Abstract

We present a set of rate equations for the modal amplitudes and carrier-inversion moments that describe the deterministic multi-mode dynamics of a semiconductor laser due to spatial hole burning. Mutual interactions among the lasing modes, induced by high- frequency modulations of the carrier distribution, are included by carrier-inversion moments for which rate equations are given as well. We derive the Bogatov effect of asymmetric gain suppression in semiconductor lasers and illustrate the potential of the model for a two and three-mode laser by numerical and analytical methods.

© 2014 Optical Society of America

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  1. M. Homar, S. Balle, M. San Miguel, “Mode competition in a Fabry-Perot semiconductor laser: travelling wave model with asymmetric dynamical gain,” Opt. Commun. 131(4–6), 380–390 (1996).
    [CrossRef]
  2. A. M. Yacomotti, L. Furfaro, X. Hachair, F. Pedaci, M. Giudici, J. Tredicce, J. Javaloyes, S. Balle, E. Viktorov, P. Mandel, “Dynamics of multimode semiconductor lasers,” Phys. Rev. A 69(5), 053816 (2004).
    [CrossRef]
  3. M. Yousefi and D. Lenstra, “Rate-equation description of multi-mode semiconductor lasers,” in SPIE Photonics West 2014, Physics and Simulation of Optoelectronic Devices XXII (2014), paper # 8980–10, to be published.
  4. M. Sargent, “Laser saturation grating phenomena,” Appl. Phys. 9(2), 127–141 (1976).
    [CrossRef]
  5. C. Serrat, C. Masoller, “Modeling spatial effects in multi-longitudinal-mode semiconductor lasers,” Phys. Rev. A 73(4), 043812 (2006).
    [CrossRef]
  6. M. Yamada, “Theoretical analysis of nonlinear optical phenomena taking into account the beating vibration of the electron density in semiconductor lasers,” J. Appl. Phys. 66(1), 81–89 (1989).
    [CrossRef]
  7. A. P. Bogatov, P. G. Eliseev, B. N. Sverdlov, “Anomalous interaction of spectral modes in a semiconductor laser,” IEEE J. Quantum Electron. 11(7), 510–515 (1975).
    [CrossRef]
  8. M. Ahmed, M. Yamada, “Influence of instantaneous mode competition on the dynamics of semiconductor lasers,” IEEE J. Quantum Electron. 38(6), 682–693 (2002).
    [CrossRef]
  9. A. T. Ryan, G. P. Agrawal, G. R. Gray, E. C. Gage, “Optical-feedback-induced chaos and its control in multimode semiconductor lasers,” IEEE J. Quantum Electron. 30(3), 668–679 (1994).
    [CrossRef]
  10. M. Homar, S. Balle, M. San Miguel, “Mode competition in a Fabry-Perot semiconductor laser: travelling wave model with asymmetric dynamical gain,” Opt. Commun. 131(4–6), 380–390 (1996).
    [CrossRef]
  11. C. Born, G. Yuan, Z. Wang, M. Sorel, S. Yu, “Lasing mode hysteresis characteristics in semiconductor ring lasers,” IEEE J. Quantum Electron. 44(12), 1171–1179 (2008).
    [CrossRef]
  12. E. G. Lariontsev, “Switching of synchronized chaotic oscillations in a modulated solid-state ring laser,” Opt. Express 2(5), 198–203 (1998).
    [CrossRef] [PubMed]
  13. C. Z. Ning, R. A. Indik, J. V. Moloney, “Effective Bloch equations for semiconductor lasers and amplifiers,” IEEE J. Quantum Electron. 33(9), 1543–1550 (1997).
    [CrossRef]
  14. M. Yousefi, D. Lenstra, G. Vemuri, “Carrier inversion noise has important influence on the dynamics of a semiconductor laser,” IEEE J. Sel. Top. Quantum Electron. 10(5), 955–960 (2004).
    [CrossRef]

2008

C. Born, G. Yuan, Z. Wang, M. Sorel, S. Yu, “Lasing mode hysteresis characteristics in semiconductor ring lasers,” IEEE J. Quantum Electron. 44(12), 1171–1179 (2008).
[CrossRef]

2006

C. Serrat, C. Masoller, “Modeling spatial effects in multi-longitudinal-mode semiconductor lasers,” Phys. Rev. A 73(4), 043812 (2006).
[CrossRef]

2004

A. M. Yacomotti, L. Furfaro, X. Hachair, F. Pedaci, M. Giudici, J. Tredicce, J. Javaloyes, S. Balle, E. Viktorov, P. Mandel, “Dynamics of multimode semiconductor lasers,” Phys. Rev. A 69(5), 053816 (2004).
[CrossRef]

M. Yousefi, D. Lenstra, G. Vemuri, “Carrier inversion noise has important influence on the dynamics of a semiconductor laser,” IEEE J. Sel. Top. Quantum Electron. 10(5), 955–960 (2004).
[CrossRef]

2002

M. Ahmed, M. Yamada, “Influence of instantaneous mode competition on the dynamics of semiconductor lasers,” IEEE J. Quantum Electron. 38(6), 682–693 (2002).
[CrossRef]

1998

1997

C. Z. Ning, R. A. Indik, J. V. Moloney, “Effective Bloch equations for semiconductor lasers and amplifiers,” IEEE J. Quantum Electron. 33(9), 1543–1550 (1997).
[CrossRef]

1996

M. Homar, S. Balle, M. San Miguel, “Mode competition in a Fabry-Perot semiconductor laser: travelling wave model with asymmetric dynamical gain,” Opt. Commun. 131(4–6), 380–390 (1996).
[CrossRef]

M. Homar, S. Balle, M. San Miguel, “Mode competition in a Fabry-Perot semiconductor laser: travelling wave model with asymmetric dynamical gain,” Opt. Commun. 131(4–6), 380–390 (1996).
[CrossRef]

1994

A. T. Ryan, G. P. Agrawal, G. R. Gray, E. C. Gage, “Optical-feedback-induced chaos and its control in multimode semiconductor lasers,” IEEE J. Quantum Electron. 30(3), 668–679 (1994).
[CrossRef]

1989

M. Yamada, “Theoretical analysis of nonlinear optical phenomena taking into account the beating vibration of the electron density in semiconductor lasers,” J. Appl. Phys. 66(1), 81–89 (1989).
[CrossRef]

1976

M. Sargent, “Laser saturation grating phenomena,” Appl. Phys. 9(2), 127–141 (1976).
[CrossRef]

1975

A. P. Bogatov, P. G. Eliseev, B. N. Sverdlov, “Anomalous interaction of spectral modes in a semiconductor laser,” IEEE J. Quantum Electron. 11(7), 510–515 (1975).
[CrossRef]

Agrawal, G. P.

A. T. Ryan, G. P. Agrawal, G. R. Gray, E. C. Gage, “Optical-feedback-induced chaos and its control in multimode semiconductor lasers,” IEEE J. Quantum Electron. 30(3), 668–679 (1994).
[CrossRef]

Ahmed, M.

M. Ahmed, M. Yamada, “Influence of instantaneous mode competition on the dynamics of semiconductor lasers,” IEEE J. Quantum Electron. 38(6), 682–693 (2002).
[CrossRef]

Balle, S.

A. M. Yacomotti, L. Furfaro, X. Hachair, F. Pedaci, M. Giudici, J. Tredicce, J. Javaloyes, S. Balle, E. Viktorov, P. Mandel, “Dynamics of multimode semiconductor lasers,” Phys. Rev. A 69(5), 053816 (2004).
[CrossRef]

M. Homar, S. Balle, M. San Miguel, “Mode competition in a Fabry-Perot semiconductor laser: travelling wave model with asymmetric dynamical gain,” Opt. Commun. 131(4–6), 380–390 (1996).
[CrossRef]

M. Homar, S. Balle, M. San Miguel, “Mode competition in a Fabry-Perot semiconductor laser: travelling wave model with asymmetric dynamical gain,” Opt. Commun. 131(4–6), 380–390 (1996).
[CrossRef]

Bogatov, A. P.

A. P. Bogatov, P. G. Eliseev, B. N. Sverdlov, “Anomalous interaction of spectral modes in a semiconductor laser,” IEEE J. Quantum Electron. 11(7), 510–515 (1975).
[CrossRef]

Born, C.

C. Born, G. Yuan, Z. Wang, M. Sorel, S. Yu, “Lasing mode hysteresis characteristics in semiconductor ring lasers,” IEEE J. Quantum Electron. 44(12), 1171–1179 (2008).
[CrossRef]

Eliseev, P. G.

A. P. Bogatov, P. G. Eliseev, B. N. Sverdlov, “Anomalous interaction of spectral modes in a semiconductor laser,” IEEE J. Quantum Electron. 11(7), 510–515 (1975).
[CrossRef]

Furfaro, L.

A. M. Yacomotti, L. Furfaro, X. Hachair, F. Pedaci, M. Giudici, J. Tredicce, J. Javaloyes, S. Balle, E. Viktorov, P. Mandel, “Dynamics of multimode semiconductor lasers,” Phys. Rev. A 69(5), 053816 (2004).
[CrossRef]

Gage, E. C.

A. T. Ryan, G. P. Agrawal, G. R. Gray, E. C. Gage, “Optical-feedback-induced chaos and its control in multimode semiconductor lasers,” IEEE J. Quantum Electron. 30(3), 668–679 (1994).
[CrossRef]

Giudici, M.

A. M. Yacomotti, L. Furfaro, X. Hachair, F. Pedaci, M. Giudici, J. Tredicce, J. Javaloyes, S. Balle, E. Viktorov, P. Mandel, “Dynamics of multimode semiconductor lasers,” Phys. Rev. A 69(5), 053816 (2004).
[CrossRef]

Gray, G. R.

A. T. Ryan, G. P. Agrawal, G. R. Gray, E. C. Gage, “Optical-feedback-induced chaos and its control in multimode semiconductor lasers,” IEEE J. Quantum Electron. 30(3), 668–679 (1994).
[CrossRef]

Hachair, X.

A. M. Yacomotti, L. Furfaro, X. Hachair, F. Pedaci, M. Giudici, J. Tredicce, J. Javaloyes, S. Balle, E. Viktorov, P. Mandel, “Dynamics of multimode semiconductor lasers,” Phys. Rev. A 69(5), 053816 (2004).
[CrossRef]

Homar, M.

M. Homar, S. Balle, M. San Miguel, “Mode competition in a Fabry-Perot semiconductor laser: travelling wave model with asymmetric dynamical gain,” Opt. Commun. 131(4–6), 380–390 (1996).
[CrossRef]

M. Homar, S. Balle, M. San Miguel, “Mode competition in a Fabry-Perot semiconductor laser: travelling wave model with asymmetric dynamical gain,” Opt. Commun. 131(4–6), 380–390 (1996).
[CrossRef]

Indik, R. A.

C. Z. Ning, R. A. Indik, J. V. Moloney, “Effective Bloch equations for semiconductor lasers and amplifiers,” IEEE J. Quantum Electron. 33(9), 1543–1550 (1997).
[CrossRef]

Javaloyes, J.

A. M. Yacomotti, L. Furfaro, X. Hachair, F. Pedaci, M. Giudici, J. Tredicce, J. Javaloyes, S. Balle, E. Viktorov, P. Mandel, “Dynamics of multimode semiconductor lasers,” Phys. Rev. A 69(5), 053816 (2004).
[CrossRef]

Lariontsev, E. G.

Lenstra, D.

M. Yousefi, D. Lenstra, G. Vemuri, “Carrier inversion noise has important influence on the dynamics of a semiconductor laser,” IEEE J. Sel. Top. Quantum Electron. 10(5), 955–960 (2004).
[CrossRef]

Mandel, P.

A. M. Yacomotti, L. Furfaro, X. Hachair, F. Pedaci, M. Giudici, J. Tredicce, J. Javaloyes, S. Balle, E. Viktorov, P. Mandel, “Dynamics of multimode semiconductor lasers,” Phys. Rev. A 69(5), 053816 (2004).
[CrossRef]

Masoller, C.

C. Serrat, C. Masoller, “Modeling spatial effects in multi-longitudinal-mode semiconductor lasers,” Phys. Rev. A 73(4), 043812 (2006).
[CrossRef]

Moloney, J. V.

C. Z. Ning, R. A. Indik, J. V. Moloney, “Effective Bloch equations for semiconductor lasers and amplifiers,” IEEE J. Quantum Electron. 33(9), 1543–1550 (1997).
[CrossRef]

Ning, C. Z.

C. Z. Ning, R. A. Indik, J. V. Moloney, “Effective Bloch equations for semiconductor lasers and amplifiers,” IEEE J. Quantum Electron. 33(9), 1543–1550 (1997).
[CrossRef]

Pedaci, F.

A. M. Yacomotti, L. Furfaro, X. Hachair, F. Pedaci, M. Giudici, J. Tredicce, J. Javaloyes, S. Balle, E. Viktorov, P. Mandel, “Dynamics of multimode semiconductor lasers,” Phys. Rev. A 69(5), 053816 (2004).
[CrossRef]

Ryan, A. T.

A. T. Ryan, G. P. Agrawal, G. R. Gray, E. C. Gage, “Optical-feedback-induced chaos and its control in multimode semiconductor lasers,” IEEE J. Quantum Electron. 30(3), 668–679 (1994).
[CrossRef]

San Miguel, M.

M. Homar, S. Balle, M. San Miguel, “Mode competition in a Fabry-Perot semiconductor laser: travelling wave model with asymmetric dynamical gain,” Opt. Commun. 131(4–6), 380–390 (1996).
[CrossRef]

M. Homar, S. Balle, M. San Miguel, “Mode competition in a Fabry-Perot semiconductor laser: travelling wave model with asymmetric dynamical gain,” Opt. Commun. 131(4–6), 380–390 (1996).
[CrossRef]

Sargent, M.

M. Sargent, “Laser saturation grating phenomena,” Appl. Phys. 9(2), 127–141 (1976).
[CrossRef]

Serrat, C.

C. Serrat, C. Masoller, “Modeling spatial effects in multi-longitudinal-mode semiconductor lasers,” Phys. Rev. A 73(4), 043812 (2006).
[CrossRef]

Sorel, M.

C. Born, G. Yuan, Z. Wang, M. Sorel, S. Yu, “Lasing mode hysteresis characteristics in semiconductor ring lasers,” IEEE J. Quantum Electron. 44(12), 1171–1179 (2008).
[CrossRef]

Sverdlov, B. N.

A. P. Bogatov, P. G. Eliseev, B. N. Sverdlov, “Anomalous interaction of spectral modes in a semiconductor laser,” IEEE J. Quantum Electron. 11(7), 510–515 (1975).
[CrossRef]

Tredicce, J.

A. M. Yacomotti, L. Furfaro, X. Hachair, F. Pedaci, M. Giudici, J. Tredicce, J. Javaloyes, S. Balle, E. Viktorov, P. Mandel, “Dynamics of multimode semiconductor lasers,” Phys. Rev. A 69(5), 053816 (2004).
[CrossRef]

Vemuri, G.

M. Yousefi, D. Lenstra, G. Vemuri, “Carrier inversion noise has important influence on the dynamics of a semiconductor laser,” IEEE J. Sel. Top. Quantum Electron. 10(5), 955–960 (2004).
[CrossRef]

Viktorov, E.

A. M. Yacomotti, L. Furfaro, X. Hachair, F. Pedaci, M. Giudici, J. Tredicce, J. Javaloyes, S. Balle, E. Viktorov, P. Mandel, “Dynamics of multimode semiconductor lasers,” Phys. Rev. A 69(5), 053816 (2004).
[CrossRef]

Wang, Z.

C. Born, G. Yuan, Z. Wang, M. Sorel, S. Yu, “Lasing mode hysteresis characteristics in semiconductor ring lasers,” IEEE J. Quantum Electron. 44(12), 1171–1179 (2008).
[CrossRef]

Yacomotti, A. M.

A. M. Yacomotti, L. Furfaro, X. Hachair, F. Pedaci, M. Giudici, J. Tredicce, J. Javaloyes, S. Balle, E. Viktorov, P. Mandel, “Dynamics of multimode semiconductor lasers,” Phys. Rev. A 69(5), 053816 (2004).
[CrossRef]

Yamada, M.

M. Ahmed, M. Yamada, “Influence of instantaneous mode competition on the dynamics of semiconductor lasers,” IEEE J. Quantum Electron. 38(6), 682–693 (2002).
[CrossRef]

M. Yamada, “Theoretical analysis of nonlinear optical phenomena taking into account the beating vibration of the electron density in semiconductor lasers,” J. Appl. Phys. 66(1), 81–89 (1989).
[CrossRef]

Yousefi, M.

M. Yousefi, D. Lenstra, G. Vemuri, “Carrier inversion noise has important influence on the dynamics of a semiconductor laser,” IEEE J. Sel. Top. Quantum Electron. 10(5), 955–960 (2004).
[CrossRef]

Yu, S.

C. Born, G. Yuan, Z. Wang, M. Sorel, S. Yu, “Lasing mode hysteresis characteristics in semiconductor ring lasers,” IEEE J. Quantum Electron. 44(12), 1171–1179 (2008).
[CrossRef]

Yuan, G.

C. Born, G. Yuan, Z. Wang, M. Sorel, S. Yu, “Lasing mode hysteresis characteristics in semiconductor ring lasers,” IEEE J. Quantum Electron. 44(12), 1171–1179 (2008).
[CrossRef]

Appl. Phys.

M. Sargent, “Laser saturation grating phenomena,” Appl. Phys. 9(2), 127–141 (1976).
[CrossRef]

IEEE J. Quantum Electron.

A. P. Bogatov, P. G. Eliseev, B. N. Sverdlov, “Anomalous interaction of spectral modes in a semiconductor laser,” IEEE J. Quantum Electron. 11(7), 510–515 (1975).
[CrossRef]

M. Ahmed, M. Yamada, “Influence of instantaneous mode competition on the dynamics of semiconductor lasers,” IEEE J. Quantum Electron. 38(6), 682–693 (2002).
[CrossRef]

A. T. Ryan, G. P. Agrawal, G. R. Gray, E. C. Gage, “Optical-feedback-induced chaos and its control in multimode semiconductor lasers,” IEEE J. Quantum Electron. 30(3), 668–679 (1994).
[CrossRef]

C. Born, G. Yuan, Z. Wang, M. Sorel, S. Yu, “Lasing mode hysteresis characteristics in semiconductor ring lasers,” IEEE J. Quantum Electron. 44(12), 1171–1179 (2008).
[CrossRef]

C. Z. Ning, R. A. Indik, J. V. Moloney, “Effective Bloch equations for semiconductor lasers and amplifiers,” IEEE J. Quantum Electron. 33(9), 1543–1550 (1997).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

M. Yousefi, D. Lenstra, G. Vemuri, “Carrier inversion noise has important influence on the dynamics of a semiconductor laser,” IEEE J. Sel. Top. Quantum Electron. 10(5), 955–960 (2004).
[CrossRef]

J. Appl. Phys.

M. Yamada, “Theoretical analysis of nonlinear optical phenomena taking into account the beating vibration of the electron density in semiconductor lasers,” J. Appl. Phys. 66(1), 81–89 (1989).
[CrossRef]

Opt. Commun.

M. Homar, S. Balle, M. San Miguel, “Mode competition in a Fabry-Perot semiconductor laser: travelling wave model with asymmetric dynamical gain,” Opt. Commun. 131(4–6), 380–390 (1996).
[CrossRef]

M. Homar, S. Balle, M. San Miguel, “Mode competition in a Fabry-Perot semiconductor laser: travelling wave model with asymmetric dynamical gain,” Opt. Commun. 131(4–6), 380–390 (1996).
[CrossRef]

Opt. Express

Phys. Rev. A

A. M. Yacomotti, L. Furfaro, X. Hachair, F. Pedaci, M. Giudici, J. Tredicce, J. Javaloyes, S. Balle, E. Viktorov, P. Mandel, “Dynamics of multimode semiconductor lasers,” Phys. Rev. A 69(5), 053816 (2004).
[CrossRef]

C. Serrat, C. Masoller, “Modeling spatial effects in multi-longitudinal-mode semiconductor lasers,” Phys. Rev. A 73(4), 043812 (2006).
[CrossRef]

Other

M. Yousefi and D. Lenstra, “Rate-equation description of multi-mode semiconductor lasers,” in SPIE Photonics West 2014, Physics and Simulation of Optoelectronic Devices XXII (2014), paper # 8980–10, to be published.

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Figures (3)

Fig. 1
Fig. 1

Transition of lasing mode due to modal interaction: the frequency spacing between the two modes is chosen as 80GHz. The system is prepared in the short wavelength mode and gradually evolves to the long wavelength mode. The time is measured in nanoseconds and the intensity is given in number of photons. Parameters are as in Table 1. The laser is pumped twice above threshold.

Fig. 2
Fig. 2

Sequential on-off switching of modes in the mode-resolved time series. (a) shows the total photon number and (b) the intensity of the individual modes. The parameters are as in Table 1. The mode spacing is Δω=2π×12  GHz. Note the period-5 dynamics.

Fig. 3
Fig. 3

Optical spectra of the dynamics depicted in Fig. 2. Plot (a) shows the spectrum of the total field E j   (t) e i ω j t ; plots (b), (c) and (d) show modes 1, 2 and 3, respectively. The relaxation oscillation frequency is ~3.6 GHz and the ~720 MHz peak fine structure corresponds to the period-5 oscillation in Fig. 2 and is caused by the system dynamics and mode competition. Note that each mode 1 and 3 contains spectral components of itself and the other, while the content of mode 2 is at the frequencies of the two side modes.

Tables (2)

Tables Icon

Table 1 Explanation and meaning of the various symbols in Eqs. (1) and (2)

Tables Icon

Table 2 Parameter values taken in Eq. (9)

Equations (1)

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  f jk;0 = δ jk ;   f jk;0m = f jk;m0 = f jk;m  , ( m );   f jk;m = 1 2 [ δ m,| kj | δ m,k+j ] ,  ( m1 )  

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