Please use this identifier to cite or link to this item: http://hdl.handle.net/11455/90440
標題: Numerical methods for solving paraxial equation with high wave number
數值方法解高波數近軸方程式
作者: 陳偉閔
Wei-Min Chen
關鍵字: 近軸方程式
局部一維法
分離法
paraxial equation
locally one dimensional method
separation method
引用: [1] G. D. Boyd and J. P. Gordon, Confocal multimode resonator for millimeter through optical wavelength masers, Bell Syst. Tech. J., 40 (1961), 489-408. [2] J.A. Fleck, Jr., Solution of the scalar Helmholtz wave equation by Lanczos reduction, Prog. Electromagn. Res., Pier 11 (1995), 103-141. [3] L. Gonzalez, S. Guha, J. W. Rogers, and Q. Sheng, An effective z-stretching method for paraxial light beam propagation simulations, J. Comput. Phys., 227 (2008), 7264-7278. [4] G. Goubau and F.Schwering, On the guided propagation of electromagnetic wave beams, IRE Trans. Antennas Propag., AP-9 (1961), 248-256. [5] S. Guha, A steepest descent approach for the eikonal equation in geometrical optics, Tech. Report AFRL/RXPJ 09-02-09 (2009) [6] G. R. Hadley, Transparent boundary condition for beam propagation, Opt. Lett., 16 (1911), 624-626. [7] H. Kogelnik and T.Li, Laser beams and resonators, Proc. IEEE, 54 (1966), 1312-1329. [8] G.I. Marchuk, Splitting and alternating direction methods, Handbook of Numerical Analysis, Volume 1, 1990, 197–462. [9] D. W. Peaceman and H. H. Rachford, Jr., The numerical solution of parabolic and elliptic differential equations, J. Soc. Indust. Appl. Math., 3.1 (1955), 28-41. [10] Q. Sheng, S. Guha, and L. Gonzalez, An exponential transformation based splitting method for fast computations of highly oscillatory solutions, J. Comput. Appl. Math., 235 (2011), 4452-4463. [11] Q. Sheng and H.-W. Sun, Exponential splitting for n-dimensional paraxial Helmholtz equation with high wavenumbers, Comput. Math. Appl., 68.10 (2014), 1341-1354. [12] A.E. Siegman, Hermite-Gaussian functions of complex argument as optical beam eigenfunctions, J. Opt. Soc. Am., 63 (1973), 1093-1094. [13] N. N. Yanenko. The method of fractional steps, Springer-Verlag, Berlin, 1971.
摘要: In this thesis, we compare the results of the numerical methods with separation method (SM) and locally one dimensional (LOD) method and numerical methods without SM for solving paraxial equation. Without SM, we solve the paraxial equation directly by forward Euler, backward Euler, and Crank-Nicolson schemes. For the second discretization with SM, we transform the paraxial equation into a coupled partial differential equations where the amplitude and phase are unknown functions, and then the numerical results are reconstructed by an inverse transformation. Our numerical results show that the errors with SM are much less than those without using SM. Forward Euler with SM performs the best, and LOD method with SM provides reasonably accurate solutions while a coarser mesh size is applied.
在這篇論文中,我們解近軸方程式比較使用分離法加局部一維法和沒有使用分離法。在沒有使用分離法,我們直接地利用向前尤拉,向後尤拉,和克蘭克-尼科爾森離散法解近軸方程式。對於第二種使用分離法的離散,我們轉換近軸方程式為以振幅和相位為未知方程式的耦合偏微分方程式,然後再利用逆轉換重建數值結果。我們的數值結果顯示使用分離法的誤差小於沒有使用分離法的誤差。有使用分離法的向前尤拉表現最佳,而使用分離法的局部一維法在粗糙網格時提供相當精確的解。
URI: http://hdl.handle.net/11455/90440
文章公開時間: 2018-07-22
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