Abstract

We apply quadratic electroabsorption spectroscopy (QES) to thin-film solid solutions of squarylium dye molecules in poly(methyl methacrylate) polymer to study the dye’s electronic excited states and to investigate the importance of these states with regard to their contribution to the third-order nonlinear-optical susceptibility. We first show that the room-temperature tensor ratio a=χ3333(3)/χ1133(3)3 throughout most of the visible region to establish that the electronic mechanism dominates. Because QES is a third-order nonlinear-optical susceptibility measurement, it can be used to identify two photon states. By obtaining good agreement between the quadratic electroabsorption spectrum and a three level model, we conclude that there are two dominant states that contribute to the near-resonant third-order susceptibility: a one-photon state and a two-photon state that are separated by less than 0.2 eV in energy. QES is thus shown to be a versatile tool for measuring the nature of excited states in a molecule. Furthermore, by applying a Kramers–Kronig transformation to determine the real part of the response, we are able to assess the two-photon all-optical device figure of merit of these materials. Such an understanding is important for designing molecules for all-optical device applications.

© 1995 Optical Society of America

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References

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  1. J. Messier, F. Kajzar, P. Prasad, and D. Ulrich, eds., Nonlinear Optical Effects in Organic Polymers, Vol. 162 of NATO ASI Series E (Kluwer, Dordrecht, The Netherlands, 1989).
    [Crossref]
  2. See, for example, S. R. Marder, J. E. Sohn, and G. D. Studky, eds., Materials for Nonlinear Optics: Chemical Perspectives,” Vol. 455 of ACS Symposium Series (American Chemical Society, Washington, D.C., 1991).
    [Crossref]
  3. M. G. Kuzyk and J. Swalen, eds., special issue of Nonlin. Opt.6(2) (1993).
  4. M. G. Kuzyk, U. C. Paek, and C. W. Dirk, Appl. Phys. Lett. 59, 902 (1991).
    [Crossref]
  5. M. G. Kuzyk and C. W. Dirk, Appl. Phys. Lett. 54, 1628 (1989).
    [Crossref]
  6. M. G. Kuzyk, J. E. Sohn, and C. W. Dirk, J. Opt. Soc. Am. B 5, 842 (1990).
    [Crossref]
  7. C. W. Dirk, L. T. Cheng, and M. G. Kuzyk, Int. J. Quantum Chem. 43, 27 (1992).
    [Crossref]
  8. J. R. Heflin, K. Y. Wong, O. Zamani-Khamiri, and A. F. Garito, Phys. Rev. B 38, 1573 (1988).
    [Crossref]
  9. Z. G. Soos and S. Ramasesha, J. Chem. Phys. 90, 1067 (1989).
    [Crossref]
  10. M. G. Kuzyk and C. W. Dirk, Phys. Rev. A 41, 5098 (1990).
    [Crossref] [PubMed]
  11. Q. L. Zhou, R. F. Shi, O. Zamani-Khamari, and A. F. Garito, Ref. 3, p. 145.
  12. S. H. Stevenson, D. S. Donald, and G. R. Meridith, Mater. Res. Soc. Proc. 109, 103 (1988).
    [Crossref]
  13. S. R. Marder, J. W. Perry, G. Bourhill, C. B. Gorman, B. G. Tiemann, and K. Mansour, Science 26, 186 (1993).
    [Crossref]
  14. C. Poga, M. G. Kuzyk, and C. W. Dirk, J. Opt. Soc. Am. B 11, 80 (1994).
    [Crossref]
  15. J. H. Andrews, J. D. V. Khaydarov, and K. D. Singer, Opt. Lett. 19, 984 (1994);“Contribution of the 2 1Ag state to the third-order optical nonlinearity in a squaraine dye: erratum,” submitted to Opt. Lett.
    [Crossref] [PubMed]
  16. Y. Z. Yu, R. F. Shi, A. F. Garito, and C. H. Grossman, Opt. Lett. 19, 786 (1994).
    [Crossref] [PubMed]
  17. C. W. Dirk, W. C. Herndon, F. Cervantes-Lee, H. Selnau, S. Martinez, A. Tan, G. Campos, M. Velez, J. Zyss, and L. -T. Cheng, “Squarylium dyes—structural factors pertaining to the negative third-order nonlinear optical response,” submitted to J. Am. Chem. Soc.
  18. C. Poga, “Mechanisms of the third-order nonlinear optical response in dye-doped polymers,” Ph.D. dissertation (Washington State University, Pullman, Wash., 1994).
  19. V. Mizrahi, K. W. DeLong, G. I. Stegeman, M. A. Saifi, and M. J. Andrejco, Opt. Lett. 14, 1140 (1989).
    [Crossref] [PubMed]

1994 (3)

1993 (1)

S. R. Marder, J. W. Perry, G. Bourhill, C. B. Gorman, B. G. Tiemann, and K. Mansour, Science 26, 186 (1993).
[Crossref]

1992 (1)

C. W. Dirk, L. T. Cheng, and M. G. Kuzyk, Int. J. Quantum Chem. 43, 27 (1992).
[Crossref]

1991 (1)

M. G. Kuzyk, U. C. Paek, and C. W. Dirk, Appl. Phys. Lett. 59, 902 (1991).
[Crossref]

1990 (2)

M. G. Kuzyk, J. E. Sohn, and C. W. Dirk, J. Opt. Soc. Am. B 5, 842 (1990).
[Crossref]

M. G. Kuzyk and C. W. Dirk, Phys. Rev. A 41, 5098 (1990).
[Crossref] [PubMed]

1989 (3)

Z. G. Soos and S. Ramasesha, J. Chem. Phys. 90, 1067 (1989).
[Crossref]

M. G. Kuzyk and C. W. Dirk, Appl. Phys. Lett. 54, 1628 (1989).
[Crossref]

V. Mizrahi, K. W. DeLong, G. I. Stegeman, M. A. Saifi, and M. J. Andrejco, Opt. Lett. 14, 1140 (1989).
[Crossref] [PubMed]

1988 (2)

J. R. Heflin, K. Y. Wong, O. Zamani-Khamiri, and A. F. Garito, Phys. Rev. B 38, 1573 (1988).
[Crossref]

S. H. Stevenson, D. S. Donald, and G. R. Meridith, Mater. Res. Soc. Proc. 109, 103 (1988).
[Crossref]

Andrejco, M. J.

Andrews, J. H.

Bourhill, G.

S. R. Marder, J. W. Perry, G. Bourhill, C. B. Gorman, B. G. Tiemann, and K. Mansour, Science 26, 186 (1993).
[Crossref]

Campos, G.

C. W. Dirk, W. C. Herndon, F. Cervantes-Lee, H. Selnau, S. Martinez, A. Tan, G. Campos, M. Velez, J. Zyss, and L. -T. Cheng, “Squarylium dyes—structural factors pertaining to the negative third-order nonlinear optical response,” submitted to J. Am. Chem. Soc.

Cervantes-Lee, F.

C. W. Dirk, W. C. Herndon, F. Cervantes-Lee, H. Selnau, S. Martinez, A. Tan, G. Campos, M. Velez, J. Zyss, and L. -T. Cheng, “Squarylium dyes—structural factors pertaining to the negative third-order nonlinear optical response,” submitted to J. Am. Chem. Soc.

Cheng, L. T.

C. W. Dirk, L. T. Cheng, and M. G. Kuzyk, Int. J. Quantum Chem. 43, 27 (1992).
[Crossref]

Cheng, L. -T.

C. W. Dirk, W. C. Herndon, F. Cervantes-Lee, H. Selnau, S. Martinez, A. Tan, G. Campos, M. Velez, J. Zyss, and L. -T. Cheng, “Squarylium dyes—structural factors pertaining to the negative third-order nonlinear optical response,” submitted to J. Am. Chem. Soc.

DeLong, K. W.

Dirk, C. W.

C. Poga, M. G. Kuzyk, and C. W. Dirk, J. Opt. Soc. Am. B 11, 80 (1994).
[Crossref]

C. W. Dirk, L. T. Cheng, and M. G. Kuzyk, Int. J. Quantum Chem. 43, 27 (1992).
[Crossref]

M. G. Kuzyk, U. C. Paek, and C. W. Dirk, Appl. Phys. Lett. 59, 902 (1991).
[Crossref]

M. G. Kuzyk, J. E. Sohn, and C. W. Dirk, J. Opt. Soc. Am. B 5, 842 (1990).
[Crossref]

M. G. Kuzyk and C. W. Dirk, Phys. Rev. A 41, 5098 (1990).
[Crossref] [PubMed]

M. G. Kuzyk and C. W. Dirk, Appl. Phys. Lett. 54, 1628 (1989).
[Crossref]

C. W. Dirk, W. C. Herndon, F. Cervantes-Lee, H. Selnau, S. Martinez, A. Tan, G. Campos, M. Velez, J. Zyss, and L. -T. Cheng, “Squarylium dyes—structural factors pertaining to the negative third-order nonlinear optical response,” submitted to J. Am. Chem. Soc.

Donald, D. S.

S. H. Stevenson, D. S. Donald, and G. R. Meridith, Mater. Res. Soc. Proc. 109, 103 (1988).
[Crossref]

Garito, A. F.

Y. Z. Yu, R. F. Shi, A. F. Garito, and C. H. Grossman, Opt. Lett. 19, 786 (1994).
[Crossref] [PubMed]

J. R. Heflin, K. Y. Wong, O. Zamani-Khamiri, and A. F. Garito, Phys. Rev. B 38, 1573 (1988).
[Crossref]

Q. L. Zhou, R. F. Shi, O. Zamani-Khamari, and A. F. Garito, Ref. 3, p. 145.

Gorman, C. B.

S. R. Marder, J. W. Perry, G. Bourhill, C. B. Gorman, B. G. Tiemann, and K. Mansour, Science 26, 186 (1993).
[Crossref]

Grossman, C. H.

Heflin, J. R.

J. R. Heflin, K. Y. Wong, O. Zamani-Khamiri, and A. F. Garito, Phys. Rev. B 38, 1573 (1988).
[Crossref]

Herndon, W. C.

C. W. Dirk, W. C. Herndon, F. Cervantes-Lee, H. Selnau, S. Martinez, A. Tan, G. Campos, M. Velez, J. Zyss, and L. -T. Cheng, “Squarylium dyes—structural factors pertaining to the negative third-order nonlinear optical response,” submitted to J. Am. Chem. Soc.

Khaydarov, J. D. V.

Kuzyk, M. G.

C. Poga, M. G. Kuzyk, and C. W. Dirk, J. Opt. Soc. Am. B 11, 80 (1994).
[Crossref]

C. W. Dirk, L. T. Cheng, and M. G. Kuzyk, Int. J. Quantum Chem. 43, 27 (1992).
[Crossref]

M. G. Kuzyk, U. C. Paek, and C. W. Dirk, Appl. Phys. Lett. 59, 902 (1991).
[Crossref]

M. G. Kuzyk, J. E. Sohn, and C. W. Dirk, J. Opt. Soc. Am. B 5, 842 (1990).
[Crossref]

M. G. Kuzyk and C. W. Dirk, Phys. Rev. A 41, 5098 (1990).
[Crossref] [PubMed]

M. G. Kuzyk and C. W. Dirk, Appl. Phys. Lett. 54, 1628 (1989).
[Crossref]

Mansour, K.

S. R. Marder, J. W. Perry, G. Bourhill, C. B. Gorman, B. G. Tiemann, and K. Mansour, Science 26, 186 (1993).
[Crossref]

Marder, S. R.

S. R. Marder, J. W. Perry, G. Bourhill, C. B. Gorman, B. G. Tiemann, and K. Mansour, Science 26, 186 (1993).
[Crossref]

Martinez, S.

C. W. Dirk, W. C. Herndon, F. Cervantes-Lee, H. Selnau, S. Martinez, A. Tan, G. Campos, M. Velez, J. Zyss, and L. -T. Cheng, “Squarylium dyes—structural factors pertaining to the negative third-order nonlinear optical response,” submitted to J. Am. Chem. Soc.

Meridith, G. R.

S. H. Stevenson, D. S. Donald, and G. R. Meridith, Mater. Res. Soc. Proc. 109, 103 (1988).
[Crossref]

Mizrahi, V.

Paek, U. C.

M. G. Kuzyk, U. C. Paek, and C. W. Dirk, Appl. Phys. Lett. 59, 902 (1991).
[Crossref]

Perry, J. W.

S. R. Marder, J. W. Perry, G. Bourhill, C. B. Gorman, B. G. Tiemann, and K. Mansour, Science 26, 186 (1993).
[Crossref]

Poga, C.

C. Poga, M. G. Kuzyk, and C. W. Dirk, J. Opt. Soc. Am. B 11, 80 (1994).
[Crossref]

C. Poga, “Mechanisms of the third-order nonlinear optical response in dye-doped polymers,” Ph.D. dissertation (Washington State University, Pullman, Wash., 1994).

Ramasesha, S.

Z. G. Soos and S. Ramasesha, J. Chem. Phys. 90, 1067 (1989).
[Crossref]

Saifi, M. A.

Selnau, H.

C. W. Dirk, W. C. Herndon, F. Cervantes-Lee, H. Selnau, S. Martinez, A. Tan, G. Campos, M. Velez, J. Zyss, and L. -T. Cheng, “Squarylium dyes—structural factors pertaining to the negative third-order nonlinear optical response,” submitted to J. Am. Chem. Soc.

Shi, R. F.

Y. Z. Yu, R. F. Shi, A. F. Garito, and C. H. Grossman, Opt. Lett. 19, 786 (1994).
[Crossref] [PubMed]

Q. L. Zhou, R. F. Shi, O. Zamani-Khamari, and A. F. Garito, Ref. 3, p. 145.

Singer, K. D.

Sohn, J. E.

M. G. Kuzyk, J. E. Sohn, and C. W. Dirk, J. Opt. Soc. Am. B 5, 842 (1990).
[Crossref]

Soos, Z. G.

Z. G. Soos and S. Ramasesha, J. Chem. Phys. 90, 1067 (1989).
[Crossref]

Stegeman, G. I.

Stevenson, S. H.

S. H. Stevenson, D. S. Donald, and G. R. Meridith, Mater. Res. Soc. Proc. 109, 103 (1988).
[Crossref]

Tan, A.

C. W. Dirk, W. C. Herndon, F. Cervantes-Lee, H. Selnau, S. Martinez, A. Tan, G. Campos, M. Velez, J. Zyss, and L. -T. Cheng, “Squarylium dyes—structural factors pertaining to the negative third-order nonlinear optical response,” submitted to J. Am. Chem. Soc.

Tiemann, B. G.

S. R. Marder, J. W. Perry, G. Bourhill, C. B. Gorman, B. G. Tiemann, and K. Mansour, Science 26, 186 (1993).
[Crossref]

Velez, M.

C. W. Dirk, W. C. Herndon, F. Cervantes-Lee, H. Selnau, S. Martinez, A. Tan, G. Campos, M. Velez, J. Zyss, and L. -T. Cheng, “Squarylium dyes—structural factors pertaining to the negative third-order nonlinear optical response,” submitted to J. Am. Chem. Soc.

Wong, K. Y.

J. R. Heflin, K. Y. Wong, O. Zamani-Khamiri, and A. F. Garito, Phys. Rev. B 38, 1573 (1988).
[Crossref]

Yu, Y. Z.

Zamani-Khamari, O.

Q. L. Zhou, R. F. Shi, O. Zamani-Khamari, and A. F. Garito, Ref. 3, p. 145.

Zamani-Khamiri, O.

J. R. Heflin, K. Y. Wong, O. Zamani-Khamiri, and A. F. Garito, Phys. Rev. B 38, 1573 (1988).
[Crossref]

Zhou, Q. L.

Q. L. Zhou, R. F. Shi, O. Zamani-Khamari, and A. F. Garito, Ref. 3, p. 145.

Zyss, J.

C. W. Dirk, W. C. Herndon, F. Cervantes-Lee, H. Selnau, S. Martinez, A. Tan, G. Campos, M. Velez, J. Zyss, and L. -T. Cheng, “Squarylium dyes—structural factors pertaining to the negative third-order nonlinear optical response,” submitted to J. Am. Chem. Soc.

Appl. Phys. Lett. (2)

M. G. Kuzyk, U. C. Paek, and C. W. Dirk, Appl. Phys. Lett. 59, 902 (1991).
[Crossref]

M. G. Kuzyk and C. W. Dirk, Appl. Phys. Lett. 54, 1628 (1989).
[Crossref]

Int. J. Quantum Chem. (1)

C. W. Dirk, L. T. Cheng, and M. G. Kuzyk, Int. J. Quantum Chem. 43, 27 (1992).
[Crossref]

J. Chem. Phys. (1)

Z. G. Soos and S. Ramasesha, J. Chem. Phys. 90, 1067 (1989).
[Crossref]

J. Opt. Soc. Am. B (2)

M. G. Kuzyk, J. E. Sohn, and C. W. Dirk, J. Opt. Soc. Am. B 5, 842 (1990).
[Crossref]

C. Poga, M. G. Kuzyk, and C. W. Dirk, J. Opt. Soc. Am. B 11, 80 (1994).
[Crossref]

Mater. Res. Soc. Proc. (1)

S. H. Stevenson, D. S. Donald, and G. R. Meridith, Mater. Res. Soc. Proc. 109, 103 (1988).
[Crossref]

Opt. Lett. (3)

Phys. Rev. A (1)

M. G. Kuzyk and C. W. Dirk, Phys. Rev. A 41, 5098 (1990).
[Crossref] [PubMed]

Phys. Rev. B (1)

J. R. Heflin, K. Y. Wong, O. Zamani-Khamiri, and A. F. Garito, Phys. Rev. B 38, 1573 (1988).
[Crossref]

Science (1)

S. R. Marder, J. W. Perry, G. Bourhill, C. B. Gorman, B. G. Tiemann, and K. Mansour, Science 26, 186 (1993).
[Crossref]

Other (6)

C. W. Dirk, W. C. Herndon, F. Cervantes-Lee, H. Selnau, S. Martinez, A. Tan, G. Campos, M. Velez, J. Zyss, and L. -T. Cheng, “Squarylium dyes—structural factors pertaining to the negative third-order nonlinear optical response,” submitted to J. Am. Chem. Soc.

C. Poga, “Mechanisms of the third-order nonlinear optical response in dye-doped polymers,” Ph.D. dissertation (Washington State University, Pullman, Wash., 1994).

Q. L. Zhou, R. F. Shi, O. Zamani-Khamari, and A. F. Garito, Ref. 3, p. 145.

J. Messier, F. Kajzar, P. Prasad, and D. Ulrich, eds., Nonlinear Optical Effects in Organic Polymers, Vol. 162 of NATO ASI Series E (Kluwer, Dordrecht, The Netherlands, 1989).
[Crossref]

See, for example, S. R. Marder, J. E. Sohn, and G. D. Studky, eds., Materials for Nonlinear Optics: Chemical Perspectives,” Vol. 455 of ACS Symposium Series (American Chemical Society, Washington, D.C., 1991).
[Crossref]

M. G. Kuzyk and J. Swalen, eds., special issue of Nonlin. Opt.6(2) (1993).

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Figures (10)

Fig. 1
Fig. 1

ISQ chromophore in cis form (ISQ4) and in trans form (ISQ6).

Fig. 2
Fig. 2

Quadratic electroabsorption experiment.

Fig. 3
Fig. 3

Absorption and fluorescence spectra for a solution of ISQ in methanol.

Fig. 4
Fig. 4

Room-temperature ratio a = χ 3333 ( 3 ) / χ 1133 ( 3 ) of two independent tensor components of the third-order nonlinear-optical response of an ISQ in PMMA as a function of wavelength.

Fig. 5
Fig. 5

Room-temperature quadratic electroabsorption spectrum (points) and limited three-level model (smooth curve).

Fig. 6
Fig. 6

Temperature dependence of the tensor ratio a at 596 and 632 nm.

Fig. 7
Fig. 7

Ratio of two-photon contribution to one-photon contribution of the third-order nonlinear-optical susceptibility as a function of wavelength.

Fig. 8
Fig. 8

One-photon and two-photon contributions to the third-order nonlinear-optical susceptibility as a function of wavelength.

Fig. 9
Fig. 9

(a) Real and imaginary parts of the third-order susceptibility, (b) the ratio of the real to the imaginary parts as a function of wavelength.

Fig. 10
Fig. 10

One-photon figure of merit as a function of wavelength.

Tables (1)

Tables Icon

Table 1 Spectroscopic Designation of the States of the ISQ Chromophore

Equations (38)

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γ K [ μ 01 4 D 11 + μ 01 2 ( Δ μ 01 ) 2 D 111 + μ 12 2 μ 01 2 D 121 ] ,
P i = N p i * i ,
χ 1133 ( 3 ) ( ω ; ω , 0 , 0 ) = N 15 [ γ z z z z * ( ω ; ω , 0 , 0 ) 2 3 α z z * ( ω , ω ) α z z * ( 0 , 0 ) k θ ] ,
χ 3333 ( 3 ) ( ω ; ω , 0 , 0 ) = N 15 [ 3 γ z z z z * ( ω ; ω , 0 , 0 ) + 4 3 α z z * ( ω , ω ) α z z * ( 0 , 0 ) k θ ] .
χ 1133 ( 3 ) ( ω ; ω , 0 , 0 ) = N 6 { 6 γ * ( ω ; ω , 0 , 0 ) × [ 1 15 11 3375 ( k θ k T ) ] + α * ( ω , ω ) α * ( 0 ; 0 ) k T × [ 2 45 + ( 1967 10125 π 2 36 ) ( k θ k T ) ] } ,
χ 3333 ( 3 ) ( ω ; ω , 0 , 0 ) = N 6 { 6 γ * ( ω ; ω , 0 , 0 ) × [ 1 5 + 298 1125 ( k θ k T ) ] + 2 α * ( ω ; ω ) α * ( 0 ; 0 ) 45 k T × [ 2 + 1591 225 ( k θ k T ) ] } ,
γ K ( μ 01 4 D 11 + μ 12 2 μ 01 2 D 121 ) ,
Im ( γ ) = μ 01 4 [ μ 12 2 μ 01 2 Im ( D 121 ) Im ( D 11 ) ] .
n i j = ( n 0 ) i j + ( n 0 ) i j 3 s i j k l 2 E k 0 E l 0 ,
( n 0 3 s ) I = λ d 4 π V rms 2 I 2 Ω I sig ,
Im [ ( χ 1133 ( 3 ) ) 3 L ] = N 15 ( f ω ) 2 ( f 0 ) 2 Im ( γ ) .
Im ( n 0 4 s ) = 3 Im [ χ ( 3 ) ] .
Δ d d = N α z z * V 2 K r d 2 .
Δ d d = V 2 8 π K r d 2 .
I = I 0 T 2 exp ( α d ) | 1 R exp ( i Φ ) | 2 = I 0 T 2 exp ( α d ) ( 1 R ) 2 × ( 1 + F sin 2 Φ 2 ) 1 ,
λ 4 π Δ I I = ( n I Δ d + d Δ n I ) [ 1 F sin 2 ( 4 π n R d λ + ϕ r 2 ) ] + F ( n R Δ d + d Δ n R ) sin ( 4 π n R d λ + ϕ r ) .
Δ I I = 4 π λ [ n I Δ d + F n R Δ d sin ( 4 π n R d λ + ϕ r ) ] .
γ l , m , n μ g l μ l m μ m n μ n g ( ω n g ω + i Γ n ) ( ω m g ω + i Γ m ) ( ω l g ω + i Γ l ) l , n μ g l μ l g μ g n μ n g ( ω l g ω + i Γ l ) 2 ( ω n + i Γ n ) + ,
γ 3 L = μ g 1 2 μ 12 2 ( ω 1 g ω + i Γ 1 ) 2 ( ω 2 + i Γ 2 )
γ L 2 = μ g 1 4 ( ω 1 g ω + i Γ 1 ) 2 ( ω 1 g + i Γ 1 ) .
γ 3 L γ 2 L = μ 12 2 μ g 1 2 ( ω 1 g + i Γ 1 ) ( ω 2 g ω 1 g + i Γ 2 ) .
γ 31 = μ g 1 2 μ 12 2 ( ω 1 g ω + i Γ 1 ) 2 ( ω 2 g ω + i Γ 2 ) ,
γ 32 = n g , 2 μ g 1 2 μ 1 n 2 ( ω 1 g ω + i Γ 1 ) 2 ( ω n g ω + i Γ n ) ,
γ 33 = n g , 1 μ g 1 μ 12 μ 2 n μ n g ( ω 1 g ω + i Γ 1 ) ( ω n g ω + i Γ n ) ( ω 2 g ω + i Γ 2 ) ,
γ 34 = m , n g , 1 μ g m μ m 2 μ 2 n μ n g ( ω n g ω + i Γ n ) ( ω m g ω + i Γ m ) ( ω 2 g ω + i Γ 2 ) ,
γ 35 = n g , 1 , m 2 μ g 1 μ 1 m μ m n μ n g ( ω n g ω + i Γ n ) ( ω m g ω + i Γ m ) ( ω 1 g ω + i Γ 1 ) .
γ 32 γ 31 = n g , 2 ( μ 1 n 2 μ 12 2 ) ( ω 2 g ω + i Γ 2 ω n g ω + i Γ n ) .
γ 32 γ 31 = i n g , 2 ( μ 1 n 2 μ 12 2 ) ( Γ 2 ω n g ω 2 g ) .
γ 33 γ 31 = n g , 1 ( μ 2 n μ 12 ) ( μ n g μ 1 g ) ( ω 1 g ω + i Γ 1 ω n g ω + i Γ n ) .
( μ n g μ 1 g ) ( μ 2 n μ 12 ) ω n g ω 1 g Γ 1 .
γ 34 γ 31 = m , n g , 1 ( μ m 2 μ 12 ) ( μ 2 n μ 12 ) ( μ g m μ g 1 ) ( μ n g μ g 1 ) × ( ω 1 g ω + i Γ 1 ω n g ω + i Γ n ) ( ω 1 g ω + i Γ 1 ω m g ω + i Γ m ) .
γ 35 γ 31 = m 2 , n g , 1 ( μ 1 m μ 12 ) ( μ m n μ 12 ) ( μ n g μ g 1 ) ( ω 1 g ω + i Γ 1 ω n g ω + i Γ n ) × ( ω 2 g ω + i Γ 2 ω m g ω + i Γ m ) .
γ 21 = μ g 1 4 ( ω 1 g ω + i Γ 1 ) 2 ( ω 1 g + i Γ 1 ) ,
γ 22 = n 1 , g μ g 1 2 μ g n 2 ( ω 1 g ω + i Γ 1 ) 2 ( ω n g + i Γ n ) .
γ 22 γ 21 = n g , 1 ( μ g n 2 μ g l 2 ) [ ( ω 1 g + i Γ 1 ) ( ω n g + i Γ n ) ] .
μ g n 2 / μ g 1 2 1.9 .
Im ( D 11 ) = 16 Γ 1 ω ω 01 ( Γ 1 4 ω 4 2 Γ 1 2 ω 01 2 + 4 ω 01 2 3 ω 01 4 ) ( ω 01 2 + Γ 1 2 ) [ ( ω 01 + ω ) 2 + Γ 1 2 ] 2 [ ( ω 01 ω ) 2 + Γ 1 2 ] 2 ,
Im ( D 121 ) = Γ 1 2 Γ 2 Γ 2 ω ω 1 + Γ 2 ω 1 2 2 Γ 1 2 Γ 2 Γ 1 ω ω 2 + 2 Γ 1 ω 1 ω 2 ( Γ 1 2 + ω 1 2 ) ( Γ 1 2 + ω 2 2 ω ω 1 + ω 1 2 ) ( Γ 2 2 + ω 2 2 ) + Γ 1 ω 2 2 Γ 1 2 Γ 2 Γ 2 ω 2 + 3 Γ 2 ω ω 1 2 Γ 2 ω 1 2 Γ 1 ω ω 2 ( Γ 1 2 + ω 1 2 ) ( Γ 1 2 + ω 2 2 ω ω 1 + ω 1 2 ) ( Γ 2 2 + ω 2 2 ω ω 2 + ω 2 2 ) + 1 ( Γ 1 2 + ω 1 2 ) ( Γ 1 2 + ω 2 2 ω ω 1 + ω 1 2 ) 2 ( Γ 2 2 + ω 2 2 ) ( Γ 2 2 + ω 2 2 ω ω 2 + ω 2 2 ) × [ 3 Γ 1 4 Γ 2 3 + Γ 1 4 Γ 2 ω 2 3 Γ 1 3 Γ 2 2 ω 2 + Γ 1 2 Γ 2 3 ω 2 + Γ 1 2 Γ 2 ω 4 Γ 1 Γ 2 2 ω 4 + 4 Γ 1 3 Γ 2 2 ω ω 1 Γ 1 2 Γ 2 ω 3 ω 1 + 4 Γ 1 Γ 2 2 ω 3 ω 1 + 2 Γ 2 3 ω 3 ω 1 + Γ 2 ω 5 ω 1 7 Γ 1 Γ 2 2 ω 2 ω 1 2 7 Γ 2 3 ω 2 ω 1 2 3 Γ 2 ω 4 ω 1 2 + 4 Γ 1 Γ 2 2 ω ω 1 3 + 8 Γ 2 3 ω ω 1 3 + 3 Γ 2 ω 3 ω 1 3 3 Γ 2 3 ω 1 4 Γ 2 ω 2 ω 1 4 2 Γ 1 4 Γ 2 ω ω 2 + 4 Γ 1 3 Γ 2 2 ω ω 2 + Γ 1 3 ω 3 ω 2 2 Γ 1 2 Γ 2 ω 3 ω 2 + 2 Γ 1 Γ 2 2 ω 3 ω 2 + Γ 1 ω 5 ω 2 6 Γ 1 3 Γ 2 2 ω 1 ω 2 2 Γ 1 3 ω 2 ω 1 ω 2 + 2 Γ 1 2 Γ 2 ω 2 ω 1 ω 2 8 Γ 1 Γ 2 2 ω 2 ω 1 ω 2 4 Γ 1 ω 4 ω 1 ω 2 2 Γ 2 ω 2 ω 1 ω 2 + 12 Γ 1 Γ 2 2 ω ω 1 2 ω 2 + 5 Γ 1 ω 3 ω 1 2 ω 2 + 6 Γ 2 ω 2 ω 1 2 ω 2 6 Γ 1 Γ 2 2 ω 1 3 ω 2 2 Γ 1 ω 2 ω 1 3 ω 2 6 Γ 2 ω 2 ω 1 3 ω 2 + 2 Γ 2 ω ω 1 4 ω 2 + 3 Γ 1 4 Γ 2 ω 2 2 5 Γ 1 3 ω 2 ω 2 2 + Γ 1 2 Γ 2 ω 2 ω 2 2 3 Γ 1 ω 4 ω 2 2 + 8 Γ 1 3 ω ω 1 ω 2 2 + 12 Γ 1 ω 3 ω 1 ω 2 2 + 2 Γ 2 ω 3 ω 1 ω 2 2 17 Γ 1 ω 2 ω 1 2 ω 2 2 7 Γ 2 ω 2 ω 1 2 ω 2 2 + 8 Γ 1 ω ω 1 3 ω 2 2 + 8 Γ 2 ω ω 1 3 ω 2 2 3 Γ 2 ω 1 4 ω 2 2 + 4 Γ 1 3 ω ω 2 3 + 2 Γ 1 ω 3 ω 2 3 6 Γ 1 3 ω 1 ω 2 3 8 Γ 1 ω 2 ω 1 ω 2 3 + 12 Γ 1 ω ω 1 2 ω 2 3 6 Γ 1 ω 1 3 ω 2 3 ] + 2 Γ 1 4 Γ 2 + 3 Γ 1 3 ω 2 + Γ 1 ω 4 + 4 Γ 1 3 ω ω 1 ( Γ 1 2 + ω 1 2 ) ( Γ 1 2 + ω 2 + 2 ω ω 1 + ω 1 2 ) 2 ( Γ 2 2 + ω 2 + 2 ω ω 2 + ω 2 2 ) + Γ 1 2 Γ 2 ω ω 1 + 4 Γ 1 ω 3 ω 1 + Γ 2 ω 3 ω 1 + 7 Γ 1 ω 2 ω 1 2 ( Γ 1 2 + ω 1 2 ) ( Γ 1 2 + ω 2 + 2 ω ω 1 + ω 1 2 ) ( Γ 2 2 + ω 2 + 2 ω ω 2 + ω 2 2 ) + 4 Γ 2 ω 3 ω 1 2 + 4 Γ 1 ω ω 1 3 + 5 Γ 2 ω ω 1 3 ( Γ 1 2 + ω 1 2 ) ( Γ 1 2 + ω 2 + 2 ω ω 1 + ω 1 2 ) 2 ( Γ 2 2 + ω 2 + 2 ω ω 2 + ω 2 2 ) + 2 Γ 2 ω 1 4 + 2 Γ 1 2 Γ 2 Γ 1 ω 2 + Γ 2 ω 2 ( Γ 1 2 + ω 1 2 ) ( Γ 1 2 + ω 2 + 2 ω ω 1 + ω 1 2 ) 2 ( Γ 2 2 + ω 2 + 2 ω ω 2 + ω 2 2 ) + 3 Γ 2 ω ω 1 + 2 Γ 2 ω 1 2 + 3 Γ 1 3 ω ω 2 + Γ 1 ω 3 ω 2 ( Γ 1 2 + ω 1 2 ) ( Γ 1 2 + ω 2 + 2 ω ω 1 + ω 1 2 ) 2 ( Γ 2 2 + ω 2 + 2 ω ω 2 + ω 2 2 ) + 4 Γ 1 3 ω 1 ω 2 + 4 Γ 1 ω 2 ω 1 ω 2 + 7 Γ 1 ω ω 1 2 ω 2 ( Γ 1 2 + ω 1 2 ) ( Γ 1 2 + ω 2 + 2 ω ω 1 + ω 1 2 ) 2 ( Γ 2 2 + ω 2 + 2 ω ω 2 + ω 2 2 ) + 4 Γ 1 ω 1 3 ω 2 Γ 1 ω ω 2 ( Γ 1 2 + ω 1 2 ) ( Γ 1 2 + ω 2 + 2 ω ω 1 + ω 1 2 ) 2 ( Γ 2 2 + ω 2 + 2 ω ω 2 + ω 2 2 ) .

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