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.

C. Vassallo, Optical Waveguide Concepts (Elsevier, Amsterdam, 1991).

C. Vassallo, “Theory and practical calculation of antireflection coatings on semiconductor laser diode optical amplifiers,” IEE Proc. J: Optoelectron. 137, 193–202 (1990).

D. Yevick, W. Bardyszewski, B. Hermansson, M. Glasner, “Lanczos reduction techniques for electric field reflection,” J. Lightwave Technol. 10, 1234–1237 (1992).
[CrossRef]

C. J. Smartt, T. M. Benson, P. C. Kendall, “Exact analysis of waveguide discontinuities: junctions and laser facets,” Electron. Lett. 29, 1352–1353 (1993).
[CrossRef]

G. H. Brooke, M. M. Kharadly, “Scattering by abrupt discontinuities on planar dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-30, 760–770 (1982).
[CrossRef]

C. Vassallo, F. Collino, “Highly efficient boundary conditions for the beam propagation method,” J. Lightwave Technol. 11, 1831–1838 (1993).

C. Vassallo, F. Collino, “Highly efficient boundary conditions for the beam propagation method,” J. Lightwave Technol. 11, 1831–1838 (1993).

C. J. Smartt, T. M. Benson, P. C. Kendall, “Exact analysis of waveguide discontinuities: junctions and laser facets,” Electron. Lett. 29, 1352–1353 (1993).
[CrossRef]

1992 (1)

D. Yevick, W. Bardyszewski, B. Hermansson, M. Glasner, “Lanczos reduction techniques for electric field reflection,” J. Lightwave Technol. 10, 1234–1237 (1992).
[CrossRef]

1990 (1)

C. Vassallo, “Theory and practical calculation of antireflection coatings on semiconductor laser diode optical amplifiers,” IEE Proc. J: Optoelectron. 137, 193–202 (1990).

G. H. Brooke, M. M. Kharadly, “Scattering by abrupt discontinuities on planar dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-30, 760–770 (1982).
[CrossRef]

Bardyszewski, W.

D. Yevick, W. Bardyszewski, B. Hermansson, M. Glasner, “Lanczos reduction techniques for electric field reflection,” J. Lightwave Technol. 10, 1234–1237 (1992).
[CrossRef]

Benson, T. M.

C. J. Smartt, T. M. Benson, P. C. Kendall, “Exact analysis of waveguide discontinuities: junctions and laser facets,” Electron. Lett. 29, 1352–1353 (1993).
[CrossRef]

Brooke, G. H.

G. H. Brooke, M. M. Kharadly, “Scattering by abrupt discontinuities on planar dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-30, 760–770 (1982).
[CrossRef]

Collino, F.

C. Vassallo, F. Collino, “Highly efficient boundary conditions for the beam propagation method,” J. Lightwave Technol. 11, 1831–1838 (1993).

D. Yevick, W. Bardyszewski, B. Hermansson, M. Glasner, “Lanczos reduction techniques for electric field reflection,” J. Lightwave Technol. 10, 1234–1237 (1992).
[CrossRef]

Hermansson, B.

D. Yevick, W. Bardyszewski, B. Hermansson, M. Glasner, “Lanczos reduction techniques for electric field reflection,” J. Lightwave Technol. 10, 1234–1237 (1992).
[CrossRef]

Kendall, P. C.

C. J. Smartt, T. M. Benson, P. C. Kendall, “Exact analysis of waveguide discontinuities: junctions and laser facets,” Electron. Lett. 29, 1352–1353 (1993).
[CrossRef]

Kharadly, M. M.

G. H. Brooke, M. M. Kharadly, “Scattering by abrupt discontinuities on planar dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-30, 760–770 (1982).
[CrossRef]

C. J. Smartt, T. M. Benson, P. C. Kendall, “Exact analysis of waveguide discontinuities: junctions and laser facets,” Electron. Lett. 29, 1352–1353 (1993).
[CrossRef]

Vassallo, C.

C. Vassallo, F. Collino, “Highly efficient boundary conditions for the beam propagation method,” J. Lightwave Technol. 11, 1831–1838 (1993).

C. Vassallo, “Theory and practical calculation of antireflection coatings on semiconductor laser diode optical amplifiers,” IEE Proc. J: Optoelectron. 137, 193–202 (1990).

D. Yevick, W. Bardyszewski, B. Hermansson, M. Glasner, “Lanczos reduction techniques for electric field reflection,” J. Lightwave Technol. 10, 1234–1237 (1992).
[CrossRef]

C. J. Smartt, T. M. Benson, P. C. Kendall, “Exact analysis of waveguide discontinuities: junctions and laser facets,” Electron. Lett. 29, 1352–1353 (1993).
[CrossRef]

IEE Proc. J: Optoelectron. (1)

C. Vassallo, “Theory and practical calculation of antireflection coatings on semiconductor laser diode optical amplifiers,” IEE Proc. J: Optoelectron. 137, 193–202 (1990).

IEEE Trans. Microwave Theory Tech. (1)

G. H. Brooke, M. M. Kharadly, “Scattering by abrupt discontinuities on planar dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-30, 760–770 (1982).
[CrossRef]

J. Lightwave Technol. (2)

C. Vassallo, F. Collino, “Highly efficient boundary conditions for the beam propagation method,” J. Lightwave Technol. 11, 1831–1838 (1993).

D. Yevick, W. Bardyszewski, B. Hermansson, M. Glasner, “Lanczos reduction techniques for electric field reflection,” J. Lightwave Technol. 10, 1234–1237 (1992).
[CrossRef]

Close-up of Fig. 3. Curve a is the same as that in Fig. 3 for eight points in the core. The regular oscillating pattern can be severely altered with other discretizations of Eq. (6), such as those for curves b and c, as explained in the text.

Plots of $|R{|}^{2}$ error, i.e., $|R{|}^{2}$ minus the exact value, through FFT computations for the bare junction JuB, showing the reduced $W$ dependence for large $W$ (compare with Figs. 3 and 6). The solid curves are obtained from an initial computation of the incident mode within the $(-4,+4)$ domain sampled with 128 points, i.e., eight points in the core; then the grid is extended up to $N=1000$ to 1232 points, depending on $W.$ Several points are lacking because the FFT algorithm used here does not accept every $N;$ this results in oscillations less regular than those in Fig. 3. The dashed curves are obtained with 1024 points by using the exact analytical form for the incident mode. The order corresponds to the approximation of V in the series (12), i.e., $\mathbf{V}=0,$ relation (18), or relation (21) for order 0, 1, or 2, respectively.

Reflectivity of the end facet of the square 3-D waveguide discussed in the text. The curves show how the computed reflectivity depends on the size of the computational domain $(W=\mathrm{half}-\mathrm{side}).$ Three sampling steps $\delta x$ are considered, as indicated. On the left the thick curves are obtained with a TBC on the computational domain boundary, as explained in Section 4; the strongly oscillating curve is obtained by removing the TBC for $\delta x/a=0.25.$ The curves on the right are obtained with the FFT method, at orders 0, 1, and 2, as indicated.

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 ${10}^{5}$