Abstract

We present the second order nonlinear optical properties of the Disperse Red 1 (DR1)-doped poly (cyano phenylene sulfide) (PCPS) - novel ferroelectric amorphous polymers. The PCPS possess self-organized long-range polarizations when they are annealed at temperatures higher than their glass transition point. From the unique nonelectrical poling effect, the second-order nonlinear susceptibility was obtained without the conventional poling procedures. The optimized conditions to create the second-order nonlinear optical susceptibility were also investigated in terms of the sample thicknesses and the annealing temperatures.

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References

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  1. Y. R. Schen, The Principles of Nonlinear Optics (John Wiley & Sons, 2003).
  2. R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic Press, 2008).
  3. V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, 1998).
  4. H. S. Nalwa, T. Watanabe, and S. Miyata, “Organic materials for nonlinear optics,” in Nonlinear Optics of Organic Molecules and Polymers, H. S. Nalwa and S. Miyata, eds. (CRC Press, 1996), pp. 89–350.
  5. A. Sugita, M. Morimoto, Y. Ishida, N. Mase, Y. Kawata, and S. Tasaka, “Linear and nonlinear optical properties of disperse red dyes in poly-(cyano phenylene sulfide),” Chem. Phys. Lett. 501(1–3), 39–43 (2010).
    [CrossRef]
  6. J. Ide, S. Tasaka, and N. Inagaki, “Nonelectrical poling in ferroelectric polycyanophenylenesulfides,” Jpn. J. Appl. Phys. 38(Part 1, No. 4A), 2049–2052 (1999).
    [CrossRef]
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    [CrossRef]
  8. L. De Boni, C. Toro, A. E. Masunov, and F. E. Hernández, “Untangling the excited states of DR1 in solution: an experimental and theoretical study,” J. Phys. Chem. A 112(17), 3886–3890 (2008).
    [CrossRef] [PubMed]
  9. C. Toro, A. Thibert, L. De Boni, A. E. Masunov, and F. E. Hernández, “Fluorescence emission of disperse Red 1 in solution at room temperature,” J. Phys. Chem. B 112(3), 929–937 (2008).
    [CrossRef] [PubMed]
  10. K. Kajikawa, H. Takezoe, and A. Fukuda, “Symmetry and second order susceptibility of hemicyanine monolayer studied by surface second harmonic generations,” Jpn. J. Appl. Phys. 30(Part 1, No. 5), 1050–1062 (1991).
    [CrossRef]
  11. F. Lagugné-Labarthet, C. Sourisseau, R. D. Schaller, R. J. Saykally, and P. Rochon, “Chromophore orientations in a nonlinear optical azopolymer diffraction grating: even and odd order parameters from far-field Raman and near-field second harmonic generation microscopies,” J. Phys. Chem. B 108(44), 17059–17068 (2004).
    [CrossRef]
  12. Y. Tu, Q. Zhang, and H. Ågren, “Electric field poled polymeric nonlinear optical systems: molecular dynamics simulations of poly(methyl methacrylate) doped with disperse red chromophores,” J. Phys. Chem. B 111(14), 3591–3598 (2007).
    [CrossRef] [PubMed]
  13. D. A. Kleinman, “Nonlinear dielectric polarization in optical media,” Phys. Rev. 126(6), 1977–1979 (1962).
    [CrossRef]
  14. T. G. Zhang, C. H. Zhang, and G. K. Wong, “Determination of molecular orientation in molecular monolayers by second-harmonic generation,” J. Opt. Soc. Am. B 7(6), 902–907 (1990).
    [CrossRef]
  15. K. Shirota, K. Kajikawa, H. Takezoe, and A. Fukuda, “Molecular orientation in mixed layers of hemicyanine and fatty acid at air/water interface studied by second harmonic generations,” Jpn. J. Appl. Phys. 29(Part 1, No. 4), 750–755 (1990).
    [CrossRef]
  16. T. Pliška, W.-R. Cho, J. Meier, A.-C. Le Duff, V. Ricci, A. Otomo, M. Canva, G. I. Stegeman, P. Raimond, and F. Kajzar, “Comparative study of nonlinear-optical polymers for guided-wave second-harmonic generation at telecommunication wavelengths,” J. Opt. Soc. Am. B 17(9), 1554–1564 (2000).
    [CrossRef]
  17. E. Bihler, K. Holdik, and W. Eisenmenger, “Polarization distributions in isotropic, stretched or annealed PVDF films,” IEEE Trans. Electr. Insul. 24(3), 541–545 (1989).
    [CrossRef]
  18. S. Bauer, G. Eberle, W. Eisenmenger, and H. Schlaich, “Second-harmonic generation with partially poled polymers,” Opt. Lett. 18(1), 16–18 (1993).
    [CrossRef] [PubMed]

2010

A. Sugita, M. Morimoto, Y. Ishida, N. Mase, Y. Kawata, and S. Tasaka, “Linear and nonlinear optical properties of disperse red dyes in poly-(cyano phenylene sulfide),” Chem. Phys. Lett. 501(1–3), 39–43 (2010).
[CrossRef]

2008

L. De Boni, C. Toro, A. E. Masunov, and F. E. Hernández, “Untangling the excited states of DR1 in solution: an experimental and theoretical study,” J. Phys. Chem. A 112(17), 3886–3890 (2008).
[CrossRef] [PubMed]

C. Toro, A. Thibert, L. De Boni, A. E. Masunov, and F. E. Hernández, “Fluorescence emission of disperse Red 1 in solution at room temperature,” J. Phys. Chem. B 112(3), 929–937 (2008).
[CrossRef] [PubMed]

2007

Y. Tu, Q. Zhang, and H. Ågren, “Electric field poled polymeric nonlinear optical systems: molecular dynamics simulations of poly(methyl methacrylate) doped with disperse red chromophores,” J. Phys. Chem. B 111(14), 3591–3598 (2007).
[CrossRef] [PubMed]

2004

F. Lagugné-Labarthet, C. Sourisseau, R. D. Schaller, R. J. Saykally, and P. Rochon, “Chromophore orientations in a nonlinear optical azopolymer diffraction grating: even and odd order parameters from far-field Raman and near-field second harmonic generation microscopies,” J. Phys. Chem. B 108(44), 17059–17068 (2004).
[CrossRef]

2000

1999

J. Ide, S. Tasaka, and N. Inagaki, “Nonelectrical poling in ferroelectric polycyanophenylenesulfides,” Jpn. J. Appl. Phys. 38(Part 1, No. 4A), 2049–2052 (1999).
[CrossRef]

1993

1991

K. Kajikawa, H. Takezoe, and A. Fukuda, “Symmetry and second order susceptibility of hemicyanine monolayer studied by surface second harmonic generations,” Jpn. J. Appl. Phys. 30(Part 1, No. 5), 1050–1062 (1991).
[CrossRef]

1990

T. G. Zhang, C. H. Zhang, and G. K. Wong, “Determination of molecular orientation in molecular monolayers by second-harmonic generation,” J. Opt. Soc. Am. B 7(6), 902–907 (1990).
[CrossRef]

K. Shirota, K. Kajikawa, H. Takezoe, and A. Fukuda, “Molecular orientation in mixed layers of hemicyanine and fatty acid at air/water interface studied by second harmonic generations,” Jpn. J. Appl. Phys. 29(Part 1, No. 4), 750–755 (1990).
[CrossRef]

1989

E. Bihler, K. Holdik, and W. Eisenmenger, “Polarization distributions in isotropic, stretched or annealed PVDF films,” IEEE Trans. Electr. Insul. 24(3), 541–545 (1989).
[CrossRef]

1965

1962

D. A. Kleinman, “Nonlinear dielectric polarization in optical media,” Phys. Rev. 126(6), 1977–1979 (1962).
[CrossRef]

Ågren, H.

Y. Tu, Q. Zhang, and H. Ågren, “Electric field poled polymeric nonlinear optical systems: molecular dynamics simulations of poly(methyl methacrylate) doped with disperse red chromophores,” J. Phys. Chem. B 111(14), 3591–3598 (2007).
[CrossRef] [PubMed]

Bauer, S.

Bihler, E.

E. Bihler, K. Holdik, and W. Eisenmenger, “Polarization distributions in isotropic, stretched or annealed PVDF films,” IEEE Trans. Electr. Insul. 24(3), 541–545 (1989).
[CrossRef]

Canva, M.

Cho, W.-R.

De Boni, L.

C. Toro, A. Thibert, L. De Boni, A. E. Masunov, and F. E. Hernández, “Fluorescence emission of disperse Red 1 in solution at room temperature,” J. Phys. Chem. B 112(3), 929–937 (2008).
[CrossRef] [PubMed]

L. De Boni, C. Toro, A. E. Masunov, and F. E. Hernández, “Untangling the excited states of DR1 in solution: an experimental and theoretical study,” J. Phys. Chem. A 112(17), 3886–3890 (2008).
[CrossRef] [PubMed]

Eberle, G.

Eisenmenger, W.

S. Bauer, G. Eberle, W. Eisenmenger, and H. Schlaich, “Second-harmonic generation with partially poled polymers,” Opt. Lett. 18(1), 16–18 (1993).
[CrossRef] [PubMed]

E. Bihler, K. Holdik, and W. Eisenmenger, “Polarization distributions in isotropic, stretched or annealed PVDF films,” IEEE Trans. Electr. Insul. 24(3), 541–545 (1989).
[CrossRef]

Fukuda, A.

K. Kajikawa, H. Takezoe, and A. Fukuda, “Symmetry and second order susceptibility of hemicyanine monolayer studied by surface second harmonic generations,” Jpn. J. Appl. Phys. 30(Part 1, No. 5), 1050–1062 (1991).
[CrossRef]

K. Shirota, K. Kajikawa, H. Takezoe, and A. Fukuda, “Molecular orientation in mixed layers of hemicyanine and fatty acid at air/water interface studied by second harmonic generations,” Jpn. J. Appl. Phys. 29(Part 1, No. 4), 750–755 (1990).
[CrossRef]

Hernández, F. E.

C. Toro, A. Thibert, L. De Boni, A. E. Masunov, and F. E. Hernández, “Fluorescence emission of disperse Red 1 in solution at room temperature,” J. Phys. Chem. B 112(3), 929–937 (2008).
[CrossRef] [PubMed]

L. De Boni, C. Toro, A. E. Masunov, and F. E. Hernández, “Untangling the excited states of DR1 in solution: an experimental and theoretical study,” J. Phys. Chem. A 112(17), 3886–3890 (2008).
[CrossRef] [PubMed]

Holdik, K.

E. Bihler, K. Holdik, and W. Eisenmenger, “Polarization distributions in isotropic, stretched or annealed PVDF films,” IEEE Trans. Electr. Insul. 24(3), 541–545 (1989).
[CrossRef]

Ide, J.

J. Ide, S. Tasaka, and N. Inagaki, “Nonelectrical poling in ferroelectric polycyanophenylenesulfides,” Jpn. J. Appl. Phys. 38(Part 1, No. 4A), 2049–2052 (1999).
[CrossRef]

Inagaki, N.

J. Ide, S. Tasaka, and N. Inagaki, “Nonelectrical poling in ferroelectric polycyanophenylenesulfides,” Jpn. J. Appl. Phys. 38(Part 1, No. 4A), 2049–2052 (1999).
[CrossRef]

Ishida, Y.

A. Sugita, M. Morimoto, Y. Ishida, N. Mase, Y. Kawata, and S. Tasaka, “Linear and nonlinear optical properties of disperse red dyes in poly-(cyano phenylene sulfide),” Chem. Phys. Lett. 501(1–3), 39–43 (2010).
[CrossRef]

Kajikawa, K.

K. Kajikawa, H. Takezoe, and A. Fukuda, “Symmetry and second order susceptibility of hemicyanine monolayer studied by surface second harmonic generations,” Jpn. J. Appl. Phys. 30(Part 1, No. 5), 1050–1062 (1991).
[CrossRef]

K. Shirota, K. Kajikawa, H. Takezoe, and A. Fukuda, “Molecular orientation in mixed layers of hemicyanine and fatty acid at air/water interface studied by second harmonic generations,” Jpn. J. Appl. Phys. 29(Part 1, No. 4), 750–755 (1990).
[CrossRef]

Kajzar, F.

Kawata, Y.

A. Sugita, M. Morimoto, Y. Ishida, N. Mase, Y. Kawata, and S. Tasaka, “Linear and nonlinear optical properties of disperse red dyes in poly-(cyano phenylene sulfide),” Chem. Phys. Lett. 501(1–3), 39–43 (2010).
[CrossRef]

Kleinman, D. A.

D. A. Kleinman, “Nonlinear dielectric polarization in optical media,” Phys. Rev. 126(6), 1977–1979 (1962).
[CrossRef]

Lagugné-Labarthet, F.

F. Lagugné-Labarthet, C. Sourisseau, R. D. Schaller, R. J. Saykally, and P. Rochon, “Chromophore orientations in a nonlinear optical azopolymer diffraction grating: even and odd order parameters from far-field Raman and near-field second harmonic generation microscopies,” J. Phys. Chem. B 108(44), 17059–17068 (2004).
[CrossRef]

Le Duff, A.-C.

Malitson, I. H.

Mase, N.

A. Sugita, M. Morimoto, Y. Ishida, N. Mase, Y. Kawata, and S. Tasaka, “Linear and nonlinear optical properties of disperse red dyes in poly-(cyano phenylene sulfide),” Chem. Phys. Lett. 501(1–3), 39–43 (2010).
[CrossRef]

Masunov, A. E.

L. De Boni, C. Toro, A. E. Masunov, and F. E. Hernández, “Untangling the excited states of DR1 in solution: an experimental and theoretical study,” J. Phys. Chem. A 112(17), 3886–3890 (2008).
[CrossRef] [PubMed]

C. Toro, A. Thibert, L. De Boni, A. E. Masunov, and F. E. Hernández, “Fluorescence emission of disperse Red 1 in solution at room temperature,” J. Phys. Chem. B 112(3), 929–937 (2008).
[CrossRef] [PubMed]

Meier, J.

Morimoto, M.

A. Sugita, M. Morimoto, Y. Ishida, N. Mase, Y. Kawata, and S. Tasaka, “Linear and nonlinear optical properties of disperse red dyes in poly-(cyano phenylene sulfide),” Chem. Phys. Lett. 501(1–3), 39–43 (2010).
[CrossRef]

Otomo, A.

Pliška, T.

Raimond, P.

Ricci, V.

Rochon, P.

F. Lagugné-Labarthet, C. Sourisseau, R. D. Schaller, R. J. Saykally, and P. Rochon, “Chromophore orientations in a nonlinear optical azopolymer diffraction grating: even and odd order parameters from far-field Raman and near-field second harmonic generation microscopies,” J. Phys. Chem. B 108(44), 17059–17068 (2004).
[CrossRef]

Saykally, R. J.

F. Lagugné-Labarthet, C. Sourisseau, R. D. Schaller, R. J. Saykally, and P. Rochon, “Chromophore orientations in a nonlinear optical azopolymer diffraction grating: even and odd order parameters from far-field Raman and near-field second harmonic generation microscopies,” J. Phys. Chem. B 108(44), 17059–17068 (2004).
[CrossRef]

Schaller, R. D.

F. Lagugné-Labarthet, C. Sourisseau, R. D. Schaller, R. J. Saykally, and P. Rochon, “Chromophore orientations in a nonlinear optical azopolymer diffraction grating: even and odd order parameters from far-field Raman and near-field second harmonic generation microscopies,” J. Phys. Chem. B 108(44), 17059–17068 (2004).
[CrossRef]

Schlaich, H.

Shirota, K.

K. Shirota, K. Kajikawa, H. Takezoe, and A. Fukuda, “Molecular orientation in mixed layers of hemicyanine and fatty acid at air/water interface studied by second harmonic generations,” Jpn. J. Appl. Phys. 29(Part 1, No. 4), 750–755 (1990).
[CrossRef]

Sourisseau, C.

F. Lagugné-Labarthet, C. Sourisseau, R. D. Schaller, R. J. Saykally, and P. Rochon, “Chromophore orientations in a nonlinear optical azopolymer diffraction grating: even and odd order parameters from far-field Raman and near-field second harmonic generation microscopies,” J. Phys. Chem. B 108(44), 17059–17068 (2004).
[CrossRef]

Stegeman, G. I.

Sugita, A.

A. Sugita, M. Morimoto, Y. Ishida, N. Mase, Y. Kawata, and S. Tasaka, “Linear and nonlinear optical properties of disperse red dyes in poly-(cyano phenylene sulfide),” Chem. Phys. Lett. 501(1–3), 39–43 (2010).
[CrossRef]

Takezoe, H.

K. Kajikawa, H. Takezoe, and A. Fukuda, “Symmetry and second order susceptibility of hemicyanine monolayer studied by surface second harmonic generations,” Jpn. J. Appl. Phys. 30(Part 1, No. 5), 1050–1062 (1991).
[CrossRef]

K. Shirota, K. Kajikawa, H. Takezoe, and A. Fukuda, “Molecular orientation in mixed layers of hemicyanine and fatty acid at air/water interface studied by second harmonic generations,” Jpn. J. Appl. Phys. 29(Part 1, No. 4), 750–755 (1990).
[CrossRef]

Tasaka, S.

A. Sugita, M. Morimoto, Y. Ishida, N. Mase, Y. Kawata, and S. Tasaka, “Linear and nonlinear optical properties of disperse red dyes in poly-(cyano phenylene sulfide),” Chem. Phys. Lett. 501(1–3), 39–43 (2010).
[CrossRef]

J. Ide, S. Tasaka, and N. Inagaki, “Nonelectrical poling in ferroelectric polycyanophenylenesulfides,” Jpn. J. Appl. Phys. 38(Part 1, No. 4A), 2049–2052 (1999).
[CrossRef]

Thibert, A.

C. Toro, A. Thibert, L. De Boni, A. E. Masunov, and F. E. Hernández, “Fluorescence emission of disperse Red 1 in solution at room temperature,” J. Phys. Chem. B 112(3), 929–937 (2008).
[CrossRef] [PubMed]

Toro, C.

C. Toro, A. Thibert, L. De Boni, A. E. Masunov, and F. E. Hernández, “Fluorescence emission of disperse Red 1 in solution at room temperature,” J. Phys. Chem. B 112(3), 929–937 (2008).
[CrossRef] [PubMed]

L. De Boni, C. Toro, A. E. Masunov, and F. E. Hernández, “Untangling the excited states of DR1 in solution: an experimental and theoretical study,” J. Phys. Chem. A 112(17), 3886–3890 (2008).
[CrossRef] [PubMed]

Tu, Y.

Y. Tu, Q. Zhang, and H. Ågren, “Electric field poled polymeric nonlinear optical systems: molecular dynamics simulations of poly(methyl methacrylate) doped with disperse red chromophores,” J. Phys. Chem. B 111(14), 3591–3598 (2007).
[CrossRef] [PubMed]

Wong, G. K.

Zhang, C. H.

Zhang, Q.

Y. Tu, Q. Zhang, and H. Ågren, “Electric field poled polymeric nonlinear optical systems: molecular dynamics simulations of poly(methyl methacrylate) doped with disperse red chromophores,” J. Phys. Chem. B 111(14), 3591–3598 (2007).
[CrossRef] [PubMed]

Zhang, T. G.

Chem. Phys. Lett.

A. Sugita, M. Morimoto, Y. Ishida, N. Mase, Y. Kawata, and S. Tasaka, “Linear and nonlinear optical properties of disperse red dyes in poly-(cyano phenylene sulfide),” Chem. Phys. Lett. 501(1–3), 39–43 (2010).
[CrossRef]

IEEE Trans. Electr. Insul.

E. Bihler, K. Holdik, and W. Eisenmenger, “Polarization distributions in isotropic, stretched or annealed PVDF films,” IEEE Trans. Electr. Insul. 24(3), 541–545 (1989).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. B

J. Phys. Chem. A

L. De Boni, C. Toro, A. E. Masunov, and F. E. Hernández, “Untangling the excited states of DR1 in solution: an experimental and theoretical study,” J. Phys. Chem. A 112(17), 3886–3890 (2008).
[CrossRef] [PubMed]

J. Phys. Chem. B

C. Toro, A. Thibert, L. De Boni, A. E. Masunov, and F. E. Hernández, “Fluorescence emission of disperse Red 1 in solution at room temperature,” J. Phys. Chem. B 112(3), 929–937 (2008).
[CrossRef] [PubMed]

F. Lagugné-Labarthet, C. Sourisseau, R. D. Schaller, R. J. Saykally, and P. Rochon, “Chromophore orientations in a nonlinear optical azopolymer diffraction grating: even and odd order parameters from far-field Raman and near-field second harmonic generation microscopies,” J. Phys. Chem. B 108(44), 17059–17068 (2004).
[CrossRef]

Y. Tu, Q. Zhang, and H. Ågren, “Electric field poled polymeric nonlinear optical systems: molecular dynamics simulations of poly(methyl methacrylate) doped with disperse red chromophores,” J. Phys. Chem. B 111(14), 3591–3598 (2007).
[CrossRef] [PubMed]

Jpn. J. Appl. Phys.

J. Ide, S. Tasaka, and N. Inagaki, “Nonelectrical poling in ferroelectric polycyanophenylenesulfides,” Jpn. J. Appl. Phys. 38(Part 1, No. 4A), 2049–2052 (1999).
[CrossRef]

K. Shirota, K. Kajikawa, H. Takezoe, and A. Fukuda, “Molecular orientation in mixed layers of hemicyanine and fatty acid at air/water interface studied by second harmonic generations,” Jpn. J. Appl. Phys. 29(Part 1, No. 4), 750–755 (1990).
[CrossRef]

K. Kajikawa, H. Takezoe, and A. Fukuda, “Symmetry and second order susceptibility of hemicyanine monolayer studied by surface second harmonic generations,” Jpn. J. Appl. Phys. 30(Part 1, No. 5), 1050–1062 (1991).
[CrossRef]

Opt. Lett.

Phys. Rev.

D. A. Kleinman, “Nonlinear dielectric polarization in optical media,” Phys. Rev. 126(6), 1977–1979 (1962).
[CrossRef]

Other

Y. R. Schen, The Principles of Nonlinear Optics (John Wiley & Sons, 2003).

R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic Press, 2008).

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, 1998).

H. S. Nalwa, T. Watanabe, and S. Miyata, “Organic materials for nonlinear optics,” in Nonlinear Optics of Organic Molecules and Polymers, H. S. Nalwa and S. Miyata, eds. (CRC Press, 1996), pp. 89–350.

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Figures (6)

Fig. 1
Fig. 1

Structure of (a) PCPS and (b) DR1.

Fig. 2
Fig. 2

(a) Linear absorption spectra of 10 w%-DR1-doped PCPS thin film (solid curve) and undoped PCPS thin film (dashed curve). (b) Linear refractive index spectrum of 10 w%-DR1-doped PCPS thin film.

Fig. 3
Fig. 3

Temperature dependence of the SH signal from 10 w% DR1-doped PCPS thin films during the annealing procedure.

Fig. 4
Fig. 4

Excitation beam polarization angle-dependence of SH signal intensity from 10 w% DR1-doped PCPS thin films.

Fig. 5
Fig. 5

(a) Definition of the molecular tilt angle of the NLO chromophores. (b) Schematics of the optical geometry for the SH measurements.

Fig. 6
Fig. 6

Dependence of (a) the SH signal intensity and (b) the calculated average molecular tilt angles on the film thickness.

Equations (5)

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

χ zzz =N L z ( 2ω ) L z ( ω ) 2 cos 3 Θ β ξξξ χ zii = 1 2 N L z ( 2ω ) L i ( ω ) 2 cosΘ sin 2 Θ β ξξξ χ izi = 1 2 N L i ( 2ω ) L z ( ω ) L i ( ω ) cosΘ sin 2 Θ β ξξξ
I SH ( γ p ,θ )=K | χ eff | 2 sin 2 ( Ψ ) I pump 2 χ eff =A χ zzz cos 2 γ p B χ xzx cos 2 γ p +C χ zxx cos 2 γ p +D χ zyy sin 2 γ p Ψ= ωL c [ n( ω )cos θ 2 ( ω )n( 2ω )cos θ 2 ( 2ω ) ]
A= 4 n 2 ( ω ) sin 2 θ 2 ( ω ) cos 2 θ 1 sin 2 θ 1 sin 2 ( θ 1 + θ 2 ( ω ) ) cos 2 ( θ 1 θ 2 ( ω ) ) B= 4 n( ω ) sin 2 θ 2 ( ω )cos θ 2 ( ω )sin θ 1 cos 2 θ 1 sin 2 ( θ 1 + θ 2 ( ω ) ) cos 2 ( θ 1 θ 2 ( ω ) ) C=4 sin 2 θ 2 ( ω ) cos 2 θ 2 ( ω ) cos 2 θ 1 sin 2 ( θ 1 + θ 2 ( ω ) ) cos 2 ( θ 1 θ 2 ( ω ) ) D=4 sin 2 θ 2 ( ω ) cos 2 θ 1 sin 2 ( θ 1 + θ 2 ( ω ) )
tan 2 Θ= 2 χ zii χ zzz .
L c = λ 4| n( ω )cos θ 2 ( ω )n( 2ω )cos θ 2 ( 2ω ) | .

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