Abstract

Transverse intensity profiles from a model for a laser with a parabolic transverse excitation profile are shown to change from symmetric to asymmetric patterns, with bistability between some patterns, as the frequency spacing between the transverse cavity modes is decreased. The parabolic excitation profile suppresses some of the nonlinear coupling of the transverse modes that predominates when the excitation profile is uniform. Nonetheless there are substantial qualitative similarities between the results for uniform and parabolic excitation, suggesting some generic governing principles for the pattern formation caused by mode competition and mode correlation.

© 1991 Optical Society of America

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References

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  1. See, for example, the overview (and the references therein) to a feature issue containing 34 recent papers on this subject, N. B. Abraham and W. J. Firth, J. Opt. Soc. Am. B 7, 947–1157 (1990).
  2. A. F. Suchkov, Zh. Eksp. Teor. Fiz. 49, 1495–1503 (1965) [Sov. Phys. JETP 22, 622–628 (1966)]; R. G. Allakhverdyan, A. N. Oraevskii, and A. F. Suchkov, Fiz. Tekh. Poluprovodn. 4, 341–346 (1970) [Sov. Phys. Semicond. 4, 277–281 (1970)].
  3. R. Hauck, F. Hollinger, and H. Weber, Opt. Commun. 47, 141 (1983); F. Hollinger and Chr. Jung, J. Opt. Soc. Am. B 2, 218 (1985).
    [CrossRef]
  4. M. L. Shih, P. W. Milonni, and J. R. Ackerhalt, J. Opt. Soc. Am. B 2, 218 (1985).
    [CrossRef]
  5. D. J. Biswas and R. G. Harrison, Phys. Rev. A 32, 3835 (1985).
    [CrossRef] [PubMed]
  6. L. A. Lugiato, C. Oldano, and L. M. Narducci, J. Opt. Soc. Am. B 5, 879 (1988).
    [CrossRef]
  7. L. A. Lugiato, G.-L. Oppo, M. A. Pernigo, J. R. Tredicce, L. M. Narducci, and D. K. Bandy, Opt. Commun. 68, 63 (1988).
    [CrossRef]
  8. L. A. Lugiato, F. Prati, L. M. Narducci, and G.-L. Oppo, Opt. Commun. 69, 387 (1989).
    [CrossRef]
  9. J. R. Tredicce, E. J. Quel, A. M. Ghazzawi, C. Green, M. A. Pernigo, L. M. Narducci, and L. A. Lugiato, Phys. Rev. Lett. 62, 1274 (1989).
    [CrossRef] [PubMed]
  10. F. Hollinger, Chr. Jung, and H. Weber, Opt. Commun. 75, 84 (1990).
    [CrossRef]
  11. C. O. Weiss, Chr. Tamm, and P. Coullet, J. Mod. Opt. 37, 1825 (1990).
    [CrossRef]
  12. F. Hollinger, Chr. Jung, and H. Weber, J. Opt. Soc. Am. B 7, 1013 (1990).
    [CrossRef]
  13. L. A. Lugiato, G. L. Oppo, J. R. Tredicce, L. M. Narducci, and M. A. Pernigo, J. Opt. Soc. Am. B 7, 1019 (1990).
    [CrossRef]
  14. L. A. Melnikov, S. A. Tatarkova, and G. N. Tatarkov, J. Opt. Soc. Am. B 7, 1286 (1990).
    [CrossRef]
  15. L.-G. Zhang, N.-P. Chen, and E.-X. Ba, J. Opt. Soc. Am. B 7, 1293 (1990).
    [CrossRef]
  16. N. Yu, R. K. DeFreez, D. J. Bossert, G. A. Wilson, R. A. Elliott, S.-S. Wang, and H. G. Winful, Appl. Opt. 30, 2503 (1991).
    [CrossRef] [PubMed]
  17. J. Buus, IEEE J. Quantum Electron. QE-15, 734 (1979).
    [CrossRef]
  18. H. Lin and N. B. Abraham, Opt. Commun. 79, 476 (1990).
    [CrossRef]
  19. H. Lin and N. B. Abraham, in Nonlinear Dynamics in Optical Systems, N. B. Abraham, E. M. Garmire, and P. Mandel, eds., Vol. 7 of 1991 OSA Meetings Proceedings Series (Optical Society of America, Washington, D.C., 1991), pp. 159–162.

1991 (1)

1990 (8)

H. Lin and N. B. Abraham, Opt. Commun. 79, 476 (1990).
[CrossRef]

F. Hollinger, Chr. Jung, and H. Weber, Opt. Commun. 75, 84 (1990).
[CrossRef]

C. O. Weiss, Chr. Tamm, and P. Coullet, J. Mod. Opt. 37, 1825 (1990).
[CrossRef]

F. Hollinger, Chr. Jung, and H. Weber, J. Opt. Soc. Am. B 7, 1013 (1990).
[CrossRef]

L. A. Lugiato, G. L. Oppo, J. R. Tredicce, L. M. Narducci, and M. A. Pernigo, J. Opt. Soc. Am. B 7, 1019 (1990).
[CrossRef]

L. A. Melnikov, S. A. Tatarkova, and G. N. Tatarkov, J. Opt. Soc. Am. B 7, 1286 (1990).
[CrossRef]

L.-G. Zhang, N.-P. Chen, and E.-X. Ba, J. Opt. Soc. Am. B 7, 1293 (1990).
[CrossRef]

See, for example, the overview (and the references therein) to a feature issue containing 34 recent papers on this subject, N. B. Abraham and W. J. Firth, J. Opt. Soc. Am. B 7, 947–1157 (1990).

1989 (2)

L. A. Lugiato, F. Prati, L. M. Narducci, and G.-L. Oppo, Opt. Commun. 69, 387 (1989).
[CrossRef]

J. R. Tredicce, E. J. Quel, A. M. Ghazzawi, C. Green, M. A. Pernigo, L. M. Narducci, and L. A. Lugiato, Phys. Rev. Lett. 62, 1274 (1989).
[CrossRef] [PubMed]

1988 (2)

L. A. Lugiato, C. Oldano, and L. M. Narducci, J. Opt. Soc. Am. B 5, 879 (1988).
[CrossRef]

L. A. Lugiato, G.-L. Oppo, M. A. Pernigo, J. R. Tredicce, L. M. Narducci, and D. K. Bandy, Opt. Commun. 68, 63 (1988).
[CrossRef]

1985 (2)

1983 (1)

R. Hauck, F. Hollinger, and H. Weber, Opt. Commun. 47, 141 (1983); F. Hollinger and Chr. Jung, J. Opt. Soc. Am. B 2, 218 (1985).
[CrossRef]

1979 (1)

J. Buus, IEEE J. Quantum Electron. QE-15, 734 (1979).
[CrossRef]

1965 (1)

A. F. Suchkov, Zh. Eksp. Teor. Fiz. 49, 1495–1503 (1965) [Sov. Phys. JETP 22, 622–628 (1966)]; R. G. Allakhverdyan, A. N. Oraevskii, and A. F. Suchkov, Fiz. Tekh. Poluprovodn. 4, 341–346 (1970) [Sov. Phys. Semicond. 4, 277–281 (1970)].

Abraham, N. B.

See, for example, the overview (and the references therein) to a feature issue containing 34 recent papers on this subject, N. B. Abraham and W. J. Firth, J. Opt. Soc. Am. B 7, 947–1157 (1990).

H. Lin and N. B. Abraham, Opt. Commun. 79, 476 (1990).
[CrossRef]

H. Lin and N. B. Abraham, in Nonlinear Dynamics in Optical Systems, N. B. Abraham, E. M. Garmire, and P. Mandel, eds., Vol. 7 of 1991 OSA Meetings Proceedings Series (Optical Society of America, Washington, D.C., 1991), pp. 159–162.

Ackerhalt, J. R.

Ba, E.-X.

Bandy, D. K.

L. A. Lugiato, G.-L. Oppo, M. A. Pernigo, J. R. Tredicce, L. M. Narducci, and D. K. Bandy, Opt. Commun. 68, 63 (1988).
[CrossRef]

Biswas, D. J.

D. J. Biswas and R. G. Harrison, Phys. Rev. A 32, 3835 (1985).
[CrossRef] [PubMed]

Bossert, D. J.

Buus, J.

J. Buus, IEEE J. Quantum Electron. QE-15, 734 (1979).
[CrossRef]

Chen, N.-P.

Coullet, P.

C. O. Weiss, Chr. Tamm, and P. Coullet, J. Mod. Opt. 37, 1825 (1990).
[CrossRef]

DeFreez, R. K.

Elliott, R. A.

Firth, W. J.

See, for example, the overview (and the references therein) to a feature issue containing 34 recent papers on this subject, N. B. Abraham and W. J. Firth, J. Opt. Soc. Am. B 7, 947–1157 (1990).

Ghazzawi, A. M.

J. R. Tredicce, E. J. Quel, A. M. Ghazzawi, C. Green, M. A. Pernigo, L. M. Narducci, and L. A. Lugiato, Phys. Rev. Lett. 62, 1274 (1989).
[CrossRef] [PubMed]

Green, C.

J. R. Tredicce, E. J. Quel, A. M. Ghazzawi, C. Green, M. A. Pernigo, L. M. Narducci, and L. A. Lugiato, Phys. Rev. Lett. 62, 1274 (1989).
[CrossRef] [PubMed]

Harrison, R. G.

D. J. Biswas and R. G. Harrison, Phys. Rev. A 32, 3835 (1985).
[CrossRef] [PubMed]

Hauck, R.

R. Hauck, F. Hollinger, and H. Weber, Opt. Commun. 47, 141 (1983); F. Hollinger and Chr. Jung, J. Opt. Soc. Am. B 2, 218 (1985).
[CrossRef]

Hollinger, F.

F. Hollinger, Chr. Jung, and H. Weber, Opt. Commun. 75, 84 (1990).
[CrossRef]

F. Hollinger, Chr. Jung, and H. Weber, J. Opt. Soc. Am. B 7, 1013 (1990).
[CrossRef]

R. Hauck, F. Hollinger, and H. Weber, Opt. Commun. 47, 141 (1983); F. Hollinger and Chr. Jung, J. Opt. Soc. Am. B 2, 218 (1985).
[CrossRef]

Jung, Chr.

F. Hollinger, Chr. Jung, and H. Weber, Opt. Commun. 75, 84 (1990).
[CrossRef]

F. Hollinger, Chr. Jung, and H. Weber, J. Opt. Soc. Am. B 7, 1013 (1990).
[CrossRef]

Lin, H.

H. Lin and N. B. Abraham, Opt. Commun. 79, 476 (1990).
[CrossRef]

H. Lin and N. B. Abraham, in Nonlinear Dynamics in Optical Systems, N. B. Abraham, E. M. Garmire, and P. Mandel, eds., Vol. 7 of 1991 OSA Meetings Proceedings Series (Optical Society of America, Washington, D.C., 1991), pp. 159–162.

Lugiato, L. A.

L. A. Lugiato, G. L. Oppo, J. R. Tredicce, L. M. Narducci, and M. A. Pernigo, J. Opt. Soc. Am. B 7, 1019 (1990).
[CrossRef]

J. R. Tredicce, E. J. Quel, A. M. Ghazzawi, C. Green, M. A. Pernigo, L. M. Narducci, and L. A. Lugiato, Phys. Rev. Lett. 62, 1274 (1989).
[CrossRef] [PubMed]

L. A. Lugiato, F. Prati, L. M. Narducci, and G.-L. Oppo, Opt. Commun. 69, 387 (1989).
[CrossRef]

L. A. Lugiato, G.-L. Oppo, M. A. Pernigo, J. R. Tredicce, L. M. Narducci, and D. K. Bandy, Opt. Commun. 68, 63 (1988).
[CrossRef]

L. A. Lugiato, C. Oldano, and L. M. Narducci, J. Opt. Soc. Am. B 5, 879 (1988).
[CrossRef]

Melnikov, L. A.

Milonni, P. W.

Narducci, L. M.

L. A. Lugiato, G. L. Oppo, J. R. Tredicce, L. M. Narducci, and M. A. Pernigo, J. Opt. Soc. Am. B 7, 1019 (1990).
[CrossRef]

L. A. Lugiato, F. Prati, L. M. Narducci, and G.-L. Oppo, Opt. Commun. 69, 387 (1989).
[CrossRef]

J. R. Tredicce, E. J. Quel, A. M. Ghazzawi, C. Green, M. A. Pernigo, L. M. Narducci, and L. A. Lugiato, Phys. Rev. Lett. 62, 1274 (1989).
[CrossRef] [PubMed]

L. A. Lugiato, G.-L. Oppo, M. A. Pernigo, J. R. Tredicce, L. M. Narducci, and D. K. Bandy, Opt. Commun. 68, 63 (1988).
[CrossRef]

L. A. Lugiato, C. Oldano, and L. M. Narducci, J. Opt. Soc. Am. B 5, 879 (1988).
[CrossRef]

Oldano, C.

Oppo, G. L.

Oppo, G.-L.

L. A. Lugiato, F. Prati, L. M. Narducci, and G.-L. Oppo, Opt. Commun. 69, 387 (1989).
[CrossRef]

L. A. Lugiato, G.-L. Oppo, M. A. Pernigo, J. R. Tredicce, L. M. Narducci, and D. K. Bandy, Opt. Commun. 68, 63 (1988).
[CrossRef]

Pernigo, M. A.

L. A. Lugiato, G. L. Oppo, J. R. Tredicce, L. M. Narducci, and M. A. Pernigo, J. Opt. Soc. Am. B 7, 1019 (1990).
[CrossRef]

J. R. Tredicce, E. J. Quel, A. M. Ghazzawi, C. Green, M. A. Pernigo, L. M. Narducci, and L. A. Lugiato, Phys. Rev. Lett. 62, 1274 (1989).
[CrossRef] [PubMed]

L. A. Lugiato, G.-L. Oppo, M. A. Pernigo, J. R. Tredicce, L. M. Narducci, and D. K. Bandy, Opt. Commun. 68, 63 (1988).
[CrossRef]

Prati, F.

L. A. Lugiato, F. Prati, L. M. Narducci, and G.-L. Oppo, Opt. Commun. 69, 387 (1989).
[CrossRef]

Quel, E. J.

J. R. Tredicce, E. J. Quel, A. M. Ghazzawi, C. Green, M. A. Pernigo, L. M. Narducci, and L. A. Lugiato, Phys. Rev. Lett. 62, 1274 (1989).
[CrossRef] [PubMed]

Shih, M. L.

Suchkov, A. F.

A. F. Suchkov, Zh. Eksp. Teor. Fiz. 49, 1495–1503 (1965) [Sov. Phys. JETP 22, 622–628 (1966)]; R. G. Allakhverdyan, A. N. Oraevskii, and A. F. Suchkov, Fiz. Tekh. Poluprovodn. 4, 341–346 (1970) [Sov. Phys. Semicond. 4, 277–281 (1970)].

Tamm, Chr.

C. O. Weiss, Chr. Tamm, and P. Coullet, J. Mod. Opt. 37, 1825 (1990).
[CrossRef]

Tatarkov, G. N.

Tatarkova, S. A.

Tredicce, J. R.

L. A. Lugiato, G. L. Oppo, J. R. Tredicce, L. M. Narducci, and M. A. Pernigo, J. Opt. Soc. Am. B 7, 1019 (1990).
[CrossRef]

J. R. Tredicce, E. J. Quel, A. M. Ghazzawi, C. Green, M. A. Pernigo, L. M. Narducci, and L. A. Lugiato, Phys. Rev. Lett. 62, 1274 (1989).
[CrossRef] [PubMed]

L. A. Lugiato, G.-L. Oppo, M. A. Pernigo, J. R. Tredicce, L. M. Narducci, and D. K. Bandy, Opt. Commun. 68, 63 (1988).
[CrossRef]

Wang, S.-S.

Weber, H.

F. Hollinger, Chr. Jung, and H. Weber, J. Opt. Soc. Am. B 7, 1013 (1990).
[CrossRef]

F. Hollinger, Chr. Jung, and H. Weber, Opt. Commun. 75, 84 (1990).
[CrossRef]

R. Hauck, F. Hollinger, and H. Weber, Opt. Commun. 47, 141 (1983); F. Hollinger and Chr. Jung, J. Opt. Soc. Am. B 2, 218 (1985).
[CrossRef]

Weiss, C. O.

C. O. Weiss, Chr. Tamm, and P. Coullet, J. Mod. Opt. 37, 1825 (1990).
[CrossRef]

Wilson, G. A.

Winful, H. G.

Yu, N.

Zhang, L.-G.

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

J. Buus, IEEE J. Quantum Electron. QE-15, 734 (1979).
[CrossRef]

J. Mod. Opt. (1)

C. O. Weiss, Chr. Tamm, and P. Coullet, J. Mod. Opt. 37, 1825 (1990).
[CrossRef]

J. Opt. Soc. Am. B (7)

Opt. Commun. (5)

F. Hollinger, Chr. Jung, and H. Weber, Opt. Commun. 75, 84 (1990).
[CrossRef]

L. A. Lugiato, G.-L. Oppo, M. A. Pernigo, J. R. Tredicce, L. M. Narducci, and D. K. Bandy, Opt. Commun. 68, 63 (1988).
[CrossRef]

L. A. Lugiato, F. Prati, L. M. Narducci, and G.-L. Oppo, Opt. Commun. 69, 387 (1989).
[CrossRef]

H. Lin and N. B. Abraham, Opt. Commun. 79, 476 (1990).
[CrossRef]

R. Hauck, F. Hollinger, and H. Weber, Opt. Commun. 47, 141 (1983); F. Hollinger and Chr. Jung, J. Opt. Soc. Am. B 2, 218 (1985).
[CrossRef]

Phys. Rev. A (1)

D. J. Biswas and R. G. Harrison, Phys. Rev. A 32, 3835 (1985).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

J. R. Tredicce, E. J. Quel, A. M. Ghazzawi, C. Green, M. A. Pernigo, L. M. Narducci, and L. A. Lugiato, Phys. Rev. Lett. 62, 1274 (1989).
[CrossRef] [PubMed]

Zh. Eksp. Teor. Fiz. (1)

A. F. Suchkov, Zh. Eksp. Teor. Fiz. 49, 1495–1503 (1965) [Sov. Phys. JETP 22, 622–628 (1966)]; R. G. Allakhverdyan, A. N. Oraevskii, and A. F. Suchkov, Fiz. Tekh. Poluprovodn. 4, 341–346 (1970) [Sov. Phys. Semicond. 4, 277–281 (1970)].

Other (1)

H. Lin and N. B. Abraham, in Nonlinear Dynamics in Optical Systems, N. B. Abraham, E. M. Garmire, and P. Mandel, eds., Vol. 7 of 1991 OSA Meetings Proceedings Series (Optical Society of America, Washington, D.C., 1991), pp. 159–162.

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Figures (5)

Fig. 1
Fig. 1

Numerical results for the uniform excitation profile and Δ = 0.1. The other parameters are κ′ = 0.1, γ′ = 1.0, N = 15, and 2C = 2.33. (a) Moduli of amplitudes of dominant modes, |fn|, versus mode spacing parameter a(1), found by perturbation of an initial spatially uniform solution. Different symbols represent amplitudes of different cavity modes as indicated by labels. (b) Frequency shift δ and (c) average total intensity, 〈I〉, of the CFL states versus a(1) with additional solutions beyond those shown in (a), found by adiabatic scanning of parameter a(1). The curves are added as guides to the eye.

Fig. 2
Fig. 2

Numerical results for the parabolic excitation profile and Δ = 0.1. The other parameters are κ′ = 0.1, γ′ = 1.0, N = 15, and 2C = 3.5. (a) Moduli of amplitudes of dominant modes, |fn|, versus mode spacing parameter a(1), found by perturbation of an initially spatially uniform solution. Different symbols represent amplitudes of different cavity modes as indicated by labels. (b) Frequency shift δ and (c) average total intensity, 〈I〉, of the CFL states versus a(1) with additional solutions beyond those shown in (a), found by adiabatic scanning of parameter a(1). The isolated branch of solutions shown in (b) and (c) corresponds to the symmetric solutions found in that region in (a). The other branch of solutions is symmetric for a(1) > 0.6 and asymmetric for a(1) < 0.6. The curves are added as guides to the eye.

Fig. 3
Fig. 3

Numerical results for the uniform excitation profile and Δ = 0.4. The other parameters are κ′ = 0.1, γ′ = 1.0, N = 15, and 2C = 2.33. (a) Moduli of amplitudes of dominant modes, |fn|, versus mode spacing parameter a(1) for solutions found by perturbation of a spatially uniform initial solution. Different symbols represent amplitudes of different cavity modes as indicated by labels. (b) Frequency shift δ and (c) average total intensity, 〈I〉, of the CFL states versus a(1) with additional solutions beyond those shown in (a), found by adiabatic scanning of the parameter a(1). The curves are added as guides to the eye.

Fig. 4
Fig. 4

Numerical results for the parabolic excitation profile and Δ = 0.4. The other parameters are κ′ = 0.1, γ′ = 1.0, N = 15, and 2C = 3.5. (a) Moduli of amplitudes of dominant modes, |fn|, found by perturbation of initial spatially uniform solutions. (b) Frequency shift δ and (c) average total intensity, 〈I〉, versus a(1) with additional solutions beyond those shown in (a), found by adiabatic scanning of the parameter a(1). CFL states and time-dependent solutions in (b) and (c) are indicated by filled and open circles, respectively. The curves are added as guides to the eye.

Fig. 5
Fig. 5

Comparison between the single-mode lasing frequency, indicated by lines, and the frequencies of the numerical solutions, indicated by filled or open circles for the parabolic excitation profile. The other parameters are Δ = 0.65, 2C = 3.5, κ′ = 0.1, γ′ = 1.0, and N = 15. The lasing frequencies of the cavity modes are labeled according to the indices of these modes. The filled circles indicate the frequencies of CFL solutions, found by numerical calculation, and the open circles indicate the two dominant frequencies in the optical power spectrum when the numerical solutions can be explained as beating between two supermodes, each dominated by one of the modes. Each open circle is the average frequency of one of the two supermodes.

Equations (17)

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E t = - κ [ ( 1 - i Δ ) E - 2 C P - i a 2 E x 2 ] ,
P t = - γ [ ( 1 + i Δ ) P - E D ] ,
D t = - γ [ D - D 0 ( x ) + 1 2 ( E P * + E * P ) ] ,
ω n c = ω 0 c + κ a ( n ) ,
D 0 ( x ) = 1.
D 0 ( x ) = 4 x ( 1 - x ) .
D 0 ( x ) = 2 3 - n = 1 8 [ 1 + ( - 1 ) n ] n 2 π 2 cos ( n π x ) ,
d f n d t = - κ [ ( 1 - i Δ ) f n - 2 C p n + i a ( n ) f n ] ,
d p n d t = - γ 1 [ ( 1 + i Δ ) p n + 1 2 ( 1 + δ n , 0 ) m q 0 * f m d q ] .
d d 0 d t = - γ [ d 0 - 2 3 + 1 8 m q 0 * ( f m p q * + f m * p q ) ] ,
d d n d t = - γ [ d n + 8 [ 1 + ( - 1 ) n ] ( n π ) 2 + 1 4 m q 0 * ( f m p q * + f m * p q ) ] ,             n > 0 ,
1 κ E t = - ( 1 - i Δ ) E [ 1 - 2 C 1 + Δ 2 + E 2 D 0 ( x ) ] + i a 2 E x 2 .
( 1 + Δ 2 + E 2 ) - 1 1 / ( 1 + Δ 2 ) - E 2 / ( 1 + Δ 2 ) 2 ,
1 κ d F n d t = ( 1 - i Δ ) [ r F n - 1 4 ( 1 + δ n , 0 ) × m q s 0 * F m F q * F s - 2 C 1 + Δ 2 1 2 ( 1 + δ n , 0 ) × k s 0 * 8 [ 1 + ( - 1 ) k ] ( k π ) 2 F s + 3 2 1 8 ( 1 + δ n , 0 ) × m q k s 0 * 8 [ 1 + ( - 1 ) k ] ( k π ) 2 F m F q * F s ] - i a ( n ) F n ,
D ( x ) = ( 1 + Δ 2 ) 1 + Δ 2 + E ( x ) 2 D 0 ( x ) ,
P ( x ) = ( 1 - i Δ ) 1 + Δ 2 + E ( x ) 2 D 0 ( x ) E ( x ) ,
i a 2 E ( x ) x 2 = ( 1 - i Δ ) 1 + Δ 2 + E 2 - 2 C D 0 ( x ) 1 + Δ 2 + E ( x ) 2 E ( x ) .

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