Abstract

The electro-optic effect can be employed to modulate the refractive index of an optical superlattice. In coupled quasi-phase matched processes, this modulation will introduce quasi-phase mismatches and result in energy redistribution among the optical waves. Numerical results indicate that an efficient third harmonic in a periodic or quasi-periodic superlattice can be achieved by varying the external dc electric field. This method provides a simple and convenient way to control the efficiencies of frequency conversion.

© 2005 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. Feng, N. B. Ming, J. F. Hong, Y. S. Yang, J. S. Zhu, Z. Yang, Y. N. Wang, “Enhancement of second-harmonic generation in LiNbO3 crystals with periodic laminar ferroelectric domains,” Appl. Phys. Lett. 37, 607–609 (1980).
    [CrossRef]
  2. M. M. Fejer, G. A. Magel, D. H. Jundt, R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
    [CrossRef]
  3. S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, H. F. Wang, C. Z. Ge, N. B. Ming, “Experimental realization of second harmonic generation in a Fibonacci optical superlattice of LiTaO3,” Phys. Rev. Lett. 78, 2752–2755 (1997).
    [CrossRef]
  4. S. N. Zhu, Y. Y. Zhu, N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278, 843–846 (1997).
    [CrossRef]
  5. R. L. Byer, “Quasi-phase-matched nonlinear interactions and devices,” J. Nonlinear Opt. Phys. Mater. 6, 549–592 (1997).
    [CrossRef]
  6. Y. Y. Zhu, N. B. Ming, “Dielectric superlattices for nonlinear optical effects,” Opt. Quantum Electron. 31, 1093–1128 (1999).
    [CrossRef]
  7. G. Z. Luo, S. N. Zhu, J. L. He, Y. Y. Zhu, H. T. Wang, Z. W. Liu, C. Zhang, N. B. Ming, “Simultaneously efficient blue and red light generations in a periodically poled LiTaO3,” Appl. Phys. Lett. 78, 3006–3008 (2001).
    [CrossRef]
  8. Z. W. Liu, S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, J. L. He, C. Zhang, H. T. Wang, N. B. Ming, X. Y. Liang, Z. Y. Xu, “Quasi-cw ultraviolet generation in a dual-periodic LiTaO3 superlattice by frequency tripling,” Jpn. J. Appl. Phys. 40, 6841–6844 (2001).
    [CrossRef]
  9. K. K. Fradkin, A. Arie, P. Urenski, G. Rosenman, “Multiple nonlinear optical interactions with arbitrary wave vector differences,” Phys. Rev. Lett. 88, 023903 (2002).
    [CrossRef]
  10. P. Xu, S. H. Ji, S. N. Zhu, X. Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, N. B. Ming, “Conical second harmonic generation in a two-dimensional χ(2) photonic crystal: a hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. 93, 133904 (2004).
    [CrossRef]
  11. Y. Q. Lu, Z. L. Wang, Q. Wang, Y. X. Xi, N. B. Ming, “Electro-optic effect of periodically poled optical superlattice LiNbO3 and its applications,” Appl. Phys. Lett. 77, 3719–3721 (2000).
    [CrossRef]
  12. Y. Q. Lu, M. Xiao, G. J. Salamo, “Wide-bandwidth high-frequency electro-optic modulator based on periodically poled LiNbO3,” Appl. Phys. Lett. 78, 1035–1037 (2001).
    [CrossRef]
  13. K. T. Gahagan, D. A. Scrymgeour, J. L. Casson, V. Gopalan, J. M. Robinson, “Integrated high-power electro-optic lens and large-angle deflector,” Appl. Opt. 40, 5638–5642 (2001).
    [CrossRef]
  14. D. A. Scrymgeour, A. Sharan, V. Gopalan, K. T. Gahagan, J. L. Casson, “Cascaded electro-optic scanning of laser light over large angles using domain microengineered ferroelectrics,” Appl. Phys. Lett. 81, 3140–3142 (2002).
    [CrossRef]
  15. Y. Q. Lu, J. J. Zheng, Y. L. Lu, N. B. Ming, “Frequency tuning of optical parametric generator in periodically poled optical superlattice LiNbO3 by electro-optic effect,” Appl. Phys. Lett. 74, 123–125 (1999).
    [CrossRef]
  16. N. O’Brien, M. Missey, P. Powers, V. Dominic, K. L. Schepler, “Electro-optic spectral tuning in a continuous-wave, asymmetric-duty-cycle, periodically poled LiNbO3 optical parametric oscillator,” Opt. Lett. 24, 1750–1752 (1999).
    [CrossRef]
  17. Y. H. Chen, F. C. Fan, Y. Y. Lin, Y. C. Huang, J. T. Shy, Y. P. Lan, Y. F. Chen, “Simultaneous amplitude modulation and wavelength conversion in an asymmetric-duty-cycle periodically poled lithium niobate,” Opt. Commun. 223, 417–423 (2003).
    [CrossRef]
  18. C. Zhang, Y. Y. Zhu, S. X. Yang, Y. Q. Qin, S. N. Zhu, Y. B. Chen, H. Liu, N. B. Ming, “Crucial effects of coupling coefficients on quasi-phase-matched harmonic generation in an optical superlattice,” Opt. Lett. 25, 436–438 (2000).
    [CrossRef]
  19. J. Feng, Y. Y. Zhu, N. B. Ming, “Harmonic generations in an optical Fibonacci superlattice,” Phys. Rev. B 41, 5578–5582 (1990).
    [CrossRef]
  20. R. K. P. Zia, W. J. Dallas, “A simple derivation of quasi-crystalline spectra,” J. Phys. A 18, L341–L345 (1985).
    [CrossRef]
  21. C. Zhang, H. Wei, Y. Y. Zhu, H. T. Wang, S. N. Zhu, N. B. Ming, “Third-harmonic generation in a general two-component quasi-periodic optical superlattice,” Opt. Lett. 26, 899–901 (2001).
    [CrossRef]
  22. J. P. Meyn, M. M. Fejer, “Tunable ultraviolet radiation by second-harmonic generation in periodically poled lithium tantalate,” Opt. Lett. 22, 1214–1216 (1997).
    [CrossRef] [PubMed]

2004

P. Xu, S. H. Ji, S. N. Zhu, X. Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, N. B. Ming, “Conical second harmonic generation in a two-dimensional χ(2) photonic crystal: a hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. 93, 133904 (2004).
[CrossRef]

2003

Y. H. Chen, F. C. Fan, Y. Y. Lin, Y. C. Huang, J. T. Shy, Y. P. Lan, Y. F. Chen, “Simultaneous amplitude modulation and wavelength conversion in an asymmetric-duty-cycle periodically poled lithium niobate,” Opt. Commun. 223, 417–423 (2003).
[CrossRef]

2002

D. A. Scrymgeour, A. Sharan, V. Gopalan, K. T. Gahagan, J. L. Casson, “Cascaded electro-optic scanning of laser light over large angles using domain microengineered ferroelectrics,” Appl. Phys. Lett. 81, 3140–3142 (2002).
[CrossRef]

K. K. Fradkin, A. Arie, P. Urenski, G. Rosenman, “Multiple nonlinear optical interactions with arbitrary wave vector differences,” Phys. Rev. Lett. 88, 023903 (2002).
[CrossRef]

2001

G. Z. Luo, S. N. Zhu, J. L. He, Y. Y. Zhu, H. T. Wang, Z. W. Liu, C. Zhang, N. B. Ming, “Simultaneously efficient blue and red light generations in a periodically poled LiTaO3,” Appl. Phys. Lett. 78, 3006–3008 (2001).
[CrossRef]

Z. W. Liu, S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, J. L. He, C. Zhang, H. T. Wang, N. B. Ming, X. Y. Liang, Z. Y. Xu, “Quasi-cw ultraviolet generation in a dual-periodic LiTaO3 superlattice by frequency tripling,” Jpn. J. Appl. Phys. 40, 6841–6844 (2001).
[CrossRef]

Y. Q. Lu, M. Xiao, G. J. Salamo, “Wide-bandwidth high-frequency electro-optic modulator based on periodically poled LiNbO3,” Appl. Phys. Lett. 78, 1035–1037 (2001).
[CrossRef]

C. Zhang, H. Wei, Y. Y. Zhu, H. T. Wang, S. N. Zhu, N. B. Ming, “Third-harmonic generation in a general two-component quasi-periodic optical superlattice,” Opt. Lett. 26, 899–901 (2001).
[CrossRef]

K. T. Gahagan, D. A. Scrymgeour, J. L. Casson, V. Gopalan, J. M. Robinson, “Integrated high-power electro-optic lens and large-angle deflector,” Appl. Opt. 40, 5638–5642 (2001).
[CrossRef]

2000

Y. Q. Lu, Z. L. Wang, Q. Wang, Y. X. Xi, N. B. Ming, “Electro-optic effect of periodically poled optical superlattice LiNbO3 and its applications,” Appl. Phys. Lett. 77, 3719–3721 (2000).
[CrossRef]

C. Zhang, Y. Y. Zhu, S. X. Yang, Y. Q. Qin, S. N. Zhu, Y. B. Chen, H. Liu, N. B. Ming, “Crucial effects of coupling coefficients on quasi-phase-matched harmonic generation in an optical superlattice,” Opt. Lett. 25, 436–438 (2000).
[CrossRef]

1999

Y. Q. Lu, J. J. Zheng, Y. L. Lu, N. B. Ming, “Frequency tuning of optical parametric generator in periodically poled optical superlattice LiNbO3 by electro-optic effect,” Appl. Phys. Lett. 74, 123–125 (1999).
[CrossRef]

Y. Y. Zhu, N. B. Ming, “Dielectric superlattices for nonlinear optical effects,” Opt. Quantum Electron. 31, 1093–1128 (1999).
[CrossRef]

N. O’Brien, M. Missey, P. Powers, V. Dominic, K. L. Schepler, “Electro-optic spectral tuning in a continuous-wave, asymmetric-duty-cycle, periodically poled LiNbO3 optical parametric oscillator,” Opt. Lett. 24, 1750–1752 (1999).
[CrossRef]

1997

S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, H. F. Wang, C. Z. Ge, N. B. Ming, “Experimental realization of second harmonic generation in a Fibonacci optical superlattice of LiTaO3,” Phys. Rev. Lett. 78, 2752–2755 (1997).
[CrossRef]

S. N. Zhu, Y. Y. Zhu, N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278, 843–846 (1997).
[CrossRef]

R. L. Byer, “Quasi-phase-matched nonlinear interactions and devices,” J. Nonlinear Opt. Phys. Mater. 6, 549–592 (1997).
[CrossRef]

J. P. Meyn, M. M. Fejer, “Tunable ultraviolet radiation by second-harmonic generation in periodically poled lithium tantalate,” Opt. Lett. 22, 1214–1216 (1997).
[CrossRef] [PubMed]

1992

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

1990

J. Feng, Y. Y. Zhu, N. B. Ming, “Harmonic generations in an optical Fibonacci superlattice,” Phys. Rev. B 41, 5578–5582 (1990).
[CrossRef]

1985

R. K. P. Zia, W. J. Dallas, “A simple derivation of quasi-crystalline spectra,” J. Phys. A 18, L341–L345 (1985).
[CrossRef]

1980

D. Feng, N. B. Ming, J. F. Hong, Y. S. Yang, J. S. Zhu, Z. Yang, Y. N. Wang, “Enhancement of second-harmonic generation in LiNbO3 crystals with periodic laminar ferroelectric domains,” Appl. Phys. Lett. 37, 607–609 (1980).
[CrossRef]

Arie, A.

K. K. Fradkin, A. Arie, P. Urenski, G. Rosenman, “Multiple nonlinear optical interactions with arbitrary wave vector differences,” Phys. Rev. Lett. 88, 023903 (2002).
[CrossRef]

Byer, R. L.

R. L. Byer, “Quasi-phase-matched nonlinear interactions and devices,” J. Nonlinear Opt. Phys. Mater. 6, 549–592 (1997).
[CrossRef]

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

Casson, J. L.

D. A. Scrymgeour, A. Sharan, V. Gopalan, K. T. Gahagan, J. L. Casson, “Cascaded electro-optic scanning of laser light over large angles using domain microengineered ferroelectrics,” Appl. Phys. Lett. 81, 3140–3142 (2002).
[CrossRef]

K. T. Gahagan, D. A. Scrymgeour, J. L. Casson, V. Gopalan, J. M. Robinson, “Integrated high-power electro-optic lens and large-angle deflector,” Appl. Opt. 40, 5638–5642 (2001).
[CrossRef]

Chen, Y. B.

Chen, Y. F.

Y. H. Chen, F. C. Fan, Y. Y. Lin, Y. C. Huang, J. T. Shy, Y. P. Lan, Y. F. Chen, “Simultaneous amplitude modulation and wavelength conversion in an asymmetric-duty-cycle periodically poled lithium niobate,” Opt. Commun. 223, 417–423 (2003).
[CrossRef]

Chen, Y. H.

Y. H. Chen, F. C. Fan, Y. Y. Lin, Y. C. Huang, J. T. Shy, Y. P. Lan, Y. F. Chen, “Simultaneous amplitude modulation and wavelength conversion in an asymmetric-duty-cycle periodically poled lithium niobate,” Opt. Commun. 223, 417–423 (2003).
[CrossRef]

Dallas, W. J.

R. K. P. Zia, W. J. Dallas, “A simple derivation of quasi-crystalline spectra,” J. Phys. A 18, L341–L345 (1985).
[CrossRef]

Dominic, V.

Fan, F. C.

Y. H. Chen, F. C. Fan, Y. Y. Lin, Y. C. Huang, J. T. Shy, Y. P. Lan, Y. F. Chen, “Simultaneous amplitude modulation and wavelength conversion in an asymmetric-duty-cycle periodically poled lithium niobate,” Opt. Commun. 223, 417–423 (2003).
[CrossRef]

Fejer, M. M.

J. P. Meyn, M. M. Fejer, “Tunable ultraviolet radiation by second-harmonic generation in periodically poled lithium tantalate,” Opt. Lett. 22, 1214–1216 (1997).
[CrossRef] [PubMed]

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

Feng, D.

D. Feng, N. B. Ming, J. F. Hong, Y. S. Yang, J. S. Zhu, Z. Yang, Y. N. Wang, “Enhancement of second-harmonic generation in LiNbO3 crystals with periodic laminar ferroelectric domains,” Appl. Phys. Lett. 37, 607–609 (1980).
[CrossRef]

Feng, J.

J. Feng, Y. Y. Zhu, N. B. Ming, “Harmonic generations in an optical Fibonacci superlattice,” Phys. Rev. B 41, 5578–5582 (1990).
[CrossRef]

Fradkin, K. K.

K. K. Fradkin, A. Arie, P. Urenski, G. Rosenman, “Multiple nonlinear optical interactions with arbitrary wave vector differences,” Phys. Rev. Lett. 88, 023903 (2002).
[CrossRef]

Gahagan, K. T.

D. A. Scrymgeour, A. Sharan, V. Gopalan, K. T. Gahagan, J. L. Casson, “Cascaded electro-optic scanning of laser light over large angles using domain microengineered ferroelectrics,” Appl. Phys. Lett. 81, 3140–3142 (2002).
[CrossRef]

K. T. Gahagan, D. A. Scrymgeour, J. L. Casson, V. Gopalan, J. M. Robinson, “Integrated high-power electro-optic lens and large-angle deflector,” Appl. Opt. 40, 5638–5642 (2001).
[CrossRef]

Ge, C. Z.

S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, H. F. Wang, C. Z. Ge, N. B. Ming, “Experimental realization of second harmonic generation in a Fibonacci optical superlattice of LiTaO3,” Phys. Rev. Lett. 78, 2752–2755 (1997).
[CrossRef]

Gopalan, V.

D. A. Scrymgeour, A. Sharan, V. Gopalan, K. T. Gahagan, J. L. Casson, “Cascaded electro-optic scanning of laser light over large angles using domain microengineered ferroelectrics,” Appl. Phys. Lett. 81, 3140–3142 (2002).
[CrossRef]

K. T. Gahagan, D. A. Scrymgeour, J. L. Casson, V. Gopalan, J. M. Robinson, “Integrated high-power electro-optic lens and large-angle deflector,” Appl. Opt. 40, 5638–5642 (2001).
[CrossRef]

He, J. L.

P. Xu, S. H. Ji, S. N. Zhu, X. Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, N. B. Ming, “Conical second harmonic generation in a two-dimensional χ(2) photonic crystal: a hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. 93, 133904 (2004).
[CrossRef]

G. Z. Luo, S. N. Zhu, J. L. He, Y. Y. Zhu, H. T. Wang, Z. W. Liu, C. Zhang, N. B. Ming, “Simultaneously efficient blue and red light generations in a periodically poled LiTaO3,” Appl. Phys. Lett. 78, 3006–3008 (2001).
[CrossRef]

Z. W. Liu, S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, J. L. He, C. Zhang, H. T. Wang, N. B. Ming, X. Y. Liang, Z. Y. Xu, “Quasi-cw ultraviolet generation in a dual-periodic LiTaO3 superlattice by frequency tripling,” Jpn. J. Appl. Phys. 40, 6841–6844 (2001).
[CrossRef]

Hong, J. F.

D. Feng, N. B. Ming, J. F. Hong, Y. S. Yang, J. S. Zhu, Z. Yang, Y. N. Wang, “Enhancement of second-harmonic generation in LiNbO3 crystals with periodic laminar ferroelectric domains,” Appl. Phys. Lett. 37, 607–609 (1980).
[CrossRef]

Huang, Y. C.

Y. H. Chen, F. C. Fan, Y. Y. Lin, Y. C. Huang, J. T. Shy, Y. P. Lan, Y. F. Chen, “Simultaneous amplitude modulation and wavelength conversion in an asymmetric-duty-cycle periodically poled lithium niobate,” Opt. Commun. 223, 417–423 (2003).
[CrossRef]

Ji, S. H.

P. Xu, S. H. Ji, S. N. Zhu, X. Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, N. B. Ming, “Conical second harmonic generation in a two-dimensional χ(2) photonic crystal: a hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. 93, 133904 (2004).
[CrossRef]

Jundt, D. H.

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

Lan, Y. P.

Y. H. Chen, F. C. Fan, Y. Y. Lin, Y. C. Huang, J. T. Shy, Y. P. Lan, Y. F. Chen, “Simultaneous amplitude modulation and wavelength conversion in an asymmetric-duty-cycle periodically poled lithium niobate,” Opt. Commun. 223, 417–423 (2003).
[CrossRef]

Liang, X. Y.

Z. W. Liu, S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, J. L. He, C. Zhang, H. T. Wang, N. B. Ming, X. Y. Liang, Z. Y. Xu, “Quasi-cw ultraviolet generation in a dual-periodic LiTaO3 superlattice by frequency tripling,” Jpn. J. Appl. Phys. 40, 6841–6844 (2001).
[CrossRef]

Lin, Y. Y.

Y. H. Chen, F. C. Fan, Y. Y. Lin, Y. C. Huang, J. T. Shy, Y. P. Lan, Y. F. Chen, “Simultaneous amplitude modulation and wavelength conversion in an asymmetric-duty-cycle periodically poled lithium niobate,” Opt. Commun. 223, 417–423 (2003).
[CrossRef]

Liu, H.

Liu, Z. W.

G. Z. Luo, S. N. Zhu, J. L. He, Y. Y. Zhu, H. T. Wang, Z. W. Liu, C. Zhang, N. B. Ming, “Simultaneously efficient blue and red light generations in a periodically poled LiTaO3,” Appl. Phys. Lett. 78, 3006–3008 (2001).
[CrossRef]

Z. W. Liu, S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, J. L. He, C. Zhang, H. T. Wang, N. B. Ming, X. Y. Liang, Z. Y. Xu, “Quasi-cw ultraviolet generation in a dual-periodic LiTaO3 superlattice by frequency tripling,” Jpn. J. Appl. Phys. 40, 6841–6844 (2001).
[CrossRef]

Lu, Y. L.

Y. Q. Lu, J. J. Zheng, Y. L. Lu, N. B. Ming, “Frequency tuning of optical parametric generator in periodically poled optical superlattice LiNbO3 by electro-optic effect,” Appl. Phys. Lett. 74, 123–125 (1999).
[CrossRef]

Lu, Y. Q.

Y. Q. Lu, M. Xiao, G. J. Salamo, “Wide-bandwidth high-frequency electro-optic modulator based on periodically poled LiNbO3,” Appl. Phys. Lett. 78, 1035–1037 (2001).
[CrossRef]

Y. Q. Lu, Z. L. Wang, Q. Wang, Y. X. Xi, N. B. Ming, “Electro-optic effect of periodically poled optical superlattice LiNbO3 and its applications,” Appl. Phys. Lett. 77, 3719–3721 (2000).
[CrossRef]

Y. Q. Lu, J. J. Zheng, Y. L. Lu, N. B. Ming, “Frequency tuning of optical parametric generator in periodically poled optical superlattice LiNbO3 by electro-optic effect,” Appl. Phys. Lett. 74, 123–125 (1999).
[CrossRef]

Luo, G. Z.

G. Z. Luo, S. N. Zhu, J. L. He, Y. Y. Zhu, H. T. Wang, Z. W. Liu, C. Zhang, N. B. Ming, “Simultaneously efficient blue and red light generations in a periodically poled LiTaO3,” Appl. Phys. Lett. 78, 3006–3008 (2001).
[CrossRef]

Magel, G. A.

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

Meyn, J. P.

Ming, N. B.

P. Xu, S. H. Ji, S. N. Zhu, X. Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, N. B. Ming, “Conical second harmonic generation in a two-dimensional χ(2) photonic crystal: a hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. 93, 133904 (2004).
[CrossRef]

G. Z. Luo, S. N. Zhu, J. L. He, Y. Y. Zhu, H. T. Wang, Z. W. Liu, C. Zhang, N. B. Ming, “Simultaneously efficient blue and red light generations in a periodically poled LiTaO3,” Appl. Phys. Lett. 78, 3006–3008 (2001).
[CrossRef]

C. Zhang, H. Wei, Y. Y. Zhu, H. T. Wang, S. N. Zhu, N. B. Ming, “Third-harmonic generation in a general two-component quasi-periodic optical superlattice,” Opt. Lett. 26, 899–901 (2001).
[CrossRef]

Z. W. Liu, S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, J. L. He, C. Zhang, H. T. Wang, N. B. Ming, X. Y. Liang, Z. Y. Xu, “Quasi-cw ultraviolet generation in a dual-periodic LiTaO3 superlattice by frequency tripling,” Jpn. J. Appl. Phys. 40, 6841–6844 (2001).
[CrossRef]

Y. Q. Lu, Z. L. Wang, Q. Wang, Y. X. Xi, N. B. Ming, “Electro-optic effect of periodically poled optical superlattice LiNbO3 and its applications,” Appl. Phys. Lett. 77, 3719–3721 (2000).
[CrossRef]

C. Zhang, Y. Y. Zhu, S. X. Yang, Y. Q. Qin, S. N. Zhu, Y. B. Chen, H. Liu, N. B. Ming, “Crucial effects of coupling coefficients on quasi-phase-matched harmonic generation in an optical superlattice,” Opt. Lett. 25, 436–438 (2000).
[CrossRef]

Y. Y. Zhu, N. B. Ming, “Dielectric superlattices for nonlinear optical effects,” Opt. Quantum Electron. 31, 1093–1128 (1999).
[CrossRef]

Y. Q. Lu, J. J. Zheng, Y. L. Lu, N. B. Ming, “Frequency tuning of optical parametric generator in periodically poled optical superlattice LiNbO3 by electro-optic effect,” Appl. Phys. Lett. 74, 123–125 (1999).
[CrossRef]

S. N. Zhu, Y. Y. Zhu, N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278, 843–846 (1997).
[CrossRef]

S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, H. F. Wang, C. Z. Ge, N. B. Ming, “Experimental realization of second harmonic generation in a Fibonacci optical superlattice of LiTaO3,” Phys. Rev. Lett. 78, 2752–2755 (1997).
[CrossRef]

J. Feng, Y. Y. Zhu, N. B. Ming, “Harmonic generations in an optical Fibonacci superlattice,” Phys. Rev. B 41, 5578–5582 (1990).
[CrossRef]

D. Feng, N. B. Ming, J. F. Hong, Y. S. Yang, J. S. Zhu, Z. Yang, Y. N. Wang, “Enhancement of second-harmonic generation in LiNbO3 crystals with periodic laminar ferroelectric domains,” Appl. Phys. Lett. 37, 607–609 (1980).
[CrossRef]

Missey, M.

O’Brien, N.

Powers, P.

Qin, Y. Q.

Z. W. Liu, S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, J. L. He, C. Zhang, H. T. Wang, N. B. Ming, X. Y. Liang, Z. Y. Xu, “Quasi-cw ultraviolet generation in a dual-periodic LiTaO3 superlattice by frequency tripling,” Jpn. J. Appl. Phys. 40, 6841–6844 (2001).
[CrossRef]

C. Zhang, Y. Y. Zhu, S. X. Yang, Y. Q. Qin, S. N. Zhu, Y. B. Chen, H. Liu, N. B. Ming, “Crucial effects of coupling coefficients on quasi-phase-matched harmonic generation in an optical superlattice,” Opt. Lett. 25, 436–438 (2000).
[CrossRef]

S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, H. F. Wang, C. Z. Ge, N. B. Ming, “Experimental realization of second harmonic generation in a Fibonacci optical superlattice of LiTaO3,” Phys. Rev. Lett. 78, 2752–2755 (1997).
[CrossRef]

Robinson, J. M.

Rosenman, G.

K. K. Fradkin, A. Arie, P. Urenski, G. Rosenman, “Multiple nonlinear optical interactions with arbitrary wave vector differences,” Phys. Rev. Lett. 88, 023903 (2002).
[CrossRef]

Salamo, G. J.

Y. Q. Lu, M. Xiao, G. J. Salamo, “Wide-bandwidth high-frequency electro-optic modulator based on periodically poled LiNbO3,” Appl. Phys. Lett. 78, 1035–1037 (2001).
[CrossRef]

Schepler, K. L.

Scrymgeour, D. A.

D. A. Scrymgeour, A. Sharan, V. Gopalan, K. T. Gahagan, J. L. Casson, “Cascaded electro-optic scanning of laser light over large angles using domain microengineered ferroelectrics,” Appl. Phys. Lett. 81, 3140–3142 (2002).
[CrossRef]

K. T. Gahagan, D. A. Scrymgeour, J. L. Casson, V. Gopalan, J. M. Robinson, “Integrated high-power electro-optic lens and large-angle deflector,” Appl. Opt. 40, 5638–5642 (2001).
[CrossRef]

Sharan, A.

D. A. Scrymgeour, A. Sharan, V. Gopalan, K. T. Gahagan, J. L. Casson, “Cascaded electro-optic scanning of laser light over large angles using domain microengineered ferroelectrics,” Appl. Phys. Lett. 81, 3140–3142 (2002).
[CrossRef]

Shy, J. T.

Y. H. Chen, F. C. Fan, Y. Y. Lin, Y. C. Huang, J. T. Shy, Y. P. Lan, Y. F. Chen, “Simultaneous amplitude modulation and wavelength conversion in an asymmetric-duty-cycle periodically poled lithium niobate,” Opt. Commun. 223, 417–423 (2003).
[CrossRef]

Sun, J.

P. Xu, S. H. Ji, S. N. Zhu, X. Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, N. B. Ming, “Conical second harmonic generation in a two-dimensional χ(2) photonic crystal: a hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. 93, 133904 (2004).
[CrossRef]

Urenski, P.

K. K. Fradkin, A. Arie, P. Urenski, G. Rosenman, “Multiple nonlinear optical interactions with arbitrary wave vector differences,” Phys. Rev. Lett. 88, 023903 (2002).
[CrossRef]

Wang, H. F.

S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, H. F. Wang, C. Z. Ge, N. B. Ming, “Experimental realization of second harmonic generation in a Fibonacci optical superlattice of LiTaO3,” Phys. Rev. Lett. 78, 2752–2755 (1997).
[CrossRef]

Wang, H. T.

P. Xu, S. H. Ji, S. N. Zhu, X. Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, N. B. Ming, “Conical second harmonic generation in a two-dimensional χ(2) photonic crystal: a hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. 93, 133904 (2004).
[CrossRef]

G. Z. Luo, S. N. Zhu, J. L. He, Y. Y. Zhu, H. T. Wang, Z. W. Liu, C. Zhang, N. B. Ming, “Simultaneously efficient blue and red light generations in a periodically poled LiTaO3,” Appl. Phys. Lett. 78, 3006–3008 (2001).
[CrossRef]

C. Zhang, H. Wei, Y. Y. Zhu, H. T. Wang, S. N. Zhu, N. B. Ming, “Third-harmonic generation in a general two-component quasi-periodic optical superlattice,” Opt. Lett. 26, 899–901 (2001).
[CrossRef]

Z. W. Liu, S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, J. L. He, C. Zhang, H. T. Wang, N. B. Ming, X. Y. Liang, Z. Y. Xu, “Quasi-cw ultraviolet generation in a dual-periodic LiTaO3 superlattice by frequency tripling,” Jpn. J. Appl. Phys. 40, 6841–6844 (2001).
[CrossRef]

Wang, Q.

Y. Q. Lu, Z. L. Wang, Q. Wang, Y. X. Xi, N. B. Ming, “Electro-optic effect of periodically poled optical superlattice LiNbO3 and its applications,” Appl. Phys. Lett. 77, 3719–3721 (2000).
[CrossRef]

Wang, Y. N.

D. Feng, N. B. Ming, J. F. Hong, Y. S. Yang, J. S. Zhu, Z. Yang, Y. N. Wang, “Enhancement of second-harmonic generation in LiNbO3 crystals with periodic laminar ferroelectric domains,” Appl. Phys. Lett. 37, 607–609 (1980).
[CrossRef]

Wang, Z. L.

Y. Q. Lu, Z. L. Wang, Q. Wang, Y. X. Xi, N. B. Ming, “Electro-optic effect of periodically poled optical superlattice LiNbO3 and its applications,” Appl. Phys. Lett. 77, 3719–3721 (2000).
[CrossRef]

Wei, H.

Xi, Y. X.

Y. Q. Lu, Z. L. Wang, Q. Wang, Y. X. Xi, N. B. Ming, “Electro-optic effect of periodically poled optical superlattice LiNbO3 and its applications,” Appl. Phys. Lett. 77, 3719–3721 (2000).
[CrossRef]

Xiao, M.

Y. Q. Lu, M. Xiao, G. J. Salamo, “Wide-bandwidth high-frequency electro-optic modulator based on periodically poled LiNbO3,” Appl. Phys. Lett. 78, 1035–1037 (2001).
[CrossRef]

Xu, P.

P. Xu, S. H. Ji, S. N. Zhu, X. Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, N. B. Ming, “Conical second harmonic generation in a two-dimensional χ(2) photonic crystal: a hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. 93, 133904 (2004).
[CrossRef]

Xu, Z. Y.

Z. W. Liu, S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, J. L. He, C. Zhang, H. T. Wang, N. B. Ming, X. Y. Liang, Z. Y. Xu, “Quasi-cw ultraviolet generation in a dual-periodic LiTaO3 superlattice by frequency tripling,” Jpn. J. Appl. Phys. 40, 6841–6844 (2001).
[CrossRef]

Yang, S. X.

Yang, Y. S.

D. Feng, N. B. Ming, J. F. Hong, Y. S. Yang, J. S. Zhu, Z. Yang, Y. N. Wang, “Enhancement of second-harmonic generation in LiNbO3 crystals with periodic laminar ferroelectric domains,” Appl. Phys. Lett. 37, 607–609 (1980).
[CrossRef]

Yang, Z.

D. Feng, N. B. Ming, J. F. Hong, Y. S. Yang, J. S. Zhu, Z. Yang, Y. N. Wang, “Enhancement of second-harmonic generation in LiNbO3 crystals with periodic laminar ferroelectric domains,” Appl. Phys. Lett. 37, 607–609 (1980).
[CrossRef]

Yu, X. Q.

P. Xu, S. H. Ji, S. N. Zhu, X. Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, N. B. Ming, “Conical second harmonic generation in a two-dimensional χ(2) photonic crystal: a hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. 93, 133904 (2004).
[CrossRef]

Zhang, C.

G. Z. Luo, S. N. Zhu, J. L. He, Y. Y. Zhu, H. T. Wang, Z. W. Liu, C. Zhang, N. B. Ming, “Simultaneously efficient blue and red light generations in a periodically poled LiTaO3,” Appl. Phys. Lett. 78, 3006–3008 (2001).
[CrossRef]

C. Zhang, H. Wei, Y. Y. Zhu, H. T. Wang, S. N. Zhu, N. B. Ming, “Third-harmonic generation in a general two-component quasi-periodic optical superlattice,” Opt. Lett. 26, 899–901 (2001).
[CrossRef]

Z. W. Liu, S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, J. L. He, C. Zhang, H. T. Wang, N. B. Ming, X. Y. Liang, Z. Y. Xu, “Quasi-cw ultraviolet generation in a dual-periodic LiTaO3 superlattice by frequency tripling,” Jpn. J. Appl. Phys. 40, 6841–6844 (2001).
[CrossRef]

C. Zhang, Y. Y. Zhu, S. X. Yang, Y. Q. Qin, S. N. Zhu, Y. B. Chen, H. Liu, N. B. Ming, “Crucial effects of coupling coefficients on quasi-phase-matched harmonic generation in an optical superlattice,” Opt. Lett. 25, 436–438 (2000).
[CrossRef]

Zheng, J. J.

Y. Q. Lu, J. J. Zheng, Y. L. Lu, N. B. Ming, “Frequency tuning of optical parametric generator in periodically poled optical superlattice LiNbO3 by electro-optic effect,” Appl. Phys. Lett. 74, 123–125 (1999).
[CrossRef]

Zhu, J. S.

D. Feng, N. B. Ming, J. F. Hong, Y. S. Yang, J. S. Zhu, Z. Yang, Y. N. Wang, “Enhancement of second-harmonic generation in LiNbO3 crystals with periodic laminar ferroelectric domains,” Appl. Phys. Lett. 37, 607–609 (1980).
[CrossRef]

Zhu, S. N.

P. Xu, S. H. Ji, S. N. Zhu, X. Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, N. B. Ming, “Conical second harmonic generation in a two-dimensional χ(2) photonic crystal: a hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. 93, 133904 (2004).
[CrossRef]

G. Z. Luo, S. N. Zhu, J. L. He, Y. Y. Zhu, H. T. Wang, Z. W. Liu, C. Zhang, N. B. Ming, “Simultaneously efficient blue and red light generations in a periodically poled LiTaO3,” Appl. Phys. Lett. 78, 3006–3008 (2001).
[CrossRef]

C. Zhang, H. Wei, Y. Y. Zhu, H. T. Wang, S. N. Zhu, N. B. Ming, “Third-harmonic generation in a general two-component quasi-periodic optical superlattice,” Opt. Lett. 26, 899–901 (2001).
[CrossRef]

Z. W. Liu, S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, J. L. He, C. Zhang, H. T. Wang, N. B. Ming, X. Y. Liang, Z. Y. Xu, “Quasi-cw ultraviolet generation in a dual-periodic LiTaO3 superlattice by frequency tripling,” Jpn. J. Appl. Phys. 40, 6841–6844 (2001).
[CrossRef]

C. Zhang, Y. Y. Zhu, S. X. Yang, Y. Q. Qin, S. N. Zhu, Y. B. Chen, H. Liu, N. B. Ming, “Crucial effects of coupling coefficients on quasi-phase-matched harmonic generation in an optical superlattice,” Opt. Lett. 25, 436–438 (2000).
[CrossRef]

S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, H. F. Wang, C. Z. Ge, N. B. Ming, “Experimental realization of second harmonic generation in a Fibonacci optical superlattice of LiTaO3,” Phys. Rev. Lett. 78, 2752–2755 (1997).
[CrossRef]

S. N. Zhu, Y. Y. Zhu, N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278, 843–846 (1997).
[CrossRef]

Zhu, Y. Y.

P. Xu, S. H. Ji, S. N. Zhu, X. Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, N. B. Ming, “Conical second harmonic generation in a two-dimensional χ(2) photonic crystal: a hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. 93, 133904 (2004).
[CrossRef]

G. Z. Luo, S. N. Zhu, J. L. He, Y. Y. Zhu, H. T. Wang, Z. W. Liu, C. Zhang, N. B. Ming, “Simultaneously efficient blue and red light generations in a periodically poled LiTaO3,” Appl. Phys. Lett. 78, 3006–3008 (2001).
[CrossRef]

C. Zhang, H. Wei, Y. Y. Zhu, H. T. Wang, S. N. Zhu, N. B. Ming, “Third-harmonic generation in a general two-component quasi-periodic optical superlattice,” Opt. Lett. 26, 899–901 (2001).
[CrossRef]

Z. W. Liu, S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, J. L. He, C. Zhang, H. T. Wang, N. B. Ming, X. Y. Liang, Z. Y. Xu, “Quasi-cw ultraviolet generation in a dual-periodic LiTaO3 superlattice by frequency tripling,” Jpn. J. Appl. Phys. 40, 6841–6844 (2001).
[CrossRef]

C. Zhang, Y. Y. Zhu, S. X. Yang, Y. Q. Qin, S. N. Zhu, Y. B. Chen, H. Liu, N. B. Ming, “Crucial effects of coupling coefficients on quasi-phase-matched harmonic generation in an optical superlattice,” Opt. Lett. 25, 436–438 (2000).
[CrossRef]

Y. Y. Zhu, N. B. Ming, “Dielectric superlattices for nonlinear optical effects,” Opt. Quantum Electron. 31, 1093–1128 (1999).
[CrossRef]

S. N. Zhu, Y. Y. Zhu, N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278, 843–846 (1997).
[CrossRef]

S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, H. F. Wang, C. Z. Ge, N. B. Ming, “Experimental realization of second harmonic generation in a Fibonacci optical superlattice of LiTaO3,” Phys. Rev. Lett. 78, 2752–2755 (1997).
[CrossRef]

J. Feng, Y. Y. Zhu, N. B. Ming, “Harmonic generations in an optical Fibonacci superlattice,” Phys. Rev. B 41, 5578–5582 (1990).
[CrossRef]

Zia, R. K. P.

R. K. P. Zia, W. J. Dallas, “A simple derivation of quasi-crystalline spectra,” J. Phys. A 18, L341–L345 (1985).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

D. Feng, N. B. Ming, J. F. Hong, Y. S. Yang, J. S. Zhu, Z. Yang, Y. N. Wang, “Enhancement of second-harmonic generation in LiNbO3 crystals with periodic laminar ferroelectric domains,” Appl. Phys. Lett. 37, 607–609 (1980).
[CrossRef]

G. Z. Luo, S. N. Zhu, J. L. He, Y. Y. Zhu, H. T. Wang, Z. W. Liu, C. Zhang, N. B. Ming, “Simultaneously efficient blue and red light generations in a periodically poled LiTaO3,” Appl. Phys. Lett. 78, 3006–3008 (2001).
[CrossRef]

Y. Q. Lu, Z. L. Wang, Q. Wang, Y. X. Xi, N. B. Ming, “Electro-optic effect of periodically poled optical superlattice LiNbO3 and its applications,” Appl. Phys. Lett. 77, 3719–3721 (2000).
[CrossRef]

Y. Q. Lu, M. Xiao, G. J. Salamo, “Wide-bandwidth high-frequency electro-optic modulator based on periodically poled LiNbO3,” Appl. Phys. Lett. 78, 1035–1037 (2001).
[CrossRef]

D. A. Scrymgeour, A. Sharan, V. Gopalan, K. T. Gahagan, J. L. Casson, “Cascaded electro-optic scanning of laser light over large angles using domain microengineered ferroelectrics,” Appl. Phys. Lett. 81, 3140–3142 (2002).
[CrossRef]

Y. Q. Lu, J. J. Zheng, Y. L. Lu, N. B. Ming, “Frequency tuning of optical parametric generator in periodically poled optical superlattice LiNbO3 by electro-optic effect,” Appl. Phys. Lett. 74, 123–125 (1999).
[CrossRef]

IEEE J. Quantum Electron.

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

J. Nonlinear Opt. Phys. Mater.

R. L. Byer, “Quasi-phase-matched nonlinear interactions and devices,” J. Nonlinear Opt. Phys. Mater. 6, 549–592 (1997).
[CrossRef]

J. Phys. A

R. K. P. Zia, W. J. Dallas, “A simple derivation of quasi-crystalline spectra,” J. Phys. A 18, L341–L345 (1985).
[CrossRef]

Jpn. J. Appl. Phys.

Z. W. Liu, S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, J. L. He, C. Zhang, H. T. Wang, N. B. Ming, X. Y. Liang, Z. Y. Xu, “Quasi-cw ultraviolet generation in a dual-periodic LiTaO3 superlattice by frequency tripling,” Jpn. J. Appl. Phys. 40, 6841–6844 (2001).
[CrossRef]

Opt. Commun.

Y. H. Chen, F. C. Fan, Y. Y. Lin, Y. C. Huang, J. T. Shy, Y. P. Lan, Y. F. Chen, “Simultaneous amplitude modulation and wavelength conversion in an asymmetric-duty-cycle periodically poled lithium niobate,” Opt. Commun. 223, 417–423 (2003).
[CrossRef]

Opt. Lett.

Opt. Quantum Electron.

Y. Y. Zhu, N. B. Ming, “Dielectric superlattices for nonlinear optical effects,” Opt. Quantum Electron. 31, 1093–1128 (1999).
[CrossRef]

Phys. Rev. B

J. Feng, Y. Y. Zhu, N. B. Ming, “Harmonic generations in an optical Fibonacci superlattice,” Phys. Rev. B 41, 5578–5582 (1990).
[CrossRef]

Phys. Rev. Lett.

K. K. Fradkin, A. Arie, P. Urenski, G. Rosenman, “Multiple nonlinear optical interactions with arbitrary wave vector differences,” Phys. Rev. Lett. 88, 023903 (2002).
[CrossRef]

P. Xu, S. H. Ji, S. N. Zhu, X. Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, N. B. Ming, “Conical second harmonic generation in a two-dimensional χ(2) photonic crystal: a hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. 93, 133904 (2004).
[CrossRef]

S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, H. F. Wang, C. Z. Ge, N. B. Ming, “Experimental realization of second harmonic generation in a Fibonacci optical superlattice of LiTaO3,” Phys. Rev. Lett. 78, 2752–2755 (1997).
[CrossRef]

Science

S. N. Zhu, Y. Y. Zhu, N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278, 843–846 (1997).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Dependence of conversion efficiencies of the pump (dotted curves), the SH (dashed curves), and the TH (solid curves) on the dc electric field for (a) sample I and (b) sample II. The total length of each sample is 20 mm, and the pump intensity is 200 MW/cm2.

Fig. 2
Fig. 2

Variation of the efficiency of TH (solid curves) and the phase difference Δφ (dashed curves) with crystal length under the action of a dc electric field (sample I). (a), (b), and (c) correspond to Ed = 0, 94, and 200 V/mm, respectively. The pump intensity is 200 MW/cm2.

Fig. 3
Fig. 3

Dependence of the improved efficiency of the TH (filled circles) and the required dc electric field (filled triangles) on the pump intensity. The efficiency of the TH in the absence of the dc electric field is also included (open circles).

Fig. 4
Fig. 4

Oscillation of r(x) with crystal length. When the length is long enough, r(x) approaches −0.50 in (a) sample I and −0.19 in (b) sample II.

Equations (8)

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

d 33 ( x ) = d 33 f ( x ) ,
n ( x ) = n 0 - ½ n 0 3 γ 33 E d f ( x ) ,
f ( x ) = { + 1 ( in positive domains ) - 1 ( in negative domains ) .
d E 1 d x = - i ω 1 d 33 ( x ) c n 1 ( x ) { E 2 E 1 * exp [ - i 0 x Δ k a ( x ) d x ] + E 3 E 2 * exp [ - i 0 x Δ k b ( x ) d x ] } , d E 2 d x = - i ω 2 d 33 ( x ) c n 2 ( x ) { E 1 2 2 exp [ i 0 x Δ k a ( x ) d x ] + E 3 E 1 * exp [ - i 0 x Δ k b ( x ) d x ] } , d E 3 d x = - i ω 3 d 33 ( x ) c n 3 ( x ) { E 1 E 2 exp [ i 0 x Δ k b ( x ) d x ] } ,
Δ k a ( x ) = k 2 ( x ) - 2 k 1 ( x ) ,             Δ k b ( x ) = k 3 ( x ) - k 2 ( x ) - k 1 ( x ) .
d E 1 d x = - i κ 1 a E 2 E 1 exp [ i β a 0 x f ( x ) d x ] - i κ 1 b E 3 E 2 exp [ i β b 0 x f ( x ) d x ] , d E 2 d x = - i κ 2 a E 1 2 2 exp [ - i β a 0 x f ( x ) d x ] - i κ 2 b E 3 E 1 * exp [ i β b 0 x f ( x ) d x ] , d E 3 d x = - i κ 3 b E 1 E 2 exp [ - i β b 0 x f ( x ) d x ] ,
κ μ ν = d 33 ω μ f ν c n μ ,             ( μ = 1 , 2 , 3 ; ν = a , b ) , β a = 2 π λ 1 ( n 20 3 - n 10 3 ) γ 33 E d , β b = π λ 1 ( 3 n 30 3 - 2 n 20 3 - n 10 3 ) γ 3 E d ,
r ( x ) = ( 1 / x ) 0 x f ( x ) d x .

Metrics