Abstract

Relatively efficient (up to 3%) third-harmonic generation of picosecond Nd:YAG laser pulses for wavelength λ = 1.06 μm in a low-density laser plasma is reported for a coherent (i.e., collinear) geometry. The third-harmonic beam produced is nearly diffraction limited and coherent and can be used in other nonlinear-optical experiments. A simple theoretical model is proposed in order to explain this result. In the model, third-harmonic generation is considered to be a process that arises from the scattering of electrons by ions in the presence of the strong laser field.

© 1991 Optical Society of America

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  1. J. L. Bobin, Phys. Lett. 122, 173 (1985).
  2. A. M. Brodnikovskii, S. M. Gladkov, N. I. Koroteev, and V. N. Zadkov, in Proceedings of the Fifth USSR Meeting on Non-resonant Interaction of Optical Radiation with Matter, Leningrad, 1981 (Nauka, Leningrad, 1981), p. 227.
  3. A. M. Brodnikovskii, S. M. Gladkov, N. I. Koroteev, and V. N. Zadkov, Pis’ma Zh. Tekh. Fiz. 8, 497 (1982).
  4. S. M. Gladkov, N. I. Koroteev, M. V. Rychev, and A. B. Fedorov, Pis’ma Zh. Tekh. Fiz. 12, 1272 (1986).
  5. S. M. Gladkov, N. I. Koroteev, M. V. Rychev, and A. B. Fedorov, Izv. Akad. Nauk SSSR Ser. Fiz. 50, 1139 (1986).
  6. S. M. Gladkov, A. M. Zheltikov, N. I. Koroteev, A. B. Fedorov, V. G. Tunkin, and M. V. Rychev, Izv. Akad. Nauk SSSR Ser. Fiz. 52, 217 (1988).
  7. S. M. Gladkov, A. M. Zheltikov, N. I. Koroteev, and A. B. Fedotov, Pis’ma Zh. Tekh. Fiz. 14, 1399 (1988).
  8. G. V. Ostrovskaya and A. H. Zaidel’, Usp. Fiz. Nauk 111, 579 (1973).
    [CrossRef]
  9. Ju. P. Raizer, Laser Spark and Discharge Propagation (Nauka, Moscow, 1974).
  10. M. I. Demchuk, S. M. Dmitriev, V. P. Mikhailov, G. A. Pribytok, and A. V. Strashko, Prib. Tekh. Eksp. 1987, 165 (1987).
  11. A. A. Demidov, G. M. Esajan, and J. L. Kalaidzidis, Prib. Tekh. Eksp. 1984, 186 (1984).
  12. S. M. Gladkov, N. I. Koroteev, M. V. Rychev, and O. Stenzel, JETP Lett. 43, 227 (1986).
  13. S. M. Gladkov, in Lasers in Atomic, Molecular, and Nuclear Physics, V. S. Letokhov, ed., Proceedings of the Fourth International School on Laser Applications in Atomic, Molecular, and Nuclear Physics, August 1987, Vilnius, USSR (World Scientific, Singapore, 1989) p. 366.
  14. A. K. Sharma, J. Appl. Phys. 55, 690 (1984).
    [CrossRef]
  15. V. L. Ginzburg, Propagation of Electromagnetic Waves in a Plasma (Nauka, Moscow, 1967).
  16. E. A. McLean, J. A. Stamper, B. H. Ripin, H. R. Griem, and J. M. McMahon, Appl. Phys. Lett. 31, 825 (1977).
    [CrossRef]
  17. N. H. Burnet, H. A. Baldis, M. C. Richardson, and G. D. Enright, Appl. Phys. Lett. 31, 172 (1977).
    [CrossRef]
  18. R. L. Carman, C. K. Rhodes, and R. F. Benjamin, Phys. Rev. A. 24, 2649 (1981).
    [CrossRef]
  19. S. A. Akhmanov, S. M. Gladkov, N. I. Koroteev, and A. M. Zheltikov, “Optical harmonics generation in the process of scattering of electrons by ions,” (Department of Physics, Moscow State University, Moscow, 1988).
  20. N. Blombergen, Nonlinear Optics (Benjamin, New York, 1964).
  21. L. V. Keldysh, Sov. Phys. JETP 20, 1307 (1965).
  22. R. B. Miles and S. E. Harris, IEEE J. Quantum Electron. QE-9, 470 (1973).
    [CrossRef]

1988 (2)

S. M. Gladkov, A. M. Zheltikov, N. I. Koroteev, A. B. Fedorov, V. G. Tunkin, and M. V. Rychev, Izv. Akad. Nauk SSSR Ser. Fiz. 52, 217 (1988).

S. M. Gladkov, A. M. Zheltikov, N. I. Koroteev, and A. B. Fedotov, Pis’ma Zh. Tekh. Fiz. 14, 1399 (1988).

1987 (1)

M. I. Demchuk, S. M. Dmitriev, V. P. Mikhailov, G. A. Pribytok, and A. V. Strashko, Prib. Tekh. Eksp. 1987, 165 (1987).

1986 (3)

S. M. Gladkov, N. I. Koroteev, M. V. Rychev, and A. B. Fedorov, Pis’ma Zh. Tekh. Fiz. 12, 1272 (1986).

S. M. Gladkov, N. I. Koroteev, M. V. Rychev, and A. B. Fedorov, Izv. Akad. Nauk SSSR Ser. Fiz. 50, 1139 (1986).

S. M. Gladkov, N. I. Koroteev, M. V. Rychev, and O. Stenzel, JETP Lett. 43, 227 (1986).

1985 (1)

J. L. Bobin, Phys. Lett. 122, 173 (1985).

1984 (2)

A. A. Demidov, G. M. Esajan, and J. L. Kalaidzidis, Prib. Tekh. Eksp. 1984, 186 (1984).

A. K. Sharma, J. Appl. Phys. 55, 690 (1984).
[CrossRef]

1982 (1)

A. M. Brodnikovskii, S. M. Gladkov, N. I. Koroteev, and V. N. Zadkov, Pis’ma Zh. Tekh. Fiz. 8, 497 (1982).

1981 (1)

R. L. Carman, C. K. Rhodes, and R. F. Benjamin, Phys. Rev. A. 24, 2649 (1981).
[CrossRef]

1977 (2)

E. A. McLean, J. A. Stamper, B. H. Ripin, H. R. Griem, and J. M. McMahon, Appl. Phys. Lett. 31, 825 (1977).
[CrossRef]

N. H. Burnet, H. A. Baldis, M. C. Richardson, and G. D. Enright, Appl. Phys. Lett. 31, 172 (1977).
[CrossRef]

1973 (2)

R. B. Miles and S. E. Harris, IEEE J. Quantum Electron. QE-9, 470 (1973).
[CrossRef]

G. V. Ostrovskaya and A. H. Zaidel’, Usp. Fiz. Nauk 111, 579 (1973).
[CrossRef]

1965 (1)

L. V. Keldysh, Sov. Phys. JETP 20, 1307 (1965).

Akhmanov, S. A.

S. A. Akhmanov, S. M. Gladkov, N. I. Koroteev, and A. M. Zheltikov, “Optical harmonics generation in the process of scattering of electrons by ions,” (Department of Physics, Moscow State University, Moscow, 1988).

Baldis, H. A.

N. H. Burnet, H. A. Baldis, M. C. Richardson, and G. D. Enright, Appl. Phys. Lett. 31, 172 (1977).
[CrossRef]

Benjamin, R. F.

R. L. Carman, C. K. Rhodes, and R. F. Benjamin, Phys. Rev. A. 24, 2649 (1981).
[CrossRef]

Blombergen, N.

N. Blombergen, Nonlinear Optics (Benjamin, New York, 1964).

Bobin, J. L.

J. L. Bobin, Phys. Lett. 122, 173 (1985).

Brodnikovskii, A. M.

A. M. Brodnikovskii, S. M. Gladkov, N. I. Koroteev, and V. N. Zadkov, Pis’ma Zh. Tekh. Fiz. 8, 497 (1982).

A. M. Brodnikovskii, S. M. Gladkov, N. I. Koroteev, and V. N. Zadkov, in Proceedings of the Fifth USSR Meeting on Non-resonant Interaction of Optical Radiation with Matter, Leningrad, 1981 (Nauka, Leningrad, 1981), p. 227.

Burnet, N. H.

N. H. Burnet, H. A. Baldis, M. C. Richardson, and G. D. Enright, Appl. Phys. Lett. 31, 172 (1977).
[CrossRef]

Carman, R. L.

R. L. Carman, C. K. Rhodes, and R. F. Benjamin, Phys. Rev. A. 24, 2649 (1981).
[CrossRef]

Demchuk, M. I.

M. I. Demchuk, S. M. Dmitriev, V. P. Mikhailov, G. A. Pribytok, and A. V. Strashko, Prib. Tekh. Eksp. 1987, 165 (1987).

Demidov, A. A.

A. A. Demidov, G. M. Esajan, and J. L. Kalaidzidis, Prib. Tekh. Eksp. 1984, 186 (1984).

Dmitriev, S. M.

M. I. Demchuk, S. M. Dmitriev, V. P. Mikhailov, G. A. Pribytok, and A. V. Strashko, Prib. Tekh. Eksp. 1987, 165 (1987).

Enright, G. D.

N. H. Burnet, H. A. Baldis, M. C. Richardson, and G. D. Enright, Appl. Phys. Lett. 31, 172 (1977).
[CrossRef]

Esajan, G. M.

A. A. Demidov, G. M. Esajan, and J. L. Kalaidzidis, Prib. Tekh. Eksp. 1984, 186 (1984).

Fedorov, A. B.

S. M. Gladkov, A. M. Zheltikov, N. I. Koroteev, A. B. Fedorov, V. G. Tunkin, and M. V. Rychev, Izv. Akad. Nauk SSSR Ser. Fiz. 52, 217 (1988).

S. M. Gladkov, N. I. Koroteev, M. V. Rychev, and A. B. Fedorov, Pis’ma Zh. Tekh. Fiz. 12, 1272 (1986).

S. M. Gladkov, N. I. Koroteev, M. V. Rychev, and A. B. Fedorov, Izv. Akad. Nauk SSSR Ser. Fiz. 50, 1139 (1986).

Fedotov, A. B.

S. M. Gladkov, A. M. Zheltikov, N. I. Koroteev, and A. B. Fedotov, Pis’ma Zh. Tekh. Fiz. 14, 1399 (1988).

Ginzburg, V. L.

V. L. Ginzburg, Propagation of Electromagnetic Waves in a Plasma (Nauka, Moscow, 1967).

Gladkov, S. M.

S. M. Gladkov, A. M. Zheltikov, N. I. Koroteev, and A. B. Fedotov, Pis’ma Zh. Tekh. Fiz. 14, 1399 (1988).

S. M. Gladkov, A. M. Zheltikov, N. I. Koroteev, A. B. Fedorov, V. G. Tunkin, and M. V. Rychev, Izv. Akad. Nauk SSSR Ser. Fiz. 52, 217 (1988).

S. M. Gladkov, N. I. Koroteev, M. V. Rychev, and A. B. Fedorov, Pis’ma Zh. Tekh. Fiz. 12, 1272 (1986).

S. M. Gladkov, N. I. Koroteev, M. V. Rychev, and A. B. Fedorov, Izv. Akad. Nauk SSSR Ser. Fiz. 50, 1139 (1986).

S. M. Gladkov, N. I. Koroteev, M. V. Rychev, and O. Stenzel, JETP Lett. 43, 227 (1986).

A. M. Brodnikovskii, S. M. Gladkov, N. I. Koroteev, and V. N. Zadkov, Pis’ma Zh. Tekh. Fiz. 8, 497 (1982).

A. M. Brodnikovskii, S. M. Gladkov, N. I. Koroteev, and V. N. Zadkov, in Proceedings of the Fifth USSR Meeting on Non-resonant Interaction of Optical Radiation with Matter, Leningrad, 1981 (Nauka, Leningrad, 1981), p. 227.

S. M. Gladkov, in Lasers in Atomic, Molecular, and Nuclear Physics, V. S. Letokhov, ed., Proceedings of the Fourth International School on Laser Applications in Atomic, Molecular, and Nuclear Physics, August 1987, Vilnius, USSR (World Scientific, Singapore, 1989) p. 366.

S. A. Akhmanov, S. M. Gladkov, N. I. Koroteev, and A. M. Zheltikov, “Optical harmonics generation in the process of scattering of electrons by ions,” (Department of Physics, Moscow State University, Moscow, 1988).

Griem, H. R.

E. A. McLean, J. A. Stamper, B. H. Ripin, H. R. Griem, and J. M. McMahon, Appl. Phys. Lett. 31, 825 (1977).
[CrossRef]

Harris, S. E.

R. B. Miles and S. E. Harris, IEEE J. Quantum Electron. QE-9, 470 (1973).
[CrossRef]

Kalaidzidis, J. L.

A. A. Demidov, G. M. Esajan, and J. L. Kalaidzidis, Prib. Tekh. Eksp. 1984, 186 (1984).

Keldysh, L. V.

L. V. Keldysh, Sov. Phys. JETP 20, 1307 (1965).

Koroteev, N. I.

S. M. Gladkov, A. M. Zheltikov, N. I. Koroteev, and A. B. Fedotov, Pis’ma Zh. Tekh. Fiz. 14, 1399 (1988).

S. M. Gladkov, A. M. Zheltikov, N. I. Koroteev, A. B. Fedorov, V. G. Tunkin, and M. V. Rychev, Izv. Akad. Nauk SSSR Ser. Fiz. 52, 217 (1988).

S. M. Gladkov, N. I. Koroteev, M. V. Rychev, and A. B. Fedorov, Izv. Akad. Nauk SSSR Ser. Fiz. 50, 1139 (1986).

S. M. Gladkov, N. I. Koroteev, M. V. Rychev, and A. B. Fedorov, Pis’ma Zh. Tekh. Fiz. 12, 1272 (1986).

S. M. Gladkov, N. I. Koroteev, M. V. Rychev, and O. Stenzel, JETP Lett. 43, 227 (1986).

A. M. Brodnikovskii, S. M. Gladkov, N. I. Koroteev, and V. N. Zadkov, Pis’ma Zh. Tekh. Fiz. 8, 497 (1982).

A. M. Brodnikovskii, S. M. Gladkov, N. I. Koroteev, and V. N. Zadkov, in Proceedings of the Fifth USSR Meeting on Non-resonant Interaction of Optical Radiation with Matter, Leningrad, 1981 (Nauka, Leningrad, 1981), p. 227.

S. A. Akhmanov, S. M. Gladkov, N. I. Koroteev, and A. M. Zheltikov, “Optical harmonics generation in the process of scattering of electrons by ions,” (Department of Physics, Moscow State University, Moscow, 1988).

McLean, E. A.

E. A. McLean, J. A. Stamper, B. H. Ripin, H. R. Griem, and J. M. McMahon, Appl. Phys. Lett. 31, 825 (1977).
[CrossRef]

McMahon, J. M.

E. A. McLean, J. A. Stamper, B. H. Ripin, H. R. Griem, and J. M. McMahon, Appl. Phys. Lett. 31, 825 (1977).
[CrossRef]

Mikhailov, V. P.

M. I. Demchuk, S. M. Dmitriev, V. P. Mikhailov, G. A. Pribytok, and A. V. Strashko, Prib. Tekh. Eksp. 1987, 165 (1987).

Miles, R. B.

R. B. Miles and S. E. Harris, IEEE J. Quantum Electron. QE-9, 470 (1973).
[CrossRef]

Ostrovskaya, G. V.

G. V. Ostrovskaya and A. H. Zaidel’, Usp. Fiz. Nauk 111, 579 (1973).
[CrossRef]

Pribytok, G. A.

M. I. Demchuk, S. M. Dmitriev, V. P. Mikhailov, G. A. Pribytok, and A. V. Strashko, Prib. Tekh. Eksp. 1987, 165 (1987).

Raizer, Ju. P.

Ju. P. Raizer, Laser Spark and Discharge Propagation (Nauka, Moscow, 1974).

Rhodes, C. K.

R. L. Carman, C. K. Rhodes, and R. F. Benjamin, Phys. Rev. A. 24, 2649 (1981).
[CrossRef]

Richardson, M. C.

N. H. Burnet, H. A. Baldis, M. C. Richardson, and G. D. Enright, Appl. Phys. Lett. 31, 172 (1977).
[CrossRef]

Ripin, B. H.

E. A. McLean, J. A. Stamper, B. H. Ripin, H. R. Griem, and J. M. McMahon, Appl. Phys. Lett. 31, 825 (1977).
[CrossRef]

Rychev, M. V.

S. M. Gladkov, A. M. Zheltikov, N. I. Koroteev, A. B. Fedorov, V. G. Tunkin, and M. V. Rychev, Izv. Akad. Nauk SSSR Ser. Fiz. 52, 217 (1988).

S. M. Gladkov, N. I. Koroteev, M. V. Rychev, and A. B. Fedorov, Pis’ma Zh. Tekh. Fiz. 12, 1272 (1986).

S. M. Gladkov, N. I. Koroteev, M. V. Rychev, and A. B. Fedorov, Izv. Akad. Nauk SSSR Ser. Fiz. 50, 1139 (1986).

S. M. Gladkov, N. I. Koroteev, M. V. Rychev, and O. Stenzel, JETP Lett. 43, 227 (1986).

Sharma, A. K.

A. K. Sharma, J. Appl. Phys. 55, 690 (1984).
[CrossRef]

Stamper, J. A.

E. A. McLean, J. A. Stamper, B. H. Ripin, H. R. Griem, and J. M. McMahon, Appl. Phys. Lett. 31, 825 (1977).
[CrossRef]

Stenzel, O.

S. M. Gladkov, N. I. Koroteev, M. V. Rychev, and O. Stenzel, JETP Lett. 43, 227 (1986).

Strashko, A. V.

M. I. Demchuk, S. M. Dmitriev, V. P. Mikhailov, G. A. Pribytok, and A. V. Strashko, Prib. Tekh. Eksp. 1987, 165 (1987).

Tunkin, V. G.

S. M. Gladkov, A. M. Zheltikov, N. I. Koroteev, A. B. Fedorov, V. G. Tunkin, and M. V. Rychev, Izv. Akad. Nauk SSSR Ser. Fiz. 52, 217 (1988).

Zadkov, V. N.

A. M. Brodnikovskii, S. M. Gladkov, N. I. Koroteev, and V. N. Zadkov, Pis’ma Zh. Tekh. Fiz. 8, 497 (1982).

A. M. Brodnikovskii, S. M. Gladkov, N. I. Koroteev, and V. N. Zadkov, in Proceedings of the Fifth USSR Meeting on Non-resonant Interaction of Optical Radiation with Matter, Leningrad, 1981 (Nauka, Leningrad, 1981), p. 227.

Zaidel’, A. H.

G. V. Ostrovskaya and A. H. Zaidel’, Usp. Fiz. Nauk 111, 579 (1973).
[CrossRef]

Zheltikov, A. M.

S. M. Gladkov, A. M. Zheltikov, N. I. Koroteev, and A. B. Fedotov, Pis’ma Zh. Tekh. Fiz. 14, 1399 (1988).

S. M. Gladkov, A. M. Zheltikov, N. I. Koroteev, A. B. Fedorov, V. G. Tunkin, and M. V. Rychev, Izv. Akad. Nauk SSSR Ser. Fiz. 52, 217 (1988).

S. A. Akhmanov, S. M. Gladkov, N. I. Koroteev, and A. M. Zheltikov, “Optical harmonics generation in the process of scattering of electrons by ions,” (Department of Physics, Moscow State University, Moscow, 1988).

Appl. Phys. Lett. (2)

E. A. McLean, J. A. Stamper, B. H. Ripin, H. R. Griem, and J. M. McMahon, Appl. Phys. Lett. 31, 825 (1977).
[CrossRef]

N. H. Burnet, H. A. Baldis, M. C. Richardson, and G. D. Enright, Appl. Phys. Lett. 31, 172 (1977).
[CrossRef]

IEEE J. Quantum Electron. (1)

R. B. Miles and S. E. Harris, IEEE J. Quantum Electron. QE-9, 470 (1973).
[CrossRef]

Izv. Akad. Nauk SSSR Ser. Fiz. (2)

S. M. Gladkov, N. I. Koroteev, M. V. Rychev, and A. B. Fedorov, Izv. Akad. Nauk SSSR Ser. Fiz. 50, 1139 (1986).

S. M. Gladkov, A. M. Zheltikov, N. I. Koroteev, A. B. Fedorov, V. G. Tunkin, and M. V. Rychev, Izv. Akad. Nauk SSSR Ser. Fiz. 52, 217 (1988).

J. Appl. Phys. (1)

A. K. Sharma, J. Appl. Phys. 55, 690 (1984).
[CrossRef]

JETP Lett. (1)

S. M. Gladkov, N. I. Koroteev, M. V. Rychev, and O. Stenzel, JETP Lett. 43, 227 (1986).

Phys. Lett. (1)

J. L. Bobin, Phys. Lett. 122, 173 (1985).

Phys. Rev. A. (1)

R. L. Carman, C. K. Rhodes, and R. F. Benjamin, Phys. Rev. A. 24, 2649 (1981).
[CrossRef]

Pis’ma Zh. Tekh. Fiz. (3)

A. M. Brodnikovskii, S. M. Gladkov, N. I. Koroteev, and V. N. Zadkov, Pis’ma Zh. Tekh. Fiz. 8, 497 (1982).

S. M. Gladkov, N. I. Koroteev, M. V. Rychev, and A. B. Fedorov, Pis’ma Zh. Tekh. Fiz. 12, 1272 (1986).

S. M. Gladkov, A. M. Zheltikov, N. I. Koroteev, and A. B. Fedotov, Pis’ma Zh. Tekh. Fiz. 14, 1399 (1988).

Prib. Tekh. Eksp. (2)

M. I. Demchuk, S. M. Dmitriev, V. P. Mikhailov, G. A. Pribytok, and A. V. Strashko, Prib. Tekh. Eksp. 1987, 165 (1987).

A. A. Demidov, G. M. Esajan, and J. L. Kalaidzidis, Prib. Tekh. Eksp. 1984, 186 (1984).

Sov. Phys. JETP (1)

L. V. Keldysh, Sov. Phys. JETP 20, 1307 (1965).

Usp. Fiz. Nauk (1)

G. V. Ostrovskaya and A. H. Zaidel’, Usp. Fiz. Nauk 111, 579 (1973).
[CrossRef]

Other (6)

Ju. P. Raizer, Laser Spark and Discharge Propagation (Nauka, Moscow, 1974).

A. M. Brodnikovskii, S. M. Gladkov, N. I. Koroteev, and V. N. Zadkov, in Proceedings of the Fifth USSR Meeting on Non-resonant Interaction of Optical Radiation with Matter, Leningrad, 1981 (Nauka, Leningrad, 1981), p. 227.

S. M. Gladkov, in Lasers in Atomic, Molecular, and Nuclear Physics, V. S. Letokhov, ed., Proceedings of the Fourth International School on Laser Applications in Atomic, Molecular, and Nuclear Physics, August 1987, Vilnius, USSR (World Scientific, Singapore, 1989) p. 366.

S. A. Akhmanov, S. M. Gladkov, N. I. Koroteev, and A. M. Zheltikov, “Optical harmonics generation in the process of scattering of electrons by ions,” (Department of Physics, Moscow State University, Moscow, 1988).

N. Blombergen, Nonlinear Optics (Benjamin, New York, 1964).

V. L. Ginzburg, Propagation of Electromagnetic Waves in a Plasma (Nauka, Moscow, 1967).

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Figures (2)

Fig. 1
Fig. 1

Experimental setup for THG study.

Fig. 2
Fig. 2

Dependence of third-harmonic intensity on time delay τ between plasma formation and THG probing.

Tables (1)

Tables Icon

Table 1 Experimental Data on Frequency-Tripling Conversion Efficiency in Plasmas

Equations (12)

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

i Ψ t = ( H 0 + V ) Ψ ,
H 0 = 1 2 m [ k - e ω E cos ( ω t ) ] 2
φ k ( r ) = 1 ( 2 π ) 3 / 2 exp { i [ k r - k 2 2 m t + e k E m ω 2 sin ( ω t ) - 1 4 m ( e E ω ) 2 ( sin ( 2 ω t ) 2 ω + t ) ] } .
ψ k 0 = ( 2 π α ) 3 / 2 π 1 / 2 φ k exp ( - α r ) ,
( ψ k 0 ) * ψ k 0 d r = 1 ,
d = ψ k 0 + ψ k e r ψ k 0 + ψ k .
d n ω = - 2 7 α 4 e 3 π | 0 s = 1 ( V s ( - ) + V s ( + ) ) × ( 1 ω k k + s ω - 1 ω k k - s ω ) q 3 d q ( α 2 + q 2 ) 3 ( 4 α 2 + q 2 ) | ,
q = k - k ,             ω k k = 2 m ( k 2 - k 2 ) , V s ( ± ) = ( e q E m ω 2 ) 2 s + n 2 2 s + n ( 2 s + n + 2 ) s ! ( s + n ) ! F 3 4 × [ 2 s + n + 1 2 , 2 s + n + 2 2 , 2 s + n + 2 2 ; s + 1 , s + n + 1 , 2 s + n + 1 , 2 s + n + 4 2 ; - ( e q E m ω 2 ) 2 ]
V s ( - ) = ( - 1 ) n - s ( e q E m ω 2 ) n 2 n ( n + 2 ) s ! ( n - s ) ! F 3 4 [ n + 1 2 , n + 2 2 , n + 2 2 ; s + 1 , n - s + 1 , n + 1 , n + 4 2 ; - ( e q E m ω 2 ) 2 ] ,
I 3 ~ N e 2 [ d ( 3 ω ) ] 2 I 2 ,
I ( Δ k , ξ , ζ ) = ζ ξ d ξ exp [ ½ i b Δ k ( ξ - ξ ) ] ( 1 + i ξ ) 2 , Δ k = k 3 ω - 3 k ω .
n = 1 - ω p 2 2 ω 2

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