Abstract

Light scattered from one counterpropagating mode into the other can drastically modify the behavior of a ring laser. We point out that, within the domain of the third-order laser theory, the effect of backscattering on the fluctuation properties of the light can be reduced simply by increasing the cavity-loss rate, so long as the increased loss can be compensated for by increased pumping of the gain medium.

© 1987 Optical Society of America

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  1. I. P. Efanova, E. G. Lariontsev, Zh. Eksp. Teor. Fiz. 55, 1532 (1968)[Sov. Phys. JETP 28, 802 (1969)].
  2. F. Aronowitz, R. J. Collins, J. Appl. Phys. 41, 130 (1970).
    [CrossRef]
  3. F. Aronowitz, Appl. Opt. 11, 405 (1972).
    [CrossRef] [PubMed]
  4. F. Aronowitz, in Laser Applications, M. Ross, ed. (Academic, New York, 1971), p. 133.
  5. L. M. Menegozzi, W. E. Lamb, Phys. Rev. A 8, 2103 (1973).
    [CrossRef]
  6. E. L. Klochan, L. S. Kornienko, N. V. Kravtsov, E. G. Lariontsev, A. N. Shelaev, Zh. Eksp. Teor. Fiz. 65, 1344 (1973) [Sov. Phys. JETP 38, 669 (1974)].
  7. F. Aronowitz, W. L. Lim, IEEE J. Quantum Electron. QE-13, 338 (1977).
    [CrossRef]
  8. V. I. Sardyko, V. N. Severikov, J. Appl. Spectrosc. 26, 592 (1977).
    [CrossRef]
  9. G. V. Perevedentseva, P. A. Khandokhin, Ya. I. Khanin, Kvantovaya Elektron. (Moscow) 7, 128 (1980) [Sov. J. Qunatum Electron. 10, 71 (1980)].
  10. D. Kühlke, G. Jetschke, Physica 106C, 287 (1981).
  11. D. Kühlke, H. Horak, Physica 111C, 111 (1981).
  12. D. Kühlke, Acta Phys. Polon. A 61, 547 (1982).
  13. J. D. Cresser, W. H. Louisell, P. Meystre, W. Schleich, M. O. Scully, Phys. Rev. A 25, 2214 (1982).
    [CrossRef]
  14. J. D. Cresser, D. Hammons, W. H. Louisell, P. Meystre, H. Risken, Phys. Rev. A 25, 2226 (1982).
    [CrossRef]
  15. W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, M. O. Scully, Rev. Mod. Phys. 57, 61 (1985).
    [CrossRef]
  16. W. R. Christian, L. Mandel, Phys. Rev. A 34, 3932 (1986).There is an error in the solution to the Fokker-Planck equation [Eqs. (14) and (15)] given in that paper, in that the solution and subsequently derived equations are correct at line center with a1 = a2 and |R1| = |R2|, which is what we assume here.
    [CrossRef] [PubMed]
  17. M. M-Tehrani, L. Mandel, Opt. Lett. 1, 196 (1977).
    [CrossRef] [PubMed]
  18. M. M-Tehrani, L. Mandel, Phys. Rev. A 17, 677, 694 (1978).
    [CrossRef]
  19. T. A. Dorschner, H. A. Haus, M. Holz, I. W. Smith, H. Statz, IEEE J. Quantum Electron. QE-16, 1376 (1980).
    [CrossRef]
  20. H. A. Haus, H. Statz, I. W. Smith, IEEE J. Quantum Electron. QE-21, 78 (1985).
    [CrossRef]
  21. W. E. Lamb, Phys. Rev. 134, A1429 (1964);see also M. Sargent, M. O. Scully, W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974).
    [CrossRef]
  22. See H. Haken, in Laser Theory, Vol. XXV 2C of Encyclopedia of Physics, S. Flügge, ed. (Springer-Verlag, Berlin, 1970).
  23. J. B. Hambenne, M. Sargent, IEEE J. Quantum Electron. QE-11, 90 (1975).
    [CrossRef]
  24. S. Singh, Phys. Rev. A 23, 837 (1981);Phys. Rep. 108, 217 (1984).
    [CrossRef]

1986 (1)

W. R. Christian, L. Mandel, Phys. Rev. A 34, 3932 (1986).There is an error in the solution to the Fokker-Planck equation [Eqs. (14) and (15)] given in that paper, in that the solution and subsequently derived equations are correct at line center with a1 = a2 and |R1| = |R2|, which is what we assume here.
[CrossRef] [PubMed]

1985 (2)

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, M. O. Scully, Rev. Mod. Phys. 57, 61 (1985).
[CrossRef]

H. A. Haus, H. Statz, I. W. Smith, IEEE J. Quantum Electron. QE-21, 78 (1985).
[CrossRef]

1982 (3)

D. Kühlke, Acta Phys. Polon. A 61, 547 (1982).

J. D. Cresser, W. H. Louisell, P. Meystre, W. Schleich, M. O. Scully, Phys. Rev. A 25, 2214 (1982).
[CrossRef]

J. D. Cresser, D. Hammons, W. H. Louisell, P. Meystre, H. Risken, Phys. Rev. A 25, 2226 (1982).
[CrossRef]

1981 (3)

D. Kühlke, G. Jetschke, Physica 106C, 287 (1981).

D. Kühlke, H. Horak, Physica 111C, 111 (1981).

S. Singh, Phys. Rev. A 23, 837 (1981);Phys. Rep. 108, 217 (1984).
[CrossRef]

1980 (2)

T. A. Dorschner, H. A. Haus, M. Holz, I. W. Smith, H. Statz, IEEE J. Quantum Electron. QE-16, 1376 (1980).
[CrossRef]

G. V. Perevedentseva, P. A. Khandokhin, Ya. I. Khanin, Kvantovaya Elektron. (Moscow) 7, 128 (1980) [Sov. J. Qunatum Electron. 10, 71 (1980)].

1978 (1)

M. M-Tehrani, L. Mandel, Phys. Rev. A 17, 677, 694 (1978).
[CrossRef]

1977 (3)

F. Aronowitz, W. L. Lim, IEEE J. Quantum Electron. QE-13, 338 (1977).
[CrossRef]

V. I. Sardyko, V. N. Severikov, J. Appl. Spectrosc. 26, 592 (1977).
[CrossRef]

M. M-Tehrani, L. Mandel, Opt. Lett. 1, 196 (1977).
[CrossRef] [PubMed]

1975 (1)

J. B. Hambenne, M. Sargent, IEEE J. Quantum Electron. QE-11, 90 (1975).
[CrossRef]

1973 (2)

L. M. Menegozzi, W. E. Lamb, Phys. Rev. A 8, 2103 (1973).
[CrossRef]

E. L. Klochan, L. S. Kornienko, N. V. Kravtsov, E. G. Lariontsev, A. N. Shelaev, Zh. Eksp. Teor. Fiz. 65, 1344 (1973) [Sov. Phys. JETP 38, 669 (1974)].

1972 (1)

1970 (1)

F. Aronowitz, R. J. Collins, J. Appl. Phys. 41, 130 (1970).
[CrossRef]

1968 (1)

I. P. Efanova, E. G. Lariontsev, Zh. Eksp. Teor. Fiz. 55, 1532 (1968)[Sov. Phys. JETP 28, 802 (1969)].

1964 (1)

W. E. Lamb, Phys. Rev. 134, A1429 (1964);see also M. Sargent, M. O. Scully, W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974).
[CrossRef]

Aronowitz, F.

F. Aronowitz, W. L. Lim, IEEE J. Quantum Electron. QE-13, 338 (1977).
[CrossRef]

F. Aronowitz, Appl. Opt. 11, 405 (1972).
[CrossRef] [PubMed]

F. Aronowitz, R. J. Collins, J. Appl. Phys. 41, 130 (1970).
[CrossRef]

F. Aronowitz, in Laser Applications, M. Ross, ed. (Academic, New York, 1971), p. 133.

Chow, W. W.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, M. O. Scully, Rev. Mod. Phys. 57, 61 (1985).
[CrossRef]

Christian, W. R.

W. R. Christian, L. Mandel, Phys. Rev. A 34, 3932 (1986).There is an error in the solution to the Fokker-Planck equation [Eqs. (14) and (15)] given in that paper, in that the solution and subsequently derived equations are correct at line center with a1 = a2 and |R1| = |R2|, which is what we assume here.
[CrossRef] [PubMed]

Collins, R. J.

F. Aronowitz, R. J. Collins, J. Appl. Phys. 41, 130 (1970).
[CrossRef]

Cresser, J. D.

J. D. Cresser, D. Hammons, W. H. Louisell, P. Meystre, H. Risken, Phys. Rev. A 25, 2226 (1982).
[CrossRef]

J. D. Cresser, W. H. Louisell, P. Meystre, W. Schleich, M. O. Scully, Phys. Rev. A 25, 2214 (1982).
[CrossRef]

Dorschner, T. A.

T. A. Dorschner, H. A. Haus, M. Holz, I. W. Smith, H. Statz, IEEE J. Quantum Electron. QE-16, 1376 (1980).
[CrossRef]

Efanova, I. P.

I. P. Efanova, E. G. Lariontsev, Zh. Eksp. Teor. Fiz. 55, 1532 (1968)[Sov. Phys. JETP 28, 802 (1969)].

Gea-Banacloche, J.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, M. O. Scully, Rev. Mod. Phys. 57, 61 (1985).
[CrossRef]

Haken, H.

See H. Haken, in Laser Theory, Vol. XXV 2C of Encyclopedia of Physics, S. Flügge, ed. (Springer-Verlag, Berlin, 1970).

Hambenne, J. B.

J. B. Hambenne, M. Sargent, IEEE J. Quantum Electron. QE-11, 90 (1975).
[CrossRef]

Hammons, D.

J. D. Cresser, D. Hammons, W. H. Louisell, P. Meystre, H. Risken, Phys. Rev. A 25, 2226 (1982).
[CrossRef]

Haus, H. A.

H. A. Haus, H. Statz, I. W. Smith, IEEE J. Quantum Electron. QE-21, 78 (1985).
[CrossRef]

T. A. Dorschner, H. A. Haus, M. Holz, I. W. Smith, H. Statz, IEEE J. Quantum Electron. QE-16, 1376 (1980).
[CrossRef]

Holz, M.

T. A. Dorschner, H. A. Haus, M. Holz, I. W. Smith, H. Statz, IEEE J. Quantum Electron. QE-16, 1376 (1980).
[CrossRef]

Horak, H.

D. Kühlke, H. Horak, Physica 111C, 111 (1981).

Jetschke, G.

D. Kühlke, G. Jetschke, Physica 106C, 287 (1981).

Khandokhin, P. A.

G. V. Perevedentseva, P. A. Khandokhin, Ya. I. Khanin, Kvantovaya Elektron. (Moscow) 7, 128 (1980) [Sov. J. Qunatum Electron. 10, 71 (1980)].

Khanin, Ya. I.

G. V. Perevedentseva, P. A. Khandokhin, Ya. I. Khanin, Kvantovaya Elektron. (Moscow) 7, 128 (1980) [Sov. J. Qunatum Electron. 10, 71 (1980)].

Klochan, E. L.

E. L. Klochan, L. S. Kornienko, N. V. Kravtsov, E. G. Lariontsev, A. N. Shelaev, Zh. Eksp. Teor. Fiz. 65, 1344 (1973) [Sov. Phys. JETP 38, 669 (1974)].

Kornienko, L. S.

E. L. Klochan, L. S. Kornienko, N. V. Kravtsov, E. G. Lariontsev, A. N. Shelaev, Zh. Eksp. Teor. Fiz. 65, 1344 (1973) [Sov. Phys. JETP 38, 669 (1974)].

Kravtsov, N. V.

E. L. Klochan, L. S. Kornienko, N. V. Kravtsov, E. G. Lariontsev, A. N. Shelaev, Zh. Eksp. Teor. Fiz. 65, 1344 (1973) [Sov. Phys. JETP 38, 669 (1974)].

Kühlke, D.

D. Kühlke, Acta Phys. Polon. A 61, 547 (1982).

D. Kühlke, G. Jetschke, Physica 106C, 287 (1981).

D. Kühlke, H. Horak, Physica 111C, 111 (1981).

Lamb, W. E.

L. M. Menegozzi, W. E. Lamb, Phys. Rev. A 8, 2103 (1973).
[CrossRef]

W. E. Lamb, Phys. Rev. 134, A1429 (1964);see also M. Sargent, M. O. Scully, W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974).
[CrossRef]

Lariontsev, E. G.

E. L. Klochan, L. S. Kornienko, N. V. Kravtsov, E. G. Lariontsev, A. N. Shelaev, Zh. Eksp. Teor. Fiz. 65, 1344 (1973) [Sov. Phys. JETP 38, 669 (1974)].

I. P. Efanova, E. G. Lariontsev, Zh. Eksp. Teor. Fiz. 55, 1532 (1968)[Sov. Phys. JETP 28, 802 (1969)].

Lim, W. L.

F. Aronowitz, W. L. Lim, IEEE J. Quantum Electron. QE-13, 338 (1977).
[CrossRef]

Louisell, W. H.

J. D. Cresser, W. H. Louisell, P. Meystre, W. Schleich, M. O. Scully, Phys. Rev. A 25, 2214 (1982).
[CrossRef]

J. D. Cresser, D. Hammons, W. H. Louisell, P. Meystre, H. Risken, Phys. Rev. A 25, 2226 (1982).
[CrossRef]

Mandel, L.

W. R. Christian, L. Mandel, Phys. Rev. A 34, 3932 (1986).There is an error in the solution to the Fokker-Planck equation [Eqs. (14) and (15)] given in that paper, in that the solution and subsequently derived equations are correct at line center with a1 = a2 and |R1| = |R2|, which is what we assume here.
[CrossRef] [PubMed]

M. M-Tehrani, L. Mandel, Phys. Rev. A 17, 677, 694 (1978).
[CrossRef]

M. M-Tehrani, L. Mandel, Opt. Lett. 1, 196 (1977).
[CrossRef] [PubMed]

Menegozzi, L. M.

L. M. Menegozzi, W. E. Lamb, Phys. Rev. A 8, 2103 (1973).
[CrossRef]

Meystre, P.

J. D. Cresser, D. Hammons, W. H. Louisell, P. Meystre, H. Risken, Phys. Rev. A 25, 2226 (1982).
[CrossRef]

J. D. Cresser, W. H. Louisell, P. Meystre, W. Schleich, M. O. Scully, Phys. Rev. A 25, 2214 (1982).
[CrossRef]

M-Tehrani, M.

M. M-Tehrani, L. Mandel, Phys. Rev. A 17, 677, 694 (1978).
[CrossRef]

M. M-Tehrani, L. Mandel, Opt. Lett. 1, 196 (1977).
[CrossRef] [PubMed]

Pedrotti, L. M.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, M. O. Scully, Rev. Mod. Phys. 57, 61 (1985).
[CrossRef]

Perevedentseva, G. V.

G. V. Perevedentseva, P. A. Khandokhin, Ya. I. Khanin, Kvantovaya Elektron. (Moscow) 7, 128 (1980) [Sov. J. Qunatum Electron. 10, 71 (1980)].

Risken, H.

J. D. Cresser, D. Hammons, W. H. Louisell, P. Meystre, H. Risken, Phys. Rev. A 25, 2226 (1982).
[CrossRef]

Sanders, V. E.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, M. O. Scully, Rev. Mod. Phys. 57, 61 (1985).
[CrossRef]

Sardyko, V. I.

V. I. Sardyko, V. N. Severikov, J. Appl. Spectrosc. 26, 592 (1977).
[CrossRef]

Sargent, M.

J. B. Hambenne, M. Sargent, IEEE J. Quantum Electron. QE-11, 90 (1975).
[CrossRef]

Schleich, W.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, M. O. Scully, Rev. Mod. Phys. 57, 61 (1985).
[CrossRef]

J. D. Cresser, W. H. Louisell, P. Meystre, W. Schleich, M. O. Scully, Phys. Rev. A 25, 2214 (1982).
[CrossRef]

Scully, M. O.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, M. O. Scully, Rev. Mod. Phys. 57, 61 (1985).
[CrossRef]

J. D. Cresser, W. H. Louisell, P. Meystre, W. Schleich, M. O. Scully, Phys. Rev. A 25, 2214 (1982).
[CrossRef]

Severikov, V. N.

V. I. Sardyko, V. N. Severikov, J. Appl. Spectrosc. 26, 592 (1977).
[CrossRef]

Shelaev, A. N.

E. L. Klochan, L. S. Kornienko, N. V. Kravtsov, E. G. Lariontsev, A. N. Shelaev, Zh. Eksp. Teor. Fiz. 65, 1344 (1973) [Sov. Phys. JETP 38, 669 (1974)].

Singh, S.

S. Singh, Phys. Rev. A 23, 837 (1981);Phys. Rep. 108, 217 (1984).
[CrossRef]

Smith, I. W.

H. A. Haus, H. Statz, I. W. Smith, IEEE J. Quantum Electron. QE-21, 78 (1985).
[CrossRef]

T. A. Dorschner, H. A. Haus, M. Holz, I. W. Smith, H. Statz, IEEE J. Quantum Electron. QE-16, 1376 (1980).
[CrossRef]

Statz, H.

H. A. Haus, H. Statz, I. W. Smith, IEEE J. Quantum Electron. QE-21, 78 (1985).
[CrossRef]

T. A. Dorschner, H. A. Haus, M. Holz, I. W. Smith, H. Statz, IEEE J. Quantum Electron. QE-16, 1376 (1980).
[CrossRef]

Acta Phys. Polon. A (1)

D. Kühlke, Acta Phys. Polon. A 61, 547 (1982).

Appl. Opt. (1)

IEEE J. Quantum Electron. (4)

J. B. Hambenne, M. Sargent, IEEE J. Quantum Electron. QE-11, 90 (1975).
[CrossRef]

F. Aronowitz, W. L. Lim, IEEE J. Quantum Electron. QE-13, 338 (1977).
[CrossRef]

T. A. Dorschner, H. A. Haus, M. Holz, I. W. Smith, H. Statz, IEEE J. Quantum Electron. QE-16, 1376 (1980).
[CrossRef]

H. A. Haus, H. Statz, I. W. Smith, IEEE J. Quantum Electron. QE-21, 78 (1985).
[CrossRef]

J. Appl. Phys. (1)

F. Aronowitz, R. J. Collins, J. Appl. Phys. 41, 130 (1970).
[CrossRef]

J. Appl. Spectrosc. (1)

V. I. Sardyko, V. N. Severikov, J. Appl. Spectrosc. 26, 592 (1977).
[CrossRef]

Kvantovaya Elektron. (Moscow) (1)

G. V. Perevedentseva, P. A. Khandokhin, Ya. I. Khanin, Kvantovaya Elektron. (Moscow) 7, 128 (1980) [Sov. J. Qunatum Electron. 10, 71 (1980)].

Opt. Lett. (1)

Phys. Rev. (1)

W. E. Lamb, Phys. Rev. 134, A1429 (1964);see also M. Sargent, M. O. Scully, W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974).
[CrossRef]

Phys. Rev. A (6)

L. M. Menegozzi, W. E. Lamb, Phys. Rev. A 8, 2103 (1973).
[CrossRef]

W. R. Christian, L. Mandel, Phys. Rev. A 34, 3932 (1986).There is an error in the solution to the Fokker-Planck equation [Eqs. (14) and (15)] given in that paper, in that the solution and subsequently derived equations are correct at line center with a1 = a2 and |R1| = |R2|, which is what we assume here.
[CrossRef] [PubMed]

M. M-Tehrani, L. Mandel, Phys. Rev. A 17, 677, 694 (1978).
[CrossRef]

J. D. Cresser, W. H. Louisell, P. Meystre, W. Schleich, M. O. Scully, Phys. Rev. A 25, 2214 (1982).
[CrossRef]

J. D. Cresser, D. Hammons, W. H. Louisell, P. Meystre, H. Risken, Phys. Rev. A 25, 2226 (1982).
[CrossRef]

S. Singh, Phys. Rev. A 23, 837 (1981);Phys. Rep. 108, 217 (1984).
[CrossRef]

Physica (2)

D. Kühlke, G. Jetschke, Physica 106C, 287 (1981).

D. Kühlke, H. Horak, Physica 111C, 111 (1981).

Rev. Mod. Phys. (1)

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, M. O. Scully, Rev. Mod. Phys. 57, 61 (1985).
[CrossRef]

Zh. Eksp. Teor. Fiz. (2)

E. L. Klochan, L. S. Kornienko, N. V. Kravtsov, E. G. Lariontsev, A. N. Shelaev, Zh. Eksp. Teor. Fiz. 65, 1344 (1973) [Sov. Phys. JETP 38, 669 (1974)].

I. P. Efanova, E. G. Lariontsev, Zh. Eksp. Teor. Fiz. 55, 1532 (1968)[Sov. Phys. JETP 28, 802 (1969)].

Other (2)

F. Aronowitz, in Laser Applications, M. Ross, ed. (Academic, New York, 1971), p. 133.

See H. Haken, in Laser Theory, Vol. XXV 2C of Encyclopedia of Physics, S. Flügge, ed. (Springer-Verlag, Berlin, 1970).

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Figures (1)

Fig. 1
Fig. 1

Plots of the normalized intensity cross correlation λ12 as a function of mean light intensity 〈I〉 for several different values of the cavity-loss rate κ in reciprocal seconds. The dotted curve corresponds to a laser without backscattering.

Equations (17)

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

˙ 1 ( t ) = [ A κ 1 B ( | 1 ( t ) | 2 + ξ | 2 ( t ) | 2 ) ] 1 ( t ) , ˙ 2 ( t ) = [ A κ 2 B ( | 2 ( t ) | 2 + ξ | 1 ( t ) | 2 ) ] 2 ( t ) ,
A D 0 | μ 12 | 2 ω 0 2 0 γ , B = 2 D 0 | μ 12 | 4 ω 0 F 0 3 γ 2 γ 11 = ( 4 | μ 12 | 2 F 2 γ 11 γ ) A G A
q i * ( t ) q j ( t ) = 4 δ i j K A δ ( t t ) ( i , j = 1 , 2 ) .
˙ 1 ( t ) = [ A κ 1 B ( | 1 | 2 + | 2 | 2 ) ] 1 + 1 2 + q 1 , ˙ ( t ) = [ A κ 2 B ( | 2 | 2 + | 1 | 2 ) ] 2 + 2 1 + q 2 .
j = ( K / G ) 1 / 4 E j ( j = 1 , 2 ) , t = ( K G ) 1 / 2 A 1 t , A κ j = ( K G ) 1 / 2 A a j ( j = 1 , 2 ) , q j = K 3 / 4 G 1 / 4 A q j ( j = 1 , 2 ) , j = ( K G ) 1 / 2 A R j ( j = 1 , 2 )
d E 1 d t = [ a 1 ( | E 1 | 2 + | E 2 | 2 ) ] E 1 + R 1 E 2 + q 1 , d E 2 d t = [ a 2 ( | E 2 | 2 + | E 1 | 2 ) ] E 2 + R 2 E 1 + q 2 .
q i * ( t ) q j ( t ) = 4 δ i j δ ( t t ) .
x 1 + i x 2 = E 1 , x 3 + i x 4 = E 2 .
p ( x , t ) t = j = 1 4 x j ( A j p ) + 1 2 j = 1 4 k = 1 4 2 x j x k ( D j k p ) ,
D j k = 2 δ j k ( j , k = 1 , 2 , 3 , 4 )
A 1 = [ a 1 r ( x 1 2 + x 2 2 + x 3 2 + x 4 2 ) ] x 1 + r 1 x 3 r 2 x 4 , A 2 = [ a 1 r ( x 1 2 + x 2 2 + x 3 2 + x 4 2 ) ] x 2 + r 1 x 4 + r 2 x 3 , A 3 = [ a 2 r ( x 3 2 + x 4 2 + x 1 2 + x 2 2 ) ] x 3 + r 3 x 1 r 4 x 2 , A 4 = [ a 2 r ( x 3 2 + x 4 2 + x 1 2 + x 2 2 ) ] x 4 + r 3 x 2 + r 4 x 1 ,
R 1 = r 1 + i r 2 , R 2 = r 3 + i r 4 .
R j = ( j / A ) 1 ( K G ) 1 / 2 ,
R j = j / κ j ( K G ) 1 / 2 .
P ( I 1 , I 2 ) = π 2 Q 1 I 0 ( R I 1 1 / 2 I 2 1 / 2 ) × exp [ a ( I 1 + I 2 ) / 2 ( 1 / 4 ) ( I 1 2 + I 2 2 + 2 I 1 I 2 ) ] ,
λ 12 I 1 I 2 I 1 I 2 1 = 1 I 1 I 2 0 I 1 I 2 P ( I 1 , I 2 ) d I 1 d I 2 1
/ ( K G ) 1 / 2 2 × 10 7 sec 1 .

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