Abstract

Second harmonic generation (SHG) from the aperiodic optical superlattice (AOS) in considering the pump depletion is investigated. It is found the domain configuration designed in undepleted pump approximation (UPA) can also be used to achieve multiple wavelength SHGs with high enough conversion efficiency for an exact solution. The applicable scope of UPA was estimated by a relative tolerance based on the related SHG conversion efficiency calculated in UPA and an exact solution. Results reveal that the relative tolerance is solely determined by the conversion efficiency, and unrelated to the sample configuration, pump intensity, incidental wavelength and nonlinear media. A model to evaluate an exact solution is proposed, and it is suggested that the SHG conversion efficiency can be easily assessed by the developed model. These results can be used to provide direct guidance for practical application, and can also make the estimation of practical samples more convenient.

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References

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  1. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev.127, 1918–1939 (1962).
    [CrossRef]
  2. J. Zhao and L. M. Zhao, “The effect of an infinite plane-wave approximation on calculations for second-harmonic generation in a one-dimensional nonlinear crystal,” Europhys. Lett.98, 44004 (2012).
    [CrossRef]
  3. L. M. Zhao, G. K. Yue, and Y. S. Zhou, “Effect of the pump depletion itself on the quasi-phase-matching for second-harmonic generation,” Europhys. Lett.99, 34002 (2012).
    [CrossRef]
  4. S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, H. F. Wang, C. Z. Ge, and N. B. Ming, “Harmonic generation in a Fibonacci optical superlattice of LiTaO3,” Phys. Rev. Lett.78, 2752–2755 (1997).
    [CrossRef]
  5. R. Lifshitz, A. Arie, and A. Bahabad, “Photonic quasicrystals for nonlinear optical frequency conversion,” Phys. Rev. Lett.95, 133901 (2005).
    [CrossRef] [PubMed]
  6. B. Y. Gu, B. Z. Dong, Y. Zhang, and G. Z. Yang, “Enhanced harmonic generation in aperiodic optical superlattices,” Appl. Phys. Lett.75, 2175–2177 (1999).
    [CrossRef]
  7. M. A. Arbore, O. Marco, and M. M. Fejer, “Pulse compression during second-harmonic generation in aperiodic quasi-phase-matching gratings,” Opt. Lett.22(12), 865–867 (1997).
    [CrossRef] [PubMed]
  8. R. Fischer, S. M. Saltiel, D. N. Neshev, W. Krolikowski, and Y. S. Kivshar, “Broadband femtosecond frequency doubling in random media,” Appl. Phys. Lett.89(19), 191105 (2006).
    [CrossRef]
  9. X. Vidal and J. Martorell, “Generation of light in media with a random distribution of nonlinear domains,” Phys. Rev. Lett.97(1), 013902 (2006).
    [CrossRef] [PubMed]
  10. K. C. Rustagi, S. C. Mehendale, and S. Meenakshi, “Optical frequency conversion in quasi-phase-matched states of nonlinear crystals,” IEEE J. Quantum Electron.18(6), 1029–1041 (1982).
    [CrossRef]
  11. L. M. Zhao, B. Y. Gu, G. Z. Yang, and Y. S Zhou, “Optimal design of aperiodic optical superlattices for achieving parameteric amplification of second harmonic generation with consideration of the depletion of pumping light power,” J. Nonlinear Opt. Phys. Mater.14, 115–131 (2005).
    [CrossRef]
  12. U. K. Sapaev, “Optimum formation of the response of aperiodic nonlinear crystals in the process of second harmonic generation,” Opt. Spectrosc.102, 939–943 (2007).
    [CrossRef]
  13. G. Bao and D. C. Dobson, “Second harmonic generation in nonlinear optical films,” J. Math. Phys.35, 1622–1633 (1994).
    [CrossRef]
  14. J. Xia, “Enhancement of second harmonic generation in one-dimensional nonlinear photonic-crystal microcavities,” Opt. Express17, 20069–20077 (2009).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  18. V. G. Dmitriev, G. G. Gurazdyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals(Springer, Berlin, 1997).
    [CrossRef]

2012

J. Zhao and L. M. Zhao, “The effect of an infinite plane-wave approximation on calculations for second-harmonic generation in a one-dimensional nonlinear crystal,” Europhys. Lett.98, 44004 (2012).
[CrossRef]

L. M. Zhao, G. K. Yue, and Y. S. Zhou, “Effect of the pump depletion itself on the quasi-phase-matching for second-harmonic generation,” Europhys. Lett.99, 34002 (2012).
[CrossRef]

2010

2009

2007

U. K. Sapaev and G. Assanto, “Femtosecond pulse synthesis by efficient second- harmonic generation in engineered quasi phase matching gratings,” Opt. Express15, 7448–7457 (2007).
[CrossRef] [PubMed]

U. K. Sapaev, “Optimum formation of the response of aperiodic nonlinear crystals in the process of second harmonic generation,” Opt. Spectrosc.102, 939–943 (2007).
[CrossRef]

2006

R. Fischer, S. M. Saltiel, D. N. Neshev, W. Krolikowski, and Y. S. Kivshar, “Broadband femtosecond frequency doubling in random media,” Appl. Phys. Lett.89(19), 191105 (2006).
[CrossRef]

X. Vidal and J. Martorell, “Generation of light in media with a random distribution of nonlinear domains,” Phys. Rev. Lett.97(1), 013902 (2006).
[CrossRef] [PubMed]

2005

L. M. Zhao, B. Y. Gu, G. Z. Yang, and Y. S Zhou, “Optimal design of aperiodic optical superlattices for achieving parameteric amplification of second harmonic generation with consideration of the depletion of pumping light power,” J. Nonlinear Opt. Phys. Mater.14, 115–131 (2005).
[CrossRef]

R. Lifshitz, A. Arie, and A. Bahabad, “Photonic quasicrystals for nonlinear optical frequency conversion,” Phys. Rev. Lett.95, 133901 (2005).
[CrossRef] [PubMed]

2000

1999

B. Y. Gu, B. Z. Dong, Y. Zhang, and G. Z. Yang, “Enhanced harmonic generation in aperiodic optical superlattices,” Appl. Phys. Lett.75, 2175–2177 (1999).
[CrossRef]

1997

M. A. Arbore, O. Marco, and M. M. Fejer, “Pulse compression during second-harmonic generation in aperiodic quasi-phase-matching gratings,” Opt. Lett.22(12), 865–867 (1997).
[CrossRef] [PubMed]

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

1994

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

1982

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

1962

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

Arbore, M. A.

Arie, A.

R. Lifshitz, A. Arie, and A. Bahabad, “Photonic quasicrystals for nonlinear optical frequency conversion,” Phys. Rev. Lett.95, 133901 (2005).
[CrossRef] [PubMed]

Armstrong, A.

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

Assanto, G.

Bahabad, A.

R. Lifshitz, A. Arie, and A. Bahabad, “Photonic quasicrystals for nonlinear optical frequency conversion,” Phys. Rev. Lett.95, 133901 (2005).
[CrossRef] [PubMed]

Bao, G.

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

Bloembergen, N.

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

Buffa, R.

Cavalieri, S.

Dmitriev, V. G.

V. G. Dmitriev, G. G. Gurazdyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals(Springer, Berlin, 1997).
[CrossRef]

Dobson, D. C.

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

Dong, B. Z.

B. Y. Gu, B. Z. Dong, Y. Zhang, and G. Z. Yang, “Enhanced harmonic generation in aperiodic optical superlattices,” Appl. Phys. Lett.75, 2175–2177 (1999).
[CrossRef]

Ducuing, J.

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

Fejer, M. M.

Fischer, R.

R. Fischer, S. M. Saltiel, D. N. Neshev, W. Krolikowski, and Y. S. Kivshar, “Broadband femtosecond frequency doubling in random media,” Appl. Phys. Lett.89(19), 191105 (2006).
[CrossRef]

Ge, C. Z.

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

Gu, B. Y.

L. M. Zhao, B. Y. Gu, G. Z. Yang, and Y. S Zhou, “Optimal design of aperiodic optical superlattices for achieving parameteric amplification of second harmonic generation with consideration of the depletion of pumping light power,” J. Nonlinear Opt. Phys. Mater.14, 115–131 (2005).
[CrossRef]

B. Y. Gu, B. Z. Dong, Y. Zhang, and G. Z. Yang, “Enhanced harmonic generation in aperiodic optical superlattices,” Appl. Phys. Lett.75, 2175–2177 (1999).
[CrossRef]

Gurazdyan, G. G.

V. G. Dmitriev, G. G. Gurazdyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals(Springer, Berlin, 1997).
[CrossRef]

Kivshar, Y. S.

R. Fischer, S. M. Saltiel, D. N. Neshev, W. Krolikowski, and Y. S. Kivshar, “Broadband femtosecond frequency doubling in random media,” Appl. Phys. Lett.89(19), 191105 (2006).
[CrossRef]

Krolikowski, W.

R. Fischer, S. M. Saltiel, D. N. Neshev, W. Krolikowski, and Y. S. Kivshar, “Broadband femtosecond frequency doubling in random media,” Appl. Phys. Lett.89(19), 191105 (2006).
[CrossRef]

Li, Z. Y.

Lifshitz, R.

R. Lifshitz, A. Arie, and A. Bahabad, “Photonic quasicrystals for nonlinear optical frequency conversion,” Phys. Rev. Lett.95, 133901 (2005).
[CrossRef] [PubMed]

Marco, O.

Martorell, J.

X. Vidal and J. Martorell, “Generation of light in media with a random distribution of nonlinear domains,” Phys. Rev. Lett.97(1), 013902 (2006).
[CrossRef] [PubMed]

Meenakshi, S.

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

Ming, N. B.

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

Neshev, D. N.

R. Fischer, S. M. Saltiel, D. N. Neshev, W. Krolikowski, and Y. S. Kivshar, “Broadband femtosecond frequency doubling in random media,” Appl. Phys. Lett.89(19), 191105 (2006).
[CrossRef]

Nikogosyan, D. N.

V. G. Dmitriev, G. G. Gurazdyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals(Springer, Berlin, 1997).
[CrossRef]

Pershan, P. S.

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

Qin, Y. Q.

S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, H. F. Wang, C. Z. Ge, and N. B. Ming, “Harmonic generation in a Fibonacci optical superlattice of LiTaO3,” Phys. Rev. Lett.78, 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 states of nonlinear crystals,” IEEE J. Quantum Electron.18(6), 1029–1041 (1982).
[CrossRef]

Saltiel, S. M.

R. Fischer, S. M. Saltiel, D. N. Neshev, W. Krolikowski, and Y. S. Kivshar, “Broadband femtosecond frequency doubling in random media,” Appl. Phys. Lett.89(19), 191105 (2006).
[CrossRef]

Sapaev, U. K.

U. K. Sapaev and G. Assanto, “Femtosecond pulse synthesis by efficient second- harmonic generation in engineered quasi phase matching gratings,” Opt. Express15, 7448–7457 (2007).
[CrossRef] [PubMed]

U. K. Sapaev, “Optimum formation of the response of aperiodic nonlinear crystals in the process of second harmonic generation,” Opt. Spectrosc.102, 939–943 (2007).
[CrossRef]

Vidal, X.

X. Vidal and J. Martorell, “Generation of light in media with a random distribution of nonlinear domains,” Phys. Rev. Lett.97(1), 013902 (2006).
[CrossRef] [PubMed]

Wang, H. F.

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

Xia, J.

Yang, G. Z.

L. M. Zhao, B. Y. Gu, G. Z. Yang, and Y. S Zhou, “Optimal design of aperiodic optical superlattices for achieving parameteric amplification of second harmonic generation with consideration of the depletion of pumping light power,” J. Nonlinear Opt. Phys. Mater.14, 115–131 (2005).
[CrossRef]

B. Y. Gu, B. Z. Dong, Y. Zhang, and G. Z. Yang, “Enhanced harmonic generation in aperiodic optical superlattices,” Appl. Phys. Lett.75, 2175–2177 (1999).
[CrossRef]

Yue, G. K.

L. M. Zhao, G. K. Yue, and Y. S. Zhou, “Effect of the pump depletion itself on the quasi-phase-matching for second-harmonic generation,” Europhys. Lett.99, 34002 (2012).
[CrossRef]

Zhang, Y.

B. Y. Gu, B. Z. Dong, Y. Zhang, and G. Z. Yang, “Enhanced harmonic generation in aperiodic optical superlattices,” Appl. Phys. Lett.75, 2175–2177 (1999).
[CrossRef]

Zhao, J.

J. Zhao and L. M. Zhao, “The effect of an infinite plane-wave approximation on calculations for second-harmonic generation in a one-dimensional nonlinear crystal,” Europhys. Lett.98, 44004 (2012).
[CrossRef]

Zhao, L. M.

J. Zhao and L. M. Zhao, “The effect of an infinite plane-wave approximation on calculations for second-harmonic generation in a one-dimensional nonlinear crystal,” Europhys. Lett.98, 44004 (2012).
[CrossRef]

L. M. Zhao, G. K. Yue, and Y. S. Zhou, “Effect of the pump depletion itself on the quasi-phase-matching for second-harmonic generation,” Europhys. Lett.99, 34002 (2012).
[CrossRef]

L. M. Zhao, B. Y. Gu, G. Z. Yang, and Y. S Zhou, “Optimal design of aperiodic optical superlattices for achieving parameteric amplification of second harmonic generation with consideration of the depletion of pumping light power,” J. Nonlinear Opt. Phys. Mater.14, 115–131 (2005).
[CrossRef]

Zhou, Y. S

L. M. Zhao, B. Y. Gu, G. Z. Yang, and Y. S Zhou, “Optimal design of aperiodic optical superlattices for achieving parameteric amplification of second harmonic generation with consideration of the depletion of pumping light power,” J. Nonlinear Opt. Phys. Mater.14, 115–131 (2005).
[CrossRef]

Zhou, Y. S.

L. M. Zhao, G. K. Yue, and Y. S. Zhou, “Effect of the pump depletion itself on the quasi-phase-matching for second-harmonic generation,” Europhys. Lett.99, 34002 (2012).
[CrossRef]

Zhu, S. N.

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

Zhu, Y. Y.

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

Appl. Phys. Lett.

B. Y. Gu, B. Z. Dong, Y. Zhang, and G. Z. Yang, “Enhanced harmonic generation in aperiodic optical superlattices,” Appl. Phys. Lett.75, 2175–2177 (1999).
[CrossRef]

R. Fischer, S. M. Saltiel, D. N. Neshev, W. Krolikowski, and Y. S. Kivshar, “Broadband femtosecond frequency doubling in random media,” Appl. Phys. Lett.89(19), 191105 (2006).
[CrossRef]

Europhys. Lett.

J. Zhao and L. M. Zhao, “The effect of an infinite plane-wave approximation on calculations for second-harmonic generation in a one-dimensional nonlinear crystal,” Europhys. Lett.98, 44004 (2012).
[CrossRef]

L. M. Zhao, G. K. Yue, and Y. S. Zhou, “Effect of the pump depletion itself on the quasi-phase-matching for second-harmonic generation,” Europhys. Lett.99, 34002 (2012).
[CrossRef]

IEEE J. Quantum Electron.

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

J. Math. Phys.

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

J. Nonlinear Opt. Phys. Mater.

L. M. Zhao, B. Y. Gu, G. Z. Yang, and Y. S Zhou, “Optimal design of aperiodic optical superlattices for achieving parameteric amplification of second harmonic generation with consideration of the depletion of pumping light power,” J. Nonlinear Opt. Phys. Mater.14, 115–131 (2005).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Express

Opt. Lett.

Opt. Spectrosc.

U. K. Sapaev, “Optimum formation of the response of aperiodic nonlinear crystals in the process of second harmonic generation,” Opt. Spectrosc.102, 939–943 (2007).
[CrossRef]

Phys. Rev.

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

Phys. Rev. Lett.

X. Vidal and J. Martorell, “Generation of light in media with a random distribution of nonlinear domains,” Phys. Rev. Lett.97(1), 013902 (2006).
[CrossRef] [PubMed]

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

R. Lifshitz, A. Arie, and A. Bahabad, “Photonic quasicrystals for nonlinear optical frequency conversion,” Phys. Rev. Lett.95, 133901 (2005).
[CrossRef] [PubMed]

Other

V. G. Dmitriev, G. G. Gurazdyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals(Springer, Berlin, 1997).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Variation of SHG conversion efficiency η with the wavelength for the pump intensity I = 1.0 × 1012W/m2.

Fig. 2
Fig. 2

u2(xn) as a function of sequence of domain n for the three different pump intensities in the two different cases: red curve for the UPA, and black curve for an exact solution (EA). Solid curve for I = 1.0 × 1010W/m2, dashed curve corresponds to I = 1.0 × 1011W/m2, and dotted curve is I = 1.0 × 1012W/m2.

Fig. 3
Fig. 3

Variation of relative tolerance σ = u ˜ 2 ( x n ) u 2 ( x n ) u 2 ( x n ) with u2(xn) for the three different pump intensities, ũ2(xn) represents the value in the case of UPA, solid curve for I = 1.0 × 1010W/m2, dashed curve corresponds to I = 1.0 × 1011W/m2, and dotted curve is I = 1.0 × 1012W/m2.

Fig. 4
Fig. 4

Variation of u2(L) as a function of input intensity I/I0, I0 = 1.0 × 109W/m2, solid curve denotes the data for exact solution and dotted curve is the fitted curve according to Eq. (14) provided that ũ2(L) in UPA is known for a specific sample.

Fig. 5
Fig. 5

Variation of u2(xn) as a function of xn, solid curve denotes the data for exact solution and dotted curve is the fitted curve according to Eq. (15).

Equations (21)

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

d E 1 ( x ) d x = i ω 1 2 χ ( 2 ) ( x ) k 1 c 2 E 2 ( x ) E 1 * ( x ) e i Δ k x ,
d E 2 ( x ) d x = i ω 2 2 χ ( 2 ) ( x ) 2 k 2 c 2 E 1 2 ( x ) e i Δ k x ,
χ ( 2 ) ( x ) = 2 | d 33 | d ˜ ( x ) ,
u α ( x ) = ( ε 0 c 2 k α 2 ω α I ) 1 / 2 ρ α ( x ) , α = 1 , 2 ,
d u 1 d x = ξ u 1 u 2 sin θ ,
d u 2 d x = ξ u 1 2 sin θ ,
d ϕ 1 d x = ξ u 2 cos θ ,
d ϕ 2 d x = ξ u 1 2 u 2 cos θ ,
d θ d x = Δ k + ξ u 1 2 u 2 cos θ 2 ξ u 2 cos θ .
cos θ ( n ) ( x ) = Δ k 2 ξ n ( u 2 ( n ) ( x ) ) 2 + Γ n ( u 1 ( n ) ( x ) ) 2 u 2 ( n ) ( x ) ,
Γ n + 1 = { Γ n , when the consecutive domains have identical polarization 2 u 2 2 ( x n ) Γ n , otherwise
( u 2 ( x n ) ) 3 + Δ k 2 | ξ n | ( u 2 ( x n ) ) 2 u 2 ( x n ) + Δ k 2 | ξ n | Γ n = 0.
u 2 ( x n ) = s n [ y , β ] ,
y = 0 w d t [ ( 1 t 2 ) ( 1 β 2 t 2 ) ] 1 / 2 , w = s n ( y , β ) .
y = ± ( u 3 c 2 u 3 a 2 ) 1 / 2 ( ζ 1 ζ 0 ) + A 0 .
A 0 = 0 u 2 ( x n 1 ) d u 2 [ 1 u 2 2 ] [ 1 ( u 3 b 2 u 3 a 2 ) ( u 3 c 2 u 3 a 2 ) 1 u 2 2 ] .
( u 2 ( n ) ) 2 ( 1 ( u 2 ( n ) ) 2 ) Δ k 2 4 ξ n 2 ( ( u 2 ( n ) ) 2 + Γ n ) 2 = 0.
( u 2 ( n ) ) 2 = ( u 2 ( n ) ) 2 u 3 a 2 u 3 b 2 u 3 a 2 .
σ = 0.39 u 2 2 ( x n ) 0.0115 u 2 ( x n ) ( u 2 ( x n ) 0.4 ) ,
σ = 0.0097 e u 2 ( x n ) / 0.217 ( 0.4 < u 2 ( x n ) < 0.9 ) .
u 2 ( x n ) = α u 1 2 ( x n 1 ) + u 2 ( x n 1 ) .

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