Abstract

Using a phase-conjugation configuration based on six-wave mixing interactions, we have investigated in detail the mechanism of photoinduced second-harmonic generation (SHG) in initially centrosymmetric materials. After some theoretical study of the process we discuss how different complementary tests (with the same experimental setup), such as investigations of the polarization properties or the lifetime dependence of the induced χ(2) grating on the solvent, can reveal the microscopic characteristics of the process. We present experimental studies performed in various solutions of an azo-dye molecule (Disperse Red 1) with the 532-nm harmonic light along with the 1.064-μm light from a picosecond-pulsed Nd:YAG laser. The results show good agreement with the theoretical predictions. It is shown moreover that the light-induced χ(2) grating is due to oriental hole burning of the molecules, whereas reorientation of the molecules in the solvent is one major cause of relaxation.

© 1994 Optical Society of America

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  1. U. Österberg and W. Margulis, Opt. Lett. 11, 516 (1986).
    [CrossRef] [PubMed]
  2. N. B. Baranova and B. Ya. Zel’dovich, Pis’ma Zh. Eksp. Theor. Fiz. 45, 562 (1987) [JETP Lett. 45, 717 (1987)].
  3. B. Ya. Zel’dovich and Yu. E. Kapitskii, Pis’ma Zh. Eksp. Theor. Fiz. 51, 389 (1990) [JETP Lett. 51, 441 (1990)].
  4. F. Charra and J. M. Nunzi, J. Opt. Soc. Am. B 8, 570 (1991).
    [CrossRef]
  5. R. H. Stolen and H. W. K. Tom, Opt. Lett. 7, 585 (1987).
    [CrossRef]
  6. F. Charra, F. Devaux, J. M. Nunzi, and P. Raimond, Phys. Rev. Lett. 68, 2440 (1992).
    [CrossRef] [PubMed]
  7. M. Schubert and B. Wilhelmi, Nonlinear Optics and Quantum Electronics (Wiley-Interscience, New York, 1986).
  8. D. S. Chemla and J. Zyss, eds., Nonlinear Optical Properties of Organic Molecules and Crystals (Academic, Boston, Mass., 1987).
  9. J. L. Oudar, J. Chem. Phys. 67, 446 (1977).
    [CrossRef]
  10. V. Mizrahi, Y. Hibino, and G. Stegeman, Opt. Commun. 78, 283 (1990).
    [CrossRef]
  11. H. J. Eichler, P. Günter, and D. W. Pohl, Laser Induced Dynamic Gratings, Vol. 50 of Springer Series in Optical Science (Springer-Verlag, Berlin, 1986), p. 52.
  12. B. J. Orr and J. F. Ward, Mol. Phys. 20, 513 (1971).
    [CrossRef]
  13. W. Liptay, Angew. Chem. Int. Ed. Engl. 8, 177 (1969).
    [CrossRef]
  14. F. Charra, F. Kajzar, J. M. Nunzi, P. Raimond, and E. Idiard, Opt. Lett. 18, 941 (1993).
    [CrossRef] [PubMed]
  15. Z. Sekkat and M. Dumont, Appl. Phys. B 54, 486 (1992).
    [CrossRef]
  16. C. Jones and S. Day, Nature (London) 351, 15 (1991).
    [CrossRef]
  17. D. Markovitsi, H. Sigal, C. Ecoffet, P. Millie, C. Fiorini, F. Charra, J. M. Nunzi, and H. Strzelecka, Chem. Phys. 182, 69 (1994).
    [CrossRef]
  18. J. M. Nunzi, F. Charra, C. Fiorini, and J. Zyss, Chem. Phys. Lett. 219, 349 (1994).
    [CrossRef]
  19. J. F. Ward, Rev. Mod. Phys. 37, 1 (1965).
    [CrossRef]
  20. D. Gegiou, K. A. Muskat, and E. Fisher, J. Am. Chem. Soc. 90, 3907 (1968).
    [CrossRef]
  21. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), p. 98.

1994 (2)

D. Markovitsi, H. Sigal, C. Ecoffet, P. Millie, C. Fiorini, F. Charra, J. M. Nunzi, and H. Strzelecka, Chem. Phys. 182, 69 (1994).
[CrossRef]

J. M. Nunzi, F. Charra, C. Fiorini, and J. Zyss, Chem. Phys. Lett. 219, 349 (1994).
[CrossRef]

1993 (1)

1992 (2)

Z. Sekkat and M. Dumont, Appl. Phys. B 54, 486 (1992).
[CrossRef]

F. Charra, F. Devaux, J. M. Nunzi, and P. Raimond, Phys. Rev. Lett. 68, 2440 (1992).
[CrossRef] [PubMed]

1991 (2)

C. Jones and S. Day, Nature (London) 351, 15 (1991).
[CrossRef]

F. Charra and J. M. Nunzi, J. Opt. Soc. Am. B 8, 570 (1991).
[CrossRef]

1990 (2)

B. Ya. Zel’dovich and Yu. E. Kapitskii, Pis’ma Zh. Eksp. Theor. Fiz. 51, 389 (1990) [JETP Lett. 51, 441 (1990)].

V. Mizrahi, Y. Hibino, and G. Stegeman, Opt. Commun. 78, 283 (1990).
[CrossRef]

1987 (2)

N. B. Baranova and B. Ya. Zel’dovich, Pis’ma Zh. Eksp. Theor. Fiz. 45, 562 (1987) [JETP Lett. 45, 717 (1987)].

R. H. Stolen and H. W. K. Tom, Opt. Lett. 7, 585 (1987).
[CrossRef]

1986 (1)

1977 (1)

J. L. Oudar, J. Chem. Phys. 67, 446 (1977).
[CrossRef]

1971 (1)

B. J. Orr and J. F. Ward, Mol. Phys. 20, 513 (1971).
[CrossRef]

1969 (1)

W. Liptay, Angew. Chem. Int. Ed. Engl. 8, 177 (1969).
[CrossRef]

1968 (1)

D. Gegiou, K. A. Muskat, and E. Fisher, J. Am. Chem. Soc. 90, 3907 (1968).
[CrossRef]

1965 (1)

J. F. Ward, Rev. Mod. Phys. 37, 1 (1965).
[CrossRef]

Baranova, N. B.

N. B. Baranova and B. Ya. Zel’dovich, Pis’ma Zh. Eksp. Theor. Fiz. 45, 562 (1987) [JETP Lett. 45, 717 (1987)].

Charra, F.

D. Markovitsi, H. Sigal, C. Ecoffet, P. Millie, C. Fiorini, F. Charra, J. M. Nunzi, and H. Strzelecka, Chem. Phys. 182, 69 (1994).
[CrossRef]

J. M. Nunzi, F. Charra, C. Fiorini, and J. Zyss, Chem. Phys. Lett. 219, 349 (1994).
[CrossRef]

F. Charra, F. Kajzar, J. M. Nunzi, P. Raimond, and E. Idiard, Opt. Lett. 18, 941 (1993).
[CrossRef] [PubMed]

F. Charra, F. Devaux, J. M. Nunzi, and P. Raimond, Phys. Rev. Lett. 68, 2440 (1992).
[CrossRef] [PubMed]

F. Charra and J. M. Nunzi, J. Opt. Soc. Am. B 8, 570 (1991).
[CrossRef]

Day, S.

C. Jones and S. Day, Nature (London) 351, 15 (1991).
[CrossRef]

Devaux, F.

F. Charra, F. Devaux, J. M. Nunzi, and P. Raimond, Phys. Rev. Lett. 68, 2440 (1992).
[CrossRef] [PubMed]

Dumont, M.

Z. Sekkat and M. Dumont, Appl. Phys. B 54, 486 (1992).
[CrossRef]

Ecoffet, C.

D. Markovitsi, H. Sigal, C. Ecoffet, P. Millie, C. Fiorini, F. Charra, J. M. Nunzi, and H. Strzelecka, Chem. Phys. 182, 69 (1994).
[CrossRef]

Eichler, H. J.

H. J. Eichler, P. Günter, and D. W. Pohl, Laser Induced Dynamic Gratings, Vol. 50 of Springer Series in Optical Science (Springer-Verlag, Berlin, 1986), p. 52.

Fiorini, C.

D. Markovitsi, H. Sigal, C. Ecoffet, P. Millie, C. Fiorini, F. Charra, J. M. Nunzi, and H. Strzelecka, Chem. Phys. 182, 69 (1994).
[CrossRef]

J. M. Nunzi, F. Charra, C. Fiorini, and J. Zyss, Chem. Phys. Lett. 219, 349 (1994).
[CrossRef]

Fisher, E.

D. Gegiou, K. A. Muskat, and E. Fisher, J. Am. Chem. Soc. 90, 3907 (1968).
[CrossRef]

Gegiou, D.

D. Gegiou, K. A. Muskat, and E. Fisher, J. Am. Chem. Soc. 90, 3907 (1968).
[CrossRef]

Günter, P.

H. J. Eichler, P. Günter, and D. W. Pohl, Laser Induced Dynamic Gratings, Vol. 50 of Springer Series in Optical Science (Springer-Verlag, Berlin, 1986), p. 52.

Hibino, Y.

V. Mizrahi, Y. Hibino, and G. Stegeman, Opt. Commun. 78, 283 (1990).
[CrossRef]

Idiard, E.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), p. 98.

Jones, C.

C. Jones and S. Day, Nature (London) 351, 15 (1991).
[CrossRef]

Kajzar, F.

Kapitskii, Yu. E.

B. Ya. Zel’dovich and Yu. E. Kapitskii, Pis’ma Zh. Eksp. Theor. Fiz. 51, 389 (1990) [JETP Lett. 51, 441 (1990)].

Liptay, W.

W. Liptay, Angew. Chem. Int. Ed. Engl. 8, 177 (1969).
[CrossRef]

Margulis, W.

Markovitsi, D.

D. Markovitsi, H. Sigal, C. Ecoffet, P. Millie, C. Fiorini, F. Charra, J. M. Nunzi, and H. Strzelecka, Chem. Phys. 182, 69 (1994).
[CrossRef]

Millie, P.

D. Markovitsi, H. Sigal, C. Ecoffet, P. Millie, C. Fiorini, F. Charra, J. M. Nunzi, and H. Strzelecka, Chem. Phys. 182, 69 (1994).
[CrossRef]

Mizrahi, V.

V. Mizrahi, Y. Hibino, and G. Stegeman, Opt. Commun. 78, 283 (1990).
[CrossRef]

Muskat, K. A.

D. Gegiou, K. A. Muskat, and E. Fisher, J. Am. Chem. Soc. 90, 3907 (1968).
[CrossRef]

Nunzi, J. M.

J. M. Nunzi, F. Charra, C. Fiorini, and J. Zyss, Chem. Phys. Lett. 219, 349 (1994).
[CrossRef]

D. Markovitsi, H. Sigal, C. Ecoffet, P. Millie, C. Fiorini, F. Charra, J. M. Nunzi, and H. Strzelecka, Chem. Phys. 182, 69 (1994).
[CrossRef]

F. Charra, F. Kajzar, J. M. Nunzi, P. Raimond, and E. Idiard, Opt. Lett. 18, 941 (1993).
[CrossRef] [PubMed]

F. Charra, F. Devaux, J. M. Nunzi, and P. Raimond, Phys. Rev. Lett. 68, 2440 (1992).
[CrossRef] [PubMed]

F. Charra and J. M. Nunzi, J. Opt. Soc. Am. B 8, 570 (1991).
[CrossRef]

Orr, B. J.

B. J. Orr and J. F. Ward, Mol. Phys. 20, 513 (1971).
[CrossRef]

Österberg, U.

Oudar, J. L.

J. L. Oudar, J. Chem. Phys. 67, 446 (1977).
[CrossRef]

Pohl, D. W.

H. J. Eichler, P. Günter, and D. W. Pohl, Laser Induced Dynamic Gratings, Vol. 50 of Springer Series in Optical Science (Springer-Verlag, Berlin, 1986), p. 52.

Raimond, P.

F. Charra, F. Kajzar, J. M. Nunzi, P. Raimond, and E. Idiard, Opt. Lett. 18, 941 (1993).
[CrossRef] [PubMed]

F. Charra, F. Devaux, J. M. Nunzi, and P. Raimond, Phys. Rev. Lett. 68, 2440 (1992).
[CrossRef] [PubMed]

Schubert, M.

M. Schubert and B. Wilhelmi, Nonlinear Optics and Quantum Electronics (Wiley-Interscience, New York, 1986).

Sekkat, Z.

Z. Sekkat and M. Dumont, Appl. Phys. B 54, 486 (1992).
[CrossRef]

Sigal, H.

D. Markovitsi, H. Sigal, C. Ecoffet, P. Millie, C. Fiorini, F. Charra, J. M. Nunzi, and H. Strzelecka, Chem. Phys. 182, 69 (1994).
[CrossRef]

Stegeman, G.

V. Mizrahi, Y. Hibino, and G. Stegeman, Opt. Commun. 78, 283 (1990).
[CrossRef]

Stolen, R. H.

R. H. Stolen and H. W. K. Tom, Opt. Lett. 7, 585 (1987).
[CrossRef]

Strzelecka, H.

D. Markovitsi, H. Sigal, C. Ecoffet, P. Millie, C. Fiorini, F. Charra, J. M. Nunzi, and H. Strzelecka, Chem. Phys. 182, 69 (1994).
[CrossRef]

Tom, H. W. K.

R. H. Stolen and H. W. K. Tom, Opt. Lett. 7, 585 (1987).
[CrossRef]

Ward, J. F.

B. J. Orr and J. F. Ward, Mol. Phys. 20, 513 (1971).
[CrossRef]

J. F. Ward, Rev. Mod. Phys. 37, 1 (1965).
[CrossRef]

Wilhelmi, B.

M. Schubert and B. Wilhelmi, Nonlinear Optics and Quantum Electronics (Wiley-Interscience, New York, 1986).

Zel’dovich, B. Ya.

B. Ya. Zel’dovich and Yu. E. Kapitskii, Pis’ma Zh. Eksp. Theor. Fiz. 51, 389 (1990) [JETP Lett. 51, 441 (1990)].

N. B. Baranova and B. Ya. Zel’dovich, Pis’ma Zh. Eksp. Theor. Fiz. 45, 562 (1987) [JETP Lett. 45, 717 (1987)].

Zyss, J.

J. M. Nunzi, F. Charra, C. Fiorini, and J. Zyss, Chem. Phys. Lett. 219, 349 (1994).
[CrossRef]

Angew. Chem. Int. Ed. Engl. (1)

W. Liptay, Angew. Chem. Int. Ed. Engl. 8, 177 (1969).
[CrossRef]

Appl. Phys. B (1)

Z. Sekkat and M. Dumont, Appl. Phys. B 54, 486 (1992).
[CrossRef]

Chem. Phys. (1)

D. Markovitsi, H. Sigal, C. Ecoffet, P. Millie, C. Fiorini, F. Charra, J. M. Nunzi, and H. Strzelecka, Chem. Phys. 182, 69 (1994).
[CrossRef]

Chem. Phys. Lett. (1)

J. M. Nunzi, F. Charra, C. Fiorini, and J. Zyss, Chem. Phys. Lett. 219, 349 (1994).
[CrossRef]

J. Am. Chem. Soc. (1)

D. Gegiou, K. A. Muskat, and E. Fisher, J. Am. Chem. Soc. 90, 3907 (1968).
[CrossRef]

J. Chem. Phys. (1)

J. L. Oudar, J. Chem. Phys. 67, 446 (1977).
[CrossRef]

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

Mol. Phys. (1)

B. J. Orr and J. F. Ward, Mol. Phys. 20, 513 (1971).
[CrossRef]

Nature (London) (1)

C. Jones and S. Day, Nature (London) 351, 15 (1991).
[CrossRef]

Opt. Commun. (1)

V. Mizrahi, Y. Hibino, and G. Stegeman, Opt. Commun. 78, 283 (1990).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. Lett. (1)

F. Charra, F. Devaux, J. M. Nunzi, and P. Raimond, Phys. Rev. Lett. 68, 2440 (1992).
[CrossRef] [PubMed]

Pis’ma Zh. Eksp. Theor. Fiz. (2)

N. B. Baranova and B. Ya. Zel’dovich, Pis’ma Zh. Eksp. Theor. Fiz. 45, 562 (1987) [JETP Lett. 45, 717 (1987)].

B. Ya. Zel’dovich and Yu. E. Kapitskii, Pis’ma Zh. Eksp. Theor. Fiz. 51, 389 (1990) [JETP Lett. 51, 441 (1990)].

Rev. Mod. Phys. (1)

J. F. Ward, Rev. Mod. Phys. 37, 1 (1965).
[CrossRef]

Other (4)

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), p. 98.

H. J. Eichler, P. Günter, and D. W. Pohl, Laser Induced Dynamic Gratings, Vol. 50 of Springer Series in Optical Science (Springer-Verlag, Berlin, 1986), p. 52.

M. Schubert and B. Wilhelmi, Nonlinear Optics and Quantum Electronics (Wiley-Interscience, New York, 1986).

D. S. Chemla and J. Zyss, eds., Nonlinear Optical Properties of Organic Molecules and Crystals (Academic, Boston, Mass., 1987).

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Figures (10)

Fig. 1
Fig. 1

Beam arrangement for phase conjugation pumped at half-frequency. Beams 1 and 2 are the counterpropagating waves at the fundamental frequency ω (with wave vectors kω and kω′, respectively); beam 3 represents the writing field at frequency 2ω, and beam 4 the phase-conjugate (PC) signal.

Fig. 2
Fig. 2

Representation of the sample cell along with the writing and reading beams in the three-dimensional space. The cell and the beams are located with respect to the reference triad (x, y, z). The reference for the propagation coordinate z is the front of the sample (where writing beams 1 and 3 are converging). The medium thus extends from z = −d to z = 0.

Fig. 3
Fig. 3

Representation of the molecular axis in a reference set of axes (x, y, z). Its polar coordinates are the Euler angles (θ, φ).

Fig. 4
Fig. 4

Block diagram of the experimental setup for phase conjugation with frequency doubling. PD, photodiode switched by a part of the readout beam (beam 2) used for the synchronization of the fast sampler, giving the phase-conjugate intensity of the signal held by the PMT. L/2’s, half-wave plates; P’s, polarizers; S’s, 50% beam splitters; M’s, mirrors. F1 and F2 are interference filters at 532 nm.

Fig. 5
Fig. 5

Top, signal; bottom, diffusion of beam 3. PMT response for a beam-2 delay of 5 ns (one small square of the oscilloscope screen represents 1 ns). The oscilloscope was externally synchronized with the laser. When the sample was replaced by a diffusing screen, the light from incident beam 3 was detected 5 ns earlier, as is shown by the lower curve.

Fig. 6
Fig. 6

Planar chemical structure of the dipolar-molecule Disperse Red One (DR1).

Fig. 7
Fig. 7

Experimental dependence of the phase-conjugate signal intensities (logarithmic scales). The continuous intensity variation of each of these beams was obtained by use of two consecutive polarizers (in order not to modify the polarization state of the beam).

Fig. 8
Fig. 8

Phase-conjugate reflectivity (logarithmic scale) as a function of beam-2 delay for a solution of DR1 in paraffin oil. The measured decay rate is τ = 500 ps. The dashed curve shows the theoretical dependence expected for such a decay rate.

Fig. 9
Fig. 9

Schematics of the three different polarization situations experimentally studied (the x and y directions, respectively, represent vertical and horizontal polarization).

Fig. 10
Fig. 10

Polarized signal amplitude (arbitrary units) as a function of beam-2 polarization angle θ (a) at zero delay and (b) at delay time t = 70 ps for a solution of DR1 in THF in the three different polarization situations shown in Fig. 9. The fitting curves are derived from Table 1.

Tables (2)

Tables Icon

Table 1 Summary of Theoretical Dependences Expected in Each of the Three Different Polarization Situations Shown in Fig. 9a

Tables Icon

Table 2 Summary of All the Different Polarization Combinations for the Fifth-Order Susceptibility Functions G(5)(ω; ω1, ω2, ω3, ω4, ω5)a

Equations (63)

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χ ( 2 ) E ( M , t ) 3 = E ω 2 ( M ) E 2 ω * ( M ) + c . c .
P 2 ω ( M ) = ½ 0 χ ( 2 ) E ω 2 .
P 2 ω ( M ) ½ 0 E ω 2 [ E ω ( M ) 2 E 2 ω * ( M ) + E ω * ( M ) 2 E 2 ω ( M ) ] .
χ ( 2 ) ( t ) = - t G ( 5 ) ( t - τ ) [ E ω ( M ) 2 E 2 ω * ( M ) + E ω * ( M ) 2 E 2 ω ( M ) ] d τ ,
G ( 5 ) ( t ) = 5 8 χ ( 5 ) f ( t ) - t f ( t - τ ) d τ ,
P 2 ω ( M ) = 1 2 0 χ ( 5 ) ( 2 ω ; ω , ω , ω , ω , - 2 ω ) × E ω 2 [ exp - ( i Δ k - · M ) ] × - t f ( t - τ ) ( E ω 2 E 2 ω * ) d τ + 1 2 0 χ ( 5 ) ( 2 ω ; ω , ω , - ω , - ω , 2 ω ) × E ω 2 [ exp - ( i Δ k + · M ) ] × - t f ( t - τ ) ( E ω * 2 E 2 ω ) d τ ,
P ˜ 2 ω ( t ) = ½ 0 χ ( 2 ) ( t ) E ˜ ω ( t ) 2 ,             with             E ˜ ω = E ˜ ω = E ˜ 1 ( t ) + E ˜ 2 ( t ) .
P ˜ 2 ω PC ( t ) = 1 2 0 [ E ˜ 2 ( t ) 2 - t G ( 5 ) ( t - τ ) E ˜ 1 ( τ ) 2 E ˜ 3 * ( τ ) d τ + E ˜ 1 ( t ) 2 - t G ( 5 ) ( t - τ ) E ˜ 2 ( τ ) 2 E 3 * ( τ ) d τ + 4 E ˜ 1 ( t ) E ˜ 2 ( t ) - t G ( 5 ) ( t - τ ) E ˜ 1 ( τ ) E ˜ 2 ( τ ) E ˜ 3 * ( τ ) d τ ] .
d E 4 d z = - α 2 E 4 - i ω n c 2 ω PC 0 ,
P 2 ω PC = 1 2 0 E 2 2 - t 0 G ( 5 ) ( t 0 - τ ) E ˜ 1 ( τ ) 2 E ˜ 3 ( τ ) * d τ ,
E 4 ( z ) = i ω E ¯ 1 2 E ¯ 2 2 E ¯ 3 * n c α exp ( - α 2 d ) × sinh [ α 2 ( z + d ) ] - t 0 G ( 5 ) ( t 0 - τ ) × E ˜ 1 ( τ ) 2 E 1 2 E ˜ 3 ( τ ) * E 3 * d τ .
r = ω [ 1 - exp ( - α d ) ] E ¯ ω 4 2 n c α × - t 0 G ( 5 ) ( t 0 - τ ) E ˜ 1 ( τ ) 2 E 1 2 E ˜ 3 ( τ ) * E 3 * d τ .
r = ω E ¯ ω 4 2 n c α - t 0 G ( 5 ) ( t 0 - τ ) E ˜ 1 ( τ ) 2 E 1 2 E ˜ 3 ( τ ) * E 3 * d τ .
E i ( M , t ) = E ¯ i exp - ( t Δ t i ) 2 exp - ( r w i ) 2
r 2 = ( E ¯ 4 E ¯ 3 ) 2 = 3 1.5 F 4 F 3 = 3 1.5 ,
χ ( 5 ) = 32 n c α 5 ω Γ π Δ t exp - ( Γ t 0 ) exp ( Γ 2 Δ t 2 16 ) r E ¯ ω 4 .
E 0 = ¾ χ ( 3 ) ( 0 ; ω , ω , - 2 ω ) E 1 2 E 3 * .
P 4 ( 2 ω ) = ³ / ɛ 0 χ ( 3 ) ( 2 ω ; ω , ω , 0 ) E 2 2 E 0 .
P 4 ( 2 ω ) = / ɛ 0 χ ( 3 ) ( 2 ω ; ω , ω , 0 ) χ ( 3 ) × ( 0 ; ω , ω , - 2 ω ) E 1 2 E 2 2 E 3 * .
r E ¯ ω 4 χ ( 5 ) α ,
α = OD ln 10 d 2 ω N Φ ( 2 ω ) N ,
r I ω 2 Δ β Δ μ Γ e ,
f ( t ) = Γ e exp ( - Γ e t ) f r ( t ) ,
G x x x x x x ( 5 ) [ 3 exp ( - 2 D t ) + ( 4 / 7 ) exp ( - 12 D t ) ] exp ( - Γ e t ) ,
G x y y x x x ( 5 ) = G x x x y y x ( 5 ) [ exp ( - 2 D t ) - ( 2 / 7 ) exp ( - 12 D t ) ] × exp ( - Γ e t ) ,
G x y y y y x ( 5 ) = G x y x y x x ( 5 ) [ ( 1 / 3 ) exp ( - 2 D t ) + ( 8 / 21 ) exp ( - 12 D t ) ] × exp ( - Γ e t ) ,
G x y y x x x ( 5 ) = G x x x y y x ( 5 ) = G x y y y y x ( 5 ) = G y y y x y x ( 5 ) = G x x y x y x ( 5 ) = G y x y y y x ( 5 ) = G y x x x y x ( 5 ) = G y x y x x x ( 5 ) = G x x x x x x ( 5 ) .
G x y y x x x ( 5 ) = G x x x y y x ( 5 ) = G y y y x y x ( 5 ) = G y x y x x x ( 5 ) = G x x x x x x ( 5 ) ,
G x y y y y x ( 5 ) = G x x y x y x ( 5 ) = G y x y y y x ( 5 ) = G y x x x y x ( 5 ) = ¹ / G x x x x x x ( 5 ) .
χ exp ( 5 ) = 6 × 10 - 37 m 4 / V 4 .
P x PC G x x x x x x ( 5 ) cos 2 θ + G x y y x x x ( 5 ) sin 2 θ .
P x PC G x x x y y x ( 5 ) cos 2 θ + G x y y y y x ( 5 ) sin 2 θ .
P y PC G y x y x x x ( 5 ) sin 2 θ .
G x y y x x x ( 5 ) = G x x x y y x ( 5 ) = G x y y y y x ( 5 ) = G y x y x x x ( 5 ) = G x x x x x x ( 5 ) ,
G x y y x x x ( 5 ) = G x x x y y x ( 5 ) = G y x y x x x ( 5 ) = G x x x x x x ( 5 ) , G x y y y y x ( 5 ) = G x y y x x x ( 5 ) = ¹ / G x x x x x x ( 5 ) .
W t = P ( t ) E * ( t ) t ,
W t = β ( 2 ω ; ω , ω ) E ω 2 E 2 ω * + 2 β ( ω ; 2 ω , - ω ) E ω * 2 E 2 ω + c . c .
P 2 ω = p ( θ , φ ) β ( 2 ω ; ω , ω ) ( E ω cos θ ) 2 ( cos θ ) d Ω = N 7 k T β ( 2 ω ; ω , ω ) [ 2 β * ( ω ; - 2 ω , ω ) + β ( 2 ω ; ω , ω ) ] E ω 4 E 2 ω * ,
χ ( 5 ) = 16 35 f ( ω ) 4 f ( 2 ω ) 2 N 0 k T β ( 2 ω ; ω , ω ) × [ 2 β * ( ω ; 2 ω , - ω ) + β ( 2 ω ; ω , ω ) ] ,
ψ = s n = 0 , 1 a n ( s ) ( t ) n ,
i d d t b n ( 0 ) = 0 , i d d t b n ( s ) = k exp ( i ω n k t ) H n k b k ( s - 1 ) , with H n k = n H k .
H = H ω + H 2 ω - e E ω · r cos ( ω t ) + e E 2 ω · r cos ( 2 ω t ) .
b 1 ( 1 ) = μ 01 · E 2 ω 2 [ 1 - exp i ( ω 01 - 2 ω ) ω 01 - 2 ω ] , b 1 ( 2 ) = ( μ 01 · E ω ) ( Δ μ · E ω ) 4 2 ω [ 1 - exp i ( ω 01 - 2 ω ) ω 01 - 2 ω ] ,
P 10 = 1 2 [ 1 4 ( μ 01 · E 2 ω ) ( μ 01 · E 2 ω ) * + ( μ 01 · E 2 ω ) ( Δ μ · E ω ) ( μ 01 · E ω ) * ( Δ μ · E ω ) * 16 ( ω ) 2 + ( μ 01 · E 2 ω ) * ( μ 01 · E ω ) ( Δ μ · E ω ) + ( μ 01 · E 2 ω ) ( μ 01 · E ω ) * ( Δ μ · E ω ) * 8 ( ω ) 2 ] * F ( ω , t ) ,
F ( ω , t ) = 2 ω ω 01 sinc 2 ( ω 01 - 2 ω 2 t ) Φ ( ω ) d ω Φ ( 2 ω ) ,
P 10 = 1 2 ( μ 01 · E 2 ω ) * ( μ 01 · E ω ) ( Δ μ · E ω ) 8 ω Φ ( 2 ω ) = 1 2 μ 01 2 Δ μ ( cos 3 θ ) E ω 2 E 2 ω * 8 ω Φ ( 2 ω ) .
P 2 ω = 1 4 π d t d Ω [ N ( cos θ ) P 01 Δ β × exp ( - Γ e t ) E ω 2 cos 2 θ ] ,
P 2 ω = N 7 1 Γ e 1 2 μ 01 2 Δ μ Δ β 8 ω Φ ( 2 ω ) E ω 4 E 2 ω * .
P 2 ω = 5 16 0 χ ( 5 ) E ω 4 E 2 ω * .
χ ( 5 ) = f ( ω ) 4 f ( 2 ω ) 2 1 Γ e 1 2 16 N 35 0 μ 01 2 Δ μ Δ β 8 ω Φ ( 2 ω ) .
G x 1 x 2 x 3 x 1 x 2 x 3 ( 5 ) ( 2 ω ; ω , ω , ω , ω , - 2 ω ) f e ( t ) l exp [ - l ( l + 1 ) D t ] Y l m ( θ , φ ) n = 1 , 3 ( x n · i ) d Ω ,
G x x x x x x ( 5 ) ( cos 3 θ ) [ exp ( - 12 D t ) Y 3 0 + 3 5 exp ( - 2 D t ) Y 1 0 ] d Ω 4 π 25 f e ( t ) [ 3 exp ( - 2 D t ) + 4 7 exp ( - 12 D t ) ] , G x y y x x x ( 5 ) cos θ sin 2 θ ( cos 2 φ ) [ exp ( - 12 D t ) Y 3 0 + 3 5 exp ( - 2 D t ) Y 1 0 ] d Ω 4 π 25 f e ( t ) [ exp ( - 2 D t ) - 2 7 exp ( - 12 D t ) ] .
ρ y y x ( θ ) = sin 2 θ cos 2 φ cos θ = Y 1 0 - ½ Y 3 0 + ½ Y 3 2 ,
G x x x y y x ( 5 ) ( cos 3 θ ) [ 1 5 Y 1 0 exp ( - 2 D t ) - 1 2 Y 3 0 exp ( - 12 D t ) + 1 2 Y 3 2 exp ( - 12 D t ) ] d Ω 4 π 25 f e ( t ) [ exp ( - 2 D t ) - 2 7 exp ( - 12 D t ) ] , G x y y y y x ( 5 ) 4 π 25 f e ( t ) [ 1 3 exp ( - 2 D t ) + 8 21 exp ( - 12 D t ) ] , G y x y y y x ( 5 ) 4 π 25 f e ( t ) [ 1 3 exp ( - 2 D t ) + 8 21 exp ( - 12 D t ) ] .
ρ x y x ( θ ) = ½ cos θ ( cos θ + sin θ cos φ ) 2 = ¼ Y 3 0 + Y 1 0 + ¼ Y 3 2 + Y 3 1 + Y 1 1 ,
G y y y x y x ( 5 ) 4 π 25 f e ( t ) [ exp ( - 2 D t ) - 2 7 exp ( - 12 D t ) ] .
P x ( 2 ω ) = i = 1 , 16 P i ( 2 ω ) i = 1 , 16 G i ( 5 ) π i ,
G 3 ( 5 ) = G 4 ( 5 ) = G 6 ( 5 ) = G 7 ( 5 ) , G 5 ( 5 ) = G 8 ( 5 )             G 10 ( 5 ) = G 11 ( 5 ) , G 13 ( 5 ) = G 14 ( 5 ) ,             G 15 ( 5 ) = G 16 ( 5 ) .
A = A x x + A y y = A x x + A y y ,
A x = A x cos θ + A y sin θ , A y = - A x sin θ + A y cos θ .
P x = P x cos θ + P y sin θ = i = 1 , 16 G i ( 5 ) π i ,
G 15 ( 5 ) = G 1 ( 5 ) - G 2 ( 5 ) - 2 G 3 ( 5 ) - G 5 ( 5 ) , 4 G 3 ( 5 ) = G 1 ( 5 ) - G 2 ( 5 ) - G 9 ( 5 ) + G 12 ( 5 ) , G 12 ( 5 ) = G 5 ( 5 ) + 2 G 3 ( 5 ) - G 10 ( 5 ) - G 13 ( 5 ) , G 13 ( 5 ) = G 5 ( 5 ) + G 10 ( 5 ) - G 15 ( 5 ) .
G 1 ( 5 ) = G x x x x x x ( 5 ) ,             G 2 ( 5 ) = G x y y x x x ( 5 ) , G 9 ( 5 ) = G x x x y y x ( 5 ) ,             G 15 ( 5 ) = G y x y x x x ( 5 ) , G 12 ( 5 ) = G x y y y y x ( 5 ) .

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