Abstract

We present the formalism for the calculation of all second- and third-order nonlinear susceptibility coefficients based on the Landau–Devonshire free-energy expansion for cubic symmetry in the high-temperature paraelectric phase and the Landau–Khalatnikov dynamical equations. Second-order phase transition and single-frequency input waves are considered. Detailed results are given for all nonvanishing tensor elements of the second- and third-order nonlinear optical effects in the paraelectric and the tetragonal and rhombohedral ferroelectric phases.

© 2002 Optical Society of America

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References

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  1. C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1986).
  2. M. E. Lines and A. M. Glass, Principles and Applications of Ferroelectrics and Related Materials (Clarendon, Oxford, UK, 1977).
  3. Y. Ishibashi and H. Orihara, “A phenomenological theory of nonlinear dielectric response,” Ferroelectrics 156, 185–188 (1994).
    [CrossRef]
  4. J. Osman, Y. Ishibashi, S.-C. Lim, and D. R. Tilley, “Nonlinear optic coefficients in the ferroelectric phase,” J. Korean Phys. Soc. 32, S446–S449 (1998).
  5. C. Haas, “Phase transitions in ferroelectric and antiferroelectric crytals,” Phys. Rev. 140, A863–A868 (1965).
    [CrossRef]
  6. J. Osman, Y. Ishibashi, and D. R. Tilley, “Calculation of nonlinear susceptibility tensor components in ferroelectrics,” Jpn. J. Appl. Phys. 37, 4887–4893 (1998).
    [CrossRef]
  7. J. Grindlay, An Introduction to the Phenomenological Theory of Ferroelectricity (Pergamon, New York, 1970).
  8. A. F. Devonshire, “Theory of barium titanate (Part 1),” Philos. Mag. 40, 1040–1063 (1949).
  9. K. Fujita and Y. Ishibashi, “Roles of the higher order aniso-tropic terms in successive structural phase transitions: The method of determination of phenomenological parameters,” Jpn. J. Appl. Phys., 36, 254–259 (1997).
    [CrossRef]
  10. Y. Ishibashi and M. Iwata, “Morphotropic phase boundary in solid solution systems of perovskite-type oxide ferroelectrics,” Jpn. J. Appl. Phys. 37, L985–L987 (1998).
    [CrossRef]
  11. P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge University, Cambridge, UK 1990).
  12. H. Orihara and Y. Ishibashi, “A phenomenological theory of nonlinear dielectric response II. Miller’s rule and nonlinear response in nonferroelectrics,” J. Phys. Soc. Jpn. 66, 242–246 (1997).
    [CrossRef]
  13. S. V. Popov, Yu. P. Svirko, and N. I. Zheludev, Susceptibility Tensors for Nonlinear Optics (Institute of Physics, London, 1995).
  14. Y. G. Wang, W. L. Zhong, and P. L. Zhang, “Size effects onthe Curie temperature of ferroelectric particles,” Solid State Commun. 92, 519–523 (1994).
    [CrossRef]
  15. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).
  16. D. L. Mills, Nonlinear Optics (Springer-Verlag, Berlin, 1991).
  17. Y. Ishibashi and M. Iwata, “A theory of morphotropic phase boundary in solid-solution systems of perovskite-type oxide ferroelectrics,” Jpn. J. Appl. Phys. 38, 800–804 (1999).
    [CrossRef]
  18. M. Iwata and Y. Ishibashi, “Theory of morphotropic phase boundary in solid solution systems of perovskite-type oxide ferroelectrics: engineered domain configurations,” Jpn. J. Appl. Phys. 39, 5156–5163 (2000).
    [CrossRef]
  19. L. Gag, J. Osman, and D. R. Tilley, “Effective-medium theory of dielectric-constant anomalies in ferroelectric composites,” Ferroelectrics 255, 59–72 (2001).
    [CrossRef]
  20. E. Fatuzzo and W. J. Merz, Ferroelectricity (North-Holland, Amsterdam, 1967).

2001

L. Gag, J. Osman, and D. R. Tilley, “Effective-medium theory of dielectric-constant anomalies in ferroelectric composites,” Ferroelectrics 255, 59–72 (2001).
[CrossRef]

2000

M. Iwata and Y. Ishibashi, “Theory of morphotropic phase boundary in solid solution systems of perovskite-type oxide ferroelectrics: engineered domain configurations,” Jpn. J. Appl. Phys. 39, 5156–5163 (2000).
[CrossRef]

1999

Y. Ishibashi and M. Iwata, “A theory of morphotropic phase boundary in solid-solution systems of perovskite-type oxide ferroelectrics,” Jpn. J. Appl. Phys. 38, 800–804 (1999).
[CrossRef]

1998

J. Osman, Y. Ishibashi, S.-C. Lim, and D. R. Tilley, “Nonlinear optic coefficients in the ferroelectric phase,” J. Korean Phys. Soc. 32, S446–S449 (1998).

J. Osman, Y. Ishibashi, and D. R. Tilley, “Calculation of nonlinear susceptibility tensor components in ferroelectrics,” Jpn. J. Appl. Phys. 37, 4887–4893 (1998).
[CrossRef]

Y. Ishibashi and M. Iwata, “Morphotropic phase boundary in solid solution systems of perovskite-type oxide ferroelectrics,” Jpn. J. Appl. Phys. 37, L985–L987 (1998).
[CrossRef]

1997

H. Orihara and Y. Ishibashi, “A phenomenological theory of nonlinear dielectric response II. Miller’s rule and nonlinear response in nonferroelectrics,” J. Phys. Soc. Jpn. 66, 242–246 (1997).
[CrossRef]

K. Fujita and Y. Ishibashi, “Roles of the higher order aniso-tropic terms in successive structural phase transitions: The method of determination of phenomenological parameters,” Jpn. J. Appl. Phys., 36, 254–259 (1997).
[CrossRef]

1994

Y. Ishibashi and H. Orihara, “A phenomenological theory of nonlinear dielectric response,” Ferroelectrics 156, 185–188 (1994).
[CrossRef]

Y. G. Wang, W. L. Zhong, and P. L. Zhang, “Size effects onthe Curie temperature of ferroelectric particles,” Solid State Commun. 92, 519–523 (1994).
[CrossRef]

1965

C. Haas, “Phase transitions in ferroelectric and antiferroelectric crytals,” Phys. Rev. 140, A863–A868 (1965).
[CrossRef]

1949

A. F. Devonshire, “Theory of barium titanate (Part 1),” Philos. Mag. 40, 1040–1063 (1949).

Devonshire, A. F.

A. F. Devonshire, “Theory of barium titanate (Part 1),” Philos. Mag. 40, 1040–1063 (1949).

Fujita, K.

K. Fujita and Y. Ishibashi, “Roles of the higher order aniso-tropic terms in successive structural phase transitions: The method of determination of phenomenological parameters,” Jpn. J. Appl. Phys., 36, 254–259 (1997).
[CrossRef]

Gag, L.

L. Gag, J. Osman, and D. R. Tilley, “Effective-medium theory of dielectric-constant anomalies in ferroelectric composites,” Ferroelectrics 255, 59–72 (2001).
[CrossRef]

Haas, C.

C. Haas, “Phase transitions in ferroelectric and antiferroelectric crytals,” Phys. Rev. 140, A863–A868 (1965).
[CrossRef]

Ishibashi, Y.

M. Iwata and Y. Ishibashi, “Theory of morphotropic phase boundary in solid solution systems of perovskite-type oxide ferroelectrics: engineered domain configurations,” Jpn. J. Appl. Phys. 39, 5156–5163 (2000).
[CrossRef]

Y. Ishibashi and M. Iwata, “A theory of morphotropic phase boundary in solid-solution systems of perovskite-type oxide ferroelectrics,” Jpn. J. Appl. Phys. 38, 800–804 (1999).
[CrossRef]

Y. Ishibashi and M. Iwata, “Morphotropic phase boundary in solid solution systems of perovskite-type oxide ferroelectrics,” Jpn. J. Appl. Phys. 37, L985–L987 (1998).
[CrossRef]

J. Osman, Y. Ishibashi, and D. R. Tilley, “Calculation of nonlinear susceptibility tensor components in ferroelectrics,” Jpn. J. Appl. Phys. 37, 4887–4893 (1998).
[CrossRef]

J. Osman, Y. Ishibashi, S.-C. Lim, and D. R. Tilley, “Nonlinear optic coefficients in the ferroelectric phase,” J. Korean Phys. Soc. 32, S446–S449 (1998).

H. Orihara and Y. Ishibashi, “A phenomenological theory of nonlinear dielectric response II. Miller’s rule and nonlinear response in nonferroelectrics,” J. Phys. Soc. Jpn. 66, 242–246 (1997).
[CrossRef]

K. Fujita and Y. Ishibashi, “Roles of the higher order aniso-tropic terms in successive structural phase transitions: The method of determination of phenomenological parameters,” Jpn. J. Appl. Phys., 36, 254–259 (1997).
[CrossRef]

Y. Ishibashi and H. Orihara, “A phenomenological theory of nonlinear dielectric response,” Ferroelectrics 156, 185–188 (1994).
[CrossRef]

Iwata, M.

M. Iwata and Y. Ishibashi, “Theory of morphotropic phase boundary in solid solution systems of perovskite-type oxide ferroelectrics: engineered domain configurations,” Jpn. J. Appl. Phys. 39, 5156–5163 (2000).
[CrossRef]

Y. Ishibashi and M. Iwata, “A theory of morphotropic phase boundary in solid-solution systems of perovskite-type oxide ferroelectrics,” Jpn. J. Appl. Phys. 38, 800–804 (1999).
[CrossRef]

Y. Ishibashi and M. Iwata, “Morphotropic phase boundary in solid solution systems of perovskite-type oxide ferroelectrics,” Jpn. J. Appl. Phys. 37, L985–L987 (1998).
[CrossRef]

Lim, S.-C.

J. Osman, Y. Ishibashi, S.-C. Lim, and D. R. Tilley, “Nonlinear optic coefficients in the ferroelectric phase,” J. Korean Phys. Soc. 32, S446–S449 (1998).

Orihara, H.

H. Orihara and Y. Ishibashi, “A phenomenological theory of nonlinear dielectric response II. Miller’s rule and nonlinear response in nonferroelectrics,” J. Phys. Soc. Jpn. 66, 242–246 (1997).
[CrossRef]

Y. Ishibashi and H. Orihara, “A phenomenological theory of nonlinear dielectric response,” Ferroelectrics 156, 185–188 (1994).
[CrossRef]

Osman, J.

L. Gag, J. Osman, and D. R. Tilley, “Effective-medium theory of dielectric-constant anomalies in ferroelectric composites,” Ferroelectrics 255, 59–72 (2001).
[CrossRef]

J. Osman, Y. Ishibashi, S.-C. Lim, and D. R. Tilley, “Nonlinear optic coefficients in the ferroelectric phase,” J. Korean Phys. Soc. 32, S446–S449 (1998).

J. Osman, Y. Ishibashi, and D. R. Tilley, “Calculation of nonlinear susceptibility tensor components in ferroelectrics,” Jpn. J. Appl. Phys. 37, 4887–4893 (1998).
[CrossRef]

Tilley, D. R.

L. Gag, J. Osman, and D. R. Tilley, “Effective-medium theory of dielectric-constant anomalies in ferroelectric composites,” Ferroelectrics 255, 59–72 (2001).
[CrossRef]

J. Osman, Y. Ishibashi, and D. R. Tilley, “Calculation of nonlinear susceptibility tensor components in ferroelectrics,” Jpn. J. Appl. Phys. 37, 4887–4893 (1998).
[CrossRef]

J. Osman, Y. Ishibashi, S.-C. Lim, and D. R. Tilley, “Nonlinear optic coefficients in the ferroelectric phase,” J. Korean Phys. Soc. 32, S446–S449 (1998).

Wang, Y. G.

Y. G. Wang, W. L. Zhong, and P. L. Zhang, “Size effects onthe Curie temperature of ferroelectric particles,” Solid State Commun. 92, 519–523 (1994).
[CrossRef]

Zhang, P. L.

Y. G. Wang, W. L. Zhong, and P. L. Zhang, “Size effects onthe Curie temperature of ferroelectric particles,” Solid State Commun. 92, 519–523 (1994).
[CrossRef]

Zhong, W. L.

Y. G. Wang, W. L. Zhong, and P. L. Zhang, “Size effects onthe Curie temperature of ferroelectric particles,” Solid State Commun. 92, 519–523 (1994).
[CrossRef]

Ferroelectrics

Y. Ishibashi and H. Orihara, “A phenomenological theory of nonlinear dielectric response,” Ferroelectrics 156, 185–188 (1994).
[CrossRef]

L. Gag, J. Osman, and D. R. Tilley, “Effective-medium theory of dielectric-constant anomalies in ferroelectric composites,” Ferroelectrics 255, 59–72 (2001).
[CrossRef]

J. Korean Phys. Soc.

J. Osman, Y. Ishibashi, S.-C. Lim, and D. R. Tilley, “Nonlinear optic coefficients in the ferroelectric phase,” J. Korean Phys. Soc. 32, S446–S449 (1998).

J. Phys. Soc. Jpn.

H. Orihara and Y. Ishibashi, “A phenomenological theory of nonlinear dielectric response II. Miller’s rule and nonlinear response in nonferroelectrics,” J. Phys. Soc. Jpn. 66, 242–246 (1997).
[CrossRef]

Jpn. J. Appl. Phys.

Y. Ishibashi and M. Iwata, “A theory of morphotropic phase boundary in solid-solution systems of perovskite-type oxide ferroelectrics,” Jpn. J. Appl. Phys. 38, 800–804 (1999).
[CrossRef]

M. Iwata and Y. Ishibashi, “Theory of morphotropic phase boundary in solid solution systems of perovskite-type oxide ferroelectrics: engineered domain configurations,” Jpn. J. Appl. Phys. 39, 5156–5163 (2000).
[CrossRef]

J. Osman, Y. Ishibashi, and D. R. Tilley, “Calculation of nonlinear susceptibility tensor components in ferroelectrics,” Jpn. J. Appl. Phys. 37, 4887–4893 (1998).
[CrossRef]

K. Fujita and Y. Ishibashi, “Roles of the higher order aniso-tropic terms in successive structural phase transitions: The method of determination of phenomenological parameters,” Jpn. J. Appl. Phys., 36, 254–259 (1997).
[CrossRef]

Y. Ishibashi and M. Iwata, “Morphotropic phase boundary in solid solution systems of perovskite-type oxide ferroelectrics,” Jpn. J. Appl. Phys. 37, L985–L987 (1998).
[CrossRef]

Philos. Mag.

A. F. Devonshire, “Theory of barium titanate (Part 1),” Philos. Mag. 40, 1040–1063 (1949).

Phys. Rev.

C. Haas, “Phase transitions in ferroelectric and antiferroelectric crytals,” Phys. Rev. 140, A863–A868 (1965).
[CrossRef]

Solid State Commun.

Y. G. Wang, W. L. Zhong, and P. L. Zhang, “Size effects onthe Curie temperature of ferroelectric particles,” Solid State Commun. 92, 519–523 (1994).
[CrossRef]

Other

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).

D. L. Mills, Nonlinear Optics (Springer-Verlag, Berlin, 1991).

S. V. Popov, Yu. P. Svirko, and N. I. Zheludev, Susceptibility Tensors for Nonlinear Optics (Institute of Physics, London, 1995).

E. Fatuzzo and W. J. Merz, Ferroelectricity (North-Holland, Amsterdam, 1967).

C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1986).

M. E. Lines and A. M. Glass, Principles and Applications of Ferroelectrics and Related Materials (Clarendon, Oxford, UK, 1977).

P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge University, Cambridge, UK 1990).

J. Grindlay, An Introduction to the Phenomenological Theory of Ferroelectricity (Pergamon, New York, 1970).

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Figures (3)

Fig. 1
Fig. 1

Plot of scaled χxzx(2)(-2ω; ω, ω). Solid curve, real; dashed curve, imaginary. Versus reduced frequency f at reduced temperature (a) t=0.76, (b) t=0.69. (c) Versus reduced temperature t at reduced frequency f=0.15.

Fig. 2
Fig. 2

Plot of scaled χxxxx(3)(-3ω; ω, ω, ω). Solid curve, real; dashed curve, imaginary. Versus reduced frequency f at reduced temperature (a) t=0.76, (b) t=0.69. (c) Versus reduced temperature t at reduced frequency f=0.15.

Fig. 3
Fig. 3

Graph of |Δkx| versus frequency ω/2π for SHG under phase-mismatching condition.

Tables (5)

Tables Icon

Table 1 Cubic Symmetry: Nonvanishing NLO Tensor Elements in the Paraelectric Phase, T>TC

Tables Icon

Table 2 Tetragonal Symmetry: Nonvanishing Second-Order, χ(2), NLO Tensor Elements in the Ferroelectric Phase, T<TC

Tables Icon

Table 3 Tetragonal Symmetry: Nonvanishing Third-Order, χ(3), NLO Tensor Elements in the Ferroelectric Phase, T<TC

Tables Icon

Table 4 Rhombohedral Symmetry: Nonvanishing Second-Order, χ(2), NLO Tensor Elements in the Ferroelectric Phase, T<TC

Tables Icon

Table 5 Rhombohedral Symmetry: Nonvanishing Third-Order, χ(3), NLO Tensor Elements in the Ferroelectric Phase, T<TC

Equations (103)

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F=α20PiPi+β1402(PiPi)2+β2202(Px2Py2+Py2Pz2+Pz2Px2),
FE=F-E·P,
OˆPi=-F/Pi+Ei=fi(P)+Ei,
Oˆ=γt(relaxationaldynamics),
=md2dt2+γddt(oscillatorydynamics).
Pi=Pi(1)+Pi(2)+Pi(3)=0χil(1)El+0χilm(2)ElEm+0χilmn(3)EiEmEn,
Ei(t)=12[Eoi exp(-iωt)+Eoi*exp(iωt)],
(Pi(n))ωσ=0K(-ωσ;ω1, ωn)χiα1α2..αn(n)(-ωσ;ω1, ωn)×(Eα1)ω1(Eα2)ω2(Eαn)ωn,
fi(P)=fil(0)Pl+12film(0)PlPm+16filmn(0)PlPmPn,
Oˆ(Pi(1)+Pi(2)+Pi(3))=Ei+fil(0)(Pl(1)+Pl(2)+Pl(3))+16filmn(0)(Pi(1)+Pl(2)+Pl(3))×(Pm(1)+Pm(2)+Pm(3))(Pn(1)+Pn(2)+Pn(3)),
fil(0)=-α0δil,
Ψilmn=-filmn(0)=6β102δinδilδim+2β202(δjnδjmδil+δknδkmδil+δimδjnδjl+δinδjmδjl+δkmδinδkl+δknδimδkl),
Pi(1)=12[r1(ω)Eoi exp(-iωt)+r1*(ω)Eoi*exp(iωt)],
r1(ω)=Oˆ+α0-1=-iωγ+α0-1(relaxationaldynamics), =-mω2-iωγ+α0-1 (oscillatorydynamics).
r1(ω)=[m(ωT2-ω2-iωγ/m)]-1.
Oˆ+α0Pi(3)=-16ΨilmnPl(1)Pm(1)Pn(1).
Oˆ+α0Pi(3)=-16Ψilmn12[r1Eol exp(-iωt)+r1*Eol*exp(iωt)]×12[r1Eom exp(-iωt)+r1*Eom*exp(iωt)]×12[r1Eon exp(-iωt)+r1*Eon*exp(iωt)],
Pi(3)=-148Ψilmn[(r3r13EolEomEon)exp(-3iωt)+(r1r12r1*EolEomEon*)exp(-iωt)+c.c.].
χilmn(3)(-3ω; ω, ω, ω)=-160Ψilmnr3r13,
χilmn(3)(-ω; -ω, ω, ω)=-16oΨilmn|r1|2r12,
f=f0+α20(Qx2+Qy2+Qz2)+β1402(Qx4+Qy4+Qz4)+β2202(Qx2Qy2+Qy2Qz2+Qz2Qx2)+α20(P02+2QzPo)+β1402(6Qz2P02+4Qz3P0+4QzP03+P04)+β2202[(Qz2+2QzP0+P02)(Qx2+Qy2)]-EP.
fil(P0)=-α0δil+3β102δizδziδilP02+β202P02×(δilδzkδkz+2δziδzkδkl),
film(P0)=-6β102P0δziδimδil+2β202P0(δilδzkδkm+δimδzkδkl+δziδkmδkl),
filmn(P0)=-6β102δinδilδim+2β202(δjnδjmδil+δknδkmδil+δimδjnδjl+δinδjmδjl+δkmδinδkl+δknδimδkl).
α+β10P02=0.
fi(P)=fi(P0)+fil(P0)Ql+12film(P0)QlQm+16filmn(P0)QlQmQn.
Oˆ(Qi(1)+Qi(2)+Qi(3))=fil(P0)(Ql(1)+Ql(2)+Ql(3))+12film(P0)×(Ql(1)+Ql(2)+Ql(3))(Qm(1)+Qm(2)+Qm(3))+16filmn(P0)(Ql(1)+Ql(2)+Ql(3))×(Qm(1)+Qm(2)+Qm(3))(Qn(1)+Qn(2)+Qn(3))+Ei.
[Oˆ-fil(P0)δil]Ql(1)=El,
fxx=fyy=-α0+β202P02,
fzz=-2β102P02.
Qi(1)=12[Bii(ω)Eoi exp(-iωt)+Bii*(ω)Eoi*exp(iωt)],
Bxx(ω)=Byy(ω)σ(ω),
Bzz(ω)s(ω),
σ(ω)=1Θ(ω)+α0+β202P02,
s(ω)=1Θ(ω)+α0+3β102P02=1Θ(ω)-2α0.
Θ(ω)=-iγω(relaxationaldynamics),
=-iγω-mω2(oscillatorydynamics).
OˆQi(2)=fil(P0)Ql(2)+12film(P0)Ql(1)Qm(1).
[Oˆ-fil(P0)δil]Qi(2)=12film(P0)12[Bll(ω)Eol exp(-iωt)+Bll*(ω)Eol*exp(iωt)]12[Bmm(ω)Eom exp(-iωt)
+Bmm*(ω)Eom*exp(iωt).
[Oˆ-fil(P0)δil]Qi(3)=12film(P0)(Ql(1)Qm(2)+Qm(2)Ql(1))+16filmn(P0)(Ql(1)Qm(1)Qn(1)).
C1=12film(P0)[Oˆ-fil(P0)δil]-1(Ql(1)Qm(2)+Qm(2)Ql(1)).
C1=filr(P0)[Oˆ-fil(P0)δil]-1Ql(1)Qr(2).
Qi1(3)=116filr(P0)frmn(P0)[Bll(ω)Eol exp(-iωt)+Bll*(ω)Eol*exp(iωt)]×[Brr(2ω)Bmm(ω)Bnn(ω)EomEon exp(-iωt)+Brr(0)Bmm*(ω)Bnn(ω)Eom*Eon+c.c.].
C2=16filmn(P0)[Oˆ-fil(P0)δil]-1Ql(1)Qm(1)Qn(1).
Qi2(3)=148filmn(P0){[Bll(ω)Bmm(ω)Bnn(ω)EolEomEon×exp(-3iωt)]+[Bll(ω)Bmm*(ω)Bnn(ω)EolEom*Eon×exp(-iωt)+c.c.]}.
χ¯iill(3)(-3ω;ω, ω, ω)=13(χiill+χilil+χilli),
χ¯ilil(3)(-ω; 0, 0, ω)=12(χilil+χiill),
fil0=-α0δil+3β102P02δil+2β202P02δil+2β202P02(δjlδij+δklδik),
film0=-6β102P0iδilδim+2β202(P0jδjmδil+Pokδkmδil+Pojδimδjl+Poiδjmδjl+Poiδkmδkl+Pokδimδkl),
filmn0=-6β102δinδilδim+2β2002(δjnδjmδil+δknδkmδil+δimδjnδjl+δinδjmδjl+δkmδinδkl+δimδklδkn),
α+10(β1+2β2)P02=0.
s(ω)=(Oˆ1-q)0[Oˆ1(Oˆ1-q)-2q2],
σ(ω)=q0[Oˆ1(Oˆ1-q)-2q2],
Oˆn=Θ(nω)-p,
p=fii=α0+P0202(3β1+2β2),
q=fij=2β202P02.
u=fiij=fiji=fjii=2β202P0,
v=fiii=6β102P0,
h=fiiii=6β102,
l=fiijj=fijij=fijji=2β202,
h1=σ2+2sσ,
h2=s2+3σ2+2sσ,
h3=σ2+4sσ,
g1=|σ|2+sσ*+s*σ,
g2=3|σ|2+|s|2+sσ*+s*σ,
τ1=14v02sχ1SHG+14u02[2σχ1SHG+2(s+σ)χ2SHG]+148h03s3+18l02σs2,
τ2=14v02σχ2SHG+14u02[(s+σ)χ1SHG+(s+3σ)χ2SHG]+148h03σ3+116l02(σs2+σ3),
τ3=14v02sχ3SHG+14u02(s+3σ)χ3SHG+116h033s2σ+18l03(σs2+σ3+sσ2),
τ4=14v02σχ3SHG+14u02(s+3σ)χ3SHG+116h03sσ2+116l03(3sσ2+2σ3+s3),
ρ1=18v02s*χ1SHG+18u02[2σ*χ1SHG+2(s*+σ*)χ2SHG]+148h03s|s|2+148l03(2s*σ2+4s|σ|2),
ρ2=18v02σ*χ2SHG+18u02[(σ*+s*)χ1SHG+(s*+3σ*)χ2SHG]+148h03σ|σ|2+148l03(σ*s2+2σ|s|2+3σ|σ|2),
ρ3=18v02s*χ3SHG+18u02(3σ*+s*)χ3SHG+148h03σ|s|2+148l03(σ*s2+σ|s|2+2σ|σ|2+s|σ|2+s*σ2),
ρ4=148h03s|σ|2+148l03(s|s|2+2s|σ|2+2σ|σ|2+s*σ2),
ρ5=148h03s|σ|2+148l03(s|σ|2+2σ|σ|2+σ|s|2+s*σ2+s2σ*),
ρ6=148h03σ|s|2+148l03(s|σ|2+2σ|σ|2+σ|s|2+s*σ2+s2σ*),
ρ7=18v02σ*χ2SHG+18u02(2σ*+2s*)χ3SHG+148h03σ2s*+148l03(2σ|σ|2+2s|σ|2+s*σ2+s|s|2),
ρ8=18u02(3σ*+s*)χ3SHG+148h03σ|σ|2+148l03(σ*s2+σ|s|2+2s|σ|2+σ|σ|2+s*σ2),
ρ9=148h02σ|σ|2+148l03(3s|σ|2+σ|σ|2+σ|s|2+s*σ2),
c1=148h03|s|2s+148l03(2s*σ2+4s|σ|2),
c2=148h03|σ|2σ+148l03(s*σ2+2σ|s|2+3|σ|2σ),
c3=148h03s2σ*+148l03(2|s|2σ+2s|σ|2+2|σ|2σ),
c4=148h03|s|2σ+148l03(|s|2σ+s|σ|2+2|σ|2σ+s2σ*+s*σ2),
c5=148h03|σ|2s+148l03(|s|2s+2s|σ|2+2|σ|2σ+s*σ2),
c6=148h03σ2s*+148l03(|s|2s+2s|σ|2+2|σ|2σ+s*σ2),
c7=148h03σ2s*+148l03(2|s|2σ+2s|σ|2+2|σ|2σ),
c8=148h03|σ|2s+148l03(s2σ*+σ|s|2+2|σ|2σ+s|σ|2+s*σ2),
d1=18v02sχ1OR+18u02[2σχ1OR+(s+2σ)χ2OR],
d2=18v02sχ1OR+18u02[σχ1OR+(s+3σ)χ2OR],
d3=18v02σχ2OR+18u02[(s+σ)χ1OR+(s+2σ)χ2OR],
d4=18v02σχ2OR+18u02[sχ1OR+(s+3σ)χ2OR],
d5=18v02σχ2OR+18u02[(s+σ)χ2OR+3σ)χ2OR],
d6=18v02sχ3OR+18u02[σχ3OR+(s+σ)χ4OR],
d7=18v02sχ3OR+18u02[2σχ3OR+(s+σ)χ4OR].
2Ei2ω1y2=μ02t2[0Ei2ω1+PiL+PiNL],i=x, z,
Ez2ω1=12[Ez2ω1(y)exp(ik2zy)exp(-i2ω1t)+c.c.].
PL=0[χxx(1)(-2ω; ω)Ex2ω1xˆ+χzz(1)(-2ω; ω)Ez2ω1zˆ],
PNL=0{[χxxz(2)(-2ω; ω, ω)Exω1Ezω1+χxzx(2)(-2ω; ω, ω)Ezω1Exω1]xˆ+[χzxx(2)(-2ω; ω, ω)Exω1Exω1+χzzz(2)(-2ω; ω, ω)Ezω1Ezω1]zˆ},
Exω1=12[Exω1exp(ik1xy)exp(-iω1t)+c.c.],
2ik2zdEz2ω1dy=-2ω12c2χzxx(2)(-2ω; ω, ω)(Exω1)2exp(iΔkxy),
Δkx=2k1x-k2z
xx=2β1β2-β1zz,
zz=1|α|+3β1P0202,xx=1|α|+β2P0202.

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