Abstract

We present here the quasi-phase-matched second-harmonic (SH) light of holographic all-optical poling measured along the off-axis direction of the wave vector of 4kωk2ω+Δk with Δk=0, where kω is the wave vector of ω writing beam, k2ω is the wave vector of 2ω writing beam, and Δk is the wave-vector mismatch whose vector is parallel to kω. As expected, Δk0 of the phase-mismatched condition gave rise to a very low SH signal. Furthermore, tensor components of χ(2) gratings were analyzed, and the contribution of the photoisomeric state of the cis configuration of azobenzene chromophore studied was discussed.

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. P. Yeh, Introduction to Photorefractive Nonlinear Optics(Wiley-Interscience, 1993), Chap. 12.
  2. Y. Fujii, B. S. Kawasaki, K. O. Hill, and D. C. Johnson, “Sum-frequency light generation in optical fibers,” Opt. Lett. 5, 48–50(1980).
    [CrossRef] [PubMed]
  3. U. Österberg and W. Margulis, “Dye laser pumped by Nd:YAG laser pulses frequency doubling in a glass optical fiber,” Opt. Lett. 11, 516–518 (1986).
    [CrossRef] [PubMed]
  4. R. H. Stolen and H. W. K. Tom, “Self-organized phase-matched harmonic generation in optical fibers,” Opt. Lett. 12, 585–587(1987).
    [CrossRef] [PubMed]
  5. C. Fiorini, F. Charra, J. M. Nunzi, and P. Raimond, “Photoinduced non centrosymmetry in azo-dye polymers,” Nonlinear Opt. 9, 339–347 (1995).
  6. C. Fiorini, F. Charra, P. Raimond, and J.-M. Nunzi, “All-optical induction of noncentrosymmetry in a transparent nonlinear polymer rod,” Opt. Lett. 22, 1846–1848 (1997).
    [CrossRef]
  7. J. Si, J. Qiu, and K. Hirao, “Controlling the noncentrosymmetry of azodye-doped polymers by nonresonant dual-frequency coherent excitation,” Appl. Phys. Lett. 77, 3887–3889 (2000).
    [CrossRef]
  8. A. Apostoluk, C. Fiorini-Debuisschert, and J.-M. Nunzi, “All optical in polymers and applications,” in Photoreactive Organic Thin Films, Z.Sekkat and W.Knoll, eds. (Academic, 2002), pp. 331–365.
    [CrossRef]
  9. S. Bidault, J. Gouya, S. Brasselet, and J. Zyss, “Encoding multipolar polarization patterns by optical poling in polymers: towards nonlinear optical memories,” Opt. Express 13, 505–510(2005).
    [CrossRef] [PubMed]
  10. K. Danel, K. Ozga, and I. V. Kityk, “Circularly light-induced electrogyration in the arylethynyl derivatives incorporated within the oligoetheracrylate photopolymer matrices,” Chem. Phys. 313, 33–38 (2005).
    [CrossRef]
  11. N. Tsutsumi, Y. Ikeyama, and W. Sakai, “Self-organized phase-matched second harmonic generation from the all optically induced χ(2) polar structures,” Nonlinear Opt. 22, 63–66 (1999).
  12. N. Tsutsumi, J. Yamamoto, and W. Sakai, “Diffraction measurement for grating formed in optically poled polymeric materials,” Jpn. J. Appl. Phys. 40, 2264–2268 (2001).
    [CrossRef]
  13. N. Tsutsumi, M. Imamura, W. Sakai, Y. Nagase, N. Nemoto, Y. Tian, and J. Abe, “All optically induced χ(2) structures and their optical anisotropy in betaine dispersed in polymer matrix,” Jpn. J. Appl. Phys. 41, 5247–5253 (2002).
    [CrossRef]
  14. N. Tsutsumi and C. Odane, “Phase-matched noncentrosymmetric polarization in a polymeric waveguide induced by all-optical poling,” J. Opt. Soc. Am. B 20, 1514–1519 (2003).
    [CrossRef]
  15. N. Tsutsumi and K. Nakatani, “χ(2) polarization induced in molecular glass of conjugated compound by all-optical poling,” Opt. Commun. 259, 852–855 (2006).
    [CrossRef]
  16. J. Si and K. Hirao, “Phase-matched second-harmonic generation in cross-linking polyurethane films by thermal-assisted optical poling,” Appl. Phys. Lett. 91, 091105 (2007).
    [CrossRef]
  17. R. Barille, A. Samac, M. Samac, B. Luther-Davies, and J.-M. Nunzi, “Stable frequency doubling by all-optical poling in dye-doped polymer optical fibers,” Opt. Lett. 35, 3595–3597(2010).
    [CrossRef] [PubMed]
  18. N. Tsutsumi and T. Shingu, “χ(2) holography induced by all-optical poling,” Chem. Phys. Lett. 403, 420–424(2005).
    [CrossRef]
  19. N. Tsutsumi and Y. Ikegami, “Second-order optical nonlinearities from χ(2) gratings induced by holographic all-optical poling,” Opt. Commun. 281, 5905–5909 (2008).
    [CrossRef]

2010

2008

N. Tsutsumi and Y. Ikegami, “Second-order optical nonlinearities from χ(2) gratings induced by holographic all-optical poling,” Opt. Commun. 281, 5905–5909 (2008).
[CrossRef]

2007

J. Si and K. Hirao, “Phase-matched second-harmonic generation in cross-linking polyurethane films by thermal-assisted optical poling,” Appl. Phys. Lett. 91, 091105 (2007).
[CrossRef]

2006

N. Tsutsumi and K. Nakatani, “χ(2) polarization induced in molecular glass of conjugated compound by all-optical poling,” Opt. Commun. 259, 852–855 (2006).
[CrossRef]

2005

S. Bidault, J. Gouya, S. Brasselet, and J. Zyss, “Encoding multipolar polarization patterns by optical poling in polymers: towards nonlinear optical memories,” Opt. Express 13, 505–510(2005).
[CrossRef] [PubMed]

N. Tsutsumi and T. Shingu, “χ(2) holography induced by all-optical poling,” Chem. Phys. Lett. 403, 420–424(2005).
[CrossRef]

K. Danel, K. Ozga, and I. V. Kityk, “Circularly light-induced electrogyration in the arylethynyl derivatives incorporated within the oligoetheracrylate photopolymer matrices,” Chem. Phys. 313, 33–38 (2005).
[CrossRef]

2003

2002

N. Tsutsumi, M. Imamura, W. Sakai, Y. Nagase, N. Nemoto, Y. Tian, and J. Abe, “All optically induced χ(2) structures and their optical anisotropy in betaine dispersed in polymer matrix,” Jpn. J. Appl. Phys. 41, 5247–5253 (2002).
[CrossRef]

2001

N. Tsutsumi, J. Yamamoto, and W. Sakai, “Diffraction measurement for grating formed in optically poled polymeric materials,” Jpn. J. Appl. Phys. 40, 2264–2268 (2001).
[CrossRef]

2000

J. Si, J. Qiu, and K. Hirao, “Controlling the noncentrosymmetry of azodye-doped polymers by nonresonant dual-frequency coherent excitation,” Appl. Phys. Lett. 77, 3887–3889 (2000).
[CrossRef]

1999

N. Tsutsumi, Y. Ikeyama, and W. Sakai, “Self-organized phase-matched second harmonic generation from the all optically induced χ(2) polar structures,” Nonlinear Opt. 22, 63–66 (1999).

1997

1995

C. Fiorini, F. Charra, J. M. Nunzi, and P. Raimond, “Photoinduced non centrosymmetry in azo-dye polymers,” Nonlinear Opt. 9, 339–347 (1995).

1987

1986

1980

Abe, J.

N. Tsutsumi, M. Imamura, W. Sakai, Y. Nagase, N. Nemoto, Y. Tian, and J. Abe, “All optically induced χ(2) structures and their optical anisotropy in betaine dispersed in polymer matrix,” Jpn. J. Appl. Phys. 41, 5247–5253 (2002).
[CrossRef]

Apostoluk, A.

A. Apostoluk, C. Fiorini-Debuisschert, and J.-M. Nunzi, “All optical in polymers and applications,” in Photoreactive Organic Thin Films, Z.Sekkat and W.Knoll, eds. (Academic, 2002), pp. 331–365.
[CrossRef]

Barille, R.

Bidault, S.

Brasselet, S.

Charra, F.

C. Fiorini, F. Charra, P. Raimond, and J.-M. Nunzi, “All-optical induction of noncentrosymmetry in a transparent nonlinear polymer rod,” Opt. Lett. 22, 1846–1848 (1997).
[CrossRef]

C. Fiorini, F. Charra, J. M. Nunzi, and P. Raimond, “Photoinduced non centrosymmetry in azo-dye polymers,” Nonlinear Opt. 9, 339–347 (1995).

Danel, K.

K. Danel, K. Ozga, and I. V. Kityk, “Circularly light-induced electrogyration in the arylethynyl derivatives incorporated within the oligoetheracrylate photopolymer matrices,” Chem. Phys. 313, 33–38 (2005).
[CrossRef]

Fiorini, C.

C. Fiorini, F. Charra, P. Raimond, and J.-M. Nunzi, “All-optical induction of noncentrosymmetry in a transparent nonlinear polymer rod,” Opt. Lett. 22, 1846–1848 (1997).
[CrossRef]

C. Fiorini, F. Charra, J. M. Nunzi, and P. Raimond, “Photoinduced non centrosymmetry in azo-dye polymers,” Nonlinear Opt. 9, 339–347 (1995).

Fiorini-Debuisschert, C.

A. Apostoluk, C. Fiorini-Debuisschert, and J.-M. Nunzi, “All optical in polymers and applications,” in Photoreactive Organic Thin Films, Z.Sekkat and W.Knoll, eds. (Academic, 2002), pp. 331–365.
[CrossRef]

Fujii, Y.

Gouya, J.

Hill, K. O.

Hirao, K.

J. Si and K. Hirao, “Phase-matched second-harmonic generation in cross-linking polyurethane films by thermal-assisted optical poling,” Appl. Phys. Lett. 91, 091105 (2007).
[CrossRef]

J. Si, J. Qiu, and K. Hirao, “Controlling the noncentrosymmetry of azodye-doped polymers by nonresonant dual-frequency coherent excitation,” Appl. Phys. Lett. 77, 3887–3889 (2000).
[CrossRef]

Ikegami, Y.

N. Tsutsumi and Y. Ikegami, “Second-order optical nonlinearities from χ(2) gratings induced by holographic all-optical poling,” Opt. Commun. 281, 5905–5909 (2008).
[CrossRef]

Ikeyama, Y.

N. Tsutsumi, Y. Ikeyama, and W. Sakai, “Self-organized phase-matched second harmonic generation from the all optically induced χ(2) polar structures,” Nonlinear Opt. 22, 63–66 (1999).

Imamura, M.

N. Tsutsumi, M. Imamura, W. Sakai, Y. Nagase, N. Nemoto, Y. Tian, and J. Abe, “All optically induced χ(2) structures and their optical anisotropy in betaine dispersed in polymer matrix,” Jpn. J. Appl. Phys. 41, 5247–5253 (2002).
[CrossRef]

Johnson, D. C.

Kawasaki, B. S.

Kityk, I. V.

K. Danel, K. Ozga, and I. V. Kityk, “Circularly light-induced electrogyration in the arylethynyl derivatives incorporated within the oligoetheracrylate photopolymer matrices,” Chem. Phys. 313, 33–38 (2005).
[CrossRef]

Luther-Davies, B.

Margulis, W.

Nagase, Y.

N. Tsutsumi, M. Imamura, W. Sakai, Y. Nagase, N. Nemoto, Y. Tian, and J. Abe, “All optically induced χ(2) structures and their optical anisotropy in betaine dispersed in polymer matrix,” Jpn. J. Appl. Phys. 41, 5247–5253 (2002).
[CrossRef]

Nakatani, K.

N. Tsutsumi and K. Nakatani, “χ(2) polarization induced in molecular glass of conjugated compound by all-optical poling,” Opt. Commun. 259, 852–855 (2006).
[CrossRef]

Nemoto, N.

N. Tsutsumi, M. Imamura, W. Sakai, Y. Nagase, N. Nemoto, Y. Tian, and J. Abe, “All optically induced χ(2) structures and their optical anisotropy in betaine dispersed in polymer matrix,” Jpn. J. Appl. Phys. 41, 5247–5253 (2002).
[CrossRef]

Nunzi, J. M.

C. Fiorini, F. Charra, J. M. Nunzi, and P. Raimond, “Photoinduced non centrosymmetry in azo-dye polymers,” Nonlinear Opt. 9, 339–347 (1995).

Nunzi, J.-M.

Odane, C.

Österberg, U.

Ozga, K.

K. Danel, K. Ozga, and I. V. Kityk, “Circularly light-induced electrogyration in the arylethynyl derivatives incorporated within the oligoetheracrylate photopolymer matrices,” Chem. Phys. 313, 33–38 (2005).
[CrossRef]

Qiu, J.

J. Si, J. Qiu, and K. Hirao, “Controlling the noncentrosymmetry of azodye-doped polymers by nonresonant dual-frequency coherent excitation,” Appl. Phys. Lett. 77, 3887–3889 (2000).
[CrossRef]

Raimond, P.

C. Fiorini, F. Charra, P. Raimond, and J.-M. Nunzi, “All-optical induction of noncentrosymmetry in a transparent nonlinear polymer rod,” Opt. Lett. 22, 1846–1848 (1997).
[CrossRef]

C. Fiorini, F. Charra, J. M. Nunzi, and P. Raimond, “Photoinduced non centrosymmetry in azo-dye polymers,” Nonlinear Opt. 9, 339–347 (1995).

Sakai, W.

N. Tsutsumi, M. Imamura, W. Sakai, Y. Nagase, N. Nemoto, Y. Tian, and J. Abe, “All optically induced χ(2) structures and their optical anisotropy in betaine dispersed in polymer matrix,” Jpn. J. Appl. Phys. 41, 5247–5253 (2002).
[CrossRef]

N. Tsutsumi, J. Yamamoto, and W. Sakai, “Diffraction measurement for grating formed in optically poled polymeric materials,” Jpn. J. Appl. Phys. 40, 2264–2268 (2001).
[CrossRef]

N. Tsutsumi, Y. Ikeyama, and W. Sakai, “Self-organized phase-matched second harmonic generation from the all optically induced χ(2) polar structures,” Nonlinear Opt. 22, 63–66 (1999).

Samac, A.

Samac, M.

Shingu, T.

N. Tsutsumi and T. Shingu, “χ(2) holography induced by all-optical poling,” Chem. Phys. Lett. 403, 420–424(2005).
[CrossRef]

Si, J.

J. Si and K. Hirao, “Phase-matched second-harmonic generation in cross-linking polyurethane films by thermal-assisted optical poling,” Appl. Phys. Lett. 91, 091105 (2007).
[CrossRef]

J. Si, J. Qiu, and K. Hirao, “Controlling the noncentrosymmetry of azodye-doped polymers by nonresonant dual-frequency coherent excitation,” Appl. Phys. Lett. 77, 3887–3889 (2000).
[CrossRef]

Stolen, R. H.

Tian, Y.

N. Tsutsumi, M. Imamura, W. Sakai, Y. Nagase, N. Nemoto, Y. Tian, and J. Abe, “All optically induced χ(2) structures and their optical anisotropy in betaine dispersed in polymer matrix,” Jpn. J. Appl. Phys. 41, 5247–5253 (2002).
[CrossRef]

Tom, H. W. K.

Tsutsumi, N.

N. Tsutsumi and Y. Ikegami, “Second-order optical nonlinearities from χ(2) gratings induced by holographic all-optical poling,” Opt. Commun. 281, 5905–5909 (2008).
[CrossRef]

N. Tsutsumi and K. Nakatani, “χ(2) polarization induced in molecular glass of conjugated compound by all-optical poling,” Opt. Commun. 259, 852–855 (2006).
[CrossRef]

N. Tsutsumi and T. Shingu, “χ(2) holography induced by all-optical poling,” Chem. Phys. Lett. 403, 420–424(2005).
[CrossRef]

N. Tsutsumi and C. Odane, “Phase-matched noncentrosymmetric polarization in a polymeric waveguide induced by all-optical poling,” J. Opt. Soc. Am. B 20, 1514–1519 (2003).
[CrossRef]

N. Tsutsumi, M. Imamura, W. Sakai, Y. Nagase, N. Nemoto, Y. Tian, and J. Abe, “All optically induced χ(2) structures and their optical anisotropy in betaine dispersed in polymer matrix,” Jpn. J. Appl. Phys. 41, 5247–5253 (2002).
[CrossRef]

N. Tsutsumi, J. Yamamoto, and W. Sakai, “Diffraction measurement for grating formed in optically poled polymeric materials,” Jpn. J. Appl. Phys. 40, 2264–2268 (2001).
[CrossRef]

N. Tsutsumi, Y. Ikeyama, and W. Sakai, “Self-organized phase-matched second harmonic generation from the all optically induced χ(2) polar structures,” Nonlinear Opt. 22, 63–66 (1999).

Yamamoto, J.

N. Tsutsumi, J. Yamamoto, and W. Sakai, “Diffraction measurement for grating formed in optically poled polymeric materials,” Jpn. J. Appl. Phys. 40, 2264–2268 (2001).
[CrossRef]

Yeh, P.

P. Yeh, Introduction to Photorefractive Nonlinear Optics(Wiley-Interscience, 1993), Chap. 12.

Zyss, J.

Appl. Phys. Lett.

J. Si, J. Qiu, and K. Hirao, “Controlling the noncentrosymmetry of azodye-doped polymers by nonresonant dual-frequency coherent excitation,” Appl. Phys. Lett. 77, 3887–3889 (2000).
[CrossRef]

J. Si and K. Hirao, “Phase-matched second-harmonic generation in cross-linking polyurethane films by thermal-assisted optical poling,” Appl. Phys. Lett. 91, 091105 (2007).
[CrossRef]

Chem. Phys.

K. Danel, K. Ozga, and I. V. Kityk, “Circularly light-induced electrogyration in the arylethynyl derivatives incorporated within the oligoetheracrylate photopolymer matrices,” Chem. Phys. 313, 33–38 (2005).
[CrossRef]

Chem. Phys. Lett.

N. Tsutsumi and T. Shingu, “χ(2) holography induced by all-optical poling,” Chem. Phys. Lett. 403, 420–424(2005).
[CrossRef]

J. Opt. Soc. Am. B

Jpn. J. Appl. Phys.

N. Tsutsumi, J. Yamamoto, and W. Sakai, “Diffraction measurement for grating formed in optically poled polymeric materials,” Jpn. J. Appl. Phys. 40, 2264–2268 (2001).
[CrossRef]

N. Tsutsumi, M. Imamura, W. Sakai, Y. Nagase, N. Nemoto, Y. Tian, and J. Abe, “All optically induced χ(2) structures and their optical anisotropy in betaine dispersed in polymer matrix,” Jpn. J. Appl. Phys. 41, 5247–5253 (2002).
[CrossRef]

Nonlinear Opt.

N. Tsutsumi, Y. Ikeyama, and W. Sakai, “Self-organized phase-matched second harmonic generation from the all optically induced χ(2) polar structures,” Nonlinear Opt. 22, 63–66 (1999).

C. Fiorini, F. Charra, J. M. Nunzi, and P. Raimond, “Photoinduced non centrosymmetry in azo-dye polymers,” Nonlinear Opt. 9, 339–347 (1995).

Opt. Commun.

N. Tsutsumi and K. Nakatani, “χ(2) polarization induced in molecular glass of conjugated compound by all-optical poling,” Opt. Commun. 259, 852–855 (2006).
[CrossRef]

N. Tsutsumi and Y. Ikegami, “Second-order optical nonlinearities from χ(2) gratings induced by holographic all-optical poling,” Opt. Commun. 281, 5905–5909 (2008).
[CrossRef]

Opt. Express

Opt. Lett.

Other

P. Yeh, Introduction to Photorefractive Nonlinear Optics(Wiley-Interscience, 1993), Chap. 12.

A. Apostoluk, C. Fiorini-Debuisschert, and J.-M. Nunzi, “All optical in polymers and applications,” in Photoreactive Organic Thin Films, Z.Sekkat and W.Knoll, eds. (Academic, 2002), pp. 331–365.
[CrossRef]

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 (5)

Fig. 1
Fig. 1

Schematic of wave-vector conservation.

Fig. 2
Fig. 2

(a) Dependence of SH intensity from photoinduced χ ( 2 ) gratings on energy of 2 ω writing beam for various holographic internal angle 2 θ between ω and 2 ω beams. (b) Plots of SH intensity as a function of holographic internal angle 2 θ between ω and 2 ω beams.

Fig. 3
Fig. 3

Schematic holographic χ ( 2 ) gratings in the sample.

Fig. 4
Fig. 4

Dependence of square root of SH signal amplitude on the polarization angle of reading ω beam. Polarization direction of back polarizer (analyzer): horizontal direction in (a), vertical direction in (b). Solid curves are theoretical fittings by Eqs. (11, 12).

Fig. 5
Fig. 5

Absorption spectrum of DR19 polymer film.

Tables (3)

Tables Icon

Table 1 Summary of χ Tensors a

Tables Icon

Table 2 Summary of First-Order Hyperpolarizabilities of Trans and Cis Forms of DR19 Chromophore Calculated by an SOS Method Using WinMopac 3.0 a

Tables Icon

Table 3 χ Tensor Ratios Calculated from Eqs. (25, 26, 27, 28, 29, 30) with First Hyperpolarizabilities Listed in Table 2 for Trans Form and Cis Form of DR19 Chromophore a

Equations (32)

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

k 2 ω SHG 1 = k ω + k ω + K G = k 2 ω ,
k 2 ω SHG 2 = k ω + k ω K G + Δ k = 4 k ω k 2 ω + Δ k = k SHG ,
Δ k = 4 π λ 2 ω ( n 2 ω cos 2 θ n ω ) ,
cos 2 θ = n ω n 2 ω
1 | k ω | cos ( β θ ) = 2 | k 2 ω | cos ( β + θ ) ,
tan β = 1 tan θ | k 2 ω | 2 | k ω | | k 2 ω | + 2 | k ω | .
Λ χ ( 2 ) = 2 π | k 2 ω | cos ( β θ ) cos ( 90 2 θ ) = π | k 2 ω | ( cos β sin θ + sin β cos θ ) .
Λ χ ( 2 ) = 2 π { sin 2 θ ( | k 2 ω | + 2 | k ω | ) 2 + cos 2 θ ( | k 2 ω | 2 | k ω | ) 2 } 1 / 2 ,
Λ χ ( 2 ) = 2 π | k 2 ω | 2 | k ω | .
Λ χ ( 2 ) = 2 π | k 2 ω | + 2 | k ω | .
I x 2 ω | χ x x x cos 2 δ + χ x y y sin 2 δ + 2 χ x x y sin δ cos δ | E ω 2 ,
I y 2 ω | χ y x x cos 2 δ + χ y y y sin 2 δ + 2 χ y x y sin δ cos δ | E ω 2 ,
χ i j k ( 2 ) = 2 d i j k = 2 N f i 2 ω f j ω f k ω b i j k ,
f L γ = ( n L γ ) 2 + 2 3 ,
b x x x = cos 3 θ β X X X + cos 2 α sin 2 θ cos θ β X Y Y + 2 cos 2 α sin 2 θ cos θ β Y X Y = L 3 ( p ) β X X X + cos 2 α ( L 1 ( p ) L 3 ( p ) ) β X Y Y + 2 cos 2 α ( L 1 ( p ) L 3 ( p ) ) β Y X Y ,
b x y y = cos 2 ϕ sin 2 θ cos θ β X X X + ( cos 2 α cos 2 ϕ cos 3 θ + sin 2 α sin 2 ϕ cos θ ) β X Y Y 2 cos 2 α cos 2 ϕ sin 2 θ cos θ β Y X Y = cos 2 ϕ ( L 1 ( p ) L 3 ( p ) ) β X X X + ( cos 2 α cos 2 ϕ L 3 ( p ) + sin 2 α sin 2 ϕ L 1 ( p ) ) β X Y Y 2 cos 2 α cos 2 ϕ ( L 1 ( p ) L 3 ( p ) ) β Y X Y ,
b y x y = cos 2 α sin 2 θ cos θ β X X X cos 2 α cos 2 ϕ sin 2 θ cos θ β X Y Y + ( cos 2 α cos 2 ϕ cos 3 θ + sin 2 α sin 2 ϕ cos θ cos 2 α cos 2 ϕ sin 2 θ cos θ β Y X Y = cos 2 α ( L 1 ( p ) L 3 ( p ) ) β X X X cos 2 α cos 2 ϕ ( L 1 ( p ) L 3 ( p ) β X Y Y + ( cos 2 α cos 2 ϕ L 3 ( p ) + sin 2 α sin 2 ϕ L 1 ( p ) cos 2 α cos 2 ϕ ( L 1 ( p ) L 3 ( p ) ) β Y X Y ,
b y x x = cos α cos ϕ ( cos 3 θ β Y X X + cos 2 α sin 2 θ cos θ β Y Y Y 2 sin 2 θ cos θ β X X Y ) = cos α cos ϕ ( L 3 ( p ) β Y X X + cos 2 α ( L 1 ( p ) L 3 ( p ) ) β Y Y Y 2 ( L 1 ( p ) L 3 ( p ) ) β X X Y ) ,
b y y y = cos α cos ϕ ( cos 2 α sin 2 θ cos θ β Y X X + cos 2 α cos 2 ϕ ( cos 3 θ + 3 cos θ ) β Y Y Y + 2 cos 2 α sin 2 θ cos θ β X X Y ) = cos α cos ϕ ( cos 2 α ( L 1 ( p ) L 3 ( p ) ) β Y X X + cos 2 α cos 2 ϕ ( L 3 ( p ) + 3 L 1 ( p ) ) β Y Y Y + 2 cos 2 α ( L 1 ( p ) L 3 ( p ) ) β X X Y ) ,
b x x y = cos α cos ϕ ( sin 2 θ cos θ β Y X X + cos 2 α sin 2 θ cos θ β Y Y Y + ( cos 3 θ sin 2 θ cos θ ) β X X Y ) = cos α cos ϕ ( ( L 1 ( p ) L 3 ( p ) ) β Y X X + cos 2 α ( L 1 ( p ) L 3 ( p ) ) β Y Y Y + ( L 3 ( p ) ( L 1 ( p ) L 3 ( p ) ) ) β X X Y ) ,
cos θ = L 1 ( p ) = coth ( p ) 1 p ,
cos 3 θ = L 3 ( p ) = ( 1 + 6 p 2 ) coth ( p ) ( 3 p + 6 p 3 ) = ( 1 + 6 p 2 ) L 1 ( p ) 2 p ,
p = μ E k T ,
cos 2 α = 1 2 π 0 2 π cos 2 α d α = 1 2 , cos 2 ϕ = 1 2 π 0 2 π cos 2 ϕ d ϕ = 1 2 , sin 2 α = 1 2 π 0 2 π sin 2 α d α = 1 2 , sin 2 ϕ = 1 2 π 0 2 π sin 2 ϕ d ϕ = 1 2
b x x x = μ E 15 k T ( 3 β X X X + β X Y Y + 2 β Y X Y ) ,
b x y y = μ E 15 k T ( β X X X + 2 β X Y Y β Y X Y ) ,
b y x y = μ E 15 k T ( β X X X 1 2 β X Y Y + 3 2 β Y X Y ) ,
b y x x = μ E 15 k T ( 3 2 β Y X X + 1 2 β Y Y Y 2 β X X Y ) ,
b y y y = μ E 15 k T ( 1 2 β Y X X + 9 4 β Y Y Y + β X X Y ) ,
b x x y = μ E 15 k T ( β Y X X + 1 2 β Y Y Y + 1 2 β X X Y ) .
b x x x = 3 b x y y = 3 b y x y
χ x x x ( 2 ) χ x y y ( 2 ) χ x x y ( 2 ) χ y x x ( 2 ) χ y y y ( 2 ) χ y x y ( 2 ) = b x x x b x y y b x x y b y x x b y y y b y x y = 3 1 0 0 0 1

Metrics