Abstract

Previously demonstrated planar thin-film optical switch employing the total internal reflection phenomenon is given a full theoretical evaluation for its possible adaptability to an optical channel waveguide system and also to polarization insensitive immultiport fiber-channel switch.

© 1978 Optical Society of America

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References

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  1. E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2071 (1969).
  2. S. Kurazono, K. Iwasaki, N. Kumagai, Electron. Commun. Jpn. 55, 103 (1972).
  3. H. F. Taylor, J. Appl. Phys. 44, 3257 (1973).
    [CrossRef]
  4. S. Somekh, E. Garmire, H. L. Garvin, R. G. Hunsperger, Appl. Opt. 13, 327 (1974).
    [CrossRef] [PubMed]
  5. K. Tada, K. Hirose, Appl. Phys. Lett. 25561 (1974).
    [CrossRef]
  6. M. Papuchon et al., Appl. Phys. Lett. 27, 289 (1975).
    [CrossRef]
  7. J. C. Campbell, F. A. Blum, D. W. Shaw, K. L. Lawley, Appl. Phys. Lett. 27, 203 (1975).
    [CrossRef]
  8. H. Kogelnik, R. V. Schmidt, IEEE J. Quantum Electron. QE-12, 396 (1976).
    [CrossRef]
  9. B. Chen, G. L. Tangonan, A. B. Lee, presented at 1977 IEEE/OSA Conference on Laser Engineering and Applications, 1–3 June, Washington, D.C., CLEA Digest;IEEE J. Quantum Electron. QE-13, 27 (1977).
  10. S. K. Sheem, C. S. Tsai, presented at the 1977 IEEE/OSA Conference on Laser Engineering and Applications, 1–3 June, Washington, D.C., CLEA Digest, IEEE J. Quantum Electron. QE-13, 27 (7D) (1977); Appl. Opt. 17, 892 (1978).
    [PubMed]
  11. R. A. Steinberg, T. G. Giallorenzi, Appl. Opt. 15, 2440 (1976).
    [CrossRef] [PubMed]
  12. R. A. Steinberg, T. G. Giallorenzi, IEEE J. Quantum Electron. QE-13, 122 (1977).
    [CrossRef]
  13. R. A. Steinberg, T. G. Giallorenzi, R. G. Priest, Appl. Opt. 16, 2166 (1977).
    [CrossRef] [PubMed]
  14. T. G. Giallorenzi, W. K. Burns, R. A. Steinberg, in Digest of Topical Meeting on Integrated and Guided Wave Optics (Optical Society of America, Washington, D.C., 1978).
  15. I. P. Kaminow, J. R. Carruthers, Appl. Phys. Lett. 22, 326 (1973).
    [CrossRef]
  16. J. M. Hammer, W. Phillips, Appl. Phys. Lett. 24, 545 (1974).
    [CrossRef]
  17. R. V. Schmidt, I. P. Kaminow, Appl. Lett. 25, 458 (1974).
    [CrossRef]
  18. J. R. Carruthers, I. P. Kaminow, L. W. Stulz, Appl. Opt. 13, 2333 (1974).
    [CrossRef] [PubMed]
  19. T. R. Raganath, S. Wang, Appl. Phys. Lett. 30, 376 (1977).
    [CrossRef]
  20. R. Ulich, R. J. Martin, Appl. Opt. 10, 2077 (1971).
    [CrossRef]
  21. R. K. Winn, J. H. Harris, IEEE Trans. Microwave Theory Tech. MTT-23, 92 (1975).
    [CrossRef]
  22. W. K. Burns, A. F. Milton, A. B. Lee, Appl. Phys. Lett. 30, 28 (1977).
    [CrossRef]
  23. For example, see S. Ramo, J. R. Whinnery, T. Van Duzer, Field and Waves in Communication Electronics (Wiley, New York, 1967).
  24. D. Marcuse, IEEE J. Quantum Electron. QE-11, 759 (1975).
    [CrossRef]
  25. W. N. Hansen, J. Opt. Soc. Am. 58, 380 (1968).
    [CrossRef]
  26. L. P. Boivin, Appl. Opt. 13, 391 (1974).
    [CrossRef] [PubMed]
  27. H. P. Hsu, A. F. Milton, Electron Lett. 12, 404 (1976).
    [CrossRef]
  28. H. P. Hsu, A. F. Milton, W. K. Burns, S. K. Sheem, in Digest of Topical Meeting on Integrated and Guided Wave Optics (Optical Society of America, Washington, D.C., 1978).
  29. P. K. Tien, R. Ulich, J. Opt. Soc. Am. 60, 1325 (1970).
    [CrossRef]
  30. R. Ulich, J. Opt. Soc. Am. 60, 1337 (1970).
    [CrossRef]
  31. J. E. Midwinter, IEEE J. Quantum Electron. QE-6, 583 (1970).
    [CrossRef]
  32. B. Hocker, W. K. Burns, IEEE J. Quantum Electron. QE-11, 270 (1975).
    [CrossRef]
  33. S. Wang, J. Appl. Phys. 44, 767 (1973).
    [CrossRef]
  34. H. Kogelnik, C. V. Shank, Appl. Phys. Lett. 18, 152 (1971).
    [CrossRef]
  35. P. Zory, Appl. Phys. Lett. 22, 125 (1973).
    [CrossRef]
  36. R. Shubert, J. Appl. Phys. 45, 209 (1974).
    [CrossRef]
  37. M. Nakamura, K. Aiki, J. Umeda, A. Yariv, Appl. Phys. Lett. 27, 403 (1975).
    [CrossRef]

1977

R. A. Steinberg, T. G. Giallorenzi, IEEE J. Quantum Electron. QE-13, 122 (1977).
[CrossRef]

R. A. Steinberg, T. G. Giallorenzi, R. G. Priest, Appl. Opt. 16, 2166 (1977).
[CrossRef] [PubMed]

T. R. Raganath, S. Wang, Appl. Phys. Lett. 30, 376 (1977).
[CrossRef]

W. K. Burns, A. F. Milton, A. B. Lee, Appl. Phys. Lett. 30, 28 (1977).
[CrossRef]

1976

H. P. Hsu, A. F. Milton, Electron Lett. 12, 404 (1976).
[CrossRef]

H. Kogelnik, R. V. Schmidt, IEEE J. Quantum Electron. QE-12, 396 (1976).
[CrossRef]

R. A. Steinberg, T. G. Giallorenzi, Appl. Opt. 15, 2440 (1976).
[CrossRef] [PubMed]

1975

M. Papuchon et al., Appl. Phys. Lett. 27, 289 (1975).
[CrossRef]

J. C. Campbell, F. A. Blum, D. W. Shaw, K. L. Lawley, Appl. Phys. Lett. 27, 203 (1975).
[CrossRef]

R. K. Winn, J. H. Harris, IEEE Trans. Microwave Theory Tech. MTT-23, 92 (1975).
[CrossRef]

D. Marcuse, IEEE J. Quantum Electron. QE-11, 759 (1975).
[CrossRef]

B. Hocker, W. K. Burns, IEEE J. Quantum Electron. QE-11, 270 (1975).
[CrossRef]

M. Nakamura, K. Aiki, J. Umeda, A. Yariv, Appl. Phys. Lett. 27, 403 (1975).
[CrossRef]

1974

R. Shubert, J. Appl. Phys. 45, 209 (1974).
[CrossRef]

L. P. Boivin, Appl. Opt. 13, 391 (1974).
[CrossRef] [PubMed]

J. M. Hammer, W. Phillips, Appl. Phys. Lett. 24, 545 (1974).
[CrossRef]

R. V. Schmidt, I. P. Kaminow, Appl. Lett. 25, 458 (1974).
[CrossRef]

J. R. Carruthers, I. P. Kaminow, L. W. Stulz, Appl. Opt. 13, 2333 (1974).
[CrossRef] [PubMed]

S. Somekh, E. Garmire, H. L. Garvin, R. G. Hunsperger, Appl. Opt. 13, 327 (1974).
[CrossRef] [PubMed]

K. Tada, K. Hirose, Appl. Phys. Lett. 25561 (1974).
[CrossRef]

1973

I. P. Kaminow, J. R. Carruthers, Appl. Phys. Lett. 22, 326 (1973).
[CrossRef]

H. F. Taylor, J. Appl. Phys. 44, 3257 (1973).
[CrossRef]

P. Zory, Appl. Phys. Lett. 22, 125 (1973).
[CrossRef]

S. Wang, J. Appl. Phys. 44, 767 (1973).
[CrossRef]

1972

S. Kurazono, K. Iwasaki, N. Kumagai, Electron. Commun. Jpn. 55, 103 (1972).

1971

R. Ulich, R. J. Martin, Appl. Opt. 10, 2077 (1971).
[CrossRef]

H. Kogelnik, C. V. Shank, Appl. Phys. Lett. 18, 152 (1971).
[CrossRef]

1970

1969

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2071 (1969).

1968

Aiki, K.

M. Nakamura, K. Aiki, J. Umeda, A. Yariv, Appl. Phys. Lett. 27, 403 (1975).
[CrossRef]

Blum, F. A.

J. C. Campbell, F. A. Blum, D. W. Shaw, K. L. Lawley, Appl. Phys. Lett. 27, 203 (1975).
[CrossRef]

Boivin, L. P.

Burns, W. K.

W. K. Burns, A. F. Milton, A. B. Lee, Appl. Phys. Lett. 30, 28 (1977).
[CrossRef]

B. Hocker, W. K. Burns, IEEE J. Quantum Electron. QE-11, 270 (1975).
[CrossRef]

H. P. Hsu, A. F. Milton, W. K. Burns, S. K. Sheem, in Digest of Topical Meeting on Integrated and Guided Wave Optics (Optical Society of America, Washington, D.C., 1978).

T. G. Giallorenzi, W. K. Burns, R. A. Steinberg, in Digest of Topical Meeting on Integrated and Guided Wave Optics (Optical Society of America, Washington, D.C., 1978).

Campbell, J. C.

J. C. Campbell, F. A. Blum, D. W. Shaw, K. L. Lawley, Appl. Phys. Lett. 27, 203 (1975).
[CrossRef]

Carruthers, J. R.

Chen, B.

B. Chen, G. L. Tangonan, A. B. Lee, presented at 1977 IEEE/OSA Conference on Laser Engineering and Applications, 1–3 June, Washington, D.C., CLEA Digest;IEEE J. Quantum Electron. QE-13, 27 (1977).

Garmire, E.

Garvin, H. L.

Giallorenzi, T. G.

R. A. Steinberg, T. G. Giallorenzi, IEEE J. Quantum Electron. QE-13, 122 (1977).
[CrossRef]

R. A. Steinberg, T. G. Giallorenzi, R. G. Priest, Appl. Opt. 16, 2166 (1977).
[CrossRef] [PubMed]

R. A. Steinberg, T. G. Giallorenzi, Appl. Opt. 15, 2440 (1976).
[CrossRef] [PubMed]

T. G. Giallorenzi, W. K. Burns, R. A. Steinberg, in Digest of Topical Meeting on Integrated and Guided Wave Optics (Optical Society of America, Washington, D.C., 1978).

Hammer, J. M.

J. M. Hammer, W. Phillips, Appl. Phys. Lett. 24, 545 (1974).
[CrossRef]

Hansen, W. N.

Harris, J. H.

R. K. Winn, J. H. Harris, IEEE Trans. Microwave Theory Tech. MTT-23, 92 (1975).
[CrossRef]

Hirose, K.

K. Tada, K. Hirose, Appl. Phys. Lett. 25561 (1974).
[CrossRef]

Hocker, B.

B. Hocker, W. K. Burns, IEEE J. Quantum Electron. QE-11, 270 (1975).
[CrossRef]

Hsu, H. P.

H. P. Hsu, A. F. Milton, Electron Lett. 12, 404 (1976).
[CrossRef]

H. P. Hsu, A. F. Milton, W. K. Burns, S. K. Sheem, in Digest of Topical Meeting on Integrated and Guided Wave Optics (Optical Society of America, Washington, D.C., 1978).

Hunsperger, R. G.

Iwasaki, K.

S. Kurazono, K. Iwasaki, N. Kumagai, Electron. Commun. Jpn. 55, 103 (1972).

Kaminow, I. P.

J. R. Carruthers, I. P. Kaminow, L. W. Stulz, Appl. Opt. 13, 2333 (1974).
[CrossRef] [PubMed]

R. V. Schmidt, I. P. Kaminow, Appl. Lett. 25, 458 (1974).
[CrossRef]

I. P. Kaminow, J. R. Carruthers, Appl. Phys. Lett. 22, 326 (1973).
[CrossRef]

Kogelnik, H.

H. Kogelnik, R. V. Schmidt, IEEE J. Quantum Electron. QE-12, 396 (1976).
[CrossRef]

H. Kogelnik, C. V. Shank, Appl. Phys. Lett. 18, 152 (1971).
[CrossRef]

Kumagai, N.

S. Kurazono, K. Iwasaki, N. Kumagai, Electron. Commun. Jpn. 55, 103 (1972).

Kurazono, S.

S. Kurazono, K. Iwasaki, N. Kumagai, Electron. Commun. Jpn. 55, 103 (1972).

Lawley, K. L.

J. C. Campbell, F. A. Blum, D. W. Shaw, K. L. Lawley, Appl. Phys. Lett. 27, 203 (1975).
[CrossRef]

Lee, A. B.

W. K. Burns, A. F. Milton, A. B. Lee, Appl. Phys. Lett. 30, 28 (1977).
[CrossRef]

B. Chen, G. L. Tangonan, A. B. Lee, presented at 1977 IEEE/OSA Conference on Laser Engineering and Applications, 1–3 June, Washington, D.C., CLEA Digest;IEEE J. Quantum Electron. QE-13, 27 (1977).

Marcatili, E. A. J.

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2071 (1969).

Marcuse, D.

D. Marcuse, IEEE J. Quantum Electron. QE-11, 759 (1975).
[CrossRef]

Martin, R. J.

Midwinter, J. E.

J. E. Midwinter, IEEE J. Quantum Electron. QE-6, 583 (1970).
[CrossRef]

Milton, A. F.

W. K. Burns, A. F. Milton, A. B. Lee, Appl. Phys. Lett. 30, 28 (1977).
[CrossRef]

H. P. Hsu, A. F. Milton, Electron Lett. 12, 404 (1976).
[CrossRef]

H. P. Hsu, A. F. Milton, W. K. Burns, S. K. Sheem, in Digest of Topical Meeting on Integrated and Guided Wave Optics (Optical Society of America, Washington, D.C., 1978).

Nakamura, M.

M. Nakamura, K. Aiki, J. Umeda, A. Yariv, Appl. Phys. Lett. 27, 403 (1975).
[CrossRef]

Papuchon, M.

M. Papuchon et al., Appl. Phys. Lett. 27, 289 (1975).
[CrossRef]

Phillips, W.

J. M. Hammer, W. Phillips, Appl. Phys. Lett. 24, 545 (1974).
[CrossRef]

Priest, R. G.

Raganath, T. R.

T. R. Raganath, S. Wang, Appl. Phys. Lett. 30, 376 (1977).
[CrossRef]

Ramo, S.

For example, see S. Ramo, J. R. Whinnery, T. Van Duzer, Field and Waves in Communication Electronics (Wiley, New York, 1967).

Schmidt, R. V.

H. Kogelnik, R. V. Schmidt, IEEE J. Quantum Electron. QE-12, 396 (1976).
[CrossRef]

R. V. Schmidt, I. P. Kaminow, Appl. Lett. 25, 458 (1974).
[CrossRef]

Shank, C. V.

H. Kogelnik, C. V. Shank, Appl. Phys. Lett. 18, 152 (1971).
[CrossRef]

Shaw, D. W.

J. C. Campbell, F. A. Blum, D. W. Shaw, K. L. Lawley, Appl. Phys. Lett. 27, 203 (1975).
[CrossRef]

Sheem, S. K.

S. K. Sheem, C. S. Tsai, presented at the 1977 IEEE/OSA Conference on Laser Engineering and Applications, 1–3 June, Washington, D.C., CLEA Digest, IEEE J. Quantum Electron. QE-13, 27 (7D) (1977); Appl. Opt. 17, 892 (1978).
[PubMed]

H. P. Hsu, A. F. Milton, W. K. Burns, S. K. Sheem, in Digest of Topical Meeting on Integrated and Guided Wave Optics (Optical Society of America, Washington, D.C., 1978).

Shubert, R.

R. Shubert, J. Appl. Phys. 45, 209 (1974).
[CrossRef]

Somekh, S.

Steinberg, R. A.

R. A. Steinberg, T. G. Giallorenzi, IEEE J. Quantum Electron. QE-13, 122 (1977).
[CrossRef]

R. A. Steinberg, T. G. Giallorenzi, R. G. Priest, Appl. Opt. 16, 2166 (1977).
[CrossRef] [PubMed]

R. A. Steinberg, T. G. Giallorenzi, Appl. Opt. 15, 2440 (1976).
[CrossRef] [PubMed]

T. G. Giallorenzi, W. K. Burns, R. A. Steinberg, in Digest of Topical Meeting on Integrated and Guided Wave Optics (Optical Society of America, Washington, D.C., 1978).

Stulz, L. W.

Tada, K.

K. Tada, K. Hirose, Appl. Phys. Lett. 25561 (1974).
[CrossRef]

Tangonan, G. L.

B. Chen, G. L. Tangonan, A. B. Lee, presented at 1977 IEEE/OSA Conference on Laser Engineering and Applications, 1–3 June, Washington, D.C., CLEA Digest;IEEE J. Quantum Electron. QE-13, 27 (1977).

Taylor, H. F.

H. F. Taylor, J. Appl. Phys. 44, 3257 (1973).
[CrossRef]

Tien, P. K.

Tsai, C. S.

S. K. Sheem, C. S. Tsai, presented at the 1977 IEEE/OSA Conference on Laser Engineering and Applications, 1–3 June, Washington, D.C., CLEA Digest, IEEE J. Quantum Electron. QE-13, 27 (7D) (1977); Appl. Opt. 17, 892 (1978).
[PubMed]

Ulich, R.

Umeda, J.

M. Nakamura, K. Aiki, J. Umeda, A. Yariv, Appl. Phys. Lett. 27, 403 (1975).
[CrossRef]

Van Duzer, T.

For example, see S. Ramo, J. R. Whinnery, T. Van Duzer, Field and Waves in Communication Electronics (Wiley, New York, 1967).

Wang, S.

T. R. Raganath, S. Wang, Appl. Phys. Lett. 30, 376 (1977).
[CrossRef]

S. Wang, J. Appl. Phys. 44, 767 (1973).
[CrossRef]

Whinnery, J. R.

For example, see S. Ramo, J. R. Whinnery, T. Van Duzer, Field and Waves in Communication Electronics (Wiley, New York, 1967).

Winn, R. K.

R. K. Winn, J. H. Harris, IEEE Trans. Microwave Theory Tech. MTT-23, 92 (1975).
[CrossRef]

Yariv, A.

M. Nakamura, K. Aiki, J. Umeda, A. Yariv, Appl. Phys. Lett. 27, 403 (1975).
[CrossRef]

Zory, P.

P. Zory, Appl. Phys. Lett. 22, 125 (1973).
[CrossRef]

Appl. Lett.

R. V. Schmidt, I. P. Kaminow, Appl. Lett. 25, 458 (1974).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

W. K. Burns, A. F. Milton, A. B. Lee, Appl. Phys. Lett. 30, 28 (1977).
[CrossRef]

H. Kogelnik, C. V. Shank, Appl. Phys. Lett. 18, 152 (1971).
[CrossRef]

P. Zory, Appl. Phys. Lett. 22, 125 (1973).
[CrossRef]

M. Nakamura, K. Aiki, J. Umeda, A. Yariv, Appl. Phys. Lett. 27, 403 (1975).
[CrossRef]

K. Tada, K. Hirose, Appl. Phys. Lett. 25561 (1974).
[CrossRef]

M. Papuchon et al., Appl. Phys. Lett. 27, 289 (1975).
[CrossRef]

J. C. Campbell, F. A. Blum, D. W. Shaw, K. L. Lawley, Appl. Phys. Lett. 27, 203 (1975).
[CrossRef]

I. P. Kaminow, J. R. Carruthers, Appl. Phys. Lett. 22, 326 (1973).
[CrossRef]

J. M. Hammer, W. Phillips, Appl. Phys. Lett. 24, 545 (1974).
[CrossRef]

T. R. Raganath, S. Wang, Appl. Phys. Lett. 30, 376 (1977).
[CrossRef]

Bell Syst. Tech. J.

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2071 (1969).

Electron Lett.

H. P. Hsu, A. F. Milton, Electron Lett. 12, 404 (1976).
[CrossRef]

Electron. Commun. Jpn.

S. Kurazono, K. Iwasaki, N. Kumagai, Electron. Commun. Jpn. 55, 103 (1972).

IEEE J. Quantum Electron.

R. A. Steinberg, T. G. Giallorenzi, IEEE J. Quantum Electron. QE-13, 122 (1977).
[CrossRef]

H. Kogelnik, R. V. Schmidt, IEEE J. Quantum Electron. QE-12, 396 (1976).
[CrossRef]

D. Marcuse, IEEE J. Quantum Electron. QE-11, 759 (1975).
[CrossRef]

J. E. Midwinter, IEEE J. Quantum Electron. QE-6, 583 (1970).
[CrossRef]

B. Hocker, W. K. Burns, IEEE J. Quantum Electron. QE-11, 270 (1975).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

R. K. Winn, J. H. Harris, IEEE Trans. Microwave Theory Tech. MTT-23, 92 (1975).
[CrossRef]

J. Appl. Phys.

H. F. Taylor, J. Appl. Phys. 44, 3257 (1973).
[CrossRef]

S. Wang, J. Appl. Phys. 44, 767 (1973).
[CrossRef]

R. Shubert, J. Appl. Phys. 45, 209 (1974).
[CrossRef]

J. Opt. Soc. Am.

Other

H. P. Hsu, A. F. Milton, W. K. Burns, S. K. Sheem, in Digest of Topical Meeting on Integrated and Guided Wave Optics (Optical Society of America, Washington, D.C., 1978).

For example, see S. Ramo, J. R. Whinnery, T. Van Duzer, Field and Waves in Communication Electronics (Wiley, New York, 1967).

B. Chen, G. L. Tangonan, A. B. Lee, presented at 1977 IEEE/OSA Conference on Laser Engineering and Applications, 1–3 June, Washington, D.C., CLEA Digest;IEEE J. Quantum Electron. QE-13, 27 (1977).

S. K. Sheem, C. S. Tsai, presented at the 1977 IEEE/OSA Conference on Laser Engineering and Applications, 1–3 June, Washington, D.C., CLEA Digest, IEEE J. Quantum Electron. QE-13, 27 (7D) (1977); Appl. Opt. 17, 892 (1978).
[PubMed]

T. G. Giallorenzi, W. K. Burns, R. A. Steinberg, in Digest of Topical Meeting on Integrated and Guided Wave Optics (Optical Society of America, Washington, D.C., 1978).

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Figures (8)

Fig. 1
Fig. 1

(a) Schematic view of the total internal reflection optical thin-film switch to be evaluated in this work, (b) The effective index of refraction for a mode being propagated in the planar thin-film waveguide shown in (a). The index is assumed to be invariant along the x direction.

Fig. 2
Fig. 2

The relation between the barrier width (g = z2 = z1) and the electric field required for switching for various crosstalk levels.

Fig. 3
Fig. 3

The switch shown in Fig. 1 would have a problem of geometrical mismatch (D0) in one of the output ports (B port here) when being operated as a four-port switch.

Fig. 4
Fig. 4

By employing a multiplicity of electrode pairs in parallel, as shown here, one would be able to increase the electric field while keeping the applied voltage the same.

Fig. 5
Fig. 5

A Gaussian beam model for the analysis of crosstalk being caused by finite beamwidth.

Fig. 6
Fig. 6

(a) A schematic view of a way to draw a planar (waveguide) beam from channel waveguide beam. The coupling between two beams is quite analogous to that of the prism coupler. (b) A four-port channel waveguide switch constructed from the combination of the switching scheme shown in Fig. 1 and the coupling scheme shown in (a).

Fig. 7
Fig. 7

A model for the effective index profile used in the analysis presented in the Appendix A.

Fig. 8
Fig. 8

Models for the analysis presented in Appendix B.

Equations (64)

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

sin θ = n 1 / n 2 ,
n 1 β 1 / k , region I ,
n 2 β 2 / k , region II .
β x = n ( z ) k sin θ ,
β z = n ( z ) k cos θ ,
n ( z ) = n 1 + δ n .
β z = [ k 2 ( n 1 + δ n ) 2 β x 2 ] 1 / 2 .
n 1 δ n ,
β z = [ ( k 2 n 1 2 β x 2 ) 2 n 1 k 2 δ n ] 1 / 2 .
Δ β mis 2 k 2 n 1 2 β x 2 .
β z = [ Δ β mis 2 2 n 1 k 2 δ n ] 1 / 2 .
δ n = ( δ n ) dc + ( δ n ) ac ,
Δ β mis 2 = 2 n 1 k 2 ( δ n ) dc .
β z = [ 2 n 1 k 2 ( δ n ) ac ] 1 / 2 .
( δ n i ) a c a [ ( n 1 3 ) / 2 ] r i 3 E z ( y = 0 ) , ( i = 1,2,3 ) .
E z ( z ) = V π [ ( g / 2 ) 2 z 2 ] 1 / 2 , y = 0 ,
z z ( z 1 + g / 2 ) .
E z = b 2 [ ( V ac ) / g ] , y = 0 .
α = exp ( 2 i β z g ) , β z 2 < 0 ,
α i = exp { [ 2 a n 1 2 k ( r i 3 ) 1 / 2 ] ( g 2 E z ) 1 / 2 } , E z > 0 ,
α i = exp { [ 2 a b n 1 2 k ( r i 3 ) 1 / 2 ] ( g V ac ) 1 / 2 } , V ac > 0 ,
2 a b n 1 2 k ( r i 3 ) 1 / 2 ( g V ac ) 1 / 2 > 4.6.
g V ac > ( r 33 ) / ( r i 3 ) × 75 , i = 1,2,3 .
g = m g 0 ,
α i = exp { [ 2 a n 1 2 k ( r i 3 ) 1 / 2 ] [ m ( E z g 0 ) ( m g 0 ) ] 1 / 2 }
α i = exp { [ 2 a b n 1 2 k ( r i 3 ) 1 / 2 ] ( m V ac g ) 1 / 2 } .
E ( ξ , η ) = A exp ( i k ξ ξ η 2 / 2 W 2 ) .
E ( ξ , η ) = ( k η ) exp ( i k ξ ξ + i k η η ) d k η ,
( k η ) = 1 2 π E ( ξ , η ) exp ( i k η η ) d η ;
k ξ 2 + k η 2 = n 2 k 2 .
| ( k η ) | 2 = 1 2 π W 2 exp [ k η / ( 2 / W ) 2 ] .
Δ θ k η / k ξ , k η k ξ .
Δ θ k W = Δ θ | k η = 2 / W = λ π n 2 ( W / 2 ) .
β z ( Δ θ ) = [ k 2 ( n 1 + δ n D ) 2 + n 2 2 k 2 sin 2 ( θ 0 + Δ θ ) ] 1 / 2 ,
β z ( Δ θ ) = [ k 2 ( n 1 2 + 2 n 1 δ n D ) k 2 ( n 2 2 sin 2 θ 0 + 2 n 2 2 sin θ 0 cos θ 0 Δ θ ) ] 1 / 2 .
N = W π k η + exp [ k η / ( 2 / W ) 2 d k η ] = 1 2 { 1 erf [ k η ( 2 / W ) ] } ,
k η = k ξ · Δ θ c .
0 = n 1 2 n 2 2 sin 2 θ o 2 n 2 2 sin θ o cos θ o Δ θ c .
0 = n 1 2 n 2 2 sin 2 θ o + 2 n 2 2 sin θ o cos θ o Δ θ c + 2 n 1 δ n D .
δ n D = [ ( 2 n 2 2 ) / n 1 ] sin θ o cos θ o Δ θ c .
δ n D 1.65 × ( 4 λ π ) ( cos θ W ) .
2 L 4 W / ( cos θ ) .
δ n D 1.65 × ( 4 λ π ) ( 2 L ) ,
α T = N + α = k η f ( k η ) d k η + k η + f ( k η ) exp [ 2 g | β z ( k η ) | ] d k η ,
f ( k η ) = W π exp [ k η / ( 2 / W ) 2 ] ,
0 > β z 2 ( k η ) = k 2 ( n 1 + δ n ) 2 n 2 2 k 2 sin 2 [ θ o + ( k η / n 2 k ) ] .
D 0 = 2 g tan θ cos θ g , for θ π / 2 .
δ n ( y , z ) = B E z ( y , z ) ,
a ( z ) δ n eff ( z ) δ n ( o , z ) .
δ n eff = a 1 ( n b n b ) + a 2 [ ( n s n s ) ( n b n b ) ] .
a 1 1.
b ( n eff 2 n b 2 ) / ( n s 2 n b 2 ) ,
V k D ( n s 2 n b 2 ) 1 / 2 .
b = A 1 V A 2 ,
A 1 = 0.17 , A 2 = 0.35 ,
A 1 = 0.96 , A 2 = 0.28 ,
a 2 d n eff d n 2 | n b = constant = n s n 1 [ 3 2 A 1 k D ( n s 2 n b 2 ) 1 / 2 A 2 ] ,
a 2 0.3
δ n eff ( n b n b ) + 0.3 [ ( n s n 2 ) ( n b n b ) ] .
a { ( n b n b ) + 0.3 [ ( n s n s ) ( n b n b ) ] / ( n s n s ) } .
0.3 < a < 1.
V ˆ a ( x ) = { V 2 ( t ˆ m 2 ) exp [ i k N m ( P ) x ] } { 2 x l m exp ( x / l m ] ,
V ˆ b ( x ) = { V 2 ( t ˆ m 2 ) exp [ i k N m ( P ) x ] } { exp [ ( x / l m ) 1 ] } ,
S ( x ) = b 1 x + b 2 ,

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