Abstract

The phase jumps in a linear coupler and a nonlinear coherent coupler are demonstrated analytically. The theoretical results show that slight changes in the ratio of input powers or a mismatch in the linear coupler slight changes in the input power in the nonlinear coupler can lead to a phase shift of π. This characteristic of the coupler may lead to electro-optical switching with an ultralow switching voltage or to all-optical switching with low switching power.

© 2001 Optical Society of America

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References

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  1. D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972).
  2. A. W. Synder and J. D. Love, Optical Waveguide Theory, Chapman & Hall, London, 1983), pp. 570 ff.
  3. A. Yariv, “The coupled-mode Formalism in guided-wave optics,” in Fiber and Integrated Optics, D. B. Ostrowsky, ed. (Plenum, New York, 1979), pp. 223–256.
  4. H. Kogelink, “Theory of dielectric waveguides,” in Integrated Optics, T. Tamir, ed. (Springer-Verlag, New York, 1975), Chap. 2.
  5. A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. LT-3, 1135–1146 (1985).
    [Crossref]
  6. E. A. J. Marcatili, “Improved coupled-mode equations for dielectric guides,” IEEE J. Quantum Electron. QE-22, 988–993 (1986).
    [Crossref]
  7. E. A. J. Marcatili, L. L. Buhi, and R. C. Alferness, “Experimental verification of the improved coupled-mode equations,” IEEE J. Quantum Electron. QE-22, 988–993 (1986).
    [Crossref]
  8. See, for example, H. Nishihara, M. Haruna, and T. Suhara, Optical Integrated Circuit (McGraw-Hill, New York, 1989).
  9. See, for example, A. B. Buckman, Guide-Wave Photonics (Saunders, Fort Worth, Tex., 1992).
  10. D. B. Mortimore, “Fiber loop reflector,” J. Lightwave Technol. 6, 1217–1224 (1988).
    [Crossref]
  11. L. F. Stokes, M. Chodorow, and H. J. Shaw, “Sensitive all-single-mode-fiber resonant ring interferometer,” J. Lightwave Technol. LT-1, 110–115 (1983).
    [Crossref]
  12. A. T. Pham and L. N. Binh, “Nonlinear optical directional coupler two-input operation,” Int. J. Optoelectron. 5, 367–380 (1990).
  13. A. T. Pham and L. N. Binh, “All-optical modulation and switching using a nonlinear-optical directional coupler,” J. Opt. Soc. Am. B 8, 1914–1931 (1991).
    [Crossref]
  14. K.-I. Kitayama and S. Wang, “Optical pulse compression by nonlinear coupling,” Appl. Phys. Lett. 43, P17–P19 (1983).
    [Crossref]
  15. B. Daino, G. Gregori, and S. Wabnitz, “Stability analysis of nonlinear coherent coupling,” J. Appl. Phys. 58, 4512–4514 (1985).
    [Crossref]
  16. R. T. Deck and C. Mapalagama, “Improved theory of nonlinear directional couplers,” in Quantum Electronics and Laser Science, Vol. 13 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 126–128.
  17. K. Yasumoto, H. Maeda, and N. Maekawa, “Coupled-mode analysis of an asymmetric nonlinear directional coupler,” J. Lightwave Technol. 14, 628–633 (1996).
    [Crossref]
  18. S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. QE-18, 1580–1583 (1981).
  19. Y. Chen, A. W. Snyder, and D. J. Mitchell, “Ideal optical switching by nonlinear multiple (parasitic) core couplers,” Electron. Lett. 26, 77–78 (1990).
    [Crossref]
  20. Y. Wang, “Nonlinear optical limiter and digital optical switch by cascaded nonlinear couplers: analysis,” J. Lightwave Technol. 17, 292–297 (1999).
    [Crossref]
  21. Y. Wang and J. Liu, “All-fiber logical devices based on the nonlinear directional coupler,” IEEE Photonics Technol. Lett. 11, 72–74 (1999).
    [Crossref]
  22. C. Thirstrup, “Optical bistability in a nonlinear directional coupler,” IEEE J. Quantum Electron. 31, 2101–2106 (1995).
    [Crossref]
  23. A. Villeneuve, C. C. Yang, P. G. J. Wigley, G. I. Stegeman, J. S. Aitchison, and C. N. Ironside, “Ultrafast all-optical switching in semiconductor nonlinear directional couplers at half the band gap,” Appl. Phys. Lett. 61, 147–149 (1992).
    [Crossref]
  24. J. S. Aitchison, A. Villeneuve, and G. I. Stegeman, “All-optical switching in two cascaded nonlinear directional couplers,” Opt. Lett. 20, 698–700 (1995).
    [Crossref] [PubMed]
  25. S. R. Friberg, Y. Silberberg, M. K. Oliver, M. J. Andrejco, M. A. Saifi, and P. W. Smith, “Ultrafast all-optical switching in a dual-core fiber nonlinear coupler,” Appl. Phys. Lett. 51, 135–137 (1987).
    [Crossref]
  26. Y. Wang and C.-K. Lee, “Phase characteristics of nonlinear optical directional coupler,” presented at the Second International Conference on Information, Communications and Signal Processing Singapore, December 1999.
  27. Y. Chen, “Twin core nonlinear couplers with saturable nonlinearity,” Electron. Lett. 26, 1374–1375 (1990).
    [Crossref]
  28. G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1989), Chap. 7.

1999 (2)

Y. Wang, “Nonlinear optical limiter and digital optical switch by cascaded nonlinear couplers: analysis,” J. Lightwave Technol. 17, 292–297 (1999).
[Crossref]

Y. Wang and J. Liu, “All-fiber logical devices based on the nonlinear directional coupler,” IEEE Photonics Technol. Lett. 11, 72–74 (1999).
[Crossref]

1996 (1)

K. Yasumoto, H. Maeda, and N. Maekawa, “Coupled-mode analysis of an asymmetric nonlinear directional coupler,” J. Lightwave Technol. 14, 628–633 (1996).
[Crossref]

1995 (2)

C. Thirstrup, “Optical bistability in a nonlinear directional coupler,” IEEE J. Quantum Electron. 31, 2101–2106 (1995).
[Crossref]

J. S. Aitchison, A. Villeneuve, and G. I. Stegeman, “All-optical switching in two cascaded nonlinear directional couplers,” Opt. Lett. 20, 698–700 (1995).
[Crossref] [PubMed]

1992 (1)

A. Villeneuve, C. C. Yang, P. G. J. Wigley, G. I. Stegeman, J. S. Aitchison, and C. N. Ironside, “Ultrafast all-optical switching in semiconductor nonlinear directional couplers at half the band gap,” Appl. Phys. Lett. 61, 147–149 (1992).
[Crossref]

1991 (1)

1990 (3)

A. T. Pham and L. N. Binh, “Nonlinear optical directional coupler two-input operation,” Int. J. Optoelectron. 5, 367–380 (1990).

Y. Chen, A. W. Snyder, and D. J. Mitchell, “Ideal optical switching by nonlinear multiple (parasitic) core couplers,” Electron. Lett. 26, 77–78 (1990).
[Crossref]

Y. Chen, “Twin core nonlinear couplers with saturable nonlinearity,” Electron. Lett. 26, 1374–1375 (1990).
[Crossref]

1988 (1)

D. B. Mortimore, “Fiber loop reflector,” J. Lightwave Technol. 6, 1217–1224 (1988).
[Crossref]

1987 (1)

S. R. Friberg, Y. Silberberg, M. K. Oliver, M. J. Andrejco, M. A. Saifi, and P. W. Smith, “Ultrafast all-optical switching in a dual-core fiber nonlinear coupler,” Appl. Phys. Lett. 51, 135–137 (1987).
[Crossref]

1986 (2)

E. A. J. Marcatili, “Improved coupled-mode equations for dielectric guides,” IEEE J. Quantum Electron. QE-22, 988–993 (1986).
[Crossref]

E. A. J. Marcatili, L. L. Buhi, and R. C. Alferness, “Experimental verification of the improved coupled-mode equations,” IEEE J. Quantum Electron. QE-22, 988–993 (1986).
[Crossref]

1985 (2)

A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. LT-3, 1135–1146 (1985).
[Crossref]

B. Daino, G. Gregori, and S. Wabnitz, “Stability analysis of nonlinear coherent coupling,” J. Appl. Phys. 58, 4512–4514 (1985).
[Crossref]

1983 (2)

K.-I. Kitayama and S. Wang, “Optical pulse compression by nonlinear coupling,” Appl. Phys. Lett. 43, P17–P19 (1983).
[Crossref]

L. F. Stokes, M. Chodorow, and H. J. Shaw, “Sensitive all-single-mode-fiber resonant ring interferometer,” J. Lightwave Technol. LT-1, 110–115 (1983).
[Crossref]

1981 (1)

S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. QE-18, 1580–1583 (1981).

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1989), Chap. 7.

Aitchison, J. S.

J. S. Aitchison, A. Villeneuve, and G. I. Stegeman, “All-optical switching in two cascaded nonlinear directional couplers,” Opt. Lett. 20, 698–700 (1995).
[Crossref] [PubMed]

A. Villeneuve, C. C. Yang, P. G. J. Wigley, G. I. Stegeman, J. S. Aitchison, and C. N. Ironside, “Ultrafast all-optical switching in semiconductor nonlinear directional couplers at half the band gap,” Appl. Phys. Lett. 61, 147–149 (1992).
[Crossref]

Alferness, R. C.

E. A. J. Marcatili, L. L. Buhi, and R. C. Alferness, “Experimental verification of the improved coupled-mode equations,” IEEE J. Quantum Electron. QE-22, 988–993 (1986).
[Crossref]

Andrejco, M. J.

S. R. Friberg, Y. Silberberg, M. K. Oliver, M. J. Andrejco, M. A. Saifi, and P. W. Smith, “Ultrafast all-optical switching in a dual-core fiber nonlinear coupler,” Appl. Phys. Lett. 51, 135–137 (1987).
[Crossref]

Binh, L. N.

A. T. Pham and L. N. Binh, “All-optical modulation and switching using a nonlinear-optical directional coupler,” J. Opt. Soc. Am. B 8, 1914–1931 (1991).
[Crossref]

A. T. Pham and L. N. Binh, “Nonlinear optical directional coupler two-input operation,” Int. J. Optoelectron. 5, 367–380 (1990).

Buckman, A. B.

See, for example, A. B. Buckman, Guide-Wave Photonics (Saunders, Fort Worth, Tex., 1992).

Buhi, L. L.

E. A. J. Marcatili, L. L. Buhi, and R. C. Alferness, “Experimental verification of the improved coupled-mode equations,” IEEE J. Quantum Electron. QE-22, 988–993 (1986).
[Crossref]

Chen, Y.

Y. Chen, A. W. Snyder, and D. J. Mitchell, “Ideal optical switching by nonlinear multiple (parasitic) core couplers,” Electron. Lett. 26, 77–78 (1990).
[Crossref]

Y. Chen, “Twin core nonlinear couplers with saturable nonlinearity,” Electron. Lett. 26, 1374–1375 (1990).
[Crossref]

Chodorow, M.

L. F. Stokes, M. Chodorow, and H. J. Shaw, “Sensitive all-single-mode-fiber resonant ring interferometer,” J. Lightwave Technol. LT-1, 110–115 (1983).
[Crossref]

Daino, B.

B. Daino, G. Gregori, and S. Wabnitz, “Stability analysis of nonlinear coherent coupling,” J. Appl. Phys. 58, 4512–4514 (1985).
[Crossref]

Deck, R. T.

R. T. Deck and C. Mapalagama, “Improved theory of nonlinear directional couplers,” in Quantum Electronics and Laser Science, Vol. 13 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 126–128.

Friberg, S. R.

S. R. Friberg, Y. Silberberg, M. K. Oliver, M. J. Andrejco, M. A. Saifi, and P. W. Smith, “Ultrafast all-optical switching in a dual-core fiber nonlinear coupler,” Appl. Phys. Lett. 51, 135–137 (1987).
[Crossref]

Gregori, G.

B. Daino, G. Gregori, and S. Wabnitz, “Stability analysis of nonlinear coherent coupling,” J. Appl. Phys. 58, 4512–4514 (1985).
[Crossref]

Hardy, A.

A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. LT-3, 1135–1146 (1985).
[Crossref]

Haruna, M.

See, for example, H. Nishihara, M. Haruna, and T. Suhara, Optical Integrated Circuit (McGraw-Hill, New York, 1989).

Ironside, C. N.

A. Villeneuve, C. C. Yang, P. G. J. Wigley, G. I. Stegeman, J. S. Aitchison, and C. N. Ironside, “Ultrafast all-optical switching in semiconductor nonlinear directional couplers at half the band gap,” Appl. Phys. Lett. 61, 147–149 (1992).
[Crossref]

Jensen, S. M.

S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. QE-18, 1580–1583 (1981).

Kitayama, K.-I.

K.-I. Kitayama and S. Wang, “Optical pulse compression by nonlinear coupling,” Appl. Phys. Lett. 43, P17–P19 (1983).
[Crossref]

Kogelink, H.

H. Kogelink, “Theory of dielectric waveguides,” in Integrated Optics, T. Tamir, ed. (Springer-Verlag, New York, 1975), Chap. 2.

Lee, C.-K.

Y. Wang and C.-K. Lee, “Phase characteristics of nonlinear optical directional coupler,” presented at the Second International Conference on Information, Communications and Signal Processing Singapore, December 1999.

Liu, J.

Y. Wang and J. Liu, “All-fiber logical devices based on the nonlinear directional coupler,” IEEE Photonics Technol. Lett. 11, 72–74 (1999).
[Crossref]

Love, J. D.

A. W. Synder and J. D. Love, Optical Waveguide Theory, Chapman & Hall, London, 1983), pp. 570 ff.

Maeda, H.

K. Yasumoto, H. Maeda, and N. Maekawa, “Coupled-mode analysis of an asymmetric nonlinear directional coupler,” J. Lightwave Technol. 14, 628–633 (1996).
[Crossref]

Maekawa, N.

K. Yasumoto, H. Maeda, and N. Maekawa, “Coupled-mode analysis of an asymmetric nonlinear directional coupler,” J. Lightwave Technol. 14, 628–633 (1996).
[Crossref]

Mapalagama, C.

R. T. Deck and C. Mapalagama, “Improved theory of nonlinear directional couplers,” in Quantum Electronics and Laser Science, Vol. 13 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 126–128.

Marcatili, E. A. J.

E. A. J. Marcatili, “Improved coupled-mode equations for dielectric guides,” IEEE J. Quantum Electron. QE-22, 988–993 (1986).
[Crossref]

E. A. J. Marcatili, L. L. Buhi, and R. C. Alferness, “Experimental verification of the improved coupled-mode equations,” IEEE J. Quantum Electron. QE-22, 988–993 (1986).
[Crossref]

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972).

Mitchell, D. J.

Y. Chen, A. W. Snyder, and D. J. Mitchell, “Ideal optical switching by nonlinear multiple (parasitic) core couplers,” Electron. Lett. 26, 77–78 (1990).
[Crossref]

Mortimore, D. B.

D. B. Mortimore, “Fiber loop reflector,” J. Lightwave Technol. 6, 1217–1224 (1988).
[Crossref]

Nishihara, H.

See, for example, H. Nishihara, M. Haruna, and T. Suhara, Optical Integrated Circuit (McGraw-Hill, New York, 1989).

Oliver, M. K.

S. R. Friberg, Y. Silberberg, M. K. Oliver, M. J. Andrejco, M. A. Saifi, and P. W. Smith, “Ultrafast all-optical switching in a dual-core fiber nonlinear coupler,” Appl. Phys. Lett. 51, 135–137 (1987).
[Crossref]

Pham, A. T.

A. T. Pham and L. N. Binh, “All-optical modulation and switching using a nonlinear-optical directional coupler,” J. Opt. Soc. Am. B 8, 1914–1931 (1991).
[Crossref]

A. T. Pham and L. N. Binh, “Nonlinear optical directional coupler two-input operation,” Int. J. Optoelectron. 5, 367–380 (1990).

Saifi, M. A.

S. R. Friberg, Y. Silberberg, M. K. Oliver, M. J. Andrejco, M. A. Saifi, and P. W. Smith, “Ultrafast all-optical switching in a dual-core fiber nonlinear coupler,” Appl. Phys. Lett. 51, 135–137 (1987).
[Crossref]

Shaw, H. J.

L. F. Stokes, M. Chodorow, and H. J. Shaw, “Sensitive all-single-mode-fiber resonant ring interferometer,” J. Lightwave Technol. LT-1, 110–115 (1983).
[Crossref]

Silberberg, Y.

S. R. Friberg, Y. Silberberg, M. K. Oliver, M. J. Andrejco, M. A. Saifi, and P. W. Smith, “Ultrafast all-optical switching in a dual-core fiber nonlinear coupler,” Appl. Phys. Lett. 51, 135–137 (1987).
[Crossref]

Smith, P. W.

S. R. Friberg, Y. Silberberg, M. K. Oliver, M. J. Andrejco, M. A. Saifi, and P. W. Smith, “Ultrafast all-optical switching in a dual-core fiber nonlinear coupler,” Appl. Phys. Lett. 51, 135–137 (1987).
[Crossref]

Snyder, A. W.

Y. Chen, A. W. Snyder, and D. J. Mitchell, “Ideal optical switching by nonlinear multiple (parasitic) core couplers,” Electron. Lett. 26, 77–78 (1990).
[Crossref]

Stegeman, G. I.

J. S. Aitchison, A. Villeneuve, and G. I. Stegeman, “All-optical switching in two cascaded nonlinear directional couplers,” Opt. Lett. 20, 698–700 (1995).
[Crossref] [PubMed]

A. Villeneuve, C. C. Yang, P. G. J. Wigley, G. I. Stegeman, J. S. Aitchison, and C. N. Ironside, “Ultrafast all-optical switching in semiconductor nonlinear directional couplers at half the band gap,” Appl. Phys. Lett. 61, 147–149 (1992).
[Crossref]

Stokes, L. F.

L. F. Stokes, M. Chodorow, and H. J. Shaw, “Sensitive all-single-mode-fiber resonant ring interferometer,” J. Lightwave Technol. LT-1, 110–115 (1983).
[Crossref]

Streifer, W.

A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. LT-3, 1135–1146 (1985).
[Crossref]

Suhara, T.

See, for example, H. Nishihara, M. Haruna, and T. Suhara, Optical Integrated Circuit (McGraw-Hill, New York, 1989).

Synder, A. W.

A. W. Synder and J. D. Love, Optical Waveguide Theory, Chapman & Hall, London, 1983), pp. 570 ff.

Thirstrup, C.

C. Thirstrup, “Optical bistability in a nonlinear directional coupler,” IEEE J. Quantum Electron. 31, 2101–2106 (1995).
[Crossref]

Villeneuve, A.

J. S. Aitchison, A. Villeneuve, and G. I. Stegeman, “All-optical switching in two cascaded nonlinear directional couplers,” Opt. Lett. 20, 698–700 (1995).
[Crossref] [PubMed]

A. Villeneuve, C. C. Yang, P. G. J. Wigley, G. I. Stegeman, J. S. Aitchison, and C. N. Ironside, “Ultrafast all-optical switching in semiconductor nonlinear directional couplers at half the band gap,” Appl. Phys. Lett. 61, 147–149 (1992).
[Crossref]

Wabnitz, S.

B. Daino, G. Gregori, and S. Wabnitz, “Stability analysis of nonlinear coherent coupling,” J. Appl. Phys. 58, 4512–4514 (1985).
[Crossref]

Wang, S.

K.-I. Kitayama and S. Wang, “Optical pulse compression by nonlinear coupling,” Appl. Phys. Lett. 43, P17–P19 (1983).
[Crossref]

Wang, Y.

Y. Wang and J. Liu, “All-fiber logical devices based on the nonlinear directional coupler,” IEEE Photonics Technol. Lett. 11, 72–74 (1999).
[Crossref]

Y. Wang, “Nonlinear optical limiter and digital optical switch by cascaded nonlinear couplers: analysis,” J. Lightwave Technol. 17, 292–297 (1999).
[Crossref]

Y. Wang and C.-K. Lee, “Phase characteristics of nonlinear optical directional coupler,” presented at the Second International Conference on Information, Communications and Signal Processing Singapore, December 1999.

Wigley, P. G. J.

A. Villeneuve, C. C. Yang, P. G. J. Wigley, G. I. Stegeman, J. S. Aitchison, and C. N. Ironside, “Ultrafast all-optical switching in semiconductor nonlinear directional couplers at half the band gap,” Appl. Phys. Lett. 61, 147–149 (1992).
[Crossref]

Yang, C. C.

A. Villeneuve, C. C. Yang, P. G. J. Wigley, G. I. Stegeman, J. S. Aitchison, and C. N. Ironside, “Ultrafast all-optical switching in semiconductor nonlinear directional couplers at half the band gap,” Appl. Phys. Lett. 61, 147–149 (1992).
[Crossref]

Yariv, A.

A. Yariv, “The coupled-mode Formalism in guided-wave optics,” in Fiber and Integrated Optics, D. B. Ostrowsky, ed. (Plenum, New York, 1979), pp. 223–256.

Yasumoto, K.

K. Yasumoto, H. Maeda, and N. Maekawa, “Coupled-mode analysis of an asymmetric nonlinear directional coupler,” J. Lightwave Technol. 14, 628–633 (1996).
[Crossref]

Appl. Phys. Lett. (3)

K.-I. Kitayama and S. Wang, “Optical pulse compression by nonlinear coupling,” Appl. Phys. Lett. 43, P17–P19 (1983).
[Crossref]

A. Villeneuve, C. C. Yang, P. G. J. Wigley, G. I. Stegeman, J. S. Aitchison, and C. N. Ironside, “Ultrafast all-optical switching in semiconductor nonlinear directional couplers at half the band gap,” Appl. Phys. Lett. 61, 147–149 (1992).
[Crossref]

S. R. Friberg, Y. Silberberg, M. K. Oliver, M. J. Andrejco, M. A. Saifi, and P. W. Smith, “Ultrafast all-optical switching in a dual-core fiber nonlinear coupler,” Appl. Phys. Lett. 51, 135–137 (1987).
[Crossref]

Electron. Lett. (2)

Y. Chen, A. W. Snyder, and D. J. Mitchell, “Ideal optical switching by nonlinear multiple (parasitic) core couplers,” Electron. Lett. 26, 77–78 (1990).
[Crossref]

Y. Chen, “Twin core nonlinear couplers with saturable nonlinearity,” Electron. Lett. 26, 1374–1375 (1990).
[Crossref]

IEEE J. Quantum Electron. (4)

C. Thirstrup, “Optical bistability in a nonlinear directional coupler,” IEEE J. Quantum Electron. 31, 2101–2106 (1995).
[Crossref]

S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. QE-18, 1580–1583 (1981).

E. A. J. Marcatili, “Improved coupled-mode equations for dielectric guides,” IEEE J. Quantum Electron. QE-22, 988–993 (1986).
[Crossref]

E. A. J. Marcatili, L. L. Buhi, and R. C. Alferness, “Experimental verification of the improved coupled-mode equations,” IEEE J. Quantum Electron. QE-22, 988–993 (1986).
[Crossref]

IEEE Photonics Technol. Lett. (1)

Y. Wang and J. Liu, “All-fiber logical devices based on the nonlinear directional coupler,” IEEE Photonics Technol. Lett. 11, 72–74 (1999).
[Crossref]

Int. J. Optoelectron. (1)

A. T. Pham and L. N. Binh, “Nonlinear optical directional coupler two-input operation,” Int. J. Optoelectron. 5, 367–380 (1990).

J. Appl. Phys. (1)

B. Daino, G. Gregori, and S. Wabnitz, “Stability analysis of nonlinear coherent coupling,” J. Appl. Phys. 58, 4512–4514 (1985).
[Crossref]

J. Lightwave Technol. (5)

A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. LT-3, 1135–1146 (1985).
[Crossref]

K. Yasumoto, H. Maeda, and N. Maekawa, “Coupled-mode analysis of an asymmetric nonlinear directional coupler,” J. Lightwave Technol. 14, 628–633 (1996).
[Crossref]

D. B. Mortimore, “Fiber loop reflector,” J. Lightwave Technol. 6, 1217–1224 (1988).
[Crossref]

L. F. Stokes, M. Chodorow, and H. J. Shaw, “Sensitive all-single-mode-fiber resonant ring interferometer,” J. Lightwave Technol. LT-1, 110–115 (1983).
[Crossref]

Y. Wang, “Nonlinear optical limiter and digital optical switch by cascaded nonlinear couplers: analysis,” J. Lightwave Technol. 17, 292–297 (1999).
[Crossref]

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

Opt. Lett. (1)

Other (9)

Y. Wang and C.-K. Lee, “Phase characteristics of nonlinear optical directional coupler,” presented at the Second International Conference on Information, Communications and Signal Processing Singapore, December 1999.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1989), Chap. 7.

R. T. Deck and C. Mapalagama, “Improved theory of nonlinear directional couplers,” in Quantum Electronics and Laser Science, Vol. 13 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 126–128.

See, for example, H. Nishihara, M. Haruna, and T. Suhara, Optical Integrated Circuit (McGraw-Hill, New York, 1989).

See, for example, A. B. Buckman, Guide-Wave Photonics (Saunders, Fort Worth, Tex., 1992).

D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972).

A. W. Synder and J. D. Love, Optical Waveguide Theory, Chapman & Hall, London, 1983), pp. 570 ff.

A. Yariv, “The coupled-mode Formalism in guided-wave optics,” in Fiber and Integrated Optics, D. B. Ostrowsky, ed. (Plenum, New York, 1979), pp. 223–256.

H. Kogelink, “Theory of dielectric waveguides,” in Integrated Optics, T. Tamir, ed. (Springer-Verlag, New York, 1975), Chap. 2.

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Figures (12)

Fig. 1
Fig. 1

Phase shift induced by coupling as a function of the length of the linear matched coupler. Solid curve, P2(0)/P1(0)=10, φ0=-0.495π; long dashed curve, P2(0)/P1(0)=1, φ0=-0.495π; short-dashed curve, P2(0)/P1(0)=1, φ0=0.495π.

Fig. 2
Fig. 2

Phase-shift ϕ1(Z)-ϕ1(0) and normalized power of the beam in waveguide 1 as functions of Z. Solid curves are phase curves and dashed curves are power curves.

Fig. 3
Fig. 3

Phase shift ϕ1(Z)-ϕ1(0), phase arctan(Im{-A1(Z)×exp[-jϕ1(0)]}/Re{A¯1 exp[-jϕ1(0)]}), and both the real part and the imaginary part of the complex A1(Z)exp[-jϕ1(0)] as function of Z.

Fig. 4
Fig. 4

Influence of the input-power ratio γ=P2(0)/P1(0) on the phase-jump characteristics of the linear coupler.

Fig. 5
Fig. 5

Phase shift ϕ1(Z)-ϕ1(0) and the output power of the beam in waveguide 1 as functions of the mismatch δ, where the solid curve is a phase curve and dashed curve is a power curve.

Fig. 6
Fig. 6

Electro-optical switching configuration.

Fig. 7
Fig. 7

Phase shifts ϕn1(Z)-ϕn1(0) and ϕn2(Z)-ϕn2(0) as well as output powers Pn1(Z) and Pn2(Z) as functions of the initial normalized power Pn1(0), where Pn2(0)=0.003, Z=1.8π and φn0=0.47π.

Fig. 8
Fig. 8

Phase shift ϕn2(Z)-ϕn2(0) as a function of the length of the coupler for various initial inputs Pn1(0), where Pn2(0)=0.003 and φn0=0.47π.

Fig. 9
Fig. 9

Phase shift induced by nonlinear coupling and power as functions of the length of the nonlinear coupler.

Fig. 10
Fig. 10

Influence of input power Pn1(0) and ratio Pn1(0)/Pn2(0) on the phase-shift characteristics, where φn0=0.47π. Long-dashed curve, Pn1(0)=0.3 and Pn2(0)=0.003; short-dashed curve, Pn1(0)=0.27 and Pn2(0)=0.0027; solid curve, Pn1(0)=0.27 and Pn2(0)=0.003.

Fig. 11
Fig. 11

Phase shift of the beam in the self-focusing waveguide induced by nonlinear coupling as a function of the power Pn1(0), where Pn2(0)=0.29, φn0=0.56π, and Z=3.2π.

Fig. 12
Fig. 12

Phase shift of the beam in waveguide 1 of the symmetric nonlinear coupler as a function of normalized input power Pn1(0), where Pn2(0)=0.1, φn0=0.31π, and Z=1.8π.

Equations (60)

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-ida1dz=β1a1+Ca2+Q1|a1|2a1+iσa1,
-ida2dz=β2a2+Ca1+Q2|a2|2a2+iσa2,
-idb1dz=β1b1+Cb2,
-idb2dz=β2b2+Cb1,
bm=Am exp[jβmz+jϕm(z)],
dA1dz=CA2 sin(φ+Δz),
dA2dz=-CA1 sin(φ+Δz),
A1dϕ1dz=CA2 cos(φ+Δz),
A2dϕ2dz=CA1 cos(φ+Δz),
Pt=P1+P2,
Γ=A1A2 cos(φ+Δz)+δA12,
dP1dZ2=P1P2-(Γ-δP1),
P1(Z)=M+H cos α1(Z),
P1(Z)=Pt-M+H cos α2(Z),
M=1+γ+2δΓ¯2(1+δ2)P1(0),
α0=sin-1P1(0)-MH,-π2α0π2,
α1(Z)=1+δ2Z+α0-π2,
α2(Z)=1+δ2Z±α0+π2,
H=[(1+γ)2+4δΓ¯(1+γ)-4Γ¯2]1/22(1+δ2)P1(0).
dϕ1(Z)dZ=Γ2P1-δ2,
dϕ2(Z)dZ=Γ-δPt2P2+δ2.
ϕ1(Z)-ϕ1(0)
=sign(Γ)tan-1Pa tanα1(Z)2-tan-1Pa tanα1(0)2+N1π-δ2Z,
ϕ2(Z)-ϕ2(0)
=sign(Γ-δPl)tan-1Pb tanα2(Z)2-tan-1Pb tanα2(0)2+N2π+δ2Z,
N1,2=intα1,2(Z)2π+0.5,Pa=M-HM+H1/2, Pb=Pt-M-HPt-M+H1/2.
M=1+γ2P1(0),
H=[(1+γ)2-4γ cos2 φ0]1/22P1(0),
α0=sin-11-γ[(1+γ)2-4γ cos2 φ0]1/2, -π2α0π2.
Zc=3π2-α0,sin φ0>0,
Zc=π2+α0,sin φ0<0.
am=PcAnm exp[jβz+jϕnm(z)],
dPn1dZ2=Pn1Pn2-(Γn+Pn22-Pn12)2,
Pn1dϕn1(Z)dZ=12(Γn+Pnt2)+2Pn12-PntPn1,
Pn2dϕn2(Z)dZ=12(Γn-Pnt2)-2Pn22+PntPn2,
Γn=[Pn1(0)Pn2(0)]1/2 cos φn0+Pn12(0)-Pn22(0),
Pn1(Z)=Mn+Hn cos(αn0±1+4Pnt2Z),
Pn2(Z)=Pnt-Mn-Hn cos(αn0±1+4Pnt2Z),
Mn=Pnt2+2ΓnPnt1+4Pnt2,
Hn=Mn2-(Γn+Pnt2)21+4Pnt21/2=(Pnt2+4Pnt4-4Γn2)1/22(1+4Pnt2),
αn0=cos-1Pn1(0)-MnHn,0αn0π.
dϕn1(Z)dZ=2Pn1-Pnt.
dϕn2(Z)dZ=Pnt-2Pn2.
ϕn1(Z)=(2Mn-Pnt)Z±2Hn1+4Pnt2[sin αn(Z)-sin αn(0)]±sign(Γn+Pnt2)×arctan|Γn+Pnt2|g1 tanαn(Z)2-arctan|Γn+Pnt2|g1 tanαn(0)2+Nπ,
ϕn2(Z)=(2Mn-Pnt)Z±2Hn1+4Pnt2[sin αn(Z)-sin αn(0)]±sign(Γn-Pnt2)×arctan-1g2|Γn-Pnt2| tanαn(Z)2-arctan-1g2|Γn-Pnt2| tanαn(0)2+Nπ,
N=Intαn(Z)2π+0.5,
αn(Z)=αn0±1+4Pnt2Z,
g1=(Mn+Hn)1+4Pnt2,
g2=(Pnt-Mn+Hn)1+4Pnt2.
dϕn1dZ=Pnt+Pn1+Γn/8Pn1,
dϕn2dZ=Pnt+Pn2+Γn/8Pn2,
Pn1(Z)=Pnt2+b0 cn[Z1±2b0Z/k1, k1],
A22Pnt2A1,
Pn1(Z)=Pnt2+b0 dn[Z2±2b0Z, k2],
Pn1(0)>Pnt2,2Pnt2>A1,
Pn1(Z)=Pnt2-b0 dn[Z2±2b0Z, k2],
Pn1(0)>Pnt2,2Pnt2>A1,
bm=A¯m exp(jβmz),
dA¯mdz=iCA¯3-m exp[(-1)mΔz].
A¯1(Z)=exp[jϕ1(0)]21+δ2 ({2R-A1(0)-A2(0)exp[-jφ(0)]}exp(-jZR+)+{2R+A1(0)+A2(0)exp[-Jφ(0)]}exp(jZR-)),

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