Abstract

Empirical relations for the propagation constant and the field profile parameters of integrated optical diffused channel waveguides have been developed. The field profile used is the evanescent secant- hyperbolic field, which has been shown earlier to be a very good approximation for diffused channel- waveguide modes. Least-square fitting has been used to obtain the empirical relations. The results show that the error in empirical relations for the propagation constant is within 2% for a broad range of waveguide parameters. The obtained empirical relations for the field profile and the propagation constant have been used, as an example, to calculate the coupling length of diffused channel- waveguide-based directional couplers.

© 2008 Optical Society of America

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  1. G. B. Hocker and W. K. Burns, “Mode dispersion in diffused channel waveguides by the effective index method,” Appl. Opt. 15, 113-118 (1977).
    [CrossRef]
  2. M. J. Adams, An Introduction to Optical Waveguides (Wiley, 1981).
  3. T. M. Benson and P. C. Kendall, “Variational techniques including effective and weighted index methods,” in Progress in Electromagnetic Research (EMW, 1995), Vol. 10, pp. 1-40.
  4. M. Kada, J. Ctyroký, I. Gregora, and J. Schröfel, “WKB analysis of guided and semileaky modes in graded-index anisotropic optical waveguides,” Opt. Commun. 28, 59-63 (1978).
  5. E. Schweig and W. B. Bridges, “Computer analysis of dielectric waveguides-A finite difference method,” IEEE Trans. Microwave Theory Tech. MTT-32, 531-541 (1984).
    [CrossRef]
  6. R. K. Lagu and R. V. Ramaswamy, “A variational finite-difference method for analyzing channel waveguides with arbitrary index profiles,” IEEE J. Quantum Electron. QE-22, 968-976 (1986).
    [CrossRef]
  7. F. A. Katsriku, B. M. A. Rahman, and K. T. V. Grattan, “Finite element analysis if diffused anisotropic optical waveguides,” J. Lightwave Technol. 14, 780-786 (1996).
    [CrossRef]
  8. J. L. Jackel, C. E. Rice, and J. J. Veselka, “Proton exchange for high-index waveguides in LiNbO3,” Appl. Phys. Lett. 41, 607-608 (1996).
    [CrossRef]
  9. J. K. Burns, P. H. Klein, E. J. West, and L. E. Plew ,“Ti diffusion in Ti:LiNb3 planar and channel waveguides,” J. Appl. Phys. 50, 6175-6182 (1979).
    [CrossRef]
  10. A. Sharma and P. Bindal, “Variational analysis of diffused planar and channel waveguides and directional couplers,” J. Opt. Soc. Am. A 11, 2244-2248 (1994).
    [CrossRef]
  11. A. Sharma and P. Bindal, “Analysis of diffused planar and channel waveguides,” IEEE J. Quantum Electron. 29, 150-153 (1993).
    [CrossRef]
  12. S. K. Korotoky, W. J. Minfold, L. L. Buhl, M. D. Divino, and R. C. Alferness, “Mode size and method of estimating the propagation constant of single mode Ti:LiNbO3 strip waveguides,” IEEE J. Quantum Electron. QE-18, 1796-1801 (1982).
    [CrossRef]
  13. P. K. Mishra and A. Sharma, “Analysis of single mode inhomogeneous planar waveguides,” J. Lightwave Technol. 4, 204-112 (1986).
    [CrossRef]
  14. A. Di Lallo, A. Cino, C. Conti, and G. Assanto, “Second harmonic generation in reverse proton exchanged lithium niobate waveguides,” Opt. Express 8, 232-237 (2001).
    [CrossRef] [PubMed]
  15. P. Baldi, P. Aschieri, S. Nouh, M. D. Micheli, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, “Modelling and experimental observation of parametric fluorescence in periodically poled lithium niobate waveguides,” IEEE J. Quantum Electron. QE-31, 997-1008 (1995).
    [CrossRef]
  16. R. C. Alferness, “Guided-wave devices for optical communication,” IEEE J. Quantum Electron. QE-17, 946-959 (1981).
    [CrossRef]
  17. I. P. Kaminow and J. R. Karruthers, “Optical waveguide layers in LiNbO3 and LiTaO3,” Appl. Phys. Lett. 22, 326-328 (1973).
    [CrossRef]
  18. E. M. Conwell, “Modes in optical waveguides formed by diffusion,” Appl. Phys. Lett. 23, 328-329 (1973).
    [CrossRef]
  19. A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. QE-9, 919-933 (1973).
    [CrossRef]
  20. A. Ghatak and K. Thyagrajan, Optical Electronics (Cambridge U. Press, 2007).
  21. J. Noda, M. Fukuma, and O. Mikami, “Design calculations for directional couplers fabricated by Ti-diffused LiNbO3 waveguides,” Appl. Opt. 20, 2284-2290 (1981).
    [CrossRef] [PubMed]

2007

A. Ghatak and K. Thyagrajan, Optical Electronics (Cambridge U. Press, 2007).

2001

1996

F. A. Katsriku, B. M. A. Rahman, and K. T. V. Grattan, “Finite element analysis if diffused anisotropic optical waveguides,” J. Lightwave Technol. 14, 780-786 (1996).
[CrossRef]

1995

T. M. Benson and P. C. Kendall, “Variational techniques including effective and weighted index methods,” in Progress in Electromagnetic Research (EMW, 1995), Vol. 10, pp. 1-40.

P. Baldi, P. Aschieri, S. Nouh, M. D. Micheli, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, “Modelling and experimental observation of parametric fluorescence in periodically poled lithium niobate waveguides,” IEEE J. Quantum Electron. QE-31, 997-1008 (1995).
[CrossRef]

1994

1993

A. Sharma and P. Bindal, “Analysis of diffused planar and channel waveguides,” IEEE J. Quantum Electron. 29, 150-153 (1993).
[CrossRef]

1986

R. K. Lagu and R. V. Ramaswamy, “A variational finite-difference method for analyzing channel waveguides with arbitrary index profiles,” IEEE J. Quantum Electron. QE-22, 968-976 (1986).
[CrossRef]

P. K. Mishra and A. Sharma, “Analysis of single mode inhomogeneous planar waveguides,” J. Lightwave Technol. 4, 204-112 (1986).
[CrossRef]

1984

E. Schweig and W. B. Bridges, “Computer analysis of dielectric waveguides-A finite difference method,” IEEE Trans. Microwave Theory Tech. MTT-32, 531-541 (1984).
[CrossRef]

1982

S. K. Korotoky, W. J. Minfold, L. L. Buhl, M. D. Divino, and R. C. Alferness, “Mode size and method of estimating the propagation constant of single mode Ti:LiNbO3 strip waveguides,” IEEE J. Quantum Electron. QE-18, 1796-1801 (1982).
[CrossRef]

1981

J. Noda, M. Fukuma, and O. Mikami, “Design calculations for directional couplers fabricated by Ti-diffused LiNbO3 waveguides,” Appl. Opt. 20, 2284-2290 (1981).
[CrossRef] [PubMed]

R. C. Alferness, “Guided-wave devices for optical communication,” IEEE J. Quantum Electron. QE-17, 946-959 (1981).
[CrossRef]

M. J. Adams, An Introduction to Optical Waveguides (Wiley, 1981).

1979

J. K. Burns, P. H. Klein, E. J. West, and L. E. Plew ,“Ti diffusion in Ti:LiNb3 planar and channel waveguides,” J. Appl. Phys. 50, 6175-6182 (1979).
[CrossRef]

1978

M. Kada, J. Ctyroký, I. Gregora, and J. Schröfel, “WKB analysis of guided and semileaky modes in graded-index anisotropic optical waveguides,” Opt. Commun. 28, 59-63 (1978).

1977

G. B. Hocker and W. K. Burns, “Mode dispersion in diffused channel waveguides by the effective index method,” Appl. Opt. 15, 113-118 (1977).
[CrossRef]

1973

I. P. Kaminow and J. R. Karruthers, “Optical waveguide layers in LiNbO3 and LiTaO3,” Appl. Phys. Lett. 22, 326-328 (1973).
[CrossRef]

E. M. Conwell, “Modes in optical waveguides formed by diffusion,” Appl. Phys. Lett. 23, 328-329 (1973).
[CrossRef]

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. QE-9, 919-933 (1973).
[CrossRef]

Adams, M. J.

M. J. Adams, An Introduction to Optical Waveguides (Wiley, 1981).

Alferness, R. C.

S. K. Korotoky, W. J. Minfold, L. L. Buhl, M. D. Divino, and R. C. Alferness, “Mode size and method of estimating the propagation constant of single mode Ti:LiNbO3 strip waveguides,” IEEE J. Quantum Electron. QE-18, 1796-1801 (1982).
[CrossRef]

R. C. Alferness, “Guided-wave devices for optical communication,” IEEE J. Quantum Electron. QE-17, 946-959 (1981).
[CrossRef]

Aschieri, P.

P. Baldi, P. Aschieri, S. Nouh, M. D. Micheli, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, “Modelling and experimental observation of parametric fluorescence in periodically poled lithium niobate waveguides,” IEEE J. Quantum Electron. QE-31, 997-1008 (1995).
[CrossRef]

Assanto, G.

Baldi, P.

P. Baldi, P. Aschieri, S. Nouh, M. D. Micheli, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, “Modelling and experimental observation of parametric fluorescence in periodically poled lithium niobate waveguides,” IEEE J. Quantum Electron. QE-31, 997-1008 (1995).
[CrossRef]

Benson, T. M.

T. M. Benson and P. C. Kendall, “Variational techniques including effective and weighted index methods,” in Progress in Electromagnetic Research (EMW, 1995), Vol. 10, pp. 1-40.

Bindal, P.

A. Sharma and P. Bindal, “Variational analysis of diffused planar and channel waveguides and directional couplers,” J. Opt. Soc. Am. A 11, 2244-2248 (1994).
[CrossRef]

A. Sharma and P. Bindal, “Analysis of diffused planar and channel waveguides,” IEEE J. Quantum Electron. 29, 150-153 (1993).
[CrossRef]

Bridges, W. B.

E. Schweig and W. B. Bridges, “Computer analysis of dielectric waveguides-A finite difference method,” IEEE Trans. Microwave Theory Tech. MTT-32, 531-541 (1984).
[CrossRef]

Buhl, L. L.

S. K. Korotoky, W. J. Minfold, L. L. Buhl, M. D. Divino, and R. C. Alferness, “Mode size and method of estimating the propagation constant of single mode Ti:LiNbO3 strip waveguides,” IEEE J. Quantum Electron. QE-18, 1796-1801 (1982).
[CrossRef]

Burns, J. K.

J. K. Burns, P. H. Klein, E. J. West, and L. E. Plew ,“Ti diffusion in Ti:LiNb3 planar and channel waveguides,” J. Appl. Phys. 50, 6175-6182 (1979).
[CrossRef]

Burns, W. K.

G. B. Hocker and W. K. Burns, “Mode dispersion in diffused channel waveguides by the effective index method,” Appl. Opt. 15, 113-118 (1977).
[CrossRef]

Cino, A.

Conti, C.

Conwell, E. M.

E. M. Conwell, “Modes in optical waveguides formed by diffusion,” Appl. Phys. Lett. 23, 328-329 (1973).
[CrossRef]

Ctyroký, J.

M. Kada, J. Ctyroký, I. Gregora, and J. Schröfel, “WKB analysis of guided and semileaky modes in graded-index anisotropic optical waveguides,” Opt. Commun. 28, 59-63 (1978).

Delacourt, D.

P. Baldi, P. Aschieri, S. Nouh, M. D. Micheli, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, “Modelling and experimental observation of parametric fluorescence in periodically poled lithium niobate waveguides,” IEEE J. Quantum Electron. QE-31, 997-1008 (1995).
[CrossRef]

Di Lallo, A.

Divino, M. D.

S. K. Korotoky, W. J. Minfold, L. L. Buhl, M. D. Divino, and R. C. Alferness, “Mode size and method of estimating the propagation constant of single mode Ti:LiNbO3 strip waveguides,” IEEE J. Quantum Electron. QE-18, 1796-1801 (1982).
[CrossRef]

Fukuma, M.

Ghatak, A.

A. Ghatak and K. Thyagrajan, Optical Electronics (Cambridge U. Press, 2007).

Grattan, K. T. V.

F. A. Katsriku, B. M. A. Rahman, and K. T. V. Grattan, “Finite element analysis if diffused anisotropic optical waveguides,” J. Lightwave Technol. 14, 780-786 (1996).
[CrossRef]

Gregora, I.

M. Kada, J. Ctyroký, I. Gregora, and J. Schröfel, “WKB analysis of guided and semileaky modes in graded-index anisotropic optical waveguides,” Opt. Commun. 28, 59-63 (1978).

Hocker, G. B.

G. B. Hocker and W. K. Burns, “Mode dispersion in diffused channel waveguides by the effective index method,” Appl. Opt. 15, 113-118 (1977).
[CrossRef]

Jackel, J. L.

J. L. Jackel, C. E. Rice, and J. J. Veselka, “Proton exchange for high-index waveguides in LiNbO3,” Appl. Phys. Lett. 41, 607-608 (1996).
[CrossRef]

Kada, M.

M. Kada, J. Ctyroký, I. Gregora, and J. Schröfel, “WKB analysis of guided and semileaky modes in graded-index anisotropic optical waveguides,” Opt. Commun. 28, 59-63 (1978).

Kaminow, I. P.

I. P. Kaminow and J. R. Karruthers, “Optical waveguide layers in LiNbO3 and LiTaO3,” Appl. Phys. Lett. 22, 326-328 (1973).
[CrossRef]

Karruthers, J. R.

I. P. Kaminow and J. R. Karruthers, “Optical waveguide layers in LiNbO3 and LiTaO3,” Appl. Phys. Lett. 22, 326-328 (1973).
[CrossRef]

Katsriku, F. A.

F. A. Katsriku, B. M. A. Rahman, and K. T. V. Grattan, “Finite element analysis if diffused anisotropic optical waveguides,” J. Lightwave Technol. 14, 780-786 (1996).
[CrossRef]

Kendall, P. C.

T. M. Benson and P. C. Kendall, “Variational techniques including effective and weighted index methods,” in Progress in Electromagnetic Research (EMW, 1995), Vol. 10, pp. 1-40.

Klein, P. H.

J. K. Burns, P. H. Klein, E. J. West, and L. E. Plew ,“Ti diffusion in Ti:LiNb3 planar and channel waveguides,” J. Appl. Phys. 50, 6175-6182 (1979).
[CrossRef]

Korotoky, S. K.

S. K. Korotoky, W. J. Minfold, L. L. Buhl, M. D. Divino, and R. C. Alferness, “Mode size and method of estimating the propagation constant of single mode Ti:LiNbO3 strip waveguides,” IEEE J. Quantum Electron. QE-18, 1796-1801 (1982).
[CrossRef]

Lagu, R. K.

R. K. Lagu and R. V. Ramaswamy, “A variational finite-difference method for analyzing channel waveguides with arbitrary index profiles,” IEEE J. Quantum Electron. QE-22, 968-976 (1986).
[CrossRef]

Micheli, M. D.

P. Baldi, P. Aschieri, S. Nouh, M. D. Micheli, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, “Modelling and experimental observation of parametric fluorescence in periodically poled lithium niobate waveguides,” IEEE J. Quantum Electron. QE-31, 997-1008 (1995).
[CrossRef]

Mikami, O.

Minfold, W. J.

S. K. Korotoky, W. J. Minfold, L. L. Buhl, M. D. Divino, and R. C. Alferness, “Mode size and method of estimating the propagation constant of single mode Ti:LiNbO3 strip waveguides,” IEEE J. Quantum Electron. QE-18, 1796-1801 (1982).
[CrossRef]

Mishra, P. K.

P. K. Mishra and A. Sharma, “Analysis of single mode inhomogeneous planar waveguides,” J. Lightwave Technol. 4, 204-112 (1986).
[CrossRef]

Noda, J.

Nouh, S.

P. Baldi, P. Aschieri, S. Nouh, M. D. Micheli, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, “Modelling and experimental observation of parametric fluorescence in periodically poled lithium niobate waveguides,” IEEE J. Quantum Electron. QE-31, 997-1008 (1995).
[CrossRef]

Ostrowsky, D. B.

P. Baldi, P. Aschieri, S. Nouh, M. D. Micheli, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, “Modelling and experimental observation of parametric fluorescence in periodically poled lithium niobate waveguides,” IEEE J. Quantum Electron. QE-31, 997-1008 (1995).
[CrossRef]

Papuchon, M.

P. Baldi, P. Aschieri, S. Nouh, M. D. Micheli, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, “Modelling and experimental observation of parametric fluorescence in periodically poled lithium niobate waveguides,” IEEE J. Quantum Electron. QE-31, 997-1008 (1995).
[CrossRef]

Plew, L. E.

J. K. Burns, P. H. Klein, E. J. West, and L. E. Plew ,“Ti diffusion in Ti:LiNb3 planar and channel waveguides,” J. Appl. Phys. 50, 6175-6182 (1979).
[CrossRef]

Rahman, B. M. A.

F. A. Katsriku, B. M. A. Rahman, and K. T. V. Grattan, “Finite element analysis if diffused anisotropic optical waveguides,” J. Lightwave Technol. 14, 780-786 (1996).
[CrossRef]

Ramaswamy, R. V.

R. K. Lagu and R. V. Ramaswamy, “A variational finite-difference method for analyzing channel waveguides with arbitrary index profiles,” IEEE J. Quantum Electron. QE-22, 968-976 (1986).
[CrossRef]

Rice, C. E.

J. L. Jackel, C. E. Rice, and J. J. Veselka, “Proton exchange for high-index waveguides in LiNbO3,” Appl. Phys. Lett. 41, 607-608 (1996).
[CrossRef]

Schröfel, J.

M. Kada, J. Ctyroký, I. Gregora, and J. Schröfel, “WKB analysis of guided and semileaky modes in graded-index anisotropic optical waveguides,” Opt. Commun. 28, 59-63 (1978).

Schweig, E.

E. Schweig and W. B. Bridges, “Computer analysis of dielectric waveguides-A finite difference method,” IEEE Trans. Microwave Theory Tech. MTT-32, 531-541 (1984).
[CrossRef]

Sharma, A.

A. Sharma and P. Bindal, “Variational analysis of diffused planar and channel waveguides and directional couplers,” J. Opt. Soc. Am. A 11, 2244-2248 (1994).
[CrossRef]

A. Sharma and P. Bindal, “Analysis of diffused planar and channel waveguides,” IEEE J. Quantum Electron. 29, 150-153 (1993).
[CrossRef]

P. K. Mishra and A. Sharma, “Analysis of single mode inhomogeneous planar waveguides,” J. Lightwave Technol. 4, 204-112 (1986).
[CrossRef]

Thyagrajan, K.

A. Ghatak and K. Thyagrajan, Optical Electronics (Cambridge U. Press, 2007).

Veselka, J. J.

J. L. Jackel, C. E. Rice, and J. J. Veselka, “Proton exchange for high-index waveguides in LiNbO3,” Appl. Phys. Lett. 41, 607-608 (1996).
[CrossRef]

West, E. J.

J. K. Burns, P. H. Klein, E. J. West, and L. E. Plew ,“Ti diffusion in Ti:LiNb3 planar and channel waveguides,” J. Appl. Phys. 50, 6175-6182 (1979).
[CrossRef]

Yariv, A.

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. QE-9, 919-933 (1973).
[CrossRef]

Appl. Opt.

G. B. Hocker and W. K. Burns, “Mode dispersion in diffused channel waveguides by the effective index method,” Appl. Opt. 15, 113-118 (1977).
[CrossRef]

J. Noda, M. Fukuma, and O. Mikami, “Design calculations for directional couplers fabricated by Ti-diffused LiNbO3 waveguides,” Appl. Opt. 20, 2284-2290 (1981).
[CrossRef] [PubMed]

Appl. Phys. Lett.

J. L. Jackel, C. E. Rice, and J. J. Veselka, “Proton exchange for high-index waveguides in LiNbO3,” Appl. Phys. Lett. 41, 607-608 (1996).
[CrossRef]

I. P. Kaminow and J. R. Karruthers, “Optical waveguide layers in LiNbO3 and LiTaO3,” Appl. Phys. Lett. 22, 326-328 (1973).
[CrossRef]

E. M. Conwell, “Modes in optical waveguides formed by diffusion,” Appl. Phys. Lett. 23, 328-329 (1973).
[CrossRef]

IEEE J. Quantum Electron.

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. QE-9, 919-933 (1973).
[CrossRef]

A. Sharma and P. Bindal, “Analysis of diffused planar and channel waveguides,” IEEE J. Quantum Electron. 29, 150-153 (1993).
[CrossRef]

S. K. Korotoky, W. J. Minfold, L. L. Buhl, M. D. Divino, and R. C. Alferness, “Mode size and method of estimating the propagation constant of single mode Ti:LiNbO3 strip waveguides,” IEEE J. Quantum Electron. QE-18, 1796-1801 (1982).
[CrossRef]

R. K. Lagu and R. V. Ramaswamy, “A variational finite-difference method for analyzing channel waveguides with arbitrary index profiles,” IEEE J. Quantum Electron. QE-22, 968-976 (1986).
[CrossRef]

P. Baldi, P. Aschieri, S. Nouh, M. D. Micheli, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, “Modelling and experimental observation of parametric fluorescence in periodically poled lithium niobate waveguides,” IEEE J. Quantum Electron. QE-31, 997-1008 (1995).
[CrossRef]

R. C. Alferness, “Guided-wave devices for optical communication,” IEEE J. Quantum Electron. QE-17, 946-959 (1981).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

E. Schweig and W. B. Bridges, “Computer analysis of dielectric waveguides-A finite difference method,” IEEE Trans. Microwave Theory Tech. MTT-32, 531-541 (1984).
[CrossRef]

J. Appl. Phys.

J. K. Burns, P. H. Klein, E. J. West, and L. E. Plew ,“Ti diffusion in Ti:LiNb3 planar and channel waveguides,” J. Appl. Phys. 50, 6175-6182 (1979).
[CrossRef]

J. Lightwave Technol.

F. A. Katsriku, B. M. A. Rahman, and K. T. V. Grattan, “Finite element analysis if diffused anisotropic optical waveguides,” J. Lightwave Technol. 14, 780-786 (1996).
[CrossRef]

P. K. Mishra and A. Sharma, “Analysis of single mode inhomogeneous planar waveguides,” J. Lightwave Technol. 4, 204-112 (1986).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

M. Kada, J. Ctyroký, I. Gregora, and J. Schröfel, “WKB analysis of guided and semileaky modes in graded-index anisotropic optical waveguides,” Opt. Commun. 28, 59-63 (1978).

Opt. Express

Other

A. Ghatak and K. Thyagrajan, Optical Electronics (Cambridge U. Press, 2007).

M. J. Adams, An Introduction to Optical Waveguides (Wiley, 1981).

T. M. Benson and P. C. Kendall, “Variational techniques including effective and weighted index methods,” in Progress in Electromagnetic Research (EMW, 1995), Vol. 10, pp. 1-40.

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Figures (4)

Fig. 1
Fig. 1

Comparison of the normalized propagation constant, b, calculated by empirical relation (dots) with the actual ESH propagation constant (solid lines). The width to diffusion depth ratio r varies from 1.5 to 2.25 in steps of 0.125 from top to bottom. (a) error–Gaussian refractive index profile waveguide, (b) error–exponential refractive index profile waveguide, (c) Gaussian–Gaussian refractive index profile waveguide.

Fig. 2
Fig. 2

Comparison of field profile obtained using empirical relation with the exact field profile. Waveguide parameters are V = 3.0 , n s = 2.203 , Δ n = 0.03 , and r = 1.5 with error–Gaussian refractive index profile along the (a) depth and (b) width of the waveguide.

Fig. 3
Fig. 3

Comparison of coupling length obtained using coupled-mode theory with the field from empirical relations with that obtained by supermode theory [10] as a function of s, half the waveguide separation. The waveguide parameters are V = 2.0 , n s = 2.203 , n c = 1 , Δ n = 0.03 , λ = 1.3 μm , and r = 1.875 ; the refractive index profile is error–Gaussian.

Fig. 4
Fig. 4

Same as Fig. 3 for parameters n s = 2.152 , n c = 1 , Δ n = 0.006 , λ = 1.152 μm , a = 4 μm , and D = 5 μm ; the profile function is Gaussian–Gaussian.

Equations (14)

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

n 2 ( x , y ) = n s 2 + 2 n s Δ n f ( x ) g ( y ) y > 0 = n c 2 y < 0 ,
β 2 = k 0 2 n 2 ( x , y ) | ψ ( x , y ) | 2 d x d y [ | ψ ( x , y ) x | 2 + | ψ ( x , y ) y | 2 ] d x d y | ψ ( x , y ) | 2 d x d y ,
f ( x ) = erf [ ( x + a ) / ( 2 D ) ] erf [ ( x a ) / ( 2 D ) ] 2 erf ( a / 2 D ) , g ( y ) = exp ( y 2 / ( 2 D 2 ) ) .
f ( x ) = erf [ ( x + a ) / ( 2 D ) ] erf [ ( x a ) / ( 2 D ) ] 2 erf ( a / 2 D ) , g ( y ) = exp ( y / ( 2 D ) ) .
f ( x ) = exp ( x 2 / ( 4 a 2 ) ) , g ( y ) = exp ( y 2 / ( 2 D 2 ) ) ,
ψ SH ( x ) = sec h δ ( x / a ) ψ SH ( y ) = sinh ( y / D ) sec h τ ( y / D ) y 0 = 0 y < 0 ,
ψ ESH ( x ) = sec h δ ( x / a ) , ψ ESH ( y ) = [ 1 + W c sinh ( y / D ) sec h τ ( y / D ) ] y 0 = 0 y < 0 ,
b = 1.57656 + 1.32514 r + 0.41271 V 0.03803 V 2 , δ = 2.77890 + 1.45556 r 0.37672 r 2 + 1.26574 V 0.09746 V 2 , τ = 3.12852 2.71429 r + 0.34709 r 2 + 1.35839 V 0.08386 V 2 .
b = 0.94338 + 0.89782 r + 0.22529 V 0.01682 V 2 , δ = 1.40535 + 0.98480 r 0.23615 r 2 + 0.56841 V , τ = 3.13301 1.25389 r + 0.18581 V + 0.08260 V 2 .
b = 1.90206 + 1.55559 r 0.3 + 0.50937 V 0.08367 V 1.75 δ = 0.61290 + 0.40857 r 0.17524 r 2 + 0.61749 V τ = 3.84422 3.33429 r + 0.55619 r 2 + 1.15407 V 0.04930 V 2
n 2 ( x , y ) = n s 2 + 2 n s Δ n g ( y ) [ f ( x + s ) + f ( x s ) ] y 0 = n c 2 y < 0 .
n 1 2 ( x , y ) = n s 2 + 2 n s Δ n g ( y ) [ f ( x + s ) ] y 0 n c 2 y < 0 .
κ = k 0 2 2 β ψ 1 * ( x , y ) [ n ( x , y ) 2 n 1 ( x , y ) 2 ] ψ 2 ( x , y ) d x d y ψ 1 * ( x , y ) ψ 1 ( x , y ) d x d y ,
L c = π / ( 2 κ ) .

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