Abstract

Modulational instability of continuous waves in transmission systems with semiconductor optical amplifiers and in-line filters was theoretically and experimentally analyzed. Stable and unstable domains were identified. Even in the normal-dispersion region, modulationally unstable domains can exist for an appropriate filter detuning.

© 2000 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Hasegawa and W. F. Brinkman, IEEE J. Quantum Electron. QE-16, 694 (1980).
    [CrossRef]
  2. A. Hasegawa, Opt. Lett. 9, 288 (1984); K. Tai, A. Hasegawa, and A. Tomita, Phys. Rev. Lett. 56, 135 (1986).
    [CrossRef] [PubMed]
  3. M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993).
    [CrossRef]
  4. A. Mecozzi, Opt. Lett. 20, 1616 (1995); S. Wabnitz, Opt. Lett. 20, 1979 (1995); S. K. Turitsyn, Phys. Rev. EPLEEE8 54, 3154 (1996).
    [CrossRef] [PubMed]
  5. V. S. Grigoryan, Opt. Lett. 21, 1882 (1996); E. V. Vanin, A. I. Korytin, A. M. Sergeev, D. Anderson, and M. Lisak, Phys. Rev. A 49, 2806 (1994); N. N. Akhmediev, M. J. Lederer, and B. Luther-Davies, Phys. Rev. E PLEEE8 57, 3664 (1998); J. M. Soto-Crespo and N. N. Akhmediev, J. Opt. Soc. Am. B JOBPDE 16, 674 (1999); M. Gölles, I. M. Uzunov, and F. Lederer, J. Opt. Soc. Am. BJOBPDE 16, 689 (1999).
    [CrossRef] [PubMed]
  6. G. Onishchukov, V. Lokhnygin, A. Shipulin, and P. Riedel, Electron. Lett. 34, 1597 (1998).
    [CrossRef]
  7. J. J. E. Reid, L. Cucala, M. Ferreira, M. Settembre, H. Haunstein, and R. C. J. Smets, in Technical Digest of European Conference on Optical Communication '98 (IEEE Lasers and Electro-Optics Society, Madrid, 1998), paper WdD01.
  8. M. Settembre, F. Matera, V. Hägele, I. Gabitov, A. W. Mattheus, and S. Turitsyn, J. Lightwave Technol. 15, 962 (1997).
    [CrossRef]
  9. G. Onishchukov, V. Lokhnygin, A. Shipulin, and M. Gölles, in Digest of Optical Amplifiers and Their Applications (Optical Society of America, Washington, D.C., 1999), paper ThD4.
  10. M. Desaix, D. Anderson, and M. Lisak, Opt. Lett. 15, 1285 (1990).
    [CrossRef] [PubMed]
  11. F. Kh. Abdullaev, S. A. Darmanyan, and P. Khabibulaev, Optical Solitons (Springer-Verlag, Berlin, 1993).
    [CrossRef]

1998

G. Onishchukov, V. Lokhnygin, A. Shipulin, and P. Riedel, Electron. Lett. 34, 1597 (1998).
[CrossRef]

1997

M. Settembre, F. Matera, V. Hägele, I. Gabitov, A. W. Mattheus, and S. Turitsyn, J. Lightwave Technol. 15, 962 (1997).
[CrossRef]

1996

V. S. Grigoryan, Opt. Lett. 21, 1882 (1996); E. V. Vanin, A. I. Korytin, A. M. Sergeev, D. Anderson, and M. Lisak, Phys. Rev. A 49, 2806 (1994); N. N. Akhmediev, M. J. Lederer, and B. Luther-Davies, Phys. Rev. E PLEEE8 57, 3664 (1998); J. M. Soto-Crespo and N. N. Akhmediev, J. Opt. Soc. Am. B JOBPDE 16, 674 (1999); M. Gölles, I. M. Uzunov, and F. Lederer, J. Opt. Soc. Am. BJOBPDE 16, 689 (1999).
[CrossRef] [PubMed]

1995

1993

M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993).
[CrossRef]

1990

1984

A. Hasegawa, Opt. Lett. 9, 288 (1984); K. Tai, A. Hasegawa, and A. Tomita, Phys. Rev. Lett. 56, 135 (1986).
[CrossRef] [PubMed]

1980

A. Hasegawa and W. F. Brinkman, IEEE J. Quantum Electron. QE-16, 694 (1980).
[CrossRef]

Abdullaev, F. Kh.

F. Kh. Abdullaev, S. A. Darmanyan, and P. Khabibulaev, Optical Solitons (Springer-Verlag, Berlin, 1993).
[CrossRef]

Akhmediev, N. N.

V. S. Grigoryan, Opt. Lett. 21, 1882 (1996); E. V. Vanin, A. I. Korytin, A. M. Sergeev, D. Anderson, and M. Lisak, Phys. Rev. A 49, 2806 (1994); N. N. Akhmediev, M. J. Lederer, and B. Luther-Davies, Phys. Rev. E PLEEE8 57, 3664 (1998); J. M. Soto-Crespo and N. N. Akhmediev, J. Opt. Soc. Am. B JOBPDE 16, 674 (1999); M. Gölles, I. M. Uzunov, and F. Lederer, J. Opt. Soc. Am. BJOBPDE 16, 689 (1999).
[CrossRef] [PubMed]

V. S. Grigoryan, Opt. Lett. 21, 1882 (1996); E. V. Vanin, A. I. Korytin, A. M. Sergeev, D. Anderson, and M. Lisak, Phys. Rev. A 49, 2806 (1994); N. N. Akhmediev, M. J. Lederer, and B. Luther-Davies, Phys. Rev. E PLEEE8 57, 3664 (1998); J. M. Soto-Crespo and N. N. Akhmediev, J. Opt. Soc. Am. B JOBPDE 16, 674 (1999); M. Gölles, I. M. Uzunov, and F. Lederer, J. Opt. Soc. Am. BJOBPDE 16, 689 (1999).
[CrossRef] [PubMed]

Anderson, D.

V. S. Grigoryan, Opt. Lett. 21, 1882 (1996); E. V. Vanin, A. I. Korytin, A. M. Sergeev, D. Anderson, and M. Lisak, Phys. Rev. A 49, 2806 (1994); N. N. Akhmediev, M. J. Lederer, and B. Luther-Davies, Phys. Rev. E PLEEE8 57, 3664 (1998); J. M. Soto-Crespo and N. N. Akhmediev, J. Opt. Soc. Am. B JOBPDE 16, 674 (1999); M. Gölles, I. M. Uzunov, and F. Lederer, J. Opt. Soc. Am. BJOBPDE 16, 689 (1999).
[CrossRef] [PubMed]

M. Desaix, D. Anderson, and M. Lisak, Opt. Lett. 15, 1285 (1990).
[CrossRef] [PubMed]

Brinkman, W. F.

A. Hasegawa and W. F. Brinkman, IEEE J. Quantum Electron. QE-16, 694 (1980).
[CrossRef]

Cross, M. C.

M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993).
[CrossRef]

Cucala, L.

J. J. E. Reid, L. Cucala, M. Ferreira, M. Settembre, H. Haunstein, and R. C. J. Smets, in Technical Digest of European Conference on Optical Communication '98 (IEEE Lasers and Electro-Optics Society, Madrid, 1998), paper WdD01.

Darmanyan, S. A.

F. Kh. Abdullaev, S. A. Darmanyan, and P. Khabibulaev, Optical Solitons (Springer-Verlag, Berlin, 1993).
[CrossRef]

Desaix, M.

Ferreira, M.

J. J. E. Reid, L. Cucala, M. Ferreira, M. Settembre, H. Haunstein, and R. C. J. Smets, in Technical Digest of European Conference on Optical Communication '98 (IEEE Lasers and Electro-Optics Society, Madrid, 1998), paper WdD01.

Gabitov, I.

M. Settembre, F. Matera, V. Hägele, I. Gabitov, A. W. Mattheus, and S. Turitsyn, J. Lightwave Technol. 15, 962 (1997).
[CrossRef]

Gölles, M.

V. S. Grigoryan, Opt. Lett. 21, 1882 (1996); E. V. Vanin, A. I. Korytin, A. M. Sergeev, D. Anderson, and M. Lisak, Phys. Rev. A 49, 2806 (1994); N. N. Akhmediev, M. J. Lederer, and B. Luther-Davies, Phys. Rev. E PLEEE8 57, 3664 (1998); J. M. Soto-Crespo and N. N. Akhmediev, J. Opt. Soc. Am. B JOBPDE 16, 674 (1999); M. Gölles, I. M. Uzunov, and F. Lederer, J. Opt. Soc. Am. BJOBPDE 16, 689 (1999).
[CrossRef] [PubMed]

G. Onishchukov, V. Lokhnygin, A. Shipulin, and M. Gölles, in Digest of Optical Amplifiers and Their Applications (Optical Society of America, Washington, D.C., 1999), paper ThD4.

Grigoryan, V. S.

V. S. Grigoryan, Opt. Lett. 21, 1882 (1996); E. V. Vanin, A. I. Korytin, A. M. Sergeev, D. Anderson, and M. Lisak, Phys. Rev. A 49, 2806 (1994); N. N. Akhmediev, M. J. Lederer, and B. Luther-Davies, Phys. Rev. E PLEEE8 57, 3664 (1998); J. M. Soto-Crespo and N. N. Akhmediev, J. Opt. Soc. Am. B JOBPDE 16, 674 (1999); M. Gölles, I. M. Uzunov, and F. Lederer, J. Opt. Soc. Am. BJOBPDE 16, 689 (1999).
[CrossRef] [PubMed]

Hägele, V.

M. Settembre, F. Matera, V. Hägele, I. Gabitov, A. W. Mattheus, and S. Turitsyn, J. Lightwave Technol. 15, 962 (1997).
[CrossRef]

Hasegawa, A.

A. Hasegawa, Opt. Lett. 9, 288 (1984); K. Tai, A. Hasegawa, and A. Tomita, Phys. Rev. Lett. 56, 135 (1986).
[CrossRef] [PubMed]

A. Hasegawa, Opt. Lett. 9, 288 (1984); K. Tai, A. Hasegawa, and A. Tomita, Phys. Rev. Lett. 56, 135 (1986).
[CrossRef] [PubMed]

A. Hasegawa and W. F. Brinkman, IEEE J. Quantum Electron. QE-16, 694 (1980).
[CrossRef]

Haunstein, H.

J. J. E. Reid, L. Cucala, M. Ferreira, M. Settembre, H. Haunstein, and R. C. J. Smets, in Technical Digest of European Conference on Optical Communication '98 (IEEE Lasers and Electro-Optics Society, Madrid, 1998), paper WdD01.

Hohenberg, P. C.

M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993).
[CrossRef]

Khabibulaev, P.

F. Kh. Abdullaev, S. A. Darmanyan, and P. Khabibulaev, Optical Solitons (Springer-Verlag, Berlin, 1993).
[CrossRef]

Korytin, A. I.

V. S. Grigoryan, Opt. Lett. 21, 1882 (1996); E. V. Vanin, A. I. Korytin, A. M. Sergeev, D. Anderson, and M. Lisak, Phys. Rev. A 49, 2806 (1994); N. N. Akhmediev, M. J. Lederer, and B. Luther-Davies, Phys. Rev. E PLEEE8 57, 3664 (1998); J. M. Soto-Crespo and N. N. Akhmediev, J. Opt. Soc. Am. B JOBPDE 16, 674 (1999); M. Gölles, I. M. Uzunov, and F. Lederer, J. Opt. Soc. Am. BJOBPDE 16, 689 (1999).
[CrossRef] [PubMed]

Lederer, F.

V. S. Grigoryan, Opt. Lett. 21, 1882 (1996); E. V. Vanin, A. I. Korytin, A. M. Sergeev, D. Anderson, and M. Lisak, Phys. Rev. A 49, 2806 (1994); N. N. Akhmediev, M. J. Lederer, and B. Luther-Davies, Phys. Rev. E PLEEE8 57, 3664 (1998); J. M. Soto-Crespo and N. N. Akhmediev, J. Opt. Soc. Am. B JOBPDE 16, 674 (1999); M. Gölles, I. M. Uzunov, and F. Lederer, J. Opt. Soc. Am. BJOBPDE 16, 689 (1999).
[CrossRef] [PubMed]

Lederer, M. J.

V. S. Grigoryan, Opt. Lett. 21, 1882 (1996); E. V. Vanin, A. I. Korytin, A. M. Sergeev, D. Anderson, and M. Lisak, Phys. Rev. A 49, 2806 (1994); N. N. Akhmediev, M. J. Lederer, and B. Luther-Davies, Phys. Rev. E PLEEE8 57, 3664 (1998); J. M. Soto-Crespo and N. N. Akhmediev, J. Opt. Soc. Am. B JOBPDE 16, 674 (1999); M. Gölles, I. M. Uzunov, and F. Lederer, J. Opt. Soc. Am. BJOBPDE 16, 689 (1999).
[CrossRef] [PubMed]

Lisak, M.

V. S. Grigoryan, Opt. Lett. 21, 1882 (1996); E. V. Vanin, A. I. Korytin, A. M. Sergeev, D. Anderson, and M. Lisak, Phys. Rev. A 49, 2806 (1994); N. N. Akhmediev, M. J. Lederer, and B. Luther-Davies, Phys. Rev. E PLEEE8 57, 3664 (1998); J. M. Soto-Crespo and N. N. Akhmediev, J. Opt. Soc. Am. B JOBPDE 16, 674 (1999); M. Gölles, I. M. Uzunov, and F. Lederer, J. Opt. Soc. Am. BJOBPDE 16, 689 (1999).
[CrossRef] [PubMed]

M. Desaix, D. Anderson, and M. Lisak, Opt. Lett. 15, 1285 (1990).
[CrossRef] [PubMed]

Lokhnygin, V.

G. Onishchukov, V. Lokhnygin, A. Shipulin, and P. Riedel, Electron. Lett. 34, 1597 (1998).
[CrossRef]

G. Onishchukov, V. Lokhnygin, A. Shipulin, and M. Gölles, in Digest of Optical Amplifiers and Their Applications (Optical Society of America, Washington, D.C., 1999), paper ThD4.

Luther-Davies, B.

V. S. Grigoryan, Opt. Lett. 21, 1882 (1996); E. V. Vanin, A. I. Korytin, A. M. Sergeev, D. Anderson, and M. Lisak, Phys. Rev. A 49, 2806 (1994); N. N. Akhmediev, M. J. Lederer, and B. Luther-Davies, Phys. Rev. E PLEEE8 57, 3664 (1998); J. M. Soto-Crespo and N. N. Akhmediev, J. Opt. Soc. Am. B JOBPDE 16, 674 (1999); M. Gölles, I. M. Uzunov, and F. Lederer, J. Opt. Soc. Am. BJOBPDE 16, 689 (1999).
[CrossRef] [PubMed]

Matera, F.

M. Settembre, F. Matera, V. Hägele, I. Gabitov, A. W. Mattheus, and S. Turitsyn, J. Lightwave Technol. 15, 962 (1997).
[CrossRef]

Mattheus, A. W.

M. Settembre, F. Matera, V. Hägele, I. Gabitov, A. W. Mattheus, and S. Turitsyn, J. Lightwave Technol. 15, 962 (1997).
[CrossRef]

Mecozzi, A.

Onishchukov, G.

G. Onishchukov, V. Lokhnygin, A. Shipulin, and P. Riedel, Electron. Lett. 34, 1597 (1998).
[CrossRef]

G. Onishchukov, V. Lokhnygin, A. Shipulin, and M. Gölles, in Digest of Optical Amplifiers and Their Applications (Optical Society of America, Washington, D.C., 1999), paper ThD4.

Reid, J. J. E.

J. J. E. Reid, L. Cucala, M. Ferreira, M. Settembre, H. Haunstein, and R. C. J. Smets, in Technical Digest of European Conference on Optical Communication '98 (IEEE Lasers and Electro-Optics Society, Madrid, 1998), paper WdD01.

Riedel, P.

G. Onishchukov, V. Lokhnygin, A. Shipulin, and P. Riedel, Electron. Lett. 34, 1597 (1998).
[CrossRef]

Sergeev, A. M.

V. S. Grigoryan, Opt. Lett. 21, 1882 (1996); E. V. Vanin, A. I. Korytin, A. M. Sergeev, D. Anderson, and M. Lisak, Phys. Rev. A 49, 2806 (1994); N. N. Akhmediev, M. J. Lederer, and B. Luther-Davies, Phys. Rev. E PLEEE8 57, 3664 (1998); J. M. Soto-Crespo and N. N. Akhmediev, J. Opt. Soc. Am. B JOBPDE 16, 674 (1999); M. Gölles, I. M. Uzunov, and F. Lederer, J. Opt. Soc. Am. BJOBPDE 16, 689 (1999).
[CrossRef] [PubMed]

Settembre, M.

M. Settembre, F. Matera, V. Hägele, I. Gabitov, A. W. Mattheus, and S. Turitsyn, J. Lightwave Technol. 15, 962 (1997).
[CrossRef]

J. J. E. Reid, L. Cucala, M. Ferreira, M. Settembre, H. Haunstein, and R. C. J. Smets, in Technical Digest of European Conference on Optical Communication '98 (IEEE Lasers and Electro-Optics Society, Madrid, 1998), paper WdD01.

Shipulin, A.

G. Onishchukov, V. Lokhnygin, A. Shipulin, and P. Riedel, Electron. Lett. 34, 1597 (1998).
[CrossRef]

G. Onishchukov, V. Lokhnygin, A. Shipulin, and M. Gölles, in Digest of Optical Amplifiers and Their Applications (Optical Society of America, Washington, D.C., 1999), paper ThD4.

Smets, R. C. J.

J. J. E. Reid, L. Cucala, M. Ferreira, M. Settembre, H. Haunstein, and R. C. J. Smets, in Technical Digest of European Conference on Optical Communication '98 (IEEE Lasers and Electro-Optics Society, Madrid, 1998), paper WdD01.

Soto-Crespo, J. M.

V. S. Grigoryan, Opt. Lett. 21, 1882 (1996); E. V. Vanin, A. I. Korytin, A. M. Sergeev, D. Anderson, and M. Lisak, Phys. Rev. A 49, 2806 (1994); N. N. Akhmediev, M. J. Lederer, and B. Luther-Davies, Phys. Rev. E PLEEE8 57, 3664 (1998); J. M. Soto-Crespo and N. N. Akhmediev, J. Opt. Soc. Am. B JOBPDE 16, 674 (1999); M. Gölles, I. M. Uzunov, and F. Lederer, J. Opt. Soc. Am. BJOBPDE 16, 689 (1999).
[CrossRef] [PubMed]

Tai, K.

A. Hasegawa, Opt. Lett. 9, 288 (1984); K. Tai, A. Hasegawa, and A. Tomita, Phys. Rev. Lett. 56, 135 (1986).
[CrossRef] [PubMed]

Tomita, A.

A. Hasegawa, Opt. Lett. 9, 288 (1984); K. Tai, A. Hasegawa, and A. Tomita, Phys. Rev. Lett. 56, 135 (1986).
[CrossRef] [PubMed]

Turitsyn, S.

M. Settembre, F. Matera, V. Hägele, I. Gabitov, A. W. Mattheus, and S. Turitsyn, J. Lightwave Technol. 15, 962 (1997).
[CrossRef]

Turitsyn, S. K.

Uzunov, I. M.

V. S. Grigoryan, Opt. Lett. 21, 1882 (1996); E. V. Vanin, A. I. Korytin, A. M. Sergeev, D. Anderson, and M. Lisak, Phys. Rev. A 49, 2806 (1994); N. N. Akhmediev, M. J. Lederer, and B. Luther-Davies, Phys. Rev. E PLEEE8 57, 3664 (1998); J. M. Soto-Crespo and N. N. Akhmediev, J. Opt. Soc. Am. B JOBPDE 16, 674 (1999); M. Gölles, I. M. Uzunov, and F. Lederer, J. Opt. Soc. Am. BJOBPDE 16, 689 (1999).
[CrossRef] [PubMed]

Vanin, E. V.

V. S. Grigoryan, Opt. Lett. 21, 1882 (1996); E. V. Vanin, A. I. Korytin, A. M. Sergeev, D. Anderson, and M. Lisak, Phys. Rev. A 49, 2806 (1994); N. N. Akhmediev, M. J. Lederer, and B. Luther-Davies, Phys. Rev. E PLEEE8 57, 3664 (1998); J. M. Soto-Crespo and N. N. Akhmediev, J. Opt. Soc. Am. B JOBPDE 16, 674 (1999); M. Gölles, I. M. Uzunov, and F. Lederer, J. Opt. Soc. Am. BJOBPDE 16, 689 (1999).
[CrossRef] [PubMed]

Wabnitz, S.

Electron. Lett.

G. Onishchukov, V. Lokhnygin, A. Shipulin, and P. Riedel, Electron. Lett. 34, 1597 (1998).
[CrossRef]

IEEE J. Quantum Electron.

A. Hasegawa and W. F. Brinkman, IEEE J. Quantum Electron. QE-16, 694 (1980).
[CrossRef]

J. Lightwave Technol.

M. Settembre, F. Matera, V. Hägele, I. Gabitov, A. W. Mattheus, and S. Turitsyn, J. Lightwave Technol. 15, 962 (1997).
[CrossRef]

Opt. Lett.

A. Hasegawa, Opt. Lett. 9, 288 (1984); K. Tai, A. Hasegawa, and A. Tomita, Phys. Rev. Lett. 56, 135 (1986).
[CrossRef] [PubMed]

V. S. Grigoryan, Opt. Lett. 21, 1882 (1996); E. V. Vanin, A. I. Korytin, A. M. Sergeev, D. Anderson, and M. Lisak, Phys. Rev. A 49, 2806 (1994); N. N. Akhmediev, M. J. Lederer, and B. Luther-Davies, Phys. Rev. E PLEEE8 57, 3664 (1998); J. M. Soto-Crespo and N. N. Akhmediev, J. Opt. Soc. Am. B JOBPDE 16, 674 (1999); M. Gölles, I. M. Uzunov, and F. Lederer, J. Opt. Soc. Am. BJOBPDE 16, 689 (1999).
[CrossRef] [PubMed]

Opt. Lett.

Rev. Mod. Phys.

M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993).
[CrossRef]

Other

J. J. E. Reid, L. Cucala, M. Ferreira, M. Settembre, H. Haunstein, and R. C. J. Smets, in Technical Digest of European Conference on Optical Communication '98 (IEEE Lasers and Electro-Optics Society, Madrid, 1998), paper WdD01.

G. Onishchukov, V. Lokhnygin, A. Shipulin, and M. Gölles, in Digest of Optical Amplifiers and Their Applications (Optical Society of America, Washington, D.C., 1999), paper ThD4.

F. Kh. Abdullaev, S. A. Darmanyan, and P. Khabibulaev, Optical Solitons (Springer-Verlag, Berlin, 1993).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

MI gain [dB/(25 km)] as a function of filter detuning λF and modulational frequency ωM, with βF=0.091 ps2/km and ρ=0.026/kmpJ for fiber input powers of (a) 1 mW and (b) 10 mW. ωN/2π=40 GHz and λN=0.25 nm.

Fig. 2
Fig. 2

MI gain [dB/(25 km)] as a function of filter detuning λF and modulational frequency ωM with βF=0.091 ps2/km, χ=3.1/W km, and ρ=0.026/kmpJ for fiber input power 1 mW (left-hand column) and 10 mW (right-hand column). ωN/2π=40 GHz and λN=0.25 nm.

Fig. 3
Fig. 3

Comparison between (top) experimental and (bottom) numerical evolution of spectral power cm2 of a 3-mW signal at the blue BPF side in the normal-dispersion regime according to Eq. (4).

Fig. 4
Fig. 4

Propagation of an amplitude-modulated cw-signal at normal dispersion, ωS=0, ωM/2π=4.5 GHz, for input powers (a) a0Q02=1 mW and (b) a0Q02=10 mW; t0=3.73 ps and za=25 km.

Equations (5)

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

iQz=-2βFωSQt+βD2+iβF2Qt2-χQ2Q+iδeQ-ρα+iQ-tQ2 exp-t-ttrdt,
iz+2βFωSt-βD2+iβF2t2+χQ02+*=-ρα+iQ02-t+*exp-t-ttrdt.
λ+βFωM22+2ρQ02tr1-iωMtrλ+βFωM2+aa-2χQ02-2ραQ02tr1-iωMtr=0,
idcmdz=iδ-βD2mωM2-iβFmωM-ωS2cm-j,k,l:m=j-k+lχ+ρα+itr1-im-jωMtrcjck*cl.
NSR20dωM exp2zRλωMHRωM0dωMHRωM

Metrics