端面抽运准二能级双包层光纤激光器的理论探讨
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摘要
与传统固体激光器相比，掺稀土光纤激光器具有其独特的优点，即具有结构紧凑、体积小巧、阈值低、光束质量好、转换效率高以及表面积／体积比率大利于散热等特性。双包层光纤激光器的问世，更加促使了高功率光纤激光器的飞速发展，并在远程光纤通信、传感、工业加工、医疗、国防和军事等领域获得了日益广泛的应用。对双包层光纤激光器的进一步研究，成为了当前相关领域研究的热点，并在实验和理论上取得了众多丰富的成果。在对光纤激光器的理论研究中，人们通常采用速率方程理论进行分析。研究表明，速率方程组正确、有效，为阐述激光器的工作原理、特点以及研究和设计激光器提供了理论指导。
由于速率方程组的解析解相对数值解具有一些明显的优点，人们便尝试在各种合理近似下对速率方程组进行解析分析和讨论。然而，由于速率方程组的非线性特性，很难对之严格求解。不过，在忽略信号光损耗系数的近似情况下，人们已得到了满足下列近似情况之一的解析解：(a)抽运光损耗系数可忽略的单包层光纤激光器件；(b)上能级粒子与抽运光相互作用的再发射可以忽略的较适合于抽运光损耗系数较大的长腔双包层光纤激光器件。
对于典型的足够长双包层光纤激光器，抽运光损耗系数通常是远大于再发射项所产生的增益，因而，忽略上能级粒子数是合理的。然而，对于光纤长度短到小于或约等于1米的双包层光纤激光器，或者当讨论单包层光纤激光器的时候，则可能会出现再发射项的贡献与抽运光损耗系数相当，甚至可能会大于抽运光损耗系数，在这种情况下，忽略抽运光损耗系数较为合理。不过，可以推算到，不管是同时忽略信号光损耗系数和抽运光损耗系数，还是同时忽略信号光损耗系数和再发射项，都会造成抽运阈值功率被低估而输出激光功率被高估的事实。前人文献的实验数据充分表明了近似条件下得出的阈值略小于实验测得的结果。
依据前人理论基础，本文采用端面抽运准二能级掺杂光纤激光器稳态速率方程组进行了解析分析。主要做了如下工作：
(1)在忽略信号光损耗系数的前提下，明确提出了要获得光纤激光器输出功率的解析表达式，关键在于求得抽运光损耗系数与抽运光场乘积的积分值。并针对典型的双包层光纤激光器的参量，在估计了上能级粒子再发射项和抽运光的影响之后，采用保留再发射贡献主导部分而只忽略更高阶小量的办法来提高可积分近似方程的精确度和可应用的腔长范围。运用简洁的方法求得了近似条件下的双包层光纤激光器输出功率的显函解析表达式，导出了激光输出功率对抽运功率的斜线表达式，以及阈值抽运功率的表达式。并给出了近似条件下的最短光纤长度表达式，求得了固定抽运功率下获得最大激光输出功率的最佳光纤长度的隐函式，并对相关的问题进行了数值模拟分析和讨论。
(2)在不忽略损耗系数的情况下，对掺准二能级稀土粒子光纤激光器的抽运阈值进行了完全解析求解。求得了抽运阈值功率的显函数解析式，同时，得到了抽运光阈值功率随光纤位置变化的隐函数表达式，以及信号光功率随光纤位置变化的解析解和上能级粒子数随光纤位置变化的解析表达式。并根据所得解，与前人的理论和实验结果进行了比较分析，结果表明，不同近似条件下得到的抽运阈值功率均比实验值要低，但本文的结果却是差别最小的。
(3)运用(1)中所用的近似解析方法，从理论上分析了由F-P滤波器实现的多波长双包层光纤激光器的输出特性。讨论了光纤出光端面和双色镜的反射率、F-P滤波器腔长以及抽运光波长等参数对多波长输出功率的影响。为我们在实验中优化设计性能优越的双包层多波长光纤激光器提供了理论依据。
Compared to the tradition solid lasers, rare earth doped fiber lasers has it's unique characteristics, owing to their compactness, small volume and weigh, lower threshold, superb beam quality, high conversion efficiency, and large ratio of area/ volume, in favor of heat dispersion. Following the invention of the double-clad fiber lasers, the higher and higher powers have been achieved from the fiber lasers. At present, the fiber lasers have been widely used in the fields of long-distance fiber-optic communications, sensor, industry, medical treatment, national defence, etc.. As a consequence, in-depth studies on double-clad fiber lasers have become an important research topic, and fruitful and abundant experimental and theoretical results have been achieved. In the theoretical research of the fiber lasers, people generally used the rate equations. The investigation indicates that, the rate equations are correct and effective, and most widely used and powerful tool to theoretically predict the lasers characteristics and it's elements of working, investigate and devise fiber laser.
Compared to the numerical solutions of the rate equations, the analytical solutions have some dearly strongpoints, so that, after certain justified approximations have been adopted, peole have attempted to derive the analytical results. Unfortunately, owing to the nonlinear nature of the rate equations, it is difficult to obtain the exact analytical expressions. Even though, after the neglection of the loss coefficient of the laser radiation, people have obtained the analytical results satisfying one of the following approximate conditions: (a) the loss coefficient of the pump light can be neglected, which is applicable to the single-clad fiber lasers; (b) the re-emission arising from the interaction between the pump photons and upper-level ions can be neglected, which is applicable to the larger loss coefficient of the pump light and longer cavity of double-dad fiber lasers.
In typical double-clad fiber lasers with a reasonable length, the loss coefficient of pump light is usually larger than the gain provided by the re-emission arising from the interaction between the pump photons and upper-level ions. However, when the double-clad fiber length reduces down to the order of～1 m, or when the fiber used is a single-clad fiber, it may happen that the contribution of the re-emission becomes comparable to, or even larger than the loss coefficient of pump radiation. Under these circumstances, neglecting the quantity of the loss coefficient of pump radiation may be a better approximation. It is conceivable that, the neglection of the loss coefficient of pump light and the loss coefficient of the laser radiation, or the loss coefficient of the laser radiation and the upper-level population, definitely resulted in an underestimation of the threshold pump power and overestimation of the laser output power. The data of experiment have indicated that the threshold predicted by the approximated theory is slightly smaller than the measured result.
In this work, based on the theoretic of the previous studies, we have made an analytical solution by using the rate equations of an end-pumped fiber lasers doped with two-energy-level ions in the steaty state. The main works are as follows:
(1) After negenecting the loss coefficient of laser radiation, it becomes apparent that the key issue of finding analytical expression for the laser output is evaluating the integrations of the loss coefficient of pump radiation multiplied by the pump radiation. Using typical parameters of double-clad fiber lasers, estimations have been made on the contribution of the re-emission affect arising from the interaction between the upper-level atoms and pump light. It is realized that the retaining of the predominance, part of the re-emission can improve the accuracy and extend the application range of the integratable approximated rate equations. As a result, an explicit analytical expression for the output of the double-clad fiber lasers, together with gradient efficiency of the pump radiation, and the analytical expression to the threshold pump power has been deduced in an economic way also. And the expression of the minimum fiber length has been deduced, and a dosed-form expression to the optimum fiber length at which the laser output power takes maxmum has been obtaind, and certain issues of concern have been discussed.
(2) The rate equations in the steaty state including loss coefficients describing threshold pumped fiber lasers doped with quasi two-energy-level rare-earth ions have been analytically and fully solved. An analytical expression to the threshold pump power has been deduced, following the derivation of a closed-form expression for the variation of the pump power along the fiber has also been obtained, together with analytical solutions to the laser power in terms of the pump power and the position dependent atom population density. A comparison has been made between the analytical result with the previous reported results, which indicates that various approximations have underestimated the threshold pump powers, while the theoretical predication of this work is the closest to the experimental result.
(3) Using the approximations analytical method used in (1), the output performance of multiwavelength double-clade fiber lasers with a Fabry-Perot (F-P) flter is analyzed. And the parameters that may influence the output performance of multiwavelength have been discussed, such as the reflectivity of fiber facets, the reflectivity of dichroic mirror, the length of the F-P filter, and the wavelength of the pump radiation. It may offer the theoretical basis in our expirement to design a multiwavelength double-clade fiber lasers with better performances.
由于速率方程组的解析解相对数值解具有一些明显的优点，人们便尝试在各种合理近似下对速率方程组进行解析分析和讨论。然而，由于速率方程组的非线性特性，很难对之严格求解。不过，在忽略信号光损耗系数的近似情况下，人们已得到了满足下列近似情况之一的解析解：(a)抽运光损耗系数可忽略的单包层光纤激光器件；(b)上能级粒子与抽运光相互作用的再发射可以忽略的较适合于抽运光损耗系数较大的长腔双包层光纤激光器件。
对于典型的足够长双包层光纤激光器，抽运光损耗系数通常是远大于再发射项所产生的增益，因而，忽略上能级粒子数是合理的。然而，对于光纤长度短到小于或约等于1米的双包层光纤激光器，或者当讨论单包层光纤激光器的时候，则可能会出现再发射项的贡献与抽运光损耗系数相当，甚至可能会大于抽运光损耗系数，在这种情况下，忽略抽运光损耗系数较为合理。不过，可以推算到，不管是同时忽略信号光损耗系数和抽运光损耗系数，还是同时忽略信号光损耗系数和再发射项，都会造成抽运阈值功率被低估而输出激光功率被高估的事实。前人文献的实验数据充分表明了近似条件下得出的阈值略小于实验测得的结果。
依据前人理论基础，本文采用端面抽运准二能级掺杂光纤激光器稳态速率方程组进行了解析分析。主要做了如下工作：
(1)在忽略信号光损耗系数的前提下，明确提出了要获得光纤激光器输出功率的解析表达式，关键在于求得抽运光损耗系数与抽运光场乘积的积分值。并针对典型的双包层光纤激光器的参量，在估计了上能级粒子再发射项和抽运光的影响之后，采用保留再发射贡献主导部分而只忽略更高阶小量的办法来提高可积分近似方程的精确度和可应用的腔长范围。运用简洁的方法求得了近似条件下的双包层光纤激光器输出功率的显函解析表达式，导出了激光输出功率对抽运功率的斜线表达式，以及阈值抽运功率的表达式。并给出了近似条件下的最短光纤长度表达式，求得了固定抽运功率下获得最大激光输出功率的最佳光纤长度的隐函式，并对相关的问题进行了数值模拟分析和讨论。
(2)在不忽略损耗系数的情况下，对掺准二能级稀土粒子光纤激光器的抽运阈值进行了完全解析求解。求得了抽运阈值功率的显函数解析式，同时，得到了抽运光阈值功率随光纤位置变化的隐函数表达式，以及信号光功率随光纤位置变化的解析解和上能级粒子数随光纤位置变化的解析表达式。并根据所得解，与前人的理论和实验结果进行了比较分析，结果表明，不同近似条件下得到的抽运阈值功率均比实验值要低，但本文的结果却是差别最小的。
(3)运用(1)中所用的近似解析方法，从理论上分析了由F-P滤波器实现的多波长双包层光纤激光器的输出特性。讨论了光纤出光端面和双色镜的反射率、F-P滤波器腔长以及抽运光波长等参数对多波长输出功率的影响。为我们在实验中优化设计性能优越的双包层多波长光纤激光器提供了理论依据。
Compared to the tradition solid lasers, rare earth doped fiber lasers has it's unique characteristics, owing to their compactness, small volume and weigh, lower threshold, superb beam quality, high conversion efficiency, and large ratio of area/ volume, in favor of heat dispersion. Following the invention of the double-clad fiber lasers, the higher and higher powers have been achieved from the fiber lasers. At present, the fiber lasers have been widely used in the fields of long-distance fiber-optic communications, sensor, industry, medical treatment, national defence, etc.. As a consequence, in-depth studies on double-clad fiber lasers have become an important research topic, and fruitful and abundant experimental and theoretical results have been achieved. In the theoretical research of the fiber lasers, people generally used the rate equations. The investigation indicates that, the rate equations are correct and effective, and most widely used and powerful tool to theoretically predict the lasers characteristics and it's elements of working, investigate and devise fiber laser.
Compared to the numerical solutions of the rate equations, the analytical solutions have some dearly strongpoints, so that, after certain justified approximations have been adopted, peole have attempted to derive the analytical results. Unfortunately, owing to the nonlinear nature of the rate equations, it is difficult to obtain the exact analytical expressions. Even though, after the neglection of the loss coefficient of the laser radiation, people have obtained the analytical results satisfying one of the following approximate conditions: (a) the loss coefficient of the pump light can be neglected, which is applicable to the single-clad fiber lasers; (b) the re-emission arising from the interaction between the pump photons and upper-level ions can be neglected, which is applicable to the larger loss coefficient of the pump light and longer cavity of double-dad fiber lasers.
In typical double-clad fiber lasers with a reasonable length, the loss coefficient of pump light is usually larger than the gain provided by the re-emission arising from the interaction between the pump photons and upper-level ions. However, when the double-clad fiber length reduces down to the order of～1 m, or when the fiber used is a single-clad fiber, it may happen that the contribution of the re-emission becomes comparable to, or even larger than the loss coefficient of pump radiation. Under these circumstances, neglecting the quantity of the loss coefficient of pump radiation may be a better approximation. It is conceivable that, the neglection of the loss coefficient of pump light and the loss coefficient of the laser radiation, or the loss coefficient of the laser radiation and the upper-level population, definitely resulted in an underestimation of the threshold pump power and overestimation of the laser output power. The data of experiment have indicated that the threshold predicted by the approximated theory is slightly smaller than the measured result.
In this work, based on the theoretic of the previous studies, we have made an analytical solution by using the rate equations of an end-pumped fiber lasers doped with two-energy-level ions in the steaty state. The main works are as follows:
(1) After negenecting the loss coefficient of laser radiation, it becomes apparent that the key issue of finding analytical expression for the laser output is evaluating the integrations of the loss coefficient of pump radiation multiplied by the pump radiation. Using typical parameters of double-clad fiber lasers, estimations have been made on the contribution of the re-emission affect arising from the interaction between the upper-level atoms and pump light. It is realized that the retaining of the predominance, part of the re-emission can improve the accuracy and extend the application range of the integratable approximated rate equations. As a result, an explicit analytical expression for the output of the double-clad fiber lasers, together with gradient efficiency of the pump radiation, and the analytical expression to the threshold pump power has been deduced in an economic way also. And the expression of the minimum fiber length has been deduced, and a dosed-form expression to the optimum fiber length at which the laser output power takes maxmum has been obtaind, and certain issues of concern have been discussed.
(2) The rate equations in the steaty state including loss coefficients describing threshold pumped fiber lasers doped with quasi two-energy-level rare-earth ions have been analytically and fully solved. An analytical expression to the threshold pump power has been deduced, following the derivation of a closed-form expression for the variation of the pump power along the fiber has also been obtained, together with analytical solutions to the laser power in terms of the pump power and the position dependent atom population density. A comparison has been made between the analytical result with the previous reported results, which indicates that various approximations have underestimated the threshold pump powers, while the theoretical predication of this work is the closest to the experimental result.
(3) Using the approximations analytical method used in (1), the output performance of multiwavelength double-clade fiber lasers with a Fabry-Perot (F-P) flter is analyzed. And the parameters that may influence the output performance of multiwavelength have been discussed, such as the reflectivity of fiber facets, the reflectivity of dichroic mirror, the length of the F-P filter, and the wavelength of the pump radiation. It may offer the theoretical basis in our expirement to design a multiwavelength double-clade fiber lasers with better performances.
引文
[1. 1] E. Snittzer. Optical maser action of Nd~(3+) in a Barium Crown Glass [J]. Physics Review Letters, 1961, 7(12): 441-446
[1. 2] J. Koester, E. Snitzer. Amplification in a Fiber Laser [J]. Applied optics, 1964, 3 (10): 1182-1186
[1. 3] J. Koester. Laser Action by Enhanced Total Internal Reflection [J]. IEEE Journal of Quantum Electronics, 1966, 2 (9): 580-584
[1．4] 黄琳，刘永智，代志勇．大功率双包层光纤激光器[J]．光电子技术，2005，25 (2)：133-137
[1. 5] Snitzer E., Hakimi Po H., Tumminelli F., et.al., double-clad, offset core Nd fiber laser [A]. Optical fiber Communication Conf [C], 1988
[1．6] 乔学光，杨和钱，贾振安等．光纤激光器的研究进展与展望[J]．光机电信息，2006，9：44-50
[1．7] 孙宏志，梁建中，胡谊梅等．高功率可调谐掺镱双包层光纤激光器研究[J]．光学学报．2002，22(11)：1372-1374
[1．8] 楼祺洪，周军，孔令峰等．高功率脉冲双包层光纤激光器的新进展[J]．量子电子学报．22 (4)．2005：510-515
[1. 9] R. J. Meats, L Reekie, S. B. Poole, et al.. Neodymium-doped silica single-mode fibre lasers [J]. Electron. Lett., 1985, 21(17): 738-740
[1．10] 马海全，刘畅，赵卫等．8字形腔锁模掺Yb~(3+)光纤激光器[J]．中国激光，2005，32(9)： 1173-1117
[1．11] 黄德修，舒学文，文韬等．一种新颖的多波长环形腔掺铒光纤激光器[J]．中国激光．2000，27(4)：313-316
[1．12] 谢春霞，吕福云，王健等．大模面积掺Yb~(3+)双包层光纤激光器的实验研究[J]，光子学报，2005，34 (5)：644-547
[1．13] 林浩佳，阮双琛，程超等．掺Yb~(3+)双包层大模面积光子晶体光纤激光器的研究[J]．光子学报，2004，33 (7)：797-799
[1．14] 周军，楼棋洪，朱健强等．采用国产大模场面积双包层光纤的714 W连续光纤激光器[J]．光学学报，2006，26 (7)：1119-1120
[1．15] 王凤懿．掺镱双包层光纤激光器理论及实验研究[J]．中国工程物理研究院学位论文．2005：18
[1. 16] Y. Jeong, J. K. Sahu, D. N. Payne et al.. Ytterbium-doped large-core fiber laser with 1.36kW continuous-wave output power [J]. Opt. Express, 2004, 12(25): 6088-6092
[1. 17] Y. Wang, C. -Q. Xu, and H. Po. Thermal effects in kilowatt fiber lasers [J]. IEEE Photonics Technology Letters. 2004, 16: 63-65
[1．18] 王凤蕊，李明中，陈吉欣等．双包层光纤激光器的抽运技术研究新进展[J]．激光与光电子学进展，2004，41(10)：26-29
[1. 19] Po H., Cao J D, Laliberte B N, et al.. High power neodymium-doped single transvers mode fibre laser [J]. Electronics Letters, 1993, 29 (17): 1500-1501
[1. 20] H. M. Pask, J. L Archambault, D. C. Hanna, etc.. Operation of cladding-pumped Yb~(3+)-doped silica fiber lasers in 1 μm region [J]. Electon. Lett., 1994, 30(11): 863-865
[1. 21] R Pasehotta, J Nilsson, A. C Tropper, et al.. Ytterbium-Doped Fiber Amplifiers [J], IEEE J. Quantum Elect., 1997, 33 (7): 1049-1056
[1．22] 赵玉辉，郑义，詹仪等．高功率掺镱双包层光纤激光器[J]．激光与红外，2006，36(9)：833-836
[1．23] 韦文楼，闫平，欧攀等．强泵浦下掺Yb~(3+)双包层光纤激光器输出特性数值分析和实验研究[J]．光学技术，2004，30 (1)：13-19
[1．24] 陈吉欣，隋展，陈福深等．温度调谐掺Yb~(3+)双包层光纤激光器研究[J]．光电工程，2005．32 (10)：77-79
[1. 25] J. Nilsson, W. A. Clarkson, R. Selvas, et al.. High-power wavelength-tunable cladding-pumped rare-earth-doped silica fiber lasers [J]. Optical Fiber Technology, 2004, 10: 5-30
[1. 26] Du W Ch. IPG. High power fiber laser and their potential applications [C]. 广州: 2002年光纤激光器研究与应用最新进展研讨会．2002，7
[1．27] 朱志武．多点泵浦掺镱双包层光纤激光器输出特性理论分析与数值模拟[D]．国防科学技术大学研究生院学位论文，2005
[1．28] 付圣贵．高功率双包层光纤激光器的理论与实验研究[D]．南开大学博士研究生学位论文，2005
[1．29] 宁鼎，黄榜才，项阳等．后腔镜对掺Yb~(3+)双包层光纤激光器性能影响的研究[J]．光学学报，2003，23 (3)：313-316
[1．30] 项阳，许兆文，宁鼎等．掺Yb~(3+)双包层光纤激光器的多波长输出[J]．中国激光，2004．31 (4)：403-406
[1. 31] Fan Y X, Lu F Y, Hu SL, et al.. Narrow-linewidth widely tunable hybrid O-switched double-clad fiber laser [J]. Optics Letters, 2003, 28 (7): 537-539
[1．32] 楼棋洪，周军，李铁军等．4．9 W掺镱双包层光纤激光器[J]．中国激光，2002，29：306
[1．33] 楼祺洪，周军，朱健强等．10瓦级双包层光纤激光器[J]．光学学报，2003，23(9)：1080-1081
[1. 34] L. Qihong, Zhoujun et al..50-watt ytterbium.doped double-clad fiber laser[J]. Chinese Optics Letters, 2003, 1 (10): 559-560
[1．35] 楼棋洪，周军，朱健强 等．百瓦级掺镜双包层光纤激光器[J]．中国激光，2003，30(12)：1064
[1．36] 楼棋洪，周军，朱健强等．单端抽运国产D形双包层光纤激光器实现输出功率200W[J]．中国激光，2004，32(9)：1029
[1．37] 楼棋洪，周军，朱健强等．国产双包层掺镜光纤实现440 W的连续高功率激光输出[J]．中国激光，2004，32：20
[1．38] 赵鸿，周寿桓，朱辰等．大功率光纤激光器输出功率超过1．2 KW[J]．激光与红外，2006．36：930
[1. 39] Ido Kelson and Amos A. Hardy. Strongly Pumped Fiber Lasers [J]. IEEE J. Quantum Elect., 1998, 34 (9): 1570-1577
[1. 40] Hardy A. and Oron R.. Signal amplification in strongly pumped fiber amplifiers [J]. IEEE J. Quantum Elect., 1997, 33 (3): 307-313
[1. 41] Jiangno Chen, X. Zhu and W. Sibbett. Derivation of the threshold pump power of erbium-doped fiber lasers [J]. Optics Letters. 1992, 17 (13): 926-928
[1. 42] J. Chen, X. Zhu and W. Sibbett. Rate-equation studies of erbium-doped fiber lasers with common pump and laser energy bands [J]. J. Opt. Soc. Am. B, 1992, 9: 1876-1882
[1．43] Guijun Hu, Chengyu Shah, Xiaoying Deng et al.. Threshold characteristics of linear cavity Yb~(3+)-doped double-clad fiber laser [J]. Optics and laser technology, 2004, 37: 3-7
[1. 44] Liping Zhang, Zhichun Duan, Jianguo Chen. Analytical solutions to rate equations of fiber lasers containing three-energy-level ions [J]. Optics Communications, 2006, 267 (1): 149-153
[1．45] 张利平，段志春，陈建国等．双包层光纤激光器泵浦阈值的解析表达式[J]．光电工程，2006，33 (11)：P101-104
[1. 46] Zhichun Duan, Liping Zhang, Jianguo Chen. Analytical characterization of an end pumped rare-earth-doped double-clad fiber laser [J]. Optical Fiber Technology, 2007, 13 (2): 143-148.
[1．47] 段志春，陈建国，张利平 等．双包层光纤激光器速率方程的解析求解问题[J]．激光技术，2007，(6月将出版)
[1. 40] Zhichun Duan, Liping Zhang, Jianguo Chen. Analytical solutions to rate equations including losses describing threshold pumped fiber lasers [J]. Optik, 2007 (to be published)
[2．1] 钱梅珍，崔一平，杨正名．激光物理(第二版)[M]．北京：电子工业出版社，2001．
[2. 2] Yuen H. Tsang, Terence A. King, Do-Kyeong Ko et al.. Output dynamics and stabilisation of a multi-mode double-clad Yb-doped silica fibre laser [J]. Optics Communications. 2006, 259: 236-241
[2. 3] N. S. Kim, T. Hamada, M. Prabhu, et al., Numefical analysis and experimental results of output performance for Nd-doped double-clad fiber lasers [J]. Opt. Commun., 2000, 180: 329-337.
[2．4] 汪帆，潘炜，罗斌 等．掺镱双包层光纤激光器典型参数对斜线效率的影响[J]．强激光与粒子数，2004，16 (12)：1509-1512
[2. 5] Jianguo Chen, X. Zhu and W. Sibbett. Derivation of the threshold pump power of erbium-doped fiber lasers [J]. Optics Letters. 1992, 17 (13): 926-928
[2. 6] J. Chen, X. Zhu and W. Sibbett. Rate-equation studies of erbium-doped fiber lasers with common pump and laser energy bands [J]. J. Opt. Soc. Am. B, 1992, 9: 1876-1882
[2．7] 吴正茂，夏光琼，陈建国．掺铒光纤激光器输出特性的解析研究[J]．光学学报，1996，16 (12)：1676-1680
[2. 8] Guijun Hu, Chengyu Shan, Xiaoying Deng et al.. Threshold characteristics of linear cavityYb~(3+)-doped double-clad fiber laser [J]. Optics and laser technology, 2004, 37: 3-7。
[2. 9] I. Kelson and A. A. Hardy. Strongly Pumped Fiber Lasers [J]. IEEE J. Quantum Electron. 1998, 34: 1570-1577
[2. 10] Limin Xiao, Ping Yan, Mall Gong et al.. An approximate analytic solution of strongly pumped Yb-doped double-clad fiber lasers without neglecting the scattering loss [J]. Optics Communications, 2004, 230: 401-410
[2. 11] A. Hardy and R. Oron: Signal amplification in strongly pumped fiber amplifiers [J]. IEEE J. Ouantum Electron. 1997, 33: 307-313
[2．12] 徐华斌，陈林．掺铒光纤激光器输出特性的研究[J]．光子学报，2004，33(7)：777-781
[2. 13] Ido Kelson and Amos Hardy. Optimization of Strongly Pumped Fiber Lasers [J]. Journal of Lightwave Technology, 1999, 17 (5): 891-897
[2. 14] E. Yahel and A. Hardy. Modeling and optimization of short Er3+-Yb3+ co-doped fiber lasers [J]. IEEE J. Quantum Electron. 2003, 39: 1444-1451
[2. 15] E Yahel and A. Hardy. Modeling high power Er3+-Yb3+ co-doped fiber lasers [J]. J. Lightwave Technol. 2003, 21: 2044-2052
[2．16] 汪帆，潘炜，罗斌 等．掺镱双包层光纤激光器多波长输出性能分析[J]．激光技术，2005，29 (5)：497-500
[2．17] 朱志武，多点泵浦掺镱双包层光纤激光器输出特性理论分析与数值模拟[D]．国防科学技术大学研究生院学位论文，2005
[3. 1] J. Chen, X. Zhu and W. Sibbett. Rate-equation studies of erbium-doped fiber lasers with common pump and laser energy bands [J]. J. Opt. Soc. Am. B, 1992, 9: 1876-1882
[3．2] 吴正茂，夏光琼，陈建国．掺铒光纤激光器输出特性的解析研究[J]．光学学报，1996，16 (12)：1676-1680
[3. 3] L Kelson and A. A. Hardy: Strongly Pumped Fiber Lasers [J]. IEEE J. Quantum Electron., 1998, 34: 1570-1577
[3. 4] Ido Kelson and Amos Hardy. Optimization of Strongly Pumped Fiber Lasers [J]. Journal of Lightwave Technology, 1999, 17 (5): 891-897
[3. 5] Guijun Hu, Chengyu Shah, Xiaoying Deng et al.. Threshold characteristics of linear cavity Yb~(3+)-doped double-clad fiber laser [J]. Optics and laser technology, 2004, 37: 3-7
[3. 6] Limin Xiao, Ping Yan, Mall Gong et al.. An approximate analytic solution of strongly pumped Yb-doped double-clad fiber lasers without neglecting the scattering loss [J]. Optics . Communications, 2004, 230: 401-410
[3．7] 徐华斌，陈林．掺铒光纤激光器输出特性的研究[J]．光子学报，2004，33(7)：777-781
[3．8] 汪帆，潘炜，罗斌 等．掺镱双包层光纤激光器典型参数对斜线效率的影响[J]．强激光与粒子数，2004，16(12)：1509-1512
[3. 9] E. Yahel and A. Hardy. Modeling and optimization of short Er3+-Yb3+ co-doped fiber lasers [J]. IEEE J. Quantum Electron., 2003, 39: 1444-1451
[3. 10] E Yahel and A. Hardy. Modeling high power Er3+-Yb3+ co-doped fiber lasers [J]. J. Lightwave Technol., 2003, 21: 2044-2052
[3．11] 宁鼎，黄榜才，项阳等．后腔镜对掺Yb~(3+)双包层光纤激光器性能影响的研究[J]．光学学报，2003，23 (3)：313-316
[3. 12] J. Nilsson, W. A. Clarkson, R. Selvas et al.. High-power wavelength-tunable cladding-pumped rare-earth-doped silica fiber lasers [J]. Optical Fiber Technology, 2004, 10: 5-30
[3. 13] Govind P. Agrawal. Fiber-optic communication systems (3rd Ed) [M]. 2002 by John Wiley & Sons, Inc.
[4.1] Guijun Hu, Chengyu Shah, Xiaoying Deng et al.. Threshold characteristics of linear cavity Yb~(3+)-doped double-clad fiber laser [J]. Optics and laser technology, 2004, 37: 3-7
[4.2] Yuen H. Tsang, Terence A. King, Do-Kyeong Ko et al.. Output dynamics and stabilisation of a multi-mode double-clad Yb-doped silica fibre laser [J]. Optics Communications. 2006, 259: 236-241
[4.3] J. Nilsson, W. A. Clarkson, R. Selvas et al.. High-power wavelength-tunable cladding-pumped rare-earth-doped silica fiber lasers [J]. Optical Fiber Technology, 2004, 10: 5-30
[4.4] I. Kelson and A. A. Hardy. Strongly Pumped Fiber Lasers [J]. IEEE J. Quantum Electron., 1998, 34: 1570-1577
[4.5] I. Kelson and A. Hardy. Optimization of Strongly Pumped Fiber Lasers [J]. J. Lightwave Technol., 1999, 17: 891-897
[4.6] Jianguo Chert, X. Zhn and W. Sibbett. Derivation of the threshold pump power of erbium-doped fiber lasers [J]. Optics Letters, 1992, 17 (13): 926-928
[5．1] 孙军强，刘德明，黄德修 等．窄线宽多波长掺铒光纤激光器[J]．中国激光，2000，27(9)：773-776
[5．2] 杨先波，刘德明，黄德修．光纤喇曼激光器及多波长输出的分析[J]．激光技术，2003，27(5)：409-414
[5．3] 迟荣华，吕可诚，陈胜平等．单一偏振的多波长环形腔掺镱光纤激光器[J]．中国激光，2003，30 (12)：1061-1064
[5．4] 汪帆，潘炜，罗斌 等．掺镱双包层光纤激光器多波长输出性能分析[J]．激光技术，2005，29 (5)：497-500
[5．5] 项阳，许兆文，宁鼎等．掺Yb~(3+)包层光纤激光器的多波长输出[J]．中国激光，2004，31 (4)：403-406
[5．6] 陈柏，陈兰荣，范薇等．掺Yb双包层光纤激光器波长调谐输出[J]．光子学报，1999，28 (9)：835-838
[5．7] 胡智勇，刘俭辉，葛春风等．基于光纤环形镜滤波的L波段可调谐光纤激光器[J]． 半导体光电，2004，25(6)：456458
[5． 8] 冯月琴，盛秋琴，陈凯等．几种常用滤波器的特性[J]．光通信技术，2002，1：46～52
[5． 9] 聂秋华．光纤激光器和放大器技术[M]．北京：电子工业出版社，1997：305-313
[5．10] 徐建华，张位在，干福熹．法布里-珀罗干涉仪外腔半导体激光器的选模和锁模特性研究[J]．中国激光，1995，22 (2)：85-90
[5. 11] J. Nilsson, W. A. Clarkson, R. Selvas, et al.. High-power wavelength-tunable cladding-pumped rare-earth-doped silica fiber lasers [J]. Optical Fiber Technology, 2004, 10: 5-30
[5. 12] I. Kelson and A. A. Hardy. Strongly Pumped Fiber Lasers [J]. IEEE J. Quantum Electron., 1998, 34: 1570-1577
[1. 2] J. Koester, E. Snitzer. Amplification in a Fiber Laser [J]. Applied optics, 1964, 3 (10): 1182-1186
[1. 3] J. Koester. Laser Action by Enhanced Total Internal Reflection [J]. IEEE Journal of Quantum Electronics, 1966, 2 (9): 580-584
[1．4] 黄琳，刘永智，代志勇．大功率双包层光纤激光器[J]．光电子技术，2005，25 (2)：133-137
[1. 5] Snitzer E., Hakimi Po H., Tumminelli F., et.al., double-clad, offset core Nd fiber laser [A]. Optical fiber Communication Conf [C], 1988
[1．6] 乔学光，杨和钱，贾振安等．光纤激光器的研究进展与展望[J]．光机电信息，2006，9：44-50
[1．7] 孙宏志，梁建中，胡谊梅等．高功率可调谐掺镱双包层光纤激光器研究[J]．光学学报．2002，22(11)：1372-1374
[1．8] 楼祺洪，周军，孔令峰等．高功率脉冲双包层光纤激光器的新进展[J]．量子电子学报．22 (4)．2005：510-515
[1. 9] R. J. Meats, L Reekie, S. B. Poole, et al.. Neodymium-doped silica single-mode fibre lasers [J]. Electron. Lett., 1985, 21(17): 738-740
[1．10] 马海全，刘畅，赵卫等．8字形腔锁模掺Yb~(3+)光纤激光器[J]．中国激光，2005，32(9)： 1173-1117
[1．11] 黄德修，舒学文，文韬等．一种新颖的多波长环形腔掺铒光纤激光器[J]．中国激光．2000，27(4)：313-316
[1．12] 谢春霞，吕福云，王健等．大模面积掺Yb~(3+)双包层光纤激光器的实验研究[J]，光子学报，2005，34 (5)：644-547
[1．13] 林浩佳，阮双琛，程超等．掺Yb~(3+)双包层大模面积光子晶体光纤激光器的研究[J]．光子学报，2004，33 (7)：797-799
[1．14] 周军，楼棋洪，朱健强等．采用国产大模场面积双包层光纤的714 W连续光纤激光器[J]．光学学报，2006，26 (7)：1119-1120
[1．15] 王凤懿．掺镱双包层光纤激光器理论及实验研究[J]．中国工程物理研究院学位论文．2005：18
[1. 16] Y. Jeong, J. K. Sahu, D. N. Payne et al.. Ytterbium-doped large-core fiber laser with 1.36kW continuous-wave output power [J]. Opt. Express, 2004, 12(25): 6088-6092
[1. 17] Y. Wang, C. -Q. Xu, and H. Po. Thermal effects in kilowatt fiber lasers [J]. IEEE Photonics Technology Letters. 2004, 16: 63-65
[1．18] 王凤蕊，李明中，陈吉欣等．双包层光纤激光器的抽运技术研究新进展[J]．激光与光电子学进展，2004，41(10)：26-29
[1. 19] Po H., Cao J D, Laliberte B N, et al.. High power neodymium-doped single transvers mode fibre laser [J]. Electronics Letters, 1993, 29 (17): 1500-1501
[1. 20] H. M. Pask, J. L Archambault, D. C. Hanna, etc.. Operation of cladding-pumped Yb~(3+)-doped silica fiber lasers in 1 μm region [J]. Electon. Lett., 1994, 30(11): 863-865
[1. 21] R Pasehotta, J Nilsson, A. C Tropper, et al.. Ytterbium-Doped Fiber Amplifiers [J], IEEE J. Quantum Elect., 1997, 33 (7): 1049-1056
[1．22] 赵玉辉，郑义，詹仪等．高功率掺镱双包层光纤激光器[J]．激光与红外，2006，36(9)：833-836
[1．23] 韦文楼，闫平，欧攀等．强泵浦下掺Yb~(3+)双包层光纤激光器输出特性数值分析和实验研究[J]．光学技术，2004，30 (1)：13-19
[1．24] 陈吉欣，隋展，陈福深等．温度调谐掺Yb~(3+)双包层光纤激光器研究[J]．光电工程，2005．32 (10)：77-79
[1. 25] J. Nilsson, W. A. Clarkson, R. Selvas, et al.. High-power wavelength-tunable cladding-pumped rare-earth-doped silica fiber lasers [J]. Optical Fiber Technology, 2004, 10: 5-30
[1. 26] Du W Ch. IPG. High power fiber laser and their potential applications [C]. 广州: 2002年光纤激光器研究与应用最新进展研讨会．2002，7
[1．27] 朱志武．多点泵浦掺镱双包层光纤激光器输出特性理论分析与数值模拟[D]．国防科学技术大学研究生院学位论文，2005
[1．28] 付圣贵．高功率双包层光纤激光器的理论与实验研究[D]．南开大学博士研究生学位论文，2005
[1．29] 宁鼎，黄榜才，项阳等．后腔镜对掺Yb~(3+)双包层光纤激光器性能影响的研究[J]．光学学报，2003，23 (3)：313-316
[1．30] 项阳，许兆文，宁鼎等．掺Yb~(3+)双包层光纤激光器的多波长输出[J]．中国激光，2004．31 (4)：403-406
[1. 31] Fan Y X, Lu F Y, Hu SL, et al.. Narrow-linewidth widely tunable hybrid O-switched double-clad fiber laser [J]. Optics Letters, 2003, 28 (7): 537-539
[1．32] 楼棋洪，周军，李铁军等．4．9 W掺镱双包层光纤激光器[J]．中国激光，2002，29：306
[1．33] 楼祺洪，周军，朱健强等．10瓦级双包层光纤激光器[J]．光学学报，2003，23(9)：1080-1081
[1. 34] L. Qihong, Zhoujun et al..50-watt ytterbium.doped double-clad fiber laser[J]. Chinese Optics Letters, 2003, 1 (10): 559-560
[1．35] 楼棋洪，周军，朱健强 等．百瓦级掺镜双包层光纤激光器[J]．中国激光，2003，30(12)：1064
[1．36] 楼棋洪，周军，朱健强等．单端抽运国产D形双包层光纤激光器实现输出功率200W[J]．中国激光，2004，32(9)：1029
[1．37] 楼棋洪，周军，朱健强等．国产双包层掺镜光纤实现440 W的连续高功率激光输出[J]．中国激光，2004，32：20
[1．38] 赵鸿，周寿桓，朱辰等．大功率光纤激光器输出功率超过1．2 KW[J]．激光与红外，2006．36：930
[1. 39] Ido Kelson and Amos A. Hardy. Strongly Pumped Fiber Lasers [J]. IEEE J. Quantum Elect., 1998, 34 (9): 1570-1577
[1. 40] Hardy A. and Oron R.. Signal amplification in strongly pumped fiber amplifiers [J]. IEEE J. Quantum Elect., 1997, 33 (3): 307-313
[1. 41] Jiangno Chen, X. Zhu and W. Sibbett. Derivation of the threshold pump power of erbium-doped fiber lasers [J]. Optics Letters. 1992, 17 (13): 926-928
[1. 42] J. Chen, X. Zhu and W. Sibbett. Rate-equation studies of erbium-doped fiber lasers with common pump and laser energy bands [J]. J. Opt. Soc. Am. B, 1992, 9: 1876-1882
[1．43] Guijun Hu, Chengyu Shah, Xiaoying Deng et al.. Threshold characteristics of linear cavity Yb~(3+)-doped double-clad fiber laser [J]. Optics and laser technology, 2004, 37: 3-7
[1. 44] Liping Zhang, Zhichun Duan, Jianguo Chen. Analytical solutions to rate equations of fiber lasers containing three-energy-level ions [J]. Optics Communications, 2006, 267 (1): 149-153
[1．45] 张利平，段志春，陈建国等．双包层光纤激光器泵浦阈值的解析表达式[J]．光电工程，2006，33 (11)：P101-104
[1. 46] Zhichun Duan, Liping Zhang, Jianguo Chen. Analytical characterization of an end pumped rare-earth-doped double-clad fiber laser [J]. Optical Fiber Technology, 2007, 13 (2): 143-148.
[1．47] 段志春，陈建国，张利平 等．双包层光纤激光器速率方程的解析求解问题[J]．激光技术，2007，(6月将出版)
[1. 40] Zhichun Duan, Liping Zhang, Jianguo Chen. Analytical solutions to rate equations including losses describing threshold pumped fiber lasers [J]. Optik, 2007 (to be published)
[2．1] 钱梅珍，崔一平，杨正名．激光物理(第二版)[M]．北京：电子工业出版社，2001．
[2. 2] Yuen H. Tsang, Terence A. King, Do-Kyeong Ko et al.. Output dynamics and stabilisation of a multi-mode double-clad Yb-doped silica fibre laser [J]. Optics Communications. 2006, 259: 236-241
[2. 3] N. S. Kim, T. Hamada, M. Prabhu, et al., Numefical analysis and experimental results of output performance for Nd-doped double-clad fiber lasers [J]. Opt. Commun., 2000, 180: 329-337.
[2．4] 汪帆，潘炜，罗斌 等．掺镱双包层光纤激光器典型参数对斜线效率的影响[J]．强激光与粒子数，2004，16 (12)：1509-1512
[2. 5] Jianguo Chen, X. Zhu and W. Sibbett. Derivation of the threshold pump power of erbium-doped fiber lasers [J]. Optics Letters. 1992, 17 (13): 926-928
[2. 6] J. Chen, X. Zhu and W. Sibbett. Rate-equation studies of erbium-doped fiber lasers with common pump and laser energy bands [J]. J. Opt. Soc. Am. B, 1992, 9: 1876-1882
[2．7] 吴正茂，夏光琼，陈建国．掺铒光纤激光器输出特性的解析研究[J]．光学学报，1996，16 (12)：1676-1680
[2. 8] Guijun Hu, Chengyu Shan, Xiaoying Deng et al.. Threshold characteristics of linear cavityYb~(3+)-doped double-clad fiber laser [J]. Optics and laser technology, 2004, 37: 3-7。
[2. 9] I. Kelson and A. A. Hardy. Strongly Pumped Fiber Lasers [J]. IEEE J. Quantum Electron. 1998, 34: 1570-1577
[2. 10] Limin Xiao, Ping Yan, Mall Gong et al.. An approximate analytic solution of strongly pumped Yb-doped double-clad fiber lasers without neglecting the scattering loss [J]. Optics Communications, 2004, 230: 401-410
[2. 11] A. Hardy and R. Oron: Signal amplification in strongly pumped fiber amplifiers [J]. IEEE J. Ouantum Electron. 1997, 33: 307-313
[2．12] 徐华斌，陈林．掺铒光纤激光器输出特性的研究[J]．光子学报，2004，33(7)：777-781
[2. 13] Ido Kelson and Amos Hardy. Optimization of Strongly Pumped Fiber Lasers [J]. Journal of Lightwave Technology, 1999, 17 (5): 891-897
[2. 14] E. Yahel and A. Hardy. Modeling and optimization of short Er3+-Yb3+ co-doped fiber lasers [J]. IEEE J. Quantum Electron. 2003, 39: 1444-1451
[2. 15] E Yahel and A. Hardy. Modeling high power Er3+-Yb3+ co-doped fiber lasers [J]. J. Lightwave Technol. 2003, 21: 2044-2052
[2．16] 汪帆，潘炜，罗斌 等．掺镱双包层光纤激光器多波长输出性能分析[J]．激光技术，2005，29 (5)：497-500
[2．17] 朱志武，多点泵浦掺镱双包层光纤激光器输出特性理论分析与数值模拟[D]．国防科学技术大学研究生院学位论文，2005
[3. 1] J. Chen, X. Zhu and W. Sibbett. Rate-equation studies of erbium-doped fiber lasers with common pump and laser energy bands [J]. J. Opt. Soc. Am. B, 1992, 9: 1876-1882
[3．2] 吴正茂，夏光琼，陈建国．掺铒光纤激光器输出特性的解析研究[J]．光学学报，1996，16 (12)：1676-1680
[3. 3] L Kelson and A. A. Hardy: Strongly Pumped Fiber Lasers [J]. IEEE J. Quantum Electron., 1998, 34: 1570-1577
[3. 4] Ido Kelson and Amos Hardy. Optimization of Strongly Pumped Fiber Lasers [J]. Journal of Lightwave Technology, 1999, 17 (5): 891-897
[3. 5] Guijun Hu, Chengyu Shah, Xiaoying Deng et al.. Threshold characteristics of linear cavity Yb~(3+)-doped double-clad fiber laser [J]. Optics and laser technology, 2004, 37: 3-7
[3. 6] Limin Xiao, Ping Yan, Mall Gong et al.. An approximate analytic solution of strongly pumped Yb-doped double-clad fiber lasers without neglecting the scattering loss [J]. Optics . Communications, 2004, 230: 401-410
[3．7] 徐华斌，陈林．掺铒光纤激光器输出特性的研究[J]．光子学报，2004，33(7)：777-781
[3．8] 汪帆，潘炜，罗斌 等．掺镱双包层光纤激光器典型参数对斜线效率的影响[J]．强激光与粒子数，2004，16(12)：1509-1512
[3. 9] E. Yahel and A. Hardy. Modeling and optimization of short Er3+-Yb3+ co-doped fiber lasers [J]. IEEE J. Quantum Electron., 2003, 39: 1444-1451
[3. 10] E Yahel and A. Hardy. Modeling high power Er3+-Yb3+ co-doped fiber lasers [J]. J. Lightwave Technol., 2003, 21: 2044-2052
[3．11] 宁鼎，黄榜才，项阳等．后腔镜对掺Yb~(3+)双包层光纤激光器性能影响的研究[J]．光学学报，2003，23 (3)：313-316
[3. 12] J. Nilsson, W. A. Clarkson, R. Selvas et al.. High-power wavelength-tunable cladding-pumped rare-earth-doped silica fiber lasers [J]. Optical Fiber Technology, 2004, 10: 5-30
[3. 13] Govind P. Agrawal. Fiber-optic communication systems (3rd Ed) [M]. 2002 by John Wiley & Sons, Inc.
[4.1] Guijun Hu, Chengyu Shah, Xiaoying Deng et al.. Threshold characteristics of linear cavity Yb~(3+)-doped double-clad fiber laser [J]. Optics and laser technology, 2004, 37: 3-7
[4.2] Yuen H. Tsang, Terence A. King, Do-Kyeong Ko et al.. Output dynamics and stabilisation of a multi-mode double-clad Yb-doped silica fibre laser [J]. Optics Communications. 2006, 259: 236-241
[4.3] J. Nilsson, W. A. Clarkson, R. Selvas et al.. High-power wavelength-tunable cladding-pumped rare-earth-doped silica fiber lasers [J]. Optical Fiber Technology, 2004, 10: 5-30
[4.4] I. Kelson and A. A. Hardy. Strongly Pumped Fiber Lasers [J]. IEEE J. Quantum Electron., 1998, 34: 1570-1577
[4.5] I. Kelson and A. Hardy. Optimization of Strongly Pumped Fiber Lasers [J]. J. Lightwave Technol., 1999, 17: 891-897
[4.6] Jianguo Chert, X. Zhn and W. Sibbett. Derivation of the threshold pump power of erbium-doped fiber lasers [J]. Optics Letters, 1992, 17 (13): 926-928
[5．1] 孙军强，刘德明，黄德修 等．窄线宽多波长掺铒光纤激光器[J]．中国激光，2000，27(9)：773-776
[5．2] 杨先波，刘德明，黄德修．光纤喇曼激光器及多波长输出的分析[J]．激光技术，2003，27(5)：409-414
[5．3] 迟荣华，吕可诚，陈胜平等．单一偏振的多波长环形腔掺镱光纤激光器[J]．中国激光，2003，30 (12)：1061-1064
[5．4] 汪帆，潘炜，罗斌 等．掺镱双包层光纤激光器多波长输出性能分析[J]．激光技术，2005，29 (5)：497-500
[5．5] 项阳，许兆文，宁鼎等．掺Yb~(3+)包层光纤激光器的多波长输出[J]．中国激光，2004，31 (4)：403-406
[5．6] 陈柏，陈兰荣，范薇等．掺Yb双包层光纤激光器波长调谐输出[J]．光子学报，1999，28 (9)：835-838
[5．7] 胡智勇，刘俭辉，葛春风等．基于光纤环形镜滤波的L波段可调谐光纤激光器[J]． 半导体光电，2004，25(6)：456458
[5． 8] 冯月琴，盛秋琴，陈凯等．几种常用滤波器的特性[J]．光通信技术，2002，1：46～52
[5． 9] 聂秋华．光纤激光器和放大器技术[M]．北京：电子工业出版社，1997：305-313
[5．10] 徐建华，张位在，干福熹．法布里-珀罗干涉仪外腔半导体激光器的选模和锁模特性研究[J]．中国激光，1995，22 (2)：85-90
[5. 11] J. Nilsson, W. A. Clarkson, R. Selvas, et al.. High-power wavelength-tunable cladding-pumped rare-earth-doped silica fiber lasers [J]. Optical Fiber Technology, 2004, 10: 5-30
[5. 12] I. Kelson and A. A. Hardy. Strongly Pumped Fiber Lasers [J]. IEEE J. Quantum Electron., 1998, 34: 1570-1577