Abstract

A calculation procedure for optical harmonic generation in multilayered structures has been developed. This formalism is based on a Green-function analysis and a transfer-matrix technique and permits a clear physical interpretation of harmonic light propagating through the structure. It is suited to computer calculation and hence applicable to arbitrary multilayered geometries. The usefulness of the method has been experimentally demonstrated in reflected second-harmonic generation from a GaP–AlP four-layered structure.

© 1995 Optical Society of America

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  1. R. Normandin, S. Létouneau, F. Chatenoud, and R. L. Williams, “Monolithic, surface-emitting, semiconductor visible lasers and spectrometers for WDM fiber communication systems,” IEEE J. Quantum Electron. 27, 1520–1530 (1991).
    [Crossref]
  2. D. Vakhshoori, R. J. Fischer, M. Hong, D. L. Sivco, G. J. Zydzik, G. N. S. Chu, and A. Y. Cho, “Blue-green surface-emitting second-harmonic generators on (111)B GaAs,” Appl. Phys. Lett. 59, 896–898 (1991).
    [Crossref]
  3. H. Takahashi, M. Ohashi, T. Kondo, N. Ogasawara, Y. Shiraki, and R. Ito, “New semiconductor second-harmonic generator based on quasi-phase-matching for cavity-enhanced fundamental standing wave,” Jpn. J. Appl. Phys. 33, L1456–L1458 (1994).
    [Crossref]
  4. See, for example, M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
    [Crossref]
  5. N. Bloembergen and P. S. Pershan, “Light waves at the boundary of nonlinear media,” Phys. Rev. 128, 606–622 (1962).
    [Crossref]
  6. D. S. Bethune, “Optical harmonic generation and mixed in multilayer media: analysis using optical transfer matrix technique,” J. Opt. Soc. Am. B 6, 910–916 (1989).
    [Crossref]
  7. D. S. Bethune, “Optical harmonic generation and mixing in multilayer media: extension of optical transfer matrix approach to include anisotropic materials,” J. Opt. Soc. Am. B 8, 367–373 (1991).
    [Crossref]
  8. N. Bloembergen, H. J. Simon, and C. H. Lee, “Total reflection phenomena in second-harmonic generation of light,” Phys. Rev. 181, 1261–1271 (1969).
    [Crossref]
  9. J. E. Sipe, “The ATR spectra of multipole surface plasmons,” Surf. Sci. 84, 75–105 (1979).
    [Crossref]
  10. J. E. Sipe, “New Green-function formalism for surface optics,” J. Opt. Soc. Am. B 4, 481–489 (1987).
    [Crossref]
  11. D. Vakhshoori, “Analysis of visible surface-emitting second-harmonic generators,” J. Appl. Phys. 70, 5205–5210 (1991).
    [Crossref]
  12. N. D. Whitbread and P. N. Robson, “Theoretical analysis of passive visible surface-emitting second-harmonic generators,” IEEE J. Quantum Electron. 30, 139–147 (1994).
    [Crossref]
  13. V. C. Y. So, R. Normandin, and G. I. Stegeman, “Field analysis of harmonic generation in thin-film integrated optics,” J. Opt. Soc. Am. 69, 1166–1171 (1979).
    [Crossref]
  14. See, for example, J. Jerphagnon and S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41, 1667–1681 (1970).
    [Crossref]
  15. N. Hashizume, T. Kondo, T. Onda, N. Ogasawara, S. Umegaki, and R. Ito, “Theoretical analysis of Čerenkov-type optical second-harmonic generation in slab waveguides,” IEEE J. Quantum Electron. 28, 1798–1815 (1992).
    [Crossref]
  16. See, for example, A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. 9, 919–933 (1973).
    [Crossref]
  17. M. Ohashi, T. Kondo, R. Ito, S. Fukatsu, Y. Shiraki, K. Kumata, and S. S. Kano, “Determination of quadratic nonlinear optical coefficients of Alx Ga1−x As system by the method of reflected second harmonics,” J. Appl. Phys. 74, 596–601 (1993).
    [Crossref]
  18. T. Kondo, S. Koh, T. Tsunoda, A. Okubo, Y. Shiraki, and R. Ito, “Reflected second-harmonic ellipsometry—a new tool for determining the nonlinear optical coefficients of thin films,” in Technical Digest of the Fifth European Quantum Electronics Conference (Institute of Electric and Electronics Engineers, Amsterdam, 1994), p. 262.
    [Crossref]
  19. M. Ohashi, T. Kondo, S. Fukatsu, S. S. Kano, Y. Shiraki, and R. Ito, “Harmonic generation from multilayered semiconductor structures,” in Sixteenth Congress of the International Commission for Optics: Optics as a Key to High Technology, Gy. Ákos, T. Lippényi, G. Lupkovics, and A. Podmaniczky, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1983, 836–837 (1993).

1994 (2)

H. Takahashi, M. Ohashi, T. Kondo, N. Ogasawara, Y. Shiraki, and R. Ito, “New semiconductor second-harmonic generator based on quasi-phase-matching for cavity-enhanced fundamental standing wave,” Jpn. J. Appl. Phys. 33, L1456–L1458 (1994).
[Crossref]

N. D. Whitbread and P. N. Robson, “Theoretical analysis of passive visible surface-emitting second-harmonic generators,” IEEE J. Quantum Electron. 30, 139–147 (1994).
[Crossref]

1993 (1)

M. Ohashi, T. Kondo, R. Ito, S. Fukatsu, Y. Shiraki, K. Kumata, and S. S. Kano, “Determination of quadratic nonlinear optical coefficients of Alx Ga1−x As system by the method of reflected second harmonics,” J. Appl. Phys. 74, 596–601 (1993).
[Crossref]

1992 (2)

N. Hashizume, T. Kondo, T. Onda, N. Ogasawara, S. Umegaki, and R. Ito, “Theoretical analysis of Čerenkov-type optical second-harmonic generation in slab waveguides,” IEEE J. Quantum Electron. 28, 1798–1815 (1992).
[Crossref]

See, for example, M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[Crossref]

1991 (4)

R. Normandin, S. Létouneau, F. Chatenoud, and R. L. Williams, “Monolithic, surface-emitting, semiconductor visible lasers and spectrometers for WDM fiber communication systems,” IEEE J. Quantum Electron. 27, 1520–1530 (1991).
[Crossref]

D. Vakhshoori, R. J. Fischer, M. Hong, D. L. Sivco, G. J. Zydzik, G. N. S. Chu, and A. Y. Cho, “Blue-green surface-emitting second-harmonic generators on (111)B GaAs,” Appl. Phys. Lett. 59, 896–898 (1991).
[Crossref]

D. S. Bethune, “Optical harmonic generation and mixing in multilayer media: extension of optical transfer matrix approach to include anisotropic materials,” J. Opt. Soc. Am. B 8, 367–373 (1991).
[Crossref]

D. Vakhshoori, “Analysis of visible surface-emitting second-harmonic generators,” J. Appl. Phys. 70, 5205–5210 (1991).
[Crossref]

1989 (1)

1987 (1)

1979 (2)

1973 (1)

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

1970 (1)

See, for example, J. Jerphagnon and S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41, 1667–1681 (1970).
[Crossref]

1969 (1)

N. Bloembergen, H. J. Simon, and C. H. Lee, “Total reflection phenomena in second-harmonic generation of light,” Phys. Rev. 181, 1261–1271 (1969).
[Crossref]

1962 (1)

N. Bloembergen and P. S. Pershan, “Light waves at the boundary of nonlinear media,” Phys. Rev. 128, 606–622 (1962).
[Crossref]

Bethune, D. S.

Bloembergen, N.

N. Bloembergen, H. J. Simon, and C. H. Lee, “Total reflection phenomena in second-harmonic generation of light,” Phys. Rev. 181, 1261–1271 (1969).
[Crossref]

N. Bloembergen and P. S. Pershan, “Light waves at the boundary of nonlinear media,” Phys. Rev. 128, 606–622 (1962).
[Crossref]

Byer, R. L.

See, for example, M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[Crossref]

Chatenoud, F.

R. Normandin, S. Létouneau, F. Chatenoud, and R. L. Williams, “Monolithic, surface-emitting, semiconductor visible lasers and spectrometers for WDM fiber communication systems,” IEEE J. Quantum Electron. 27, 1520–1530 (1991).
[Crossref]

Cho, A. Y.

D. Vakhshoori, R. J. Fischer, M. Hong, D. L. Sivco, G. J. Zydzik, G. N. S. Chu, and A. Y. Cho, “Blue-green surface-emitting second-harmonic generators on (111)B GaAs,” Appl. Phys. Lett. 59, 896–898 (1991).
[Crossref]

Chu, G. N. S.

D. Vakhshoori, R. J. Fischer, M. Hong, D. L. Sivco, G. J. Zydzik, G. N. S. Chu, and A. Y. Cho, “Blue-green surface-emitting second-harmonic generators on (111)B GaAs,” Appl. Phys. Lett. 59, 896–898 (1991).
[Crossref]

Fejer, M. M.

See, for example, M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[Crossref]

Fischer, R. J.

D. Vakhshoori, R. J. Fischer, M. Hong, D. L. Sivco, G. J. Zydzik, G. N. S. Chu, and A. Y. Cho, “Blue-green surface-emitting second-harmonic generators on (111)B GaAs,” Appl. Phys. Lett. 59, 896–898 (1991).
[Crossref]

Fukatsu, S.

M. Ohashi, T. Kondo, R. Ito, S. Fukatsu, Y. Shiraki, K. Kumata, and S. S. Kano, “Determination of quadratic nonlinear optical coefficients of Alx Ga1−x As system by the method of reflected second harmonics,” J. Appl. Phys. 74, 596–601 (1993).
[Crossref]

M. Ohashi, T. Kondo, S. Fukatsu, S. S. Kano, Y. Shiraki, and R. Ito, “Harmonic generation from multilayered semiconductor structures,” in Sixteenth Congress of the International Commission for Optics: Optics as a Key to High Technology, Gy. Ákos, T. Lippényi, G. Lupkovics, and A. Podmaniczky, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1983, 836–837 (1993).

Hashizume, N.

N. Hashizume, T. Kondo, T. Onda, N. Ogasawara, S. Umegaki, and R. Ito, “Theoretical analysis of Čerenkov-type optical second-harmonic generation in slab waveguides,” IEEE J. Quantum Electron. 28, 1798–1815 (1992).
[Crossref]

Hong, M.

D. Vakhshoori, R. J. Fischer, M. Hong, D. L. Sivco, G. J. Zydzik, G. N. S. Chu, and A. Y. Cho, “Blue-green surface-emitting second-harmonic generators on (111)B GaAs,” Appl. Phys. Lett. 59, 896–898 (1991).
[Crossref]

Ito, R.

H. Takahashi, M. Ohashi, T. Kondo, N. Ogasawara, Y. Shiraki, and R. Ito, “New semiconductor second-harmonic generator based on quasi-phase-matching for cavity-enhanced fundamental standing wave,” Jpn. J. Appl. Phys. 33, L1456–L1458 (1994).
[Crossref]

M. Ohashi, T. Kondo, R. Ito, S. Fukatsu, Y. Shiraki, K. Kumata, and S. S. Kano, “Determination of quadratic nonlinear optical coefficients of Alx Ga1−x As system by the method of reflected second harmonics,” J. Appl. Phys. 74, 596–601 (1993).
[Crossref]

N. Hashizume, T. Kondo, T. Onda, N. Ogasawara, S. Umegaki, and R. Ito, “Theoretical analysis of Čerenkov-type optical second-harmonic generation in slab waveguides,” IEEE J. Quantum Electron. 28, 1798–1815 (1992).
[Crossref]

M. Ohashi, T. Kondo, S. Fukatsu, S. S. Kano, Y. Shiraki, and R. Ito, “Harmonic generation from multilayered semiconductor structures,” in Sixteenth Congress of the International Commission for Optics: Optics as a Key to High Technology, Gy. Ákos, T. Lippényi, G. Lupkovics, and A. Podmaniczky, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1983, 836–837 (1993).

T. Kondo, S. Koh, T. Tsunoda, A. Okubo, Y. Shiraki, and R. Ito, “Reflected second-harmonic ellipsometry—a new tool for determining the nonlinear optical coefficients of thin films,” in Technical Digest of the Fifth European Quantum Electronics Conference (Institute of Electric and Electronics Engineers, Amsterdam, 1994), p. 262.
[Crossref]

Jerphagnon, J.

See, for example, J. Jerphagnon and S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41, 1667–1681 (1970).
[Crossref]

Jundt, D. H.

See, for example, M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[Crossref]

Kano, S. S.

M. Ohashi, T. Kondo, R. Ito, S. Fukatsu, Y. Shiraki, K. Kumata, and S. S. Kano, “Determination of quadratic nonlinear optical coefficients of Alx Ga1−x As system by the method of reflected second harmonics,” J. Appl. Phys. 74, 596–601 (1993).
[Crossref]

M. Ohashi, T. Kondo, S. Fukatsu, S. S. Kano, Y. Shiraki, and R. Ito, “Harmonic generation from multilayered semiconductor structures,” in Sixteenth Congress of the International Commission for Optics: Optics as a Key to High Technology, Gy. Ákos, T. Lippényi, G. Lupkovics, and A. Podmaniczky, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1983, 836–837 (1993).

Koh, S.

T. Kondo, S. Koh, T. Tsunoda, A. Okubo, Y. Shiraki, and R. Ito, “Reflected second-harmonic ellipsometry—a new tool for determining the nonlinear optical coefficients of thin films,” in Technical Digest of the Fifth European Quantum Electronics Conference (Institute of Electric and Electronics Engineers, Amsterdam, 1994), p. 262.
[Crossref]

Kondo, T.

H. Takahashi, M. Ohashi, T. Kondo, N. Ogasawara, Y. Shiraki, and R. Ito, “New semiconductor second-harmonic generator based on quasi-phase-matching for cavity-enhanced fundamental standing wave,” Jpn. J. Appl. Phys. 33, L1456–L1458 (1994).
[Crossref]

M. Ohashi, T. Kondo, R. Ito, S. Fukatsu, Y. Shiraki, K. Kumata, and S. S. Kano, “Determination of quadratic nonlinear optical coefficients of Alx Ga1−x As system by the method of reflected second harmonics,” J. Appl. Phys. 74, 596–601 (1993).
[Crossref]

N. Hashizume, T. Kondo, T. Onda, N. Ogasawara, S. Umegaki, and R. Ito, “Theoretical analysis of Čerenkov-type optical second-harmonic generation in slab waveguides,” IEEE J. Quantum Electron. 28, 1798–1815 (1992).
[Crossref]

T. Kondo, S. Koh, T. Tsunoda, A. Okubo, Y. Shiraki, and R. Ito, “Reflected second-harmonic ellipsometry—a new tool for determining the nonlinear optical coefficients of thin films,” in Technical Digest of the Fifth European Quantum Electronics Conference (Institute of Electric and Electronics Engineers, Amsterdam, 1994), p. 262.
[Crossref]

M. Ohashi, T. Kondo, S. Fukatsu, S. S. Kano, Y. Shiraki, and R. Ito, “Harmonic generation from multilayered semiconductor structures,” in Sixteenth Congress of the International Commission for Optics: Optics as a Key to High Technology, Gy. Ákos, T. Lippényi, G. Lupkovics, and A. Podmaniczky, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1983, 836–837 (1993).

Kumata, K.

M. Ohashi, T. Kondo, R. Ito, S. Fukatsu, Y. Shiraki, K. Kumata, and S. S. Kano, “Determination of quadratic nonlinear optical coefficients of Alx Ga1−x As system by the method of reflected second harmonics,” J. Appl. Phys. 74, 596–601 (1993).
[Crossref]

Kurtz, S. K.

See, for example, J. Jerphagnon and S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41, 1667–1681 (1970).
[Crossref]

Lee, C. H.

N. Bloembergen, H. J. Simon, and C. H. Lee, “Total reflection phenomena in second-harmonic generation of light,” Phys. Rev. 181, 1261–1271 (1969).
[Crossref]

Létouneau, S.

R. Normandin, S. Létouneau, F. Chatenoud, and R. L. Williams, “Monolithic, surface-emitting, semiconductor visible lasers and spectrometers for WDM fiber communication systems,” IEEE J. Quantum Electron. 27, 1520–1530 (1991).
[Crossref]

Magel, G. A.

See, for example, M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[Crossref]

Normandin, R.

R. Normandin, S. Létouneau, F. Chatenoud, and R. L. Williams, “Monolithic, surface-emitting, semiconductor visible lasers and spectrometers for WDM fiber communication systems,” IEEE J. Quantum Electron. 27, 1520–1530 (1991).
[Crossref]

V. C. Y. So, R. Normandin, and G. I. Stegeman, “Field analysis of harmonic generation in thin-film integrated optics,” J. Opt. Soc. Am. 69, 1166–1171 (1979).
[Crossref]

Ogasawara, N.

H. Takahashi, M. Ohashi, T. Kondo, N. Ogasawara, Y. Shiraki, and R. Ito, “New semiconductor second-harmonic generator based on quasi-phase-matching for cavity-enhanced fundamental standing wave,” Jpn. J. Appl. Phys. 33, L1456–L1458 (1994).
[Crossref]

N. Hashizume, T. Kondo, T. Onda, N. Ogasawara, S. Umegaki, and R. Ito, “Theoretical analysis of Čerenkov-type optical second-harmonic generation in slab waveguides,” IEEE J. Quantum Electron. 28, 1798–1815 (1992).
[Crossref]

Ohashi, M.

H. Takahashi, M. Ohashi, T. Kondo, N. Ogasawara, Y. Shiraki, and R. Ito, “New semiconductor second-harmonic generator based on quasi-phase-matching for cavity-enhanced fundamental standing wave,” Jpn. J. Appl. Phys. 33, L1456–L1458 (1994).
[Crossref]

M. Ohashi, T. Kondo, R. Ito, S. Fukatsu, Y. Shiraki, K. Kumata, and S. S. Kano, “Determination of quadratic nonlinear optical coefficients of Alx Ga1−x As system by the method of reflected second harmonics,” J. Appl. Phys. 74, 596–601 (1993).
[Crossref]

M. Ohashi, T. Kondo, S. Fukatsu, S. S. Kano, Y. Shiraki, and R. Ito, “Harmonic generation from multilayered semiconductor structures,” in Sixteenth Congress of the International Commission for Optics: Optics as a Key to High Technology, Gy. Ákos, T. Lippényi, G. Lupkovics, and A. Podmaniczky, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1983, 836–837 (1993).

Okubo, A.

T. Kondo, S. Koh, T. Tsunoda, A. Okubo, Y. Shiraki, and R. Ito, “Reflected second-harmonic ellipsometry—a new tool for determining the nonlinear optical coefficients of thin films,” in Technical Digest of the Fifth European Quantum Electronics Conference (Institute of Electric and Electronics Engineers, Amsterdam, 1994), p. 262.
[Crossref]

Onda, T.

N. Hashizume, T. Kondo, T. Onda, N. Ogasawara, S. Umegaki, and R. Ito, “Theoretical analysis of Čerenkov-type optical second-harmonic generation in slab waveguides,” IEEE J. Quantum Electron. 28, 1798–1815 (1992).
[Crossref]

Pershan, P. S.

N. Bloembergen and P. S. Pershan, “Light waves at the boundary of nonlinear media,” Phys. Rev. 128, 606–622 (1962).
[Crossref]

Robson, P. N.

N. D. Whitbread and P. N. Robson, “Theoretical analysis of passive visible surface-emitting second-harmonic generators,” IEEE J. Quantum Electron. 30, 139–147 (1994).
[Crossref]

Shiraki, Y.

H. Takahashi, M. Ohashi, T. Kondo, N. Ogasawara, Y. Shiraki, and R. Ito, “New semiconductor second-harmonic generator based on quasi-phase-matching for cavity-enhanced fundamental standing wave,” Jpn. J. Appl. Phys. 33, L1456–L1458 (1994).
[Crossref]

M. Ohashi, T. Kondo, R. Ito, S. Fukatsu, Y. Shiraki, K. Kumata, and S. S. Kano, “Determination of quadratic nonlinear optical coefficients of Alx Ga1−x As system by the method of reflected second harmonics,” J. Appl. Phys. 74, 596–601 (1993).
[Crossref]

T. Kondo, S. Koh, T. Tsunoda, A. Okubo, Y. Shiraki, and R. Ito, “Reflected second-harmonic ellipsometry—a new tool for determining the nonlinear optical coefficients of thin films,” in Technical Digest of the Fifth European Quantum Electronics Conference (Institute of Electric and Electronics Engineers, Amsterdam, 1994), p. 262.
[Crossref]

M. Ohashi, T. Kondo, S. Fukatsu, S. S. Kano, Y. Shiraki, and R. Ito, “Harmonic generation from multilayered semiconductor structures,” in Sixteenth Congress of the International Commission for Optics: Optics as a Key to High Technology, Gy. Ákos, T. Lippényi, G. Lupkovics, and A. Podmaniczky, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1983, 836–837 (1993).

Simon, H. J.

N. Bloembergen, H. J. Simon, and C. H. Lee, “Total reflection phenomena in second-harmonic generation of light,” Phys. Rev. 181, 1261–1271 (1969).
[Crossref]

Sipe, J. E.

J. E. Sipe, “New Green-function formalism for surface optics,” J. Opt. Soc. Am. B 4, 481–489 (1987).
[Crossref]

J. E. Sipe, “The ATR spectra of multipole surface plasmons,” Surf. Sci. 84, 75–105 (1979).
[Crossref]

Sivco, D. L.

D. Vakhshoori, R. J. Fischer, M. Hong, D. L. Sivco, G. J. Zydzik, G. N. S. Chu, and A. Y. Cho, “Blue-green surface-emitting second-harmonic generators on (111)B GaAs,” Appl. Phys. Lett. 59, 896–898 (1991).
[Crossref]

So, V. C. Y.

Stegeman, G. I.

Takahashi, H.

H. Takahashi, M. Ohashi, T. Kondo, N. Ogasawara, Y. Shiraki, and R. Ito, “New semiconductor second-harmonic generator based on quasi-phase-matching for cavity-enhanced fundamental standing wave,” Jpn. J. Appl. Phys. 33, L1456–L1458 (1994).
[Crossref]

Tsunoda, T.

T. Kondo, S. Koh, T. Tsunoda, A. Okubo, Y. Shiraki, and R. Ito, “Reflected second-harmonic ellipsometry—a new tool for determining the nonlinear optical coefficients of thin films,” in Technical Digest of the Fifth European Quantum Electronics Conference (Institute of Electric and Electronics Engineers, Amsterdam, 1994), p. 262.
[Crossref]

Umegaki, S.

N. Hashizume, T. Kondo, T. Onda, N. Ogasawara, S. Umegaki, and R. Ito, “Theoretical analysis of Čerenkov-type optical second-harmonic generation in slab waveguides,” IEEE J. Quantum Electron. 28, 1798–1815 (1992).
[Crossref]

Vakhshoori, D.

D. Vakhshoori, R. J. Fischer, M. Hong, D. L. Sivco, G. J. Zydzik, G. N. S. Chu, and A. Y. Cho, “Blue-green surface-emitting second-harmonic generators on (111)B GaAs,” Appl. Phys. Lett. 59, 896–898 (1991).
[Crossref]

D. Vakhshoori, “Analysis of visible surface-emitting second-harmonic generators,” J. Appl. Phys. 70, 5205–5210 (1991).
[Crossref]

Whitbread, N. D.

N. D. Whitbread and P. N. Robson, “Theoretical analysis of passive visible surface-emitting second-harmonic generators,” IEEE J. Quantum Electron. 30, 139–147 (1994).
[Crossref]

Williams, R. L.

R. Normandin, S. Létouneau, F. Chatenoud, and R. L. Williams, “Monolithic, surface-emitting, semiconductor visible lasers and spectrometers for WDM fiber communication systems,” IEEE J. Quantum Electron. 27, 1520–1530 (1991).
[Crossref]

Yariv, A.

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

Zydzik, G. J.

D. Vakhshoori, R. J. Fischer, M. Hong, D. L. Sivco, G. J. Zydzik, G. N. S. Chu, and A. Y. Cho, “Blue-green surface-emitting second-harmonic generators on (111)B GaAs,” Appl. Phys. Lett. 59, 896–898 (1991).
[Crossref]

Appl. Phys. Lett. (1)

D. Vakhshoori, R. J. Fischer, M. Hong, D. L. Sivco, G. J. Zydzik, G. N. S. Chu, and A. Y. Cho, “Blue-green surface-emitting second-harmonic generators on (111)B GaAs,” Appl. Phys. Lett. 59, 896–898 (1991).
[Crossref]

IEEE J. Quantum Electron. (5)

R. Normandin, S. Létouneau, F. Chatenoud, and R. L. Williams, “Monolithic, surface-emitting, semiconductor visible lasers and spectrometers for WDM fiber communication systems,” IEEE J. Quantum Electron. 27, 1520–1530 (1991).
[Crossref]

See, for example, M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[Crossref]

N. Hashizume, T. Kondo, T. Onda, N. Ogasawara, S. Umegaki, and R. Ito, “Theoretical analysis of Čerenkov-type optical second-harmonic generation in slab waveguides,” IEEE J. Quantum Electron. 28, 1798–1815 (1992).
[Crossref]

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

N. D. Whitbread and P. N. Robson, “Theoretical analysis of passive visible surface-emitting second-harmonic generators,” IEEE J. Quantum Electron. 30, 139–147 (1994).
[Crossref]

J. Appl. Phys. (3)

See, for example, J. Jerphagnon and S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41, 1667–1681 (1970).
[Crossref]

M. Ohashi, T. Kondo, R. Ito, S. Fukatsu, Y. Shiraki, K. Kumata, and S. S. Kano, “Determination of quadratic nonlinear optical coefficients of Alx Ga1−x As system by the method of reflected second harmonics,” J. Appl. Phys. 74, 596–601 (1993).
[Crossref]

D. Vakhshoori, “Analysis of visible surface-emitting second-harmonic generators,” J. Appl. Phys. 70, 5205–5210 (1991).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (3)

Jpn. J. Appl. Phys. (1)

H. Takahashi, M. Ohashi, T. Kondo, N. Ogasawara, Y. Shiraki, and R. Ito, “New semiconductor second-harmonic generator based on quasi-phase-matching for cavity-enhanced fundamental standing wave,” Jpn. J. Appl. Phys. 33, L1456–L1458 (1994).
[Crossref]

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[Crossref]

N. Bloembergen, H. J. Simon, and C. H. Lee, “Total reflection phenomena in second-harmonic generation of light,” Phys. Rev. 181, 1261–1271 (1969).
[Crossref]

Surf. Sci. (1)

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[Crossref]

Other (2)

T. Kondo, S. Koh, T. Tsunoda, A. Okubo, Y. Shiraki, and R. Ito, “Reflected second-harmonic ellipsometry—a new tool for determining the nonlinear optical coefficients of thin films,” in Technical Digest of the Fifth European Quantum Electronics Conference (Institute of Electric and Electronics Engineers, Amsterdam, 1994), p. 262.
[Crossref]

M. Ohashi, T. Kondo, S. Fukatsu, S. S. Kano, Y. Shiraki, and R. Ito, “Harmonic generation from multilayered semiconductor structures,” in Sixteenth Congress of the International Commission for Optics: Optics as a Key to High Technology, Gy. Ákos, T. Lippényi, G. Lupkovics, and A. Podmaniczky, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1983, 836–837 (1993).

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Figures (7)

Fig. 1
Fig. 1

Schematic of the multilayered structure considered in this paper.

Fig. 2
Fig. 2

Outline of the calculation procedure.

Fig. 3
Fig. 3

Nonlinear polarization sheet at z = z′, the radiated harmonic waves from the sheet, and the orthonormal triads (ŝ, k ^ ( + ) , p ^ ( + )) and (ŝ k ^ ( - ) , p ^ ( - )) associated with the radiated harmonic waves.

Fig. 4
Fig. 4

Schematic of the self harmonic wave E i self, which represents a harmonic field that is generated within region i and would be present if no reflection were to take place at the boundaries, and the additional harmonic wave E i add, which consists of harmonic fields that are generated in region i and are reflected from the boundaries at zi−1 and z plus those coming from other regions.

Fig. 5
Fig. 5

Nonlinear three-layered geometry.

Fig. 6
Fig. 6

Air–GaP–AlP–GaP four-layered structure.

Fig. 7
Fig. 7

Dependence of relative SH intensity on AlP layer thickness for two samples with (a) 7-nm and (b) 28-nm cap layer. In both graphs, the open circles are experimental results, and the solid curves are theoretical calculations.

Equations (86)

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E ω h ( r , t ) = 1 2 E ω h ( r ) exp ( - i ω h t ) + c . c .
D ω h ( r ) = ɛ 0 E ω h ( r ) + P linear ω h ( r ) + P nonlinear ω h ( r ) = ɛ ω h E ω h ( r ) + P nonlinear ω h ( r ) ,
[ 2 + ω h 2 c 2 ( n ω h ) 2 ] E ω h ( r ) = ( - μ 0 ω h 2 - 1 ɛ ω h · ) P nonlinear ω h ( r ) ,
E i total = E i driven + E i free ,
E i total = E i self + E i add ,
P nonlinear ω h ( r ) = P δ ( z - z ) exp ( i β x ) .
( 2 + ω h 2 c 2 n 2 ) E ( r ) = 0.
E ( ± ) ( r ) = E ( ± ) exp ( i k ( ± ) · r ) ,
k ( ± ) = β x ^ ± κ z ^ .
E ( ± ) = E s ( ± ) s ^ + E p ( ± ) p ^ ( ± ) ,
s ^ z ^ × x ^ ( = y ^ ) ,
p ^ ( ± ) s ^ × k ^ ( ± ) = 1 k ( - β z ^ ± κ x ^ ) ,
E ( r ) = E ( z ) exp ( i β x ) ,
E ( z ) = E ( + ) exp [ i κ ( z - z ) ] θ ( z - z ) + E ( - ) exp [ - i κ ( z - z ) ] θ ( z - z ) + E b δ ( z - z ) ,
θ ( z ) = { 1 ( z > 0 ) 0 ( z < 0 ) ,
E ( ± ) = i ω h 2 2 ɛ 0 c 2 κ [ s ^ s ^ · P + p ^ ( ± ) p ^ ( ± ) · P ] ,
E b = - 1 ɛ z ^ z ^ · P .
P ( r ) = P ( z ) exp ( i β x ) = - + P ( z ) δ ( z - z ) exp ( i β x ) d z .
E ( z ) = - + G ( z , z ) · P ( z ) d z ,
G ( z , z ) = i ω h 2 2 ɛ 0 c 2 κ [ s ^ s ^ + p ^ ( + ) p ^ ( + ) ] exp [ i κ ( z - z ) ] θ ( z - z ) + i ω h 2 2 ɛ 0 c 2 κ [ s ^ s ^ + p ^ ( - ) p ^ ( - ) ] exp [ - i κ ( z - z ) ] × θ ( z - z ) - 1 ɛ z ^ z ^ δ ( z - z ) .
( d 2 d z 2 + κ 2 ) E ( z ) = ( - μ 0 ω h 2 U - 1 ɛ qq ) · P ( z ) ,
q = i β x ^ + z ^ d d z ,
( d 2 d z 2 + κ 2 ) G ( z , z ) = ( - μ 0 ω h 2 U - 1 ɛ qq ) δ ( z - z ) ,
E i self ( z ) = E i r ( z ) + E i b ( z ) ,
E i r ( z ) = { i ω h 2 2 ɛ 0 c 2 κ i [ s ^ s ^ + p ^ i ( + ) p ^ i ( + ) ] · z i - 1 z P i ( z ) × exp ( - i κ i z ) d z } exp ( i κ i z ) + { i ω h 2 2 ɛ 0 c 2 κ i [ s ^ s ^ + p ^ i ( - ) p ^ i ( - ) ] · z z i P i ( z ) × exp ( i κ i z ) d z } exp ( - i κ i z ) ,
E i b ( z ) = - 1 ɛ i z ^ z ^ · P i ( z ) .
E i λ ( z ) = E i λ ( + ) λ ^ i ( + ) exp ( i κ i z ) + E i λ ( - ) γ ^ i ( - ) exp ( - i κ i z )             ( λ = s , p ) ,
E i λ ( z ) = [ E i λ ( + ) exp ( i κ i z ) E i λ ( - ) exp ( - i κ i z ) ] .
E i + 1 , λ ( z i ) = M i + 1 , i λ E i λ ( z i ) ,
M i j λ = 1 t i j λ [ 1 r i j λ r i j λ 1 ] .
E i λ ( z a ) = Φ i ( z a - z b ) E i λ ( z b ) ,             ( z i - 1 z b < z a z i ) ,
Φ i ( z ) = [ exp ( i κ i z ) 0 0 exp ( - i κ i z ) ] .
E i + 1 add ( z i ) = M i + 1 , i [ E i add ( z i ) + E i r ( z i ) ] ,
E i add ( z i - 1 ) + E i r ( z i - 1 ) = M i , i - 1 E i - 1 add ( z i - 1 ) ,
E i + 1 add ( z i ) = M i + 1 , i [ Φ i M i , i - 1 E i - 1 add ( z i - 1 ) + S i ] ,
S i = E i r ( z i ) - Φ i E i r ( z i - 1 ) ,             ( i = 1 , 2 , , N ) ,
E R add ( z N ) = T R L E L add ( z N 0 ) + T R i S i .
T i j = M i , i - 1 Φ i - 1 M i - 1 , i - 2 Φ j + 1 M j + 1 , j ,             ( j < i ) ,
T i j = M i , i + 1 Φ ¯ i + 1 M i + 1 , i + 2 Φ ¯ j - 1 M j - 1 , j ,             ( j > i ) .
E R add ( z N ) = T R L E L add ( z 0 ) + S R L ,
S R L = T R L E L r ( z 0 ) - E R r ( z N ) + i = 1 N T R i S i ,
E L add ( z 0 ) = T L R E R add ( z N ) + S L R ,
S L R = T L R E R r ( z N ) - E L r ( z 0 ) - i = 1 N T L i Φ ¯ i S i .
E R add ( z N ) = [ E R add ( + ) exp ( i κ R z N ) 0 ] ,
E L add ( z 0 ) = [ 0 E R add ( - ) exp ( - i κ L z 0 ) ] .
E i add ( z ) = Φ i ( z - z i - 1 ) [ T i L E L add ( z 0 ) + S i L ] = Φ ¯ i ( z i - z ) [ T i R E R add ( z N ) + S i R ]             ( z i - 1 < z < z i ) ,
S i L = T i L E L r ( z 0 ) - E i z ( z i - 1 ) + j = 1 i - 1 T i j S j ,
S i R = T i R E R r ( z N ) - E i r ( z i ) - j = i + 1 N T i j Φ ¯ j S j .
T = M 32 Φ 2 M 21 = 1 t 32 t 21 [ exp ( i κ 2 D ) + r 32 r 21 exp ( - i κ 2 D ) r 21 exp ( i κ 2 D ) + r 32 exp ( - i κ 2 D ) r 21 exp ( - i κ 2 D ) + r 32 exp ( i κ 2 D ) exp ( - i κ 2 D ) + r 32 r 21 exp ( i κ 2 D ) ] ,
E 3 add ( D ) = T E 1 add ( 0 ) + S ,
S = T E 1 r ( 0 ) - M 32 Φ 2 E 2 r ( 0 ) + M 32 E 2 r ( D ) - E 3 r ( D ) .
[ E 3 add ( + ) exp ( i κ 3 D ) E 1 add ( - ) ] = [ 1 - T 12 / T 22 0 - 1 / T 22 ] [ S ( + ) S ( - ) ] ,
S ( + ) = T 11 E 1 r ( + ) ( 0 ) - r 32 exp ( - i κ 2 D ) t 32 E 2 r ( - ) ( 0 ) + 1 t 32 E 2 r ( + ) ( D ) ,
S ( - ) = T 21 E 1 r ( + ) ( 0 ) - exp ( - i κ 2 D ) t 32 E 2 r ( - ) ( 0 ) + r 32 t 32 E 2 r ( + ) ( D ) + E 3 r ( - ) ( D ) ,
E 1 add ( - ) = i ω h 2 2 ɛ 0 c 2 { [ r 12 + F t 12 r 23 t 21 exp ( 2 i κ 2 D ) ] × 1 κ 1 - 0 λ ^ 1 ( + ) · P 1 ( z ) exp ( - i κ 1 z ) d z + F r 21 1 κ 2 0 D λ ^ 2 ( - ) · P 2 ( z ) exp ( i κ 2 z ) d z + F t 23 t 21 exp ( i κ 2 D ) 1 κ 2 0 D λ ^ 2 ( + ) · P 2 ( z ) × exp [ - i κ 2 ( z - D ) ] d z + F t 32 t 21 exp ( i κ 2 D ) × 1 κ 3 D λ ^ 3 ( - ) · P 3 ( z ) exp [ i κ 3 ( z - D ) ] d z } ,
E 3 add ( + ) = i ω h 2 2 ɛ 0 c 2 exp ( - i κ 3 D ) { F t 12 t 23 exp ( i κ 2 D ) × 1 κ 1 - 0 λ ^ 1 ( + ) · P 1 ( z ) exp ( - i κ 1 z ) d z + F r 21 t 23 exp ( i κ 2 D ) 1 κ 2 0 D λ ^ 2 ( 1 ) · P 2 ( z ) × exp ( i κ 2 z ) d z + F t 23 1 κ 2 0 D λ ^ 2 ( + ) · P 2 ( z ) × exp [ - i κ 2 ( z - D ) ] d z + [ r 32 + F t 32 r 21 t 23 exp ( 2 i κ 2 D ) ] × 1 κ 3 D γ ^ 3 ( - ) · P 3 ( z ) exp [ i κ 3 ( z - D ) ] d z } ,
F = 1 1 - r 21 r 23 exp ( 2 i κ 2 D ) ,
E i f ω ( r ) = γ ^ [ E i f ( + ) exp ( i κ i ω z ) + E i f ( - ) exp ( - i κ i ω z ) ] exp ( i β i ω x ) .
[ E 3 f ( + ) exp ( i κ 3 ω D ) 0 ] = T ω [ E 1 f ( + ) E 1 f ( - ) ] ,
[ E 2 f ( + ) E 2 f ( - ) ] = M 21 ω [ E 1 f ( + ) E 1 f ( - ) ] = Φ 2 ω M 23 ω [ E 3 f ( + ) exp ( i κ 3 ω D ) 0 ] ,
E 1 f ( - ) = [ r 12 ω + F ω t 12 ω r 23 ω t 21 ω exp ( 2 i κ 2 ω D ) ] E 1 f ( + ) ,
E 2 f ( + ) = F ω t 12 ω E 1 f ( + ) ,
E 2 f ( - ) = F ω t 12 ω r 23 ω exp ( 2 i κ 2 ω D ) E 1 f ( + ) ,
E 3 f ( + ) = F ω t 12 ω t 23 ω exp ( i κ 2 ω D ) E 1 f ( + ) exp ( - i κ 3 ω D ) ,
P i 2 ω ( r ) = y ^ P i ( z ) exp ( 2 i β ω x ) ,
P i ( z ) = P i ( + ) ( z ) + P i ( - ) ( z ) + P i ( 0 ) ( z ) ,
P i ( ± ) ( z ) = ɛ 0 d i [ E i f ( ± ) ] 2 exp ( ± 2 i κ i ω z ) , P i ( 0 ) ( z ) = 2 ɛ 0 d i E i f ( + ) E i f ( - ) .
E 1 SH add ( - ) = E film ( - ) + E sub ( - ) ,
E film ( - ) = i ( 2 ω c ) 2 1 2 κ 2 2 ω ( F ω t 12 ω ) 2 F 2 ω t 21 2 ω d 2 [ E 1 f ( + ) ] 2 I film ,
E sub ( - ) = i ( 2 ω c ) 2 1 2 κ 3 2 ω [ F ω t 12 ω t 23 ω exp ( i κ 2 ω D ) ] 2 × F 2 ω t 32 2 ω t 21 2 ω exp ( i κ 2 2 ω D ) d 3 [ E 1 f ( + ) ] 2 I sub ,
I film = 0 D ( { exp ( i κ 2 ω z ) + r 23 ω exp [ - i κ 2 ω ( z - 2 D ) ] } 2 × { exp ( i κ 2 2 ω z ) + r 23 2 ω exp [ - i κ 2 2 ω ( z - 2 D ) ] } ) d z ,
I sub = D exp [ i ( 2 κ 3 ω + κ 3 2 ω ) ( z - D ) ] d z ,
E 3 SH add ( + ) exp ( i κ 3 2 ω D ) = E film ( + ) + E sub ( + ) ,
E film ( + ) = i ( 2 ω c ) 2 1 2 κ 2 2 ω ( F ω t 12 ω ) 2 F 2 ω t 23 2 ω exp ( i κ 2 2 ω D ) d 2 × [ E 1 f ( + ) ] 2 I film ,
E sub ( + ) = i ( 2 ω c ) 2 1 2 κ 3 2 ω [ F ω t 12 ω t 23 ω exp ( i κ 2 ω D ) ] 2 × [ r 32 2 ω + F 2 ω t 32 2 ω r 21 2 ω t 23 2 ω exp ( 2 i κ 2 2 ω D ) ] d 3 [ E 1 f ( + ) ] 2 I sub ,
I film = 0 D ( { exp ( i κ 2 ω z ) + r 23 ω exp [ - i κ 2 ω ( z - 2 D ) ] } 2 × [ r 21 2 ω exp ( i κ 2 2 ω z ) + exp ( - i κ 2 2 ω z ) ] ) d z ,
I sub = I sub .
0 exp ( i κ z ) d z = lim α 0 [ lim L 0 L exp ( i κ - α ) z d z ] = - 1 i κ ,             ( κ 0 ) ,
P i 2 ω ( r ) = - z ^ P i ( z ) exp ( 2 i β ω x ) ,
D ω h ( r ) = E ω h ( r ) + 4 π P linear ω h ( r ) + 4 π P nonlinear ω h ( r ) = ɛ ω h E ω h ( r ) + 4 π P nonlinear ω h ( r ) ,
[ 2 + ω h 2 c 2 ( n ω h ) 2 ] E ω h ( r ) = ( - 4 π ω h 2 c 2 - 4 π ɛ ω h · ) × P nonlinear ω h ( r ) ,
G ( z , z ) = i 2 π ω h 2 c 2 κ [ s ^ s ^ + p ^ ( + ) p ^ ( + ) ] exp [ i κ ( z - z ) ] θ ( z - z ) + i 2 π ω h 2 c 2 κ [ s ^ s ^ + p ^ ( - ) p ^ ( - ) ] exp [ - i κ ( z - z ) ] × θ ( z - z ) - 4 π ɛ z ^ z ^ δ ( z - z ) .
E i r ( z ) = { i 2 π ω h 2 c 2 κ i [ s ^ s ^ + p ^ i ( + ) p ^ i ( + ) ] · z i - 1 z P i ( z ) × exp ( - i κ i z ) d z } exp ( i κ i z ) + { i 2 π ω h 2 c 2 κ i [ s ^ s ^ + p ^ i ( - ) p ^ i ( - ) ] · z z i P i ( z ) × exp ( i κ i z ) d z } exp ( - i κ i z ) ,
E i b ( z ) = - 4 π ɛ i z ^ z ^ · P i ( z ) .
E i + 1 free ( z i ) = M i + 1 , i Φ i [ M i , i - 1 E i - 1 free ( z i - 1 ) + S i ] ,
S i = ( Φ ¯ i M i NL Φ i NL - M i NL ) E i driven ( z i - 1 ) ,             ( i = 1 , 2 , , N ) ,

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