σ X; 2 P3∕2 → 2 P1∕2 is the room temperature binary collisional cross section with X species (X stands for Et or He), and vX−rel is the Rb-X relative velocity, given by s 8kB T 1 1 cm ; · 100 π M X M Rb s
σ He−broadened A D2 h i · core ; 2λ−λD2 1 ΔλFWHM Aclad D2
1.33
vHe−rel nHe T 1.71 1.48 · 10−32 · n2He 294
vX−rel
σ D2;eff λ
(5)
where Mx is the unit mass of the X species. Here, we assume that 300 Torr Et and 7600 Torr (10 Atm) He are introduced into the Kagome HC-PCF fiber. The fiber is held at a temperature of 150°C, providing an evenly distributed Rb density of nRb 8.4 · 1013 cm−3 . The effective wavelength-dependent D2 Rb absorption cross section in double-clad fiber is provided by
Fig. 1. (a) Gain versus the Kagome fiber’s d∕Λ structural property (d is hole diameter and Λ is the spacing between the air holes) for Λ 18 μm [8]. (b) BPM simulated mode profile at the output of the fiber laser. The dashed line represents the hollow core boundary.
is the He broadened atomic D1 cross where σ He−broadened D1 section. λD1 and ΔλFWHM are the D1 centerline waveD1 length and the He-broadened (FWHM) linewidth, respectively. These cross sections are homogeneously broadened according to the broadening values provided by Romalis et al. [14] and as described in [9]. The geometrical and spectroscopic parameters used in the simulation are given in Table 1. The laser simulation relies on the same rate equations provided in [9], taking into consideration the temperature-dependent binary and three-body bidirectional collisional rate in Eq. (4) and the effective cross sections in Eqs. (6) and (7). The transit time broadening due to the narrow core is considered negligible [5]: dn1 1 1 −Γp ΓL n2 Q21 n3 Q31 ; dt τD1 τD2 dn2 −ΓL γ 2 P3∕2 →2 P1∕2 T n3 − n2 dt ΔE 1 − 1 n2 − n2 Q21 ; − 2 exp − kB T τD1 dn3 ΓP − γ 2 P3∕2 →2 P1∕2 T n3 − n2 dt ΔE 1 − 1 n2 − n3 Q31 ; − 2 exp − kB T τD2 Z dP p λ λ η Γp del dλ dλ hc VL n × 1 − exp − n1 − 3 σ D2;eff λL ; 2 n3 σ λL ; × 1 Rp exp − n1 − 2 D2;eff P λ ROC fexpn2 − n1 σ D1;eff λL − 1g; ΓL L D1 V L hc 1 − ROC × f1 T 2l expn2 − n1 σ D1;eff λLg; nRb n1 n2 n3 ; He Q31 Q21 nEt vEt−rel σ Et quenching nHe vHe−rel σ quenching ; (8)
August 15, 2014 / Vol. 39, No. 16 / OPTICS LETTERS
4657
Table 1. Model Parameters Parameter Description
Parameter Symbol
Parameter Value
Roc L d dc VL Tl ηmode Rp T
0.05 150 cm 63 · 10−4 cm 125 · 10−4 cm 3.12 · 10−3 cm3 0.8 0.9 0.9 150°C
Laser Cavity Parameters Output coupler reflectivity Fiber length Hollow core diameter Pump clad diameter Hollow core volume Cavity transmission per one pass Mode overlap with hollow core Pump reflection Ambient temperature Rubidium Parameters D1, D2 transition wavelength D1, D2 radiative lifetime He-broadened D1, D2 cross section at 1 Atm He He-broadened D1, D2 FWHM linewidth at 1 Atm He He collisional mixing cross section at room temperature Et collisional mixing cross section at room temperature He, Et 52 P1∕2 and 52 P3∕2 quenching cross section
λD1 , τD1 , σ He-broadened , D1 ΔλFWHM , D1
λD2 τD2 σ He-broadened D2 ΔλFWHM D2
795 nm, 780.2 nm [11] 27.7 ns, 26.24 ns [11] 2.84 · 10−13 cm2 , 5.26 · 10−13 cm2 [14] 0.3 nm, 0.29 nm [9]
σ He , 2 P3∕2 → 2 P1∕2
1 · 10−17 cm2 [9]
σ Et , 2 P3∕2 → 2 P1∕2
7.7 · 10−15 cm2 [9]
Et σ He quenching , σ quenching
1 · 10−22 cm2 [15], 3.3 · 10−18 cm2 [16]
4 He,
Buffer Gas Et(C2 H6 ) pressure
PHe , PEt
7600 Torr, 300 Torr
Pump Parameters Pump power, Pump delivery efficiency Pump center wavelength, Pump linewidth FWHM
Pp , ηdel λp , Δλp
1.250 W, 0.9 780.2 nm, 1.5 nm
where n1 , n2 , and n3 are the population density of the 52 S1∕2 , 52 P1∕2 , and 52 P3∕2 Rb energy levels, respectively. Γp and ΓL are the pump absorption and laser emission induced transition rates. τD1 and τD2 are the 52 P1∕2 and 52 P3∕2 levels lifetime, respectively. Q21 and Q31 are the 52 P1∕2 and 52 P3∕2 quenching rates. σ Xquenching is the quenching cross section of the X species (X stands for Et or He). VL and L are the core volume and fiber length, respectively. ΔE 237 cm−1 is the energy gap between 52 P3∕2 and 52 P1∕2 . Rp and ROC are the pump reflection and output coupler reflection, respectively. PL is the output laser power. Pp λ is the wavelength-dependent diode laser pump power given by 2λ − λp 2 2 P p λ P ∕=1 ; Δλp πΔλp p
through the fiber are depicted for 60 W pump power. Even though the pump spectral profile is much wider than the absorption line, the high wing absorption, aside from the line center and the homogeneous broadening nature of the Rb, along with the relatively long interaction length in the fiber, allow for ∼60% of the power to be absorbed in the relatively low pump-power range up to ∼60 W.
(9)
where P p , λp , and Δλp are the pump power, pump wavelength, and the pump linewidth, respectively. ηdel is the pump delivery efficiency. Steady-state solutions of n1 , n2 , n3 , Γp , and ΓL can be found by utilizing the semi-numerical algorithm offered in [9] by setting all time derivatives in Eq. (8) equal to zero. From these one can find the laser-power dependence on pump power and the laser parameters given in Table 1. Figure 2 shows the output power, absorbed power in the fiber, and the laser efficiency relative to the launched pump power. In addition, the normalized spectral profiles of the pump power after the first and second pass
Fig. 2. Absorbed pump power, output power, and laser efficiency (relative to launched power) versus the diode pump power. The pump spectral profiles after a single pass and double pass through the fiber are shown in the small graph for 60 W pump power.
4658
OPTICS LETTERS / Vol. 39, No. 16 / August 15, 2014
Fig. 3. Longitudinally averaged population densities n1 , n2 , n3 (energy levels 52 S1∕2 , 52 P1∕2 , and 52 P3∕2 , respectively) versus pump power. n1 depletion occurs at high pump power due to the low n3 − n2 collisional rate.
From Fig. 2 it can be seen that laser-output power threshold is quite low compared with equivalent freespace lasers. This beneficial feature of fiber lasers [4] is known for other types of three-level lasers as well. In addition, saturation occurs at pump-power levels above ∼60 W. From the absorbed pump response to the pump power, it is clear that the main reason for this is the relatively low collisional mixing rate γ 2 P3∕2 →2 P1∕2 , which under steady-state conditions requires a highly nonequilibrium n2 –n3 distribution. This nonequilibrium results in a depletion of the ground state n1 . This phenomenon is unique for low-aperture fiber lasers in which the power flux and saturation effects are high, relative to the large aperture free-space lasers. This phenomenon can be clearly seen in Fig. 3, which shows the longitudinally averaged Rb population density at all energy levels versus pump power. It can be seen that at low pump power n3 − n2 are in a close equilibrium state, due to the relatively low pump transition and stimulated induced transition rates, Γp and ΓL . At high pump power Γp and ΓL become high enough to create a nonequilibrium n3 − n2 distribution, due to the relatively low mixing rate constant, resulting in reduced pump absorption due to the ground state depletion. The ground state depletion can be overcome by increasing the temperature and thereby increasing the Rb density. Figure 4 shows the increase in absorbed pump power and the output power versus temperature rise and consequently Rb density rise, at 200 W pump power.
Fig. 4. Output power and absorbed pump power versus ambient temperature at 200 W pump power. The Rb density rise with ambient temperature is shown to improve the laser efficiency.
However, as can be seen in Fig. 4, the laser efficiency, compared with the absorbed pump is mildly reduced with temperature rise. This is due to a small reduction in γ 2 P3∕2 →2 P1∕2 , threshold rise due to the elevated Rb population and mildly thermally depopulated upper laser-level (52 P1∕2 ). In conclusion we have shown that an alkali-based fiber laser, where Rb vapor is introduced into a core of a potentially double-clad Kagome fiber, shows beneficial performance as an efficient high-power, high-beam quality source at ∼800 nm. Main limitation is the depletion of the 52 S1∕2 population, due to the high pump-power flux, which reduces the absorbed power fraction and consequently the laser efficiency. The main bottleneck is the collisional mixing constant γ 2 P3∕2 →2 P1∕2 , which needs to be elevated in order to increase the collisional mixing rate. Increasing the ambient temperature is a possible solution. Increasing the He density is another option, but this poses a practical challenge on the implementation of such a laser. It is still required to prove the possibility to achieve the required alkali atom’s density in a hollow core of a Kagome fiber. The possibility to produce a double-clad Kagome fiber needs to be checked as well. The implementation of these two challenging tasks will pave the way to a high-power fiber-laser source at 800 nm. References 1. W. F. Krupke, “Diode pumped alkali laser,” U.S. patent No. 6,643,311 (November 4, 2003). 2. A. V. Bogachev, S. G. Garanin, A. M. Dudov, V. A. Yeroshenko, S. M. Kulikov, G. T. Mikaelian, V. A. Panarin, V. O. Pautov, A. V. Rus, and S. A. Sukharev, Quantum Electron. 42, 95 (2012). 3. S. A. Payne, R. J. Beach, J. w. Dawson, and W. F. Krupke, “Diode pumped alkali vapor fiber laser,” U.S. patent No. 7,082,148B2 (July 25, 2006). 4. D. J. Richardson, J. Nilsson, and W. A. Clarkson, J. Opt. Soc. Am. B 27, B63 (2010). 5. A. D. Slepkov, A. R. Bhagwat, V. Venkataraman, P. Londero, and A. L. Gaeta, Opt. Express 16, 18976 (2008). 6. S. Ghosh, A. R. Bhagwat, C. K. Renshaw, S. Goh, A. L. Gaeta, and B. J. Kirby, Phys. Rev. Lett. 97, 023603 (2006). 7. W. F. Krupke, R. J. Beach, V. K. Kanz, and S. A. Payne, Opt. Lett. 28, 2336 (2003). 8. Y. Y. Wang, N. V. Wheeler, F. Couny, P. J. Roberts, and F. Benabid, Opt. Lett. 36, 669 (2011). 9. R. J. Beach, W. F. Krupke, V. K. Kanz, S. A. Payne, M. A. Dubinskii, and L. D. Merkle, J. Opt. Soc. Am. B 21, 2151 (2004). 10. C. C. Davis, Lasers and Electro-Optics (Cambridge, 1996), Chap. II. 11. D. A. Steck, “Rubidium 85 D Line Data,” available online at http://steck.us/alkalidata/rubidium85numbers.pdf. 12. M. A. Gearba, J. F. Sell, B. M. Patterson, R. Lloyd, J. Plyler, and R. J. Knize, Opt. Lett. 37, 1637 (2012). 13. G. D. Hager and G. P. Perram, Appl. Phys. B 101, 45 (2010). 14. M. V. Romalis, E. Miron, and G. D. Gates, Phys. Rev. A 56, 4569 (1997). 15. E. Speller, B. Staudenmayer, and V. Kempter, Z. Phys. A 291, 311 (1979). 16. N. D. Zameroski, W. Rudolph, G. D. Hager, and D. A. Hostutler, J. Phys. B 42, 245401 (2009).
4655
Prospects for diode-pumped alkali-atom-based hollow-core photonic-crystal fiber lasers Yoav Sintov,1,2,* Dror Malka,1 and Zeev Zalevsky1 1
Faculty of Engineering, Bar Ilan University, Ramat-Gan 52900, Israel 2 Soreq NRC, Applied Physics Division, Yavne 81800, Israel *Corresponding author: [email protected]
Received May 27, 2014; revised June 19, 2014; accepted June 23, 2014; posted June 24, 2014 (Doc. ID 212922); published August 5, 2014 By employing large hollow-core Kagome fiber in a double-clad configuration, the performance of a potentially rubidium vapor-based fiber laser is explored. The absorbed power and laser efficiency versus pump power are calculated utilizing a simple laser model. Our results show that a Kagome-based high-power fiber laser is feasible provided that the value of the collisional fine-structure mixing rate will be elevated by increasing the ambient temperature or by increasing the helium pressure. © 2014 Optical Society of America OCIS codes: (140.3510) Lasers, fiber; (060.4005) Microstructured fibers; (140.1340) Atomic gas lasers; (140.3480) Lasers, diode-pumped. http://dx.doi.org/10.1364/OL.39.004655
In the last 10 years there has been a marked advance in the field of a diode-pumped alkali lasers (DPALs). Since the introduction of the concept of DPAL by Krupke [1] in 2001, there has been an effort to exploit the beneficial characteristics of alkali lasers for generating high power at ∼800 nm. Even though one of the beneficial characteristics of DPALs is low quantum defect, the major limitation in achieving high power in DPALs is the waste heat generated in the gain medium. One of the ways to leverage the heat effect is to flow the gas through the gain media. Recently, a 1 kW DPAL laser was demonstrated with an optical efficiency of 48% [2]. Another way of removing the heat in a sealed fiberlaser system is by embedding the alkali gas in a hollowcore photonic-crystal fiber (HC-PCF), which was first suggested by Payne et al. [3]. By utilizing the beneficial characteristics of high surface to volume ratio of optical fibers [4], one may exploit the low quantum defect of DPAL to generate high power without circulating the gas in a compact fiber-laser design. The possibility of introducing rubidium (Rb) atoms in a sufficient density (∼1013 cm−3 ) into a standard hollowcore fiber was demonstrated by Slepkov et al. [5]. By using light-induced atomic desorption (LIAD) technique [6], Slepkov et al. demonstrated optical depths greater than 1200, which implies sufficient Rb density for laser operation, as noted in previously reported works [7]. In this Letter we report on the possibility of achieving laser operation in hollow-core Rb-vapor doped Kagome HC-PCF [8], with a core diameter of 63 μm, in a doubleclad potential version, with an inner-clad diameter of 125 μm. The proposed design and optimization in this Letter were not reported before, to our knowledge, and they are to be used as part of the experimental realization of high-power alkali fiber lasers. A simple laser model as suggested by Beach et al. [9] is used to demonstrate the laser feasibility. It is assumed that a linear laser-cavity configuration is used in which the output coupler reflectivity at 795 nm is Roc 5%. The cavity transmission per one pass through the fiber is Tl 80% and the back mirror has 100% reflectivity 0146-9592/14/164655-04$15.00/0
at 795 nm and 90% pump reflectivity at 780.2 nm (Rp 0.9). Under these assumptions the double-pass gain required for the laser operation is given by G exp2 gL
T 2l
1 ; · Roc
(1)
where G is the double pass gain, g is the gain per unit length, and L is the active fiber length. Assuming L 150 cm, one can get from Eq. (1) the gain per unit length at the Rb D1 line resonance gν ν0;D1 0.012 cm−1 . Using this assumption, one can derive the effective refractive index of the gain material at ν ν0;D1 to be [10] gνν0;D1 c ; nν ν0;D1 ≅ n0 1 − i 4n0 πν0
(2)
where ν0;D1 is the D1 line frequency, c 3 · 1010 cm∕s is the speed of light, and n0 ≈ 1 is the unexcited core refractive index. The gain is represented by the right imaginary term in Eq. (2). By applying the effective refractive index into a beam propagating method (BPM) simulation for 63 μm core Kagome HC-PCF [8], a numerical calculation for the gain versus the fiber’s d∕Λ structural property (d is a hole diameter and Λ is the spacing between the air holes) is shown in Fig. 1(a) for Λ 18 μm [8]. It can be seen that the maximum gain is achieved for d∕Λ 0.91, at which the core losses are minimal. From the gain optimized transverse mode profile, resulting from the BPM analysis [Fig. 1(b)], the overlap integral between the amplified signal and the doped core is ηmode 0.9. The Rb density (nRb ) in the HC-PCF is given by [11] 4040 133.32 · 10 1.193− T cm−3 ; nRb T (3) kB T where T is the temperature and kB is the Boltzmann constant. As mentioned in previous works [1,2,7,9], the addition of He as a buffer gas plays an important role in the laser © 2014 Optical Society of America
4656
OPTICS LETTERS / Vol. 39, No. 16 / August 15, 2014
performance in terms of D1 and D2 linewidth broadening, as well as in collisional mixing of the Rb 52 P3∕2 and 52 P1∕2 fine structure levels. In addition, it provides good heat conduction from the excited Rb atoms to the fiber’s core walls. The He collisional mixing cross section of Rb fine structure levels is provided in the Beach model [9] with a binary approximation. However, recently it was proven [12] that the three-body collisional mixing rate may be of the same value as the binary rate, especially at high He densities and high operating temperatures. The addition of small hydrocarbon molecules like Et expedites the collisional mixing and compensates for the relatively low He collisional mixing in low He pressure setups. The two-direction collisional mixing rate dependence on temperature and He and Et density are [12,13] γ 2 P3∕2 →2 P1∕2 T σ He;2 P3∕2 →2 P1∕2
T 294
σ Et;2 P3∕2 →2 P1∕2 vEt−rel nEt ; ΔE ; γ 2 P3∕2 ←2 P1∕2 T γ 2 P3∕2 →2 P1∕2 · 2 exp − kB T
(6)
where Acore and Aclad are the core area and the innerclad area of the double-clad Kagome fiber. σ He−broadened D2 is the He broadened atomic D2 cross section. λD2 and ΔλFWHM are the D2 centerline wavelength and the D2 He-broadened full width at half-maximum (FWHM) linewidth, respectively. In addition, the D1 Rb line effective wavelengthdependent cross section includes the calculated mode overlap integral with the Rb-doped core, ηmode 0.9: σ D1;eff λ
σ He−broadened D1 h i · ηmode ; D1 1 2λ−λ FWHM Δλ
(7)
D1
(4)
where nHe and nEt are the He and Et densities, respectively. σ X; 2 P3∕2 → 2 P1∕2 is the room temperature binary collisional cross section with X species (X stands for Et or He), and vX−rel is the Rb-X relative velocity, given by s 8kB T 1 1 cm ; · 100 π M X M Rb s
σ He−broadened A D2 h i · core ; 2λ−λD2 1 ΔλFWHM Aclad D2
1.33
vHe−rel nHe T 1.71 1.48 · 10−32 · n2He 294
vX−rel
σ D2;eff λ
(5)
where Mx is the unit mass of the X species. Here, we assume that 300 Torr Et and 7600 Torr (10 Atm) He are introduced into the Kagome HC-PCF fiber. The fiber is held at a temperature of 150°C, providing an evenly distributed Rb density of nRb 8.4 · 1013 cm−3 . The effective wavelength-dependent D2 Rb absorption cross section in double-clad fiber is provided by
Fig. 1. (a) Gain versus the Kagome fiber’s d∕Λ structural property (d is hole diameter and Λ is the spacing between the air holes) for Λ 18 μm [8]. (b) BPM simulated mode profile at the output of the fiber laser. The dashed line represents the hollow core boundary.
is the He broadened atomic D1 cross where σ He−broadened D1 section. λD1 and ΔλFWHM are the D1 centerline waveD1 length and the He-broadened (FWHM) linewidth, respectively. These cross sections are homogeneously broadened according to the broadening values provided by Romalis et al. [14] and as described in [9]. The geometrical and spectroscopic parameters used in the simulation are given in Table 1. The laser simulation relies on the same rate equations provided in [9], taking into consideration the temperature-dependent binary and three-body bidirectional collisional rate in Eq. (4) and the effective cross sections in Eqs. (6) and (7). The transit time broadening due to the narrow core is considered negligible [5]: dn1 1 1 −Γp ΓL n2 Q21 n3 Q31 ; dt τD1 τD2 dn2 −ΓL γ 2 P3∕2 →2 P1∕2 T n3 − n2 dt ΔE 1 − 1 n2 − n2 Q21 ; − 2 exp − kB T τD1 dn3 ΓP − γ 2 P3∕2 →2 P1∕2 T n3 − n2 dt ΔE 1 − 1 n2 − n3 Q31 ; − 2 exp − kB T τD2 Z dP p λ λ η Γp del dλ dλ hc VL n × 1 − exp − n1 − 3 σ D2;eff λL ; 2 n3 σ λL ; × 1 Rp exp − n1 − 2 D2;eff P λ ROC fexpn2 − n1 σ D1;eff λL − 1g; ΓL L D1 V L hc 1 − ROC × f1 T 2l expn2 − n1 σ D1;eff λLg; nRb n1 n2 n3 ; He Q31 Q21 nEt vEt−rel σ Et quenching nHe vHe−rel σ quenching ; (8)
August 15, 2014 / Vol. 39, No. 16 / OPTICS LETTERS
4657
Table 1. Model Parameters Parameter Description
Parameter Symbol
Parameter Value
Roc L d dc VL Tl ηmode Rp T
0.05 150 cm 63 · 10−4 cm 125 · 10−4 cm 3.12 · 10−3 cm3 0.8 0.9 0.9 150°C
Laser Cavity Parameters Output coupler reflectivity Fiber length Hollow core diameter Pump clad diameter Hollow core volume Cavity transmission per one pass Mode overlap with hollow core Pump reflection Ambient temperature Rubidium Parameters D1, D2 transition wavelength D1, D2 radiative lifetime He-broadened D1, D2 cross section at 1 Atm He He-broadened D1, D2 FWHM linewidth at 1 Atm He He collisional mixing cross section at room temperature Et collisional mixing cross section at room temperature He, Et 52 P1∕2 and 52 P3∕2 quenching cross section
λD1 , τD1 , σ He-broadened , D1 ΔλFWHM , D1
λD2 τD2 σ He-broadened D2 ΔλFWHM D2
795 nm, 780.2 nm [11] 27.7 ns, 26.24 ns [11] 2.84 · 10−13 cm2 , 5.26 · 10−13 cm2 [14] 0.3 nm, 0.29 nm [9]
σ He , 2 P3∕2 → 2 P1∕2
1 · 10−17 cm2 [9]
σ Et , 2 P3∕2 → 2 P1∕2
7.7 · 10−15 cm2 [9]
Et σ He quenching , σ quenching
1 · 10−22 cm2 [15], 3.3 · 10−18 cm2 [16]
4 He,
Buffer Gas Et(C2 H6 ) pressure
PHe , PEt
7600 Torr, 300 Torr
Pump Parameters Pump power, Pump delivery efficiency Pump center wavelength, Pump linewidth FWHM
Pp , ηdel λp , Δλp
1.250 W, 0.9 780.2 nm, 1.5 nm
where n1 , n2 , and n3 are the population density of the 52 S1∕2 , 52 P1∕2 , and 52 P3∕2 Rb energy levels, respectively. Γp and ΓL are the pump absorption and laser emission induced transition rates. τD1 and τD2 are the 52 P1∕2 and 52 P3∕2 levels lifetime, respectively. Q21 and Q31 are the 52 P1∕2 and 52 P3∕2 quenching rates. σ Xquenching is the quenching cross section of the X species (X stands for Et or He). VL and L are the core volume and fiber length, respectively. ΔE 237 cm−1 is the energy gap between 52 P3∕2 and 52 P1∕2 . Rp and ROC are the pump reflection and output coupler reflection, respectively. PL is the output laser power. Pp λ is the wavelength-dependent diode laser pump power given by 2λ − λp 2 2 P p λ P ∕=1 ; Δλp πΔλp p
through the fiber are depicted for 60 W pump power. Even though the pump spectral profile is much wider than the absorption line, the high wing absorption, aside from the line center and the homogeneous broadening nature of the Rb, along with the relatively long interaction length in the fiber, allow for ∼60% of the power to be absorbed in the relatively low pump-power range up to ∼60 W.
(9)
where P p , λp , and Δλp are the pump power, pump wavelength, and the pump linewidth, respectively. ηdel is the pump delivery efficiency. Steady-state solutions of n1 , n2 , n3 , Γp , and ΓL can be found by utilizing the semi-numerical algorithm offered in [9] by setting all time derivatives in Eq. (8) equal to zero. From these one can find the laser-power dependence on pump power and the laser parameters given in Table 1. Figure 2 shows the output power, absorbed power in the fiber, and the laser efficiency relative to the launched pump power. In addition, the normalized spectral profiles of the pump power after the first and second pass
Fig. 2. Absorbed pump power, output power, and laser efficiency (relative to launched power) versus the diode pump power. The pump spectral profiles after a single pass and double pass through the fiber are shown in the small graph for 60 W pump power.
4658
OPTICS LETTERS / Vol. 39, No. 16 / August 15, 2014
Fig. 3. Longitudinally averaged population densities n1 , n2 , n3 (energy levels 52 S1∕2 , 52 P1∕2 , and 52 P3∕2 , respectively) versus pump power. n1 depletion occurs at high pump power due to the low n3 − n2 collisional rate.
From Fig. 2 it can be seen that laser-output power threshold is quite low compared with equivalent freespace lasers. This beneficial feature of fiber lasers [4] is known for other types of three-level lasers as well. In addition, saturation occurs at pump-power levels above ∼60 W. From the absorbed pump response to the pump power, it is clear that the main reason for this is the relatively low collisional mixing rate γ 2 P3∕2 →2 P1∕2 , which under steady-state conditions requires a highly nonequilibrium n2 –n3 distribution. This nonequilibrium results in a depletion of the ground state n1 . This phenomenon is unique for low-aperture fiber lasers in which the power flux and saturation effects are high, relative to the large aperture free-space lasers. This phenomenon can be clearly seen in Fig. 3, which shows the longitudinally averaged Rb population density at all energy levels versus pump power. It can be seen that at low pump power n3 − n2 are in a close equilibrium state, due to the relatively low pump transition and stimulated induced transition rates, Γp and ΓL . At high pump power Γp and ΓL become high enough to create a nonequilibrium n3 − n2 distribution, due to the relatively low mixing rate constant, resulting in reduced pump absorption due to the ground state depletion. The ground state depletion can be overcome by increasing the temperature and thereby increasing the Rb density. Figure 4 shows the increase in absorbed pump power and the output power versus temperature rise and consequently Rb density rise, at 200 W pump power.
Fig. 4. Output power and absorbed pump power versus ambient temperature at 200 W pump power. The Rb density rise with ambient temperature is shown to improve the laser efficiency.
However, as can be seen in Fig. 4, the laser efficiency, compared with the absorbed pump is mildly reduced with temperature rise. This is due to a small reduction in γ 2 P3∕2 →2 P1∕2 , threshold rise due to the elevated Rb population and mildly thermally depopulated upper laser-level (52 P1∕2 ). In conclusion we have shown that an alkali-based fiber laser, where Rb vapor is introduced into a core of a potentially double-clad Kagome fiber, shows beneficial performance as an efficient high-power, high-beam quality source at ∼800 nm. Main limitation is the depletion of the 52 S1∕2 population, due to the high pump-power flux, which reduces the absorbed power fraction and consequently the laser efficiency. The main bottleneck is the collisional mixing constant γ 2 P3∕2 →2 P1∕2 , which needs to be elevated in order to increase the collisional mixing rate. Increasing the ambient temperature is a possible solution. Increasing the He density is another option, but this poses a practical challenge on the implementation of such a laser. It is still required to prove the possibility to achieve the required alkali atom’s density in a hollow core of a Kagome fiber. The possibility to produce a double-clad Kagome fiber needs to be checked as well. The implementation of these two challenging tasks will pave the way to a high-power fiber-laser source at 800 nm. References 1. W. F. Krupke, “Diode pumped alkali laser,” U.S. patent No. 6,643,311 (November 4, 2003). 2. A. V. Bogachev, S. G. Garanin, A. M. Dudov, V. A. Yeroshenko, S. M. Kulikov, G. T. Mikaelian, V. A. Panarin, V. O. Pautov, A. V. Rus, and S. A. Sukharev, Quantum Electron. 42, 95 (2012). 3. S. A. Payne, R. J. Beach, J. w. Dawson, and W. F. Krupke, “Diode pumped alkali vapor fiber laser,” U.S. patent No. 7,082,148B2 (July 25, 2006). 4. D. J. Richardson, J. Nilsson, and W. A. Clarkson, J. Opt. Soc. Am. B 27, B63 (2010). 5. A. D. Slepkov, A. R. Bhagwat, V. Venkataraman, P. Londero, and A. L. Gaeta, Opt. Express 16, 18976 (2008). 6. S. Ghosh, A. R. Bhagwat, C. K. Renshaw, S. Goh, A. L. Gaeta, and B. J. Kirby, Phys. Rev. Lett. 97, 023603 (2006). 7. W. F. Krupke, R. J. Beach, V. K. Kanz, and S. A. Payne, Opt. Lett. 28, 2336 (2003). 8. Y. Y. Wang, N. V. Wheeler, F. Couny, P. J. Roberts, and F. Benabid, Opt. Lett. 36, 669 (2011). 9. R. J. Beach, W. F. Krupke, V. K. Kanz, S. A. Payne, M. A. Dubinskii, and L. D. Merkle, J. Opt. Soc. Am. B 21, 2151 (2004). 10. C. C. Davis, Lasers and Electro-Optics (Cambridge, 1996), Chap. II. 11. D. A. Steck, “Rubidium 85 D Line Data,” available online at http://steck.us/alkalidata/rubidium85numbers.pdf. 12. M. A. Gearba, J. F. Sell, B. M. Patterson, R. Lloyd, J. Plyler, and R. J. Knize, Opt. Lett. 37, 1637 (2012). 13. G. D. Hager and G. P. Perram, Appl. Phys. B 101, 45 (2010). 14. M. V. Romalis, E. Miron, and G. D. Gates, Phys. Rev. A 56, 4569 (1997). 15. E. Speller, B. Staudenmayer, and V. Kempter, Z. Phys. A 291, 311 (1979). 16. N. D. Zameroski, W. Rudolph, G. D. Hager, and D. A. Hostutler, J. Phys. B 42, 245401 (2009).
