Charles Vassallo, "Finite-difference derivation of the reflectivity at output facets of dielectric waveguides with a highly diverging output beam," J. Opt. Soc. Am. A 15, 717-726 (1998)

Brute force numerical computations are considered for the reflectivity of the output facet of a dielectric waveguide with or without an antireflection coating. Only scalar fields are considered, mainly in two-dimensional systems, through both exact equations and a series expansion, discretized with finite differences. A fair approximation of the series can be obtained with fast Fourier transforms (FFT’s) only, with neither matrix inversion nor diagonalization. Junctions with a highly diverging output beam require a very large computational domain that can be coped with FFT methods only. This is emphasized through the detailed analysis of a three-dimensional example.

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The table also gives the half-width $W$ of computational domains and the number $N$ of grid points; these values were chosen after the convergence analysis in Section 3, and they will be used in the computations in Tables 3–5.

Table 3

Same as Table 2, but with the Approximate V Operators Given by Eq. (18) or Eq. (21)

The two figures for each junction with two or three terms in the series correspond to first-order and second-order approximations, respectively. There is a single value in the one-term row because there is then no V correction. The figures would not be changed by considering more terms in the series. The discretization parameters ($W$ and $N$) are the same as those in Table 2.

Table 4

Same as Table 3, but for the Additional Display of the Deviation of $|R{|}^{2}$ with Respect to the Solution of the Full Eq. (6)

It should be emphasized that these deviations do not measure the absolute accuracy of the computation, since the reference values in this row are not the exact values given in Table 1.

Table 5

Same as Table 4, for Junctions with AR Coatings, Except That the Reflectivities and the Deviations are Multiplied by ${10}^{5}$

The table also gives the half-width $W$ of computational domains and the number $N$ of grid points; these values were chosen after the convergence analysis in Section 3, and they will be used in the computations in Tables 3–5.

Table 3

Same as Table 2, but with the Approximate V Operators Given by Eq. (18) or Eq. (21)

The two figures for each junction with two or three terms in the series correspond to first-order and second-order approximations, respectively. There is a single value in the one-term row because there is then no V correction. The figures would not be changed by considering more terms in the series. The discretization parameters ($W$ and $N$) are the same as those in Table 2.

Table 4

Same as Table 3, but for the Additional Display of the Deviation of $|R{|}^{2}$ with Respect to the Solution of the Full Eq. (6)

It should be emphasized that these deviations do not measure the absolute accuracy of the computation, since the reference values in this row are not the exact values given in Table 1.

Table 5

Same as Table 4, for Junctions with AR Coatings, Except That the Reflectivities and the Deviations are Multiplied by ${10}^{5}$