Abstract

A versatile and accurate approach that combines a numerical iteration technique and a transfer-matrix method (TMM) is developed to solve the general problem of second harmonic generation (SHG) with pump depletion in quasi-phase-matched (QPM) nonlinear optical structures. We derive the iterative formulae from the nonlinear coupled wave equations and obtain the intensity distribution of fundamental wave and second harmonic wave by TMM. The approach shows quick numerical convergence of iteration and maintains perfect conservation of total energy. The simulation results show that the model coincides with the one under undepleted pump approximation very well when the SHG efficiency is small (well below 15%) and agrees very well with the effective nonlinear susceptibility model in handling general SHG problems even when the conversion efficiency is high up to 100%. Our method is applicable to general nonlinear optical structures, such as periodic, quasi-periodic, and aperiodic QPM structures, photonic crystals, and micro-cavities that might involve complicated modulation on the linear and nonlinear susceptibility.

© 2010 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127(6), 1918–1939 (1962).
    [CrossRef]
  2. 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(24), 1834–1836 (1997).
    [CrossRef]
  3. S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, H. F. Wang, Ch. Zh. Ge, and N. B. Ming, “Experimental Realization of Second Harmonic Generation in a Fibonacci Optical Superlattice of LiTaO3,” Phys. Rev. Lett. 78(14), 2752–2755 (1997).
    [CrossRef]
  4. V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett. 81(19), 4136–4139 (1998).
    [CrossRef]
  5. P. Ni, B. Ma, X. Wang, B. Cheng, and D. Zhang, “Second-harmonic generation in two-dimensional periodically poled lithium niobate using second-order quasiphase matching,” Appl. Phys. Lett. 82(24), 4230–4232 (2003).
    [CrossRef]
  6. B. Ma, T. Wang, Y. Sheng, P. Ni, Y. Wang, B. Cheng, and D. Zhang, “Quasiphase matched harmonic generation in a two-dimensional octagonal photonic superlattice,” Appl. Phys. Lett. 87(25), 251103 (2005).
    [CrossRef]
  7. A. M. Schober, G. Imeshev, and M. M. Fejer, “Tunable-chirp pulse compression in quasi-phase-matched second-harmonic generation,” Opt. Lett. 27(13), 1129–1131 (2002).
    [CrossRef]
  8. C. B. Clausen, O. Bang, and Y. S. Kivshar, “Spatial Solitons and Induced Kerr Effects in Quasi-Phase-Matched Quadratic Media,” Phys. Rev. Lett. 78(25), 4749–4752 (1997).
    [CrossRef]
  9. P. Xu, S. H. Ji, S. N. Zhu, X. Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, and N. B. Ming, “Conical second harmonic generation in a two-dimensional χ(2) photonic crystal: a hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. 93(13), 133904 (2004).
    [CrossRef] [PubMed]
  10. P. Xu, S. N. Zhu, X. Q. Yu, S. H. Ji, Z. D. Gao, G. Zhao, Y. Y. Zhu, and N. B. Ming, “Experimental studies of enhanced Raman scattering from a hexagonally poled LiTaO3 crystal,” Phys. Rev. B 72(6), 064307 (2005).
    [CrossRef]
  11. Y. Q. Qin, Ch. Zhang, Y. Y. Zhu, X. P. Hu, and G. Zhao, “Wave-front engineering by Huygens-Fresnel principle for nonlinear optical interactions in domain engineered structures,” Phys. Rev. Lett. 100(6), 063902 (2008).
    [CrossRef] [PubMed]
  12. M. J. A. de Dood, W. T. M. Irvine, and D. Bouwmeester, “Nonlinear photonic crystals as a source of entangled photons,” Phys. Rev. Lett. 93(4), 040504 (2004).
    [CrossRef] [PubMed]
  13. T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3(7), 395–398 (2009).
    [CrossRef]
  14. J. D. McMullen, “Optical parametric interactions in isotropic materials using a phase-corrected stack of nonlinear dielectric plates,” J. Appl. Phys. 46(7), 3076–3081 (1975).
    [CrossRef]
  15. 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(11), 2631–2654 (1992).
    [CrossRef]
  16. X. H. Wang and B. Y. Gu, “Nonlinear frequency conversion in 2D χ(2) photonic crystals and novel nonlinear double-circle construction,” Eur. Phys. J. B 24(3), 323–326 (2001).
    [CrossRef]
  17. D. S. Bethune, “Optical harmonic generation and mixing in multilayer media: analysis using optical transfer matrix techniques,” J. Opt. Soc. Am. B 6(5), 910–916 (1989).
    [CrossRef]
  18. Y. Jeong and B. Lee, “Matrix analysis for layered quasi-phase-matched media considering multiple reflection and pump wave depletion,” IEEE J. Quantum Electron. 35(2), 162–178 (1999).
    [CrossRef]
  19. J. J. Li, Z. Y. Li, and D. Z. Zhang, “Second harmonic generation in one-dimensional nonlinear photonic crystals solved by the transfer matrix method,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(5), 056606 (2007).
    [CrossRef] [PubMed]
  20. J. J. Li, Z. Y. Li, Y. Sheng, and D. Z. Zhang, “Giant enhancement of second harmonic generation in poled ferroelectric crystals,” Appl. Phys. Lett. 91(2), 022903 (2007).
    [CrossRef]
  21. J. J. Li, Z. Y. Li, and D. Z. Zhang, “Nonlinear frequency conversion in two-dimensional nonlinear photonic crystals solved by a plane-wave-based transfer-matrix method,” Phys. Rev. B 77(19), 195127 (2008).
    [CrossRef]
  22. K. C. Rustagi, S. C. Mehendale, and S. Meenakshi, “Optical frequency conversion in quasi-phase-matched stacks of nonlinear crystals,” IEEE J. Quantum Electron. 18(6), 1029–1041 (1982).
    [CrossRef]
  23. H. M. Masoudi and J. M. Arnold, “Modeling second-order nonlinear effects in optical waveguides using a parallel-processing beam propagation method,” IEEE J. Quantum Electron. 31(12), 2107–2113 (1995).
    [CrossRef]
  24. G. Bao and D. C. Dobson, “Second harmonic generation in nonlinear optical films,” J. Math. Phys. 35(4), 1622–1633 (1994).
    [CrossRef]
  25. J. Yuan, “Computing for second harmonic generation in one-dimensional nonlinear photonic crystals,” Opt. Commun. 282(13), 2628–2633 (2009).
    [CrossRef]
  26. J. Xia, “Enhancement of second harmonic generation in one-dimensional nonlinear photonic-crystal microcavities,” Opt. Express 17(22), 20069–20077 (2009).
    [CrossRef] [PubMed]
  27. Z. Y. Li and L. L. Lin, “Photonic band structures solved by a plane-wave-based transfer-matrix method,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(4), 046607 (2003).
    [CrossRef] [PubMed]
  28. L. L. Lin, Z. Y. Li, and K. M. Ho, “Lattice symmetry applied in transfer-matrix methods for photonic crystals,” J. Appl. Phys. 94(2), 811–821 (2003).
    [CrossRef]
  29. V. G. Dmitriev, G. G. Gurazdyan, and D. N. Nikogosyan, Handbook of nonlinear optical crystals (Springer, Berlin, 1997), Vol. 64, p. 125.
  30. G. J. Edwards and M. Lawrence, “A temperature-dependent dispersion equation for congruently grown lithium niobate,” Opt. Quantum Electron. 16(4), 373–375 (1984).
    [CrossRef]
  31. M. L. Ren and Z. Y. Li, “Giant enhancement of second harmonic generation in nonlinear photonic crystals with distributed Bragg reflector mirrors,” Opt. Express 17(17), 14502–14510 (2009).
    [CrossRef] [PubMed]

2009

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3(7), 395–398 (2009).
[CrossRef]

J. Yuan, “Computing for second harmonic generation in one-dimensional nonlinear photonic crystals,” Opt. Commun. 282(13), 2628–2633 (2009).
[CrossRef]

J. Xia, “Enhancement of second harmonic generation in one-dimensional nonlinear photonic-crystal microcavities,” Opt. Express 17(22), 20069–20077 (2009).
[CrossRef] [PubMed]

M. L. Ren and Z. Y. Li, “Giant enhancement of second harmonic generation in nonlinear photonic crystals with distributed Bragg reflector mirrors,” Opt. Express 17(17), 14502–14510 (2009).
[CrossRef] [PubMed]

2008

J. J. Li, Z. Y. Li, and D. Z. Zhang, “Nonlinear frequency conversion in two-dimensional nonlinear photonic crystals solved by a plane-wave-based transfer-matrix method,” Phys. Rev. B 77(19), 195127 (2008).
[CrossRef]

Y. Q. Qin, Ch. Zhang, Y. Y. Zhu, X. P. Hu, and G. Zhao, “Wave-front engineering by Huygens-Fresnel principle for nonlinear optical interactions in domain engineered structures,” Phys. Rev. Lett. 100(6), 063902 (2008).
[CrossRef] [PubMed]

2007

J. J. Li, Z. Y. Li, and D. Z. Zhang, “Second harmonic generation in one-dimensional nonlinear photonic crystals solved by the transfer matrix method,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(5), 056606 (2007).
[CrossRef] [PubMed]

J. J. Li, Z. Y. Li, Y. Sheng, and D. Z. Zhang, “Giant enhancement of second harmonic generation in poled ferroelectric crystals,” Appl. Phys. Lett. 91(2), 022903 (2007).
[CrossRef]

2005

P. Xu, S. N. Zhu, X. Q. Yu, S. H. Ji, Z. D. Gao, G. Zhao, Y. Y. Zhu, and N. B. Ming, “Experimental studies of enhanced Raman scattering from a hexagonally poled LiTaO3 crystal,” Phys. Rev. B 72(6), 064307 (2005).
[CrossRef]

B. Ma, T. Wang, Y. Sheng, P. Ni, Y. Wang, B. Cheng, and D. Zhang, “Quasiphase matched harmonic generation in a two-dimensional octagonal photonic superlattice,” Appl. Phys. Lett. 87(25), 251103 (2005).
[CrossRef]

2004

P. Xu, S. H. Ji, S. N. Zhu, X. Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, and N. B. Ming, “Conical second harmonic generation in a two-dimensional χ(2) photonic crystal: a hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. 93(13), 133904 (2004).
[CrossRef] [PubMed]

M. J. A. de Dood, W. T. M. Irvine, and D. Bouwmeester, “Nonlinear photonic crystals as a source of entangled photons,” Phys. Rev. Lett. 93(4), 040504 (2004).
[CrossRef] [PubMed]

2003

P. Ni, B. Ma, X. Wang, B. Cheng, and D. Zhang, “Second-harmonic generation in two-dimensional periodically poled lithium niobate using second-order quasiphase matching,” Appl. Phys. Lett. 82(24), 4230–4232 (2003).
[CrossRef]

Z. Y. Li and L. L. Lin, “Photonic band structures solved by a plane-wave-based transfer-matrix method,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(4), 046607 (2003).
[CrossRef] [PubMed]

L. L. Lin, Z. Y. Li, and K. M. Ho, “Lattice symmetry applied in transfer-matrix methods for photonic crystals,” J. Appl. Phys. 94(2), 811–821 (2003).
[CrossRef]

2002

2001

X. H. Wang and B. Y. Gu, “Nonlinear frequency conversion in 2D χ(2) photonic crystals and novel nonlinear double-circle construction,” Eur. Phys. J. B 24(3), 323–326 (2001).
[CrossRef]

1999

Y. Jeong and B. Lee, “Matrix analysis for layered quasi-phase-matched media considering multiple reflection and pump wave depletion,” IEEE J. Quantum Electron. 35(2), 162–178 (1999).
[CrossRef]

1998

V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett. 81(19), 4136–4139 (1998).
[CrossRef]

1997

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(24), 1834–1836 (1997).
[CrossRef]

S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, H. F. Wang, Ch. Zh. Ge, and N. B. Ming, “Experimental Realization of Second Harmonic Generation in a Fibonacci Optical Superlattice of LiTaO3,” Phys. Rev. Lett. 78(14), 2752–2755 (1997).
[CrossRef]

C. B. Clausen, O. Bang, and Y. S. Kivshar, “Spatial Solitons and Induced Kerr Effects in Quasi-Phase-Matched Quadratic Media,” Phys. Rev. Lett. 78(25), 4749–4752 (1997).
[CrossRef]

1995

H. M. Masoudi and J. M. Arnold, “Modeling second-order nonlinear effects in optical waveguides using a parallel-processing beam propagation method,” IEEE J. Quantum Electron. 31(12), 2107–2113 (1995).
[CrossRef]

1994

G. Bao and D. C. Dobson, “Second harmonic generation in nonlinear optical films,” J. Math. Phys. 35(4), 1622–1633 (1994).
[CrossRef]

1992

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(11), 2631–2654 (1992).
[CrossRef]

1989

1984

G. J. Edwards and M. Lawrence, “A temperature-dependent dispersion equation for congruently grown lithium niobate,” Opt. Quantum Electron. 16(4), 373–375 (1984).
[CrossRef]

1982

K. C. Rustagi, S. C. Mehendale, and S. Meenakshi, “Optical frequency conversion in quasi-phase-matched stacks of nonlinear crystals,” IEEE J. Quantum Electron. 18(6), 1029–1041 (1982).
[CrossRef]

1975

J. D. McMullen, “Optical parametric interactions in isotropic materials using a phase-corrected stack of nonlinear dielectric plates,” J. Appl. Phys. 46(7), 3076–3081 (1975).
[CrossRef]

1962

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127(6), 1918–1939 (1962).
[CrossRef]

Arie, A.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3(7), 395–398 (2009).
[CrossRef]

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127(6), 1918–1939 (1962).
[CrossRef]

Arnold, J. M.

H. M. Masoudi and J. M. Arnold, “Modeling second-order nonlinear effects in optical waveguides using a parallel-processing beam propagation method,” IEEE J. Quantum Electron. 31(12), 2107–2113 (1995).
[CrossRef]

Bang, O.

C. B. Clausen, O. Bang, and Y. S. Kivshar, “Spatial Solitons and Induced Kerr Effects in Quasi-Phase-Matched Quadratic Media,” Phys. Rev. Lett. 78(25), 4749–4752 (1997).
[CrossRef]

Bao, G.

G. Bao and D. C. Dobson, “Second harmonic generation in nonlinear optical films,” J. Math. Phys. 35(4), 1622–1633 (1994).
[CrossRef]

Batchko, R. G.

Berger, V.

V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett. 81(19), 4136–4139 (1998).
[CrossRef]

Bethune, D. S.

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127(6), 1918–1939 (1962).
[CrossRef]

Bouwmeester, D.

M. J. A. de Dood, W. T. M. Irvine, and D. Bouwmeester, “Nonlinear photonic crystals as a source of entangled photons,” Phys. Rev. Lett. 93(4), 040504 (2004).
[CrossRef] [PubMed]

Byer, R. L.

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(24), 1834–1836 (1997).
[CrossRef]

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(11), 2631–2654 (1992).
[CrossRef]

Cheng, B.

B. Ma, T. Wang, Y. Sheng, P. Ni, Y. Wang, B. Cheng, and D. Zhang, “Quasiphase matched harmonic generation in a two-dimensional octagonal photonic superlattice,” Appl. Phys. Lett. 87(25), 251103 (2005).
[CrossRef]

P. Ni, B. Ma, X. Wang, B. Cheng, and D. Zhang, “Second-harmonic generation in two-dimensional periodically poled lithium niobate using second-order quasiphase matching,” Appl. Phys. Lett. 82(24), 4230–4232 (2003).
[CrossRef]

Clausen, C. B.

C. B. Clausen, O. Bang, and Y. S. Kivshar, “Spatial Solitons and Induced Kerr Effects in Quasi-Phase-Matched Quadratic Media,” Phys. Rev. Lett. 78(25), 4749–4752 (1997).
[CrossRef]

de Dood, M. J. A.

M. J. A. de Dood, W. T. M. Irvine, and D. Bouwmeester, “Nonlinear photonic crystals as a source of entangled photons,” Phys. Rev. Lett. 93(4), 040504 (2004).
[CrossRef] [PubMed]

Dobson, D. C.

G. Bao and D. C. Dobson, “Second harmonic generation in nonlinear optical films,” J. Math. Phys. 35(4), 1622–1633 (1994).
[CrossRef]

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127(6), 1918–1939 (1962).
[CrossRef]

Edwards, G. J.

G. J. Edwards and M. Lawrence, “A temperature-dependent dispersion equation for congruently grown lithium niobate,” Opt. Quantum Electron. 16(4), 373–375 (1984).
[CrossRef]

Ellenbogen, T.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3(7), 395–398 (2009).
[CrossRef]

Fejer, M. M.

Ganany-Padowicz, A.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3(7), 395–398 (2009).
[CrossRef]

Gao, Z. D.

P. Xu, S. N. Zhu, X. Q. Yu, S. H. Ji, Z. D. Gao, G. Zhao, Y. Y. Zhu, and N. B. Ming, “Experimental studies of enhanced Raman scattering from a hexagonally poled LiTaO3 crystal,” Phys. Rev. B 72(6), 064307 (2005).
[CrossRef]

Ge, Ch. Zh.

S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, H. F. Wang, Ch. Zh. Ge, and N. B. Ming, “Experimental Realization of Second Harmonic Generation in a Fibonacci Optical Superlattice of LiTaO3,” Phys. Rev. Lett. 78(14), 2752–2755 (1997).
[CrossRef]

Gu, B. Y.

X. H. Wang and B. Y. Gu, “Nonlinear frequency conversion in 2D χ(2) photonic crystals and novel nonlinear double-circle construction,” Eur. Phys. J. B 24(3), 323–326 (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, and N. B. Ming, “Conical second harmonic generation in a two-dimensional χ(2) photonic crystal: a hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. 93(13), 133904 (2004).
[CrossRef] [PubMed]

Ho, K. M.

L. L. Lin, Z. Y. Li, and K. M. Ho, “Lattice symmetry applied in transfer-matrix methods for photonic crystals,” J. Appl. Phys. 94(2), 811–821 (2003).
[CrossRef]

Hu, X. P.

Y. Q. Qin, Ch. Zhang, Y. Y. Zhu, X. P. Hu, and G. Zhao, “Wave-front engineering by Huygens-Fresnel principle for nonlinear optical interactions in domain engineered structures,” Phys. Rev. Lett. 100(6), 063902 (2008).
[CrossRef] [PubMed]

Imeshev, G.

Irvine, W. T. M.

M. J. A. de Dood, W. T. M. Irvine, and D. Bouwmeester, “Nonlinear photonic crystals as a source of entangled photons,” Phys. Rev. Lett. 93(4), 040504 (2004).
[CrossRef] [PubMed]

Jeong, Y.

Y. Jeong and B. Lee, “Matrix analysis for layered quasi-phase-matched media considering multiple reflection and pump wave depletion,” IEEE J. Quantum Electron. 35(2), 162–178 (1999).
[CrossRef]

Ji, S. H.

P. Xu, S. N. Zhu, X. Q. Yu, S. H. Ji, Z. D. Gao, G. Zhao, Y. Y. Zhu, and N. B. Ming, “Experimental studies of enhanced Raman scattering from a hexagonally poled LiTaO3 crystal,” Phys. Rev. B 72(6), 064307 (2005).
[CrossRef]

P. Xu, S. H. Ji, S. N. Zhu, X. Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, and N. B. Ming, “Conical second harmonic generation in a two-dimensional χ(2) photonic crystal: a hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. 93(13), 133904 (2004).
[CrossRef] [PubMed]

Jundt, D. H.

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(11), 2631–2654 (1992).
[CrossRef]

Kivshar, Y. S.

C. B. Clausen, O. Bang, and Y. S. Kivshar, “Spatial Solitons and Induced Kerr Effects in Quasi-Phase-Matched Quadratic Media,” Phys. Rev. Lett. 78(25), 4749–4752 (1997).
[CrossRef]

Lawrence, M.

G. J. Edwards and M. Lawrence, “A temperature-dependent dispersion equation for congruently grown lithium niobate,” Opt. Quantum Electron. 16(4), 373–375 (1984).
[CrossRef]

Lee, B.

Y. Jeong and B. Lee, “Matrix analysis for layered quasi-phase-matched media considering multiple reflection and pump wave depletion,” IEEE J. Quantum Electron. 35(2), 162–178 (1999).
[CrossRef]

Li, J. J.

J. J. Li, Z. Y. Li, and D. Z. Zhang, “Nonlinear frequency conversion in two-dimensional nonlinear photonic crystals solved by a plane-wave-based transfer-matrix method,” Phys. Rev. B 77(19), 195127 (2008).
[CrossRef]

J. J. Li, Z. Y. Li, and D. Z. Zhang, “Second harmonic generation in one-dimensional nonlinear photonic crystals solved by the transfer matrix method,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(5), 056606 (2007).
[CrossRef] [PubMed]

J. J. Li, Z. Y. Li, Y. Sheng, and D. Z. Zhang, “Giant enhancement of second harmonic generation in poled ferroelectric crystals,” Appl. Phys. Lett. 91(2), 022903 (2007).
[CrossRef]

Li, Z. Y.

M. L. Ren and Z. Y. Li, “Giant enhancement of second harmonic generation in nonlinear photonic crystals with distributed Bragg reflector mirrors,” Opt. Express 17(17), 14502–14510 (2009).
[CrossRef] [PubMed]

J. J. Li, Z. Y. Li, and D. Z. Zhang, “Nonlinear frequency conversion in two-dimensional nonlinear photonic crystals solved by a plane-wave-based transfer-matrix method,” Phys. Rev. B 77(19), 195127 (2008).
[CrossRef]

J. J. Li, Z. Y. Li, Y. Sheng, and D. Z. Zhang, “Giant enhancement of second harmonic generation in poled ferroelectric crystals,” Appl. Phys. Lett. 91(2), 022903 (2007).
[CrossRef]

J. J. Li, Z. Y. Li, and D. Z. Zhang, “Second harmonic generation in one-dimensional nonlinear photonic crystals solved by the transfer matrix method,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(5), 056606 (2007).
[CrossRef] [PubMed]

L. L. Lin, Z. Y. Li, and K. M. Ho, “Lattice symmetry applied in transfer-matrix methods for photonic crystals,” J. Appl. Phys. 94(2), 811–821 (2003).
[CrossRef]

Z. Y. Li and L. L. Lin, “Photonic band structures solved by a plane-wave-based transfer-matrix method,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(4), 046607 (2003).
[CrossRef] [PubMed]

Lin, L. L.

Z. Y. Li and L. L. Lin, “Photonic band structures solved by a plane-wave-based transfer-matrix method,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(4), 046607 (2003).
[CrossRef] [PubMed]

L. L. Lin, Z. Y. Li, and K. M. Ho, “Lattice symmetry applied in transfer-matrix methods for photonic crystals,” J. Appl. Phys. 94(2), 811–821 (2003).
[CrossRef]

Ma, B.

B. Ma, T. Wang, Y. Sheng, P. Ni, Y. Wang, B. Cheng, and D. Zhang, “Quasiphase matched harmonic generation in a two-dimensional octagonal photonic superlattice,” Appl. Phys. Lett. 87(25), 251103 (2005).
[CrossRef]

P. Ni, B. Ma, X. Wang, B. Cheng, and D. Zhang, “Second-harmonic generation in two-dimensional periodically poled lithium niobate using second-order quasiphase matching,” Appl. Phys. Lett. 82(24), 4230–4232 (2003).
[CrossRef]

Magel, G. A.

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(11), 2631–2654 (1992).
[CrossRef]

Masoudi, H. M.

H. M. Masoudi and J. M. Arnold, “Modeling second-order nonlinear effects in optical waveguides using a parallel-processing beam propagation method,” IEEE J. Quantum Electron. 31(12), 2107–2113 (1995).
[CrossRef]

McMullen, J. D.

J. D. McMullen, “Optical parametric interactions in isotropic materials using a phase-corrected stack of nonlinear dielectric plates,” J. Appl. Phys. 46(7), 3076–3081 (1975).
[CrossRef]

Meenakshi, S.

K. C. Rustagi, S. C. Mehendale, and S. Meenakshi, “Optical frequency conversion in quasi-phase-matched stacks of nonlinear crystals,” IEEE J. Quantum Electron. 18(6), 1029–1041 (1982).
[CrossRef]

Mehendale, S. C.

K. C. Rustagi, S. C. Mehendale, and S. Meenakshi, “Optical frequency conversion in quasi-phase-matched stacks of nonlinear crystals,” IEEE J. Quantum Electron. 18(6), 1029–1041 (1982).
[CrossRef]

Miller, G. D.

Ming, N. B.

P. Xu, S. N. Zhu, X. Q. Yu, S. H. Ji, Z. D. Gao, G. Zhao, Y. Y. Zhu, and N. B. Ming, “Experimental studies of enhanced Raman scattering from a hexagonally poled LiTaO3 crystal,” Phys. Rev. B 72(6), 064307 (2005).
[CrossRef]

P. Xu, S. H. Ji, S. N. Zhu, X. Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, and N. B. Ming, “Conical second harmonic generation in a two-dimensional χ(2) photonic crystal: a hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. 93(13), 133904 (2004).
[CrossRef] [PubMed]

S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, H. F. Wang, Ch. Zh. Ge, and N. B. Ming, “Experimental Realization of Second Harmonic Generation in a Fibonacci Optical Superlattice of LiTaO3,” Phys. Rev. Lett. 78(14), 2752–2755 (1997).
[CrossRef]

Ni, P.

B. Ma, T. Wang, Y. Sheng, P. Ni, Y. Wang, B. Cheng, and D. Zhang, “Quasiphase matched harmonic generation in a two-dimensional octagonal photonic superlattice,” Appl. Phys. Lett. 87(25), 251103 (2005).
[CrossRef]

P. Ni, B. Ma, X. Wang, B. Cheng, and D. Zhang, “Second-harmonic generation in two-dimensional periodically poled lithium niobate using second-order quasiphase matching,” Appl. Phys. Lett. 82(24), 4230–4232 (2003).
[CrossRef]

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127(6), 1918–1939 (1962).
[CrossRef]

Qin, Y. Q.

Y. Q. Qin, Ch. Zhang, Y. Y. Zhu, X. P. Hu, and G. Zhao, “Wave-front engineering by Huygens-Fresnel principle for nonlinear optical interactions in domain engineered structures,” Phys. Rev. Lett. 100(6), 063902 (2008).
[CrossRef] [PubMed]

S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, H. F. Wang, Ch. Zh. Ge, and N. B. Ming, “Experimental Realization of Second Harmonic Generation in a Fibonacci Optical Superlattice of LiTaO3,” Phys. Rev. Lett. 78(14), 2752–2755 (1997).
[CrossRef]

Ren, M. L.

Rustagi, K. C.

K. C. Rustagi, S. C. Mehendale, and S. Meenakshi, “Optical frequency conversion in quasi-phase-matched stacks of nonlinear crystals,” IEEE J. Quantum Electron. 18(6), 1029–1041 (1982).
[CrossRef]

Schober, A. M.

Sheng, Y.

J. J. Li, Z. Y. Li, Y. Sheng, and D. Z. Zhang, “Giant enhancement of second harmonic generation in poled ferroelectric crystals,” Appl. Phys. Lett. 91(2), 022903 (2007).
[CrossRef]

B. Ma, T. Wang, Y. Sheng, P. Ni, Y. Wang, B. Cheng, and D. Zhang, “Quasiphase matched harmonic generation in a two-dimensional octagonal photonic superlattice,” Appl. Phys. Lett. 87(25), 251103 (2005).
[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, and N. B. Ming, “Conical second harmonic generation in a two-dimensional χ(2) photonic crystal: a hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. 93(13), 133904 (2004).
[CrossRef] [PubMed]

Tulloch, W. M.

Voloch-Bloch, N.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3(7), 395–398 (2009).
[CrossRef]

Wang, H. F.

S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, H. F. Wang, Ch. Zh. Ge, and N. B. Ming, “Experimental Realization of Second Harmonic Generation in a Fibonacci Optical Superlattice of LiTaO3,” Phys. Rev. Lett. 78(14), 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, and N. B. Ming, “Conical second harmonic generation in a two-dimensional χ(2) photonic crystal: a hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. 93(13), 133904 (2004).
[CrossRef] [PubMed]

Wang, T.

B. Ma, T. Wang, Y. Sheng, P. Ni, Y. Wang, B. Cheng, and D. Zhang, “Quasiphase matched harmonic generation in a two-dimensional octagonal photonic superlattice,” Appl. Phys. Lett. 87(25), 251103 (2005).
[CrossRef]

Wang, X.

P. Ni, B. Ma, X. Wang, B. Cheng, and D. Zhang, “Second-harmonic generation in two-dimensional periodically poled lithium niobate using second-order quasiphase matching,” Appl. Phys. Lett. 82(24), 4230–4232 (2003).
[CrossRef]

Wang, X. H.

X. H. Wang and B. Y. Gu, “Nonlinear frequency conversion in 2D χ(2) photonic crystals and novel nonlinear double-circle construction,” Eur. Phys. J. B 24(3), 323–326 (2001).
[CrossRef]

Wang, Y.

B. Ma, T. Wang, Y. Sheng, P. Ni, Y. Wang, B. Cheng, and D. Zhang, “Quasiphase matched harmonic generation in a two-dimensional octagonal photonic superlattice,” Appl. Phys. Lett. 87(25), 251103 (2005).
[CrossRef]

Weise, D. R.

Xia, J.

Xu, P.

P. Xu, S. N. Zhu, X. Q. Yu, S. H. Ji, Z. D. Gao, G. Zhao, Y. Y. Zhu, and N. B. Ming, “Experimental studies of enhanced Raman scattering from a hexagonally poled LiTaO3 crystal,” Phys. Rev. B 72(6), 064307 (2005).
[CrossRef]

P. Xu, S. H. Ji, S. N. Zhu, X. Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, and N. B. Ming, “Conical second harmonic generation in a two-dimensional χ(2) photonic crystal: a hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. 93(13), 133904 (2004).
[CrossRef] [PubMed]

Yu, X. Q.

P. Xu, S. N. Zhu, X. Q. Yu, S. H. Ji, Z. D. Gao, G. Zhao, Y. Y. Zhu, and N. B. Ming, “Experimental studies of enhanced Raman scattering from a hexagonally poled LiTaO3 crystal,” Phys. Rev. B 72(6), 064307 (2005).
[CrossRef]

P. Xu, S. H. Ji, S. N. Zhu, X. Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, and N. B. Ming, “Conical second harmonic generation in a two-dimensional χ(2) photonic crystal: a hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. 93(13), 133904 (2004).
[CrossRef] [PubMed]

Yuan, J.

J. Yuan, “Computing for second harmonic generation in one-dimensional nonlinear photonic crystals,” Opt. Commun. 282(13), 2628–2633 (2009).
[CrossRef]

Zhang, Ch.

Y. Q. Qin, Ch. Zhang, Y. Y. Zhu, X. P. Hu, and G. Zhao, “Wave-front engineering by Huygens-Fresnel principle for nonlinear optical interactions in domain engineered structures,” Phys. Rev. Lett. 100(6), 063902 (2008).
[CrossRef] [PubMed]

Zhang, D.

B. Ma, T. Wang, Y. Sheng, P. Ni, Y. Wang, B. Cheng, and D. Zhang, “Quasiphase matched harmonic generation in a two-dimensional octagonal photonic superlattice,” Appl. Phys. Lett. 87(25), 251103 (2005).
[CrossRef]

P. Ni, B. Ma, X. Wang, B. Cheng, and D. Zhang, “Second-harmonic generation in two-dimensional periodically poled lithium niobate using second-order quasiphase matching,” Appl. Phys. Lett. 82(24), 4230–4232 (2003).
[CrossRef]

Zhang, D. Z.

J. J. Li, Z. Y. Li, and D. Z. Zhang, “Nonlinear frequency conversion in two-dimensional nonlinear photonic crystals solved by a plane-wave-based transfer-matrix method,” Phys. Rev. B 77(19), 195127 (2008).
[CrossRef]

J. J. Li, Z. Y. Li, Y. Sheng, and D. Z. Zhang, “Giant enhancement of second harmonic generation in poled ferroelectric crystals,” Appl. Phys. Lett. 91(2), 022903 (2007).
[CrossRef]

J. J. Li, Z. Y. Li, and D. Z. Zhang, “Second harmonic generation in one-dimensional nonlinear photonic crystals solved by the transfer matrix method,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(5), 056606 (2007).
[CrossRef] [PubMed]

Zhao, G.

Y. Q. Qin, Ch. Zhang, Y. Y. Zhu, X. P. Hu, and G. Zhao, “Wave-front engineering by Huygens-Fresnel principle for nonlinear optical interactions in domain engineered structures,” Phys. Rev. Lett. 100(6), 063902 (2008).
[CrossRef] [PubMed]

P. Xu, S. N. Zhu, X. Q. Yu, S. H. Ji, Z. D. Gao, G. Zhao, Y. Y. Zhu, and N. B. Ming, “Experimental studies of enhanced Raman scattering from a hexagonally poled LiTaO3 crystal,” Phys. Rev. B 72(6), 064307 (2005).
[CrossRef]

Zhu, S. N.

P. Xu, S. N. Zhu, X. Q. Yu, S. H. Ji, Z. D. Gao, G. Zhao, Y. Y. Zhu, and N. B. Ming, “Experimental studies of enhanced Raman scattering from a hexagonally poled LiTaO3 crystal,” Phys. Rev. B 72(6), 064307 (2005).
[CrossRef]

P. Xu, S. H. Ji, S. N. Zhu, X. Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, and N. B. Ming, “Conical second harmonic generation in a two-dimensional χ(2) photonic crystal: a hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. 93(13), 133904 (2004).
[CrossRef] [PubMed]

S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, H. F. Wang, Ch. Zh. Ge, and N. B. Ming, “Experimental Realization of Second Harmonic Generation in a Fibonacci Optical Superlattice of LiTaO3,” Phys. Rev. Lett. 78(14), 2752–2755 (1997).
[CrossRef]

Zhu, Y. Y.

Y. Q. Qin, Ch. Zhang, Y. Y. Zhu, X. P. Hu, and G. Zhao, “Wave-front engineering by Huygens-Fresnel principle for nonlinear optical interactions in domain engineered structures,” Phys. Rev. Lett. 100(6), 063902 (2008).
[CrossRef] [PubMed]

P. Xu, S. N. Zhu, X. Q. Yu, S. H. Ji, Z. D. Gao, G. Zhao, Y. Y. Zhu, and N. B. Ming, “Experimental studies of enhanced Raman scattering from a hexagonally poled LiTaO3 crystal,” Phys. Rev. B 72(6), 064307 (2005).
[CrossRef]

P. Xu, S. H. Ji, S. N. Zhu, X. Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, and N. B. Ming, “Conical second harmonic generation in a two-dimensional χ(2) photonic crystal: a hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. 93(13), 133904 (2004).
[CrossRef] [PubMed]

S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, H. F. Wang, Ch. Zh. Ge, and N. B. Ming, “Experimental Realization of Second Harmonic Generation in a Fibonacci Optical Superlattice of LiTaO3,” Phys. Rev. Lett. 78(14), 2752–2755 (1997).
[CrossRef]

Appl. Phys. Lett.

P. Ni, B. Ma, X. Wang, B. Cheng, and D. Zhang, “Second-harmonic generation in two-dimensional periodically poled lithium niobate using second-order quasiphase matching,” Appl. Phys. Lett. 82(24), 4230–4232 (2003).
[CrossRef]

B. Ma, T. Wang, Y. Sheng, P. Ni, Y. Wang, B. Cheng, and D. Zhang, “Quasiphase matched harmonic generation in a two-dimensional octagonal photonic superlattice,” Appl. Phys. Lett. 87(25), 251103 (2005).
[CrossRef]

J. J. Li, Z. Y. Li, Y. Sheng, and D. Z. Zhang, “Giant enhancement of second harmonic generation in poled ferroelectric crystals,” Appl. Phys. Lett. 91(2), 022903 (2007).
[CrossRef]

Eur. Phys. J. B

X. H. Wang and B. Y. Gu, “Nonlinear frequency conversion in 2D χ(2) photonic crystals and novel nonlinear double-circle construction,” Eur. Phys. J. B 24(3), 323–326 (2001).
[CrossRef]

IEEE J. Quantum Electron.

Y. Jeong and B. Lee, “Matrix analysis for layered quasi-phase-matched media considering multiple reflection and pump wave depletion,” IEEE J. Quantum Electron. 35(2), 162–178 (1999).
[CrossRef]

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(11), 2631–2654 (1992).
[CrossRef]

K. C. Rustagi, S. C. Mehendale, and S. Meenakshi, “Optical frequency conversion in quasi-phase-matched stacks of nonlinear crystals,” IEEE J. Quantum Electron. 18(6), 1029–1041 (1982).
[CrossRef]

H. M. Masoudi and J. M. Arnold, “Modeling second-order nonlinear effects in optical waveguides using a parallel-processing beam propagation method,” IEEE J. Quantum Electron. 31(12), 2107–2113 (1995).
[CrossRef]

J. Appl. Phys.

L. L. Lin, Z. Y. Li, and K. M. Ho, “Lattice symmetry applied in transfer-matrix methods for photonic crystals,” J. Appl. Phys. 94(2), 811–821 (2003).
[CrossRef]

J. D. McMullen, “Optical parametric interactions in isotropic materials using a phase-corrected stack of nonlinear dielectric plates,” J. Appl. Phys. 46(7), 3076–3081 (1975).
[CrossRef]

J. Math. Phys.

G. Bao and D. C. Dobson, “Second harmonic generation in nonlinear optical films,” J. Math. Phys. 35(4), 1622–1633 (1994).
[CrossRef]

J. Opt. Soc. Am. B

Nat. Photonics

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3(7), 395–398 (2009).
[CrossRef]

Opt. Commun.

J. Yuan, “Computing for second harmonic generation in one-dimensional nonlinear photonic crystals,” Opt. Commun. 282(13), 2628–2633 (2009).
[CrossRef]

Opt. Express

Opt. Lett.

Opt. Quantum Electron.

G. J. Edwards and M. Lawrence, “A temperature-dependent dispersion equation for congruently grown lithium niobate,” Opt. Quantum Electron. 16(4), 373–375 (1984).
[CrossRef]

Phys. Rev.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127(6), 1918–1939 (1962).
[CrossRef]

Phys. Rev. B

P. Xu, S. N. Zhu, X. Q. Yu, S. H. Ji, Z. D. Gao, G. Zhao, Y. Y. Zhu, and N. B. Ming, “Experimental studies of enhanced Raman scattering from a hexagonally poled LiTaO3 crystal,” Phys. Rev. B 72(6), 064307 (2005).
[CrossRef]

J. J. Li, Z. Y. Li, and D. Z. Zhang, “Nonlinear frequency conversion in two-dimensional nonlinear photonic crystals solved by a plane-wave-based transfer-matrix method,” Phys. Rev. B 77(19), 195127 (2008).
[CrossRef]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys.

Z. Y. Li and L. L. Lin, “Photonic band structures solved by a plane-wave-based transfer-matrix method,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(4), 046607 (2003).
[CrossRef] [PubMed]

J. J. Li, Z. Y. Li, and D. Z. Zhang, “Second harmonic generation in one-dimensional nonlinear photonic crystals solved by the transfer matrix method,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(5), 056606 (2007).
[CrossRef] [PubMed]

Phys. Rev. Lett.

Y. Q. Qin, Ch. Zhang, Y. Y. Zhu, X. P. Hu, and G. Zhao, “Wave-front engineering by Huygens-Fresnel principle for nonlinear optical interactions in domain engineered structures,” Phys. Rev. Lett. 100(6), 063902 (2008).
[CrossRef] [PubMed]

M. J. A. de Dood, W. T. M. Irvine, and D. Bouwmeester, “Nonlinear photonic crystals as a source of entangled photons,” Phys. Rev. Lett. 93(4), 040504 (2004).
[CrossRef] [PubMed]

S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, H. F. Wang, Ch. Zh. Ge, and N. B. Ming, “Experimental Realization of Second Harmonic Generation in a Fibonacci Optical Superlattice of LiTaO3,” Phys. Rev. Lett. 78(14), 2752–2755 (1997).
[CrossRef]

V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett. 81(19), 4136–4139 (1998).
[CrossRef]

C. B. Clausen, O. Bang, and Y. S. Kivshar, “Spatial Solitons and Induced Kerr Effects in Quasi-Phase-Matched Quadratic Media,” Phys. Rev. Lett. 78(25), 4749–4752 (1997).
[CrossRef]

P. Xu, S. H. Ji, S. N. Zhu, X. Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, and N. B. Ming, “Conical second harmonic generation in a two-dimensional χ(2) photonic crystal: a hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. 93(13), 133904 (2004).
[CrossRef] [PubMed]

Other

V. G. Dmitriev, G. G. Gurazdyan, and D. N. Nikogosyan, Handbook of nonlinear optical crystals (Springer, Berlin, 1997), Vol. 64, p. 125.

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 (8)

Fig. 1
Fig. 1

Comparison of the UPA model (line) and our model (hollow circle) for SHG in a homogeneous LN film placed in the background of an index-matched homogeneous medium.

Fig. 2
Fig. 2

Calculated conversion efficiency for (a) forward and (b) backward SHW versus the sample length for a LN film placed in the air background.

Fig. 3
Fig. 3

Calculated conversion efficiency of SHW versus the period number of an index-matched QPM structure utilizing the ESMUP (line with triangle) and our model (line with circle).

Fig. 4
Fig. 4

Calculated conversion efficiency of SHW versus the period number of an index-matched QPM structure by means of the ESMPD (line) and our model (solid circle).

Fig. 5
Fig. 5

Calculated efficiency of FW and SHW versus period number of an index-matched QPM structure by means of our model. The line with triangle stands for the transmission coefficient of FW, while the line with circle stands for the conversion efficiency of SHW.

Fig. 6
Fig. 6

Calculated conversion efficiency of SHW versus the period number of an index-matched QPM structure with different domain lengths as 3.4016153 μm (line and solid circle), 3.402 μm (line and triangle) and 3.4 μm (line and hollow circle)

Fig. 7
Fig. 7

Calculated conversion efficiency of SHW versus the incident wavelength in sample 2. The total period number is set as 2400.

Fig. 8
Fig. 8

Calculation results of the transmission and reflectivity of FW and the forward and backward efficiency of SHW as a function of the period number of a QPM structure that is placed in the air background.

Equations (31)

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

d 2 E 1 n d z 2 + β 1 n 2 E 1 n = 2 k 10 2 χ n ( 2 ) E 1 n E 2 n ,
d 2 E 2 n d z 2 + β 2 n 2 E 2 n = k 20 2 χ n ( 2 ) E 1 n 2 .
d 2 E 1 n d z 2 + β 1 n 2 E 1 n = 0.
E 1 n ( z ) = Ω 1 n ( + ) exp [ i β 1 n ( z z n 1 ) ] + Ω 1 n ( ) exp [ i β 1 n ( z z n 1 ) ] ,
E 2 n ( z ) = Ω 2 n ( + ) exp [ i β 2 n ( z z n 1 ) ] + Ω 2 n ( ) exp [ i β 2 n ( z z n 1 ) ] + A n + exp [ 2 i β 1 n ( z z n 1 ) ] + A n exp [ 2 i β 1 n ( z z n 1 ) ] + C n ,
E 1 n ( i ) ( z ) = a 1 n ( i ) exp [ i ( β 1 n + Δ β n ) ( z z n 1 ) ] + a 2 n ( i ) exp [ i β 1 n ( z z n 1 ) ]        + a 3 n ( i ) exp [ i ( β 1 n Δ β n ) ( z z n 1 ) ] + b 1 n ( i ) exp [ i ( β 1 n + Δ β n ) ( z z n 1 ) ]        + b 2 n ( i ) exp [ i β 1 n ( z z n 1 ) ] + b 3 n ( i ) exp [ i ( β 1 n Δ β n ) ( z z n 1 ) ] ,
E 2 n ( i ) ( z ) = g 1 n ( i ) exp [ i ( β 2 n + Δ β n ) ( z z n 1 ) ] + g 2 n ( i ) exp [ i β 2 n ( z z n 1 ) ]        + g 3 n ( i ) exp [ i ( β 2 n Δ β n ) ( z z n 1 ) ] + f 1 n ( i ) exp [ i ( β 2 n + Δ β n ) ( z z n 1 ) ]        + f 2 n ( i ) exp [ i β 2 n ( z z n 1 ) ] + f 3 n ( i ) exp [ i ( β 2 n Δ β n ) ( z z n 1 ) ]        + h 1 n ( i ) exp [ i Δ β n ( z z n 1 ) ] + h 2 n ( i ) + h 3 n ( i ) exp [ i Δ β n ( z z n 1 ) ] .
a 1 n ( 0 ) = a 3 n ( 0 ) = b 1 n ( 0 ) = b 3 n ( 0 ) = 0.
g 1 n ( i ) = a 1 n ( i ) 2 u 1 n ,
g 3 n ( i ) = [ a 2 n ( i ) 2 + 2 a 1 n ( i ) a 3 n ( i ) ] u 2 n ,
f 1 n ( i ) = b 1 n ( i ) 2 u 1 n ,
f 3 n ( i ) = [ b 2 n ( i ) 2 + 2 b 1 n ( i ) b 3 n ( i ) ] u 2 n ,
h 1 n ( i ) = 2 ( a 1 n ( i ) b 2 n ( i ) + a 2 n ( i ) b 3 n ( i ) ) v 1 n ,
h 2 n ( i ) = 2 ( a 1 n ( i ) b 1 n ( i ) + a 2 n ( i ) b 2 n ( i ) + a 3 n ( i ) b 3 n ( i ) ) v 2 n ,
h 3 n ( i ) = 2 ( a 2 n ( i ) b 1 n ( i ) + a 3 n ( i ) b 2 n ( i ) ) v 3 n ,
a 1 n ( i + 1 ) = [ g 1 n ( i ) a 1 n ( i ) + g 2 n ( i ) a 2 n ( i ) + g 3 n ( i ) a 3 n ( i ) + h 1 n ( i ) b 2 n ( i ) + h 2 n ( i ) b 1 n ( i ) ] w 1 n ,
a 3 n ( i + 1 ) = [ g 3 n ( i ) a 1 n ( i ) + h 3 n ( i ) b 2 n ( i ) + h 2 n ( i ) b 3 n ( i ) ] w 2 n ,
b 1 n ( i + 1 ) = [ f 1 n ( i ) b 1 n ( i ) + f 2 n ( i ) b 2 n ( i ) + f 3 n ( i ) b 3 n ( i ) + h 3 n ( i ) a 2 n ( i ) + h 2 n ( i ) a 1 n ( i ) ] w 1 n ,
b 3 n ( i + 1 ) = [ f 3 n ( i ) b 1 n ( i ) + h 1 n ( i ) a 2 n ( i ) + h 2 n ( i ) a 3 n ( i ) ] w 2 n .
w 1 n = 2 k 10 2 χ n ( 2 ) β 1 n 2 ( β 1 n + Δ β n ) 2 ,
w 2 n = 2 k 10 2 χ n ( 2 ) β 1 n 2 ( β 1 n Δ β n ) 2 ,
u 1 n = k 20 2 χ n ( 2 ) β 2 n 2 ( β 2 n + Δ β n ) 2 ,
u 2 n = k 20 2 χ n ( 2 ) β 2 n 2 ( β 2 n Δ β n ) 2 ,
v 1 n = k 20 2 χ n ( 2 ) β 2 n 2 Δ β n 2 ,
v 2 n = k 20 2 χ n ( 2 ) β 2 n 2 ,
v 3 n = k 20 2 χ n ( 2 ) β 2 n 2 Δ β n 2 .
| I 2 n ( i + 1 ) ( z ) I 2 n ( i ) ( z ) | I 2 n ( i ) ( z ) ε ,
η 2 f ( i ) = I 2 N ( i ) + I 0 , η 2 b ( i ) = I 20 ( i ) I 0 ,
η 1 f ( i ) = I 1 N ( i ) + I 0 , η 1 b ( i ) = I 10 ( i ) I 0 ,
η 2 f ( i ) + η 2 b ( i ) + η 1 f ( i ) + η 1 b ( i ) = 1.
η 2 f ( i ) + η 1 f ( i ) = 1.

Metrics