Modified finite-difference formula for the analysis of semivectorial modes in step-index optical waveguides

Abstract

Abstract-The modified finite-difference formula is presented for the second derivative of a semivectorial field in a step-index optical waveguide. The present formula achieves a truncation error of O(1x 2 ) provided the discontinuity coincides with a mesh point or lies midway between two mesh points. Furthermore, the formula allows a general position of the interface, when used with the beam-propagation method (BPM). To demonstrate the effectiveness of the formula, asymmetric step-index waveguides are analyzed using the imaginary-distance BPM.

Norm-conserving finite-difference beam-propagation method for TM wave analysis in step-index optical waveguides

“…With the Helmholtz equation, the second term in the right-hand side (RHS) in (8) becomes for a discontinuity between and or for a discontinuity between and . The alternative technique based on the Fresnel equation is suitable if no information on is available [12]. Although all the data in the following are generated by the technique based on the Helmholtz equation, the effects of the third-derivative term are much smaller than those caused by the discretization error in the staircase approximation.…”

“…Using the continuity relations at the interface, we obtain [10] (8) where in which , , and . Incidentally, the formula for the first derivative, which will be required to calculate the adaptive reference index in (12), is given by (9) where Similarly, we can also derive modified FD formulas when the discontinuity lies between points and . In this case, the coefficients are changed to in which , , and .…”

“…6(b). The third derivative in (8) can be evaluated using either the Helmholtz or Fresnel equation [12]. With the Helmholtz equation, the second term in the right-hand side (RHS) in (8) becomes for a discontinuity between and or for a discontinuity between and .…”

Semivector Finite-Difference Formula for the Analysis of a Step-Index Waveguide With a Tilted Interface

“…Taking into account the boundary condition of the field components, we substitute the formula for the component normal to the interface into , while that for the tangential component into . Since the relations among , , and are given by and , can be written as (2) Note that the superscripts and , respectively, means the normal and tangential components. The coefficients, , , and are defined as follows: where (6) (7) (8) in which [8], where is the free-space wavenumber.…”

“…The second derivative is evaluated by the modified FD formula developed in [2]. Taking into account the boundary condition of the field components, we substitute the formula for the component normal to the interface into , while that for the tangential component into .…”