物理学报Acta Phys.Sin.Vol.64,No.14(2015)144205 A simple and exact method to analyze optical waveguide with graded index profile* Zhang Meng-Ruo Chen Kai-Xint (Key Laboratory of Optical Fiber Sensing And Communication Ministry of Education,School of Communication and Information Engineering,University of Electronic Science and Technology of China,Chengdu 611731,China) Received 27 January 2015;revised manuscript received 25 February 2015 Abstract A simple analytical method is proposed to obtain the exact propagation constants and distributions of electric field intensity of optical waveguides with graded refractive index profile.The method is based on the Wenzel-Kramers- Brillouin (WKB)solution,variational method,modified eigen-equations and discretized scalar wave equation for planar optical waveguide.The expressions of the distribution of electric field intensity based on the conventional WKB method, which diverge around the turning point,have been demonstrated to be very exact in the region beyond the turning point where the refractive index profile varies slowly.The proposed method uses the conventional WKB method to calculate the values of electric field intensity at two adjacent positions beyond the turning point and then the electric field intensity profile for the whole region is obtained by making use of the two calculated values.Two simple and explicit formulas are deduced from the discretized scalar wave equation,which provide a relationship among the values of electric field intensity at three adjacent positions.If the effective refractive index of optical waveguide and the refractive index profile for the whole region are known,we can obtain the value of electric field intensity at any position according to the corresponding values at the adjacent positions by using the two formulas aforementioned.By using the two values calculated by WKB method,the electric field intensity profile for the whole region can be determined through the iterative use of the two formulas.The accuracy of the electric field intensity profile determined by the proposed method is greatly dependent on the accuracy of the applied value of the effective refractive index.To achieve exact propagation constants and distributions of electric field intensity,the variational method and modified eigen-equations are employed in the proposed method.Variational method is a very useful method to improve the accuracy of the propagation constants in the analysis of optical waveguides with step-asymmetrical graded refractive index profile.By combining the traditional variational method and calculation of electric field intensity profile by the proposed method,the improved variational method is presented to obtain the exact propagation constants of optical waveguides.The value of propagation constant calculated by WKB method and the corresponding electric intensity field profile determined by the proposed method are chosen as the initial trial value and trial function in the variational method.Propagation constant and the corresponding electric field intensity profile with better accuracy can be achieved by the variational calculation and then are regarded as the new trial value and trial function.By the iterative use of the variational method and calculation of electric field intensity profile by the proposed method at finite times,quite accurate results are obtained.The modified eigen-equations in combination with the proposed method is another approach to calculating accurate propagation constants of optical waveguides with both the step-asymmetrical and symmetrical graded index profile.In comparison with other published methods,the proposed method has the advantages of the simplicity and considerable accuracy. Keywords:graded refractive index profile,optical waveguides,Wentzel-Kramers-Brillouin method, variational method PACS:42.82.Et,03.65.Ge,03.65.Sq D0I:10.7498/aps.64.144205 Project supported by the National Natural Science Foundation of China (Grant Nos.61177054) Corresponding author.E-mail:chenkxuestc.edu.cn 144205-7 ?1994-2015 China Academic Journal Electronic Publishing House.All rights reserved.http://www.cnki.net物 理 学 报 Acta Phys. Sin. Vol. 64, No. 14 (2015) 144205 A simple and exact method to analyze optical waveguide with graded index profile∗ Zhang Meng-Ruo Chen Kai-Xin† (Key Laboratory of Optical Fiber Sensing And Communication Ministry of Education, School of Communication and Information Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China) ( Received 27 January 2015; revised manuscript received 25 February 2015 ) Abstract A simple analytical method is proposed to obtain the exact propagation constants and distributions of electric field intensity of optical waveguides with graded refractive index profile. The method is based on the Wenzel-Kramers￾Brillouin (WKB) solution, variational method, modified eigen-equations and discretized scalar wave equation for planar optical waveguide. The expressions of the distribution of electric field intensity based on the conventional WKB method, which diverge around the turning point, have been demonstrated to be very exact in the region beyond the turning point where the refractive index profile varies slowly. The proposed method uses the conventional WKB method to calculate the values of electric field intensity at two adjacent positions beyond the turning point and then the electric field intensity profile for the whole region is obtained by making use of the two calculated values. Two simple and explicit formulas are deduced from the discretized scalar wave equation, which provide a relationship among the values of electric field intensity at three adjacent positions. If the effective refractive index of optical waveguide and the refractive index profile for the whole region are known, we can obtain the value of electric field intensity at any position according to the corresponding values at the adjacent positions by using the two formulas aforementioned. By using the two values calculated by WKB method, the electric field intensity profile for the whole region can be determined through the iterative use of the two formulas. The accuracy of the electric field intensity profile determined by the proposed method is greatly dependent on the accuracy of the applied value of the effective refractive index. To achieve exact propagation constants and distributions of electric field intensity, the variational method and modified eigen-equations are employed in the proposed method. Variational method is a very useful method to improve the accuracy of the propagation constants in the analysis of optical waveguides with step-asymmetrical graded refractive index profile. By combining the traditional variational method and calculation of electric field intensity profile by the proposed method, the improved variational method is presented to obtain the exact propagation constants of optical waveguides. The value of propagation constant calculated by WKB method and the corresponding electric intensity field profile determined by the proposed method are chosen as the initial trial value and trial function in the variational method. Propagation constant and the corresponding electric field intensity profile with better accuracy can be achieved by the variational calculation and then are regarded as the new trial value and trial function. By the iterative use of the variational method and calculation of electric field intensity profile by the proposed method at finite times, quite accurate results are obtained. The modified eigen-equations in combination with the proposed method is another approach to calculating accurate propagation constants of optical waveguides with both the step-asymmetrical and symmetrical graded index profile. In comparison with other published methods, the proposed method has the advantages of the simplicity and considerable accuracy. Keywords: graded refractive index profile, optical waveguides, Wentzel-Kramers-Brillouin method, variational method PACS: 42.82.Et, 03.65.Ge, 03.65.Sq DOI: 10.7498/aps.64.144205 * Project supported by the National Natural Science Foundation of China (Grant Nos. 61177054) † Corresponding author. E-mail: chenkx@uestc.edu.cn 144205-7