Abstract

We present here a possible high-flux photon-pair source constructed by single lithium niobate optical superlattice (OSL) with a combined quasi-periodically and periodically poled structure, which is from the principle of electrically induced parametric down conversion (PDC) after second-harmonic generation (SHG), predicted by the united theory developed in this paper, in which SHG, PDC and electro-optic (EO) effect are comparably treated as two-order nonlinear effects. In the OSL, the e-polarized fundamental frequency photons are first converted to double frequency ones with the same polarization; then the PDC process is triggered by EO effect when the fundamental frequency photons are almost exhausted; finally, the double frequency photons are converted again to a series of two-photon pair of fundamental wave. It is demonstrated that at 100 °C, in a 20.2mm long OSL with a 30V / mm applied electric field, a 100MW/cm2, 1080 nm laser beam can be translated to a flux of high-purity two-photon pairs with a conversion efficiency close to 100%; and for a longer OSL the pump intensity can be further lowered. The device can also act as an ultra-low field electro-optic switch.

© 2007 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  3. C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, "Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels," Phys. Rev. Lett. 70, 1895 (1993).
    [CrossRef] [PubMed]
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  5. A. Barenco, D. Deutsch, A. Ekert, and R. Jozsa, "Conditional quantum dynamics and logic gates," Phys. Rev. Lett. 74, 4083 (1995).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  7. Z. Y. Ou, L. Mandel, "Violation of Bell’s inequality and classical probability in a two-photon correlation experiment," Phys. Rev. Lett. 61, 50 (1988).
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    [CrossRef] [PubMed]
  11. P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, "Ultrabright source of polarization-entangled photons," Phys. Rev. A 60, R773 (1999).
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  13. C. E. Kuklewicz, M. Fiorentino, G. Messin, F. N. C. Wong, and J. H. Shapiro, "High-flux source of polarization-entangled photons from a periodically poled KTiOPO4 parametric down-converter," Phys. Rev. A 69, 013807 (2004).
    [CrossRef]
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    [CrossRef]
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  17. J. Jing, J. Zhang, Y. Yan, F. Zhao, C. Xie, K. Peng, "Experimental demonstration of tripartite entanglement and controlled dense coding for continuous variables," Phys. Rev. Lett. 90, 167903 (2003).
    [CrossRef] [PubMed]
  18. A. Arie, G. Rosenman, V. Mahal, A. Skliar, M. Oron, M. Katz, and D. Eger, "Green and ultraviolet quasi-phase-matched second harmonic generation in bulk periodically-poled KTiOPO4," Opt. Commun. 142, 265 (1997).
    [CrossRef]
  19. G. D. Miller, R. G. Batchko, W. M. Tulloch, D. R. Weise, M. M. Fejer, and R. L. Byer, "42%-efficient single-pass cw second-harmonic generation in periodically poled lithium niobate," Opt. Lett. 22, 1834 (1997).
    [CrossRef]
  20. S. Wang, V. Pasiskevicius, F. Laurell, and H. Karlsson, "Ultraviolet generation by first-order frequency doubling in periodically poled KTiOPO4," Opt. Lett. 23, 1883 (1998).
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  21. I. Yokohama, M. Asobe, A. Yokoo, H. Itoh, and T. Kaino, "All-optical switching by use of cascading of phase-matched sum-frequency generation and difference-frequency generation processes," J. Opt. Soc. Am. B 14, 3368 (1997).
    [CrossRef]
  22. L. E. Myers and W. R. Bosenberg, "Periodically poled lithium niobate and quasi-phase-matched optical parametric oscillators," IEEE J. Quantum Electron. 33, 1663 (1997).
    [CrossRef]
  23. K. El Hadi, M. Sundheimer, P. Aschieri, P. Baldi, M. P. De Micheli, D. B. Ostrowsky, F. Laurell, "Quasi-phase-matched parametric interactions in proton-exchanged lithium niobate waveguides," J. Opt. Soc. Am. B 14, 3197 (1997).
    [CrossRef]
  24. M. Fujimura, T. Suhara, and H. Nishihara, "Periodically domain-inverted LiNbO3 for waveguide auasi-phase-matched nonlinear optic wavelength conversion devices," Bull. Mater. Sci. 22, 413 (1999).
    [CrossRef]
  25. S. Zhu, Y. Zhu, and N. Ming, "Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice," Science 278, 843 (1997).
    [CrossRef]
  26. C. Zhang, H. Wei, Y. Zhu, H. Wang, S. Zhu and N. Ming, "Third-harmonic generation in a general two-component quasi-periodic optical superlattice," Opt. Lett. 26, 899 (2001).
    [CrossRef]
  27. C. J. K. Virmani, Plasma Phys. 15, 1039 (1973).
    [CrossRef]
  28. G. Blau, M. Cairone, P. A. Chollet, F. Kajzar, "Electro-optic modulation and second-harmonic generation through grating-induced resonant excitation of guided modes," Proc. SPIE 2852, 237 (1996).
    [CrossRef]
  29. N. O’Brien, M. Missey, P. Powers, V. Dominic, and K. L. Schepler, "Electro-optical spectral tuning in a continuous-wave, asymmetric-duty-cycle, periodically poled LiNbO3 optical parametric oscillator," Opt. Lett. 24, 1750 (1999).
    [CrossRef]
  30. K. Chang, A. Chiang, T. Lin, B. Wong, Y. Chen, and Y. Huang, "Simultaneous wavelength conversion and amplitude modulation in a monolithic periodically-poled lithium niobate," Opt. Commun. 203, 163 (2002).
    [CrossRef]
  31. Y. Chen, F. Fan, Y. Lin, Y. Huang, J. Shy, Y. Lan, and Y. Chen, "Simultaneous amplitude modulation and wavelength conversion in an asymmetric-duty-cycle periodically poled lithium niobate," Opt. Commun. 223, 417 (2003).
    [CrossRef]
  32. F. Xu, J. Liao, X. Zhang, J. He, H. Wang, N. Ming, "Complete conversion of sum-frequency generation enhanced by controllable linear gratings induced by an electro-optic effect in a periodic optical superlattice," Phys. Rev. A 68, 033808 (2003).
    [CrossRef]
  33. F. Xu, J. Liao, C. Guo, J. He, H. Wang, S. Zhu, Z. Wang, Y. Zhu, N. Ming, "Highly efficient direct third-harmonic generation based on control of the electro-optic effect in quasi-periodic optical superlattices," Opt. Lett. 28, 429 (2003).
    [CrossRef] [PubMed]
  34. C. Huang, Q. Wang, Y. Zhu, "Cascaded frequency doubling and electro-optic coupling in a single optical superlattice," Appl. Phys. B 80, 741 (2005).
    [CrossRef]
  35. S. Tanzilli, H. De Riedmatten, W. Tittel, H. Zbinden, P. Baldi, M. De Micheli, D. B. Ostrowski, and N. Gisin, "Highly efficient photo-pair source using periodically poled lithium niobate waveguide," Electron. Lett. 37, 26 (2001).
    [CrossRef]
  36. W. She, W. Lee, "Wave coupling theory of linear electrooptic effect," Opt. Commun. 195, 303 (2001).
    [CrossRef]
  37. G. Zheng, H. Wang, W. She, "Wave coupling theory of quasi-phase-matched linear electro-optic effect," Opt. Exp. 14, 5535 (2006).
    [CrossRef]
  38. Y. Lu, Z. Wan, Q. Wang, Y. Xi, and N. Ming, "Electro-optic effect of periodically poled optical superlattice LiNbO3 and its applications," Appl. Phys. Lett. 77, 3719 (2000).
    [CrossRef]
  39. C. Huang, Y. Wang, Y. Zhu, "Effect of electro-optic modulation on coupled quasi-phase-matched frequency conversion," Appl. Opt. 44, 4980 (2005).
    [CrossRef] [PubMed]
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  41. G. Luo, S. Zhu, J. He, Y. Zhu, H. Wang, Z. Liu, C. Zhang, and N. Ming, "Simultaneously efficient blue and red light generations in a periodically poled LiTaO3," Appl. Phys. Lett. 78, 3006 (2001).
    [CrossRef]
  42. Keren Fradkin-Kashi and Ady Arie, "Multiple-wavelength quasi-phase-matched nonlinear interactions," IEEE J. Quantum Electron. 35, 1649 (1999).
    [CrossRef]
  43. G. J.  Edwards and M.  Lawrence, "A temperature-dependent dispersion equation for congruently grown lithium niobate," Opt. and Quant. Electron. 16, 373 (1984).
    [CrossRef]
  44. L. Mandel and E. Wolf, Optical coherence and quantum optics (Cambridge University Press, 1995) p.1071.
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    [CrossRef]
  46. Y. H. Chen and Y. C. Huang, "Actively Q-switched Nd:YVO4 laser using an electro-optic periodically poled lithium niobate crystal as a laser Q-switch," Opt. Lett. 28, 1460 (2003).
    [CrossRef] [PubMed]
  47. K. S. Abedin, T. Tsuritani, M. Sato, H. Ito, K. Shimamura, and T. Fukuda, "Integrated electro-optic Q switching in a domain-inverted Nd:LiTaO3," Opt. Lett. 20, 1985 (1995).
    [CrossRef] [PubMed]

2006

G. Zheng, H. Wang, W. She, "Wave coupling theory of quasi-phase-matched linear electro-optic effect," Opt. Exp. 14, 5535 (2006).
[CrossRef]

2005

C. Huang, Q. Wang, Y. Zhu, "Cascaded frequency doubling and electro-optic coupling in a single optical superlattice," Appl. Phys. B 80, 741 (2005).
[CrossRef]

C. Huang, Y. Wang, Y. Zhu, "Effect of electro-optic modulation on coupled quasi-phase-matched frequency conversion," Appl. Opt. 44, 4980 (2005).
[CrossRef] [PubMed]

2004

C. E. Kuklewicz, M. Fiorentino, G. Messin, F. N. C. Wong, and J. H. Shapiro, "High-flux source of polarization-entangled photons from a periodically poled KTiOPO4 parametric down-converter," Phys. Rev. A 69, 013807 (2004).
[CrossRef]

M. Pelton, P. Marsden, D. Ljunggren, M. Tengner, A. Karlsson, A. Fragemann, C. Canalias, and F. Laurell, "Bright, single-spatial-mode source of frequency non-degenerate, polarization-entangled photon pairs using periodically poled KTP," Opt. Exp. 12, 3573 (2004).
[CrossRef]

M. Fiorentino, G. Messin, C. E. Kuklewicz, F. N. C. Wong, and J. H. Shapiro, "Generation of ultrabright tunable polarization entanglement without spatial, spectral, or temporal constraints," Phys. Rev. A 69, 041801(R) (2004).
[CrossRef]

2003

J. Jing, J. Zhang, Y. Yan, F. Zhao, C. Xie, K. Peng, "Experimental demonstration of tripartite entanglement and controlled dense coding for continuous variables," Phys. Rev. Lett. 90, 167903 (2003).
[CrossRef] [PubMed]

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

F. Xu, J. Liao, X. Zhang, J. He, H. Wang, N. Ming, "Complete conversion of sum-frequency generation enhanced by controllable linear gratings induced by an electro-optic effect in a periodic optical superlattice," Phys. Rev. A 68, 033808 (2003).
[CrossRef]

F. Xu, J. Liao, C. Guo, J. He, H. Wang, S. Zhu, Z. Wang, Y. Zhu, N. Ming, "Highly efficient direct third-harmonic generation based on control of the electro-optic effect in quasi-periodic optical superlattices," Opt. Lett. 28, 429 (2003).
[CrossRef] [PubMed]

Y. H. Chen and Y. C. Huang, "Actively Q-switched Nd:YVO4 laser using an electro-optic periodically poled lithium niobate crystal as a laser Q-switch," Opt. Lett. 28, 1460 (2003).
[CrossRef] [PubMed]

2002

K. Chang, A. Chiang, T. Lin, B. Wong, Y. Chen, and Y. Huang, "Simultaneous wavelength conversion and amplitude modulation in a monolithic periodically-poled lithium niobate," Opt. Commun. 203, 163 (2002).
[CrossRef]

X. Li, Q. Pan, J. Jing, J. Zhang, C. Xie, and K. Peng, "Quantum dense coding exploiting a bright Einstein-Podolsky-Rosen beam," Phys. Rev. Lett. 88, 047904 (2002).
[CrossRef] [PubMed]

2001

K. Banaszek, A. B. U’Ren, and I. A. Walmsley, "Generation of correlated photons in controlled spatial modes by downconversion in nonlinear waveguides," Opt. Lett. 26, 1367(2001).
[CrossRef]

S. Tanzilli, H. De Riedmatten, W. Tittel, H. Zbinden, P. Baldi, M. De Micheli, D. B. Ostrowski, and N. Gisin, "Highly efficient photo-pair source using periodically poled lithium niobate waveguide," Electron. Lett. 37, 26 (2001).
[CrossRef]

W. She, W. Lee, "Wave coupling theory of linear electrooptic effect," Opt. Commun. 195, 303 (2001).
[CrossRef]

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

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

2000

Y. Lu, Z. Wan, Q. Wang, Y. Xi, and N. Ming, "Electro-optic effect of periodically poled optical superlattice LiNbO3 and its applications," Appl. Phys. Lett. 77, 3719 (2000).
[CrossRef]

1999

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

M. Fujimura, T. Suhara, and H. Nishihara, "Periodically domain-inverted LiNbO3 for waveguide auasi-phase-matched nonlinear optic wavelength conversion devices," Bull. Mater. Sci. 22, 413 (1999).
[CrossRef]

P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, "Ultrabright source of polarization-entangled photons," Phys. Rev. A 60, R773 (1999).
[CrossRef]

Keren Fradkin-Kashi and Ady Arie, "Multiple-wavelength quasi-phase-matched nonlinear interactions," IEEE J. Quantum Electron. 35, 1649 (1999).
[CrossRef]

1998

W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, "Violation of Bell inequalities by photons more than 10 km apart, " Phys. Rev. Lett. 81, 3563 (1998).
[CrossRef]

F. De Martini, "Amplification of quantum entanglement," Phys. Rev. Lett. 81, 2842 (1998).
[CrossRef]

S. Wang, V. Pasiskevicius, F. Laurell, and H. Karlsson, "Ultraviolet generation by first-order frequency doubling in periodically poled KTiOPO4," Opt. Lett. 23, 1883 (1998).
[CrossRef]

1997

I. Yokohama, M. Asobe, A. Yokoo, H. Itoh, and T. Kaino, "All-optical switching by use of cascading of phase-matched sum-frequency generation and difference-frequency generation processes," J. Opt. Soc. Am. B 14, 3368 (1997).
[CrossRef]

L. E. Myers and W. R. Bosenberg, "Periodically poled lithium niobate and quasi-phase-matched optical parametric oscillators," IEEE J. Quantum Electron. 33, 1663 (1997).
[CrossRef]

K. El Hadi, M. Sundheimer, P. Aschieri, P. Baldi, M. P. De Micheli, D. B. Ostrowsky, F. Laurell, "Quasi-phase-matched parametric interactions in proton-exchanged lithium niobate waveguides," J. Opt. Soc. Am. B 14, 3197 (1997).
[CrossRef]

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

A. Arie, G. Rosenman, V. Mahal, A. Skliar, M. Oron, M. Katz, and D. Eger, "Green and ultraviolet quasi-phase-matched second harmonic generation in bulk periodically-poled KTiOPO4," Opt. Commun. 142, 265 (1997).
[CrossRef]

G. D. Miller, R. G. Batchko, W. M. Tulloch, D. R. Weise, M. M. Fejer, and R. L. Byer, "42%-efficient single-pass cw second-harmonic generation in periodically poled lithium niobate," Opt. Lett. 22, 1834 (1997).
[CrossRef]

1996

G. Blau, M. Cairone, P. A. Chollet, F. Kajzar, "Electro-optic modulation and second-harmonic generation through grating-induced resonant excitation of guided modes," Proc. SPIE 2852, 237 (1996).
[CrossRef]

1995

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, "New high-intensity source of polarization-entangled photon pairs," Phys. Rev. Lett. 75, 4337 (1995).
[CrossRef] [PubMed]

A. Barenco, D. Deutsch, A. Ekert, and R. Jozsa, "Conditional quantum dynamics and logic gates," Phys. Rev. Lett. 74, 4083 (1995).
[CrossRef] [PubMed]

K. S. Abedin, T. Tsuritani, M. Sato, H. Ito, K. Shimamura, and T. Fukuda, "Integrated electro-optic Q switching in a domain-inverted Nd:LiTaO3," Opt. Lett. 20, 1985 (1995).
[CrossRef] [PubMed]

1993

C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, "Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels," Phys. Rev. Lett. 70, 1895 (1993).
[CrossRef] [PubMed]

M. Zukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert, "Event-ready-detectors Bell experiment via entanglement swapping," Phys. Rev. Lett. 71, 4287 (1993).
[CrossRef] [PubMed]

1992

C. H. Bennett and S. J. Wiesner, "Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states," Phys. Rev. Lett. 69, 2881 (1992).
[CrossRef] [PubMed]

1991

A. K. Ekert, "Quantum cryptography based on Bell’s theorem," Phys. Rev. Lett. 67, 661 (1991).
[CrossRef] [PubMed]

1990

J. G. Rarity, P. R. Tapster, "Experimental violation of Bell’s inequality based on phase and momentum," Phys. Rev. Lett. 64, 2495 (1990).
[CrossRef] [PubMed]

1988

Y. H. Shih, C. O. Alley, "New type of Einstein-Podolsky-Rosen-Bohm experiment using pairs of light quanta produced by optical parametric down conversion," Phys. Rev. Lett. 61, 2921 (1988).
[CrossRef] [PubMed]

Z. Y. Ou, L. Mandel, "Violation of Bell’s inequality and classical probability in a two-photon correlation experiment," Phys. Rev. Lett. 61, 50 (1988).
[CrossRef] [PubMed]

1984

G. J.  Edwards and M.  Lawrence, "A temperature-dependent dispersion equation for congruently grown lithium niobate," Opt. and Quant. Electron. 16, 373 (1984).
[CrossRef]

1973

C. J. K. Virmani, Plasma Phys. 15, 1039 (1973).
[CrossRef]

Appl. Opt.

Appl. Phys. B

C. Huang, Q. Wang, Y. Zhu, "Cascaded frequency doubling and electro-optic coupling in a single optical superlattice," Appl. Phys. B 80, 741 (2005).
[CrossRef]

Appl. Phys. Lett.

Y. Lu, Z. Wan, Q. Wang, Y. Xi, and N. Ming, "Electro-optic effect of periodically poled optical superlattice LiNbO3 and its applications," Appl. Phys. Lett. 77, 3719 (2000).
[CrossRef]

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

Bull. Mater. Sci.

M. Fujimura, T. Suhara, and H. Nishihara, "Periodically domain-inverted LiNbO3 for waveguide auasi-phase-matched nonlinear optic wavelength conversion devices," Bull. Mater. Sci. 22, 413 (1999).
[CrossRef]

Electron. Lett.

S. Tanzilli, H. De Riedmatten, W. Tittel, H. Zbinden, P. Baldi, M. De Micheli, D. B. Ostrowski, and N. Gisin, "Highly efficient photo-pair source using periodically poled lithium niobate waveguide," Electron. Lett. 37, 26 (2001).
[CrossRef]

IEEE J. Quantum Electron.

L. E. Myers and W. R. Bosenberg, "Periodically poled lithium niobate and quasi-phase-matched optical parametric oscillators," IEEE J. Quantum Electron. 33, 1663 (1997).
[CrossRef]

Keren Fradkin-Kashi and Ady Arie, "Multiple-wavelength quasi-phase-matched nonlinear interactions," IEEE J. Quantum Electron. 35, 1649 (1999).
[CrossRef]

J. Opt. Soc. Am. B

Opt. and Quant. Electron.

G. J.  Edwards and M.  Lawrence, "A temperature-dependent dispersion equation for congruently grown lithium niobate," Opt. and Quant. Electron. 16, 373 (1984).
[CrossRef]

Opt. Commun.

W. She, W. Lee, "Wave coupling theory of linear electrooptic effect," Opt. Commun. 195, 303 (2001).
[CrossRef]

K. Chang, A. Chiang, T. Lin, B. Wong, Y. Chen, and Y. Huang, "Simultaneous wavelength conversion and amplitude modulation in a monolithic periodically-poled lithium niobate," Opt. Commun. 203, 163 (2002).
[CrossRef]

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

A. Arie, G. Rosenman, V. Mahal, A. Skliar, M. Oron, M. Katz, and D. Eger, "Green and ultraviolet quasi-phase-matched second harmonic generation in bulk periodically-poled KTiOPO4," Opt. Commun. 142, 265 (1997).
[CrossRef]

Opt. Exp.

M. Pelton, P. Marsden, D. Ljunggren, M. Tengner, A. Karlsson, A. Fragemann, C. Canalias, and F. Laurell, "Bright, single-spatial-mode source of frequency non-degenerate, polarization-entangled photon pairs using periodically poled KTP," Opt. Exp. 12, 3573 (2004).
[CrossRef]

G. Zheng, H. Wang, W. She, "Wave coupling theory of quasi-phase-matched linear electro-optic effect," Opt. Exp. 14, 5535 (2006).
[CrossRef]

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Figures (2)

Fig. 1.
Fig. 1.

Schematic diagram of the SHG, PDC in the OSL controlled by an applied electric field. x , y and z represent three axes of the crystal. The arrows indicate the polarized directions. F is a filter, only allowing e-polarized fundamental wave to pass. Section 1 and 2 represent QPPLN and PPLN, respectively.

Fig. 2.
Fig. 2.

Dependence of normalized intensities of e-polarized pump fundamental wave, e-polarized second harmonic and o-polarized fundamental wave respectively on the length of the crystal with different external electric field. The total length of the OSL is 20.2mm , and the pump intensity is 100MW / cm 2 . Red, green and brown lines represent the intensities of e-polarized pump wave, e-polarized second harmonic and o-polarized fundamental wave, respectively. And the normalized intensity of o-polarized fundamental wave is enlarged by 1000 times.

Equations (25)

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d E 1 y ( x ) dx = id 1 ( x ) E 1 z ( x ) e i Δ k a x id 2 ( x ) E 1 y ( x )
+ i ω 1 cn 1 y ( x ) [ d 22 ( x ) E 1 y * ( x ) E 2 y ( x ) e i Δ k c x + d 24 ( x ) E 1 y * ( x ) E 2 z ( x ) e i Δ k d x + d 24 ( x ) E 1 z * ( x ) E 2 y ( x ) e i Δ k e x ] ,
d E 1 z ( x ) dx = id 3 ( x ) E 1 y ( x ) e i Δ k a x id 4 ( x ) E 1 z ( x )
+ i ω 1 cn 1 z ( x ) [ d 32 ( x ) E 1 y * ( x ) E 2 y ( x ) e i Δ k e x + d 33 ( x ) E 1 z * ( x ) E 2 z ( x ) e i Δ k f x ] ,
d E 2 y ( x ) dx = 2 id 5 ( x ) E 2 z ( x ) e i Δ k b x 2 id 6 ( x ) E 2 y ( x )
+ i ω 2 cn 2 y ( x ) [ 1 2 d 22 ( x ) E 1 y ( x ) E 1 y ( x ) e i Δ k c x + d 24 ( x ) E 1 y ( x ) E 1 z ( x ) e i Δ k e x ] ,
d E 2 z ( x ) dx = 2 id 7 ( x ) E 2 y ( x ) e i Δ k b x 2 id 8 ( x ) E 2 z ( x )
+ i ω 2 cn 2 z ( x ) [ 1 2 d 32 ( x ) E 1 y ( x ) E 1 y ( x ) e i Δ k d x + 1 2 d 33 ( x ) E 1 z ( x ) E 1 z ( x ) e i Δ k f x ] ,
Δ k a = k 1 y k 1 z = 2 π λ 1 ( n 1 y n 1 z ) , Δ k b = k 2 y k 2 z = 4 π λ 1 ( n 2 y n 2 z ) ,
Δ k c = 2 k 1 y k 2 y = 4 π λ 1 ( n 1 y n 2 y ) , Δ k d = 2 k 1 y k 2 z = 4 π λ 1 ( n 1 y n 2 z ) ,
Δ k e = k 1 y + k 1 z k 2 y = 2 π λ 1 ( n 1 y + n 1 z 2 n 2 y ) , Δ k f = 2 k 1 z k 2 z = 4 π λ 1 ( n 1 z n 2 z ) ,
d 1 ( x ) = k 0 n 1 y n 1 z 2 r 42 2 E 0 f ( x ) , d 2 ( x ) = k 0 n 1 y 3 r 22 2 E 0 f ( x ) ,
d 3 ( x ) = k 0 n 1 y 2 n 1 z r 42 2 E 0 f ( x ) , d 4 ( x ) = k 0 n 1 z 3 r 32 2 E 0 f ( x ) = 0 ,
d 5 ( x ) = k 0 n 2 y n 2 z 2 r 42 2 E 0 f ( x ) , d 6 ( x ) = k 0 n 2 y 3 r 22 2 E 0 f ( x ) ,
d 7 ( x ) = k 0 n 2 y 2 n 2 z r 42 2 E 0 f ( x ) , d 8 ( x ) = k 0 n 2 z 3 r 32 2 E 0 f ( x ) , = 0
d 22 ( x ) = d 22 f ( x ) , d 24 ( x ) = d 24 f ( x ) , d 32 ( x ) = d 32 f ( x ) , d 33 ( x ) = d 33 f ( x ) ,
dE 1 y ( x ) dx = i k 0 n 1 y n 1 z 2 r 42 E 0 f a 2 E 1 z ( x ) i k 0 n 1 y 3 r 22 E 0 2 f ( x ) E 1 y ( x ) ,
dE 1 z ( x ) dx = i k 0 n 1 y 2 n 1 z r 42 E 0 f a 2 E 1 y ( x ) + 2 f E 1 z * ( x ) E 2 z ( x ) ,
dE 2 z ( x ) dx = i 1 2 κ 4 f E 1 z ( x ) E 1 z ( x ) ,
κ 1 d = d 24 ω 1 c n 1 y , κ 2 f = d 33 ω 1 f f c n 1 z , κ 4 d = d 32 ω 2 c n 2 z , κ 4 f = d 33 ω 2 f f c n 2 z ,
f a = { f 1,0 for QPPLN 0 for PPLN and f f = { f 1,1 for QPPLN f 1 for PPLN ,
d A ˆ ω dt = 2 ig A ˆ ω + A ˆ 2 ω
d A ˆ 2 ω dt = ig A ˆ ω 2 ,
( d < n ̑ ω > dt ) Δ t = 2 ( d < n ˆ 2 ω > dt ) Δ t ,
ħ ω 1 z + ħ ω 1 z = ħ ω 2 z .

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