离轴非稳—波导混合腔腔内光场特性研究
|
摘要
高功率扩散冷却板条CO2激光器被广泛地应用于工业激光加工领域。离轴非稳—波导混合腔是其核心部件之一。本论文针对离轴非稳—波导混合腔腔内光场的空间分布特性作出了系统细致的研究,尤其是利用渐进的方法对非稳方向上场的分布特性进行了详细的论述,包括最低阶模式的虚源法,特征函数法。讨论了模式交叉以及过量噪声因子特性。其主要研究内容如下：(1)研究了平板波导内场沿正向和反向传输的传输特性。利用几何球面波的近似,分别求解了正向传输的均匀平面行波和负向柱面行波在平板波导中的传输特性,指出正向行波在波导内以TE波或TM波的形式存在,且两者不能共存。分析了不同波导材料对传输损耗的影响,指出正向行波损耗最小的材料为氧化铝。分析了负支离轴共焦腔中“热点”处场在波导方向上的分布特性。指出在“热点”处场的六个分量中仅有一个沿波导方向的分量不为0,并分析了电极材料以及电极间距对电极表面场振幅大小的影响。(2)分析了虚源法的数学基础和物理意义。由分析可知虚源法误差来源主要包括：观察点离边界点轴向距离非常近,或者横向距离为零；未考虑各孔阑边界所发出的球面波在观察平面的几何阴影区边界不同；虚源点坐标的几何近似对场分布精细结构的影响。将虚源法扩展到离轴共焦非稳腔的最低阶光场光强模式计算中,给出所取孔阑数在满足自再现模条件下的经验取值。将其应用到千瓦级射频板条激光器所采用的大菲涅尔数离轴腔的模式计算中,得到比较精确的结果。(3)利用均匀渐进的公式,找到了一组特征函数,利用其线性叠加可以比较精确的描述腔内的横模,并给出了自相似分形法的数学基础,得到最低阶模式的近似解析表达式。通过求解一个由该组特征函数构成的特征值多项式就可以得到各阶特征值和对应的本征横模。(4)通过特征向量法的数值计算,指出离轴共焦非稳腔中各阶横模的交叉特性：正支离轴腔模式交叉发生在有效菲涅尔数取为s+0.08处(s为正整数)；而负支离轴腔的交叉发生在s-0.125附近。正支离轴共焦非稳腔中,其最低阶模式并非传统意义上的偶数阶模,而其衍射损耗也明显高于几何衍射损耗值。通过由均匀渐进方法得到的横模特征值多项式,分析了特征值的简并特性,找到了放大率在的区间[1.1,1.9]时对应的复平面内最接近实轴的有效菲涅尔数的取值,其实部大约在75.375附近,并不与模式交叉点重合,而虚部为0。(5)讨论了离轴共焦非稳腔中光场的横向分布与输出激光线宽的问题。说明描述耦合到任意给定模式自发辐射噪声(朗之万噪声)项的统计平均可以用横模的双正交性(即过量噪声K因子)与特征值特性来描述。利用衍射积分方法得到了在放大率一定的条件下,K因子的大小随有效菲涅尔数的变化情况,指出K因子在模式交叉处不存在峰值现象,而是表现出非连续性。通过分析场在腔内的渡越特性,将已有的混合几何法改进,使之能应用于离轴非稳腔,并修正了计算高阶模K因子存在的数量级误差。
The high power diffusion-cool slab CO2 laser is widely used for industrial applications. The off-axis unstable-waveguide hybrid resonator is a core component of the laser. Characteristics of the field distribution in the resonator have been studied in this thesis. The transverse modes are analysed by the asymptotic theory, including the virtual source method for the lowest transverse mode and the eigen-function method for all order modes. The mode crossing and the excess noise are also discussed in detail. The main contents are classified as follows.(1) The transmission characteristics of both travelling waves along forward and backward direction have been studied. The approximate solutions for both waves are obtained with the spherical wave theory. The results show that the eigen-solution for forward travelling wave is TE mode or TM mode, the two modes can not exist simultaneously. The effect on transmission loss caused by the dielectric constant of waveguide materials is also discussed; alumina is the optimal material on the loss account. The characteristics of the field distribution along the narrow direction at the "hot spot" in the negative branch unstable resonator are analysed with the formula for the cylindrical wave. At the "hot spot", there is only one component which is along the waveguide direction. The effects on the filed amplitudes on the surface of the electrode caused by the dielectric constant of electrode material and the distance between the electrodes are studied.(2) The mathematic formula and physical meaning of the virtual source method are presented. The analysis shows that the errors are caused by three cases, the axial distance between the observation plane and the aperture, or the transverse distance between the aperture edge and the observation point is too small, the different width of the geometric shadow of the apertures are not considered, the effects on the fine structure of distribution caused by the geometric approximation of the virtual source location are ignored. The virtual source method is extended to the calculation of the lowest mode in the off-axis unstable resonators, the errors are also treated carefully, and the number of the apertures should be small as possible under the condition of the self-produce mode. The numerical result for the kilowatt diffusion-cool slab laser shows good agreement between the virtual source method and FFT.(3) A group of eigen functions have been found by the uniform asymptotic theory. The transverse modes are presented by the linear superposition of the functions. The mathematical validity of the self similar fractal method is proved and the analytic formula for the lowest mode is presented by the functions. To calculate high order modes, a polynomial about the scaled-eigenvalue is obtained.(4) The numerical results by the eigen-vector method show that the mode crossing between the lowest-and the next-lowest-loss modes occurs when the equivalent Fresnel number Neq is close to s+0.08 (s positive integer) for positive branch resonators and is close to s-0.125 for negative branch resonators, the lowest mode in positive branch is not the traditional even mode, the diffraction loss is much larger than the value predicted by the geometric optics. The degeneracy is also determined by the polynomial. The results show that the real part of Neq which is most close to the real axis is equal to about 78.4.(5) The relationship between the laser linewidth and the transverse distribution of the field is discussed. The statistical average of the Langevin noise that describes the spontaneous radiation noise coupling into a given mode is determined by the Biorthogonality of the transverses modes (K factor) and their eigenvalues. The values of K factor varying as a function of Neq are calculated numerically by the eigen-vector method with a fixed magnification. The profile shows discontinuity at mode crossing point instead of the peak. The hybrid-geometry method is modified to calculate the K factor of all order modes, and the magnitude error is corrected.
The high power diffusion-cool slab CO2 laser is widely used for industrial applications. The off-axis unstable-waveguide hybrid resonator is a core component of the laser. Characteristics of the field distribution in the resonator have been studied in this thesis. The transverse modes are analysed by the asymptotic theory, including the virtual source method for the lowest transverse mode and the eigen-function method for all order modes. The mode crossing and the excess noise are also discussed in detail. The main contents are classified as follows.(1) The transmission characteristics of both travelling waves along forward and backward direction have been studied. The approximate solutions for both waves are obtained with the spherical wave theory. The results show that the eigen-solution for forward travelling wave is TE mode or TM mode, the two modes can not exist simultaneously. The effect on transmission loss caused by the dielectric constant of waveguide materials is also discussed; alumina is the optimal material on the loss account. The characteristics of the field distribution along the narrow direction at the "hot spot" in the negative branch unstable resonator are analysed with the formula for the cylindrical wave. At the "hot spot", there is only one component which is along the waveguide direction. The effects on the filed amplitudes on the surface of the electrode caused by the dielectric constant of electrode material and the distance between the electrodes are studied.(2) The mathematic formula and physical meaning of the virtual source method are presented. The analysis shows that the errors are caused by three cases, the axial distance between the observation plane and the aperture, or the transverse distance between the aperture edge and the observation point is too small, the different width of the geometric shadow of the apertures are not considered, the effects on the fine structure of distribution caused by the geometric approximation of the virtual source location are ignored. The virtual source method is extended to the calculation of the lowest mode in the off-axis unstable resonators, the errors are also treated carefully, and the number of the apertures should be small as possible under the condition of the self-produce mode. The numerical result for the kilowatt diffusion-cool slab laser shows good agreement between the virtual source method and FFT.(3) A group of eigen functions have been found by the uniform asymptotic theory. The transverse modes are presented by the linear superposition of the functions. The mathematical validity of the self similar fractal method is proved and the analytic formula for the lowest mode is presented by the functions. To calculate high order modes, a polynomial about the scaled-eigenvalue is obtained.(4) The numerical results by the eigen-vector method show that the mode crossing between the lowest-and the next-lowest-loss modes occurs when the equivalent Fresnel number Neq is close to s+0.08 (s positive integer) for positive branch resonators and is close to s-0.125 for negative branch resonators, the lowest mode in positive branch is not the traditional even mode, the diffraction loss is much larger than the value predicted by the geometric optics. The degeneracy is also determined by the polynomial. The results show that the real part of Neq which is most close to the real axis is equal to about 78.4.(5) The relationship between the laser linewidth and the transverse distribution of the field is discussed. The statistical average of the Langevin noise that describes the spontaneous radiation noise coupling into a given mode is determined by the Biorthogonality of the transverses modes (K factor) and their eigenvalues. The values of K factor varying as a function of Neq are calculated numerically by the eigen-vector method with a fixed magnification. The profile shows discontinuity at mode crossing point instead of the peak. The hybrid-geometry method is modified to calculate the K factor of all order modes, and the magnitude error is corrected.
引文
[1] K. M. Abramski, A. D. Colley, H. J. Baker, et al. Power scaling of large-area transverse radio frequency discharge CO2 lasers. Appl. Phys. Lett.,1989,54(19): 1833～1835.
[2] R. Nowack, H. Opower, U. Schaefer, et al. High power CO2 waveguide laser of the 1 kW category. SPIE,1990.18～28
[3] A. D. Colley, H. J. Baker, D. R. Hall, et al. Planar waveguide:1 KW CW, carbon dioxide laser excited by a single transverse RF discharge. Appl. Phys. Lett.,1992, 81(2):136～138
[4] 徐晵阳,王新兵.高功率连续cO2激光器.(第一版).北京：国防工业出版社,2000.149-150
[5] C. J. Walsh. An rf excited circular waveguide CO2 laser. Rev. Sci. instrum.,1990, 61(9):2309～2313
[6] G. D. Santiago, A. L. Peuriot, C. A. Rosito. Extension of the self-filtering unstable resonatorto a ring cavity. Opt. Eng.,2001,40(10):2215～2219
[7] Y. Lin, Y. Huang. Annular coaxial waveguide CO2 laser with RF discharge and diffusion cooling. SPIE,2002.367～370
[8] Y. Li, M. Chen, J. Guo, et al Self-reproducing ray-cluster and misalignment of gas laser resonator having a coaxial discharge chamber. Opt.& laser Tech.,2009,41 217～223
[9] D. Ehrlichmann, U. Habich, H. Plum. High-Power CO2 Laser with Coaxial Waveguide and Diffusion Cooling. J. Quantum Electron.,1993,29(7):2211～2219
[10] D. Ehrlichmann, U. Habich, H. Plum. Simple annular resonators with toric and helical mirrors. SPIE,1993.77～84
[11] D. Ehrlichmann, U. Habich, H. Plum, et al. Azimuthally Unstable Resonators for High-Power CO2 Lasers with Annular Gain Media. J. Quantum Electron.,1994, 30(6):1441～1447
[12] E. F. Yelden, H. J. J. Seguin, C. E. Capjack, et al. Multichannel slab discharge for
C02 laser excitation. Appl. Phys. Lett.,1991,58(7):693-695.
[13] H. J. J. Seguin. Power scaling of diffusion-cooled lasers. Opt.& Laser Tech.,1998, 30:331～336.
[14] L. A. Newman, R. A. Hart, J. T. Kennedy, et al. High power coupled CO2 waveguide laser array. Appl. Phys. Lett.,1986,48 (25):1701～1703
[15] K. M. Abramski, A. D. Colley, H. J. Baker, et al. Phase-locked CO2 laser array using diagonal coupling of waveguide channels. Appl. Phys. Lett.1992,60 (5):530～532
[16] K. M. Abramski, A. D. Colley, H. J. Baker, et al. Single-mode selection using coherent imaging within a slab waveguide CO2 laser. Appl. Phys. Lett.,1992,60 (20):2469～2471
[17] A. M. Hornby, H. J. Baker, A. D. Colley, et al. Phase locking of linear arrays of CO2 waveguide lasers by the waveguide confined Talbot effect. Appl. Phys. Lett.1993, 63(19):2591～2593
[18] H. J. Baker, D. R. Hall, A. M. Hornby, et al. Propagation Characteristics of Coherent Array Beams from Carbon Dioxide Waveguide Lasers. J. Quantum Electron.,1996, 32(3):400～407
[19] X. Tang, G. Xin. Phase coupling of optical phase-shifting array resonator slab waveguide CO2 lasers. SPIE,1998.162～172
[20] K. M. Abramski, A. D. Colley, H. J. Baker, et al. High-Power Two-Dimensional Waveguide CO2 Laser Arrays. J. Quantum Electron.,1996,32(2):340～349
[21] F. J. Villarreal, H. J. Baker, R. H. Abram, et al. Beam Reformatting of One-and Two-Dimensional Arrays of CO2 Waveguide Lasers. J. Quantum Electron.,1999, 35(3):267～272
[22]辛建国,方高瞻,彭雪云.射频激励扩散型冷却层叠式板条波导千瓦CO2激光器.光学学报,1996,16(6)：877～880
[23]周双全,王智勇,辛建国.射频激励层叠式波导CO2激光器.光学学报,1996,16(2)：240～243
[24]辛建国,张旺,焦文涛.射频激励扩散型冷却板条波导千瓦C02激光器.光学学报,2000,20(15)：714-716
[25]高允贵,吴春雷,姜云云等.射频板条C02激光器放电特性的数值研究.量子
电子学报,1999,16(4)：329～337
[26]苏红新,高允贵.射频板条C02激光器波导耦合损耗的理论研究.量子电子学报.2000,17(3)：226～230
[27]高允贵,朱永祥,丁义国.射频板条CO2激光器输出光束的光学变换.光学学报,2008,28(s1)：98～101
[28]柳娟,唐霞辉,彭浩等.高功率连续C02激光器脉冲调制特性研究及特殊熔覆应用.中国激光,2009,7(56)：1575～1580
[29] Du Wang, Gen Li, Yingxiong Qin, et al. Output beam characteristics of the radio frequency-excited slab CO2 laser with unstable-waveguide hybrid resonator. Opt. Engineering,2011,50(9):094204
[30] P. E. Jackson, H. J. Baker, D. R. Hall, et al. CO2 large-area discharge laser using an unstable-waveguide hybrid resonator. Appl. Phys. Lett.,1989,54(20):1950-1952
[31] A. I. Dutov, N. A. Novoselov, A. A. Kuleshov, et al. Investigations of output power and beam divergence of slab waveguide RF-pumping CO2 laser. SPIE,1996. 103～106
[32] L.Serri, V. Fantini, S. De Silvestri, et al. Theoretical and experimental study of hybrid unstable-guided resonator for diffusion cooled CO2 laser. SPIE,1996.24-34
[33] V. M. Cherezov, S. I. Mol'kov, E. F. Shishkanov, et al. Characteristics of a slab RF-excited CO2 laser using unstable-waveguide resonator. SPIE,2000.157-168
[34] A. P. Mineev, S. M. Nefedov, P. P. Pashinin, et al. RF excited planar CO2 laser with hybrid waveguide-unstable resonator cavities. SPIE,1999.35～42
[35] A.I.Dutov, A.A.Kuleshov. RF-excited slab CO2 laser with intra-cavity mode selection. SPIE,2003.84～86
[36]唐晓军,赵宗海,辛建国.射频激励相移阵列谐振腔波导C02激光器输出特性研究.物理学报,1999,48(7)：1236-1247
[37] K. D. Laakmann, W. H. Steier. Waveguides:characteristic modes of hollow rectangular dielectric waveguides. Appl. Opt,1976,15(5):1334～1340
[38] S. Avrillier, J. Verdonck. coupling losses in laser resonators containing a hollow rectangular dielectric waveguide. Appl. Phys.,1977,48(12):4937～4941
[39] J. L. Boulnois, G. P. Agrawal. Mode discrimination and coupling losses in rectangular waveguide resonators with conventional and phase-conjugate mirrors. Opt. Soc. Am.,1982,72(7):853～860
[40] A. I. Dutov, A. A. Kuleshov, N. A. Novoselov, et al. New approach to the formation of the basic waveguide radiation mode in a high-power slab-type CO2 lasers. SPIE, 2005.464～468
[41] A. Lapucci, A. Labate, F. Rossetti, et al. Hybrid stable-unstable resonators for diffusion-cooled CO2 slab lasers. Appl. Opt.,1996,35(18):3185-3192.
[42] A. E. Siegman. Stable-unstable Resonator Design for a Wide-Tuning-Range Free-Electron Laser. J. Quantum Electron.,1992,28(5):1243～1247
[43] Koji Yasui, Yushi Takenaka. Advantages of negative branch compared with positive branch one-dimensional unstable resonators. Appl. Opt.,2001,40(21):3547-3551
[44] T. Hall. Numerical studies on hybrid resonators for a medium-sized chemical oxygen iodine laser. Opt. Engineering,2005,44(11):114201
[45] A. E. Siegman. Unstable optical resonators. Appl. Opt.,1974,13(2):353～367
[46] K. Peterman. Calculated Spontaneous Emission Factor for Double-Heterostructure Injection Lasers with Gain-Induced Waveguiding. J. Quantum Electron.,1979, 15(7):566～570
[47] A. E. Siegman. Excess spontaneous emission in non-Hermitian optical systems. I. Laser amplifiers. Phys. Rev. A,1989,39(3):1253～1263
[48] A. E. Siegman. Excess spontaneous emission in non-Hermitian optical systems. II. Laser oscillators. Phys. Rev. A,1989,39(3):1264～1268
[49] W. A. Hamel, J. P. Woerdman. Observation of enhanced fundamental linewidth of a laser due to nonorthogonality of its longitudinal eigenmodes. Phys. Rev. Lett.,1990, 64(13):1506～1509
[50] G Yao, Y. C. Chen. Excess spontaneous-emission factor in unstable-resonator lasers. Opt. Lett,1992,17(17):1207～1209
[51] Yuh-Jen Cheng, P. L. Mussche, A. E. Siegman. Measurement of laser quantum frequency fluctuations using a Pound-Drever stabilization system. J. Quantum Electron.,1994,30(6):1498～1504.
[52] Yuh-Jen Cheng, C. G. Fanning, A. E. Siegman. Experimental observation of a large
excess quantum noise factor in the linewidth of a Laser oscillator having nonorthogonal modes. Phys. Rev. Lett.,1996,74(4):627～630
[53] M. A. van Eijkelenborg, A. M. Lindberg, M. S. Thijssen, et al. Resonance of quantum noise in an unstable cavity laser. Phys. Rev. Lett.,1996,77(21) 4314～4317
[54] M. A. van Eijkelenborg, A. M. Lindberg, M. S. Thijssen, et al. Influence of transverse resonator symmetry on excess noise. Opt. Commun.,1997,137 303～307
[55] M. A. van Eijkelenborg, M. P. V. Exter, J. P. Woerdman, et al. Threshold characteristics and intensity fluctuations of lasers with excess quantum noise. Phys. Rev. A,1998,57(1):571～579
[56] A. M. van der Lee, N. J. van Druten, A. Mieremet, et al. Excess Quantum Noise due to Nonorthogonal Polarization Modes. Phys. Rev. Lett.,1997,79(22):4357～4360
[57] G. H. C. New. The origin of excess noise. J. Mod. Opt.,1995,42(4):799～810
[58] P. J. Bardroff, S. Stenholm. Quantum theory of excess noise. Phys. Rev. A,1999, 60(3):2529～2533
[59] P. J. Bardroff, S. Stenholm. Quantum Langevin theory of excess noise in lasers. Phys. Rev. A,2000,61(2):023806-1～023806-7
[60] Yuh-Jen Cheng. Generalized radiation-field quantization method and the Petermann excess-noise factor. Phys. Rev. A,2003,68(4):043808-1～043808-8
[61] C. Lamprecht, H. Ritsch. Quantized atom-field dynamics in unstable cavities. Phys. Rev. Lett.,1999,82(19):3787～3790
[62] C. Lamprecht, H. Ritsch. Unexpected role of excess noise in spontaneous emission. Phys. Rev. A.2002,65(2):023803-1～023803-12
[63] C. Lamprecht, H. Ritsch. Theory of excess noise in unstable resonator lasers. Phys. Rev. A,2002,66(5):053808-1～053808-4
[64] G. S. McDonald, G. H. C. New, J. P. Woerdman, et al. Excess noise in low Fresnel number unstable resonators. Opt. Commun.,1999,164:285～295
[65] G. P. Karman, G S. McDonald, J. P. Woerdman, et al. Excess-noise dependence on intracavity aperture shape. Appl. Opt.,1999,38(33):6874～6878
[66] G. P. Karman, A. M. Lindberg, J. P.Woerdman, et al. Observed factorization of
excess quantum noise that is due to both polarization and spatial mode nonorthogonality. Opt. Lett.,1998,23(21):1698-1900
[67] M. A. van Eijkelenborg, A. M. Lindberg, M. S. Thijssen. Unstable-resonator diffraction losses and the excess-noise factor. Phys. Rev. A,1997,55(6) 4556～4562
[68] S. Anikitchev, R. Hill. Striplinelaser. U. S. Patent专利公开号：US00625633281,2001.3-5
[69]吕百达,冯国英,蔡邦维.板条激光器用光腔的模式计算快速傅里叶变换法.激光技术,1993,17(6)：335～339
[70]王宁,陆雨田,李代林.利用快速傅立叶算法模拟离轴混合腔的光场分布.科学技术与工程,2008,8(19)：5512～5515
[71]秦应雄,唐霞辉,钟如涛等.基于传输矩阵的激光谐振腔模式计算.中国激光,2008,35(10)：1463～1468
[72] Y. Cheng, Y. Wang, J. Hu, et al. An eigenvector method for optical field simulation. Optics Commun.,2004,234:1～6
[73] B. Jiang, B. Li, Y. Cheng, et al. simulation of optical field in laser resonators cavity by eigenvector method. Opt.& Laser Tech.,2007,39(3):490～499
[74] P. Horwitz. Asymptotic theory of unstable resonator modes. Opt. Soc. Am.,1973,63: 1528～1543
[75] W. H. Southwell. Virtual-source theory of unstable resonator modes. Opt. Lett., 1981,6:487～489
[76] W. H. Southwell. Unstable resonator mode derivation using virtual-source theory. Opt. Soc. Am. A.,1986,3(11):1885～1891
[77] M. A. Yates, G.H.C. New. Fractal dimension of unstable resonator modes. Opt. Commun.,2002,208:377～380
[78] G. H. C. New, M. A. Yates, J. P. Woerdman, et al. Diffractive origin of fractal resonator modes. Optics Commun.,2001,193:261～266
[79] M. A. Yates, G H. C. New, T. Albaho. Calculating higher-order modes of one-dimensional unstable laser resonators. J. Mod. Opt.,2004,51(5):657～667
[80] J. Courtial, M. J. Padgett. Monitor-Outside-a-Monitor Effect and Self-Similar
Fractal Structure in the Eigenmodes of Unstable Optical Resonators. Phys. Rev. Lett.,2000,85(25):5320～5323
[81] G P. Karman, J. P. Woerdman. Fractal structure of eigenmodes of unstable-cavity lasers. Opt. Lett,1998,23(24):1909～1911
[82] G P. Karman, G S. McDonald, G H. C. New, et al. fractal modes in unstable resonators. Nature,1999,402(11):138
[83] G S. McDonald, G. P. Karman, G H. C. New, et al. Kaleidoscope laser. Opt. Soc. Am. B.,2000,17(4):524～529
[84] J. G Huang, J. M. Christian, G S. McDonald, et al. Fresnel diffraction and fractal patterns from polygonal apertures. Opt. Soc. Am. A.,2006,23(11):2769～2774
[85] M. V. Berry, C. Storm, W. van Saarloos. Theory of unstable laser modes:edge waves and fractality. Opt. Commun.,2001,197:393-402
[86] M. V. Berry. Mode degeneracies and the Petermann excess-noise factor for unstable lasers. J. Mod. Opt.,2003,50(1):63～81
[87] M. Born, E. Wolf.光学原理[下册],杨葭荪译.(第七版).北京：北京电子工业出版社,2006.526-546
[88] C. M. G Watterson, M. J. Padgett, J. Courtial. Classic-fractal eigenmodes of unstable canonical resonators. Opt. Commun.,2003,223:17～23
[89] S. Guru, H. R. Hiziroglu. Electromagnetic Field Theory Fundamentals(第一版).北京：机械工业出版社,2002.436
[90]李根,唐霞辉,秦应雄等.电极膜层对大功率射频板条CO2激光器腔内光场模式的影响.应用激光,2011,32(3)：21～25
[91] M. A. Ordal, L. L. Long, R. J. Bell, et al. Optical properties of the metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W in the infrared and far infrared. Appl. Opt., 1983,22(7):1099～1120
[92] R. L. Kozodoy, J. A. Harrington. Solgel alumina coating for hollow waveguide delivery of CO2 laser radiation. Appl. Opt.,1995,34(34):7840～7849.
[93] E. D. Palik. Handbook of Optical Constants of Solids. (lst Edition). San Diego: Academic press,1985.733
[94] P. Horwitz. Modes in misaligned unstable resonators. Appl. Opt.,1976,15:
167～178
[95] M. Abramowitz, I. A. Stegun. Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables. (lst Edition). D. C. Washington:National Bureau of Standards,1970.302.
[96] A. E. Siegman, H. Y. Miller. Unstable Optical Resonator Loss Calculations Using the Prony Method. Appl. Opt.,1970,9(12):2729～2736
[97] R.Wang. Asymptotic Approximations to Integrals. (2nd Edition). New York: Academic Press,2001.76～84
[98] H. A. Haus, S. Kawakami. On the "Excess Spontaneous Emission Factor" in Gain-Guided Laser Amplifiers. J. Quantum Electron.,1985,21(1):63～69
[99] I. H. Deutsch, J. C. Garrison, E. M. Wright. Excess noise in gain-guided amplifiers. J. Opt. Soc. Am. B,1991,8(6):1244～1251
[100] W. J Firth, A. M. Yao. Giant Excess Noise and Transient Gain in Misaligned Laser Cavities. Phys. Rev. Lett.,2005,95:073903-1～073903-4
[101] M. A. Lauder, G. H. C. NEW. BIORTHOGONALITY PROPERTIES AND EXCESS NOISE FACTORS. Opt. Commun.,1988,65(5):343～348
[102] Gen Li, Xiahui Tang, Du Wang, et al. Excess noise in strip off-axis confocal unstable resonators. J. Mod. Opt.2012,59(3):235～240
[2] R. Nowack, H. Opower, U. Schaefer, et al. High power CO2 waveguide laser of the 1 kW category. SPIE,1990.18～28
[3] A. D. Colley, H. J. Baker, D. R. Hall, et al. Planar waveguide:1 KW CW, carbon dioxide laser excited by a single transverse RF discharge. Appl. Phys. Lett.,1992, 81(2):136～138
[4] 徐晵阳,王新兵.高功率连续cO2激光器.(第一版).北京：国防工业出版社,2000.149-150
[5] C. J. Walsh. An rf excited circular waveguide CO2 laser. Rev. Sci. instrum.,1990, 61(9):2309～2313
[6] G. D. Santiago, A. L. Peuriot, C. A. Rosito. Extension of the self-filtering unstable resonatorto a ring cavity. Opt. Eng.,2001,40(10):2215～2219
[7] Y. Lin, Y. Huang. Annular coaxial waveguide CO2 laser with RF discharge and diffusion cooling. SPIE,2002.367～370
[8] Y. Li, M. Chen, J. Guo, et al Self-reproducing ray-cluster and misalignment of gas laser resonator having a coaxial discharge chamber. Opt.& laser Tech.,2009,41 217～223
[9] D. Ehrlichmann, U. Habich, H. Plum. High-Power CO2 Laser with Coaxial Waveguide and Diffusion Cooling. J. Quantum Electron.,1993,29(7):2211～2219
[10] D. Ehrlichmann, U. Habich, H. Plum. Simple annular resonators with toric and helical mirrors. SPIE,1993.77～84
[11] D. Ehrlichmann, U. Habich, H. Plum, et al. Azimuthally Unstable Resonators for High-Power CO2 Lasers with Annular Gain Media. J. Quantum Electron.,1994, 30(6):1441～1447
[12] E. F. Yelden, H. J. J. Seguin, C. E. Capjack, et al. Multichannel slab discharge for
C02 laser excitation. Appl. Phys. Lett.,1991,58(7):693-695.
[13] H. J. J. Seguin. Power scaling of diffusion-cooled lasers. Opt.& Laser Tech.,1998, 30:331～336.
[14] L. A. Newman, R. A. Hart, J. T. Kennedy, et al. High power coupled CO2 waveguide laser array. Appl. Phys. Lett.,1986,48 (25):1701～1703
[15] K. M. Abramski, A. D. Colley, H. J. Baker, et al. Phase-locked CO2 laser array using diagonal coupling of waveguide channels. Appl. Phys. Lett.1992,60 (5):530～532
[16] K. M. Abramski, A. D. Colley, H. J. Baker, et al. Single-mode selection using coherent imaging within a slab waveguide CO2 laser. Appl. Phys. Lett.,1992,60 (20):2469～2471
[17] A. M. Hornby, H. J. Baker, A. D. Colley, et al. Phase locking of linear arrays of CO2 waveguide lasers by the waveguide confined Talbot effect. Appl. Phys. Lett.1993, 63(19):2591～2593
[18] H. J. Baker, D. R. Hall, A. M. Hornby, et al. Propagation Characteristics of Coherent Array Beams from Carbon Dioxide Waveguide Lasers. J. Quantum Electron.,1996, 32(3):400～407
[19] X. Tang, G. Xin. Phase coupling of optical phase-shifting array resonator slab waveguide CO2 lasers. SPIE,1998.162～172
[20] K. M. Abramski, A. D. Colley, H. J. Baker, et al. High-Power Two-Dimensional Waveguide CO2 Laser Arrays. J. Quantum Electron.,1996,32(2):340～349
[21] F. J. Villarreal, H. J. Baker, R. H. Abram, et al. Beam Reformatting of One-and Two-Dimensional Arrays of CO2 Waveguide Lasers. J. Quantum Electron.,1999, 35(3):267～272
[22]辛建国,方高瞻,彭雪云.射频激励扩散型冷却层叠式板条波导千瓦CO2激光器.光学学报,1996,16(6)：877～880
[23]周双全,王智勇,辛建国.射频激励层叠式波导CO2激光器.光学学报,1996,16(2)：240～243
[24]辛建国,张旺,焦文涛.射频激励扩散型冷却板条波导千瓦C02激光器.光学学报,2000,20(15)：714-716
[25]高允贵,吴春雷,姜云云等.射频板条C02激光器放电特性的数值研究.量子
电子学报,1999,16(4)：329～337
[26]苏红新,高允贵.射频板条C02激光器波导耦合损耗的理论研究.量子电子学报.2000,17(3)：226～230
[27]高允贵,朱永祥,丁义国.射频板条CO2激光器输出光束的光学变换.光学学报,2008,28(s1)：98～101
[28]柳娟,唐霞辉,彭浩等.高功率连续C02激光器脉冲调制特性研究及特殊熔覆应用.中国激光,2009,7(56)：1575～1580
[29] Du Wang, Gen Li, Yingxiong Qin, et al. Output beam characteristics of the radio frequency-excited slab CO2 laser with unstable-waveguide hybrid resonator. Opt. Engineering,2011,50(9):094204
[30] P. E. Jackson, H. J. Baker, D. R. Hall, et al. CO2 large-area discharge laser using an unstable-waveguide hybrid resonator. Appl. Phys. Lett.,1989,54(20):1950-1952
[31] A. I. Dutov, N. A. Novoselov, A. A. Kuleshov, et al. Investigations of output power and beam divergence of slab waveguide RF-pumping CO2 laser. SPIE,1996. 103～106
[32] L.Serri, V. Fantini, S. De Silvestri, et al. Theoretical and experimental study of hybrid unstable-guided resonator for diffusion cooled CO2 laser. SPIE,1996.24-34
[33] V. M. Cherezov, S. I. Mol'kov, E. F. Shishkanov, et al. Characteristics of a slab RF-excited CO2 laser using unstable-waveguide resonator. SPIE,2000.157-168
[34] A. P. Mineev, S. M. Nefedov, P. P. Pashinin, et al. RF excited planar CO2 laser with hybrid waveguide-unstable resonator cavities. SPIE,1999.35～42
[35] A.I.Dutov, A.A.Kuleshov. RF-excited slab CO2 laser with intra-cavity mode selection. SPIE,2003.84～86
[36]唐晓军,赵宗海,辛建国.射频激励相移阵列谐振腔波导C02激光器输出特性研究.物理学报,1999,48(7)：1236-1247
[37] K. D. Laakmann, W. H. Steier. Waveguides:characteristic modes of hollow rectangular dielectric waveguides. Appl. Opt,1976,15(5):1334～1340
[38] S. Avrillier, J. Verdonck. coupling losses in laser resonators containing a hollow rectangular dielectric waveguide. Appl. Phys.,1977,48(12):4937～4941
[39] J. L. Boulnois, G. P. Agrawal. Mode discrimination and coupling losses in rectangular waveguide resonators with conventional and phase-conjugate mirrors. Opt. Soc. Am.,1982,72(7):853～860
[40] A. I. Dutov, A. A. Kuleshov, N. A. Novoselov, et al. New approach to the formation of the basic waveguide radiation mode in a high-power slab-type CO2 lasers. SPIE, 2005.464～468
[41] A. Lapucci, A. Labate, F. Rossetti, et al. Hybrid stable-unstable resonators for diffusion-cooled CO2 slab lasers. Appl. Opt.,1996,35(18):3185-3192.
[42] A. E. Siegman. Stable-unstable Resonator Design for a Wide-Tuning-Range Free-Electron Laser. J. Quantum Electron.,1992,28(5):1243～1247
[43] Koji Yasui, Yushi Takenaka. Advantages of negative branch compared with positive branch one-dimensional unstable resonators. Appl. Opt.,2001,40(21):3547-3551
[44] T. Hall. Numerical studies on hybrid resonators for a medium-sized chemical oxygen iodine laser. Opt. Engineering,2005,44(11):114201
[45] A. E. Siegman. Unstable optical resonators. Appl. Opt.,1974,13(2):353～367
[46] K. Peterman. Calculated Spontaneous Emission Factor for Double-Heterostructure Injection Lasers with Gain-Induced Waveguiding. J. Quantum Electron.,1979, 15(7):566～570
[47] A. E. Siegman. Excess spontaneous emission in non-Hermitian optical systems. I. Laser amplifiers. Phys. Rev. A,1989,39(3):1253～1263
[48] A. E. Siegman. Excess spontaneous emission in non-Hermitian optical systems. II. Laser oscillators. Phys. Rev. A,1989,39(3):1264～1268
[49] W. A. Hamel, J. P. Woerdman. Observation of enhanced fundamental linewidth of a laser due to nonorthogonality of its longitudinal eigenmodes. Phys. Rev. Lett.,1990, 64(13):1506～1509
[50] G Yao, Y. C. Chen. Excess spontaneous-emission factor in unstable-resonator lasers. Opt. Lett,1992,17(17):1207～1209
[51] Yuh-Jen Cheng, P. L. Mussche, A. E. Siegman. Measurement of laser quantum frequency fluctuations using a Pound-Drever stabilization system. J. Quantum Electron.,1994,30(6):1498～1504.
[52] Yuh-Jen Cheng, C. G. Fanning, A. E. Siegman. Experimental observation of a large
excess quantum noise factor in the linewidth of a Laser oscillator having nonorthogonal modes. Phys. Rev. Lett.,1996,74(4):627～630
[53] M. A. van Eijkelenborg, A. M. Lindberg, M. S. Thijssen, et al. Resonance of quantum noise in an unstable cavity laser. Phys. Rev. Lett.,1996,77(21) 4314～4317
[54] M. A. van Eijkelenborg, A. M. Lindberg, M. S. Thijssen, et al. Influence of transverse resonator symmetry on excess noise. Opt. Commun.,1997,137 303～307
[55] M. A. van Eijkelenborg, M. P. V. Exter, J. P. Woerdman, et al. Threshold characteristics and intensity fluctuations of lasers with excess quantum noise. Phys. Rev. A,1998,57(1):571～579
[56] A. M. van der Lee, N. J. van Druten, A. Mieremet, et al. Excess Quantum Noise due to Nonorthogonal Polarization Modes. Phys. Rev. Lett.,1997,79(22):4357～4360
[57] G. H. C. New. The origin of excess noise. J. Mod. Opt.,1995,42(4):799～810
[58] P. J. Bardroff, S. Stenholm. Quantum theory of excess noise. Phys. Rev. A,1999, 60(3):2529～2533
[59] P. J. Bardroff, S. Stenholm. Quantum Langevin theory of excess noise in lasers. Phys. Rev. A,2000,61(2):023806-1～023806-7
[60] Yuh-Jen Cheng. Generalized radiation-field quantization method and the Petermann excess-noise factor. Phys. Rev. A,2003,68(4):043808-1～043808-8
[61] C. Lamprecht, H. Ritsch. Quantized atom-field dynamics in unstable cavities. Phys. Rev. Lett.,1999,82(19):3787～3790
[62] C. Lamprecht, H. Ritsch. Unexpected role of excess noise in spontaneous emission. Phys. Rev. A.2002,65(2):023803-1～023803-12
[63] C. Lamprecht, H. Ritsch. Theory of excess noise in unstable resonator lasers. Phys. Rev. A,2002,66(5):053808-1～053808-4
[64] G. S. McDonald, G. H. C. New, J. P. Woerdman, et al. Excess noise in low Fresnel number unstable resonators. Opt. Commun.,1999,164:285～295
[65] G. P. Karman, G S. McDonald, J. P. Woerdman, et al. Excess-noise dependence on intracavity aperture shape. Appl. Opt.,1999,38(33):6874～6878
[66] G. P. Karman, A. M. Lindberg, J. P.Woerdman, et al. Observed factorization of
excess quantum noise that is due to both polarization and spatial mode nonorthogonality. Opt. Lett.,1998,23(21):1698-1900
[67] M. A. van Eijkelenborg, A. M. Lindberg, M. S. Thijssen. Unstable-resonator diffraction losses and the excess-noise factor. Phys. Rev. A,1997,55(6) 4556～4562
[68] S. Anikitchev, R. Hill. Striplinelaser. U. S. Patent专利公开号：US00625633281,2001.3-5
[69]吕百达,冯国英,蔡邦维.板条激光器用光腔的模式计算快速傅里叶变换法.激光技术,1993,17(6)：335～339
[70]王宁,陆雨田,李代林.利用快速傅立叶算法模拟离轴混合腔的光场分布.科学技术与工程,2008,8(19)：5512～5515
[71]秦应雄,唐霞辉,钟如涛等.基于传输矩阵的激光谐振腔模式计算.中国激光,2008,35(10)：1463～1468
[72] Y. Cheng, Y. Wang, J. Hu, et al. An eigenvector method for optical field simulation. Optics Commun.,2004,234:1～6
[73] B. Jiang, B. Li, Y. Cheng, et al. simulation of optical field in laser resonators cavity by eigenvector method. Opt.& Laser Tech.,2007,39(3):490～499
[74] P. Horwitz. Asymptotic theory of unstable resonator modes. Opt. Soc. Am.,1973,63: 1528～1543
[75] W. H. Southwell. Virtual-source theory of unstable resonator modes. Opt. Lett., 1981,6:487～489
[76] W. H. Southwell. Unstable resonator mode derivation using virtual-source theory. Opt. Soc. Am. A.,1986,3(11):1885～1891
[77] M. A. Yates, G.H.C. New. Fractal dimension of unstable resonator modes. Opt. Commun.,2002,208:377～380
[78] G. H. C. New, M. A. Yates, J. P. Woerdman, et al. Diffractive origin of fractal resonator modes. Optics Commun.,2001,193:261～266
[79] M. A. Yates, G H. C. New, T. Albaho. Calculating higher-order modes of one-dimensional unstable laser resonators. J. Mod. Opt.,2004,51(5):657～667
[80] J. Courtial, M. J. Padgett. Monitor-Outside-a-Monitor Effect and Self-Similar
Fractal Structure in the Eigenmodes of Unstable Optical Resonators. Phys. Rev. Lett.,2000,85(25):5320～5323
[81] G P. Karman, J. P. Woerdman. Fractal structure of eigenmodes of unstable-cavity lasers. Opt. Lett,1998,23(24):1909～1911
[82] G P. Karman, G S. McDonald, G H. C. New, et al. fractal modes in unstable resonators. Nature,1999,402(11):138
[83] G S. McDonald, G. P. Karman, G H. C. New, et al. Kaleidoscope laser. Opt. Soc. Am. B.,2000,17(4):524～529
[84] J. G Huang, J. M. Christian, G S. McDonald, et al. Fresnel diffraction and fractal patterns from polygonal apertures. Opt. Soc. Am. A.,2006,23(11):2769～2774
[85] M. V. Berry, C. Storm, W. van Saarloos. Theory of unstable laser modes:edge waves and fractality. Opt. Commun.,2001,197:393-402
[86] M. V. Berry. Mode degeneracies and the Petermann excess-noise factor for unstable lasers. J. Mod. Opt.,2003,50(1):63～81
[87] M. Born, E. Wolf.光学原理[下册],杨葭荪译.(第七版).北京：北京电子工业出版社,2006.526-546
[88] C. M. G Watterson, M. J. Padgett, J. Courtial. Classic-fractal eigenmodes of unstable canonical resonators. Opt. Commun.,2003,223:17～23
[89] S. Guru, H. R. Hiziroglu. Electromagnetic Field Theory Fundamentals(第一版).北京：机械工业出版社,2002.436
[90]李根,唐霞辉,秦应雄等.电极膜层对大功率射频板条CO2激光器腔内光场模式的影响.应用激光,2011,32(3)：21～25
[91] M. A. Ordal, L. L. Long, R. J. Bell, et al. Optical properties of the metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W in the infrared and far infrared. Appl. Opt., 1983,22(7):1099～1120
[92] R. L. Kozodoy, J. A. Harrington. Solgel alumina coating for hollow waveguide delivery of CO2 laser radiation. Appl. Opt.,1995,34(34):7840～7849.
[93] E. D. Palik. Handbook of Optical Constants of Solids. (lst Edition). San Diego: Academic press,1985.733
[94] P. Horwitz. Modes in misaligned unstable resonators. Appl. Opt.,1976,15:
167～178
[95] M. Abramowitz, I. A. Stegun. Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables. (lst Edition). D. C. Washington:National Bureau of Standards,1970.302.
[96] A. E. Siegman, H. Y. Miller. Unstable Optical Resonator Loss Calculations Using the Prony Method. Appl. Opt.,1970,9(12):2729～2736
[97] R.Wang. Asymptotic Approximations to Integrals. (2nd Edition). New York: Academic Press,2001.76～84
[98] H. A. Haus, S. Kawakami. On the "Excess Spontaneous Emission Factor" in Gain-Guided Laser Amplifiers. J. Quantum Electron.,1985,21(1):63～69
[99] I. H. Deutsch, J. C. Garrison, E. M. Wright. Excess noise in gain-guided amplifiers. J. Opt. Soc. Am. B,1991,8(6):1244～1251
[100] W. J Firth, A. M. Yao. Giant Excess Noise and Transient Gain in Misaligned Laser Cavities. Phys. Rev. Lett.,2005,95:073903-1～073903-4
[101] M. A. Lauder, G. H. C. NEW. BIORTHOGONALITY PROPERTIES AND EXCESS NOISE FACTORS. Opt. Commun.,1988,65(5):343～348
[102] Gen Li, Xiahui Tang, Du Wang, et al. Excess noise in strip off-axis confocal unstable resonators. J. Mod. Opt.2012,59(3):235～240