波导中模的渐近解及其在波传播计算中的应用
|
摘要
本文主要论述了以声波导、光波导为背景的Helmholtz方程的数值计算问题,包括渐近解的计算及波的传播计算,其中渐近解的计算包括开放波导泄漏模渐近解的计算和使用了完美匹配层(PML)后泄漏模及Berenger模渐近解的计算。
高精度的泄漏模在波的传播计算中非常重要,尤其是对于光互联等比较精密的仪器。当声速及折射率为分段常数时,泄漏模的渐近解可以通过一个解析的非线性方程得到;但是对于声速随深度变化的声波导以及折射率随横向变化的光波导,无法得到关于特征值的非线性方程。本文首先给出了开放的平板光波导及海洋声波导中泄漏模的渐近解。在这种波导中,运用W.K.B方法,通过对原方程的特征问题进行分析,得到了关于特征值的一个近似的的非线性方程,再对这个方程进行进一步分析得到了泄漏模的渐近解。数值模拟表明,这些渐近解比较接近于泄漏模的精确解,并且随着模的增大,误差越来越小,因此可以作为Rayleigh商迭代的初值求解更精确的泄漏模。
开放的波导结构在数值计算时的空间区域是没有边界的,现在比较有效的方法是采用完美匹配层把开放的波导区域截断为有界的区域。在加了完美匹配层之后,除了传播模和泄漏模之外,还会产生一种模,称之为Berenger模。为了分析完美匹配层对波的各种模式的影响,本文还讨论了加了完美匹配层之后对传播计算比较重要的泄漏模的渐近解,此外,粗略地推导了Berenger模的渐近公式。结果表明,在采用了完美匹配层之后,对于声波导和光波导的TE模,略去高阶无穷小量,泄漏模的渐近公式和开放的波导是一样的;对于TM模,略去高阶无穷小量,泄漏模的渐近公式和开放的波导会有所不同。对于光波导,上下包层的折射率若是一样的话,Berenger模的轨迹只有一条;若不一样的话,Berenger模的轨迹有两条。数值模拟表明,这些泄漏模及Berenger模的渐近解在其精确解附近,因此可以作为Rayleigh商迭代的初值来求解更精确的模。
本文最后做了一些把以上所得的渐近解应用于波的传播计算的工作,主要做了两种情况下的波传播的模拟:一种是声波的波数及光波的折射率仅随一个方向变化的情况,另一种是波数及折射率随两个方向而变化的情况。数值模拟表明,把渐近解应用于波的传播计算中得到的结果和以前的方法相比,结果得到明显改善,并且计算速度快很多。对于光波导,本文还比较了未选取Berenger模和选取了Berenger模时波的传播。数值模拟结果表明,Berenger模对波的传播会产生轻微的影响。
Leaky modes play an important role in wave propagation computation of optical and acoustic waveguide ,especially in the on-chip optical interconnection proposed re-cently. They can be used to partially represent the wave field related to the continuous spectrum of the radiation and evanescent modes.
For two-dimensional step-index waveguides,the propagation constants satisfy an analysis nonlinear equation,from which asymptotic solutions of leaky modes have been given. B-ut when the refractive index n is varied with the transverse variable x in optical waveguide or acoustic velocity c is varied with z in acoustic waveguide respectively,analysis nonlinear equation of the leaky modes can not be obtained. Asymptotic solutions of leaky modes in unbounded waveguide are given first. In order to get asymptotic solutions of the leaky modes, W.K.B. method is implemented to create an approximate nonlinear equa-tion. Numerical simulations show that our asymptotic solutions are very close to exact leaky modes and the errors become smaller as the norm of the modes become larger. Our asymptotic solutions can be used as initial guesses for computing the more accurate leaky modes by Rayleigh Quotient iteration .
PML as an efficient artificial absorbing material is widely used to solve unbounded wave propagation problem, in which Berenger modes(the eigenmodes in a waveguide terminated by a finite PML) appear. High order asymptotic solutions for the Leaky modes and Berenger modes are derived in waveguide with PML. Asymptotic solutions of the modes are useful,because they can be used as initial guesses for solving the exact eigenvalues numerically.
Wave propagation computation is implemented when Leaky modes are obtained by asymptotic solutions which are used as initial guesses for Rayleigh Quotation itera-tion. Two cases are considered: first, refractive index is varied with transverse variable and acoustic velocity c is varied with depth,second, they are all varied with two directions. Numerical simulations show that results from this method are better than the methods from Lapack with highly computation efficiently.
For optical waveguide,wave propagation computation are compared with or without Berenger modes. Numerical simulations show that results from these two ways will have a slight difference.
高精度的泄漏模在波的传播计算中非常重要,尤其是对于光互联等比较精密的仪器。当声速及折射率为分段常数时,泄漏模的渐近解可以通过一个解析的非线性方程得到;但是对于声速随深度变化的声波导以及折射率随横向变化的光波导,无法得到关于特征值的非线性方程。本文首先给出了开放的平板光波导及海洋声波导中泄漏模的渐近解。在这种波导中,运用W.K.B方法,通过对原方程的特征问题进行分析,得到了关于特征值的一个近似的的非线性方程,再对这个方程进行进一步分析得到了泄漏模的渐近解。数值模拟表明,这些渐近解比较接近于泄漏模的精确解,并且随着模的增大,误差越来越小,因此可以作为Rayleigh商迭代的初值求解更精确的泄漏模。
开放的波导结构在数值计算时的空间区域是没有边界的,现在比较有效的方法是采用完美匹配层把开放的波导区域截断为有界的区域。在加了完美匹配层之后,除了传播模和泄漏模之外,还会产生一种模,称之为Berenger模。为了分析完美匹配层对波的各种模式的影响,本文还讨论了加了完美匹配层之后对传播计算比较重要的泄漏模的渐近解,此外,粗略地推导了Berenger模的渐近公式。结果表明,在采用了完美匹配层之后,对于声波导和光波导的TE模,略去高阶无穷小量,泄漏模的渐近公式和开放的波导是一样的;对于TM模,略去高阶无穷小量,泄漏模的渐近公式和开放的波导会有所不同。对于光波导,上下包层的折射率若是一样的话,Berenger模的轨迹只有一条;若不一样的话,Berenger模的轨迹有两条。数值模拟表明,这些泄漏模及Berenger模的渐近解在其精确解附近,因此可以作为Rayleigh商迭代的初值来求解更精确的模。
本文最后做了一些把以上所得的渐近解应用于波的传播计算的工作,主要做了两种情况下的波传播的模拟:一种是声波的波数及光波的折射率仅随一个方向变化的情况,另一种是波数及折射率随两个方向而变化的情况。数值模拟表明,把渐近解应用于波的传播计算中得到的结果和以前的方法相比,结果得到明显改善,并且计算速度快很多。对于光波导,本文还比较了未选取Berenger模和选取了Berenger模时波的传播。数值模拟结果表明,Berenger模对波的传播会产生轻微的影响。
Leaky modes play an important role in wave propagation computation of optical and acoustic waveguide ,especially in the on-chip optical interconnection proposed re-cently. They can be used to partially represent the wave field related to the continuous spectrum of the radiation and evanescent modes.
For two-dimensional step-index waveguides,the propagation constants satisfy an analysis nonlinear equation,from which asymptotic solutions of leaky modes have been given. B-ut when the refractive index n is varied with the transverse variable x in optical waveguide or acoustic velocity c is varied with z in acoustic waveguide respectively,analysis nonlinear equation of the leaky modes can not be obtained. Asymptotic solutions of leaky modes in unbounded waveguide are given first. In order to get asymptotic solutions of the leaky modes, W.K.B. method is implemented to create an approximate nonlinear equa-tion. Numerical simulations show that our asymptotic solutions are very close to exact leaky modes and the errors become smaller as the norm of the modes become larger. Our asymptotic solutions can be used as initial guesses for computing the more accurate leaky modes by Rayleigh Quotient iteration .
PML as an efficient artificial absorbing material is widely used to solve unbounded wave propagation problem, in which Berenger modes(the eigenmodes in a waveguide terminated by a finite PML) appear. High order asymptotic solutions for the Leaky modes and Berenger modes are derived in waveguide with PML. Asymptotic solutions of the modes are useful,because they can be used as initial guesses for solving the exact eigenvalues numerically.
Wave propagation computation is implemented when Leaky modes are obtained by asymptotic solutions which are used as initial guesses for Rayleigh Quotation itera-tion. Two cases are considered: first, refractive index is varied with transverse variable and acoustic velocity c is varied with depth,second, they are all varied with two directions. Numerical simulations show that results from this method are better than the methods from Lapack with highly computation efficiently.
For optical waveguide,wave propagation computation are compared with or without Berenger modes. Numerical simulations show that results from these two ways will have a slight difference.
引文
[1] A.Arnold and M.Ehrhaedt. Discrete transparent boundary conditions for wide angel parabolic equations in underwater acoustics. J.Comput.Phys. 1998,145(2):611-638.
[2] A. D. Pierce. Extension of the method of normal modes to sound propagation in an almost stratified medium. J. Acoust. Soc. Am., 1965,37(1): 19-27.
[3] AW.Snyder and JD.Love. Optical waveguide theory. London,UK:Chapman and Hall,1983.
[4] B.Chen,D.G.Fang and B.H.Zhou.Modified Berenger PML absorbing boundary layers. IEEE Microwave and Guided Wave Letters,1995,5(11):399-401.
[5] B.Chen and D.G.Fang. Modified Berenger PML absorbing boundary condition for FD-TD meshes. IEEE Microwave and Guided Wave Letters, 1995,5(11):399-401.
[6] B.N.Parlett.The rayleigh quotient iteration and some generalizations for nonnormal matrices. Mathematics of computation,1974,28(127):679-693.
[7] C.A.Boyles. Coupled mode solution for a cylindricall symmetric oceanic waveguide with a range and depth dependent refractive index and a time varying rough sea surface[J]. Jounal of Acoustical Society of America, 1989,86(4): 1097-1102.
[8] Cheng-kuei Jen,Safaai Jazi and Gerald W. farnell. Leaky modes in weakly guiding fiber acoustic waveguides. IEEE Transactions on ultrasonics.Ferroeletrics.and Frequency control,1986,33(6):634-643.
[9] C.M.Rappaport. Perfectly matched absorbing boundary-conditions based on anisotropic lossy mapping of space. IEEE Microwave and Guided Wave Letters, 1995,5(3):90-92.
[10] C.Vassallo and F.Collino. Highly efficient absorbing boundary conditions for the beam propagation method. Journal of Lightwave Technology, 1996,14(6): 1570-1577.
[11] D.C. Stickler and E.Ammicht. Uniform asymptotic evaluation of the continuous spectrum for the pekeris model. J.Acoust.Soc.Am.,1980,67:2018-2024.
[12]D.J.Thomson.Wide-angle parabolic equation solutions to two range-dependent benchmark problems,J.Acoust.Soc.Am,1990,87(4),1514-1520.
[13]D.Yevick and D.J.Thomson.Impedence-matched absorbers for finite-difference parabolic equation algorithms.J.Acoust.Soc.Am.,2000,107(3):1226-1234.
[14]D.Yevick,D.J.Thomson.Nonlocal boundary conditions for finite-difference parabolic equation solvers.J.Acoust.Soc.Am,1999,106(1):143-150.
[15]D.Yevick,D.J.Yu and F.Schmidt.Analytic studies of absorbing and impedance-matched boundary layers.IEEE Photonics of technology Letters,1997,9(1):73-75.
[16]D.Yevick and D.J.Thomson.Complex padé approximants for wide-angel acoustic propagators.J.Acous.Soc.Am,2000(b),108(6),2784-2790.
[17]E.Ammicht and D.C.Stickler.Uniform asymptotic evaluation of the continuous spectrum contribution for a stratified ocean.J.Acoust.Soc.Am.,1994,76:186-191.
[18]E.K.Westwood and R.A.Koch.Elimination of branch cuts from the normal-mode solutions using gradient half spaces.J.Acoust.Soc.Am.1999,106(5):2513-2523.
[19]F.A.Milinazzo,C.A.Zala and G.H.Brooke.Rational square-root approxiamtions for parabolic equation-algorithms.J.Acoust.Soc.Am.1997,101:760-766.
[20]F.B.Jensen.On the use of stair steps to approximate bathymetry changes in ocean waveguides,in:D.Lee,et al.(Eds).Theoretical and Computational Acoustics'95,World Scientific,Singapore,1995.
[21]F.B.Jensen and C.M.Ferla.Numerical solutions of range-dependent benchmark problem.J.Acoust.Soc.Am.1990,87:1499-1510.
[22]F.B.Jensen,W.A.Kuperman,M.B.Porter and H.Schmidt.Computational Ocean Acoustics.American Institute of Physics,1994.
[23]F.D.Tappert.The parabolic approximation method.Wave propagation and underwater acoustics,Springer Verlag,New York,1977,70:224-287.
[24]Francis Collino.Perfectly matched absorbing layers for the paraxial equations.Journal of Computational Physics,1997,131:164-180.
[25]F.Schmidt,T.Friese and D.Yevick.Transparent boundary conditions for split-step Padé approximations of the one-way Helmholtz equation.J.Comput.Phys.2001,170(2):696-719.
[26] G.V.Frisk. Ocean and Seabed Acoustics:A theory of wave propagation. Prentice Hall,Engelwood Cliffs,NJ,1994.
[27] G.Zhang. High order approximation one-way wave equations. J.Comput.Math.1985,3:90-97 .
[28] H.Derudder,D.DeZutter and F.Olyslager. Analysis of waveguide discontinuities using perfectly matched layers. Electronics Letters, 1998,34(22):2138-2140.
[29] H.Derudder,F.Olyslager and D.De Zutter. An efficient series expansion for 2-D Green's function of a microstrip substrate using perfectly matched layers. IEEE Microwave Guided Wave Lett, 1999,9(12):505-507.
[30] H.Rogier and D.DeZutter. Berenger and leaky modes in microstrip substrates terminated by a perfectly matched layer. IEEE Transactions on Microwave Theory and Techniques,2001,49(4):712-715.
[31] H.Rogier and D.DeZutter. Convergence behavior and accelaration of the berenger and leaky modes series composing the 2-D Green's function for the microstrip substrate. IEEE Transactions on Microwave Theory and Techniques,2002,50(7):1696-1704.
[32] H.Rogier ,L.Knockaert and D.Dezutter.Fast calculation of the propagation constants of leaky and Berenger modes of planar and circular dielectric waveguides terminited by a perfectly matched layer. Microwave Optical Technol.Letter,2003,37(3):161-171.
[33] H.Rogier and D.DeZutter. Berenger and leaky modes in optical fibers terminated with a perfectly matched layer. Journal of Lightwave and Technology,2002,20(7):l 141-1148.
[34] H.Wong,V.Filip,CK.Wong and PS.Chung. Silicon integrated photonic begins to revolution. Microelectronics Reliability,2007,47:1-10.
[35] J.A.DeSanto. Scalar wave theory-Green's functions and applications. Springer Ver-lag,New York,1992.
[36] J.P.Beneger. A Perfectly matched layer for the absorption of electromagnetic-waves. J. Comput.Phys., 1994,114(2):185-200.
[37]J.S.Papadakis,M.I.Taroudakis,P.J.Papadakis and B.Mayfield.A new method for a realistic treatment of the sea bottom in the parabolic approximation.J.Acoust.Soc.Am,1992,92(4):2030-2038.
[38]Jianxin Zhu,Ya Yan Lu.Validity of one-way models in the weak range dependence limit.Journal of Computational Acoustics,2004,12(1):55-66.
[39]J.X.Zhu and Q.X.Zhou.Eigenvalue computation in slab waveguides with some perfectly matched layers.Proc SPIE,20055279:172-180.
[40]J.X.Zhu and Y.Y.Lu.Asymptotic solutions of the leaky modes and PML modes in a pekeris waveguide,wave motion 2008,45:207-216.
[41]J.X.Zhu and Y.Y.Lu.Leaky modes of slab waveguides-asymptotic solutions.IEEE J.Lightwave.Technol.2006,24(3):1619-1623.
[42]J.X.Zhu and Y.Y.Lu.Large range step method for acoustic waveguide with two layer media.Progress in natural science,2002,12(11):820-825.
[43]J.Y.Fang and Z.H.Wu.Generalized perfectly matched layer - an extension of Berengers perfectly matched layer boundary-condition.IEEE Microwave and Guided Wave Letters,1995,5(12):451-453.
[44]L.Fishman.One-way wave propagation methods in direct and inverse scalar wave propagation modeling.Radio Science,1993,28(5),865-876.
[45]L.Fishman,A.K.Gautesen and Z.Sun.Uniform high-frequency approximation of the square root Helmholtz operator symbol.Wave Motion,1997,26(2):127-161.
[46]L.Knockaert,H.Rogier and D.DeZutter.An FFT-based signal identification approach for obtaining the propagation constants of the leaky modes in layered media.AEU,Int.J.Electron Commun,2005,59(4):230-238.
[47]Louis Fishman,Maarten V.de Hoop,and Mattheus J.N.van Stralen.Exact constructions of square-root Helmholtz operator symbols:The focusing quadratic profile.Journal of Mathematical Physics,2000,41(7):4881-4938.
[48]M.A.Pedersen and D.F.Gordon.Theoretical investigation of a double family of normal modes in an underwater acoustic surface duct.The Journal of the Acoustical Society of America,1970,47(1,2):304-326.
[49] M. B. Porter, C. M. Ferla and F. B. Jensen. The problem of energy conservation in one-way equations. J. Acoust. Soc. Am. 1991,89:1058-1067.
[50] M. D. Collins. A split-step Pade solution for the parabolic equation method. J.Acoust. Soc. Am., 1993,93(4): 1736-1742.
[51] M. D. Collins. Applications and time-domain solutions of higher-order parabolic equations in underwater acoustics. J. Acoust. Soc. Am., 1989,86(4): 1097-1102.
[52] M.Lassas and E.Somersalo. On the existence and convergence of the solution of PML equations. Computing, 1998,60:229-241.
[53] O.Hassan,K.Morgan,J.Jones and B.Larwood. Parallel 3D time domain electromagnetic scattering simulations on unstructured meshes. Computer Modeling in Engineering and Science,2004,5(5):383-394.
[54] P. Bienstman and R.Baets. Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers. Optical and Quantum Electronics,2001,33(4-5):327-341.
[55] P.L.Ho and Y.Y.LU. A mode-preserving perfectly matched layer for optical waveguides, IEEE Photonics Technology Letters,2003,15(9):1234-1236.
[56] P.L.Ho and Y.Y.LU. Improving the beam propagation method for TM polarization. Opt.Quantum Electron,2003,35(4):507-519.
[57] P. Monk and D. Wang. A least squares method for the Helmholtz equation. Computer Methods in Applied Mechanics and Engineering, 1999 ,175:121-136.
[58] R.A.Stephen. Solutions to range-dependent benchmark problems by the finite-difference method. J.Acoust.SocAm., 1990,87:1527-1534.
[59] R.B.Evans. A coupled mode solution for acoustic propagation in a waveguide with stepwise depth variations of penetrable bottom. Journal of the Acoustical Society of America, 1983,74(1): 188-195.
[60] R.Evans. A coupled mode solution for acoustic propogation in waveguide with stepwise variations of penetrable bottom. Jounal of Acoustical Society of America,1983,74:188-195.
[61] R.M.Corless,G.H.Gonnet,DEG.Hare,D.J.Jeffrey and D.E.Knuth. On the Lambert W function. Adv Comput Math,1996,5(4):329-359.
[62]R.R.Greene.The rational approximation to the acoustic-wave equation with bottom interaction.J.Acoust.Soc.Am.,1984,76(6):1764-1773.
[63]RZL,Ye and D.Yevick.Noniterative caculation of complex propagation constants in planar waveguides.J Opt Soc Am A,Opt Image Sci,2001,18(11):2819-2822.
[64]Saul Abarbanel and David Gottieb.A mathematical analysis of the PML method.Journal of Computational Physics,1997,134:357-363.
[65]SB.Gaal,HJWM.Hoekstra and PV.Lambeck.Determining PML modes in 2-D stratified media.J Lightwave Technol,2003,21(1):293-298.
[66]T.Friese,F.Schmidt and D.Yevick.Transparent boundary conditions for a wide-angel approximation of the one-way Helmholtz equation.J.Comput.Phys,2000,165(2):645-659.
[67]T.Ha,S.Seo and D.Sbeen.Parallel iterative procedures for computational electromagnetic modelling based on a nonconforming mixed finite element method.Computer Modeling in Engineering and Science,2006,14(1):57-76.
[68]V.V.Shevchenko.The expansion of the fields of open waveguides in proper and improper modes.Radio Phys.Quantum Electron,1974,14:972-977.
[69]W.C.Chew and W.H.Weedon.A 3D perfectly matched medium from modified Maxwells equations with stretched coordinates.Microwave and Optical Technology Letters,1994,7(13):599-604.
[70]W.P.Huang,C.L.Xu,W.Lui and K.Yokoyama.The perfectly matched layer(PML)boundary condition for the beam propagation method.IEEE Photonics Technology Letters,1996,8(5):649-651.
[71]Ya Yan Lu,Jianxin Zhu.A Local Orthogonal Transform for Acoustic Waveguides with an Internal Interface.Journal of Computational Acoustics,2004,12(1):37-53.
[72]Ya Yan Lu.A complex coefficient rational approximation of(1+x)~(1/2).Applied Numerical Mathematics,1998,27(2):141-154.
[73]Ya Yan Lu.Computing a matrix function for exponential integrators.Journal of Computational and Applied Mathematics,2003(12),161(1):203-216.
[74]Ya Yan Lu.Exact one-way methods for acoustics waveguides.Mathematics and Computers in simulation 1999(50):377-391.
[75]Ya Yan Lu.Fourth order conservative exponential methods for linear evolution equations.BIT,2003,40(4):762-774.
[76]Ya Yan Lu.A fourth order magnus scheme for Helmholtz equation.Journal of Computational and Applied Mathematic,2005,173:247-258.
[77]Ya Yan Lu,Jinyanghuang,Joyce McLaughin.Local orthogonal transformation and one-way methods for acoustics waveguides,wave motion 2001(34):193-207.
[78]Ya Yan Lu.One-way large range step methods for Helmholtz waveguides.Journal of Computational Physics 1999,152:231-250.
[79]Ya Yan Lu and Pui Lin Ho.A Single Scatter Improvement for Beam Propagation Methods.IEEE Photon ics Technology Letters,2002:8,14(8):1103-1105.
[80]Ya Yan Lu,JOyce R.McLaughin.The Riccati method for the Helmholtz equation.Journal of the Acoustical society of America,1996,100(3):1432-1446.
[81]Y.Y.Lu.One-way large range step methods for Helmholtz waveguides.Journal of Computational Physics,1999,152(1):231-250.
[82]Y.Y.Lu and J.X.Zhu.Large range step method for acoustic waveguide with two layer media.Progress in natural science,2002,12(11):820-825.
[83]Y.Y.Lu and J.X.Zhu.Propagating modes in optical waveguides terminated by Perfectly Matched Layers.IEEE Photonics Technology Letters,2005,17(12):2601-2603.
[84]Y.Y.Lu and P.L.ho.Beam propagation method using a[(p-1)/p]pad é approximation of the propagator.Optics Letters,2002,27(9):683-685.
[85]Y.Y.Lu and J.X.Zhu.Perfectly matched layer for acoustic waveguide modelingbenchmark calculations and perturbation analysis.Computer Modeling in Engineering and Sciences,2007,22(3):235-247.
[86]董光昌,陈仲慈,汤国桢,数学物理方程(第一版),浙江大学出版社。
[87]G.H.戈卢布,C.F.范洛恩,矩阵计算,科学出版社。
[88]黄越夏,沈瑞敏,求解Helmholtz方程的一类非线性局部正交变换。浙江大学学报理学版,2004:9,31(5):498-502。
[89]李荣华,冯国忱,偏微分方程数值解法(第三版)。高等教育出版社,1996。
[90]刘伯胜,雷家煜,水声学原理,哈尔滨工程大学出版社。
[91]徐树方,矩阵计算的理论和方法,北京大学出版社。
[2] A. D. Pierce. Extension of the method of normal modes to sound propagation in an almost stratified medium. J. Acoust. Soc. Am., 1965,37(1): 19-27.
[3] AW.Snyder and JD.Love. Optical waveguide theory. London,UK:Chapman and Hall,1983.
[4] B.Chen,D.G.Fang and B.H.Zhou.Modified Berenger PML absorbing boundary layers. IEEE Microwave and Guided Wave Letters,1995,5(11):399-401.
[5] B.Chen and D.G.Fang. Modified Berenger PML absorbing boundary condition for FD-TD meshes. IEEE Microwave and Guided Wave Letters, 1995,5(11):399-401.
[6] B.N.Parlett.The rayleigh quotient iteration and some generalizations for nonnormal matrices. Mathematics of computation,1974,28(127):679-693.
[7] C.A.Boyles. Coupled mode solution for a cylindricall symmetric oceanic waveguide with a range and depth dependent refractive index and a time varying rough sea surface[J]. Jounal of Acoustical Society of America, 1989,86(4): 1097-1102.
[8] Cheng-kuei Jen,Safaai Jazi and Gerald W. farnell. Leaky modes in weakly guiding fiber acoustic waveguides. IEEE Transactions on ultrasonics.Ferroeletrics.and Frequency control,1986,33(6):634-643.
[9] C.M.Rappaport. Perfectly matched absorbing boundary-conditions based on anisotropic lossy mapping of space. IEEE Microwave and Guided Wave Letters, 1995,5(3):90-92.
[10] C.Vassallo and F.Collino. Highly efficient absorbing boundary conditions for the beam propagation method. Journal of Lightwave Technology, 1996,14(6): 1570-1577.
[11] D.C. Stickler and E.Ammicht. Uniform asymptotic evaluation of the continuous spectrum for the pekeris model. J.Acoust.Soc.Am.,1980,67:2018-2024.
[12]D.J.Thomson.Wide-angle parabolic equation solutions to two range-dependent benchmark problems,J.Acoust.Soc.Am,1990,87(4),1514-1520.
[13]D.Yevick and D.J.Thomson.Impedence-matched absorbers for finite-difference parabolic equation algorithms.J.Acoust.Soc.Am.,2000,107(3):1226-1234.
[14]D.Yevick,D.J.Thomson.Nonlocal boundary conditions for finite-difference parabolic equation solvers.J.Acoust.Soc.Am,1999,106(1):143-150.
[15]D.Yevick,D.J.Yu and F.Schmidt.Analytic studies of absorbing and impedance-matched boundary layers.IEEE Photonics of technology Letters,1997,9(1):73-75.
[16]D.Yevick and D.J.Thomson.Complex padé approximants for wide-angel acoustic propagators.J.Acous.Soc.Am,2000(b),108(6),2784-2790.
[17]E.Ammicht and D.C.Stickler.Uniform asymptotic evaluation of the continuous spectrum contribution for a stratified ocean.J.Acoust.Soc.Am.,1994,76:186-191.
[18]E.K.Westwood and R.A.Koch.Elimination of branch cuts from the normal-mode solutions using gradient half spaces.J.Acoust.Soc.Am.1999,106(5):2513-2523.
[19]F.A.Milinazzo,C.A.Zala and G.H.Brooke.Rational square-root approxiamtions for parabolic equation-algorithms.J.Acoust.Soc.Am.1997,101:760-766.
[20]F.B.Jensen.On the use of stair steps to approximate bathymetry changes in ocean waveguides,in:D.Lee,et al.(Eds).Theoretical and Computational Acoustics'95,World Scientific,Singapore,1995.
[21]F.B.Jensen and C.M.Ferla.Numerical solutions of range-dependent benchmark problem.J.Acoust.Soc.Am.1990,87:1499-1510.
[22]F.B.Jensen,W.A.Kuperman,M.B.Porter and H.Schmidt.Computational Ocean Acoustics.American Institute of Physics,1994.
[23]F.D.Tappert.The parabolic approximation method.Wave propagation and underwater acoustics,Springer Verlag,New York,1977,70:224-287.
[24]Francis Collino.Perfectly matched absorbing layers for the paraxial equations.Journal of Computational Physics,1997,131:164-180.
[25]F.Schmidt,T.Friese and D.Yevick.Transparent boundary conditions for split-step Padé approximations of the one-way Helmholtz equation.J.Comput.Phys.2001,170(2):696-719.
[26] G.V.Frisk. Ocean and Seabed Acoustics:A theory of wave propagation. Prentice Hall,Engelwood Cliffs,NJ,1994.
[27] G.Zhang. High order approximation one-way wave equations. J.Comput.Math.1985,3:90-97 .
[28] H.Derudder,D.DeZutter and F.Olyslager. Analysis of waveguide discontinuities using perfectly matched layers. Electronics Letters, 1998,34(22):2138-2140.
[29] H.Derudder,F.Olyslager and D.De Zutter. An efficient series expansion for 2-D Green's function of a microstrip substrate using perfectly matched layers. IEEE Microwave Guided Wave Lett, 1999,9(12):505-507.
[30] H.Rogier and D.DeZutter. Berenger and leaky modes in microstrip substrates terminated by a perfectly matched layer. IEEE Transactions on Microwave Theory and Techniques,2001,49(4):712-715.
[31] H.Rogier and D.DeZutter. Convergence behavior and accelaration of the berenger and leaky modes series composing the 2-D Green's function for the microstrip substrate. IEEE Transactions on Microwave Theory and Techniques,2002,50(7):1696-1704.
[32] H.Rogier ,L.Knockaert and D.Dezutter.Fast calculation of the propagation constants of leaky and Berenger modes of planar and circular dielectric waveguides terminited by a perfectly matched layer. Microwave Optical Technol.Letter,2003,37(3):161-171.
[33] H.Rogier and D.DeZutter. Berenger and leaky modes in optical fibers terminated with a perfectly matched layer. Journal of Lightwave and Technology,2002,20(7):l 141-1148.
[34] H.Wong,V.Filip,CK.Wong and PS.Chung. Silicon integrated photonic begins to revolution. Microelectronics Reliability,2007,47:1-10.
[35] J.A.DeSanto. Scalar wave theory-Green's functions and applications. Springer Ver-lag,New York,1992.
[36] J.P.Beneger. A Perfectly matched layer for the absorption of electromagnetic-waves. J. Comput.Phys., 1994,114(2):185-200.
[37]J.S.Papadakis,M.I.Taroudakis,P.J.Papadakis and B.Mayfield.A new method for a realistic treatment of the sea bottom in the parabolic approximation.J.Acoust.Soc.Am,1992,92(4):2030-2038.
[38]Jianxin Zhu,Ya Yan Lu.Validity of one-way models in the weak range dependence limit.Journal of Computational Acoustics,2004,12(1):55-66.
[39]J.X.Zhu and Q.X.Zhou.Eigenvalue computation in slab waveguides with some perfectly matched layers.Proc SPIE,20055279:172-180.
[40]J.X.Zhu and Y.Y.Lu.Asymptotic solutions of the leaky modes and PML modes in a pekeris waveguide,wave motion 2008,45:207-216.
[41]J.X.Zhu and Y.Y.Lu.Leaky modes of slab waveguides-asymptotic solutions.IEEE J.Lightwave.Technol.2006,24(3):1619-1623.
[42]J.X.Zhu and Y.Y.Lu.Large range step method for acoustic waveguide with two layer media.Progress in natural science,2002,12(11):820-825.
[43]J.Y.Fang and Z.H.Wu.Generalized perfectly matched layer - an extension of Berengers perfectly matched layer boundary-condition.IEEE Microwave and Guided Wave Letters,1995,5(12):451-453.
[44]L.Fishman.One-way wave propagation methods in direct and inverse scalar wave propagation modeling.Radio Science,1993,28(5),865-876.
[45]L.Fishman,A.K.Gautesen and Z.Sun.Uniform high-frequency approximation of the square root Helmholtz operator symbol.Wave Motion,1997,26(2):127-161.
[46]L.Knockaert,H.Rogier and D.DeZutter.An FFT-based signal identification approach for obtaining the propagation constants of the leaky modes in layered media.AEU,Int.J.Electron Commun,2005,59(4):230-238.
[47]Louis Fishman,Maarten V.de Hoop,and Mattheus J.N.van Stralen.Exact constructions of square-root Helmholtz operator symbols:The focusing quadratic profile.Journal of Mathematical Physics,2000,41(7):4881-4938.
[48]M.A.Pedersen and D.F.Gordon.Theoretical investigation of a double family of normal modes in an underwater acoustic surface duct.The Journal of the Acoustical Society of America,1970,47(1,2):304-326.
[49] M. B. Porter, C. M. Ferla and F. B. Jensen. The problem of energy conservation in one-way equations. J. Acoust. Soc. Am. 1991,89:1058-1067.
[50] M. D. Collins. A split-step Pade solution for the parabolic equation method. J.Acoust. Soc. Am., 1993,93(4): 1736-1742.
[51] M. D. Collins. Applications and time-domain solutions of higher-order parabolic equations in underwater acoustics. J. Acoust. Soc. Am., 1989,86(4): 1097-1102.
[52] M.Lassas and E.Somersalo. On the existence and convergence of the solution of PML equations. Computing, 1998,60:229-241.
[53] O.Hassan,K.Morgan,J.Jones and B.Larwood. Parallel 3D time domain electromagnetic scattering simulations on unstructured meshes. Computer Modeling in Engineering and Science,2004,5(5):383-394.
[54] P. Bienstman and R.Baets. Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers. Optical and Quantum Electronics,2001,33(4-5):327-341.
[55] P.L.Ho and Y.Y.LU. A mode-preserving perfectly matched layer for optical waveguides, IEEE Photonics Technology Letters,2003,15(9):1234-1236.
[56] P.L.Ho and Y.Y.LU. Improving the beam propagation method for TM polarization. Opt.Quantum Electron,2003,35(4):507-519.
[57] P. Monk and D. Wang. A least squares method for the Helmholtz equation. Computer Methods in Applied Mechanics and Engineering, 1999 ,175:121-136.
[58] R.A.Stephen. Solutions to range-dependent benchmark problems by the finite-difference method. J.Acoust.SocAm., 1990,87:1527-1534.
[59] R.B.Evans. A coupled mode solution for acoustic propagation in a waveguide with stepwise depth variations of penetrable bottom. Journal of the Acoustical Society of America, 1983,74(1): 188-195.
[60] R.Evans. A coupled mode solution for acoustic propogation in waveguide with stepwise variations of penetrable bottom. Jounal of Acoustical Society of America,1983,74:188-195.
[61] R.M.Corless,G.H.Gonnet,DEG.Hare,D.J.Jeffrey and D.E.Knuth. On the Lambert W function. Adv Comput Math,1996,5(4):329-359.
[62]R.R.Greene.The rational approximation to the acoustic-wave equation with bottom interaction.J.Acoust.Soc.Am.,1984,76(6):1764-1773.
[63]RZL,Ye and D.Yevick.Noniterative caculation of complex propagation constants in planar waveguides.J Opt Soc Am A,Opt Image Sci,2001,18(11):2819-2822.
[64]Saul Abarbanel and David Gottieb.A mathematical analysis of the PML method.Journal of Computational Physics,1997,134:357-363.
[65]SB.Gaal,HJWM.Hoekstra and PV.Lambeck.Determining PML modes in 2-D stratified media.J Lightwave Technol,2003,21(1):293-298.
[66]T.Friese,F.Schmidt and D.Yevick.Transparent boundary conditions for a wide-angel approximation of the one-way Helmholtz equation.J.Comput.Phys,2000,165(2):645-659.
[67]T.Ha,S.Seo and D.Sbeen.Parallel iterative procedures for computational electromagnetic modelling based on a nonconforming mixed finite element method.Computer Modeling in Engineering and Science,2006,14(1):57-76.
[68]V.V.Shevchenko.The expansion of the fields of open waveguides in proper and improper modes.Radio Phys.Quantum Electron,1974,14:972-977.
[69]W.C.Chew and W.H.Weedon.A 3D perfectly matched medium from modified Maxwells equations with stretched coordinates.Microwave and Optical Technology Letters,1994,7(13):599-604.
[70]W.P.Huang,C.L.Xu,W.Lui and K.Yokoyama.The perfectly matched layer(PML)boundary condition for the beam propagation method.IEEE Photonics Technology Letters,1996,8(5):649-651.
[71]Ya Yan Lu,Jianxin Zhu.A Local Orthogonal Transform for Acoustic Waveguides with an Internal Interface.Journal of Computational Acoustics,2004,12(1):37-53.
[72]Ya Yan Lu.A complex coefficient rational approximation of(1+x)~(1/2).Applied Numerical Mathematics,1998,27(2):141-154.
[73]Ya Yan Lu.Computing a matrix function for exponential integrators.Journal of Computational and Applied Mathematics,2003(12),161(1):203-216.
[74]Ya Yan Lu.Exact one-way methods for acoustics waveguides.Mathematics and Computers in simulation 1999(50):377-391.
[75]Ya Yan Lu.Fourth order conservative exponential methods for linear evolution equations.BIT,2003,40(4):762-774.
[76]Ya Yan Lu.A fourth order magnus scheme for Helmholtz equation.Journal of Computational and Applied Mathematic,2005,173:247-258.
[77]Ya Yan Lu,Jinyanghuang,Joyce McLaughin.Local orthogonal transformation and one-way methods for acoustics waveguides,wave motion 2001(34):193-207.
[78]Ya Yan Lu.One-way large range step methods for Helmholtz waveguides.Journal of Computational Physics 1999,152:231-250.
[79]Ya Yan Lu and Pui Lin Ho.A Single Scatter Improvement for Beam Propagation Methods.IEEE Photon ics Technology Letters,2002:8,14(8):1103-1105.
[80]Ya Yan Lu,JOyce R.McLaughin.The Riccati method for the Helmholtz equation.Journal of the Acoustical society of America,1996,100(3):1432-1446.
[81]Y.Y.Lu.One-way large range step methods for Helmholtz waveguides.Journal of Computational Physics,1999,152(1):231-250.
[82]Y.Y.Lu and J.X.Zhu.Large range step method for acoustic waveguide with two layer media.Progress in natural science,2002,12(11):820-825.
[83]Y.Y.Lu and J.X.Zhu.Propagating modes in optical waveguides terminated by Perfectly Matched Layers.IEEE Photonics Technology Letters,2005,17(12):2601-2603.
[84]Y.Y.Lu and P.L.ho.Beam propagation method using a[(p-1)/p]pad é approximation of the propagator.Optics Letters,2002,27(9):683-685.
[85]Y.Y.Lu and J.X.Zhu.Perfectly matched layer for acoustic waveguide modelingbenchmark calculations and perturbation analysis.Computer Modeling in Engineering and Sciences,2007,22(3):235-247.
[86]董光昌,陈仲慈,汤国桢,数学物理方程(第一版),浙江大学出版社。
[87]G.H.戈卢布,C.F.范洛恩,矩阵计算,科学出版社。
[88]黄越夏,沈瑞敏,求解Helmholtz方程的一类非线性局部正交变换。浙江大学学报理学版,2004:9,31(5):498-502。
[89]李荣华,冯国忱,偏微分方程数值解法(第三版)。高等教育出版社,1996。
[90]刘伯胜,雷家煜,水声学原理,哈尔滨工程大学出版社。
[91]徐树方,矩阵计算的理论和方法,北京大学出版社。