Abstract

We demonstrate the existence of χ(5) nonlinearities induced in a transparent glass by intense femtosecond pulses at 620 nm. Polarization selectivity associated with a transient-grating technique allows us to discriminate this contribution from the electronic third-order one. A value of 5 is measured for the ratio between symmetric and nonsymmetric elements for this fifth-order susceptibility. This value corresponds to the expected value deduced from the anharmonic model presented and is definitely attributed to a direct six-wave mixing process.

© 1993 Optical Society of America

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  1. A. Maruani, J. L. Oudar, E. Batifol, and D. S. Chemla, “Nonlinear spectroscopy of biexcitons in CuCl by resonant coherent scattering,” Phys. Rev. Lett. 41, 1372–1375 (1978).
    [Crossref]
  2. R. K. Raj, Q. F. Gao, D. Bloch, and M. Ducloy, “Direct observation of higher-order optical susceptibilities via angularly-resolved multiwave mixing,” Opt. Commun. 51, 117–120 (1984).
    [Crossref]
  3. H. Saito and E. O. Gobel, “Mott-transition-induced period doubling in transient-grating experiments in CdS,” Opt. Lett. 11, 354–356 (1986).
    [Crossref] [PubMed]
  4. I. Thomazeau, J. Etchepare, G. Grillon, and A. Migus, “Electronic nonlinear optical susceptibilities of silicate glasses,” Opt. Lett. 10, 223–225 (1985).
    [Crossref] [PubMed]
  5. J. Etchepare, G. Grillon, J. P. Chambaret, G. Hamoniaux, and A. Orszag, “Polarization selectivity in time resolved transient phase grating,” Opt. Commun. 63, 329–334 (1987).
    [Crossref]
  6. J. Etchepare, G. Grillon, I. Thomazeau, and A. Orszag, “Phase grating approach to susceptibility tensors: determination in isotropic media,” Ultrafast Phenomena V (Springer-Verlag, Berlin, 1986), pp. 504–509.
    [Crossref]
  7. F. A. Hopf and G. I. Stegeman, Applied Classical Electrodynamics, Vol. 2: Nonlinear Optics (Wiley, New York, 1985), p. 95.
  8. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).
  9. H. E. Lorentz, The Theory of Electrons, 2nd ed. (Dover, New York, 1952).
  10. J. A. Armstrong, N. Bloemberger, J. Ducuing, and P. S. Pershan, “Interaction between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
    [Crossref]
  11. A. Owyoung, “The origins of the nonlinear refractive indices of liquids and glasses,” Ph.D. dissertation (California Institute of Technology, Pasadena, Calif., 1971).
  12. L. H. Acioli, A. S. L. Gomez, and J. R. Rios Leite, “Measurement of high-order optical nonlinear susceptibilities in semiconductor-doped glasses,” Appl. Phys. Lett. 53, 1788–1790 (1988).
    [Crossref]
  13. A. Blouin, P. Galarneau, and M. M. Denariez-Roberge, “Degenerate six-wave-mixing using high order Bragg diffraction in semiconductor-doped glass,” Opt. Commun. 72, 249–252 (1989).
    [Crossref]
  14. G. Manneberg, M. Gustafsson, P. Unsbo, and F. Saeidi, “Image transfer using six-wave mixing with large-wavelength conversion,” J. Opt. Soc. Am. B 10, 454–458 (1993).
    [Crossref]
  15. L. L. Chase, M. L. Claude, D. Hulin, and A. Mysyrowicz, “Optical phase conjugation based on the giant two-photon resonance of the biexcitonic molecule in CuCl,” Phys. Rev. A 28, 3696–3698 (1983).
    [Crossref]
  16. M. Ducloy and D. Bloch, “Polarization properties of phase conjugate mirrors,” Phys. Rev. A 30, 3107–3122 (1984).
    [Crossref]
  17. M. S. Malcuit, D. J. Gauthier, and R. W. Boyd, “Vector phase conjugation by two-photon-resonant degenerate four-wave mixing,” Opt. Lett. 13, 663–665 (1988).
    [Crossref] [PubMed]

1993 (1)

1989 (1)

A. Blouin, P. Galarneau, and M. M. Denariez-Roberge, “Degenerate six-wave-mixing using high order Bragg diffraction in semiconductor-doped glass,” Opt. Commun. 72, 249–252 (1989).
[Crossref]

1988 (2)

L. H. Acioli, A. S. L. Gomez, and J. R. Rios Leite, “Measurement of high-order optical nonlinear susceptibilities in semiconductor-doped glasses,” Appl. Phys. Lett. 53, 1788–1790 (1988).
[Crossref]

M. S. Malcuit, D. J. Gauthier, and R. W. Boyd, “Vector phase conjugation by two-photon-resonant degenerate four-wave mixing,” Opt. Lett. 13, 663–665 (1988).
[Crossref] [PubMed]

1987 (1)

J. Etchepare, G. Grillon, J. P. Chambaret, G. Hamoniaux, and A. Orszag, “Polarization selectivity in time resolved transient phase grating,” Opt. Commun. 63, 329–334 (1987).
[Crossref]

1986 (1)

1985 (1)

1984 (2)

R. K. Raj, Q. F. Gao, D. Bloch, and M. Ducloy, “Direct observation of higher-order optical susceptibilities via angularly-resolved multiwave mixing,” Opt. Commun. 51, 117–120 (1984).
[Crossref]

M. Ducloy and D. Bloch, “Polarization properties of phase conjugate mirrors,” Phys. Rev. A 30, 3107–3122 (1984).
[Crossref]

1983 (1)

L. L. Chase, M. L. Claude, D. Hulin, and A. Mysyrowicz, “Optical phase conjugation based on the giant two-photon resonance of the biexcitonic molecule in CuCl,” Phys. Rev. A 28, 3696–3698 (1983).
[Crossref]

1978 (1)

A. Maruani, J. L. Oudar, E. Batifol, and D. S. Chemla, “Nonlinear spectroscopy of biexcitons in CuCl by resonant coherent scattering,” Phys. Rev. Lett. 41, 1372–1375 (1978).
[Crossref]

1962 (1)

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

Acioli, L. H.

L. H. Acioli, A. S. L. Gomez, and J. R. Rios Leite, “Measurement of high-order optical nonlinear susceptibilities in semiconductor-doped glasses,” Appl. Phys. Lett. 53, 1788–1790 (1988).
[Crossref]

Armstrong, J. A.

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

Batifol, E.

A. Maruani, J. L. Oudar, E. Batifol, and D. S. Chemla, “Nonlinear spectroscopy of biexcitons in CuCl by resonant coherent scattering,” Phys. Rev. Lett. 41, 1372–1375 (1978).
[Crossref]

Bloch, D.

R. K. Raj, Q. F. Gao, D. Bloch, and M. Ducloy, “Direct observation of higher-order optical susceptibilities via angularly-resolved multiwave mixing,” Opt. Commun. 51, 117–120 (1984).
[Crossref]

M. Ducloy and D. Bloch, “Polarization properties of phase conjugate mirrors,” Phys. Rev. A 30, 3107–3122 (1984).
[Crossref]

Bloemberger, N.

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

Blouin, A.

A. Blouin, P. Galarneau, and M. M. Denariez-Roberge, “Degenerate six-wave-mixing using high order Bragg diffraction in semiconductor-doped glass,” Opt. Commun. 72, 249–252 (1989).
[Crossref]

Boyd, R. W.

Chambaret, J. P.

J. Etchepare, G. Grillon, J. P. Chambaret, G. Hamoniaux, and A. Orszag, “Polarization selectivity in time resolved transient phase grating,” Opt. Commun. 63, 329–334 (1987).
[Crossref]

Chase, L. L.

L. L. Chase, M. L. Claude, D. Hulin, and A. Mysyrowicz, “Optical phase conjugation based on the giant two-photon resonance of the biexcitonic molecule in CuCl,” Phys. Rev. A 28, 3696–3698 (1983).
[Crossref]

Chemla, D. S.

A. Maruani, J. L. Oudar, E. Batifol, and D. S. Chemla, “Nonlinear spectroscopy of biexcitons in CuCl by resonant coherent scattering,” Phys. Rev. Lett. 41, 1372–1375 (1978).
[Crossref]

Claude, M. L.

L. L. Chase, M. L. Claude, D. Hulin, and A. Mysyrowicz, “Optical phase conjugation based on the giant two-photon resonance of the biexcitonic molecule in CuCl,” Phys. Rev. A 28, 3696–3698 (1983).
[Crossref]

Denariez-Roberge, M. M.

A. Blouin, P. Galarneau, and M. M. Denariez-Roberge, “Degenerate six-wave-mixing using high order Bragg diffraction in semiconductor-doped glass,” Opt. Commun. 72, 249–252 (1989).
[Crossref]

Ducloy, M.

R. K. Raj, Q. F. Gao, D. Bloch, and M. Ducloy, “Direct observation of higher-order optical susceptibilities via angularly-resolved multiwave mixing,” Opt. Commun. 51, 117–120 (1984).
[Crossref]

M. Ducloy and D. Bloch, “Polarization properties of phase conjugate mirrors,” Phys. Rev. A 30, 3107–3122 (1984).
[Crossref]

Ducuing, J.

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

Etchepare, J.

J. Etchepare, G. Grillon, J. P. Chambaret, G. Hamoniaux, and A. Orszag, “Polarization selectivity in time resolved transient phase grating,” Opt. Commun. 63, 329–334 (1987).
[Crossref]

I. Thomazeau, J. Etchepare, G. Grillon, and A. Migus, “Electronic nonlinear optical susceptibilities of silicate glasses,” Opt. Lett. 10, 223–225 (1985).
[Crossref] [PubMed]

J. Etchepare, G. Grillon, I. Thomazeau, and A. Orszag, “Phase grating approach to susceptibility tensors: determination in isotropic media,” Ultrafast Phenomena V (Springer-Verlag, Berlin, 1986), pp. 504–509.
[Crossref]

Galarneau, P.

A. Blouin, P. Galarneau, and M. M. Denariez-Roberge, “Degenerate six-wave-mixing using high order Bragg diffraction in semiconductor-doped glass,” Opt. Commun. 72, 249–252 (1989).
[Crossref]

Gao, Q. F.

R. K. Raj, Q. F. Gao, D. Bloch, and M. Ducloy, “Direct observation of higher-order optical susceptibilities via angularly-resolved multiwave mixing,” Opt. Commun. 51, 117–120 (1984).
[Crossref]

Gauthier, D. J.

Gobel, E. O.

Gomez, A. S. L.

L. H. Acioli, A. S. L. Gomez, and J. R. Rios Leite, “Measurement of high-order optical nonlinear susceptibilities in semiconductor-doped glasses,” Appl. Phys. Lett. 53, 1788–1790 (1988).
[Crossref]

Grillon, G.

J. Etchepare, G. Grillon, J. P. Chambaret, G. Hamoniaux, and A. Orszag, “Polarization selectivity in time resolved transient phase grating,” Opt. Commun. 63, 329–334 (1987).
[Crossref]

I. Thomazeau, J. Etchepare, G. Grillon, and A. Migus, “Electronic nonlinear optical susceptibilities of silicate glasses,” Opt. Lett. 10, 223–225 (1985).
[Crossref] [PubMed]

J. Etchepare, G. Grillon, I. Thomazeau, and A. Orszag, “Phase grating approach to susceptibility tensors: determination in isotropic media,” Ultrafast Phenomena V (Springer-Verlag, Berlin, 1986), pp. 504–509.
[Crossref]

Gustafsson, M.

Hamoniaux, G.

J. Etchepare, G. Grillon, J. P. Chambaret, G. Hamoniaux, and A. Orszag, “Polarization selectivity in time resolved transient phase grating,” Opt. Commun. 63, 329–334 (1987).
[Crossref]

Hopf, F. A.

F. A. Hopf and G. I. Stegeman, Applied Classical Electrodynamics, Vol. 2: Nonlinear Optics (Wiley, New York, 1985), p. 95.

Hulin, D.

L. L. Chase, M. L. Claude, D. Hulin, and A. Mysyrowicz, “Optical phase conjugation based on the giant two-photon resonance of the biexcitonic molecule in CuCl,” Phys. Rev. A 28, 3696–3698 (1983).
[Crossref]

Lorentz, H. E.

H. E. Lorentz, The Theory of Electrons, 2nd ed. (Dover, New York, 1952).

Malcuit, M. S.

Manneberg, G.

Maruani, A.

A. Maruani, J. L. Oudar, E. Batifol, and D. S. Chemla, “Nonlinear spectroscopy of biexcitons in CuCl by resonant coherent scattering,” Phys. Rev. Lett. 41, 1372–1375 (1978).
[Crossref]

Migus, A.

Mysyrowicz, A.

L. L. Chase, M. L. Claude, D. Hulin, and A. Mysyrowicz, “Optical phase conjugation based on the giant two-photon resonance of the biexcitonic molecule in CuCl,” Phys. Rev. A 28, 3696–3698 (1983).
[Crossref]

Orszag, A.

J. Etchepare, G. Grillon, J. P. Chambaret, G. Hamoniaux, and A. Orszag, “Polarization selectivity in time resolved transient phase grating,” Opt. Commun. 63, 329–334 (1987).
[Crossref]

J. Etchepare, G. Grillon, I. Thomazeau, and A. Orszag, “Phase grating approach to susceptibility tensors: determination in isotropic media,” Ultrafast Phenomena V (Springer-Verlag, Berlin, 1986), pp. 504–509.
[Crossref]

Oudar, J. L.

A. Maruani, J. L. Oudar, E. Batifol, and D. S. Chemla, “Nonlinear spectroscopy of biexcitons in CuCl by resonant coherent scattering,” Phys. Rev. Lett. 41, 1372–1375 (1978).
[Crossref]

Owyoung, A.

A. Owyoung, “The origins of the nonlinear refractive indices of liquids and glasses,” Ph.D. dissertation (California Institute of Technology, Pasadena, Calif., 1971).

Pershan, P. S.

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

Raj, R. K.

R. K. Raj, Q. F. Gao, D. Bloch, and M. Ducloy, “Direct observation of higher-order optical susceptibilities via angularly-resolved multiwave mixing,” Opt. Commun. 51, 117–120 (1984).
[Crossref]

Rios Leite, J. R.

L. H. Acioli, A. S. L. Gomez, and J. R. Rios Leite, “Measurement of high-order optical nonlinear susceptibilities in semiconductor-doped glasses,” Appl. Phys. Lett. 53, 1788–1790 (1988).
[Crossref]

Saeidi, F.

Saito, H.

Shen, Y. R.

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

Stegeman, G. I.

F. A. Hopf and G. I. Stegeman, Applied Classical Electrodynamics, Vol. 2: Nonlinear Optics (Wiley, New York, 1985), p. 95.

Thomazeau, I.

I. Thomazeau, J. Etchepare, G. Grillon, and A. Migus, “Electronic nonlinear optical susceptibilities of silicate glasses,” Opt. Lett. 10, 223–225 (1985).
[Crossref] [PubMed]

J. Etchepare, G. Grillon, I. Thomazeau, and A. Orszag, “Phase grating approach to susceptibility tensors: determination in isotropic media,” Ultrafast Phenomena V (Springer-Verlag, Berlin, 1986), pp. 504–509.
[Crossref]

Unsbo, P.

Appl. Phys. Lett. (1)

L. H. Acioli, A. S. L. Gomez, and J. R. Rios Leite, “Measurement of high-order optical nonlinear susceptibilities in semiconductor-doped glasses,” Appl. Phys. Lett. 53, 1788–1790 (1988).
[Crossref]

J. Opt. Soc. Am. B (1)

Opt. Commun. (3)

R. K. Raj, Q. F. Gao, D. Bloch, and M. Ducloy, “Direct observation of higher-order optical susceptibilities via angularly-resolved multiwave mixing,” Opt. Commun. 51, 117–120 (1984).
[Crossref]

A. Blouin, P. Galarneau, and M. M. Denariez-Roberge, “Degenerate six-wave-mixing using high order Bragg diffraction in semiconductor-doped glass,” Opt. Commun. 72, 249–252 (1989).
[Crossref]

J. Etchepare, G. Grillon, J. P. Chambaret, G. Hamoniaux, and A. Orszag, “Polarization selectivity in time resolved transient phase grating,” Opt. Commun. 63, 329–334 (1987).
[Crossref]

Opt. Lett. (3)

Phys. Rev. (1)

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

Phys. Rev. A (2)

L. L. Chase, M. L. Claude, D. Hulin, and A. Mysyrowicz, “Optical phase conjugation based on the giant two-photon resonance of the biexcitonic molecule in CuCl,” Phys. Rev. A 28, 3696–3698 (1983).
[Crossref]

M. Ducloy and D. Bloch, “Polarization properties of phase conjugate mirrors,” Phys. Rev. A 30, 3107–3122 (1984).
[Crossref]

Phys. Rev. Lett. (1)

A. Maruani, J. L. Oudar, E. Batifol, and D. S. Chemla, “Nonlinear spectroscopy of biexcitons in CuCl by resonant coherent scattering,” Phys. Rev. Lett. 41, 1372–1375 (1978).
[Crossref]

Other (5)

A. Owyoung, “The origins of the nonlinear refractive indices of liquids and glasses,” Ph.D. dissertation (California Institute of Technology, Pasadena, Calif., 1971).

J. Etchepare, G. Grillon, I. Thomazeau, and A. Orszag, “Phase grating approach to susceptibility tensors: determination in isotropic media,” Ultrafast Phenomena V (Springer-Verlag, Berlin, 1986), pp. 504–509.
[Crossref]

F. A. Hopf and G. I. Stegeman, Applied Classical Electrodynamics, Vol. 2: Nonlinear Optics (Wiley, New York, 1985), p. 95.

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

H. E. Lorentz, The Theory of Electrons, 2nd ed. (Dover, New York, 1952).

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Figures (9)

Fig. 1
Fig. 1

Definition of the geometrical arrangements used in this study. The positions of the beams in a plane perpendicular to the propagation direction are shown as follows: ■, pump beams; ●, probe beams; ○, diffracted beams.

Fig. 2
Fig. 2

Log–log plot of the diffraction efficiency of a probe pulse in a Bragg configuration and at zero time delay against the intensity of the pump pulses. The (linear) polarization directions are defined in the inset.

Fig. 3
Fig. 3

Temporal behavior of the diffracted signal s (Bragg configuration) for two different directions of polarization of the analyzer (Wollaston) and for polarization of the pump and the probe beams as in Fig. 2: a, along β = +19°; b, along β = −71°.

Fig. 4
Fig. 4

Efficiency of the first order of diffraction (Bragg configuration) as a function of the analyzer angle β. The polarization configuration is the same as in Fig. 2.

Fig. 5
Fig. 5

Log–log plot of the diffracted efficiency (Bragg configuration) as a function of pump fluence at zero time delay after cancellation of the χ(3) process by polarization selectivity (as in inset of Fig. 2 with β = −71°).

Fig. 6
Fig. 6

y-polarized component of the diffracted beam (Bragg configuration) in the polarization configuration depicted in the inset, with the pump along ±α with respect to the x axis and the probe along x.

Fig. 7
Fig. 7

Efficiency of the diffraction against the intensity of one pump pulse relative to the other and with polarization directions as defined in Fig. 6. Signal a, polarization along the x direction, and b, polarization along y.

Fig. 8
Fig. 8

Log–log plot of the intensity of the second order of diffraction s2 (Raman–Nath configuration) against the intensity of the pump pulses. The polarization configuration is the same as in Fig. 2.

Fig. 9
Fig. 9

Efficiency of the second order of diffraction s2 (Raman–Nath configuration) as a function of analyzer angle β for the polarization directions of Fig. 2.

Equations (29)

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| Δ k n | = | k t | | k t + n ( k 1 k 2 ) | = k t | 1 ( 1 + 2 n 2 m 2 sin i 2 ) 1 / 2 | ,
S ( t d ) = | χ eff ( 3 ) | 2 + I p 2 ( t ) I t ( t t d ) d t .
S 1 ( β ) = [ χ 1111 ( 3 ) cos β + χ 1122 ( 3 ) sin β ] 2 I p 2 I t .
S ( t d ) + I p 4 ( t ) I t ( t t d ) d t
P y ( 3 ) = [ χ 1221 ( 3 ) χ 1212 ( 3 ) ] sin α ( cos α ) ( E 1 E 2 * E t ) .
P 1 ( 5 ) = χ ( 5 ) ( ω s , ω t , ω 1 , ω 1 , ω 1 , ω 2 ) | E 1 | 2 E 1 E 2 * E t ,
P 2 ( 5 ) = χ ( 5 ) ( ω s , ω t , ω 1 , ω 2 , ω 2 , ω 2 ) | E 2 | 2 E 1 E 2 * E t .
P 1 y ( 5 ) = G ( α , θ ) | E 1 | 2 E 1 E 2 * E t , P 2 y ( 5 ) = G ( θ , α ) | E 2 | 2 E 1 E 2 * E t ,
G ( η , ξ ) = χ 211112 ( 5 ) cos 3 ξ sin η + [ χ 211222 ( 5 ) + χ 212122 ( 5 ) + χ 212212 ( 5 ) ] × sin 2 ξ cos ξ sin η + χ 212221 ( 5 ) sin 3 ξ cos α + [ χ 212111 ( 5 ) + χ 211211 ( 5 ) + χ 211121 ( 5 ) ] cos 2 ξ sin ξ cos η .
P y ( 5 ) = 1 2 χ 112222 ( 5 ) sin 4 α ( | E 1 | 2 | E 2 | 2 ) E 1 E 2 * E t .
S 2 ( β ) = | i = c , d χ 111111 ( 5 ) , i cos β + χ 112222 ( 5 ) , i sin β | 2 I p 4 I t ,
Δ E i ( t ) ( ω ) c 2 2 E i ( t ) t 2 = 1 0 c 2 2 P i ( t ) t 2 ,
d 2 r d t 2 + γ d r d t + ω 0 2 r = e E m .
d 2 r d t 2 + γ d r d t + ω 0 2 r + k r · rr + h r · rr · rr = e E m .
r ( t ) = r 0 ( t ) + k r 1 ( t ) + k 2 r 2 ( t ) + h r 3 ( t ) + h 2 r 4 ( t ) + .
r 0 ( t ) = 1 N e + E ( τ ) χ ˜ ( t τ ) d τ ,
k r 1 ( t ) = k m N 4 e 5 + χ ˜ ( τ ) χ ˜ ( t 1 ) χ ˜ ( t 2 ) χ ˜ ( t 3 ) E ( t τ t 1 ) × E ( t τ t 2 ) E ( t τ t 3 ) d τ d t 1 d t 2 d t 3 ,
χ ( ω ) = N e 2 ω ( ω 0 2 ω 2 i γ ω ) 1 .
d 2 ( k 2 r 2 ) d t 2 + γ d ( k 2 r 2 ) d t + ω 0 2 k 2 r 2 = k r 0 · r 0 r 1 .
k 2 r 2 ( t ) = k m N e 2 + χ ( τ ) r 0 ( t τ ) · r 0 ( t τ ) r 1 ( t τ ) d τ .
χ i j k l m n ( 5 ) ( ω = ω 1 + ω 2 + ω 3 + ω 4 + ω 5 ) = k m 2 3 2 N 6 e 8 χ ( ω ) χ ( ω 4 ) χ ( ω 5 ) χ ( ω ) χ ( ω 1 ) χ ( ω 2 ) χ ( ω 3 ) × ( δ i j δ k x + δ i k δ x j + δ i x δ j k ) ( δ x l δ m n + δ x m δ ln + δ x n δ l m ) ,
χ x l m n ( 3 ) ( ω = ω 1 + ω 2 + ω 3 ) χ ( ω ) χ ( ω 1 ) χ ( ω 2 ) χ ( ω 3 ) × ( δ x l δ m n + δ x m δ l n + δ x n δ l m ) ;
P ( 5 ) = σ cas E ( t ) · E ( t ) E ( t ) · E ( t ) E ( t ) ,
d 2 ( h r 3 ) d t 2 + γ d ( h r 3 ) d t + ω 0 2 h r 3 = h r 0 · r 0 r 0 · r 0 r 0 ,
h r 3 ( t ) = h m N e 2 + χ ( t τ ) r 0 ( τ ) · r 0 ( τ ) r 0 ( τ ) · r 0 ( τ ) r 0 ( τ ) d τ .
χ i j k l m n ( 5 ) ( ω = ω 1 + ω 2 + ω 3 + ω 4 + ω 5 ) = h m 5 ! N 5 e 6 χ ( ω ) χ ( ω 1 ) χ ( ω 2 ) χ ( ω 3 ) χ ( ω 4 ) χ ( ω 5 ) S δ i j δ k l δ m n ,
χ i j k l m n ( 5 ) ( ω = ω 1 + ω 2 + ω 3 + ω 4 + ω 5 ) = h m 5 ! N 5 e 6 χ 6 ( ω ) S δ i j δ k l δ m n .
P ( 5 ) ( t ) = σ dir E ( t ) · E ( t ) E ( t ) · E ( t ) E ( t ) .
χ i i i i i i ( 5 ) = 5 χ i i j j j j ( 5 ) and χ i i i i i i ( 5 ) = S χ i j i j j j ( 5 ) .

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