Abstract

A novel intensity modulator for second-harmonic waveguide lasers is proposed and theoretically analyzed. Noise caused by spectrum broadening in a diode-laser direct-modulation scheme is avoided in the suggested structure by using the electro-optic effect to modulate the phase mismatch in the frequency-doubling process. In the analysis the influence of waveguide parameters and electrode dimensions on the switching voltage, the output power, and the modulation bandwidth is evaluated. The results suggest that a switching voltage below the 5-V transistor-transistor logic level in digital systems is feasible and that, in a traveling-wave strip-electrode configuration, a modulation bandwidth in the gigahertz range can be obtained.

© 1993 Optical Society of America

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  1. M. A. Haase, J. Qiu, J. M. DePuydt, and H. Cheng, “Blue-green laser diodes,” Appl. Phys. Lett. 59, 1272–1274 (1991).
    [CrossRef]
  2. K. Tatsuno, T. Andou, S. Nakatsuka, T. Miyai, M. Takahashi, and S. Helmfrid, “Highly efficient and stable green micro laser consisting of Nd:YVO4with intracavity KTP for optical storage,” in Conference on Lasers and Electro-Optics, Vol. 12 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper CWQ8.
  3. C. J. van der Poel, J. D. Bierlein, and J. B. Brown, “Efficient type I blue second-harmonic generation in periodically segmented KTiOPO4waveguides,” Appl. Phys. Lett. 57, 2074–2076 (1990).
    [CrossRef]
  4. K. Yamamoto, K. Mizuuchi, and T. Taniuchi, “Milliwatt-order blue-light generation in a periodically domain-inverted LiTaO3waveguide,” Opt. Lett. 16, 1156–1158 (1991).
    [CrossRef] [PubMed]
  5. J. Webjörn, F. Laurell, and G. Arvidsson, “Blue light generated by frequency doubling of laser diode light in a lithium niobate channel waveguide,” IEEE Photon. Technol. Lett. 1, 316–318 (1989).
    [CrossRef]
  6. E. J. Lim, M. M. Fejer, R. L. Byer, and W. J. Kozlovsky, “Blue light generation by frequency doubling in periodically poled lithium niobate channel waveguide,” Electron. Lett. 25, 731–732 (1989).
    [CrossRef]
  7. K. Kishino, S. Aoki, and Y. Suematsu, “Wavelength variation of 1.6 μ m wavelength buried heterostructure GaInAsP/InP lasers due to direct modulation,” IEEE J. Quantum Electron. QE-18, 343–351 (1982).
    [CrossRef]
  8. R. C. Alferness, “Waveguide electrooptic modulators,” IEEE Trans. Microwave Theory Tech. MTT-30, 1121–1137 (1982).
    [CrossRef]
  9. M. Izutsu, Y. Yamane, and T. Sueta, “Broad-band traveling-wave modulator using a LiNbO3optical waveguide,” IEEE J. Quantum Electron. QE-13, 287–290 (1977).
    [CrossRef]
  10. K. Kubota, J. Noda, and O. Mikami, “Traveling wave optical modulator using a directional coupler LiNbO3waveguide,” IEEE J. Quantum Electron. QE-16, 754–760 (1980).
    [CrossRef]
  11. I. P. Kaminow, L. W. Stulz, and E. H. Turner, “Efficient strip-waveguide modulator,” Appl. Phys. Lett. 27, 555–557 (1975).
    [CrossRef]
  12. N. Uesugi, K. Daikoku, and K. Kubota, “Electric field tuning of second-harmonic generation in a three-dimensional LiNbO3optical waveguide,” Appl. Phys. Lett. 34, 60–62 (1979).
    [CrossRef]
  13. E. M. Zolotov, A. M. Prokhorov, and V. A. Chernykh, “Electrooptic and temperature tuning of phase matching in second-harmonic generation in Ti:LiNbO3channel waveguides,” Sov. Tech. Phys. Lett. 7, 129–130 (1981).
  14. J. D. Bierlein, D. B. Laubacher, and J. D. Brown, “Balanced phase matching in segmented KTiOPO4waveguides,” Appl. Phys. Lett. 56, 1725–1727 (1990).
    [CrossRef]
  15. M. A. Title and S. H. Lee, “Modeling and characterization of embedded electrode performance in transverse electrooptic modulators,” Appl. Opt. 29, 85–98 (1990), Figs. 5 and 6a.
    [CrossRef] [PubMed]
  16. O. G. Ramer, “Integrated optic electrooptic modulator electrode analysis,” IEEE J. Quantum Electron. QE-18, 386–392 (1982).
    [CrossRef]
  17. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Applied Mathematics Series (National Bureau of Standards, Gaithersburg, Md., 1964), Vol. 55, p. 590.
  18. R. Plonsey and R. E. Collins, Principles and Applications of Electromagnetic Fields (McGraw-Hill, New York, 1961).
  19. N. Mabaya, P. E. Lagasse, and P. Vandenbulcke, “Finite element analysis of optical waveguides,” IEEE Trans. Microwave Theory Tech. MTT-29, 600–605 (1981).
    [CrossRef]
  20. M. Koshiba, K. Hayata, and M. Suzuki, “Approximate scalar finite-element analysis of anisotropic optical waveguides,” Electron. Lett. 18, 411–413 (1982).
    [CrossRef]
  21. R.-B. Wu and C. H. Chen, “A scalar variational conformal mapping technique for weakly guided dielectric waveguides,” IEEE J. Quantum Electron. QE-22, 603–609 (1986).
  22. J. L. Jackel, C. E. Rice, and J. J. Veselka, “Proton exchange for high index waveguides in LiNbO3,” Appl. Phys. Lett. 41, 607–608 (1982).
    [CrossRef]
  23. M. L. Bortz and M. M. Fejer, “Annealed proton-exchanged LiNbO3waveguides,” Opt. Lett. 16, 1844–1846 (1991).
    [CrossRef] [PubMed]
  24. M. M. Howerton, W. K. Burns, P. R. Skeath, and A. S. Greenblatt, “Dependence of refractive index on hydrogen concentration in proton exchanged LiNbO3,” IEEE J. Quantum Electron. 27, 593–601 (1991).
    [CrossRef]
  25. T. Yuhara, K. Tada, and Y.-S. Li, “Anomalous refractive index change and recovery of electro-optic coefficient r33in proton exchanged LiTaO3optical waveguides after annealing,” J. Appl. Phys. 71, 3966–3974 (1992).
    [CrossRef]
  26. H. Kogelnik, “Theory of dielectric waveguides,” in Integrated Optics, T. Tamir, ed. (Springer-Verlag, Berlin, 1985), p. 40.
  27. R. C. Miller, “Optical second harmonic generation in piezoelectric crystals,” Appl. Phys. Lett. 5, 17–19 (1964).
    [CrossRef]
  28. A. Yariv, Optical Electronics (Holt, Rhinehart & Winston, New York, 1985).
  29. K. Yamamoto, K. Mizuuchi, K. Takeshige, Y. Sasai, and T. Taniuchi, “Characteristics of periodically domain-inverted LiNbO3and LiTaO3waveguides for second-harmonic generation,” J. Appl. Phys. 70, 1947–1951 (1991).
    [CrossRef]
  30. K. Mizuuchi, K. Yamamoto, and T. Taniuchi, “Fabrication of first-order periodically domain-inverted structure in LiTaO3,” Appl. Phys. Lett. 59, 1538–1540 (1991).
    [CrossRef]
  31. G. Arvidsson and B. Jaskorzynska, “Periodically domain-inverted waveguides in lithium niobate for second-harmonic generation: influence of the shape of the domain boundary on the conversion efficiency,” in International Conference on Materials for Non-linear and Electro-optics, Cambridge, U.K., 1989, Int. Phys. Conf. Ser. 103, 47–52 (1989).
  32. O. G. Ramer, C. Nelson, and C. Mohr, “Experimental integrated optical circuit losses and fiber pigtailing of chips,” IEEE J. Quantum Electron. QE-17, 970–974 (1981).
    [CrossRef]
  33. E. K. Sharma, M. P. Singh, and P. C. Kendall, “Exact multilayer waveguide design including absorbing or metal layers,” Electron. Lett. 27, 408–410 (1991).
    [CrossRef]
  34. S. X. She, “Characteristic analysis of metal-clad and absorptive dielectric waveguides by a simple and accurate perturbation method,” Opt. Quantum Electron. 23, 1045–1054 (1991).
    [CrossRef]
  35. M. Izutsu, T. Itoh, and T. Sueta, “10 GHz bandwidth traveling-wave LiNbO3optical waveguide modulator,” IEEE J. Quantum Electron. QE-14, 394–395 (1978).
    [CrossRef]
  36. C. M. Gee, G. D. Thurmond, and H. W. Yen, “Traveling-wave electro-optic modulator,” Appl. Opt. 22, 2034–2037 (1983).
    [CrossRef]
  37. D. Marcuse, “Optimal electrode design for integrated optics modulators,” IEEE J. Quantum Electron. QE-18, 393–398 (1982).
    [CrossRef]

1992 (1)

T. Yuhara, K. Tada, and Y.-S. Li, “Anomalous refractive index change and recovery of electro-optic coefficient r33in proton exchanged LiTaO3optical waveguides after annealing,” J. Appl. Phys. 71, 3966–3974 (1992).
[CrossRef]

1991 (8)

M. L. Bortz and M. M. Fejer, “Annealed proton-exchanged LiNbO3waveguides,” Opt. Lett. 16, 1844–1846 (1991).
[CrossRef] [PubMed]

M. M. Howerton, W. K. Burns, P. R. Skeath, and A. S. Greenblatt, “Dependence of refractive index on hydrogen concentration in proton exchanged LiNbO3,” IEEE J. Quantum Electron. 27, 593–601 (1991).
[CrossRef]

K. Yamamoto, K. Mizuuchi, K. Takeshige, Y. Sasai, and T. Taniuchi, “Characteristics of periodically domain-inverted LiNbO3and LiTaO3waveguides for second-harmonic generation,” J. Appl. Phys. 70, 1947–1951 (1991).
[CrossRef]

K. Mizuuchi, K. Yamamoto, and T. Taniuchi, “Fabrication of first-order periodically domain-inverted structure in LiTaO3,” Appl. Phys. Lett. 59, 1538–1540 (1991).
[CrossRef]

E. K. Sharma, M. P. Singh, and P. C. Kendall, “Exact multilayer waveguide design including absorbing or metal layers,” Electron. Lett. 27, 408–410 (1991).
[CrossRef]

S. X. She, “Characteristic analysis of metal-clad and absorptive dielectric waveguides by a simple and accurate perturbation method,” Opt. Quantum Electron. 23, 1045–1054 (1991).
[CrossRef]

K. Yamamoto, K. Mizuuchi, and T. Taniuchi, “Milliwatt-order blue-light generation in a periodically domain-inverted LiTaO3waveguide,” Opt. Lett. 16, 1156–1158 (1991).
[CrossRef] [PubMed]

M. A. Haase, J. Qiu, J. M. DePuydt, and H. Cheng, “Blue-green laser diodes,” Appl. Phys. Lett. 59, 1272–1274 (1991).
[CrossRef]

1990 (3)

C. J. van der Poel, J. D. Bierlein, and J. B. Brown, “Efficient type I blue second-harmonic generation in periodically segmented KTiOPO4waveguides,” Appl. Phys. Lett. 57, 2074–2076 (1990).
[CrossRef]

J. D. Bierlein, D. B. Laubacher, and J. D. Brown, “Balanced phase matching in segmented KTiOPO4waveguides,” Appl. Phys. Lett. 56, 1725–1727 (1990).
[CrossRef]

M. A. Title and S. H. Lee, “Modeling and characterization of embedded electrode performance in transverse electrooptic modulators,” Appl. Opt. 29, 85–98 (1990), Figs. 5 and 6a.
[CrossRef] [PubMed]

1989 (3)

J. Webjörn, F. Laurell, and G. Arvidsson, “Blue light generated by frequency doubling of laser diode light in a lithium niobate channel waveguide,” IEEE Photon. Technol. Lett. 1, 316–318 (1989).
[CrossRef]

E. J. Lim, M. M. Fejer, R. L. Byer, and W. J. Kozlovsky, “Blue light generation by frequency doubling in periodically poled lithium niobate channel waveguide,” Electron. Lett. 25, 731–732 (1989).
[CrossRef]

G. Arvidsson and B. Jaskorzynska, “Periodically domain-inverted waveguides in lithium niobate for second-harmonic generation: influence of the shape of the domain boundary on the conversion efficiency,” in International Conference on Materials for Non-linear and Electro-optics, Cambridge, U.K., 1989, Int. Phys. Conf. Ser. 103, 47–52 (1989).

1986 (1)

R.-B. Wu and C. H. Chen, “A scalar variational conformal mapping technique for weakly guided dielectric waveguides,” IEEE J. Quantum Electron. QE-22, 603–609 (1986).

1983 (1)

1982 (6)

D. Marcuse, “Optimal electrode design for integrated optics modulators,” IEEE J. Quantum Electron. QE-18, 393–398 (1982).
[CrossRef]

M. Koshiba, K. Hayata, and M. Suzuki, “Approximate scalar finite-element analysis of anisotropic optical waveguides,” Electron. Lett. 18, 411–413 (1982).
[CrossRef]

J. L. Jackel, C. E. Rice, and J. J. Veselka, “Proton exchange for high index waveguides in LiNbO3,” Appl. Phys. Lett. 41, 607–608 (1982).
[CrossRef]

O. G. Ramer, “Integrated optic electrooptic modulator electrode analysis,” IEEE J. Quantum Electron. QE-18, 386–392 (1982).
[CrossRef]

K. Kishino, S. Aoki, and Y. Suematsu, “Wavelength variation of 1.6 μ m wavelength buried heterostructure GaInAsP/InP lasers due to direct modulation,” IEEE J. Quantum Electron. QE-18, 343–351 (1982).
[CrossRef]

R. C. Alferness, “Waveguide electrooptic modulators,” IEEE Trans. Microwave Theory Tech. MTT-30, 1121–1137 (1982).
[CrossRef]

1981 (3)

N. Mabaya, P. E. Lagasse, and P. Vandenbulcke, “Finite element analysis of optical waveguides,” IEEE Trans. Microwave Theory Tech. MTT-29, 600–605 (1981).
[CrossRef]

E. M. Zolotov, A. M. Prokhorov, and V. A. Chernykh, “Electrooptic and temperature tuning of phase matching in second-harmonic generation in Ti:LiNbO3channel waveguides,” Sov. Tech. Phys. Lett. 7, 129–130 (1981).

O. G. Ramer, C. Nelson, and C. Mohr, “Experimental integrated optical circuit losses and fiber pigtailing of chips,” IEEE J. Quantum Electron. QE-17, 970–974 (1981).
[CrossRef]

1980 (1)

K. Kubota, J. Noda, and O. Mikami, “Traveling wave optical modulator using a directional coupler LiNbO3waveguide,” IEEE J. Quantum Electron. QE-16, 754–760 (1980).
[CrossRef]

1979 (1)

N. Uesugi, K. Daikoku, and K. Kubota, “Electric field tuning of second-harmonic generation in a three-dimensional LiNbO3optical waveguide,” Appl. Phys. Lett. 34, 60–62 (1979).
[CrossRef]

1978 (1)

M. Izutsu, T. Itoh, and T. Sueta, “10 GHz bandwidth traveling-wave LiNbO3optical waveguide modulator,” IEEE J. Quantum Electron. QE-14, 394–395 (1978).
[CrossRef]

1977 (1)

M. Izutsu, Y. Yamane, and T. Sueta, “Broad-band traveling-wave modulator using a LiNbO3optical waveguide,” IEEE J. Quantum Electron. QE-13, 287–290 (1977).
[CrossRef]

1975 (1)

I. P. Kaminow, L. W. Stulz, and E. H. Turner, “Efficient strip-waveguide modulator,” Appl. Phys. Lett. 27, 555–557 (1975).
[CrossRef]

1964 (1)

R. C. Miller, “Optical second harmonic generation in piezoelectric crystals,” Appl. Phys. Lett. 5, 17–19 (1964).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Applied Mathematics Series (National Bureau of Standards, Gaithersburg, Md., 1964), Vol. 55, p. 590.

Alferness, R. C.

R. C. Alferness, “Waveguide electrooptic modulators,” IEEE Trans. Microwave Theory Tech. MTT-30, 1121–1137 (1982).
[CrossRef]

Andou, T.

K. Tatsuno, T. Andou, S. Nakatsuka, T. Miyai, M. Takahashi, and S. Helmfrid, “Highly efficient and stable green micro laser consisting of Nd:YVO4with intracavity KTP for optical storage,” in Conference on Lasers and Electro-Optics, Vol. 12 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper CWQ8.

Aoki, S.

K. Kishino, S. Aoki, and Y. Suematsu, “Wavelength variation of 1.6 μ m wavelength buried heterostructure GaInAsP/InP lasers due to direct modulation,” IEEE J. Quantum Electron. QE-18, 343–351 (1982).
[CrossRef]

Arvidsson, G.

J. Webjörn, F. Laurell, and G. Arvidsson, “Blue light generated by frequency doubling of laser diode light in a lithium niobate channel waveguide,” IEEE Photon. Technol. Lett. 1, 316–318 (1989).
[CrossRef]

G. Arvidsson and B. Jaskorzynska, “Periodically domain-inverted waveguides in lithium niobate for second-harmonic generation: influence of the shape of the domain boundary on the conversion efficiency,” in International Conference on Materials for Non-linear and Electro-optics, Cambridge, U.K., 1989, Int. Phys. Conf. Ser. 103, 47–52 (1989).

Bierlein, J. D.

C. J. van der Poel, J. D. Bierlein, and J. B. Brown, “Efficient type I blue second-harmonic generation in periodically segmented KTiOPO4waveguides,” Appl. Phys. Lett. 57, 2074–2076 (1990).
[CrossRef]

J. D. Bierlein, D. B. Laubacher, and J. D. Brown, “Balanced phase matching in segmented KTiOPO4waveguides,” Appl. Phys. Lett. 56, 1725–1727 (1990).
[CrossRef]

Bortz, M. L.

Brown, J. B.

C. J. van der Poel, J. D. Bierlein, and J. B. Brown, “Efficient type I blue second-harmonic generation in periodically segmented KTiOPO4waveguides,” Appl. Phys. Lett. 57, 2074–2076 (1990).
[CrossRef]

Brown, J. D.

J. D. Bierlein, D. B. Laubacher, and J. D. Brown, “Balanced phase matching in segmented KTiOPO4waveguides,” Appl. Phys. Lett. 56, 1725–1727 (1990).
[CrossRef]

Burns, W. K.

M. M. Howerton, W. K. Burns, P. R. Skeath, and A. S. Greenblatt, “Dependence of refractive index on hydrogen concentration in proton exchanged LiNbO3,” IEEE J. Quantum Electron. 27, 593–601 (1991).
[CrossRef]

Byer, R. L.

E. J. Lim, M. M. Fejer, R. L. Byer, and W. J. Kozlovsky, “Blue light generation by frequency doubling in periodically poled lithium niobate channel waveguide,” Electron. Lett. 25, 731–732 (1989).
[CrossRef]

Chen, C. H.

R.-B. Wu and C. H. Chen, “A scalar variational conformal mapping technique for weakly guided dielectric waveguides,” IEEE J. Quantum Electron. QE-22, 603–609 (1986).

Cheng, H.

M. A. Haase, J. Qiu, J. M. DePuydt, and H. Cheng, “Blue-green laser diodes,” Appl. Phys. Lett. 59, 1272–1274 (1991).
[CrossRef]

Chernykh, V. A.

E. M. Zolotov, A. M. Prokhorov, and V. A. Chernykh, “Electrooptic and temperature tuning of phase matching in second-harmonic generation in Ti:LiNbO3channel waveguides,” Sov. Tech. Phys. Lett. 7, 129–130 (1981).

Collins, R. E.

R. Plonsey and R. E. Collins, Principles and Applications of Electromagnetic Fields (McGraw-Hill, New York, 1961).

Daikoku, K.

N. Uesugi, K. Daikoku, and K. Kubota, “Electric field tuning of second-harmonic generation in a three-dimensional LiNbO3optical waveguide,” Appl. Phys. Lett. 34, 60–62 (1979).
[CrossRef]

DePuydt, J. M.

M. A. Haase, J. Qiu, J. M. DePuydt, and H. Cheng, “Blue-green laser diodes,” Appl. Phys. Lett. 59, 1272–1274 (1991).
[CrossRef]

Fejer, M. M.

M. L. Bortz and M. M. Fejer, “Annealed proton-exchanged LiNbO3waveguides,” Opt. Lett. 16, 1844–1846 (1991).
[CrossRef] [PubMed]

E. J. Lim, M. M. Fejer, R. L. Byer, and W. J. Kozlovsky, “Blue light generation by frequency doubling in periodically poled lithium niobate channel waveguide,” Electron. Lett. 25, 731–732 (1989).
[CrossRef]

Gee, C. M.

Greenblatt, A. S.

M. M. Howerton, W. K. Burns, P. R. Skeath, and A. S. Greenblatt, “Dependence of refractive index on hydrogen concentration in proton exchanged LiNbO3,” IEEE J. Quantum Electron. 27, 593–601 (1991).
[CrossRef]

Haase, M. A.

M. A. Haase, J. Qiu, J. M. DePuydt, and H. Cheng, “Blue-green laser diodes,” Appl. Phys. Lett. 59, 1272–1274 (1991).
[CrossRef]

Hayata, K.

M. Koshiba, K. Hayata, and M. Suzuki, “Approximate scalar finite-element analysis of anisotropic optical waveguides,” Electron. Lett. 18, 411–413 (1982).
[CrossRef]

Helmfrid, S.

K. Tatsuno, T. Andou, S. Nakatsuka, T. Miyai, M. Takahashi, and S. Helmfrid, “Highly efficient and stable green micro laser consisting of Nd:YVO4with intracavity KTP for optical storage,” in Conference on Lasers and Electro-Optics, Vol. 12 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper CWQ8.

Howerton, M. M.

M. M. Howerton, W. K. Burns, P. R. Skeath, and A. S. Greenblatt, “Dependence of refractive index on hydrogen concentration in proton exchanged LiNbO3,” IEEE J. Quantum Electron. 27, 593–601 (1991).
[CrossRef]

Itoh, T.

M. Izutsu, T. Itoh, and T. Sueta, “10 GHz bandwidth traveling-wave LiNbO3optical waveguide modulator,” IEEE J. Quantum Electron. QE-14, 394–395 (1978).
[CrossRef]

Izutsu, M.

M. Izutsu, T. Itoh, and T. Sueta, “10 GHz bandwidth traveling-wave LiNbO3optical waveguide modulator,” IEEE J. Quantum Electron. QE-14, 394–395 (1978).
[CrossRef]

M. Izutsu, Y. Yamane, and T. Sueta, “Broad-band traveling-wave modulator using a LiNbO3optical waveguide,” IEEE J. Quantum Electron. QE-13, 287–290 (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 (1982).
[CrossRef]

Jaskorzynska, B.

G. Arvidsson and B. Jaskorzynska, “Periodically domain-inverted waveguides in lithium niobate for second-harmonic generation: influence of the shape of the domain boundary on the conversion efficiency,” in International Conference on Materials for Non-linear and Electro-optics, Cambridge, U.K., 1989, Int. Phys. Conf. Ser. 103, 47–52 (1989).

Kaminow, I. P.

I. P. Kaminow, L. W. Stulz, and E. H. Turner, “Efficient strip-waveguide modulator,” Appl. Phys. Lett. 27, 555–557 (1975).
[CrossRef]

Kendall, P. C.

E. K. Sharma, M. P. Singh, and P. C. Kendall, “Exact multilayer waveguide design including absorbing or metal layers,” Electron. Lett. 27, 408–410 (1991).
[CrossRef]

Kishino, K.

K. Kishino, S. Aoki, and Y. Suematsu, “Wavelength variation of 1.6 μ m wavelength buried heterostructure GaInAsP/InP lasers due to direct modulation,” IEEE J. Quantum Electron. QE-18, 343–351 (1982).
[CrossRef]

Kogelnik, H.

H. Kogelnik, “Theory of dielectric waveguides,” in Integrated Optics, T. Tamir, ed. (Springer-Verlag, Berlin, 1985), p. 40.

Koshiba, M.

M. Koshiba, K. Hayata, and M. Suzuki, “Approximate scalar finite-element analysis of anisotropic optical waveguides,” Electron. Lett. 18, 411–413 (1982).
[CrossRef]

Kozlovsky, W. J.

E. J. Lim, M. M. Fejer, R. L. Byer, and W. J. Kozlovsky, “Blue light generation by frequency doubling in periodically poled lithium niobate channel waveguide,” Electron. Lett. 25, 731–732 (1989).
[CrossRef]

Kubota, K.

K. Kubota, J. Noda, and O. Mikami, “Traveling wave optical modulator using a directional coupler LiNbO3waveguide,” IEEE J. Quantum Electron. QE-16, 754–760 (1980).
[CrossRef]

N. Uesugi, K. Daikoku, and K. Kubota, “Electric field tuning of second-harmonic generation in a three-dimensional LiNbO3optical waveguide,” Appl. Phys. Lett. 34, 60–62 (1979).
[CrossRef]

Lagasse, P. E.

N. Mabaya, P. E. Lagasse, and P. Vandenbulcke, “Finite element analysis of optical waveguides,” IEEE Trans. Microwave Theory Tech. MTT-29, 600–605 (1981).
[CrossRef]

Laubacher, D. B.

J. D. Bierlein, D. B. Laubacher, and J. D. Brown, “Balanced phase matching in segmented KTiOPO4waveguides,” Appl. Phys. Lett. 56, 1725–1727 (1990).
[CrossRef]

Laurell, F.

J. Webjörn, F. Laurell, and G. Arvidsson, “Blue light generated by frequency doubling of laser diode light in a lithium niobate channel waveguide,” IEEE Photon. Technol. Lett. 1, 316–318 (1989).
[CrossRef]

Lee, S. H.

Li, Y.-S.

T. Yuhara, K. Tada, and Y.-S. Li, “Anomalous refractive index change and recovery of electro-optic coefficient r33in proton exchanged LiTaO3optical waveguides after annealing,” J. Appl. Phys. 71, 3966–3974 (1992).
[CrossRef]

Lim, E. J.

E. J. Lim, M. M. Fejer, R. L. Byer, and W. J. Kozlovsky, “Blue light generation by frequency doubling in periodically poled lithium niobate channel waveguide,” Electron. Lett. 25, 731–732 (1989).
[CrossRef]

Mabaya, N.

N. Mabaya, P. E. Lagasse, and P. Vandenbulcke, “Finite element analysis of optical waveguides,” IEEE Trans. Microwave Theory Tech. MTT-29, 600–605 (1981).
[CrossRef]

Marcuse, D.

D. Marcuse, “Optimal electrode design for integrated optics modulators,” IEEE J. Quantum Electron. QE-18, 393–398 (1982).
[CrossRef]

Mikami, O.

K. Kubota, J. Noda, and O. Mikami, “Traveling wave optical modulator using a directional coupler LiNbO3waveguide,” IEEE J. Quantum Electron. QE-16, 754–760 (1980).
[CrossRef]

Miller, R. C.

R. C. Miller, “Optical second harmonic generation in piezoelectric crystals,” Appl. Phys. Lett. 5, 17–19 (1964).
[CrossRef]

Miyai, T.

K. Tatsuno, T. Andou, S. Nakatsuka, T. Miyai, M. Takahashi, and S. Helmfrid, “Highly efficient and stable green micro laser consisting of Nd:YVO4with intracavity KTP for optical storage,” in Conference on Lasers and Electro-Optics, Vol. 12 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper CWQ8.

Mizuuchi, K.

K. Yamamoto, K. Mizuuchi, and T. Taniuchi, “Milliwatt-order blue-light generation in a periodically domain-inverted LiTaO3waveguide,” Opt. Lett. 16, 1156–1158 (1991).
[CrossRef] [PubMed]

K. Yamamoto, K. Mizuuchi, K. Takeshige, Y. Sasai, and T. Taniuchi, “Characteristics of periodically domain-inverted LiNbO3and LiTaO3waveguides for second-harmonic generation,” J. Appl. Phys. 70, 1947–1951 (1991).
[CrossRef]

K. Mizuuchi, K. Yamamoto, and T. Taniuchi, “Fabrication of first-order periodically domain-inverted structure in LiTaO3,” Appl. Phys. Lett. 59, 1538–1540 (1991).
[CrossRef]

Mohr, C.

O. G. Ramer, C. Nelson, and C. Mohr, “Experimental integrated optical circuit losses and fiber pigtailing of chips,” IEEE J. Quantum Electron. QE-17, 970–974 (1981).
[CrossRef]

Nakatsuka, S.

K. Tatsuno, T. Andou, S. Nakatsuka, T. Miyai, M. Takahashi, and S. Helmfrid, “Highly efficient and stable green micro laser consisting of Nd:YVO4with intracavity KTP for optical storage,” in Conference on Lasers and Electro-Optics, Vol. 12 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper CWQ8.

Nelson, C.

O. G. Ramer, C. Nelson, and C. Mohr, “Experimental integrated optical circuit losses and fiber pigtailing of chips,” IEEE J. Quantum Electron. QE-17, 970–974 (1981).
[CrossRef]

Noda, J.

K. Kubota, J. Noda, and O. Mikami, “Traveling wave optical modulator using a directional coupler LiNbO3waveguide,” IEEE J. Quantum Electron. QE-16, 754–760 (1980).
[CrossRef]

Plonsey, R.

R. Plonsey and R. E. Collins, Principles and Applications of Electromagnetic Fields (McGraw-Hill, New York, 1961).

Prokhorov, A. M.

E. M. Zolotov, A. M. Prokhorov, and V. A. Chernykh, “Electrooptic and temperature tuning of phase matching in second-harmonic generation in Ti:LiNbO3channel waveguides,” Sov. Tech. Phys. Lett. 7, 129–130 (1981).

Qiu, J.

M. A. Haase, J. Qiu, J. M. DePuydt, and H. Cheng, “Blue-green laser diodes,” Appl. Phys. Lett. 59, 1272–1274 (1991).
[CrossRef]

Ramer, O. G.

O. G. Ramer, “Integrated optic electrooptic modulator electrode analysis,” IEEE J. Quantum Electron. QE-18, 386–392 (1982).
[CrossRef]

O. G. Ramer, C. Nelson, and C. Mohr, “Experimental integrated optical circuit losses and fiber pigtailing of chips,” IEEE J. Quantum Electron. QE-17, 970–974 (1981).
[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 (1982).
[CrossRef]

Sasai, Y.

K. Yamamoto, K. Mizuuchi, K. Takeshige, Y. Sasai, and T. Taniuchi, “Characteristics of periodically domain-inverted LiNbO3and LiTaO3waveguides for second-harmonic generation,” J. Appl. Phys. 70, 1947–1951 (1991).
[CrossRef]

Sharma, E. K.

E. K. Sharma, M. P. Singh, and P. C. Kendall, “Exact multilayer waveguide design including absorbing or metal layers,” Electron. Lett. 27, 408–410 (1991).
[CrossRef]

She, S. X.

S. X. She, “Characteristic analysis of metal-clad and absorptive dielectric waveguides by a simple and accurate perturbation method,” Opt. Quantum Electron. 23, 1045–1054 (1991).
[CrossRef]

Singh, M. P.

E. K. Sharma, M. P. Singh, and P. C. Kendall, “Exact multilayer waveguide design including absorbing or metal layers,” Electron. Lett. 27, 408–410 (1991).
[CrossRef]

Skeath, P. R.

M. M. Howerton, W. K. Burns, P. R. Skeath, and A. S. Greenblatt, “Dependence of refractive index on hydrogen concentration in proton exchanged LiNbO3,” IEEE J. Quantum Electron. 27, 593–601 (1991).
[CrossRef]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Applied Mathematics Series (National Bureau of Standards, Gaithersburg, Md., 1964), Vol. 55, p. 590.

Stulz, L. W.

I. P. Kaminow, L. W. Stulz, and E. H. Turner, “Efficient strip-waveguide modulator,” Appl. Phys. Lett. 27, 555–557 (1975).
[CrossRef]

Suematsu, Y.

K. Kishino, S. Aoki, and Y. Suematsu, “Wavelength variation of 1.6 μ m wavelength buried heterostructure GaInAsP/InP lasers due to direct modulation,” IEEE J. Quantum Electron. QE-18, 343–351 (1982).
[CrossRef]

Sueta, T.

M. Izutsu, T. Itoh, and T. Sueta, “10 GHz bandwidth traveling-wave LiNbO3optical waveguide modulator,” IEEE J. Quantum Electron. QE-14, 394–395 (1978).
[CrossRef]

M. Izutsu, Y. Yamane, and T. Sueta, “Broad-band traveling-wave modulator using a LiNbO3optical waveguide,” IEEE J. Quantum Electron. QE-13, 287–290 (1977).
[CrossRef]

Suzuki, M.

M. Koshiba, K. Hayata, and M. Suzuki, “Approximate scalar finite-element analysis of anisotropic optical waveguides,” Electron. Lett. 18, 411–413 (1982).
[CrossRef]

Tada, K.

T. Yuhara, K. Tada, and Y.-S. Li, “Anomalous refractive index change and recovery of electro-optic coefficient r33in proton exchanged LiTaO3optical waveguides after annealing,” J. Appl. Phys. 71, 3966–3974 (1992).
[CrossRef]

Takahashi, M.

K. Tatsuno, T. Andou, S. Nakatsuka, T. Miyai, M. Takahashi, and S. Helmfrid, “Highly efficient and stable green micro laser consisting of Nd:YVO4with intracavity KTP for optical storage,” in Conference on Lasers and Electro-Optics, Vol. 12 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper CWQ8.

Takeshige, K.

K. Yamamoto, K. Mizuuchi, K. Takeshige, Y. Sasai, and T. Taniuchi, “Characteristics of periodically domain-inverted LiNbO3and LiTaO3waveguides for second-harmonic generation,” J. Appl. Phys. 70, 1947–1951 (1991).
[CrossRef]

Taniuchi, T.

K. Yamamoto, K. Mizuuchi, K. Takeshige, Y. Sasai, and T. Taniuchi, “Characteristics of periodically domain-inverted LiNbO3and LiTaO3waveguides for second-harmonic generation,” J. Appl. Phys. 70, 1947–1951 (1991).
[CrossRef]

K. Mizuuchi, K. Yamamoto, and T. Taniuchi, “Fabrication of first-order periodically domain-inverted structure in LiTaO3,” Appl. Phys. Lett. 59, 1538–1540 (1991).
[CrossRef]

K. Yamamoto, K. Mizuuchi, and T. Taniuchi, “Milliwatt-order blue-light generation in a periodically domain-inverted LiTaO3waveguide,” Opt. Lett. 16, 1156–1158 (1991).
[CrossRef] [PubMed]

Tatsuno, K.

K. Tatsuno, T. Andou, S. Nakatsuka, T. Miyai, M. Takahashi, and S. Helmfrid, “Highly efficient and stable green micro laser consisting of Nd:YVO4with intracavity KTP for optical storage,” in Conference on Lasers and Electro-Optics, Vol. 12 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper CWQ8.

Thurmond, G. D.

Title, M. A.

Turner, E. H.

I. P. Kaminow, L. W. Stulz, and E. H. Turner, “Efficient strip-waveguide modulator,” Appl. Phys. Lett. 27, 555–557 (1975).
[CrossRef]

Uesugi, N.

N. Uesugi, K. Daikoku, and K. Kubota, “Electric field tuning of second-harmonic generation in a three-dimensional LiNbO3optical waveguide,” Appl. Phys. Lett. 34, 60–62 (1979).
[CrossRef]

van der Poel, C. J.

C. J. van der Poel, J. D. Bierlein, and J. B. Brown, “Efficient type I blue second-harmonic generation in periodically segmented KTiOPO4waveguides,” Appl. Phys. Lett. 57, 2074–2076 (1990).
[CrossRef]

Vandenbulcke, P.

N. Mabaya, P. E. Lagasse, and P. Vandenbulcke, “Finite element analysis of optical waveguides,” IEEE Trans. Microwave Theory Tech. MTT-29, 600–605 (1981).
[CrossRef]

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 (1982).
[CrossRef]

Webjörn, J.

J. Webjörn, F. Laurell, and G. Arvidsson, “Blue light generated by frequency doubling of laser diode light in a lithium niobate channel waveguide,” IEEE Photon. Technol. Lett. 1, 316–318 (1989).
[CrossRef]

Wu, R.-B.

R.-B. Wu and C. H. Chen, “A scalar variational conformal mapping technique for weakly guided dielectric waveguides,” IEEE J. Quantum Electron. QE-22, 603–609 (1986).

Yamamoto, K.

K. Yamamoto, K. Mizuuchi, and T. Taniuchi, “Milliwatt-order blue-light generation in a periodically domain-inverted LiTaO3waveguide,” Opt. Lett. 16, 1156–1158 (1991).
[CrossRef] [PubMed]

K. Yamamoto, K. Mizuuchi, K. Takeshige, Y. Sasai, and T. Taniuchi, “Characteristics of periodically domain-inverted LiNbO3and LiTaO3waveguides for second-harmonic generation,” J. Appl. Phys. 70, 1947–1951 (1991).
[CrossRef]

K. Mizuuchi, K. Yamamoto, and T. Taniuchi, “Fabrication of first-order periodically domain-inverted structure in LiTaO3,” Appl. Phys. Lett. 59, 1538–1540 (1991).
[CrossRef]

Yamane, Y.

M. Izutsu, Y. Yamane, and T. Sueta, “Broad-band traveling-wave modulator using a LiNbO3optical waveguide,” IEEE J. Quantum Electron. QE-13, 287–290 (1977).
[CrossRef]

Yariv, A.

A. Yariv, Optical Electronics (Holt, Rhinehart & Winston, New York, 1985).

Yen, H. W.

Yuhara, T.

T. Yuhara, K. Tada, and Y.-S. Li, “Anomalous refractive index change and recovery of electro-optic coefficient r33in proton exchanged LiTaO3optical waveguides after annealing,” J. Appl. Phys. 71, 3966–3974 (1992).
[CrossRef]

Zolotov, E. M.

E. M. Zolotov, A. M. Prokhorov, and V. A. Chernykh, “Electrooptic and temperature tuning of phase matching in second-harmonic generation in Ti:LiNbO3channel waveguides,” Sov. Tech. Phys. Lett. 7, 129–130 (1981).

Appl. Opt. (2)

Appl. Phys. Lett. (8)

I. P. Kaminow, L. W. Stulz, and E. H. Turner, “Efficient strip-waveguide modulator,” Appl. Phys. Lett. 27, 555–557 (1975).
[CrossRef]

N. Uesugi, K. Daikoku, and K. Kubota, “Electric field tuning of second-harmonic generation in a three-dimensional LiNbO3optical waveguide,” Appl. Phys. Lett. 34, 60–62 (1979).
[CrossRef]

J. D. Bierlein, D. B. Laubacher, and J. D. Brown, “Balanced phase matching in segmented KTiOPO4waveguides,” Appl. Phys. Lett. 56, 1725–1727 (1990).
[CrossRef]

M. A. Haase, J. Qiu, J. M. DePuydt, and H. Cheng, “Blue-green laser diodes,” Appl. Phys. Lett. 59, 1272–1274 (1991).
[CrossRef]

C. J. van der Poel, J. D. Bierlein, and J. B. Brown, “Efficient type I blue second-harmonic generation in periodically segmented KTiOPO4waveguides,” Appl. Phys. Lett. 57, 2074–2076 (1990).
[CrossRef]

J. L. Jackel, C. E. Rice, and J. J. Veselka, “Proton exchange for high index waveguides in LiNbO3,” Appl. Phys. Lett. 41, 607–608 (1982).
[CrossRef]

R. C. Miller, “Optical second harmonic generation in piezoelectric crystals,” Appl. Phys. Lett. 5, 17–19 (1964).
[CrossRef]

K. Mizuuchi, K. Yamamoto, and T. Taniuchi, “Fabrication of first-order periodically domain-inverted structure in LiTaO3,” Appl. Phys. Lett. 59, 1538–1540 (1991).
[CrossRef]

Electron. Lett. (3)

E. K. Sharma, M. P. Singh, and P. C. Kendall, “Exact multilayer waveguide design including absorbing or metal layers,” Electron. Lett. 27, 408–410 (1991).
[CrossRef]

M. Koshiba, K. Hayata, and M. Suzuki, “Approximate scalar finite-element analysis of anisotropic optical waveguides,” Electron. Lett. 18, 411–413 (1982).
[CrossRef]

E. J. Lim, M. M. Fejer, R. L. Byer, and W. J. Kozlovsky, “Blue light generation by frequency doubling in periodically poled lithium niobate channel waveguide,” Electron. Lett. 25, 731–732 (1989).
[CrossRef]

IEEE J. Quantum Electron. (9)

K. Kishino, S. Aoki, and Y. Suematsu, “Wavelength variation of 1.6 μ m wavelength buried heterostructure GaInAsP/InP lasers due to direct modulation,” IEEE J. Quantum Electron. QE-18, 343–351 (1982).
[CrossRef]

M. Izutsu, Y. Yamane, and T. Sueta, “Broad-band traveling-wave modulator using a LiNbO3optical waveguide,” IEEE J. Quantum Electron. QE-13, 287–290 (1977).
[CrossRef]

K. Kubota, J. Noda, and O. Mikami, “Traveling wave optical modulator using a directional coupler LiNbO3waveguide,” IEEE J. Quantum Electron. QE-16, 754–760 (1980).
[CrossRef]

O. G. Ramer, “Integrated optic electrooptic modulator electrode analysis,” IEEE J. Quantum Electron. QE-18, 386–392 (1982).
[CrossRef]

R.-B. Wu and C. H. Chen, “A scalar variational conformal mapping technique for weakly guided dielectric waveguides,” IEEE J. Quantum Electron. QE-22, 603–609 (1986).

M. M. Howerton, W. K. Burns, P. R. Skeath, and A. S. Greenblatt, “Dependence of refractive index on hydrogen concentration in proton exchanged LiNbO3,” IEEE J. Quantum Electron. 27, 593–601 (1991).
[CrossRef]

O. G. Ramer, C. Nelson, and C. Mohr, “Experimental integrated optical circuit losses and fiber pigtailing of chips,” IEEE J. Quantum Electron. QE-17, 970–974 (1981).
[CrossRef]

M. Izutsu, T. Itoh, and T. Sueta, “10 GHz bandwidth traveling-wave LiNbO3optical waveguide modulator,” IEEE J. Quantum Electron. QE-14, 394–395 (1978).
[CrossRef]

D. Marcuse, “Optimal electrode design for integrated optics modulators,” IEEE J. Quantum Electron. QE-18, 393–398 (1982).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

J. Webjörn, F. Laurell, and G. Arvidsson, “Blue light generated by frequency doubling of laser diode light in a lithium niobate channel waveguide,” IEEE Photon. Technol. Lett. 1, 316–318 (1989).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

R. C. Alferness, “Waveguide electrooptic modulators,” IEEE Trans. Microwave Theory Tech. MTT-30, 1121–1137 (1982).
[CrossRef]

N. Mabaya, P. E. Lagasse, and P. Vandenbulcke, “Finite element analysis of optical waveguides,” IEEE Trans. Microwave Theory Tech. MTT-29, 600–605 (1981).
[CrossRef]

International Conference on Materials for Non-linear and Electro-optics, Cambridge, U.K., 1989 (1)

G. Arvidsson and B. Jaskorzynska, “Periodically domain-inverted waveguides in lithium niobate for second-harmonic generation: influence of the shape of the domain boundary on the conversion efficiency,” in International Conference on Materials for Non-linear and Electro-optics, Cambridge, U.K., 1989, Int. Phys. Conf. Ser. 103, 47–52 (1989).

J. Appl. Phys. (2)

K. Yamamoto, K. Mizuuchi, K. Takeshige, Y. Sasai, and T. Taniuchi, “Characteristics of periodically domain-inverted LiNbO3and LiTaO3waveguides for second-harmonic generation,” J. Appl. Phys. 70, 1947–1951 (1991).
[CrossRef]

T. Yuhara, K. Tada, and Y.-S. Li, “Anomalous refractive index change and recovery of electro-optic coefficient r33in proton exchanged LiTaO3optical waveguides after annealing,” J. Appl. Phys. 71, 3966–3974 (1992).
[CrossRef]

Opt. Lett. (2)

Opt. Quantum Electron. (1)

S. X. She, “Characteristic analysis of metal-clad and absorptive dielectric waveguides by a simple and accurate perturbation method,” Opt. Quantum Electron. 23, 1045–1054 (1991).
[CrossRef]

Sov. Tech. Phys. Lett. (1)

E. M. Zolotov, A. M. Prokhorov, and V. A. Chernykh, “Electrooptic and temperature tuning of phase matching in second-harmonic generation in Ti:LiNbO3channel waveguides,” Sov. Tech. Phys. Lett. 7, 129–130 (1981).

Other (5)

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Applied Mathematics Series (National Bureau of Standards, Gaithersburg, Md., 1964), Vol. 55, p. 590.

R. Plonsey and R. E. Collins, Principles and Applications of Electromagnetic Fields (McGraw-Hill, New York, 1961).

K. Tatsuno, T. Andou, S. Nakatsuka, T. Miyai, M. Takahashi, and S. Helmfrid, “Highly efficient and stable green micro laser consisting of Nd:YVO4with intracavity KTP for optical storage,” in Conference on Lasers and Electro-Optics, Vol. 12 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper CWQ8.

H. Kogelnik, “Theory of dielectric waveguides,” in Integrated Optics, T. Tamir, ed. (Springer-Verlag, Berlin, 1985), p. 40.

A. Yariv, Optical Electronics (Holt, Rhinehart & Winston, New York, 1985).

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Figures (11)

Fig. 1
Fig. 1

Electrode configuration (a) from above and (b) from the front end. Two coplanar strip electrodes are deposited on the substrate and shunted at the end with a 50-Ω resistor. In (b) the coordinate axes are defined. The origin of the coordinate system is located at the inner edge of the strip electrode. Strip-electrode width, b; gap between ground electrode and strip electrode, g; relative permittivities in LiTaO3, y = 33 = 42.6, x = 11 = 42.8; relative permittivities in LiNbO3; y = 33 = 27.9, x = 11 = 44.3.

Fig. 2
Fig. 2

Shape of the boundary between domain-inverted and nondomain-inverted regions in LiTaO3 and LiNbO3 quasi-phase-matching waveguides. The domain-inverted regions are hatched, and the nondomain-inverted part of the waveguide region is shaded. The depth of the domain-inverted regions in LiTaO3, d in the upper figure, is defined as the distance from the crystal surface. The depth position of the saw-toothed boundary in LiNbO3, d in the lower figure, is defined as the distance from the crystal surface to a point halfway between the highest point and the lowest point in the saw-toothed curve.

Fig. 3
Fig. 3

Characteristic impedance of strip electrodes in ohms as a function of width-to-gap ratio for symmetric and asymmetric electrode configurations. In the asymmetric configuration the width refers to the width of the strip electrode (not to the width of the ground electrode). The plotted relations are defined by Eqs. (7) and (8).

Fig. 4
Fig. 4

Vertical component of electric field as function of x coordinate for different depth positions. Asymmetric electrodes are assumed. Applied voltage, 1 V. Width of the strip electrode, 21 μm; gap between the strip electrode and the ground electrode, 9 μm.

Fig. 5
Fig. 5

Effective-index change of the second harmonic and the fundamental modes as a function of waveguide position, when the influence of the sign reversal of the electro-optic coefficient is not taken into account. We also show the difference in effective-index change between the two modes.

Fig. 6
Fig. 6

Switching voltage in a quasi-phase-matching LiTaO3 modulator as function of waveguide position (defined in Fig. 1). Length and width of the waveguide, 1 cm and 4 μm, respectively. Depth of the domain-inverted regions, 2.5 μm. Length of the domain-inverted regions, 5.8 μm (at the crystal surface).

Fig. 7
Fig. 7

Switching voltage and output power in a quasi-phase-matching LiTaO3 modulator as a function of the length of the domain-inverted regions. The output power was normalized by the output power in a second-order quasi-phase-matching grating with laminar domains and a 3:1 ratio between the length of the domain-inverted and the nondomain-inverted regions. The waveguide position was optimal, x = 0.7 μm. We assumed a depth of the domain-inverted regions that was a linearly growing function of the corresponding length. The depth for 2.0- and 6.5-μm lengths was 1.5 and 2.7 μm, respectively; the length and the width of the waveguide were 1 cm and 4 μm, respectively. SH, second harmonic.

Fig. 8
Fig. 8

Switching voltage and output power in a quasi-phase-matching LiTaO3 modulator as function of depth of the domain-inverted regions. Length and width of the waveguides, 1 cm and 4 μm, respectively. SH, second harmonic.

Fig. 9
Fig. 9

Domain length that maximizes the output power as function of domain depth in a quasi-phase-matching LiTaO3 waveguide.

Fig. 10
Fig. 10

Switching voltage in a quasi-phase-matching LiNbO3 modulator as a function of waveguide position. Length and width of the waveguide, 1 cm and 4 μm, respectively; depth position of the boundary between domain-inverted and nondomain-inverted regions, 1.675 μm.

Fig. 11
Fig. 11

Switching voltage and output power in a quasi-phase-matching LiNbO3 modulator as function of depth position of the boundary between the domain-inverted and nondomain-inverted regions. The optimum waveguide position was assumed in the calculation. Length and width of the waveguide, 1 cm and 4 μm, respectively.

Equations (27)

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x 2 V ( x , y ) x 2 + y 2 V ( x , y ) y 2 = 0             for y < 0 ,
2 V ( x , y ) x 2 + 2 V ( x , y ) y 2 = 0             for y > 0
V ( x , y ) = V hom ( x , α y ) ,             α = x / y             for y < 0 ,
V ( x , y ) = V hom ( x , y )             for y > 0.
d w d z = ( b + g ) 1 / 2 2 1 ( z + g ) 1 / 2 1 z 1 / 2 1 ( z - b ) 1 / 2 ,
z = x + i y ,
w = u + i v ,
w ( z = 0 ) = 0 ,
V w ( w ) = U Re ( w ) / w s ,             0 Re ( w ) w s .
E x = - V w [ w ( x , y ) ] x = - U w s Re ( d w d z ) ,
E y = - V w [ w ( x , y ) ] y = U w s Im ( d w d z ) ( x y ) 1 / 2 .
C = 2 0 eff w l / w s ,
Z 0 = 1 2 × K [ g / ( g + b ) ] K [ b / ( g + b ) ] ( μ 0 eff ) 1 / 2 .
Z 0 = K [ g / ( g + 2 b ) ] K [ 2 b / ( g + b ) ] ( μ 0 eff ) 1 / 2 .
a i j ξ j = β 2 b i j ξ j ,
a i j = e Ω e [ k 0 2 f i ( r ) f i ( r ) - 1 n y ( r ) 2 f i ( r ) x f j ( r ) x - 1 n z ( r ) 2 f i ( r ) y f j ( r ) y ] d x d y ,
b i j = e Ω e 1 n y ( r ) 2 f i ( r ) f j ( r ) d x d y ,
1 n z 2 × H x y
δ β P = ω - [ δ E ( E ) * ] d x d y ,
P = - S z d x d y ,
S = E × H * + E * × H ,
δ n eff = n eff - H x ( x , y ) 2 δ n y ( x , y ) / n y ( x , y ) 3 d x d y - H x ( x , y ) 2 / n y ( x , y ) 2 d x d y ,
δ n = ½ n y ( x , y ) 3 r 33 ( x , y ) ( x , y )
δ n eff = n eff 0 Λ [ - H x ( x , y ) 2 δ n y ( x , y , z ) / n y ( x , y ) 3 d x d y ] d z Λ - H x ( x , y ) 2 / n y ( x , y ) 2 d x d y ,
V sw = 0.3211 λ F l δ n s - δ n F ,
| 0 1 exp [ i 0 z V 0 exp ( - α ζ ) d ζ ] d ζ | 2 = 0.20.
I 0 = 3 Λ d / 4 [ d 2 - ( y max ) 2 ] 1 / 2 ,

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