Abstract

We predict that self-sustained oscillations can be generated in a semiconductor nanocavity in which the dominant effect is band-filling dispersion. Linear stability analysis, validated through numerical integration of a mean-field model, leads us to envisage oscillations typically at tens of gigahertz, with onset at much lower input power (100μW) compared with those driven by two-photon absorption.

© 2013 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. K. Ikeda and O. Akimoto, Phys. Rev. Lett. 48, 617 (1982).
    [CrossRef]
  2. K. Ikeda, H. Daido, and O. Akimoto, Phys. Rev. Lett. 45, 709 (1980).
    [CrossRef]
  3. L. A. Lugiato, L. M. Narducci, D. K. Bandy, and C. A. Pennise, Opt. Commun. 43, 281 (1982).
    [CrossRef]
  4. S. Malaguti, G. Bellanca, A. de Rossi, S. Combrié, and S. Trillo, Phys. Rev. A 83, 051802(R) (2011).
    [CrossRef]
  5. A. Armaroli, S. Malaguti, G. Bellanca, S. Trillo, A. de Rossi, and S. Combrié, Phys. Rev. A 84, 053816 (2011).
    [CrossRef]
  6. V. Grigoriev and F. Biancalana, Phys. Rev. A 83, 043816 (2011).
    [CrossRef]
  7. Y. Dumeige and P. Féron, Phys. Rev. A 84, 043847 (2011).
    [CrossRef]
  8. M. Soltani, S. Yegnanarayanan, Q. Li, A. A. Eftekhar, and A. Adibi, Phys. Rev. A 85, 053819 (2012).
    [CrossRef]
  9. M. Brunstein, A. M. Yacomotti, I. Sagnes, F. Raineri, L. Bigot, and A. Levenson, Phys. Rev. A 85, 031803(R) (2012).
    [CrossRef]
  10. T. Gu, N. Petrone, J. F. McMillan, A. van der Zande, M. Yu, G. Q. Lo, D. L. Kwong, J. Hone, and C. W. Wong, Nat. Photonics 6, 554 (2012).
    [CrossRef]
  11. N. Cazier, X. Checoury, L. D. Haret, and P. Boucaud, Opt. Express 21, 13626 (2013).
    [CrossRef]
  12. A. M. Yacomotti, S. Haddadi, and S. Barbay, Phys. Rev. A 87, 041804(R) (2013).
    [CrossRef]
  13. B. R. Bennet, R. A. Soref, and J.-A. Del Alamo, IEEE J. Quantum Electron. 26, 113 (1990).
    [CrossRef]
  14. K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, Nat. Photonics 4, 477 (2010).
    [CrossRef]
  15. K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, Y. Kawaguchi, R. Takahashi, and M. Notomi, Nat. Photonics 6, 248 (2012).
    [CrossRef]
  16. K. Nozaki, A. Shinya, S. Matsuo, T. Sato, E. Kuramochi, and M. Notomi, Opt. Express 21, 11877 (2013).
    [CrossRef]

2013

2012

K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, Y. Kawaguchi, R. Takahashi, and M. Notomi, Nat. Photonics 6, 248 (2012).
[CrossRef]

M. Soltani, S. Yegnanarayanan, Q. Li, A. A. Eftekhar, and A. Adibi, Phys. Rev. A 85, 053819 (2012).
[CrossRef]

M. Brunstein, A. M. Yacomotti, I. Sagnes, F. Raineri, L. Bigot, and A. Levenson, Phys. Rev. A 85, 031803(R) (2012).
[CrossRef]

T. Gu, N. Petrone, J. F. McMillan, A. van der Zande, M. Yu, G. Q. Lo, D. L. Kwong, J. Hone, and C. W. Wong, Nat. Photonics 6, 554 (2012).
[CrossRef]

2011

S. Malaguti, G. Bellanca, A. de Rossi, S. Combrié, and S. Trillo, Phys. Rev. A 83, 051802(R) (2011).
[CrossRef]

A. Armaroli, S. Malaguti, G. Bellanca, S. Trillo, A. de Rossi, and S. Combrié, Phys. Rev. A 84, 053816 (2011).
[CrossRef]

V. Grigoriev and F. Biancalana, Phys. Rev. A 83, 043816 (2011).
[CrossRef]

Y. Dumeige and P. Féron, Phys. Rev. A 84, 043847 (2011).
[CrossRef]

2010

K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, Nat. Photonics 4, 477 (2010).
[CrossRef]

1990

B. R. Bennet, R. A. Soref, and J.-A. Del Alamo, IEEE J. Quantum Electron. 26, 113 (1990).
[CrossRef]

1982

L. A. Lugiato, L. M. Narducci, D. K. Bandy, and C. A. Pennise, Opt. Commun. 43, 281 (1982).
[CrossRef]

K. Ikeda and O. Akimoto, Phys. Rev. Lett. 48, 617 (1982).
[CrossRef]

1980

K. Ikeda, H. Daido, and O. Akimoto, Phys. Rev. Lett. 45, 709 (1980).
[CrossRef]

Adibi, A.

M. Soltani, S. Yegnanarayanan, Q. Li, A. A. Eftekhar, and A. Adibi, Phys. Rev. A 85, 053819 (2012).
[CrossRef]

Akimoto, O.

K. Ikeda and O. Akimoto, Phys. Rev. Lett. 48, 617 (1982).
[CrossRef]

K. Ikeda, H. Daido, and O. Akimoto, Phys. Rev. Lett. 45, 709 (1980).
[CrossRef]

Armaroli, A.

A. Armaroli, S. Malaguti, G. Bellanca, S. Trillo, A. de Rossi, and S. Combrié, Phys. Rev. A 84, 053816 (2011).
[CrossRef]

Bandy, D. K.

L. A. Lugiato, L. M. Narducci, D. K. Bandy, and C. A. Pennise, Opt. Commun. 43, 281 (1982).
[CrossRef]

Barbay, S.

A. M. Yacomotti, S. Haddadi, and S. Barbay, Phys. Rev. A 87, 041804(R) (2013).
[CrossRef]

Bellanca, G.

S. Malaguti, G. Bellanca, A. de Rossi, S. Combrié, and S. Trillo, Phys. Rev. A 83, 051802(R) (2011).
[CrossRef]

A. Armaroli, S. Malaguti, G. Bellanca, S. Trillo, A. de Rossi, and S. Combrié, Phys. Rev. A 84, 053816 (2011).
[CrossRef]

Bennet, B. R.

B. R. Bennet, R. A. Soref, and J.-A. Del Alamo, IEEE J. Quantum Electron. 26, 113 (1990).
[CrossRef]

Biancalana, F.

V. Grigoriev and F. Biancalana, Phys. Rev. A 83, 043816 (2011).
[CrossRef]

Bigot, L.

M. Brunstein, A. M. Yacomotti, I. Sagnes, F. Raineri, L. Bigot, and A. Levenson, Phys. Rev. A 85, 031803(R) (2012).
[CrossRef]

Boucaud, P.

Brunstein, M.

M. Brunstein, A. M. Yacomotti, I. Sagnes, F. Raineri, L. Bigot, and A. Levenson, Phys. Rev. A 85, 031803(R) (2012).
[CrossRef]

Cazier, N.

Checoury, X.

Combrié, S.

A. Armaroli, S. Malaguti, G. Bellanca, S. Trillo, A. de Rossi, and S. Combrié, Phys. Rev. A 84, 053816 (2011).
[CrossRef]

S. Malaguti, G. Bellanca, A. de Rossi, S. Combrié, and S. Trillo, Phys. Rev. A 83, 051802(R) (2011).
[CrossRef]

Daido, H.

K. Ikeda, H. Daido, and O. Akimoto, Phys. Rev. Lett. 45, 709 (1980).
[CrossRef]

de Rossi, A.

S. Malaguti, G. Bellanca, A. de Rossi, S. Combrié, and S. Trillo, Phys. Rev. A 83, 051802(R) (2011).
[CrossRef]

A. Armaroli, S. Malaguti, G. Bellanca, S. Trillo, A. de Rossi, and S. Combrié, Phys. Rev. A 84, 053816 (2011).
[CrossRef]

Del Alamo, J.-A.

B. R. Bennet, R. A. Soref, and J.-A. Del Alamo, IEEE J. Quantum Electron. 26, 113 (1990).
[CrossRef]

Dumeige, Y.

Y. Dumeige and P. Féron, Phys. Rev. A 84, 043847 (2011).
[CrossRef]

Eftekhar, A. A.

M. Soltani, S. Yegnanarayanan, Q. Li, A. A. Eftekhar, and A. Adibi, Phys. Rev. A 85, 053819 (2012).
[CrossRef]

Féron, P.

Y. Dumeige and P. Féron, Phys. Rev. A 84, 043847 (2011).
[CrossRef]

Grigoriev, V.

V. Grigoriev and F. Biancalana, Phys. Rev. A 83, 043816 (2011).
[CrossRef]

Gu, T.

T. Gu, N. Petrone, J. F. McMillan, A. van der Zande, M. Yu, G. Q. Lo, D. L. Kwong, J. Hone, and C. W. Wong, Nat. Photonics 6, 554 (2012).
[CrossRef]

Haddadi, S.

A. M. Yacomotti, S. Haddadi, and S. Barbay, Phys. Rev. A 87, 041804(R) (2013).
[CrossRef]

Haret, L. D.

Hone, J.

T. Gu, N. Petrone, J. F. McMillan, A. van der Zande, M. Yu, G. Q. Lo, D. L. Kwong, J. Hone, and C. W. Wong, Nat. Photonics 6, 554 (2012).
[CrossRef]

Ikeda, K.

K. Ikeda and O. Akimoto, Phys. Rev. Lett. 48, 617 (1982).
[CrossRef]

K. Ikeda, H. Daido, and O. Akimoto, Phys. Rev. Lett. 45, 709 (1980).
[CrossRef]

Kawaguchi, Y.

K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, Y. Kawaguchi, R. Takahashi, and M. Notomi, Nat. Photonics 6, 248 (2012).
[CrossRef]

Kuramochi, E.

Kwong, D. L.

T. Gu, N. Petrone, J. F. McMillan, A. van der Zande, M. Yu, G. Q. Lo, D. L. Kwong, J. Hone, and C. W. Wong, Nat. Photonics 6, 554 (2012).
[CrossRef]

Levenson, A.

M. Brunstein, A. M. Yacomotti, I. Sagnes, F. Raineri, L. Bigot, and A. Levenson, Phys. Rev. A 85, 031803(R) (2012).
[CrossRef]

Li, Q.

M. Soltani, S. Yegnanarayanan, Q. Li, A. A. Eftekhar, and A. Adibi, Phys. Rev. A 85, 053819 (2012).
[CrossRef]

Lo, G. Q.

T. Gu, N. Petrone, J. F. McMillan, A. van der Zande, M. Yu, G. Q. Lo, D. L. Kwong, J. Hone, and C. W. Wong, Nat. Photonics 6, 554 (2012).
[CrossRef]

Lugiato, L. A.

L. A. Lugiato, L. M. Narducci, D. K. Bandy, and C. A. Pennise, Opt. Commun. 43, 281 (1982).
[CrossRef]

Malaguti, S.

A. Armaroli, S. Malaguti, G. Bellanca, S. Trillo, A. de Rossi, and S. Combrié, Phys. Rev. A 84, 053816 (2011).
[CrossRef]

S. Malaguti, G. Bellanca, A. de Rossi, S. Combrié, and S. Trillo, Phys. Rev. A 83, 051802(R) (2011).
[CrossRef]

Matsuo, S.

K. Nozaki, A. Shinya, S. Matsuo, T. Sato, E. Kuramochi, and M. Notomi, Opt. Express 21, 11877 (2013).
[CrossRef]

K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, Y. Kawaguchi, R. Takahashi, and M. Notomi, Nat. Photonics 6, 248 (2012).
[CrossRef]

K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, Nat. Photonics 4, 477 (2010).
[CrossRef]

McMillan, J. F.

T. Gu, N. Petrone, J. F. McMillan, A. van der Zande, M. Yu, G. Q. Lo, D. L. Kwong, J. Hone, and C. W. Wong, Nat. Photonics 6, 554 (2012).
[CrossRef]

Narducci, L. M.

L. A. Lugiato, L. M. Narducci, D. K. Bandy, and C. A. Pennise, Opt. Commun. 43, 281 (1982).
[CrossRef]

Notomi, M.

K. Nozaki, A. Shinya, S. Matsuo, T. Sato, E. Kuramochi, and M. Notomi, Opt. Express 21, 11877 (2013).
[CrossRef]

K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, Y. Kawaguchi, R. Takahashi, and M. Notomi, Nat. Photonics 6, 248 (2012).
[CrossRef]

K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, Nat. Photonics 4, 477 (2010).
[CrossRef]

Nozaki, K.

K. Nozaki, A. Shinya, S. Matsuo, T. Sato, E. Kuramochi, and M. Notomi, Opt. Express 21, 11877 (2013).
[CrossRef]

K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, Y. Kawaguchi, R. Takahashi, and M. Notomi, Nat. Photonics 6, 248 (2012).
[CrossRef]

K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, Nat. Photonics 4, 477 (2010).
[CrossRef]

Pennise, C. A.

L. A. Lugiato, L. M. Narducci, D. K. Bandy, and C. A. Pennise, Opt. Commun. 43, 281 (1982).
[CrossRef]

Petrone, N.

T. Gu, N. Petrone, J. F. McMillan, A. van der Zande, M. Yu, G. Q. Lo, D. L. Kwong, J. Hone, and C. W. Wong, Nat. Photonics 6, 554 (2012).
[CrossRef]

Raineri, F.

M. Brunstein, A. M. Yacomotti, I. Sagnes, F. Raineri, L. Bigot, and A. Levenson, Phys. Rev. A 85, 031803(R) (2012).
[CrossRef]

Sagnes, I.

M. Brunstein, A. M. Yacomotti, I. Sagnes, F. Raineri, L. Bigot, and A. Levenson, Phys. Rev. A 85, 031803(R) (2012).
[CrossRef]

Sato, T.

K. Nozaki, A. Shinya, S. Matsuo, T. Sato, E. Kuramochi, and M. Notomi, Opt. Express 21, 11877 (2013).
[CrossRef]

K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, Y. Kawaguchi, R. Takahashi, and M. Notomi, Nat. Photonics 6, 248 (2012).
[CrossRef]

K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, Nat. Photonics 4, 477 (2010).
[CrossRef]

Segawa, T.

K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, Y. Kawaguchi, R. Takahashi, and M. Notomi, Nat. Photonics 6, 248 (2012).
[CrossRef]

Shinya, A.

K. Nozaki, A. Shinya, S. Matsuo, T. Sato, E. Kuramochi, and M. Notomi, Opt. Express 21, 11877 (2013).
[CrossRef]

K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, Y. Kawaguchi, R. Takahashi, and M. Notomi, Nat. Photonics 6, 248 (2012).
[CrossRef]

K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, Nat. Photonics 4, 477 (2010).
[CrossRef]

Soltani, M.

M. Soltani, S. Yegnanarayanan, Q. Li, A. A. Eftekhar, and A. Adibi, Phys. Rev. A 85, 053819 (2012).
[CrossRef]

Soref, R. A.

B. R. Bennet, R. A. Soref, and J.-A. Del Alamo, IEEE J. Quantum Electron. 26, 113 (1990).
[CrossRef]

Suzaki, Y.

K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, Y. Kawaguchi, R. Takahashi, and M. Notomi, Nat. Photonics 6, 248 (2012).
[CrossRef]

Takahashi, R.

K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, Y. Kawaguchi, R. Takahashi, and M. Notomi, Nat. Photonics 6, 248 (2012).
[CrossRef]

Tanabe, T.

K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, Nat. Photonics 4, 477 (2010).
[CrossRef]

Taniyama, H.

K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, Nat. Photonics 4, 477 (2010).
[CrossRef]

Trillo, S.

A. Armaroli, S. Malaguti, G. Bellanca, S. Trillo, A. de Rossi, and S. Combrié, Phys. Rev. A 84, 053816 (2011).
[CrossRef]

S. Malaguti, G. Bellanca, A. de Rossi, S. Combrié, and S. Trillo, Phys. Rev. A 83, 051802(R) (2011).
[CrossRef]

van der Zande, A.

T. Gu, N. Petrone, J. F. McMillan, A. van der Zande, M. Yu, G. Q. Lo, D. L. Kwong, J. Hone, and C. W. Wong, Nat. Photonics 6, 554 (2012).
[CrossRef]

Wong, C. W.

T. Gu, N. Petrone, J. F. McMillan, A. van der Zande, M. Yu, G. Q. Lo, D. L. Kwong, J. Hone, and C. W. Wong, Nat. Photonics 6, 554 (2012).
[CrossRef]

Yacomotti, A. M.

A. M. Yacomotti, S. Haddadi, and S. Barbay, Phys. Rev. A 87, 041804(R) (2013).
[CrossRef]

M. Brunstein, A. M. Yacomotti, I. Sagnes, F. Raineri, L. Bigot, and A. Levenson, Phys. Rev. A 85, 031803(R) (2012).
[CrossRef]

Yegnanarayanan, S.

M. Soltani, S. Yegnanarayanan, Q. Li, A. A. Eftekhar, and A. Adibi, Phys. Rev. A 85, 053819 (2012).
[CrossRef]

Yu, M.

T. Gu, N. Petrone, J. F. McMillan, A. van der Zande, M. Yu, G. Q. Lo, D. L. Kwong, J. Hone, and C. W. Wong, Nat. Photonics 6, 554 (2012).
[CrossRef]

IEEE J. Quantum Electron.

B. R. Bennet, R. A. Soref, and J.-A. Del Alamo, IEEE J. Quantum Electron. 26, 113 (1990).
[CrossRef]

Nat. Photonics

K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, Nat. Photonics 4, 477 (2010).
[CrossRef]

K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, Y. Kawaguchi, R. Takahashi, and M. Notomi, Nat. Photonics 6, 248 (2012).
[CrossRef]

T. Gu, N. Petrone, J. F. McMillan, A. van der Zande, M. Yu, G. Q. Lo, D. L. Kwong, J. Hone, and C. W. Wong, Nat. Photonics 6, 554 (2012).
[CrossRef]

Opt. Commun.

L. A. Lugiato, L. M. Narducci, D. K. Bandy, and C. A. Pennise, Opt. Commun. 43, 281 (1982).
[CrossRef]

Opt. Express

Phys. Rev. A

S. Malaguti, G. Bellanca, A. de Rossi, S. Combrié, and S. Trillo, Phys. Rev. A 83, 051802(R) (2011).
[CrossRef]

A. Armaroli, S. Malaguti, G. Bellanca, S. Trillo, A. de Rossi, and S. Combrié, Phys. Rev. A 84, 053816 (2011).
[CrossRef]

V. Grigoriev and F. Biancalana, Phys. Rev. A 83, 043816 (2011).
[CrossRef]

Y. Dumeige and P. Féron, Phys. Rev. A 84, 043847 (2011).
[CrossRef]

M. Soltani, S. Yegnanarayanan, Q. Li, A. A. Eftekhar, and A. Adibi, Phys. Rev. A 85, 053819 (2012).
[CrossRef]

M. Brunstein, A. M. Yacomotti, I. Sagnes, F. Raineri, L. Bigot, and A. Levenson, Phys. Rev. A 85, 031803(R) (2012).
[CrossRef]

A. M. Yacomotti, S. Haddadi, and S. Barbay, Phys. Rev. A 87, 041804(R) (2013).
[CrossRef]

Phys. Rev. Lett.

K. Ikeda and O. Akimoto, Phys. Rev. Lett. 48, 617 (1982).
[CrossRef]

K. Ikeda, H. Daido, and O. Akimoto, Phys. Rev. Lett. 45, 709 (1980).
[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 (5)

Fig. 1.
Fig. 1.

Steady-state response E versus P for (a), (c) δ=9 and (b), (d) δ=15, and τ as indicated. Solid (dotted) lines stand for stable (unstable) branches. The blue and red curves superimposed on the right side show the dependence on E of the real λR and imaginary λI parts of the unstable eigenvalues, respectively. The dashed vertical lines stands for λR=0 or λI=0, and bifurcation points λR=0 are highlighted by empty circles (negative-slope branch) and triangles (SP). The horizontal shaded areas labeled BI (light green) and SP (light yellow) highlight the negative-slope branch of the bistable response and the SP-unstable region, respectively. Four different scenarios are shown: (a) SP occurs at PH above the knee value Pb and is unbounded for increasing P; (b) SP occurs at PH below the bistable knee Pb, still being unbounded; (c) and (d) are as in (a) and (b), respectively, though SP occurs in a finite range of powers (below the value PH+). The vertical shaded pink and pale blue regions highlight the range of powers where either bistable up-switching to a stable state is permitted or SP coexists with a negative-slope branch of the steady response.

Fig. 2.
Fig. 2.

SP threshold energies EH± as a function of τ for fixed detuning δ=9. The shaded red area corresponds to the domain (EH<E<EH+) where SP occurs, whereas the shaded blue area corresponds to the energies of the negative-slope branch (Eb<E<Eb+) of the stationary response.

Fig. 3.
Fig. 3.

Color level plots of the SP (a) “on” threshold energy EH and (b) “off” threshold energy EH+, in the parameter plane (τ,δ). Bistability (BI) occurs below the horizontal line δ=δb=4.75. The curve P(EH)=P(Eb) (black solid) divides the bistable region into a domain BI+SP where SP occurs only at powers P above the bistable knee for up-switching, and a domain SP, where up-switching always occurs toward a SP-unstable state.

Fig. 4.
Fig. 4.

(a) Temporal evolution of the intracavity energy |a(t)|2 and carrier density n(t) undergoing SP, as obtained from Eqs. (1) and (2). Here δ=9 and τ=0.2. (b) Phase-space picture associated with the optical field highlighting the stability of the attracting limit cycle (black circle). (c) Spectrum of the intracavity energy reported versus real-world values of frequency.

Fig. 5.
Fig. 5.

(a) Comparison between the steady-state responses (upper-branch portion) and the corresponding SP-unstable domains (above triangles setting the Hopf bifurcation points) with (SP domain in blue) and without (SP domain in red) TPA contribution, respectively. Here we set δ=9 and τ=0.1. (b) Similar comparison for the SP-unstable domain versus τ for fixed detuning δ=9 (red, without TPA; blue, with TPA).

Equations (7)

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

at=i(δ+n)a(1+α1p+α2p|a|2)a+P,
nt=|a|2+μ2p|a|4nτ.
P=E[(1+α1p+α2pE)2+(δ+τE+τμ2pE2)2],
dϵdt=Mϵ,M=(α^+iδ^α2pA2iAα2p(A*)2α^iδ^iA*A*+2μ2p|A|2A*A+2μ2p|A|2A1τ),
Eb±=2δ±δ23(1+α1p2)3τ,
λ3+a2λ2+a1λ+a0=0,
EH±=δ(1τ2α^)±F(τ,δ)2(α^τ1),

Metrics