Abstract

The effects of the phase mismatch due to cubic nonlinearity in the equations for second-harmonic generation are investigated. We show that the phase mismatch induced by the nonlinear refractive index of a doubling crystal can dramatically reduce the conversion efficiency of high-peak-power laser pulses. Simple, analytic expressions are derived for the conversion efficiency of cw radiation and for the estimation of the dispersion in the nonlinear refractive index of the doubling crystal, which is quite important in the determination of the magnitude of the nonlinear effects on maximum conversion. The consequences that these nonlinearities have on the frequency doubling of ultrashort (≤100 fs) pulses, including the additional effects of group-velocity walk-off between the pulses, are then numerically calculated.

© 1996 Optical Society of America

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References

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  1. P. A. Franken, A. E. Hill, C. W. Peters, and G. W. Weinreigh, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118 (1961).
    [CrossRef]
  2. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interaction between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918 (1962).
    [CrossRef]
  3. S. A. Akhmanov and R. V. Khokhlov, Problems of Nonlinear Optics (Gordon and Breach, New York, 1972).
  4. D. J. Harter and D. C. Brown, “Effects of higher order nonlinearities on second-order frequency mixing,” IEEE J. Quantum Electron. QE-18, 1146 (1982).
    [CrossRef]
  5. T. B. Razumikhina, L. S. Telegin, A. I. Kholodnykh, and A. S. Chirkin, “Three-frequency interactions of high-intensity light waves in media with quadratic and cubic nonlinearities,” Sov. J. Quantum Electron. 14, 1358 (1984).
    [CrossRef]
  6. L. S. Telegin and A. S. Chirkin, “Interaction in frequency doubling of ultrashort laser pulses,” Sov. J. Quantum Electron. 12, 1358 (1982).
    [CrossRef]
  7. W. Choe, P. P. Banerjee, and F. C. Caimi, “Second-harmonic generation in an optical medium with second- and third-order nonlinear susceptibilities,” J. Opt. Soc. Am. B 8, 1013 (1991).
    [CrossRef]
  8. C. J. McKinstrie and X. D. Cao, “Nonlinear detuning of three-wave interactions,” J. Opt. Soc. Am. B 10, 898 (1993).
    [CrossRef]
  9. V. E. Zakharov, S. L. Musher, and A. M. Rubenchik, Phys. Rep. 129, 286 (1985).
    [CrossRef]
  10. S. A. Akhmanov, A. S. Chirkin, K. N. Drabovich, A. I. Kovrigin, R. V. Khokhlov, and A. P. Sukhorukov, “Nonstationary nonlinear optical effects and ultrashort light pulse formation,” IEEE J. Quantum Electron. QE-4, 598 (1968).
    [CrossRef]
  11. J. E. Midwinter and J. Warner, “The effects of phase matching method and of uniaxial symmetry on the polar distribution of second order non-linear optical polarization,” Br. J. Appl. Phys. 16, 1135 (1965).
    [CrossRef]
  12. F. Zernike, “Refractive indices of ammonium dihydrogen phosphate and potassium dihydrogen phosphate between 2000 Å and 1.5 μ m,” J. Opt. Soc. Am. 54, 1215 (1964).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  15. H. B. Bebb and A. Gold, “Multiphoton ionization of hydrogen and rare-gas atoms,” Phys. Rev. 143, 1 (1966).
    [CrossRef]
  16. N. L. Boling, A. J. Glass, and A. Owyoung, “Empirical relationships for predicting nonlinear refractive index changes in optical solids,” IEEE J. Quantum Electron. QE-14, 601 (1978).
    [CrossRef]
  17. R. Adair, L. L. Chase, and S. A. Payne, “Nonlinear refractive index of optical crystals,” Phys. Rev. B 39, 3337 (1989).
    [CrossRef]
  18. L. D. Noordam, H. J. Bakker, M. P. de Boer, and H. B. van Linden van den Heuvell, “Second-harmonic generation of femtosecond pulses: observation of phase-mismatch effects,” Opt. Lett. 15, 1464 (1990).
    [CrossRef] [PubMed]

1995 (1)

1993 (1)

1992 (1)

R. Adair, L. L. Chase, and S. A. Payne, “Dispersion of the nonlinear refractive index of optical crystals,” Opt. Mater. 1, 185 (1992).
[CrossRef]

1991 (1)

1990 (1)

1989 (1)

R. Adair, L. L. Chase, and S. A. Payne, “Nonlinear refractive index of optical crystals,” Phys. Rev. B 39, 3337 (1989).
[CrossRef]

1985 (1)

V. E. Zakharov, S. L. Musher, and A. M. Rubenchik, Phys. Rep. 129, 286 (1985).
[CrossRef]

1984 (1)

T. B. Razumikhina, L. S. Telegin, A. I. Kholodnykh, and A. S. Chirkin, “Three-frequency interactions of high-intensity light waves in media with quadratic and cubic nonlinearities,” Sov. J. Quantum Electron. 14, 1358 (1984).
[CrossRef]

1982 (2)

L. S. Telegin and A. S. Chirkin, “Interaction in frequency doubling of ultrashort laser pulses,” Sov. J. Quantum Electron. 12, 1358 (1982).
[CrossRef]

D. J. Harter and D. C. Brown, “Effects of higher order nonlinearities on second-order frequency mixing,” IEEE J. Quantum Electron. QE-18, 1146 (1982).
[CrossRef]

1978 (1)

N. L. Boling, A. J. Glass, and A. Owyoung, “Empirical relationships for predicting nonlinear refractive index changes in optical solids,” IEEE J. Quantum Electron. QE-14, 601 (1978).
[CrossRef]

1968 (1)

S. A. Akhmanov, A. S. Chirkin, K. N. Drabovich, A. I. Kovrigin, R. V. Khokhlov, and A. P. Sukhorukov, “Nonstationary nonlinear optical effects and ultrashort light pulse formation,” IEEE J. Quantum Electron. QE-4, 598 (1968).
[CrossRef]

1966 (1)

H. B. Bebb and A. Gold, “Multiphoton ionization of hydrogen and rare-gas atoms,” Phys. Rev. 143, 1 (1966).
[CrossRef]

1965 (1)

J. E. Midwinter and J. Warner, “The effects of phase matching method and of uniaxial symmetry on the polar distribution of second order non-linear optical polarization,” Br. J. Appl. Phys. 16, 1135 (1965).
[CrossRef]

1964 (1)

1962 (1)

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

1961 (1)

P. A. Franken, A. E. Hill, C. W. Peters, and G. W. Weinreigh, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118 (1961).
[CrossRef]

Adair, R.

R. Adair, L. L. Chase, and S. A. Payne, “Dispersion of the nonlinear refractive index of optical crystals,” Opt. Mater. 1, 185 (1992).
[CrossRef]

R. Adair, L. L. Chase, and S. A. Payne, “Nonlinear refractive index of optical crystals,” Phys. Rev. B 39, 3337 (1989).
[CrossRef]

Akhmanov, S. A.

S. A. Akhmanov, A. S. Chirkin, K. N. Drabovich, A. I. Kovrigin, R. V. Khokhlov, and A. P. Sukhorukov, “Nonstationary nonlinear optical effects and ultrashort light pulse formation,” IEEE J. Quantum Electron. QE-4, 598 (1968).
[CrossRef]

S. A. Akhmanov and R. V. Khokhlov, Problems of Nonlinear Optics (Gordon and Breach, New York, 1972).

Armstrong, J. A.

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

Bakker, H. J.

Banerjee, P. P.

Bebb, H. B.

H. B. Bebb and A. Gold, “Multiphoton ionization of hydrogen and rare-gas atoms,” Phys. Rev. 143, 1 (1966).
[CrossRef]

Bloembergen, N.

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

Boling, N. L.

N. L. Boling, A. J. Glass, and A. Owyoung, “Empirical relationships for predicting nonlinear refractive index changes in optical solids,” IEEE J. Quantum Electron. QE-14, 601 (1978).
[CrossRef]

Brown, D. C.

D. J. Harter and D. C. Brown, “Effects of higher order nonlinearities on second-order frequency mixing,” IEEE J. Quantum Electron. QE-18, 1146 (1982).
[CrossRef]

Caimi, F. C.

Cao, X. D.

Chase, L. L.

R. Adair, L. L. Chase, and S. A. Payne, “Dispersion of the nonlinear refractive index of optical crystals,” Opt. Mater. 1, 185 (1992).
[CrossRef]

R. Adair, L. L. Chase, and S. A. Payne, “Nonlinear refractive index of optical crystals,” Phys. Rev. B 39, 3337 (1989).
[CrossRef]

Chien, C. Y.

Chirkin, A. S.

T. B. Razumikhina, L. S. Telegin, A. I. Kholodnykh, and A. S. Chirkin, “Three-frequency interactions of high-intensity light waves in media with quadratic and cubic nonlinearities,” Sov. J. Quantum Electron. 14, 1358 (1984).
[CrossRef]

L. S. Telegin and A. S. Chirkin, “Interaction in frequency doubling of ultrashort laser pulses,” Sov. J. Quantum Electron. 12, 1358 (1982).
[CrossRef]

S. A. Akhmanov, A. S. Chirkin, K. N. Drabovich, A. I. Kovrigin, R. V. Khokhlov, and A. P. Sukhorukov, “Nonstationary nonlinear optical effects and ultrashort light pulse formation,” IEEE J. Quantum Electron. QE-4, 598 (1968).
[CrossRef]

Choe, W.

Coe, J. S.

Craxton, R. S.

de Boer, M. P.

Drabovich, K. N.

S. A. Akhmanov, A. S. Chirkin, K. N. Drabovich, A. I. Kovrigin, R. V. Khokhlov, and A. P. Sukhorukov, “Nonstationary nonlinear optical effects and ultrashort light pulse formation,” IEEE J. Quantum Electron. QE-4, 598 (1968).
[CrossRef]

Ducuing, J.

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

Franken, P. A.

P. A. Franken, A. E. Hill, C. W. Peters, and G. W. Weinreigh, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118 (1961).
[CrossRef]

Glass, A. J.

N. L. Boling, A. J. Glass, and A. Owyoung, “Empirical relationships for predicting nonlinear refractive index changes in optical solids,” IEEE J. Quantum Electron. QE-14, 601 (1978).
[CrossRef]

Gold, A.

H. B. Bebb and A. Gold, “Multiphoton ionization of hydrogen and rare-gas atoms,” Phys. Rev. 143, 1 (1966).
[CrossRef]

Harter, D. J.

D. J. Harter and D. C. Brown, “Effects of higher order nonlinearities on second-order frequency mixing,” IEEE J. Quantum Electron. QE-18, 1146 (1982).
[CrossRef]

Hill, A. E.

P. A. Franken, A. E. Hill, C. W. Peters, and G. W. Weinreigh, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118 (1961).
[CrossRef]

Khokhlov, R. V.

S. A. Akhmanov, A. S. Chirkin, K. N. Drabovich, A. I. Kovrigin, R. V. Khokhlov, and A. P. Sukhorukov, “Nonstationary nonlinear optical effects and ultrashort light pulse formation,” IEEE J. Quantum Electron. QE-4, 598 (1968).
[CrossRef]

S. A. Akhmanov and R. V. Khokhlov, Problems of Nonlinear Optics (Gordon and Breach, New York, 1972).

Kholodnykh, A. I.

T. B. Razumikhina, L. S. Telegin, A. I. Kholodnykh, and A. S. Chirkin, “Three-frequency interactions of high-intensity light waves in media with quadratic and cubic nonlinearities,” Sov. J. Quantum Electron. 14, 1358 (1984).
[CrossRef]

Korn, G.

Kovrigin, A. I.

S. A. Akhmanov, A. S. Chirkin, K. N. Drabovich, A. I. Kovrigin, R. V. Khokhlov, and A. P. Sukhorukov, “Nonstationary nonlinear optical effects and ultrashort light pulse formation,” IEEE J. Quantum Electron. QE-4, 598 (1968).
[CrossRef]

McKinstrie, C. J.

Midwinter, J. E.

J. E. Midwinter and J. Warner, “The effects of phase matching method and of uniaxial symmetry on the polar distribution of second order non-linear optical polarization,” Br. J. Appl. Phys. 16, 1135 (1965).
[CrossRef]

Mourou, G.

Musher, S. L.

V. E. Zakharov, S. L. Musher, and A. M. Rubenchik, Phys. Rep. 129, 286 (1985).
[CrossRef]

Noordam, L. D.

Owyoung, A.

N. L. Boling, A. J. Glass, and A. Owyoung, “Empirical relationships for predicting nonlinear refractive index changes in optical solids,” IEEE J. Quantum Electron. QE-14, 601 (1978).
[CrossRef]

Payne, S. A.

R. Adair, L. L. Chase, and S. A. Payne, “Dispersion of the nonlinear refractive index of optical crystals,” Opt. Mater. 1, 185 (1992).
[CrossRef]

R. Adair, L. L. Chase, and S. A. Payne, “Nonlinear refractive index of optical crystals,” Phys. Rev. B 39, 3337 (1989).
[CrossRef]

Pershan, P. S.

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

Peters, C. W.

P. A. Franken, A. E. Hill, C. W. Peters, and G. W. Weinreigh, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118 (1961).
[CrossRef]

Razumikhina, T. B.

T. B. Razumikhina, L. S. Telegin, A. I. Kholodnykh, and A. S. Chirkin, “Three-frequency interactions of high-intensity light waves in media with quadratic and cubic nonlinearities,” Sov. J. Quantum Electron. 14, 1358 (1984).
[CrossRef]

Rubenchik, A. M.

V. E. Zakharov, S. L. Musher, and A. M. Rubenchik, Phys. Rep. 129, 286 (1985).
[CrossRef]

Squier, J.

Sukhorukov, A. P.

S. A. Akhmanov, A. S. Chirkin, K. N. Drabovich, A. I. Kovrigin, R. V. Khokhlov, and A. P. Sukhorukov, “Nonstationary nonlinear optical effects and ultrashort light pulse formation,” IEEE J. Quantum Electron. QE-4, 598 (1968).
[CrossRef]

Telegin, L. S.

T. B. Razumikhina, L. S. Telegin, A. I. Kholodnykh, and A. S. Chirkin, “Three-frequency interactions of high-intensity light waves in media with quadratic and cubic nonlinearities,” Sov. J. Quantum Electron. 14, 1358 (1984).
[CrossRef]

L. S. Telegin and A. S. Chirkin, “Interaction in frequency doubling of ultrashort laser pulses,” Sov. J. Quantum Electron. 12, 1358 (1982).
[CrossRef]

van Linden van den Heuvell, H. B.

Warner, J.

J. E. Midwinter and J. Warner, “The effects of phase matching method and of uniaxial symmetry on the polar distribution of second order non-linear optical polarization,” Br. J. Appl. Phys. 16, 1135 (1965).
[CrossRef]

Weinreigh, G. W.

P. A. Franken, A. E. Hill, C. W. Peters, and G. W. Weinreigh, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118 (1961).
[CrossRef]

Zakharov, V. E.

V. E. Zakharov, S. L. Musher, and A. M. Rubenchik, Phys. Rep. 129, 286 (1985).
[CrossRef]

Zernike, F.

Br. J. Appl. Phys. (1)

J. E. Midwinter and J. Warner, “The effects of phase matching method and of uniaxial symmetry on the polar distribution of second order non-linear optical polarization,” Br. J. Appl. Phys. 16, 1135 (1965).
[CrossRef]

IEEE J. Quantum Electron. (3)

N. L. Boling, A. J. Glass, and A. Owyoung, “Empirical relationships for predicting nonlinear refractive index changes in optical solids,” IEEE J. Quantum Electron. QE-14, 601 (1978).
[CrossRef]

D. J. Harter and D. C. Brown, “Effects of higher order nonlinearities on second-order frequency mixing,” IEEE J. Quantum Electron. QE-18, 1146 (1982).
[CrossRef]

S. A. Akhmanov, A. S. Chirkin, K. N. Drabovich, A. I. Kovrigin, R. V. Khokhlov, and A. P. Sukhorukov, “Nonstationary nonlinear optical effects and ultrashort light pulse formation,” IEEE J. Quantum Electron. QE-4, 598 (1968).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Lett. (2)

Opt. Mater. (1)

R. Adair, L. L. Chase, and S. A. Payne, “Dispersion of the nonlinear refractive index of optical crystals,” Opt. Mater. 1, 185 (1992).
[CrossRef]

Phys. Rep. (1)

V. E. Zakharov, S. L. Musher, and A. M. Rubenchik, Phys. Rep. 129, 286 (1985).
[CrossRef]

Phys. Rev. (2)

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

H. B. Bebb and A. Gold, “Multiphoton ionization of hydrogen and rare-gas atoms,” Phys. Rev. 143, 1 (1966).
[CrossRef]

Phys. Rev. B (1)

R. Adair, L. L. Chase, and S. A. Payne, “Nonlinear refractive index of optical crystals,” Phys. Rev. B 39, 3337 (1989).
[CrossRef]

Phys. Rev. Lett. (1)

P. A. Franken, A. E. Hill, C. W. Peters, and G. W. Weinreigh, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118 (1961).
[CrossRef]

Sov. J. Quantum Electron. (2)

T. B. Razumikhina, L. S. Telegin, A. I. Kholodnykh, and A. S. Chirkin, “Three-frequency interactions of high-intensity light waves in media with quadratic and cubic nonlinearities,” Sov. J. Quantum Electron. 14, 1358 (1984).
[CrossRef]

L. S. Telegin and A. S. Chirkin, “Interaction in frequency doubling of ultrashort laser pulses,” Sov. J. Quantum Electron. 12, 1358 (1982).
[CrossRef]

Other (1)

S. A. Akhmanov and R. V. Khokhlov, Problems of Nonlinear Optics (Gordon and Breach, New York, 1972).

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Figures (5)

Fig. 1
Fig. 1

Numerical solution of Eq. (12) for the maximum conversion efficiency for 800-nm light in KDP (solid curve). The approximate perturbation-theory formula, Eq. (13), is also shown for comparison (dashed curve).

Fig. 2
Fig. 2

Energy conversion efficiency calculated for a square pulse as a function of the distance propagated in a KDP crystal in the absence of group-velocity walk-off (η = 0). The incident intensity was 500 GW/cm2, and the pulse wavelength was 800 nm. Curve 1 shows the case for harmonic conversion with no cubic nonlinearities (βij = 0), and curve 2 represents the efficiency when nonlinearities are included.

Fig. 3
Fig. 3

Results for a Gaussian pulse shape: (a) The calculated conversion efficiency as a function of distance in a KDP crystal for a Gaussian pulse envelope with no group-velocity walk-off. The peak intensity of the incident pulse is 500 GW/cm2. Curve 1 (solid) represents the case of no cubic nonlinearity, curve 2 (dashed) represents the case with the nonlinear phase mismatch included, and curve 3 (dotted) represents the case with the nonlinear phase mismatch and the inclusion of additional phase mismatch to compensate for the cubic phase slip at the peak intensity of the pulse. (b) Pulse envelopes of the second-harmonic pulse for the three cases calculated in (a), shown at the peak of the pulse-conversion curves. These positions are, for curve 1, z = ∞; for curve 2, z = 1.3 mm; and for curve 3, z = 1.7 mm.

Fig. 4
Fig. 4

(a) Calculated conversion efficiency for a 100-fs Gaussian pulse with a peak intensity of 400 GW/cm2 plotted as a function of the distance traversed in KDP. Curve 1 represents the case with no walk-off and no cubic nonlinearity. Curve 2 represents the case with a group-velocity walk-off (η = 77 fs/mm) but no nonlinear phase. Curve 3 represents the case with no walk-off but with the nonlinear phase included. Curve 4 illustrates the effects of both group-velocity walk-off and nonlinear phase. (b) Curve 1 represents the pulse envelope of the incident fundamental pulse for the calculation from (a). Curve 2 represents the second-harmonic pulse envelope after a distance travelled of 2 mm into the crystal when cubic nonlinearities are included but no group-velocity walk-off is present, and curve 3 represents the pulse shape when both effects are included.

Fig. 5
Fig. 5

Calculated conversion efficiency for an 800-nm, 100-fs pulse in KDP plotted as a function of the initial peak intensity for crystal lengths of 1 mm (curve 1), 2 mm (curve 2), and 3 mm (curve 3).

Equations (23)

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2 E i ( x , t ) + ε i c 2 2 t 2 E i ( x , t ) = 4 π c 2 2 t 2 P i NL ( x , t ) ,
P 1 NL ( x , t ) = 4 d eff E 2 E 1 * + 3 χ ( 3 ) ( ω 1 , ω 1 , ω 1 ) | E 1 | 2 E 1 + 6 χ ( 3 ) ( ω 2 , ω 2 , ω 1 ) | E 2 | 2 E 1 + c.c. ,
P 2 NL ( x , t ) = 2 d eff E 1 2 + 3 χ ( 3 ) ( ω 2 , ω 2 , ω 2 ) | E 2 | 2 E 2 + 6 χ ( 3 ) ( ω 1 , ω 1 , ω 2 ) | E 1 | 2 E 2 + c.c. .
a 1 z = i α a 2 a 1 * exp ( i Δ k z ) + i ( β 11 | a 1 | 2 a 1 + β 21 | a 2 | 2 a 1 ) ,
a 2 z + η a 2 t = i α a 1 2 exp ( i Δ k z ) + i ( β 22 | a 2 | 2 a 2 + β 12 | a 1 | 2 a 2 ) .
β i j 2 | i j | 6 π ω j 2 χ ( 3 ) ( ω i , ω i , ω j ) c 2 k j .
b 1 s = 2 i b 2 b 1 * + i Λ b 1 + i ( T 11 | b 1 | 2 b 1 + T 21 | b 2 | 2 b 1 ) ,
b 2 s = i b 1 2 + i ( T 22 | b 2 | 2 b 2 + T 12 | b 1 | 2 b 2 ) ,
b 1 s = i δ H δ b 1 * , b 2 s = i δ H δ b 2 * .
H = b 2 b 1 * 2 + b 1 2 b 2 * + Λ | b 1 | 2 + 1 2 T 11 | b 1 | 4 + T 12 | b 1 | 2 | b 2 | 2 + 1 2 T 22 | b 2 | 4 .
| b 1 | 2 + 2 | b 2 | 2 = const .
b 1 ( x = 0 ) = b 0 ,
b 2 ( x = 0 ) = 0.
H = 1 2 T 11 | b 0 | 4 + Λ | b 0 | 2 ,
| b 1 | = | b 0 | 2 2 | b 2 | 2 .
b 1 = | b 1 | exp [ i ϕ 1 ] , b 2 = | b 2 | exp [ i ϕ 2 ] ,
| b 2 | s = | b 1 | 2 sin ϕ .
2 | b 2 | | b 1 | 2 cos ϕ + Λ | b 1 | 2 + 1 2 T 11 | b 1 | 4 + 1 2 T 22 | b 2 | 4 + T 12 | b 1 | 2 | b 2 | 2 = Λ b 0 2 + 1 2 T 11 b 0 4 ,
| b 2 | max 2 1 2 b 0 2 1 2 | ( 1 4 T 22 T 11 ) b 0 3 2 Λ b 0 | .
Λ = 1 2 ( 1 4 T 22 T 11 ) b 0 2 .
χ ( 3 ) ( ω i , ω i , ω j ) = e 4 N 6 ћ 3 ω 0 3 ( r 4 { 4 ( 1 x i 2 ) ( 1 x j 2 ) + 2 ( 1 x i ) ( 1 x j ) [ 1 ( x i x j ) ] + 2 ( 1 x i ) ( 1 + x j ) [ 1 + ( x i x j ) ] + 2 ( 1 + x i ) ( 1 + x j ) [ 1 + ( x i + x j ) ] + 2 ( 1 x i ) [ ( 1 x j ) ( 1 ( x i x j ) ] + 2 ( 1 x j ) [ ( 1 x j ) 2 x i 2 ] + 2 ( 1 + x j ) [ ( 1 + x j ) 2 x i 2 ] + 2 ( 1 x i ) [ ( 1 x i ) 2 x j 2 ] + 2 ( 1 + x i ) [ ( 1 + x i ) 2 x j 2 ] } r 2 4 [ 2 ( x i 2 + x j 2 ) ] ( 1 x i 2 ) 2 ( 1 x j 2 ) 2 ) ,
n ( x i ) 2 + 2 n ( x i ) 2 1 = 3 ћ ω 0 8 π e 2 N r 2 ( 1 x i 2 ) .
ζ = β | a 0 ( p ) | 2 τ p η ,

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