Abstract

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.

© 1998 Optical Society of America

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References

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  1. C. Vassallo, “Reflectivity of multidielectric coatings deposited on the end facet of a weakly dielectric slab waveguide,” J. Opt. Soc. Am. A 5, 1918–1928 (1988).
    [CrossRef]
  2. C. Vassallo, Optical Waveguide Concepts (Elsevier, Amsterdam, 1991).
  3. C. Vassallo, “Theory and practical calculation of antireflection coatings on semiconductor laser diode optical amplifiers,” IEE Proc. J: Optoelectron. 137, 193–202 (1990).
  4. D. Yevick, W. Bardyszewski, B. Hermansson, M. Glasner, “Lanczos reduction techniques for electric field reflection,” J. Lightwave Technol. 10, 1234–1237 (1992).
    [CrossRef]
  5. 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]
  6. J. Xu, D. Yevick, M. Gallant, “Approximate methods for modal reflectivity at optical waveguide facets,” J. Opt. Soc. Am. A 12, 725–728 (1995).
    [CrossRef]
  7. C. Yu, D. Yevick, “Application of the bidirectional parabolic equation method to optical waveguide facets,” J. Opt. Soc. Am. A 14, 1448–1450 (1997).
    [CrossRef]
  8. 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]
  9. C. Vassallo, F. Collino, “Highly efficient boundary conditions for the beam propagation method,” J. Lightwave Technol. 11, 1831–1838 (1993).
  10. L. Qiao, S. She, “Analysis of propagation characteristics of diffused channel waveguides: weighted residual method,” Opt. Lett. 13, 167–168 (1988).
    [CrossRef] [PubMed]

1997

1995

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

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

1990

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

1988

1982

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).

Gallant, M.

Glasner, M.

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]

Qiao, L.

She, S.

Smartt, C. J.

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).

C. Vassallo, “Reflectivity of multidielectric coatings deposited on the end facet of a weakly dielectric slab waveguide,” J. Opt. Soc. Am. A 5, 1918–1928 (1988).
[CrossRef]

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

Xu, J.

Yevick, D.

Yu, C.

Electron. Lett.

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.

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.

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.

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]

J. Opt. Soc. Am. A

Opt. Lett.

Other

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

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Figures (10)

Fig. 1
Fig. 1

Schematics of the investigated system.

Fig. 2
Fig. 2

Convergence of |R|2 for the junction JuB for a fixed computational domain (W=8a) and an increasing number of grid points.

Fig. 3
Fig. 3

Dependence of |R|2 on the width of the computational domain for the bare junction JuB. The dashed line is at the exact value.

Fig. 4
Fig. 4

Same as Fig. 3, but for junction JuD with an AR layer.

Fig. 5
Fig. 5

Same as Figs. 3 and 4, but for junctions (a) JuA and (b) JuB with AR layers.

Fig. 6
Fig. 6

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.

Fig. 7
Fig. 7

Same as Figs. 4 and 5, but with a transparent boundary condition (TBC).

Fig. 8
Fig. 8

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., V=0, relation (18), or relation (21) for order 0, 1, or 2, respectively.

Fig. 9
Fig. 9

Same as Fig. 8, but for junction JuB with a perfect AR layer, so that ideally |R|2 should be zero.

Fig. 10
Fig. 10

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=half-side). Three sampling steps δx 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 δx/a=0.25. The curves on the right are obtained with the FFT method, at orders 0, 1, and 2, as indicated.

Tables (5)

Tables Icon

Table 1 Scalar 2-D Junctions Discussed in This Paper

Tables Icon

Table 2 Quadratic Field Error for Bare Junctions, for Various Numbers of Terms in the V Power Series in Eq. (12)

Tables Icon

Table 3 Same as Table 2, but with the Approximate V Operators Given by Eq. (18) or Eq. (21)

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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)

Tables Icon

Table 5 Same as Table 4, for Junctions with AR Coatings, Except That the Reflectivities and the Deviations are Multiplied by 105

Equations (25)

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(Ya+Yb)·E=2ha1,
uz×epa·hqa=δpq,
Ya·E=phpuz×E·hp.
Ya·ψ=-izψ=pγpaϕpaϕpaψ.
Yb·ψ=-izψ=pγpbϕpbϕpbψ,
(Ya+Yb)·E=2γ1aϕ1a.
E=(1+Ya-1·Yb)-1·Ya-1·2γ1aϕ1a=(1-v+v2-)·Ya-1·2γ1aϕ1a,
(γ˜pa)2=(γpb)2+k2(na2-1),
Y˜a·Φ=pγ˜paϕpbΦϕpb,
(Y˜a+Yb)·E=2γ1aϕ1a.
V=(Y˜a+Yb)-1·(Ya-Y˜a),
E=(1-V+V2-)·(Y˜a+Yb)-1·2γ1aϕ1a.
R=Eϕ1a-1.
(x2ϕ)N=1δx2{ϕN-1-[2-exp(-ikδx)]ϕN}.
Q=(|Eapprox-Eexact|2)1/2,
Ya=(k2na2+x2+δn2)1/2kna1+x2+δn22k2na2-(x2+δn2)28k4na4,
Ya-Y˜a=kδn22na,
Vp 1γpb+γ˜paϕpb×ϕpb·kδn22na,
Ya-Y˜a=4k2na24k2na2+x2·2knaδn24na2+δn2·4k2na24k2na2+x2,
4k2na24k2na2+x22knakna+(k2na2+x2)1/2=p 2knakna+γ˜paϕpb×ϕpb.
Vp 2kna(kna+γ˜pa)(γpb+γ˜pa)ϕpb×ϕpb·2knaδn24na2+δn2·p 2knakna+γ˜paϕpb×ϕpb,
E=(Y˜a+Yb)-1·2γ1aϕ1a,
(ar1):n=1.9277, l=0.2342forδx/a=0.0625,
(ar2):n=1.9315, l=0.2342forδx/a=0.125,
(ar3):n=1.9490, l=0.2331forδx/a=0.25.

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