光波导及光纤光栅数值模拟算法与相关波导特性研究
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摘要
本文中使用数值模拟算法对平板光波导、光纤和光纤光栅进行了数值模拟,并对相关的波导特性进行了数值分析。此外,对于所采用的数值模拟算法也进行了一定的改进和创新。具体包括以下内容:
1.用两维FDBPM算法对平板波导中进行了数值模拟,计算了波导中的传导模式场,并用矩阵束法对其中的导模传播常数进行了数值分析,导模及传播常数与解析理论相比误差在10-4数量级;使用FDBPM结合保形变换对弯曲平板波导进行了模拟,计算了弯曲损耗,并与相关的解析推导结果进行了对比。
2.使用三维ADI-FDBPM算法对光纤中的光传导进行了模拟。对ADI-BPM的计算过程进行优化,提出了一种高效地求解一类特殊带状线性方程组的算法,并采用所提出的算法对光纤以及弯曲光纤中的光传导进行了模拟。在模拟的基础上,计算了光纤中的传导模式与传播常数,分析了弯曲光纤的弯曲损耗,并与相关的解析理论进行了对比和验证。
3.使用三维FDTD算法对光纤光栅中的光传导和散射进行了数值模拟。对于模拟中由于光纤光栅的设定长度较短,所产生的反射比较弱,提出了一种方法从相对较强的背景中计算微弱的反射场。此外,还对倾斜光纤光栅中的光传播与反射进行了数值模拟。
本文所包含的创新点包括:提出了一种高效求解特定类型的带状线性方程组的算法,并将该算法应用于三维ADI-FDBPM,对光纤进行了模拟;首次应用三维FDTD算法对光纤光栅进行了数值模拟,并且提出了一种模拟微弱散射电磁场的方法。
除了以上的创新点之外,本文中还根据采用的数值模拟算法及数值分析算法编程,对相关的光波导进行了模拟并与相关波导理论进行了对比,验证了算法的正确性和有效性;这些研究工作的意义在于,尝试了通过计算机模拟来对光波导的特性进行分析和评估,为将来从事光波导器件的计算机设计及优化工作奠定了良好的基础。
In this thesis, optical slab waveguide, optical fiber and fiber grating is simulated with numerical simulation algorithms respectively. To improve the computation efficiency, some modification and innovation were made to the simulation algorithms. The thesis is organized as follows.
In Chapter 1, a short introduction about numerical simulation of optical waveguide is presented, and a brief development history of numerical simulation of optical waveguide is outlined.
In Chapter 2, a single mode slab waveguide is simulated with the two-dimensional FDBPM. With the simulation, the guided mode of the slab waveguide is calculated and compared with the analytical theory. The Matrix Pencil Method (MPM) is used to calculate the propagation constant of the guided mode. The difference of the guided mode and the propagation constant between the numerical simulation and analytical theory is about 10-4, which clarified the correctness of the numerical simulation. With conformal mapping and FDBPM, bent slab waveguide is also simulated, and the bending loss is calculated. Compared with analytical theory, the error of the bending loss is about 3%.
In Chapter 3, three-dimensional ADI-FDBPM is used to simulate light propagation in optical fiber. To improve the computation efficiency, a modified algorithm which is derived by extending the Thomas method is proposed to solve a specific bend linear system during the computation with ADI-FDBPM. With the proposed algorithm, Corning SMF28 optical fiber is simulated, and the guided mode and propagation constant is calculated. The errors between the numerical and analytical result is about 10-4. Bending optical fiber is also simulated with conformal mapping and ADI-FDBPM.
In Chapter 4, fiber grating is simulated with three-dimensional FDTD method. As the simulated fiber grating is very short, which contains only 16 periods, the reflected light wave at the fiber grating is quite weak. So, a method that can make the faint reflected wave distinct and legible is proposed. The proposed method is to run two simulations synchronously, one with grating, and the other with no grating; the difference between the electric field of the two simulations shows the reflected light wave at the fiber grating. With this method, light wave reflection and radiation in fiber grating and tilted fiber grating is exhibited respectively, in numerical simulation.
1.用两维FDBPM算法对平板波导中进行了数值模拟,计算了波导中的传导模式场,并用矩阵束法对其中的导模传播常数进行了数值分析,导模及传播常数与解析理论相比误差在10-4数量级;使用FDBPM结合保形变换对弯曲平板波导进行了模拟,计算了弯曲损耗,并与相关的解析推导结果进行了对比。
2.使用三维ADI-FDBPM算法对光纤中的光传导进行了模拟。对ADI-BPM的计算过程进行优化,提出了一种高效地求解一类特殊带状线性方程组的算法,并采用所提出的算法对光纤以及弯曲光纤中的光传导进行了模拟。在模拟的基础上,计算了光纤中的传导模式与传播常数,分析了弯曲光纤的弯曲损耗,并与相关的解析理论进行了对比和验证。
3.使用三维FDTD算法对光纤光栅中的光传导和散射进行了数值模拟。对于模拟中由于光纤光栅的设定长度较短,所产生的反射比较弱,提出了一种方法从相对较强的背景中计算微弱的反射场。此外,还对倾斜光纤光栅中的光传播与反射进行了数值模拟。
本文所包含的创新点包括:提出了一种高效求解特定类型的带状线性方程组的算法,并将该算法应用于三维ADI-FDBPM,对光纤进行了模拟;首次应用三维FDTD算法对光纤光栅进行了数值模拟,并且提出了一种模拟微弱散射电磁场的方法。
除了以上的创新点之外,本文中还根据采用的数值模拟算法及数值分析算法编程,对相关的光波导进行了模拟并与相关波导理论进行了对比,验证了算法的正确性和有效性;这些研究工作的意义在于,尝试了通过计算机模拟来对光波导的特性进行分析和评估,为将来从事光波导器件的计算机设计及优化工作奠定了良好的基础。
In this thesis, optical slab waveguide, optical fiber and fiber grating is simulated with numerical simulation algorithms respectively. To improve the computation efficiency, some modification and innovation were made to the simulation algorithms. The thesis is organized as follows.
In Chapter 1, a short introduction about numerical simulation of optical waveguide is presented, and a brief development history of numerical simulation of optical waveguide is outlined.
In Chapter 2, a single mode slab waveguide is simulated with the two-dimensional FDBPM. With the simulation, the guided mode of the slab waveguide is calculated and compared with the analytical theory. The Matrix Pencil Method (MPM) is used to calculate the propagation constant of the guided mode. The difference of the guided mode and the propagation constant between the numerical simulation and analytical theory is about 10-4, which clarified the correctness of the numerical simulation. With conformal mapping and FDBPM, bent slab waveguide is also simulated, and the bending loss is calculated. Compared with analytical theory, the error of the bending loss is about 3%.
In Chapter 3, three-dimensional ADI-FDBPM is used to simulate light propagation in optical fiber. To improve the computation efficiency, a modified algorithm which is derived by extending the Thomas method is proposed to solve a specific bend linear system during the computation with ADI-FDBPM. With the proposed algorithm, Corning SMF28 optical fiber is simulated, and the guided mode and propagation constant is calculated. The errors between the numerical and analytical result is about 10-4. Bending optical fiber is also simulated with conformal mapping and ADI-FDBPM.
In Chapter 4, fiber grating is simulated with three-dimensional FDTD method. As the simulated fiber grating is very short, which contains only 16 periods, the reflected light wave at the fiber grating is quite weak. So, a method that can make the faint reflected wave distinct and legible is proposed. The proposed method is to run two simulations synchronously, one with grating, and the other with no grating; the difference between the electric field of the two simulations shows the reflected light wave at the fiber grating. With this method, light wave reflection and radiation in fiber grating and tilted fiber grating is exhibited respectively, in numerical simulation.
引文
[1] Stern M S. Semivectorial polarized finite difference method for optical waveguides with arbitrary index profiles[J]. IEE Proceedings, 1988, 135(1): 56-63.
[2] Stern M S. Semivectorial polarized H field solutions for dielectric waveguides with arbitrary index profiles[J]. IEE proceedings, 1988, 135(5): 333-338.
[3] Seki S, Yamanaka T, Yokoyama K. Two-dimensional analysis of optical waveguides with a nonuniform finite difference method[J]. IEE Proceedings, 1991, 138(2): 123-127.
[4] Yeh C, Dong S B, Oliner W. Arbitrarily shaped inhomogeneous optical fiber or integrated optical waveguides[J]. J. Appl. Phys., 1975, 46(5): 2125-2129.
[5] Yeh C, Ha K, Dong S B, Brown W P. Single mode optical waveguides[J]. Appl. Opt., 1979, 18(10): 1490– 1504.
[6] Mabaya N, Lagasse P E, Vendenbulcke P. Finite element analysis of optical waveguides[J]. IEEE Transactions on Microwave theory and techniques, 1981, 29(6): 600-605.
[7] Rahman B M A, Davies J B. Finite Element Solution of integrated optical waveguides[J]. Journal of Lightwave Technology, 1984, 2(5): 682-688.
[8] Koshiba M, Hayata K, Suzuki M. Vectorial finite element formulation without spurious solutions for dielectric waveguide problems[J]. Electron Lett., 1984, 20: 409-410.
[9] Koshiba M, Saitoh H, Eguchi M, Hirayama K. Simple scalar finite element approach to optical rib waveguides[J]. IEE Proceedings, 1991, 139(2): 166-171.
[10] Feit M D, Fleck J A. Light propagation in graded-index optical fibers[J]. Appl. Opt., 1978, 17(24): 3990-3998.
[11] Feit M D, Fleck J A. Computation of mode properties in optical fiber waveguides by a propagating beam method[J]. Appl. Opt., 1980, 19(4): 1154-1164.
[12] Feit M D, Fleck J A. Computation of mode eigenfunctions in graded-index optical fibers bye the propagating beam method[J]. Appl. Opt., 1978, 19(13): 2240-2246.
[13] Hendow S T, Shakir S A. Recursive numerical solution for nonlinear wave propagation in fibers and cylindrically symmetric systems [J]. Appl. Opt., 1986, 25(11): 1759-1764.
[14] Chung Y, Dagli N. An assessment of Finite Difference Beam Propagation Method[J]. IEEE Journal of Quantum Electronics, 1990, 26(8): 1335-1339.
[15] Hadley G R. Transparent boundary condition for beam propagation[J]. Opt. Lett., 1991, 16(9): 624-626.
[16] Yamamuchi J, Ando T, Nakano H. Beam-propagation analysis of optical fibers by alternating direction implicit method[J]. Electron. Lett., 1991, 27(18): 1663-1665.
[17] Hadley G R. Wide-angle beam propagation using Padéapproximant operators[J]. Opt. Lett., 1992, 17(20): 1426-1428.
[18] Hadley G R. Multistep method for wide-angle beam propagation[J]. Opt. Lett.,1992, 17(24): 1743-1745.
[19] Liu J M, Gomelsky L. Vectorial beam propagation method[J]. J. Opt. Soc. Am. A, 1992, 9(9): 1574-1585.
[20] Chan R Y, Liu J M. Time-Domain Wave Propagation in optical Structures.
[21] Yee K S. Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media[J]. IEEE transactions on antennas and propagation, 1966, 14(3): 302-307.
[22] Taflove A., Brodwin M E. Numerical solutions of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations[J]. IEEE transactions on microwave theory and techniques, 1975, 23(8): 623-630.
[23] Taflove A. Application of the finite-difference time-domain method to sinusoidal steady state electromagnetic penetration problems[J]. IEEE transactions on electromagnetic compatibility, 1980, 22: 191-202.
[24] Mur G. Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic field[J]. IEEE transactions on electromagnetic compatibility, 1980, 23: 377-382
[25] Berenger J. A perfectly matched layer for the absorption of electromagnetic waves"[J]. Journal of Computational Physics, 1994, 114: 185–200.
[26] Berenger J. Three-Dimensioal perfectly matched layer for the absorption of electromagnetic waves"[J]. Journal of Computational Physics, 1996, 127: 363–379.
[27] Painter O, Lee R K, Scherer A, Yariv J D, O’Brien J D, Dapkus P D, Kim I. Tow-dimensional photonic band-gap defect mode laser[J]. Science, 1999, 284: 1819-1831.
[28] Taflove A, Hagness S C. Computational electrodynamics: the finite-difference time-domain method[M]. 2nd ed. Norwood: Artech House Inc, 2000.
[29] Cao Y H, Zheng J, Zhang Y S. Numerical modeling of fiber grating[J]. Opt. Int. J. Light Electron. Opt., 2008, doi:10.1016/j.ijleo.2008.02.027
[30] Kenji Kawano, Tsutomu Kitoh, Introduction to Optical Waveguide Analysis: Solving Maxwell’s Equations[M], New York: John Wiley & Sons Inc., 2001.
[31] Dietrich Marcuse, Theory of Dielectric Optical Waveguides[M], New York: Academic Press, 1974.
[32] Hadley G R. Transparent Boundary Condition for the Beam Propagation Method[J]. IEEE Journal of Quantum Electronics, 1992, 28(1): 363-370.
[33] Press W H, Teukosky S A, Vetterling W T, Flannery B P. Numerical Recipes in Fortran 90[M]. 2nd ed. New York: Cambridge University Press, 1996.
[34]张凯院,徐仲.数值代数[M].西安:西北工业大学出版社, 2000.
[35] Hua Y G, Sarkar T K. Matrix Pencil Method for Estimating Parameters of Exponentially Damped/Undamped Sinusoids in Noise[J]. IEEE Transactions on Acoustics, Speech, and Signal Processing, 1990, 38(5): 814-824.
[36] Sarkar T K, Pereira O. Using the Matrix Pencil Method to Estimate the parameters of a Sum of Complex Exponentials[J]. IEEE Antennas and Propagation Magazine, 1995, 37(1): 48-55.
[37] Kalman D. A Singularly Valuable Decomposition: the SVD of a Matrix[J]. The College Mathematics Journal, 1996, 27(1): 2-23.
[38] Takuma Y, Miyagi M, Kawakami S. Bent asymmetric dielectric slab waveguides: a detailed analysis[J]. Applied Optics, 1981, 20(13): 2291-2298.
[39] Saijonmaa J, Yevick D. Beam-propagation analysis of loss in bent optical waveguides and fibers[J]. J. Opt. Soc. Am., 1983, 73(13): 1785-1791.
[40] Koai K T, Liu P L. Modeling of Ti:LiNbO3 Waveguide Devices: Part II—S-Shaped Channel Waveguide Bends[J]. Journal of Lightwave Technology, 1989, 7(7): 1016-1021.
[41] Yamauchi J, Kikuchi S, Hirooka T, Nakano H. Beam-Propagation analysis of bent step-index slab waveguides[J]. Electronics Letters, 1990, 26(12): 822-824.
[42] Rivera M. A Finite Difference BPM analysis of bent dielectric waveguides[J]. Journal of Lightwave Technology, 1995, 13(2): 233-238.
[43] Wang P, Wang Q, Farrell G, et el. Investigation of macrobending losses of standard single mode fiber with small bend radii[J]. Microwave and Optical Technology Letters, 2007, 49(9): 2133-2138.
[44] Heiblem M, Harris J H. Analysis of curved optical waveguides by conformal transformation[J]. IEEE Journal of Quantum Electronics, 1975, 11: 75-83.
[45] Marcuse D. Light transmission optics[M]. 2nd ed. New York: Van Nostrand Reinhold Company, 1982.
[46] Liu P L, Li B J. Study of form birefringence in waveguide devices using the semivectorial beam propagation method[J]. IEEE Photon. Technol. Lett., 1991, 3: 913-915.
[47] Marcuse D. Curvature loss formula for optical fibers[J]. J. Opt. Soc. Am., 1976, 66(3): 216-220.
[48] Hill K. Fiber Bragg grating technology fundamentals and over view[J]. J. Lightwave Technol., 1997, 15: 1263-1276.
[49] Mizrahi V, Sipe J E. Optical properties of photosensitive fiber phase gratings[J]. J. Lightwave Technol., 1993, 11: 1513-1517.
[50] Erdogan T. Fiber grating spectra[J]. J. Lightwave Technol., 1997, 15: 1277-1294.
[51] Erdogan T, Sipe J E. Tilted fiber phase gratings[J]. J. Opt. Soc. Am. A, 1996, 13(2): 296-313.
[52] Bennion I, Williams J A R, Zhang L, Sugden K, Doran N J. UV-written in-fibre Bragg gratings[J]. Optical and Quantum Electronics, 1996, 28: 93-135.
[53] Lebref R, Landousies B, Georges T, Delevaque E. Theoretical Study of the Gain Equalization of a Stabilized Gain EDFA for WDM Applications[J]. Journal of Lightwave Technology, 1997, 15(5): 766-770.
[54] Litchinitser N M, Patterson D B. Analysis of Fiber Bragg Gratings for Dispersion Compensation in Reflective and Transmissive Geoetries[J]. Journal of Lightwave Technology, 1997, 15(8): 1323-1328
[2] Stern M S. Semivectorial polarized H field solutions for dielectric waveguides with arbitrary index profiles[J]. IEE proceedings, 1988, 135(5): 333-338.
[3] Seki S, Yamanaka T, Yokoyama K. Two-dimensional analysis of optical waveguides with a nonuniform finite difference method[J]. IEE Proceedings, 1991, 138(2): 123-127.
[4] Yeh C, Dong S B, Oliner W. Arbitrarily shaped inhomogeneous optical fiber or integrated optical waveguides[J]. J. Appl. Phys., 1975, 46(5): 2125-2129.
[5] Yeh C, Ha K, Dong S B, Brown W P. Single mode optical waveguides[J]. Appl. Opt., 1979, 18(10): 1490– 1504.
[6] Mabaya N, Lagasse P E, Vendenbulcke P. Finite element analysis of optical waveguides[J]. IEEE Transactions on Microwave theory and techniques, 1981, 29(6): 600-605.
[7] Rahman B M A, Davies J B. Finite Element Solution of integrated optical waveguides[J]. Journal of Lightwave Technology, 1984, 2(5): 682-688.
[8] Koshiba M, Hayata K, Suzuki M. Vectorial finite element formulation without spurious solutions for dielectric waveguide problems[J]. Electron Lett., 1984, 20: 409-410.
[9] Koshiba M, Saitoh H, Eguchi M, Hirayama K. Simple scalar finite element approach to optical rib waveguides[J]. IEE Proceedings, 1991, 139(2): 166-171.
[10] Feit M D, Fleck J A. Light propagation in graded-index optical fibers[J]. Appl. Opt., 1978, 17(24): 3990-3998.
[11] Feit M D, Fleck J A. Computation of mode properties in optical fiber waveguides by a propagating beam method[J]. Appl. Opt., 1980, 19(4): 1154-1164.
[12] Feit M D, Fleck J A. Computation of mode eigenfunctions in graded-index optical fibers bye the propagating beam method[J]. Appl. Opt., 1978, 19(13): 2240-2246.
[13] Hendow S T, Shakir S A. Recursive numerical solution for nonlinear wave propagation in fibers and cylindrically symmetric systems [J]. Appl. Opt., 1986, 25(11): 1759-1764.
[14] Chung Y, Dagli N. An assessment of Finite Difference Beam Propagation Method[J]. IEEE Journal of Quantum Electronics, 1990, 26(8): 1335-1339.
[15] Hadley G R. Transparent boundary condition for beam propagation[J]. Opt. Lett., 1991, 16(9): 624-626.
[16] Yamamuchi J, Ando T, Nakano H. Beam-propagation analysis of optical fibers by alternating direction implicit method[J]. Electron. Lett., 1991, 27(18): 1663-1665.
[17] Hadley G R. Wide-angle beam propagation using Padéapproximant operators[J]. Opt. Lett., 1992, 17(20): 1426-1428.
[18] Hadley G R. Multistep method for wide-angle beam propagation[J]. Opt. Lett.,1992, 17(24): 1743-1745.
[19] Liu J M, Gomelsky L. Vectorial beam propagation method[J]. J. Opt. Soc. Am. A, 1992, 9(9): 1574-1585.
[20] Chan R Y, Liu J M. Time-Domain Wave Propagation in optical Structures.
[21] Yee K S. Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media[J]. IEEE transactions on antennas and propagation, 1966, 14(3): 302-307.
[22] Taflove A., Brodwin M E. Numerical solutions of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations[J]. IEEE transactions on microwave theory and techniques, 1975, 23(8): 623-630.
[23] Taflove A. Application of the finite-difference time-domain method to sinusoidal steady state electromagnetic penetration problems[J]. IEEE transactions on electromagnetic compatibility, 1980, 22: 191-202.
[24] Mur G. Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic field[J]. IEEE transactions on electromagnetic compatibility, 1980, 23: 377-382
[25] Berenger J. A perfectly matched layer for the absorption of electromagnetic waves"[J]. Journal of Computational Physics, 1994, 114: 185–200.
[26] Berenger J. Three-Dimensioal perfectly matched layer for the absorption of electromagnetic waves"[J]. Journal of Computational Physics, 1996, 127: 363–379.
[27] Painter O, Lee R K, Scherer A, Yariv J D, O’Brien J D, Dapkus P D, Kim I. Tow-dimensional photonic band-gap defect mode laser[J]. Science, 1999, 284: 1819-1831.
[28] Taflove A, Hagness S C. Computational electrodynamics: the finite-difference time-domain method[M]. 2nd ed. Norwood: Artech House Inc, 2000.
[29] Cao Y H, Zheng J, Zhang Y S. Numerical modeling of fiber grating[J]. Opt. Int. J. Light Electron. Opt., 2008, doi:10.1016/j.ijleo.2008.02.027
[30] Kenji Kawano, Tsutomu Kitoh, Introduction to Optical Waveguide Analysis: Solving Maxwell’s Equations[M], New York: John Wiley & Sons Inc., 2001.
[31] Dietrich Marcuse, Theory of Dielectric Optical Waveguides[M], New York: Academic Press, 1974.
[32] Hadley G R. Transparent Boundary Condition for the Beam Propagation Method[J]. IEEE Journal of Quantum Electronics, 1992, 28(1): 363-370.
[33] Press W H, Teukosky S A, Vetterling W T, Flannery B P. Numerical Recipes in Fortran 90[M]. 2nd ed. New York: Cambridge University Press, 1996.
[34]张凯院,徐仲.数值代数[M].西安:西北工业大学出版社, 2000.
[35] Hua Y G, Sarkar T K. Matrix Pencil Method for Estimating Parameters of Exponentially Damped/Undamped Sinusoids in Noise[J]. IEEE Transactions on Acoustics, Speech, and Signal Processing, 1990, 38(5): 814-824.
[36] Sarkar T K, Pereira O. Using the Matrix Pencil Method to Estimate the parameters of a Sum of Complex Exponentials[J]. IEEE Antennas and Propagation Magazine, 1995, 37(1): 48-55.
[37] Kalman D. A Singularly Valuable Decomposition: the SVD of a Matrix[J]. The College Mathematics Journal, 1996, 27(1): 2-23.
[38] Takuma Y, Miyagi M, Kawakami S. Bent asymmetric dielectric slab waveguides: a detailed analysis[J]. Applied Optics, 1981, 20(13): 2291-2298.
[39] Saijonmaa J, Yevick D. Beam-propagation analysis of loss in bent optical waveguides and fibers[J]. J. Opt. Soc. Am., 1983, 73(13): 1785-1791.
[40] Koai K T, Liu P L. Modeling of Ti:LiNbO3 Waveguide Devices: Part II—S-Shaped Channel Waveguide Bends[J]. Journal of Lightwave Technology, 1989, 7(7): 1016-1021.
[41] Yamauchi J, Kikuchi S, Hirooka T, Nakano H. Beam-Propagation analysis of bent step-index slab waveguides[J]. Electronics Letters, 1990, 26(12): 822-824.
[42] Rivera M. A Finite Difference BPM analysis of bent dielectric waveguides[J]. Journal of Lightwave Technology, 1995, 13(2): 233-238.
[43] Wang P, Wang Q, Farrell G, et el. Investigation of macrobending losses of standard single mode fiber with small bend radii[J]. Microwave and Optical Technology Letters, 2007, 49(9): 2133-2138.
[44] Heiblem M, Harris J H. Analysis of curved optical waveguides by conformal transformation[J]. IEEE Journal of Quantum Electronics, 1975, 11: 75-83.
[45] Marcuse D. Light transmission optics[M]. 2nd ed. New York: Van Nostrand Reinhold Company, 1982.
[46] Liu P L, Li B J. Study of form birefringence in waveguide devices using the semivectorial beam propagation method[J]. IEEE Photon. Technol. Lett., 1991, 3: 913-915.
[47] Marcuse D. Curvature loss formula for optical fibers[J]. J. Opt. Soc. Am., 1976, 66(3): 216-220.
[48] Hill K. Fiber Bragg grating technology fundamentals and over view[J]. J. Lightwave Technol., 1997, 15: 1263-1276.
[49] Mizrahi V, Sipe J E. Optical properties of photosensitive fiber phase gratings[J]. J. Lightwave Technol., 1993, 11: 1513-1517.
[50] Erdogan T. Fiber grating spectra[J]. J. Lightwave Technol., 1997, 15: 1277-1294.
[51] Erdogan T, Sipe J E. Tilted fiber phase gratings[J]. J. Opt. Soc. Am. A, 1996, 13(2): 296-313.
[52] Bennion I, Williams J A R, Zhang L, Sugden K, Doran N J. UV-written in-fibre Bragg gratings[J]. Optical and Quantum Electronics, 1996, 28: 93-135.
[53] Lebref R, Landousies B, Georges T, Delevaque E. Theoretical Study of the Gain Equalization of a Stabilized Gain EDFA for WDM Applications[J]. Journal of Lightwave Technology, 1997, 15(5): 766-770.
[54] Litchinitser N M, Patterson D B. Analysis of Fiber Bragg Gratings for Dispersion Compensation in Reflective and Transmissive Geoetries[J]. Journal of Lightwave Technology, 1997, 15(8): 1323-1328
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