A simple and exact method to analyze optical waveguide with graded index profile

Abstract

A simple analytical method is proposed to obtain the exact propagation constant and distribution 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 constant and distribution 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 constant in the analysis of optical waveguide 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 constant of optical waveguide. 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.