Abstract

Coupling efficiency between the fundamental modes of two different graded-index waveguides, which are fabricated in glass or lithium niobate, is calculated by numerical and variational methods. Analytical results are derived by the variational approach, which supplies a systematic method for studying the influence of modal coupling on the design of integrated optical components.

© 1992 Optical Society of America

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References

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  1. R. Ulrich, “Theory of prism-film coupler by plane-wave analysis,” J. Opt. Soc. Am. 60, 1337–1350 (1970).
    [CrossRef]
  2. H. Stoll, A. Yariv, “Coupled mode analysis of periodic dielectric waveguides,” Opt. Commun. 8, 5–8 (1973).
    [CrossRef]
  3. A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
    [CrossRef]
  4. R. P. Kenan, “Theory of diffraction of guided optical waves by thick holograms,” J. Appl. Phys. 46, 4545–4551 (1975).
    [CrossRef]
  5. D. Marcuse, “Coupled-mode theory for anisotropic optical waveguides,” Bell Syst. Tech. J. 54, 985–995 (1975).
  6. P. G. Suchoski, R. V. Ramaswamy, “Design of single-mode step-tapered waveguide sections,” IEEE J. Quantum Electron. QE-23, 205–211 (1987).
    [CrossRef]
  7. H. Zhenguang, R. Srivastava, R. V. Ramaswamy, “Low-loss small-mode passive waveguides and near-adiabatic tapers in BK7 glass,” IEEE J. Lightwave Technol. LT-7, 1590–1596 (1989).
    [CrossRef]
  8. G. P. Bava, P. Rosina, I. Montrosset, “Numerical analysis of planar Fresnel lenses,” J. Mod. Opt. 35, 863–869 (1988).
    [CrossRef]
  9. C. W. Pitt, S. Reid, S. Reynolds, J. Skinner, “Waveguide Fresnel lenses: modelling and fabrication,” J. Mod. Opt. 35, 1079–1111 (1988).
    [CrossRef]
  10. G. C. Righini, G. Belli, M. Varasi, A. Vannucci, “Waveguide Fresnel lenses for integrated optical processors,” in Integrated Optics and Optoelectronics, L. McCaughan, M. A. Mentzer, S. Peng, H. J. Wojtunik, K. Wong, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1177, 209–215 (1989).
  11. G. C. Righini, J. Linares, J. E. Alvarellos, “Modal coupling optimization of integrated optical devices in LiNbO3,” in Optics in Complex Systems, F. Lanzl, H. Preuss, G. Weigelt, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1319, 110–111 (1990).
  12. G. Perrone, I. Montrosset, “A correction to the two-dimensional BPM for the analysis of waveguide lenses,” J. Mod. Opt. (to be published).
  13. J. Linares, J. E. Alvarellos, G. C. Righini, “Efficiency of modal coupling between graded-index optical waveguides,” J. Mod. Opt. 38, 2177–2187 (1991).
    [CrossRef]
  14. G. H. Chartier, P. J. R. Laybourn, A. Girod, “Masking process for double-ion-exchanged glass optical waveguides,” Electron. Lett. 22, 925–926 (1986).
    [CrossRef]
  15. G. C. Righini, R. Shen, G. Belli, P. Boffi, A. Losacco, P. Mazzoldi, G. Battaglin, “Integrated optical components fabricated by two-step ion-exchange,” in Glasses for Optoelectronics, G. C. Righini, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1128, 103–109 (1989).
  16. D. Y. Zang, C. S. Tsai, “Single-mode waveguide microlenses and microlens array fabrication in LiNbO3 using titanium indiffused proton exchanged technique,” Appl. Phys. Lett. 46, 703–705 (1985).
    [CrossRef]
  17. C. S. Tsai, “Integrated-optical device modules in LiNbO3 for computing and signal processing,” J. Mod. Opt. 35, 965–977 (1988).
    [CrossRef]
  18. S. A. Reid, M. Varasi, S. Reynolds, “Double dilute melt proton exchange Fresnel lenses for LiNbO3 optical waveguides,” J. Opt. Commun. 10, 67–73 (1989).
  19. K. R. Lagu, R. V. Ramaswamy, “Process and waveguide parameter relationships for the design of planar, silver ion-exchanged glass waveguides,” IEEE J. Lightwave Technol. LT-4, 176–181 (1986).
    [CrossRef]
  20. S. E. Koonin, Computational Physics (Benjamin/Cumming, Menlo Park, Calif., 1985).
  21. S. K. Korotki, W. J. Minford, L. L. Buhl, M. D. Divino, R. C. Alferness, “Mode size and method for estimating the propagation constant of single-mode Ti:LiNbO3 strip waveguides,” IEEE J. Quantum Electron. QE-18, 1796–1801 (1982).
    [CrossRef]
  22. J. Linares, R. de la Fuente, “Optimization of the optical interconnection between microlens and channel waveguide arrays,” Jpn. J. Appl. Phys. 29, L1335–L1337 (1990).
    [CrossRef]

1991 (1)

J. Linares, J. E. Alvarellos, G. C. Righini, “Efficiency of modal coupling between graded-index optical waveguides,” J. Mod. Opt. 38, 2177–2187 (1991).
[CrossRef]

1990 (1)

J. Linares, R. de la Fuente, “Optimization of the optical interconnection between microlens and channel waveguide arrays,” Jpn. J. Appl. Phys. 29, L1335–L1337 (1990).
[CrossRef]

1989 (2)

S. A. Reid, M. Varasi, S. Reynolds, “Double dilute melt proton exchange Fresnel lenses for LiNbO3 optical waveguides,” J. Opt. Commun. 10, 67–73 (1989).

H. Zhenguang, R. Srivastava, R. V. Ramaswamy, “Low-loss small-mode passive waveguides and near-adiabatic tapers in BK7 glass,” IEEE J. Lightwave Technol. LT-7, 1590–1596 (1989).
[CrossRef]

1988 (3)

G. P. Bava, P. Rosina, I. Montrosset, “Numerical analysis of planar Fresnel lenses,” J. Mod. Opt. 35, 863–869 (1988).
[CrossRef]

C. W. Pitt, S. Reid, S. Reynolds, J. Skinner, “Waveguide Fresnel lenses: modelling and fabrication,” J. Mod. Opt. 35, 1079–1111 (1988).
[CrossRef]

C. S. Tsai, “Integrated-optical device modules in LiNbO3 for computing and signal processing,” J. Mod. Opt. 35, 965–977 (1988).
[CrossRef]

1987 (1)

P. G. Suchoski, R. V. Ramaswamy, “Design of single-mode step-tapered waveguide sections,” IEEE J. Quantum Electron. QE-23, 205–211 (1987).
[CrossRef]

1986 (2)

K. R. Lagu, R. V. Ramaswamy, “Process and waveguide parameter relationships for the design of planar, silver ion-exchanged glass waveguides,” IEEE J. Lightwave Technol. LT-4, 176–181 (1986).
[CrossRef]

G. H. Chartier, P. J. R. Laybourn, A. Girod, “Masking process for double-ion-exchanged glass optical waveguides,” Electron. Lett. 22, 925–926 (1986).
[CrossRef]

1985 (1)

D. Y. Zang, C. S. Tsai, “Single-mode waveguide microlenses and microlens array fabrication in LiNbO3 using titanium indiffused proton exchanged technique,” Appl. Phys. Lett. 46, 703–705 (1985).
[CrossRef]

1982 (1)

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

1975 (2)

R. P. Kenan, “Theory of diffraction of guided optical waves by thick holograms,” J. Appl. Phys. 46, 4545–4551 (1975).
[CrossRef]

D. Marcuse, “Coupled-mode theory for anisotropic optical waveguides,” Bell Syst. Tech. J. 54, 985–995 (1975).

1973 (2)

H. Stoll, A. Yariv, “Coupled mode analysis of periodic dielectric waveguides,” Opt. Commun. 8, 5–8 (1973).
[CrossRef]

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

1970 (1)

Alferness, R. C.

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

Alvarellos, J. E.

J. Linares, J. E. Alvarellos, G. C. Righini, “Efficiency of modal coupling between graded-index optical waveguides,” J. Mod. Opt. 38, 2177–2187 (1991).
[CrossRef]

G. C. Righini, J. Linares, J. E. Alvarellos, “Modal coupling optimization of integrated optical devices in LiNbO3,” in Optics in Complex Systems, F. Lanzl, H. Preuss, G. Weigelt, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1319, 110–111 (1990).

Battaglin, G.

G. C. Righini, R. Shen, G. Belli, P. Boffi, A. Losacco, P. Mazzoldi, G. Battaglin, “Integrated optical components fabricated by two-step ion-exchange,” in Glasses for Optoelectronics, G. C. Righini, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1128, 103–109 (1989).

Bava, G. P.

G. P. Bava, P. Rosina, I. Montrosset, “Numerical analysis of planar Fresnel lenses,” J. Mod. Opt. 35, 863–869 (1988).
[CrossRef]

Belli, G.

G. C. Righini, R. Shen, G. Belli, P. Boffi, A. Losacco, P. Mazzoldi, G. Battaglin, “Integrated optical components fabricated by two-step ion-exchange,” in Glasses for Optoelectronics, G. C. Righini, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1128, 103–109 (1989).

G. C. Righini, G. Belli, M. Varasi, A. Vannucci, “Waveguide Fresnel lenses for integrated optical processors,” in Integrated Optics and Optoelectronics, L. McCaughan, M. A. Mentzer, S. Peng, H. J. Wojtunik, K. Wong, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1177, 209–215 (1989).

Boffi, P.

G. C. Righini, R. Shen, G. Belli, P. Boffi, A. Losacco, P. Mazzoldi, G. Battaglin, “Integrated optical components fabricated by two-step ion-exchange,” in Glasses for Optoelectronics, G. C. Righini, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1128, 103–109 (1989).

Buhl, L. L.

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

Chartier, G. H.

G. H. Chartier, P. J. R. Laybourn, A. Girod, “Masking process for double-ion-exchanged glass optical waveguides,” Electron. Lett. 22, 925–926 (1986).
[CrossRef]

de la Fuente, R.

J. Linares, R. de la Fuente, “Optimization of the optical interconnection between microlens and channel waveguide arrays,” Jpn. J. Appl. Phys. 29, L1335–L1337 (1990).
[CrossRef]

Divino, M. D.

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

Girod, A.

G. H. Chartier, P. J. R. Laybourn, A. Girod, “Masking process for double-ion-exchanged glass optical waveguides,” Electron. Lett. 22, 925–926 (1986).
[CrossRef]

Kenan, R. P.

R. P. Kenan, “Theory of diffraction of guided optical waves by thick holograms,” J. Appl. Phys. 46, 4545–4551 (1975).
[CrossRef]

Koonin, S. E.

S. E. Koonin, Computational Physics (Benjamin/Cumming, Menlo Park, Calif., 1985).

Korotki, S. K.

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

Lagu, K. R.

K. R. Lagu, R. V. Ramaswamy, “Process and waveguide parameter relationships for the design of planar, silver ion-exchanged glass waveguides,” IEEE J. Lightwave Technol. LT-4, 176–181 (1986).
[CrossRef]

Laybourn, P. J. R.

G. H. Chartier, P. J. R. Laybourn, A. Girod, “Masking process for double-ion-exchanged glass optical waveguides,” Electron. Lett. 22, 925–926 (1986).
[CrossRef]

Linares, J.

J. Linares, J. E. Alvarellos, G. C. Righini, “Efficiency of modal coupling between graded-index optical waveguides,” J. Mod. Opt. 38, 2177–2187 (1991).
[CrossRef]

J. Linares, R. de la Fuente, “Optimization of the optical interconnection between microlens and channel waveguide arrays,” Jpn. J. Appl. Phys. 29, L1335–L1337 (1990).
[CrossRef]

G. C. Righini, J. Linares, J. E. Alvarellos, “Modal coupling optimization of integrated optical devices in LiNbO3,” in Optics in Complex Systems, F. Lanzl, H. Preuss, G. Weigelt, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1319, 110–111 (1990).

Losacco, A.

G. C. Righini, R. Shen, G. Belli, P. Boffi, A. Losacco, P. Mazzoldi, G. Battaglin, “Integrated optical components fabricated by two-step ion-exchange,” in Glasses for Optoelectronics, G. C. Righini, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1128, 103–109 (1989).

Marcuse, D.

D. Marcuse, “Coupled-mode theory for anisotropic optical waveguides,” Bell Syst. Tech. J. 54, 985–995 (1975).

Mazzoldi, P.

G. C. Righini, R. Shen, G. Belli, P. Boffi, A. Losacco, P. Mazzoldi, G. Battaglin, “Integrated optical components fabricated by two-step ion-exchange,” in Glasses for Optoelectronics, G. C. Righini, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1128, 103–109 (1989).

Minford, W. J.

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

Montrosset, I.

G. P. Bava, P. Rosina, I. Montrosset, “Numerical analysis of planar Fresnel lenses,” J. Mod. Opt. 35, 863–869 (1988).
[CrossRef]

G. Perrone, I. Montrosset, “A correction to the two-dimensional BPM for the analysis of waveguide lenses,” J. Mod. Opt. (to be published).

Perrone, G.

G. Perrone, I. Montrosset, “A correction to the two-dimensional BPM for the analysis of waveguide lenses,” J. Mod. Opt. (to be published).

Pitt, C. W.

C. W. Pitt, S. Reid, S. Reynolds, J. Skinner, “Waveguide Fresnel lenses: modelling and fabrication,” J. Mod. Opt. 35, 1079–1111 (1988).
[CrossRef]

Ramaswamy, R. V.

H. Zhenguang, R. Srivastava, R. V. Ramaswamy, “Low-loss small-mode passive waveguides and near-adiabatic tapers in BK7 glass,” IEEE J. Lightwave Technol. LT-7, 1590–1596 (1989).
[CrossRef]

P. G. Suchoski, R. V. Ramaswamy, “Design of single-mode step-tapered waveguide sections,” IEEE J. Quantum Electron. QE-23, 205–211 (1987).
[CrossRef]

K. R. Lagu, R. V. Ramaswamy, “Process and waveguide parameter relationships for the design of planar, silver ion-exchanged glass waveguides,” IEEE J. Lightwave Technol. LT-4, 176–181 (1986).
[CrossRef]

Reid, S.

C. W. Pitt, S. Reid, S. Reynolds, J. Skinner, “Waveguide Fresnel lenses: modelling and fabrication,” J. Mod. Opt. 35, 1079–1111 (1988).
[CrossRef]

Reid, S. A.

S. A. Reid, M. Varasi, S. Reynolds, “Double dilute melt proton exchange Fresnel lenses for LiNbO3 optical waveguides,” J. Opt. Commun. 10, 67–73 (1989).

Reynolds, S.

S. A. Reid, M. Varasi, S. Reynolds, “Double dilute melt proton exchange Fresnel lenses for LiNbO3 optical waveguides,” J. Opt. Commun. 10, 67–73 (1989).

C. W. Pitt, S. Reid, S. Reynolds, J. Skinner, “Waveguide Fresnel lenses: modelling and fabrication,” J. Mod. Opt. 35, 1079–1111 (1988).
[CrossRef]

Righini, G. C.

J. Linares, J. E. Alvarellos, G. C. Righini, “Efficiency of modal coupling between graded-index optical waveguides,” J. Mod. Opt. 38, 2177–2187 (1991).
[CrossRef]

G. C. Righini, J. Linares, J. E. Alvarellos, “Modal coupling optimization of integrated optical devices in LiNbO3,” in Optics in Complex Systems, F. Lanzl, H. Preuss, G. Weigelt, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1319, 110–111 (1990).

G. C. Righini, G. Belli, M. Varasi, A. Vannucci, “Waveguide Fresnel lenses for integrated optical processors,” in Integrated Optics and Optoelectronics, L. McCaughan, M. A. Mentzer, S. Peng, H. J. Wojtunik, K. Wong, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1177, 209–215 (1989).

G. C. Righini, R. Shen, G. Belli, P. Boffi, A. Losacco, P. Mazzoldi, G. Battaglin, “Integrated optical components fabricated by two-step ion-exchange,” in Glasses for Optoelectronics, G. C. Righini, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1128, 103–109 (1989).

Rosina, P.

G. P. Bava, P. Rosina, I. Montrosset, “Numerical analysis of planar Fresnel lenses,” J. Mod. Opt. 35, 863–869 (1988).
[CrossRef]

Shen, R.

G. C. Righini, R. Shen, G. Belli, P. Boffi, A. Losacco, P. Mazzoldi, G. Battaglin, “Integrated optical components fabricated by two-step ion-exchange,” in Glasses for Optoelectronics, G. C. Righini, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1128, 103–109 (1989).

Skinner, J.

C. W. Pitt, S. Reid, S. Reynolds, J. Skinner, “Waveguide Fresnel lenses: modelling and fabrication,” J. Mod. Opt. 35, 1079–1111 (1988).
[CrossRef]

Srivastava, R.

H. Zhenguang, R. Srivastava, R. V. Ramaswamy, “Low-loss small-mode passive waveguides and near-adiabatic tapers in BK7 glass,” IEEE J. Lightwave Technol. LT-7, 1590–1596 (1989).
[CrossRef]

Stoll, H.

H. Stoll, A. Yariv, “Coupled mode analysis of periodic dielectric waveguides,” Opt. Commun. 8, 5–8 (1973).
[CrossRef]

Suchoski, P. G.

P. G. Suchoski, R. V. Ramaswamy, “Design of single-mode step-tapered waveguide sections,” IEEE J. Quantum Electron. QE-23, 205–211 (1987).
[CrossRef]

Tsai, C. S.

C. S. Tsai, “Integrated-optical device modules in LiNbO3 for computing and signal processing,” J. Mod. Opt. 35, 965–977 (1988).
[CrossRef]

D. Y. Zang, C. S. Tsai, “Single-mode waveguide microlenses and microlens array fabrication in LiNbO3 using titanium indiffused proton exchanged technique,” Appl. Phys. Lett. 46, 703–705 (1985).
[CrossRef]

Ulrich, R.

Vannucci, A.

G. C. Righini, G. Belli, M. Varasi, A. Vannucci, “Waveguide Fresnel lenses for integrated optical processors,” in Integrated Optics and Optoelectronics, L. McCaughan, M. A. Mentzer, S. Peng, H. J. Wojtunik, K. Wong, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1177, 209–215 (1989).

Varasi, M.

S. A. Reid, M. Varasi, S. Reynolds, “Double dilute melt proton exchange Fresnel lenses for LiNbO3 optical waveguides,” J. Opt. Commun. 10, 67–73 (1989).

G. C. Righini, G. Belli, M. Varasi, A. Vannucci, “Waveguide Fresnel lenses for integrated optical processors,” in Integrated Optics and Optoelectronics, L. McCaughan, M. A. Mentzer, S. Peng, H. J. Wojtunik, K. Wong, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1177, 209–215 (1989).

Yariv, A.

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

H. Stoll, A. Yariv, “Coupled mode analysis of periodic dielectric waveguides,” Opt. Commun. 8, 5–8 (1973).
[CrossRef]

Zang, D. Y.

D. Y. Zang, C. S. Tsai, “Single-mode waveguide microlenses and microlens array fabrication in LiNbO3 using titanium indiffused proton exchanged technique,” Appl. Phys. Lett. 46, 703–705 (1985).
[CrossRef]

Zhenguang, H.

H. Zhenguang, R. Srivastava, R. V. Ramaswamy, “Low-loss small-mode passive waveguides and near-adiabatic tapers in BK7 glass,” IEEE J. Lightwave Technol. LT-7, 1590–1596 (1989).
[CrossRef]

Appl. Phys. Lett. (1)

D. Y. Zang, C. S. Tsai, “Single-mode waveguide microlenses and microlens array fabrication in LiNbO3 using titanium indiffused proton exchanged technique,” Appl. Phys. Lett. 46, 703–705 (1985).
[CrossRef]

Bell Syst. Tech. J. (1)

D. Marcuse, “Coupled-mode theory for anisotropic optical waveguides,” Bell Syst. Tech. J. 54, 985–995 (1975).

Electron. Lett. (1)

G. H. Chartier, P. J. R. Laybourn, A. Girod, “Masking process for double-ion-exchanged glass optical waveguides,” Electron. Lett. 22, 925–926 (1986).
[CrossRef]

IEEE J. Lightwave Technol. (2)

K. R. Lagu, R. V. Ramaswamy, “Process and waveguide parameter relationships for the design of planar, silver ion-exchanged glass waveguides,” IEEE J. Lightwave Technol. LT-4, 176–181 (1986).
[CrossRef]

H. Zhenguang, R. Srivastava, R. V. Ramaswamy, “Low-loss small-mode passive waveguides and near-adiabatic tapers in BK7 glass,” IEEE J. Lightwave Technol. LT-7, 1590–1596 (1989).
[CrossRef]

IEEE J. Quantum Electron. (3)

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

P. G. Suchoski, R. V. Ramaswamy, “Design of single-mode step-tapered waveguide sections,” IEEE J. Quantum Electron. QE-23, 205–211 (1987).
[CrossRef]

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

J. Appl. Phys. (1)

R. P. Kenan, “Theory of diffraction of guided optical waves by thick holograms,” J. Appl. Phys. 46, 4545–4551 (1975).
[CrossRef]

J. Mod. Opt. (4)

G. P. Bava, P. Rosina, I. Montrosset, “Numerical analysis of planar Fresnel lenses,” J. Mod. Opt. 35, 863–869 (1988).
[CrossRef]

C. W. Pitt, S. Reid, S. Reynolds, J. Skinner, “Waveguide Fresnel lenses: modelling and fabrication,” J. Mod. Opt. 35, 1079–1111 (1988).
[CrossRef]

C. S. Tsai, “Integrated-optical device modules in LiNbO3 for computing and signal processing,” J. Mod. Opt. 35, 965–977 (1988).
[CrossRef]

J. Linares, J. E. Alvarellos, G. C. Righini, “Efficiency of modal coupling between graded-index optical waveguides,” J. Mod. Opt. 38, 2177–2187 (1991).
[CrossRef]

J. Opt. Commun. (1)

S. A. Reid, M. Varasi, S. Reynolds, “Double dilute melt proton exchange Fresnel lenses for LiNbO3 optical waveguides,” J. Opt. Commun. 10, 67–73 (1989).

J. Opt. Soc. Am. (1)

Jpn. J. Appl. Phys. (1)

J. Linares, R. de la Fuente, “Optimization of the optical interconnection between microlens and channel waveguide arrays,” Jpn. J. Appl. Phys. 29, L1335–L1337 (1990).
[CrossRef]

Opt. Commun. (1)

H. Stoll, A. Yariv, “Coupled mode analysis of periodic dielectric waveguides,” Opt. Commun. 8, 5–8 (1973).
[CrossRef]

Other (5)

G. C. Righini, G. Belli, M. Varasi, A. Vannucci, “Waveguide Fresnel lenses for integrated optical processors,” in Integrated Optics and Optoelectronics, L. McCaughan, M. A. Mentzer, S. Peng, H. J. Wojtunik, K. Wong, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1177, 209–215 (1989).

G. C. Righini, J. Linares, J. E. Alvarellos, “Modal coupling optimization of integrated optical devices in LiNbO3,” in Optics in Complex Systems, F. Lanzl, H. Preuss, G. Weigelt, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1319, 110–111 (1990).

G. Perrone, I. Montrosset, “A correction to the two-dimensional BPM for the analysis of waveguide lenses,” J. Mod. Opt. (to be published).

S. E. Koonin, Computational Physics (Benjamin/Cumming, Menlo Park, Calif., 1985).

G. C. Righini, R. Shen, G. Belli, P. Boffi, A. Losacco, P. Mazzoldi, G. Battaglin, “Integrated optical components fabricated by two-step ion-exchange,” in Glasses for Optoelectronics, G. C. Righini, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1128, 103–109 (1989).

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Figures (5)

Fig. 1
Fig. 1

Typical IOPC (a waveguide lens) and the index profiles of the two waveguides outside and inside the lens area, respectively.

Fig. 2
Fig. 2

Schematic representation of the cross section (xz plane) of the interface between the single-mode n1 outer waveguide and the multimode n2 inner waveguide with effective depths d1 and d2, respectively.

Fig. 3
Fig. 3

Comparison of numerical (solid curve) and variational (dashed curve) results for coupling between the fundamental modes (TE0 → TE0) of two waveguides in a glass substrate, both with a Gaussian-index profile (the G-G case) for δn1 = 0.015 and δn2 = 0.08.

Fig. 4
Fig. 4

Comparison of numerical and variational results for modal coupling (TE0 → TE0) between an erfc and a Gaussian waveguide (the E-G case) in glass with δn1 = 0.03 and δn2 = 0.08.

Fig. 5
Fig. 5

Comparison of numerical and variational results for modal coupling (TE0 → TE0) between a GRIN (Gaussian profile) Ti-indiffused and a step-index TIPE waveguide in LiNbO3 with δn1 = 0.012 and δn2 = 0.107.

Tables (2)

Tables Icon

Table 1 Numerical Results for the Maximum Coupling Efficiency between the TE0 Modes of Two GRIN Waveguides in Glass

Tables Icon

Table 2 Comparison of Numerical and Variational (′) Results for Maximun Coupling Efficiency in the Same Four Cases Considered in Table 1

Equations (33)

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n 2 ( x ) = n 0 2 + 2 n 0 δ n F ( x ) ,
β 2 = k 2 n 2 ( x ) Ψ 0 t 2 d x - d Ψ 0 t / d x 2 d x ,
Ψ 0 t = 2 x α 3 / π 2 1 / 4 exp ( - x 2 / 2 α 2 ) .
F g ( x ) = exp ( - x 2 / d g 2 ) ,
F e ( x ) = erfc ( x / d e ) .
β 2 / α = 0 ,
α g 2 = w 0 g 2 ( 1 + α g 2 / d g 2 ) 5 / 4 ,
α e 2 = w 0 e 2 ( 1 + α e 2 / d e 2 ) 4 / 3 ,
w 0 g 2 = d g ( 2 k 2 n 0 δ n g ) - 1 / 2 ,
w 0 e 2 = ( 3 π d e ) 2 / 3 ( 8 k 2 n 0 δ n e ) - 2 / 3 .
α g 2 w 0 g 2 ,
α g 2 = w 0 g 2 ( 1 + ( w 0 g 2 / d g 2 ) { 1 + ( w 0 g 2 / d g 2 ) × [ 1 + ( w 0 g 2 / d g 2 ) [ ] 5 / 4 ] 5 / 4 } 5 / 4 ) 5 / 4 ,
α e 2 = w 0 e 2 ( 1 + ( w 0 e 2 / d e 2 ) { 1 + ( w 0 e 2 / d e 2 ) × [ 1 + ( w 0 e 2 / d e 2 ) [ ] 2 / 3 ] 2 / 3 } 2 / 3 ) 2 / 3 .
E 0 = [ 2 ( w a w b ) w a 2 + w b 2 ] 3 ,
β g 2 = 1 d g 2 { ( k d g ) 2 ( n 0 2 + 2 n 0 δ n g ) - V g 2 - 3 d g 2 2 w g 2 + V g 2 [ 1 + ( w g d g ) 2 ] - 3 / 2 } ,
β e 2 = 1 d e 2 { ( k d e ) 2 ( n 0 2 + 2 n 0 δ n e ) - V e 2 - 3 d e 2 2 w e 2 + 2 V e 2 π t n - 1 ( d e w e ) - w e d e [ 1 + ( w e d e ) 2 ] - 1 } ,
V ( g , e ) = d ( g , e ) k [ 2 n 0 δ n ( g , e ) ] 1 / 2 .
w e 2 = w 0 g 2 ( 1 + w e 2 / d g 2 ) 5 / 4 .
d g = w e 2 A g ( 1 + w e 2 / d g 2 ) - 5 / 4 ,
A g = ( 2 k 2 n 0 δ n g ) 1 / 2 .
d g w e 2 A g = d 0 g .
d g = d 0 g ( 1 + 1 / ( w e 2 A g 2 ) { 1 + 1 / ( w e 2 A g 2 ) × [ 1 + 1 / ( w e 2 A g 2 ) ] 5 / 4 } 5 / 4 ) - 5 / 4 .
n ( x ) = 1 x < 0 n 2 0 < x < d s , n 0 x > d s ,
β s 2 = k 2 n 0 2 - ( 2 k 2 Δ n s / π 1 / 2 ) ( d s / α s ) exp ( - d s 2 / α s 2 ) + k 2 Δ n s erfc ( d s / α s ) - 3 / ( 2 α s 2 ) ,
Δ n s = n 2 2 - n 0 2 .
α s 2 = [ d s 2 / ln ( c ) ] { 1 + [ α s 2 ln ( 9 α s 2 / d s 2 ) ] / ( 2 d s 2 ) } ,
c = ( 4 k 2 Δ n s d s 2 ) / π 1 / 2 .
w 0 s 2 = d s 2 / ln ( c )
w s 2 = w 0 s 2 { 1 + [ w 2 s 2 ln ( 9 w 2 s 2 / d s 2 ) ] / [ 2 d s 2 ] } ,
w 2 s 2 = w 0 s 2 { 1 + [ w 1 s 2 ln ( 9 w 1 s 2 / d s 2 ) ] / [ 2 d s 2 ] } ,
w 1 s 2 = w 0 s 2 { 1 + [ w 0 s 2 ln ( 9 w 0 s 2 / d s 2 ) ] / [ 2 d s 2 ] } .
d g = d 0 g / ( 1 + w s 2 / d g 2 ) - 5 / 4 ,
d 0 g = A g w s 2 .

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