Abstract

We compare several rapid approximation methods for the modal reflection coefficients of uncoated and coated discontinuous slab-waveguide interfaces. We further extend our previously proposed split-operator technique to the modal reflectivities of semiconductor ridge waveguides.

© 1995 Optical Society of America

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References

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  1. G. H. Brooke, M. M. Z. Kharadly, “Scattering by abrupt discontinuities on planar dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-30, 760–770 (1982).
    [CrossRef]
  2. C. Vassallo, “Rigorous and approximate calculations of antireflection layer parameters for traveling-wave diode laser amplifiers,” Electron. Lett. 21, 333–334 (1985).
    [CrossRef]
  3. C. Vassallo, “Antireflection coatings for optical semiconductor amplifiers: justification of heuristic analysis,” Electron. Lett. 24, 62–64 (1988).
    [CrossRef]
  4. C. Vassallo, “Reflectivity of multidielectric coatings deposited on the end facet of a weakly guiding dielectric slab waveguide,” J. Opt. Soc. Am. A 5, 1918–1928 (1988).
    [CrossRef]
  5. P. C. Kendall, D. A. Roberts, P. N. Robson, M. J. Adams, M. J. Robertson, “Semiconductor laser facet reflectivities using free-space radiation modes,” Proc. Inst. Electr. Eng. Part J 140, 49–55 (1993).
  6. D. Yevick, W. Bardyszewski, B. Hermansson, M. Glasner, “Split-operator electric field reflection techniques,” Photon. Technol. Lett. 3, 527–529 (1991).
    [CrossRef]
  7. D. Yevick, J. Xu, W. Bardyszewski, “Split-operator calculations of reflected electric field profiles,” Photon. Technol. Lett. 4, 1383–1386 (1992).
    [CrossRef]
  8. J. Xu, D. Yevick, M. Gallant, “Approximate methods for electric-field reflection at waveguide interfaces,” in OSA Annual Meeting, Vol. 16 of OSA 1993 Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 36.
  9. M. C. Farries, J. Buus, D. J. Robbins, “Analysis of antireflection coatings on angled facet semiconductor laser amplifiers,” Electron. Lett. 26, 381–382 (1990).
    [CrossRef]
  10. J. Buus, M. C. Farries, D. J. Robbins, “Reflectivity of coated and tilted semiconductor facets,” IEEE J. Quantum Electron. 27, 1837–1842 (1991).
    [CrossRef]
  11. D. Handelman, A. Hardy, A. Katzir, “Reflectivity of TE modes at the facets of buried heterostructure injection lasers,” IEEE J. Quantum Electron. QE-22, 498–500 (1986).
    [CrossRef]

1993 (1)

P. C. Kendall, D. A. Roberts, P. N. Robson, M. J. Adams, M. J. Robertson, “Semiconductor laser facet reflectivities using free-space radiation modes,” Proc. Inst. Electr. Eng. Part J 140, 49–55 (1993).

1992 (1)

D. Yevick, J. Xu, W. Bardyszewski, “Split-operator calculations of reflected electric field profiles,” Photon. Technol. Lett. 4, 1383–1386 (1992).
[CrossRef]

1991 (2)

D. Yevick, W. Bardyszewski, B. Hermansson, M. Glasner, “Split-operator electric field reflection techniques,” Photon. Technol. Lett. 3, 527–529 (1991).
[CrossRef]

J. Buus, M. C. Farries, D. J. Robbins, “Reflectivity of coated and tilted semiconductor facets,” IEEE J. Quantum Electron. 27, 1837–1842 (1991).
[CrossRef]

1990 (1)

M. C. Farries, J. Buus, D. J. Robbins, “Analysis of antireflection coatings on angled facet semiconductor laser amplifiers,” Electron. Lett. 26, 381–382 (1990).
[CrossRef]

1988 (2)

C. Vassallo, “Antireflection coatings for optical semiconductor amplifiers: justification of heuristic analysis,” Electron. Lett. 24, 62–64 (1988).
[CrossRef]

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

1986 (1)

D. Handelman, A. Hardy, A. Katzir, “Reflectivity of TE modes at the facets of buried heterostructure injection lasers,” IEEE J. Quantum Electron. QE-22, 498–500 (1986).
[CrossRef]

1985 (1)

C. Vassallo, “Rigorous and approximate calculations of antireflection layer parameters for traveling-wave diode laser amplifiers,” Electron. Lett. 21, 333–334 (1985).
[CrossRef]

1982 (1)

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

Adams, M. J.

P. C. Kendall, D. A. Roberts, P. N. Robson, M. J. Adams, M. J. Robertson, “Semiconductor laser facet reflectivities using free-space radiation modes,” Proc. Inst. Electr. Eng. Part J 140, 49–55 (1993).

Bardyszewski, W.

D. Yevick, J. Xu, W. Bardyszewski, “Split-operator calculations of reflected electric field profiles,” Photon. Technol. Lett. 4, 1383–1386 (1992).
[CrossRef]

D. Yevick, W. Bardyszewski, B. Hermansson, M. Glasner, “Split-operator electric field reflection techniques,” Photon. Technol. Lett. 3, 527–529 (1991).
[CrossRef]

Brooke, G. H.

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

Buus, J.

J. Buus, M. C. Farries, D. J. Robbins, “Reflectivity of coated and tilted semiconductor facets,” IEEE J. Quantum Electron. 27, 1837–1842 (1991).
[CrossRef]

M. C. Farries, J. Buus, D. J. Robbins, “Analysis of antireflection coatings on angled facet semiconductor laser amplifiers,” Electron. Lett. 26, 381–382 (1990).
[CrossRef]

Farries, M. C.

J. Buus, M. C. Farries, D. J. Robbins, “Reflectivity of coated and tilted semiconductor facets,” IEEE J. Quantum Electron. 27, 1837–1842 (1991).
[CrossRef]

M. C. Farries, J. Buus, D. J. Robbins, “Analysis of antireflection coatings on angled facet semiconductor laser amplifiers,” Electron. Lett. 26, 381–382 (1990).
[CrossRef]

Gallant, M.

J. Xu, D. Yevick, M. Gallant, “Approximate methods for electric-field reflection at waveguide interfaces,” in OSA Annual Meeting, Vol. 16 of OSA 1993 Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 36.

Glasner, M.

D. Yevick, W. Bardyszewski, B. Hermansson, M. Glasner, “Split-operator electric field reflection techniques,” Photon. Technol. Lett. 3, 527–529 (1991).
[CrossRef]

Handelman, D.

D. Handelman, A. Hardy, A. Katzir, “Reflectivity of TE modes at the facets of buried heterostructure injection lasers,” IEEE J. Quantum Electron. QE-22, 498–500 (1986).
[CrossRef]

Hardy, A.

D. Handelman, A. Hardy, A. Katzir, “Reflectivity of TE modes at the facets of buried heterostructure injection lasers,” IEEE J. Quantum Electron. QE-22, 498–500 (1986).
[CrossRef]

Hermansson, B.

D. Yevick, W. Bardyszewski, B. Hermansson, M. Glasner, “Split-operator electric field reflection techniques,” Photon. Technol. Lett. 3, 527–529 (1991).
[CrossRef]

Katzir, A.

D. Handelman, A. Hardy, A. Katzir, “Reflectivity of TE modes at the facets of buried heterostructure injection lasers,” IEEE J. Quantum Electron. QE-22, 498–500 (1986).
[CrossRef]

Kendall, P. C.

P. C. Kendall, D. A. Roberts, P. N. Robson, M. J. Adams, M. J. Robertson, “Semiconductor laser facet reflectivities using free-space radiation modes,” Proc. Inst. Electr. Eng. Part J 140, 49–55 (1993).

Kharadly, M. M. Z.

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

Robbins, D. J.

J. Buus, M. C. Farries, D. J. Robbins, “Reflectivity of coated and tilted semiconductor facets,” IEEE J. Quantum Electron. 27, 1837–1842 (1991).
[CrossRef]

M. C. Farries, J. Buus, D. J. Robbins, “Analysis of antireflection coatings on angled facet semiconductor laser amplifiers,” Electron. Lett. 26, 381–382 (1990).
[CrossRef]

Roberts, D. A.

P. C. Kendall, D. A. Roberts, P. N. Robson, M. J. Adams, M. J. Robertson, “Semiconductor laser facet reflectivities using free-space radiation modes,” Proc. Inst. Electr. Eng. Part J 140, 49–55 (1993).

Robertson, M. J.

P. C. Kendall, D. A. Roberts, P. N. Robson, M. J. Adams, M. J. Robertson, “Semiconductor laser facet reflectivities using free-space radiation modes,” Proc. Inst. Electr. Eng. Part J 140, 49–55 (1993).

Robson, P. N.

P. C. Kendall, D. A. Roberts, P. N. Robson, M. J. Adams, M. J. Robertson, “Semiconductor laser facet reflectivities using free-space radiation modes,” Proc. Inst. Electr. Eng. Part J 140, 49–55 (1993).

Vassallo, C.

C. Vassallo, “Antireflection coatings for optical semiconductor amplifiers: justification of heuristic analysis,” Electron. Lett. 24, 62–64 (1988).
[CrossRef]

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

C. Vassallo, “Rigorous and approximate calculations of antireflection layer parameters for traveling-wave diode laser amplifiers,” Electron. Lett. 21, 333–334 (1985).
[CrossRef]

Xu, J.

D. Yevick, J. Xu, W. Bardyszewski, “Split-operator calculations of reflected electric field profiles,” Photon. Technol. Lett. 4, 1383–1386 (1992).
[CrossRef]

J. Xu, D. Yevick, M. Gallant, “Approximate methods for electric-field reflection at waveguide interfaces,” in OSA Annual Meeting, Vol. 16 of OSA 1993 Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 36.

Yevick, D.

D. Yevick, J. Xu, W. Bardyszewski, “Split-operator calculations of reflected electric field profiles,” Photon. Technol. Lett. 4, 1383–1386 (1992).
[CrossRef]

D. Yevick, W. Bardyszewski, B. Hermansson, M. Glasner, “Split-operator electric field reflection techniques,” Photon. Technol. Lett. 3, 527–529 (1991).
[CrossRef]

J. Xu, D. Yevick, M. Gallant, “Approximate methods for electric-field reflection at waveguide interfaces,” in OSA Annual Meeting, Vol. 16 of OSA 1993 Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 36.

Electron. Lett. (3)

C. Vassallo, “Rigorous and approximate calculations of antireflection layer parameters for traveling-wave diode laser amplifiers,” Electron. Lett. 21, 333–334 (1985).
[CrossRef]

C. Vassallo, “Antireflection coatings for optical semiconductor amplifiers: justification of heuristic analysis,” Electron. Lett. 24, 62–64 (1988).
[CrossRef]

M. C. Farries, J. Buus, D. J. Robbins, “Analysis of antireflection coatings on angled facet semiconductor laser amplifiers,” Electron. Lett. 26, 381–382 (1990).
[CrossRef]

IEEE J. Quantum Electron. (2)

J. Buus, M. C. Farries, D. J. Robbins, “Reflectivity of coated and tilted semiconductor facets,” IEEE J. Quantum Electron. 27, 1837–1842 (1991).
[CrossRef]

D. Handelman, A. Hardy, A. Katzir, “Reflectivity of TE modes at the facets of buried heterostructure injection lasers,” IEEE J. Quantum Electron. QE-22, 498–500 (1986).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

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

J. Opt. Soc. Am. A (1)

Photon. Technol. Lett. (2)

D. Yevick, W. Bardyszewski, B. Hermansson, M. Glasner, “Split-operator electric field reflection techniques,” Photon. Technol. Lett. 3, 527–529 (1991).
[CrossRef]

D. Yevick, J. Xu, W. Bardyszewski, “Split-operator calculations of reflected electric field profiles,” Photon. Technol. Lett. 4, 1383–1386 (1992).
[CrossRef]

Proc. Inst. Electr. Eng. Part J (1)

P. C. Kendall, D. A. Roberts, P. N. Robson, M. J. Adams, M. J. Robertson, “Semiconductor laser facet reflectivities using free-space radiation modes,” Proc. Inst. Electr. Eng. Part J 140, 49–55 (1993).

Other (1)

J. Xu, D. Yevick, M. Gallant, “Approximate methods for electric-field reflection at waveguide interfaces,” in OSA Annual Meeting, Vol. 16 of OSA 1993 Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 36.

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Figures (5)

Fig. 1
Fig. 1

Slab waveguide: (a) without coating, (b) with multilayer coating.

Fig. 2
Fig. 2

Reflection coefficient for the lowest-order TE mode of a waveguide with ncore = 3.52, nclad = 3.20, and width a = 0.15 μm at a planar air interface for λ0 = 1310 nm as a function of the length of the computational window in the transverse direction, Lw, determined by a mode-matching method and zero boundary conditions.

Fig. 3
Fig. 3

Four slab-waveguide cases considered in Section 3: (a) low R facet, moderate step; (b) uncoated facet, moderate step; (c) low R facet, large step; (d) uncoated facet, large step.

Fig. 4
Fig. 4

Ridge-waveguide geometry.

Fig. 5
Fig. 5

Variation of modal reflectivity with ridge width.

Tables (1)

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Table 1 Results for Slab-Waveguide TE0 Mode

Equations (9)

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n 0 ( x ) = { n core | x | a / 2 n clad | x | > a / 2 ,
n eff 2 = | E in ( x ) | 2 n 2 ( x ) d x | E in ( x ) | 2 d x .
Φ rad = F ( s ) exp ( j γ z ) .
Φ = Φ I ( s ) exp ( j β z ) + R Φ I ( s ) exp ( j β z ) + F ( s ) exp ( j γ z ) .
Φ = G ( s ) exp [ j Γ ( s ) z ] ,
F ( s ) = ( β Γ ) ( γ + Γ ) Φ I ( β + Γ ) ( γ + Γ ) R Φ I .
R = ( β Γ ) / ( γ + Γ ) Φ I ( s ) Φ I * ( s ) d s ( β + Γ ) / ( γ + Γ ) Φ I ( s ) Φ I * ( s ) d s .
P ( m ) E ( x ) = { E ( x ) x X m 0 otherwise .
O ι = E total * ( P , S ) r TE ( P , S ) E ι + ( P , S ) d P d S ,

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