Abstract

The problem of the in-plane scattering of a TE guided wave by both waveguidesurface roughness and refractive-index fluctuations has been analyzed. Simple formulas emerge for the relative strength of the two scattering mechanisms and for the power collected by an off-axis detector. The theory is illustrated by applying it to the case of a step-index slab waveguide.

© 1985 Optical Society of America

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References

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  1. M. C. Hamilton, D. A. Wille, W. J. Micheli, “An integrated optical RF spectrum analyzer,” in Proceedings of the IEEE 1976 Ultrasonics Symposium (Institute of Electrical and Electronics Engineers, New York, 1976).
    [CrossRef]
  2. D. Mergerian, E. C. Malarkey, R. P. Pautienus, J. C. Bradley, G. E. Marx, L. D. Hutcheson, A. L. Kellner, “Operational integrated optical R.F. spectrum analyzer,” Appl. Opt. 19, 3033 (1980).
    [CrossRef] [PubMed]
  3. D. Marcuse, “Mode conversion caused by surface imperfections of a dielectric slab waveguide,” Bell Syst. Tech. J. 48, 3187 (1969).
  4. Y. Suematsu, K. Furuya, M. Hakuta, K. Chiba, “Far-field radiation pattern caused by random wall distortion of dielectric waveguides and determination of correlation length,” Electron. Commun. Jpn. 56-C, 62 (1973).
  5. M. Gottlieb, G. B. Brandt, J. J. Conroy, “Out-of-plane scattering in optical waveguides,”IEEE Trans. Circuits Syst. CAS-26, 1029 (1979).
    [CrossRef]
  6. T. L. Tsai, H. S. Tuan, “Reflection and scattering by a single groovein integrated optics,” IEEE J. Quantum Electron. QE-10, 326 (1974).
    [CrossRef]
  7. D. G. Hall, “Comparison of two approaches to the waveguide scattering problem,” Appl. Opt. 19, 1732 (1980).
    [CrossRef] [PubMed]
  8. D. J. Walter, J. Houghton, “Attenuation in thin film optical waveguides due to roughness-induced mode-coupling,” Thin Solid Films 52, 461 (1978).
    [CrossRef]
  9. S. Mianaga, M. Imai, T. Asakura, “Radiation pattern of light scattering from the core region of dielectric-slab-optical waveguides,” IEEE J. Quantum Electron. QE-14, 30 (1978).
    [CrossRef]
  10. M. Imai, M. Koseki, Y. Ohtsuka, “Light scattering from a glass thin-film optical waveguide,” J. Appl. Phys. 52, 6506 (1981).
    [CrossRef]
  11. E. Bradley, D. G. Hall, “Out-of-plane scattering from glass waveguides: comparison of theory and experiment,” Opt. Lett. 7, 235 (1982).
    [CrossRef] [PubMed]
  12. D. J. Walter, “The roughness parameters of glass films,” Vacuum 27, 7 (1976).
    [CrossRef]
  13. P. K. Tien, “Light waves in thin films and integrated optics,” Appl. Opt. 10, 2395 (1971).
    [CrossRef] [PubMed]
  14. T. Findakly, E. Garmire, H. T. Moon, “Comparison of losses in imperfect surface-diffused and buried optical waveguides,” Opt. Lett. 4, 149 (1979).
    [CrossRef] [PubMed]
  15. T. Findakly, E. Garmire, “Reduction and control of optical waveguide losses in glass,” Appl. Phys. Lett. 37, 855 (1980).
    [CrossRef]
  16. S. Dutta, H. E. Jackson, J. T. Boyd, F. S. Hickernell, R. L. Davis, “Scattering loss reduction in ZnO optical waveguides by laser annealing,” Appl. Phys. Lett. 39, 206 (1981).
    [CrossRef]
  17. S. Dutta, H. E. Jackson, J. T. Boyd, “Extremely low-loss glass thin-film optical waveguide utilizing surface coating and laser annealing,” J. Appl. Phys. 52, 3873 (1981).
    [CrossRef]
  18. G. H. Ames, D. G. Hall, “Attenuation in planar optical waveguides: comparison of theory and experiment,” IEEE J. Quantum Electron. QE-19, 845 (1983).
    [CrossRef]
  19. D. G. Hall, “Scattering of optical guided-waves by waveguide surface roughness: three-dimensional treatment,” Opt. Lett. 6, 601 (1981).
    [CrossRef] [PubMed]
  20. J. T. Boyd, D. B. Anderson, “Effect of waveguide optical scattering on the integrated optical spectral analyzer dynamic range,” IEEE J. Quantum Electron. QE-14, 437 (1978).
    [CrossRef]
  21. D. W. Vahey, “Optical waveguide scattering reduction,” (Air Force Avionics Laboratory, Wright-Patterson Air Force Base, Ohio 45433, June1979).
  22. D. W. Vahey, N. F. Hartman, R. C. Sherman, “Optical waveguide scattering reduction II,” (Air Force Avionics Laboratory, Wright-Patterson Air Force Base, Ohio 45433, December1980).
  23. G. B. Brandt, “In-plane scattering in glass and niobium oxide waveguides,” Opt. Eng. 20, 150 (1981).
    [CrossRef]
  24. F. K. Hopkins, H. E. Jackson, J. T. Boyd, “In-plane scattering in a planar optical waveguide by an integrated technique,” Appl. Opt. 20, 2761 (1981).
    [CrossRef]
  25. R. A. Modavis, D. G. Hall, “In-plane scattering in planar optical waveguides,” Opt. Lett. 9, 96 (1984).
    [CrossRef] [PubMed]
  26. See, for example, G. Arfken, Mathematical Methods for Physicists, 2nd ed. (Academic, New York, 1970), p. 760.
  27. H. Kogelnik, “Theory of dielectric waveguides,” in Integrated Optics, T. Tamir, ed. (Springer-Verlag, New York, 1979).
  28. J. M. Elson, J. M. Bennett, “Relation between the angular dependence of scattering and the statistical properties of optical surfaces,”J. Opt. Soc. Am. 69, 31 (1979).
    [CrossRef]
  29. J. M. Elson, J. M. Bennett, “Vector scattering theory,” Opt. Eng. 18, 116 (1979).
    [CrossRef]
  30. W. Panofsky, M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, Reading, Mass., 1955), Chap. 12.

1984 (1)

1983 (1)

G. H. Ames, D. G. Hall, “Attenuation in planar optical waveguides: comparison of theory and experiment,” IEEE J. Quantum Electron. QE-19, 845 (1983).
[CrossRef]

1982 (1)

1981 (6)

F. K. Hopkins, H. E. Jackson, J. T. Boyd, “In-plane scattering in a planar optical waveguide by an integrated technique,” Appl. Opt. 20, 2761 (1981).
[CrossRef]

S. Dutta, H. E. Jackson, J. T. Boyd, F. S. Hickernell, R. L. Davis, “Scattering loss reduction in ZnO optical waveguides by laser annealing,” Appl. Phys. Lett. 39, 206 (1981).
[CrossRef]

S. Dutta, H. E. Jackson, J. T. Boyd, “Extremely low-loss glass thin-film optical waveguide utilizing surface coating and laser annealing,” J. Appl. Phys. 52, 3873 (1981).
[CrossRef]

G. B. Brandt, “In-plane scattering in glass and niobium oxide waveguides,” Opt. Eng. 20, 150 (1981).
[CrossRef]

D. G. Hall, “Scattering of optical guided-waves by waveguide surface roughness: three-dimensional treatment,” Opt. Lett. 6, 601 (1981).
[CrossRef] [PubMed]

M. Imai, M. Koseki, Y. Ohtsuka, “Light scattering from a glass thin-film optical waveguide,” J. Appl. Phys. 52, 6506 (1981).
[CrossRef]

1980 (3)

1979 (4)

1978 (3)

D. J. Walter, J. Houghton, “Attenuation in thin film optical waveguides due to roughness-induced mode-coupling,” Thin Solid Films 52, 461 (1978).
[CrossRef]

S. Mianaga, M. Imai, T. Asakura, “Radiation pattern of light scattering from the core region of dielectric-slab-optical waveguides,” IEEE J. Quantum Electron. QE-14, 30 (1978).
[CrossRef]

J. T. Boyd, D. B. Anderson, “Effect of waveguide optical scattering on the integrated optical spectral analyzer dynamic range,” IEEE J. Quantum Electron. QE-14, 437 (1978).
[CrossRef]

1976 (1)

D. J. Walter, “The roughness parameters of glass films,” Vacuum 27, 7 (1976).
[CrossRef]

1974 (1)

T. L. Tsai, H. S. Tuan, “Reflection and scattering by a single groovein integrated optics,” IEEE J. Quantum Electron. QE-10, 326 (1974).
[CrossRef]

1973 (1)

Y. Suematsu, K. Furuya, M. Hakuta, K. Chiba, “Far-field radiation pattern caused by random wall distortion of dielectric waveguides and determination of correlation length,” Electron. Commun. Jpn. 56-C, 62 (1973).

1971 (1)

1969 (1)

D. Marcuse, “Mode conversion caused by surface imperfections of a dielectric slab waveguide,” Bell Syst. Tech. J. 48, 3187 (1969).

Ames, G. H.

G. H. Ames, D. G. Hall, “Attenuation in planar optical waveguides: comparison of theory and experiment,” IEEE J. Quantum Electron. QE-19, 845 (1983).
[CrossRef]

Anderson, D. B.

J. T. Boyd, D. B. Anderson, “Effect of waveguide optical scattering on the integrated optical spectral analyzer dynamic range,” IEEE J. Quantum Electron. QE-14, 437 (1978).
[CrossRef]

Arfken, G.

See, for example, G. Arfken, Mathematical Methods for Physicists, 2nd ed. (Academic, New York, 1970), p. 760.

Asakura, T.

S. Mianaga, M. Imai, T. Asakura, “Radiation pattern of light scattering from the core region of dielectric-slab-optical waveguides,” IEEE J. Quantum Electron. QE-14, 30 (1978).
[CrossRef]

Bennett, J. M.

Boyd, J. T.

F. K. Hopkins, H. E. Jackson, J. T. Boyd, “In-plane scattering in a planar optical waveguide by an integrated technique,” Appl. Opt. 20, 2761 (1981).
[CrossRef]

S. Dutta, H. E. Jackson, J. T. Boyd, “Extremely low-loss glass thin-film optical waveguide utilizing surface coating and laser annealing,” J. Appl. Phys. 52, 3873 (1981).
[CrossRef]

S. Dutta, H. E. Jackson, J. T. Boyd, F. S. Hickernell, R. L. Davis, “Scattering loss reduction in ZnO optical waveguides by laser annealing,” Appl. Phys. Lett. 39, 206 (1981).
[CrossRef]

J. T. Boyd, D. B. Anderson, “Effect of waveguide optical scattering on the integrated optical spectral analyzer dynamic range,” IEEE J. Quantum Electron. QE-14, 437 (1978).
[CrossRef]

Bradley, E.

Bradley, J. C.

Brandt, G. B.

G. B. Brandt, “In-plane scattering in glass and niobium oxide waveguides,” Opt. Eng. 20, 150 (1981).
[CrossRef]

M. Gottlieb, G. B. Brandt, J. J. Conroy, “Out-of-plane scattering in optical waveguides,”IEEE Trans. Circuits Syst. CAS-26, 1029 (1979).
[CrossRef]

Chiba, K.

Y. Suematsu, K. Furuya, M. Hakuta, K. Chiba, “Far-field radiation pattern caused by random wall distortion of dielectric waveguides and determination of correlation length,” Electron. Commun. Jpn. 56-C, 62 (1973).

Conroy, J. J.

M. Gottlieb, G. B. Brandt, J. J. Conroy, “Out-of-plane scattering in optical waveguides,”IEEE Trans. Circuits Syst. CAS-26, 1029 (1979).
[CrossRef]

Davis, R. L.

S. Dutta, H. E. Jackson, J. T. Boyd, F. S. Hickernell, R. L. Davis, “Scattering loss reduction in ZnO optical waveguides by laser annealing,” Appl. Phys. Lett. 39, 206 (1981).
[CrossRef]

Dutta, S.

S. Dutta, H. E. Jackson, J. T. Boyd, “Extremely low-loss glass thin-film optical waveguide utilizing surface coating and laser annealing,” J. Appl. Phys. 52, 3873 (1981).
[CrossRef]

S. Dutta, H. E. Jackson, J. T. Boyd, F. S. Hickernell, R. L. Davis, “Scattering loss reduction in ZnO optical waveguides by laser annealing,” Appl. Phys. Lett. 39, 206 (1981).
[CrossRef]

Elson, J. M.

Findakly, T.

T. Findakly, E. Garmire, “Reduction and control of optical waveguide losses in glass,” Appl. Phys. Lett. 37, 855 (1980).
[CrossRef]

T. Findakly, E. Garmire, H. T. Moon, “Comparison of losses in imperfect surface-diffused and buried optical waveguides,” Opt. Lett. 4, 149 (1979).
[CrossRef] [PubMed]

Furuya, K.

Y. Suematsu, K. Furuya, M. Hakuta, K. Chiba, “Far-field radiation pattern caused by random wall distortion of dielectric waveguides and determination of correlation length,” Electron. Commun. Jpn. 56-C, 62 (1973).

Garmire, E.

T. Findakly, E. Garmire, “Reduction and control of optical waveguide losses in glass,” Appl. Phys. Lett. 37, 855 (1980).
[CrossRef]

T. Findakly, E. Garmire, H. T. Moon, “Comparison of losses in imperfect surface-diffused and buried optical waveguides,” Opt. Lett. 4, 149 (1979).
[CrossRef] [PubMed]

Gottlieb, M.

M. Gottlieb, G. B. Brandt, J. J. Conroy, “Out-of-plane scattering in optical waveguides,”IEEE Trans. Circuits Syst. CAS-26, 1029 (1979).
[CrossRef]

Hakuta, M.

Y. Suematsu, K. Furuya, M. Hakuta, K. Chiba, “Far-field radiation pattern caused by random wall distortion of dielectric waveguides and determination of correlation length,” Electron. Commun. Jpn. 56-C, 62 (1973).

Hall, D. G.

Hamilton, M. C.

M. C. Hamilton, D. A. Wille, W. J. Micheli, “An integrated optical RF spectrum analyzer,” in Proceedings of the IEEE 1976 Ultrasonics Symposium (Institute of Electrical and Electronics Engineers, New York, 1976).
[CrossRef]

Hartman, N. F.

D. W. Vahey, N. F. Hartman, R. C. Sherman, “Optical waveguide scattering reduction II,” (Air Force Avionics Laboratory, Wright-Patterson Air Force Base, Ohio 45433, December1980).

Hickernell, F. S.

S. Dutta, H. E. Jackson, J. T. Boyd, F. S. Hickernell, R. L. Davis, “Scattering loss reduction in ZnO optical waveguides by laser annealing,” Appl. Phys. Lett. 39, 206 (1981).
[CrossRef]

Hopkins, F. K.

Houghton, J.

D. J. Walter, J. Houghton, “Attenuation in thin film optical waveguides due to roughness-induced mode-coupling,” Thin Solid Films 52, 461 (1978).
[CrossRef]

Hutcheson, L. D.

Imai, M.

M. Imai, M. Koseki, Y. Ohtsuka, “Light scattering from a glass thin-film optical waveguide,” J. Appl. Phys. 52, 6506 (1981).
[CrossRef]

S. Mianaga, M. Imai, T. Asakura, “Radiation pattern of light scattering from the core region of dielectric-slab-optical waveguides,” IEEE J. Quantum Electron. QE-14, 30 (1978).
[CrossRef]

Jackson, H. E.

S. Dutta, H. E. Jackson, J. T. Boyd, F. S. Hickernell, R. L. Davis, “Scattering loss reduction in ZnO optical waveguides by laser annealing,” Appl. Phys. Lett. 39, 206 (1981).
[CrossRef]

S. Dutta, H. E. Jackson, J. T. Boyd, “Extremely low-loss glass thin-film optical waveguide utilizing surface coating and laser annealing,” J. Appl. Phys. 52, 3873 (1981).
[CrossRef]

F. K. Hopkins, H. E. Jackson, J. T. Boyd, “In-plane scattering in a planar optical waveguide by an integrated technique,” Appl. Opt. 20, 2761 (1981).
[CrossRef]

Kellner, A. L.

Kogelnik, H.

H. Kogelnik, “Theory of dielectric waveguides,” in Integrated Optics, T. Tamir, ed. (Springer-Verlag, New York, 1979).

Koseki, M.

M. Imai, M. Koseki, Y. Ohtsuka, “Light scattering from a glass thin-film optical waveguide,” J. Appl. Phys. 52, 6506 (1981).
[CrossRef]

Malarkey, E. C.

Marcuse, D.

D. Marcuse, “Mode conversion caused by surface imperfections of a dielectric slab waveguide,” Bell Syst. Tech. J. 48, 3187 (1969).

Marx, G. E.

Mergerian, D.

Mianaga, S.

S. Mianaga, M. Imai, T. Asakura, “Radiation pattern of light scattering from the core region of dielectric-slab-optical waveguides,” IEEE J. Quantum Electron. QE-14, 30 (1978).
[CrossRef]

Micheli, W. J.

M. C. Hamilton, D. A. Wille, W. J. Micheli, “An integrated optical RF spectrum analyzer,” in Proceedings of the IEEE 1976 Ultrasonics Symposium (Institute of Electrical and Electronics Engineers, New York, 1976).
[CrossRef]

Modavis, R. A.

Moon, H. T.

Ohtsuka, Y.

M. Imai, M. Koseki, Y. Ohtsuka, “Light scattering from a glass thin-film optical waveguide,” J. Appl. Phys. 52, 6506 (1981).
[CrossRef]

Panofsky, W.

W. Panofsky, M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, Reading, Mass., 1955), Chap. 12.

Pautienus, R. P.

Phillips, M.

W. Panofsky, M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, Reading, Mass., 1955), Chap. 12.

Sherman, R. C.

D. W. Vahey, N. F. Hartman, R. C. Sherman, “Optical waveguide scattering reduction II,” (Air Force Avionics Laboratory, Wright-Patterson Air Force Base, Ohio 45433, December1980).

Suematsu, Y.

Y. Suematsu, K. Furuya, M. Hakuta, K. Chiba, “Far-field radiation pattern caused by random wall distortion of dielectric waveguides and determination of correlation length,” Electron. Commun. Jpn. 56-C, 62 (1973).

Tien, P. K.

Tsai, T. L.

T. L. Tsai, H. S. Tuan, “Reflection and scattering by a single groovein integrated optics,” IEEE J. Quantum Electron. QE-10, 326 (1974).
[CrossRef]

Tuan, H. S.

T. L. Tsai, H. S. Tuan, “Reflection and scattering by a single groovein integrated optics,” IEEE J. Quantum Electron. QE-10, 326 (1974).
[CrossRef]

Vahey, D. W.

D. W. Vahey, “Optical waveguide scattering reduction,” (Air Force Avionics Laboratory, Wright-Patterson Air Force Base, Ohio 45433, June1979).

D. W. Vahey, N. F. Hartman, R. C. Sherman, “Optical waveguide scattering reduction II,” (Air Force Avionics Laboratory, Wright-Patterson Air Force Base, Ohio 45433, December1980).

Walter, D. J.

D. J. Walter, J. Houghton, “Attenuation in thin film optical waveguides due to roughness-induced mode-coupling,” Thin Solid Films 52, 461 (1978).
[CrossRef]

D. J. Walter, “The roughness parameters of glass films,” Vacuum 27, 7 (1976).
[CrossRef]

Wille, D. A.

M. C. Hamilton, D. A. Wille, W. J. Micheli, “An integrated optical RF spectrum analyzer,” in Proceedings of the IEEE 1976 Ultrasonics Symposium (Institute of Electrical and Electronics Engineers, New York, 1976).
[CrossRef]

Appl. Opt. (4)

Appl. Phys. Lett. (2)

T. Findakly, E. Garmire, “Reduction and control of optical waveguide losses in glass,” Appl. Phys. Lett. 37, 855 (1980).
[CrossRef]

S. Dutta, H. E. Jackson, J. T. Boyd, F. S. Hickernell, R. L. Davis, “Scattering loss reduction in ZnO optical waveguides by laser annealing,” Appl. Phys. Lett. 39, 206 (1981).
[CrossRef]

Bell Syst. Tech. J. (1)

D. Marcuse, “Mode conversion caused by surface imperfections of a dielectric slab waveguide,” Bell Syst. Tech. J. 48, 3187 (1969).

Electron. Commun. Jpn. (1)

Y. Suematsu, K. Furuya, M. Hakuta, K. Chiba, “Far-field radiation pattern caused by random wall distortion of dielectric waveguides and determination of correlation length,” Electron. Commun. Jpn. 56-C, 62 (1973).

IEEE J. Quantum Electron. (4)

T. L. Tsai, H. S. Tuan, “Reflection and scattering by a single groovein integrated optics,” IEEE J. Quantum Electron. QE-10, 326 (1974).
[CrossRef]

S. Mianaga, M. Imai, T. Asakura, “Radiation pattern of light scattering from the core region of dielectric-slab-optical waveguides,” IEEE J. Quantum Electron. QE-14, 30 (1978).
[CrossRef]

G. H. Ames, D. G. Hall, “Attenuation in planar optical waveguides: comparison of theory and experiment,” IEEE J. Quantum Electron. QE-19, 845 (1983).
[CrossRef]

J. T. Boyd, D. B. Anderson, “Effect of waveguide optical scattering on the integrated optical spectral analyzer dynamic range,” IEEE J. Quantum Electron. QE-14, 437 (1978).
[CrossRef]

IEEE Trans. Circuits Syst. (1)

M. Gottlieb, G. B. Brandt, J. J. Conroy, “Out-of-plane scattering in optical waveguides,”IEEE Trans. Circuits Syst. CAS-26, 1029 (1979).
[CrossRef]

J. Appl. Phys. (2)

M. Imai, M. Koseki, Y. Ohtsuka, “Light scattering from a glass thin-film optical waveguide,” J. Appl. Phys. 52, 6506 (1981).
[CrossRef]

S. Dutta, H. E. Jackson, J. T. Boyd, “Extremely low-loss glass thin-film optical waveguide utilizing surface coating and laser annealing,” J. Appl. Phys. 52, 3873 (1981).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Eng. (2)

G. B. Brandt, “In-plane scattering in glass and niobium oxide waveguides,” Opt. Eng. 20, 150 (1981).
[CrossRef]

J. M. Elson, J. M. Bennett, “Vector scattering theory,” Opt. Eng. 18, 116 (1979).
[CrossRef]

Opt. Lett. (4)

Thin Solid Films (1)

D. J. Walter, J. Houghton, “Attenuation in thin film optical waveguides due to roughness-induced mode-coupling,” Thin Solid Films 52, 461 (1978).
[CrossRef]

Vacuum (1)

D. J. Walter, “The roughness parameters of glass films,” Vacuum 27, 7 (1976).
[CrossRef]

Other (6)

M. C. Hamilton, D. A. Wille, W. J. Micheli, “An integrated optical RF spectrum analyzer,” in Proceedings of the IEEE 1976 Ultrasonics Symposium (Institute of Electrical and Electronics Engineers, New York, 1976).
[CrossRef]

W. Panofsky, M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, Reading, Mass., 1955), Chap. 12.

See, for example, G. Arfken, Mathematical Methods for Physicists, 2nd ed. (Academic, New York, 1970), p. 760.

H. Kogelnik, “Theory of dielectric waveguides,” in Integrated Optics, T. Tamir, ed. (Springer-Verlag, New York, 1979).

D. W. Vahey, “Optical waveguide scattering reduction,” (Air Force Avionics Laboratory, Wright-Patterson Air Force Base, Ohio 45433, June1979).

D. W. Vahey, N. F. Hartman, R. C. Sherman, “Optical waveguide scattering reduction II,” (Air Force Avionics Laboratory, Wright-Patterson Air Force Base, Ohio 45433, December1980).

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Figures (7)

Fig. 1
Fig. 1

Slab-waveguide geometry.

Fig. 2
Fig. 2

Top view of waveguide showing the on-axis detector, labeled 1, and the off-axis detector, labeled 2. Detector 2 collects power scattered out of the incident wave by the scattering area (W by D) located directly in front of detector 1.

Fig. 3
Fig. 3

Volume-to-surface scattering ratio R1 as a function of waveguide thickness h0 for ns = 1.47, nf = 1.56, nc = 1.0, and two wavelengths: λ = 0.6328 μm and λ = 0.83 μm. The curve is plotted from Eq. (41), which treats the case a = σ and δ/heff = M/nf.

Fig. 4
Fig. 4

R = s/i|s, the fraction of the power collected by detector 1 that is collected by detector 2, as a function of the rms surface roughness δ. Curves are shown for s/D = 10−3 and s/D = 10−2. The parameters used for these plots are ns = 1.47, nf = 1.56, nc = 1.0, h0 = 0.9 μm, λ = 0.84 μm, heff = 1.344 μm, N = 1.53, Γ = 0.9, w = 10 μm, and σ = 1.0 μm.

Fig. 5
Fig. 5

Rn = s/i|n as a function of the rms refractive-index fluctuation M for s/D = 10−2 and s/D = 10−3. The correlation length is a = 1.0 μm, and the rest of the parameters are as given in Fig. 4.

Fig. 6
Fig. 6

Rs = s/i|s as a function of s/D for σ = 0.5 μm and σ = 1.0 μm with the rms surface roughness δ = 10 Å. The parameters appear in the caption of Fig. 4.

Fig. 7
Fig. 7

R = s/i|n as a function of s/D for a = 0.5 μm and a = 1.0 μm with the rms index fluctuation M = 1 × 10−5. The parameters appear in the caption of Fig. 4.

Equations (45)

Equations on this page are rendered with MathJax. Learn more.

2 E + n 2 ( ω 2 / c 2 ) E = - μ ω 2 P ,
P = ( Δ ) E ,
E = [ A exp ( i β z ) e ^ 1 + a ( y , z ) e ^ 2 ] E ^ ( x ) ,
- E ( x ) E * ( x ) d x = 2 μ ω / β ,
[ ( 2 / y 2 ) + ( 2 / z 2 ) + β 2 ] a ( y , z ) = - β ( ω / 2 ) - P · e ^ 2 E * ( x ) d x .
P · e ^ 2 E * ( x ) = ( Δ ) [ A exp ( i β z ) e ^ 1 · e ^ 2 + a ( y , z ) ] E ( x ) E * ( x ) .
[ ( 2 / y 2 ) + ( 2 / z 2 ) + β 2 ] a ( y , z ) = - 2 β K ( y , z ) A exp ( i β z ) ,
K ( y , z ) = ( ω / 4 ) cos θ - ( Δ ) E ( x ) E * ( x ) d x .
a ( y 1 , z 1 ) = G ( y 1 , z 1 ; y 2 , z 2 ) [ - 2 β K ( y 2 , z 2 ) A × exp ( i β z 2 ) ] d y 2 d z 2 ,
G ( y 1 , z 1 ; y 2 , z 2 ) = ( i / 4 ) H 0 ( 1 ) ( β r 1 - r 2 ) ,
G ( y 1 , z 1 ; y 2 , z 2 ) = ( i / 4 ) [ 2 / ( π β r 12 ) ] 1 / 2 exp [ i ( β r 12 - π / 4 ) ] ,
β r 1 - r 2 β r 1 - β ( r 1 / r 1 ) · r 2 β r 1 - β · r 2 ,
a ( y 1 , z 1 ) = - i [ β / ( 2 π ) ] 1 / 2 A exp [ i ( β r 1 - π / 4 ) ] r 1 Q ( q y , q z ) ,
Q ( q y , q z ) = K ( y 2 , z 2 ) exp [ - i ( q y y 2 + q z z 2 ) ] d y 2 d z 2 ,
h = h 0 + f ( y , z ) ,
K ( y , z ) ( ω / 4 ) E c 2 ( n f 2 - n c 2 ) 0 f ( y , z ) cos θ ,
n f = n f + δ n ( y , z ) ,
Δ = { 2 n f 0 δ n ( y , z ) 0 x h 0 , 0 otherwise
K ( y , z ) = ( ω / 4 ) 2 n f 0 δ n ( y , z ) cos θ 0 h 0 E ( x ) E * ( x ) d x ,
K ( y , z ) = ( 2 π / λ ) ( Γ n f / N ) cos θ δ n ( y , z ) ,
E s = a ( y 1 , z 1 ) E ( x ) e ^ 2 ,
P s = β 2 π r 1 | - ( L / 2 ) ( L / 2 ) - ( L / 2 ) ( L / 2 ) K ( y 2 , z 2 ) × exp [ - i ( q y y 2 + q z z 2 ) ] d y 2 d z 2 | 2 .
P s = β 2 π r 1 | - L / 2 L / 2 K ( y 2 , z 2 ) × exp [ - i ( q y y 2 + q z z 2 ) ] d y 2 d z 2 | 2 ,
P s = β L 2 2 π r 1 - ( L / 2 ) L / 2 C ( u , v ) exp [ - i ( q y u + q z v ) ] d u d v ,
C ( y 2 - y 3 , z 2 - z 3 ) = K ( y 2 , z 2 ) K ( y 3 , z 3 )
P s / L 2 β 2 π r 1 - C ( u , v ) exp [ - i ( q y u + q z v ) ] d u d v ,
P s / P i = w s θ min π / 2 ( P s / L 2 ) cos θ sin 2 θ d θ ,
C ( u , v ) = [ ( ω / 4 ) E c 2 ( n f 2 - n c 2 ) 0 cos θ ] 2 f ( y 2 , z 2 ) f ( y 3 , z 3 ) ,
C ( u , v ) = [ ( 2 π / λ ) ( Γ n f / N ) cos θ ] 2 δ n ( y 2 , z 2 ) δ n ( y 3 , z 3 ) ,
g ( q y , q z ) = - exp [ - i ( q y u + q z v ) ] d u d v ,
g n ( q y , q z ) = 2 π a 2 M 2 [ 1 + 4 β 2 a 2 sin 2 ( θ / 2 ) ] 3 / 2 ,
g s ( q y , q z ) = 2 π δ 2 σ 2 [ 1 + 4 β 2 σ 2 sin 2 ( θ / 2 ) ] 3 / 2 ,
P s / P i s = 8 π 3 w δ 2 σ 2 ( n f 2 - N 2 ) 2 N h eff 2 λ 3 × θ min π / 2 ( cos 3 θ ) / sin θ [ 1 + 4 β 2 σ 2 sin 2 ( θ / 2 ) ] 3 / 2 d θ
P s / P i n = 8 π 3 w M 2 a 2 n f 2 Γ 2 N λ 3 × θ min π / 2 ( cos 3 θ ) / sin θ [ 1 + 4 β 2 a 2 sin 2 ( θ / 2 ) ] 3 / 2 d θ
E c 2 = 4 ( μ 0 / 0 ) 1 / 2 ( n f 2 - N 2 n f 2 - n c 2 ) / ( N h eff ) ,
P s s / L 2 = 4 π 2 ( n f 2 - N 2 ) 2 cos 2 θ N h eff 2 r 1 λ 3 g s ( q y , q z ) ,
P s n / L 2 = 4 π 2 n f 2 cos 2 θ Γ 2 N r 1 λ 3 g n ( q y , q z ) .
R 1 = P s n / L 2 P s s / L 2 ,
R 1 = ( n f h eff Γ n f 2 - N 2 ) 2 g n ( q y , q z ) g s ( q y , q z ) .
R 1 ( a = σ ) = ( n f 2 n f 2 - N 2 ) 2 [ Γ ( M / n f ) / ( δ / h eff ) ] 2 .
R 1 ( a = σ ) = ( n f 2 n f 2 - N 2 ) 2 Γ 2 ,
β a 1 ,             P s / P i n Γ 2 / ( N λ 3 ) ;
β α D / s ,             P s / P i n Γ 2 / N 4 ;
β σ 1 ,             P s / P i s ( n f 2 - N 2 ) 2 / ( N h eff 2 λ 3 ) ;
β σ D / s ,             P s / P i s ( n f 2 - N 2 ) 2 / ( N 4 h eff 2 ) .

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