Abstract

Second-harmonic generation using high-intensity ultrashort pulses is studied numerically. In a highly dispersive medium and for ultrashort (subpicosecond to femtosecond) fundamental pulses the effective group-velocity dispersion (GVD) as well as the group-velocity mismatch should be taken into account. The numerical results indicate that in this case the conversion efficiency is reduced, compared with the case when only the group-velocity mismatch is taken into account, and for somewhat larger values of effective GVD parameters it exhibits an oscillatory behavior. Fundamental and harmonic pulse shapes inside the medium are also calculated. The cases with the effective GVD taken into account show that inside the medium both of the pulses split into two or more pulses.

© 1988 Optical Society of America

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  1. P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, Phys. Rev. Lett. 7, 118 (1961).
    [Crossref]
  2. J. A. Giordmaine, Phys. Rev. Lett. 8, 19 (1962);P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, Phys. Rev. Lett. 8, 21 (1962).See also P. A. Franken and J. F. Ward, Rev. Mod. Phys. 35, 23 (1963).
    [Crossref]
  3. J. E. Geusic, H. J. Levinstein, S. Singh, R. G. Smith, and L. G. Van Uitert, Appl. Phys. Lett. 12, 306 (1968).
    [Crossref]
  4. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys. Rev. 127, 1918 (1962).
    [Crossref]
  5. R. L. Fork, B. I. Greene, and C. V. Shank, Appl. Phys. Lett. 38, 671 (1981);C. V. Shank, R. L. Fork, R. Yen, R. H. Stolen, and W. J. Tomlinson, Appl. Phys. Lett. 40, 761 (1982);W. H. Knox, R. L. Fork, M. C. Downer, R. H. Stolen, C. W. Shank, and J. A. Valdmanis, Appl. Phys. Lett. 46, 1120 (1985).
    [Crossref]
  6. S. A. Akhmanov, A. I. Kovrigin, A. P. Sukhorukov, R. V. Khokhlov, and A. S. Chirkin, JETP Lett. 7, 182 (1968);S. A. Akhanov, A. S. Chirkin, A. I. Kovrigin, R. V. Khokhlov, and A. P. Sukhorukov, IEEE J. Quantum Electron. QE-4, 598 (1968).
    [Crossref]
  7. J. Comly and E. Garmire, Appl. Phys. Lett. 12, 7 (1968).
    [Crossref]
  8. R. C. Miller, Phys. Lett. 26A, 177 (1968).
  9. E. Garmire and A. Yariv, IEEE J. Quantum Electron. QE-3, 222 (1967).
    [Crossref]
  10. J. Ducuing and C. Flytzanis, in Optical Properties of Solids, F. Abeles, ed. (North-Holland, Amsterdam, 1972), p. 863.
  11. A. Bamberger, C. Flytzanis, and D. Jennevé, rapport interne no. 149, Centre de Mathématiques Appliquées [Ecole Polytechni-que, Paris, June 1986 (in French)].
  12. S. A. Akhmanov, A. I. Kovrygin, and A. P. Sukhorukov, Nonlinear Optics, Vol. 1 of Quantum Electronics, H. Rabin and C. L. Tang, eds. (Academic, New York, 1975), Part B, pp. 475–586.
  13. R. A. Fisher and W. K. Bischel, J. Appl. Phys. 46, 4921 (1975).
    [Crossref]

1981 (1)

R. L. Fork, B. I. Greene, and C. V. Shank, Appl. Phys. Lett. 38, 671 (1981);C. V. Shank, R. L. Fork, R. Yen, R. H. Stolen, and W. J. Tomlinson, Appl. Phys. Lett. 40, 761 (1982);W. H. Knox, R. L. Fork, M. C. Downer, R. H. Stolen, C. W. Shank, and J. A. Valdmanis, Appl. Phys. Lett. 46, 1120 (1985).
[Crossref]

1975 (1)

R. A. Fisher and W. K. Bischel, J. Appl. Phys. 46, 4921 (1975).
[Crossref]

1968 (4)

S. A. Akhmanov, A. I. Kovrigin, A. P. Sukhorukov, R. V. Khokhlov, and A. S. Chirkin, JETP Lett. 7, 182 (1968);S. A. Akhanov, A. S. Chirkin, A. I. Kovrigin, R. V. Khokhlov, and A. P. Sukhorukov, IEEE J. Quantum Electron. QE-4, 598 (1968).
[Crossref]

J. Comly and E. Garmire, Appl. Phys. Lett. 12, 7 (1968).
[Crossref]

R. C. Miller, Phys. Lett. 26A, 177 (1968).

J. E. Geusic, H. J. Levinstein, S. Singh, R. G. Smith, and L. G. Van Uitert, Appl. Phys. Lett. 12, 306 (1968).
[Crossref]

1967 (1)

E. Garmire and A. Yariv, IEEE J. Quantum Electron. QE-3, 222 (1967).
[Crossref]

1962 (2)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys. Rev. 127, 1918 (1962).
[Crossref]

J. A. Giordmaine, Phys. Rev. Lett. 8, 19 (1962);P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, Phys. Rev. Lett. 8, 21 (1962).See also P. A. Franken and J. F. Ward, Rev. Mod. Phys. 35, 23 (1963).
[Crossref]

1961 (1)

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, Phys. Rev. Lett. 7, 118 (1961).
[Crossref]

Akhmanov, S. A.

S. A. Akhmanov, A. I. Kovrigin, A. P. Sukhorukov, R. V. Khokhlov, and A. S. Chirkin, JETP Lett. 7, 182 (1968);S. A. Akhanov, A. S. Chirkin, A. I. Kovrigin, R. V. Khokhlov, and A. P. Sukhorukov, IEEE J. Quantum Electron. QE-4, 598 (1968).
[Crossref]

S. A. Akhmanov, A. I. Kovrygin, and A. P. Sukhorukov, Nonlinear Optics, Vol. 1 of Quantum Electronics, H. Rabin and C. L. Tang, eds. (Academic, New York, 1975), Part B, pp. 475–586.

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys. Rev. 127, 1918 (1962).
[Crossref]

Bamberger, A.

A. Bamberger, C. Flytzanis, and D. Jennevé, rapport interne no. 149, Centre de Mathématiques Appliquées [Ecole Polytechni-que, Paris, June 1986 (in French)].

Bischel, W. K.

R. A. Fisher and W. K. Bischel, J. Appl. Phys. 46, 4921 (1975).
[Crossref]

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys. Rev. 127, 1918 (1962).
[Crossref]

Chirkin, A. S.

S. A. Akhmanov, A. I. Kovrigin, A. P. Sukhorukov, R. V. Khokhlov, and A. S. Chirkin, JETP Lett. 7, 182 (1968);S. A. Akhanov, A. S. Chirkin, A. I. Kovrigin, R. V. Khokhlov, and A. P. Sukhorukov, IEEE J. Quantum Electron. QE-4, 598 (1968).
[Crossref]

Comly, J.

J. Comly and E. Garmire, Appl. Phys. Lett. 12, 7 (1968).
[Crossref]

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys. Rev. 127, 1918 (1962).
[Crossref]

J. Ducuing and C. Flytzanis, in Optical Properties of Solids, F. Abeles, ed. (North-Holland, Amsterdam, 1972), p. 863.

Fisher, R. A.

R. A. Fisher and W. K. Bischel, J. Appl. Phys. 46, 4921 (1975).
[Crossref]

Flytzanis, C.

J. Ducuing and C. Flytzanis, in Optical Properties of Solids, F. Abeles, ed. (North-Holland, Amsterdam, 1972), p. 863.

A. Bamberger, C. Flytzanis, and D. Jennevé, rapport interne no. 149, Centre de Mathématiques Appliquées [Ecole Polytechni-que, Paris, June 1986 (in French)].

Fork, R. L.

R. L. Fork, B. I. Greene, and C. V. Shank, Appl. Phys. Lett. 38, 671 (1981);C. V. Shank, R. L. Fork, R. Yen, R. H. Stolen, and W. J. Tomlinson, Appl. Phys. Lett. 40, 761 (1982);W. H. Knox, R. L. Fork, M. C. Downer, R. H. Stolen, C. W. Shank, and J. A. Valdmanis, Appl. Phys. Lett. 46, 1120 (1985).
[Crossref]

Franken, P. A.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, Phys. Rev. Lett. 7, 118 (1961).
[Crossref]

Garmire, E.

J. Comly and E. Garmire, Appl. Phys. Lett. 12, 7 (1968).
[Crossref]

E. Garmire and A. Yariv, IEEE J. Quantum Electron. QE-3, 222 (1967).
[Crossref]

Geusic, J. E.

J. E. Geusic, H. J. Levinstein, S. Singh, R. G. Smith, and L. G. Van Uitert, Appl. Phys. Lett. 12, 306 (1968).
[Crossref]

Giordmaine, J. A.

J. A. Giordmaine, Phys. Rev. Lett. 8, 19 (1962);P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, Phys. Rev. Lett. 8, 21 (1962).See also P. A. Franken and J. F. Ward, Rev. Mod. Phys. 35, 23 (1963).
[Crossref]

Greene, B. I.

R. L. Fork, B. I. Greene, and C. V. Shank, Appl. Phys. Lett. 38, 671 (1981);C. V. Shank, R. L. Fork, R. Yen, R. H. Stolen, and W. J. Tomlinson, Appl. Phys. Lett. 40, 761 (1982);W. H. Knox, R. L. Fork, M. C. Downer, R. H. Stolen, C. W. Shank, and J. A. Valdmanis, Appl. Phys. Lett. 46, 1120 (1985).
[Crossref]

Hill, A. E.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, Phys. Rev. Lett. 7, 118 (1961).
[Crossref]

Jennevé, D.

A. Bamberger, C. Flytzanis, and D. Jennevé, rapport interne no. 149, Centre de Mathématiques Appliquées [Ecole Polytechni-que, Paris, June 1986 (in French)].

Khokhlov, R. V.

S. A. Akhmanov, A. I. Kovrigin, A. P. Sukhorukov, R. V. Khokhlov, and A. S. Chirkin, JETP Lett. 7, 182 (1968);S. A. Akhanov, A. S. Chirkin, A. I. Kovrigin, R. V. Khokhlov, and A. P. Sukhorukov, IEEE J. Quantum Electron. QE-4, 598 (1968).
[Crossref]

Kovrigin, A. I.

S. A. Akhmanov, A. I. Kovrigin, A. P. Sukhorukov, R. V. Khokhlov, and A. S. Chirkin, JETP Lett. 7, 182 (1968);S. A. Akhanov, A. S. Chirkin, A. I. Kovrigin, R. V. Khokhlov, and A. P. Sukhorukov, IEEE J. Quantum Electron. QE-4, 598 (1968).
[Crossref]

Kovrygin, A. I.

S. A. Akhmanov, A. I. Kovrygin, and A. P. Sukhorukov, Nonlinear Optics, Vol. 1 of Quantum Electronics, H. Rabin and C. L. Tang, eds. (Academic, New York, 1975), Part B, pp. 475–586.

Levinstein, H. J.

J. E. Geusic, H. J. Levinstein, S. Singh, R. G. Smith, and L. G. Van Uitert, Appl. Phys. Lett. 12, 306 (1968).
[Crossref]

Miller, R. C.

R. C. Miller, Phys. Lett. 26A, 177 (1968).

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys. Rev. 127, 1918 (1962).
[Crossref]

Peters, C. W.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, Phys. Rev. Lett. 7, 118 (1961).
[Crossref]

Shank, C. V.

R. L. Fork, B. I. Greene, and C. V. Shank, Appl. Phys. Lett. 38, 671 (1981);C. V. Shank, R. L. Fork, R. Yen, R. H. Stolen, and W. J. Tomlinson, Appl. Phys. Lett. 40, 761 (1982);W. H. Knox, R. L. Fork, M. C. Downer, R. H. Stolen, C. W. Shank, and J. A. Valdmanis, Appl. Phys. Lett. 46, 1120 (1985).
[Crossref]

Singh, S.

J. E. Geusic, H. J. Levinstein, S. Singh, R. G. Smith, and L. G. Van Uitert, Appl. Phys. Lett. 12, 306 (1968).
[Crossref]

Smith, R. G.

J. E. Geusic, H. J. Levinstein, S. Singh, R. G. Smith, and L. G. Van Uitert, Appl. Phys. Lett. 12, 306 (1968).
[Crossref]

Sukhorukov, A. P.

S. A. Akhmanov, A. I. Kovrigin, A. P. Sukhorukov, R. V. Khokhlov, and A. S. Chirkin, JETP Lett. 7, 182 (1968);S. A. Akhanov, A. S. Chirkin, A. I. Kovrigin, R. V. Khokhlov, and A. P. Sukhorukov, IEEE J. Quantum Electron. QE-4, 598 (1968).
[Crossref]

S. A. Akhmanov, A. I. Kovrygin, and A. P. Sukhorukov, Nonlinear Optics, Vol. 1 of Quantum Electronics, H. Rabin and C. L. Tang, eds. (Academic, New York, 1975), Part B, pp. 475–586.

Van Uitert, L. G.

J. E. Geusic, H. J. Levinstein, S. Singh, R. G. Smith, and L. G. Van Uitert, Appl. Phys. Lett. 12, 306 (1968).
[Crossref]

Weinreich, G.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, Phys. Rev. Lett. 7, 118 (1961).
[Crossref]

Yariv, A.

E. Garmire and A. Yariv, IEEE J. Quantum Electron. QE-3, 222 (1967).
[Crossref]

Appl. Phys. Lett. (3)

J. E. Geusic, H. J. Levinstein, S. Singh, R. G. Smith, and L. G. Van Uitert, Appl. Phys. Lett. 12, 306 (1968).
[Crossref]

R. L. Fork, B. I. Greene, and C. V. Shank, Appl. Phys. Lett. 38, 671 (1981);C. V. Shank, R. L. Fork, R. Yen, R. H. Stolen, and W. J. Tomlinson, Appl. Phys. Lett. 40, 761 (1982);W. H. Knox, R. L. Fork, M. C. Downer, R. H. Stolen, C. W. Shank, and J. A. Valdmanis, Appl. Phys. Lett. 46, 1120 (1985).
[Crossref]

J. Comly and E. Garmire, Appl. Phys. Lett. 12, 7 (1968).
[Crossref]

IEEE J. Quantum Electron. (1)

E. Garmire and A. Yariv, IEEE J. Quantum Electron. QE-3, 222 (1967).
[Crossref]

J. Appl. Phys. (1)

R. A. Fisher and W. K. Bischel, J. Appl. Phys. 46, 4921 (1975).
[Crossref]

JETP Lett. (1)

S. A. Akhmanov, A. I. Kovrigin, A. P. Sukhorukov, R. V. Khokhlov, and A. S. Chirkin, JETP Lett. 7, 182 (1968);S. A. Akhanov, A. S. Chirkin, A. I. Kovrigin, R. V. Khokhlov, and A. P. Sukhorukov, IEEE J. Quantum Electron. QE-4, 598 (1968).
[Crossref]

Phys. Lett. (1)

R. C. Miller, Phys. Lett. 26A, 177 (1968).

Phys. Rev. (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys. Rev. 127, 1918 (1962).
[Crossref]

Phys. Rev. Lett. (2)

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, Phys. Rev. Lett. 7, 118 (1961).
[Crossref]

J. A. Giordmaine, Phys. Rev. Lett. 8, 19 (1962);P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, Phys. Rev. Lett. 8, 21 (1962).See also P. A. Franken and J. F. Ward, Rev. Mod. Phys. 35, 23 (1963).
[Crossref]

Other (3)

J. Ducuing and C. Flytzanis, in Optical Properties of Solids, F. Abeles, ed. (North-Holland, Amsterdam, 1972), p. 863.

A. Bamberger, C. Flytzanis, and D. Jennevé, rapport interne no. 149, Centre de Mathématiques Appliquées [Ecole Polytechni-que, Paris, June 1986 (in French)].

S. A. Akhmanov, A. I. Kovrygin, and A. P. Sukhorukov, Nonlinear Optics, Vol. 1 of Quantum Electronics, H. Rabin and C. L. Tang, eds. (Academic, New York, 1975), Part B, pp. 475–586.

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Figures (10)

Fig. 1
Fig. 1

Plots of efficiency η(ξ) versus ξ for (i) the stationary case, (ii) the nonstationary case with only wave-vector dispersion, and (iii) the nonstationary case with both wave-vector dispersion and the effective GVD taken into account. In case (iii) α = 0.1 and β = 0; the efficiency is further reduced compared with that for the case (ii).

Fig. 2
Fig. 2

Intensity profiles of the fundamental (labeled A) and SH (labeled B) waves for the nonstationary cases (ii) and (iii) of Fig. 1. When the effective GVD is taken into account in case (iii), both the pulses have split into two.

Fig. 3
Fig. 3

Plots of efficiency η(ξ) versus ξ for three different cases: (i) α = 0, β = 0.1, (ii) α = 0.1, β = −0.1, (iii) α = β = 0.1. For nonzero values of either α or β, efficiency plots in all the cases indicate oscillatory behavior.

Fig. 4
Fig. 4

Plots of efficiency η(ξ) versus ξ for four different cases: (i) α = 0.5, β = 0; (ii) α = 0, β = 0.5; (iii) α = β = 0.5; (iv) α = 0.5, β = −0.5. The curves labeled A are the plots of 1 − η. The oscillatory behavior of the efficiency in all the cases is highly pronounced for the somewhat larger values of α and β chosen for these plots.

Fig. 5
Fig. 5

Intensity profiles of the fundamental (labeled A) and the SH (labeled B) waves for the three cases of Fig. 3. In all the cases, the pulses have split into two.

Fig. 6
Fig. 6

Intensity profiles of the fundamental (labeled A) and the SH (labeled B) waves for the last two cases [(iii) and (iv)] of Fig. 4. In these two cases, splitting of the pulses into two or more pulses is seen for somewhat larger values of the effective GVD parameters.

Fig. 7
Fig. 7

Plots of efficiency η(ξ) versus ξ for α = β = 0 and (i) θ = π/4, (ii) 0 = π/2, and (iii) θ = π. The curves labeled A are the plots of 1 − η. The oscillatory behavior in all the three cases is somewhat similar to the cases for which θ = 0, α = 0, and β ≠ 0 [see Fig. 4(ii)].

Fig. 8
Fig. 8

Intensity profiles of the fundamental (labeled A) and the SH (labeled B) waves for cases (i) and (ii) of Fig. 7. Both the pulses show splitting as in the previous cases shown in Fig. (6).

Fig. 9
Fig. 9

Plots of efficiency η(ξ) versus ξ for α = 0.1, β = 0, and (i) θ = π/4, (ii) 0 = π/2, and (iii) θ = π. The curves labeled A are the plots of 1 − η. The efficiency behavior in all the cases is qualitatively similar to those in Figs. 4(i)−4(iv). However, the initial oscillatory behavior tends to smooth out asymptotically for large ξ.

Fig. 10
Fig. 10

Same as in Fig. 9 except that α = 0, β = 0.1. The plots show an identical behavior to those in Fig. 7.

Equations (30)

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

E = A 1 ( r , z , t ) exp [ i ( k 1 z ω 1 t ) ] + A 2 ( r , z , t ) × exp [ i ( k 2 z ω 2 t ) ] ,
2 E = 1 0 c 2 2 D L t 2 = 1 0 c 2 2 P NL t 2 .
P NL ( ω 1 ) = 0 [ 2 χ ( 2 ) φ 2 φ 1 * exp [ i ( k 2 k 1 ) z + χ ( 3 ) ( | φ 1 | 2 + 2 | φ 2 | 2 ) φ 1 exp ( i k 1 z ) ] exp ( i ω 1 t ) ,
P NL ( ω 2 ) = 0 [ χ ( 2 ) φ 1 2 exp ( 2 i k 1 z ) + χ ( 3 ) ( | φ 2 | 2 + 2 | φ 1 | 2 ) φ 2 exp ( i k 2 z ) ] exp ( i ω 2 t ) .
A 1 , 2 ( r , z , t ) = R ( r ) φ 1 , 2 ( z , t ) .
i ( L 1 ξ + k 1 k T τ ) q 1 ( k 1 ) 2 ( k ) 2 + k 1 k 1 2 k 1 T 2 2 q 1 τ 2 + A ω 1 2 2 k 1 c 2 2 χ ( 2 ) q 2 q 1 * + A 2 ω 1 2 2 k 1 c 2 χ ( 3 ) ( | q 1 | 2 + 2 | q 2 | 2 ) q 1 = 0 ,
i ( L 1 ξ + k 2 k T τ ) q 2 ( k 2 ) 2 ( k ) 2 + k 2 k 2 2 k 2 T 2 2 q 2 τ 2 + A ω 2 2 2 k 2 c 2 χ ( 2 ) q 1 2 + A 2 ω 2 2 2 k 2 c 2 χ ( 3 ) ( | q 2 | 2 + 2 | q 1 | 2 ) q 2 = 0 .
k 1 , 2 = k 1 , 2 ω | ω 1 , 2
k 1 , 2 = 2 k 1 , 2 ω 2 | ω 1 , 2
α = ( k 1 ) 2 ( k ) 2 + k 1 k 1 2 k 1 T 2 L = ( k 1 ) 2 ( k ) 2 + k 1 k 1 k 1 T ( k 2 k 1 ) ,
β = ( k 2 ) 2 ( k ) 2 + k 2 k 2 2 k 1 T 2 L = ( k 2 ) 2 ( k ) 2 + k 2 k 2 k 2 T ( k 2 k 1 ) ,
γ = A ω 1 2 2 k 1 c 2 2 χ ( 2 ) L = A ω 2 2 2 k 2 c 2 χ ( 2 ) L ,
ν = A 2 ω 1 2 2 k 1 c 2 χ ( 3 ) L = 1 2 A 2 ω 2 2 2 k 2 c 2 χ ( 3 ) L .
q 1 ξ q 1 τ + i α 2 q 1 τ 2 i γ q 2 q 1 * i ν ( | q 1 | 2 + 2 | q 2 | 2 ) q 1 = 0 ,
q 2 ξ + q 2 τ + i β 2 q 2 τ 2 i γ q 1 2 i 2 ν ( | q 2 | 2 + 2 | q 1 | 2 ) q 2 = 0 .
d q 1 d ξ = i γ q 1 * q 2 , d q 2 d ξ = i γ q 1 2
q 1 ξ = q 1 τ i α 2 q 1 τ 2 ,
q 2 ξ = q 2 τ i β 2 q 2 τ 2
q 1 ( ξ = 0 , τ ) = 1 c h τ exp ( i θ c h 2 τ ) ,
q 2 ( ξ = 0 , τ ) = 0 .
η ( ξ ) = | q 2 ( ξ , τ ) | 2 d τ | q 1 ( ξ , τ ) | 2 d τ + | q 2 ( ξ , τ ) | 2 d τ .
k j ω | ω j ( j = 1 , 2 )
q 1 ξ + i α 2 q 1 τ 2 i γ q 2 q 1 * = 0 ,
q 2 ξ + i β 2 q 2 τ 2 i γ q 1 2 = 0 ,
α = k 1 2 T 2 L ,
β = k 2 2 T 2 L .
q 1 ( τ , ξ ) = 2 Ω γ 1 c h [ τ ( Ω / α ) 1 / 2 ] exp ( i Ω ξ ) ,
q 2 ( τ , ξ ) = 2 Ω γ 1 c h [ τ ( Ω / α ) 1 / 2 ] exp ( 2 i Ω ξ ) ,
η = x 1 + x ,
x = + c h 4 t d t + c h 2 t d t = 2 3 .

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