Abstract

The mode competition in a dual-wavelength quarter-wave coupled-cavity semiconductor laser is analyzed using Langevin noise-driven rate equations. The laser can achieve simultaneous emission of two optical modes as a doublet with the same threshold, while the frequency difference of the two lasing wavelengths can be tuned by the injection currents. The analytical expressions for the spectral positions of the doublet and the threshold gain are derived, and a numerical model based on Langevin noise-driven rate equations is established for the mode competition process with enhanced physical insight.

© 2010 Optical Society of America

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References

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  1. K. E. Razavi and P. A. Davies, “Semiconductor laser sources for the generation of millimeter-wave signals,” in Proceedings of IEEE Conference on Optoelectronics (IEEE, 1998), pp. 159-163.
    [CrossRef]
  2. J. J. O'Reilly, P. M. Lane, R. Heidemann, and R. Hofstetter, “Optical generation of very narrow linewidth millimeter wave signals,” Electron. Lett. 28, 2309-2310 (1992).
  3. X. Liu, “A novel dual-wavelength DFB fiber laser based on symmetrical FBG structure,” IEEE Photonics Technol. Lett. 19, 632-634 (2007).
    [CrossRef]
  4. J. Sun, Y. Dai, Y. Zhang, X. Chen, and S. Xie, “Dual-wavelength DFB fiber laser based on unequalized phase shifts,” IEEE Photonics Technol. Lett. 18, 2493-2495 (2006).
    [CrossRef]
  5. F. Pozzi, R. M. DeLa Rue, and M. Sorel, “Dual-wavelength InAlGaAs-InP laterally coupled distributed feedback laser,” IEEE Photonics Technol. Lett. 18, 2563-2565 (2006).
    [CrossRef]
  6. J. J. Huang, C. C. Yang, and D. W. Huang, “Carrier capture competition between two different quantum wells in dual-wavelength semiconductor lasers,” IEEE Photonics Technol. Lett. 8, 752-755 (1996).
    [CrossRef]
  7. L. A. Coldren and T. L. Koch, “Analysis and design of coupled-cavity lasers--Part I: Threshold gain analysis and design guidelines,” IEEE J. Quantum Electron. 20, 659-670 (1984).
    [CrossRef]
  8. G. P. Agrawal and N. K. Dutta, “Coupled-cavity semiconductor lasers,” in Long-Wavelength Semiconductor Lasers. (Van Nostrand Reinhold, 1986), pp. 333-371.
  9. T. L. Koch and L. A. Coldren, “Optimum coupling junction and cavity lengths for coupled-cavity semiconductor lasers,” J. Appl. Phys. 57, 740-754 (1985).
    [CrossRef]
  10. R. J. Lang and A. Yariv, “An exact formulation of coupled-mode theory for coupled-cavity lasers,” IEEE J. Quantum Electron. 24, 66-72 (1988).
    [CrossRef]
  11. L. A. Coldren, B. I. Miller, K. Iga, and J. A. Rentschler, “Monolithic two-section GaInAsP/InP active-optical-resonator devices formed by reactive-ion-etching,” Appl. Phys. Lett. 38, 315-317 (1981).
    [CrossRef]
  12. W. T. Tsang, “The cleaved-coupled-cavity (C3) laser,” in Semiconductors and Semimetals, Vol. 22, part B, R. K. Willardson, A. C. Beer, and W. T. Tsang, eds. (Academic, 1985), pp. 258-369.
  13. G. Bjork and O. Nilsson, “A new exact and efficient numerical matrix theory of complicated laser structures: properties of asymmetric phase-shifted DFB lasers,” J. Lightwave Technol. 5, 140-146 (1987).
    [CrossRef]
  14. D. Marcuse and T. P. Lee, “Rate equation model of a coupled-cavity laser,” IEEE J. Quantum Electron. 20, 166-176 (1984).
    [CrossRef]
  15. J. F. Lepage and N. McCarthy, “Analysis of dual-wavelength oscillation in a broad-area diode laser operated with an external cavity,” Appl. Opt. 41, 4347-4355 (2002).
    [CrossRef] [PubMed]
  16. Y. Suematsu and A. R. Adams, Handbook of Semiconductor Lasers and Photonic Integrated Circuits (Chapman and Hall, 1994).
  17. A. Gearba and G. Cone, “Numerical analysis of laser mode competition and stability,” Phys. Lett. A 269, 112-119 (2000).
    [CrossRef]
  18. D. Marcuse, “Computer simulation of laser photon fluctuations: theory of single-cavity laser,” IEEE J. Quantum Electron. 20, 1139-1148 (1984).
    [CrossRef]
  19. D. Marcuse, “Computer simulation of laser photon fluctuations: single-cavity laser results,” IEEE J. Quantum Electron. 20, 1148-1155 (1984).
    [CrossRef]
  20. D. Marcuse, “Computer simulation of laser photon fluctuations: coupled-cavity lasers,” IEEE J. Quantum Electron. 21, 154-161 (1985).
    [CrossRef]
  21. David Wake, Claudio R. Lima, and Phillip A. Davies, “Optical generation of millimeter--wave signals for fiber-radio systems using a dual-mode DFB semiconductor laser,” IEEE Trans. Microwave Theory Tech. 43, 2270-2276 (1995).
    [CrossRef]

2007 (1)

X. Liu, “A novel dual-wavelength DFB fiber laser based on symmetrical FBG structure,” IEEE Photonics Technol. Lett. 19, 632-634 (2007).
[CrossRef]

2006 (2)

J. Sun, Y. Dai, Y. Zhang, X. Chen, and S. Xie, “Dual-wavelength DFB fiber laser based on unequalized phase shifts,” IEEE Photonics Technol. Lett. 18, 2493-2495 (2006).
[CrossRef]

F. Pozzi, R. M. DeLa Rue, and M. Sorel, “Dual-wavelength InAlGaAs-InP laterally coupled distributed feedback laser,” IEEE Photonics Technol. Lett. 18, 2563-2565 (2006).
[CrossRef]

2002 (1)

2000 (1)

A. Gearba and G. Cone, “Numerical analysis of laser mode competition and stability,” Phys. Lett. A 269, 112-119 (2000).
[CrossRef]

1996 (1)

J. J. Huang, C. C. Yang, and D. W. Huang, “Carrier capture competition between two different quantum wells in dual-wavelength semiconductor lasers,” IEEE Photonics Technol. Lett. 8, 752-755 (1996).
[CrossRef]

1995 (1)

David Wake, Claudio R. Lima, and Phillip A. Davies, “Optical generation of millimeter--wave signals for fiber-radio systems using a dual-mode DFB semiconductor laser,” IEEE Trans. Microwave Theory Tech. 43, 2270-2276 (1995).
[CrossRef]

1992 (1)

J. J. O'Reilly, P. M. Lane, R. Heidemann, and R. Hofstetter, “Optical generation of very narrow linewidth millimeter wave signals,” Electron. Lett. 28, 2309-2310 (1992).

1988 (1)

R. J. Lang and A. Yariv, “An exact formulation of coupled-mode theory for coupled-cavity lasers,” IEEE J. Quantum Electron. 24, 66-72 (1988).
[CrossRef]

1987 (1)

G. Bjork and O. Nilsson, “A new exact and efficient numerical matrix theory of complicated laser structures: properties of asymmetric phase-shifted DFB lasers,” J. Lightwave Technol. 5, 140-146 (1987).
[CrossRef]

1985 (2)

D. Marcuse, “Computer simulation of laser photon fluctuations: coupled-cavity lasers,” IEEE J. Quantum Electron. 21, 154-161 (1985).
[CrossRef]

T. L. Koch and L. A. Coldren, “Optimum coupling junction and cavity lengths for coupled-cavity semiconductor lasers,” J. Appl. Phys. 57, 740-754 (1985).
[CrossRef]

1984 (4)

L. A. Coldren and T. L. Koch, “Analysis and design of coupled-cavity lasers--Part I: Threshold gain analysis and design guidelines,” IEEE J. Quantum Electron. 20, 659-670 (1984).
[CrossRef]

D. Marcuse and T. P. Lee, “Rate equation model of a coupled-cavity laser,” IEEE J. Quantum Electron. 20, 166-176 (1984).
[CrossRef]

D. Marcuse, “Computer simulation of laser photon fluctuations: theory of single-cavity laser,” IEEE J. Quantum Electron. 20, 1139-1148 (1984).
[CrossRef]

D. Marcuse, “Computer simulation of laser photon fluctuations: single-cavity laser results,” IEEE J. Quantum Electron. 20, 1148-1155 (1984).
[CrossRef]

1981 (1)

L. A. Coldren, B. I. Miller, K. Iga, and J. A. Rentschler, “Monolithic two-section GaInAsP/InP active-optical-resonator devices formed by reactive-ion-etching,” Appl. Phys. Lett. 38, 315-317 (1981).
[CrossRef]

Adams, A. R.

Y. Suematsu and A. R. Adams, Handbook of Semiconductor Lasers and Photonic Integrated Circuits (Chapman and Hall, 1994).

Agrawal, G. P.

G. P. Agrawal and N. K. Dutta, “Coupled-cavity semiconductor lasers,” in Long-Wavelength Semiconductor Lasers. (Van Nostrand Reinhold, 1986), pp. 333-371.

Beer, A. C.

W. T. Tsang, “The cleaved-coupled-cavity (C3) laser,” in Semiconductors and Semimetals, Vol. 22, part B, R. K. Willardson, A. C. Beer, and W. T. Tsang, eds. (Academic, 1985), pp. 258-369.

Bjork, G.

G. Bjork and O. Nilsson, “A new exact and efficient numerical matrix theory of complicated laser structures: properties of asymmetric phase-shifted DFB lasers,” J. Lightwave Technol. 5, 140-146 (1987).
[CrossRef]

Chen, X.

J. Sun, Y. Dai, Y. Zhang, X. Chen, and S. Xie, “Dual-wavelength DFB fiber laser based on unequalized phase shifts,” IEEE Photonics Technol. Lett. 18, 2493-2495 (2006).
[CrossRef]

Coldren, L. A.

T. L. Koch and L. A. Coldren, “Optimum coupling junction and cavity lengths for coupled-cavity semiconductor lasers,” J. Appl. Phys. 57, 740-754 (1985).
[CrossRef]

L. A. Coldren and T. L. Koch, “Analysis and design of coupled-cavity lasers--Part I: Threshold gain analysis and design guidelines,” IEEE J. Quantum Electron. 20, 659-670 (1984).
[CrossRef]

L. A. Coldren, B. I. Miller, K. Iga, and J. A. Rentschler, “Monolithic two-section GaInAsP/InP active-optical-resonator devices formed by reactive-ion-etching,” Appl. Phys. Lett. 38, 315-317 (1981).
[CrossRef]

Cone, G.

A. Gearba and G. Cone, “Numerical analysis of laser mode competition and stability,” Phys. Lett. A 269, 112-119 (2000).
[CrossRef]

Dai, Y.

J. Sun, Y. Dai, Y. Zhang, X. Chen, and S. Xie, “Dual-wavelength DFB fiber laser based on unequalized phase shifts,” IEEE Photonics Technol. Lett. 18, 2493-2495 (2006).
[CrossRef]

Davies, P. A.

K. E. Razavi and P. A. Davies, “Semiconductor laser sources for the generation of millimeter-wave signals,” in Proceedings of IEEE Conference on Optoelectronics (IEEE, 1998), pp. 159-163.
[CrossRef]

Davies, Phillip A.

David Wake, Claudio R. Lima, and Phillip A. Davies, “Optical generation of millimeter--wave signals for fiber-radio systems using a dual-mode DFB semiconductor laser,” IEEE Trans. Microwave Theory Tech. 43, 2270-2276 (1995).
[CrossRef]

DeLa Rue, R. M.

F. Pozzi, R. M. DeLa Rue, and M. Sorel, “Dual-wavelength InAlGaAs-InP laterally coupled distributed feedback laser,” IEEE Photonics Technol. Lett. 18, 2563-2565 (2006).
[CrossRef]

Dutta, N. K.

G. P. Agrawal and N. K. Dutta, “Coupled-cavity semiconductor lasers,” in Long-Wavelength Semiconductor Lasers. (Van Nostrand Reinhold, 1986), pp. 333-371.

Gearba, A.

A. Gearba and G. Cone, “Numerical analysis of laser mode competition and stability,” Phys. Lett. A 269, 112-119 (2000).
[CrossRef]

Heidemann, R.

J. J. O'Reilly, P. M. Lane, R. Heidemann, and R. Hofstetter, “Optical generation of very narrow linewidth millimeter wave signals,” Electron. Lett. 28, 2309-2310 (1992).

Hofstetter, R.

J. J. O'Reilly, P. M. Lane, R. Heidemann, and R. Hofstetter, “Optical generation of very narrow linewidth millimeter wave signals,” Electron. Lett. 28, 2309-2310 (1992).

Huang, D. W.

J. J. Huang, C. C. Yang, and D. W. Huang, “Carrier capture competition between two different quantum wells in dual-wavelength semiconductor lasers,” IEEE Photonics Technol. Lett. 8, 752-755 (1996).
[CrossRef]

Huang, J. J.

J. J. Huang, C. C. Yang, and D. W. Huang, “Carrier capture competition between two different quantum wells in dual-wavelength semiconductor lasers,” IEEE Photonics Technol. Lett. 8, 752-755 (1996).
[CrossRef]

Iga, K.

L. A. Coldren, B. I. Miller, K. Iga, and J. A. Rentschler, “Monolithic two-section GaInAsP/InP active-optical-resonator devices formed by reactive-ion-etching,” Appl. Phys. Lett. 38, 315-317 (1981).
[CrossRef]

Koch, T. L.

T. L. Koch and L. A. Coldren, “Optimum coupling junction and cavity lengths for coupled-cavity semiconductor lasers,” J. Appl. Phys. 57, 740-754 (1985).
[CrossRef]

L. A. Coldren and T. L. Koch, “Analysis and design of coupled-cavity lasers--Part I: Threshold gain analysis and design guidelines,” IEEE J. Quantum Electron. 20, 659-670 (1984).
[CrossRef]

Lane, P. M.

J. J. O'Reilly, P. M. Lane, R. Heidemann, and R. Hofstetter, “Optical generation of very narrow linewidth millimeter wave signals,” Electron. Lett. 28, 2309-2310 (1992).

Lang, R. J.

R. J. Lang and A. Yariv, “An exact formulation of coupled-mode theory for coupled-cavity lasers,” IEEE J. Quantum Electron. 24, 66-72 (1988).
[CrossRef]

Lee, T. P.

D. Marcuse and T. P. Lee, “Rate equation model of a coupled-cavity laser,” IEEE J. Quantum Electron. 20, 166-176 (1984).
[CrossRef]

Lepage, J. F.

Lima, Claudio R.

David Wake, Claudio R. Lima, and Phillip A. Davies, “Optical generation of millimeter--wave signals for fiber-radio systems using a dual-mode DFB semiconductor laser,” IEEE Trans. Microwave Theory Tech. 43, 2270-2276 (1995).
[CrossRef]

Liu, X.

X. Liu, “A novel dual-wavelength DFB fiber laser based on symmetrical FBG structure,” IEEE Photonics Technol. Lett. 19, 632-634 (2007).
[CrossRef]

Marcuse, D.

D. Marcuse, “Computer simulation of laser photon fluctuations: coupled-cavity lasers,” IEEE J. Quantum Electron. 21, 154-161 (1985).
[CrossRef]

D. Marcuse, “Computer simulation of laser photon fluctuations: theory of single-cavity laser,” IEEE J. Quantum Electron. 20, 1139-1148 (1984).
[CrossRef]

D. Marcuse and T. P. Lee, “Rate equation model of a coupled-cavity laser,” IEEE J. Quantum Electron. 20, 166-176 (1984).
[CrossRef]

D. Marcuse, “Computer simulation of laser photon fluctuations: single-cavity laser results,” IEEE J. Quantum Electron. 20, 1148-1155 (1984).
[CrossRef]

McCarthy, N.

Miller, B. I.

L. A. Coldren, B. I. Miller, K. Iga, and J. A. Rentschler, “Monolithic two-section GaInAsP/InP active-optical-resonator devices formed by reactive-ion-etching,” Appl. Phys. Lett. 38, 315-317 (1981).
[CrossRef]

Nilsson, O.

G. Bjork and O. Nilsson, “A new exact and efficient numerical matrix theory of complicated laser structures: properties of asymmetric phase-shifted DFB lasers,” J. Lightwave Technol. 5, 140-146 (1987).
[CrossRef]

O'Reilly, J. J.

J. J. O'Reilly, P. M. Lane, R. Heidemann, and R. Hofstetter, “Optical generation of very narrow linewidth millimeter wave signals,” Electron. Lett. 28, 2309-2310 (1992).

Pozzi, F.

F. Pozzi, R. M. DeLa Rue, and M. Sorel, “Dual-wavelength InAlGaAs-InP laterally coupled distributed feedback laser,” IEEE Photonics Technol. Lett. 18, 2563-2565 (2006).
[CrossRef]

Razavi, K. E.

K. E. Razavi and P. A. Davies, “Semiconductor laser sources for the generation of millimeter-wave signals,” in Proceedings of IEEE Conference on Optoelectronics (IEEE, 1998), pp. 159-163.
[CrossRef]

Rentschler, J. A.

L. A. Coldren, B. I. Miller, K. Iga, and J. A. Rentschler, “Monolithic two-section GaInAsP/InP active-optical-resonator devices formed by reactive-ion-etching,” Appl. Phys. Lett. 38, 315-317 (1981).
[CrossRef]

Sorel, M.

F. Pozzi, R. M. DeLa Rue, and M. Sorel, “Dual-wavelength InAlGaAs-InP laterally coupled distributed feedback laser,” IEEE Photonics Technol. Lett. 18, 2563-2565 (2006).
[CrossRef]

Suematsu, Y.

Y. Suematsu and A. R. Adams, Handbook of Semiconductor Lasers and Photonic Integrated Circuits (Chapman and Hall, 1994).

Sun, J.

J. Sun, Y. Dai, Y. Zhang, X. Chen, and S. Xie, “Dual-wavelength DFB fiber laser based on unequalized phase shifts,” IEEE Photonics Technol. Lett. 18, 2493-2495 (2006).
[CrossRef]

Tsang, W. T.

W. T. Tsang, “The cleaved-coupled-cavity (C3) laser,” in Semiconductors and Semimetals, Vol. 22, part B, R. K. Willardson, A. C. Beer, and W. T. Tsang, eds. (Academic, 1985), pp. 258-369.

W. T. Tsang, “The cleaved-coupled-cavity (C3) laser,” in Semiconductors and Semimetals, Vol. 22, part B, R. K. Willardson, A. C. Beer, and W. T. Tsang, eds. (Academic, 1985), pp. 258-369.

Wake, David

David Wake, Claudio R. Lima, and Phillip A. Davies, “Optical generation of millimeter--wave signals for fiber-radio systems using a dual-mode DFB semiconductor laser,” IEEE Trans. Microwave Theory Tech. 43, 2270-2276 (1995).
[CrossRef]

Willardson, R. K.

W. T. Tsang, “The cleaved-coupled-cavity (C3) laser,” in Semiconductors and Semimetals, Vol. 22, part B, R. K. Willardson, A. C. Beer, and W. T. Tsang, eds. (Academic, 1985), pp. 258-369.

Xie, S.

J. Sun, Y. Dai, Y. Zhang, X. Chen, and S. Xie, “Dual-wavelength DFB fiber laser based on unequalized phase shifts,” IEEE Photonics Technol. Lett. 18, 2493-2495 (2006).
[CrossRef]

Yang, C. C.

J. J. Huang, C. C. Yang, and D. W. Huang, “Carrier capture competition between two different quantum wells in dual-wavelength semiconductor lasers,” IEEE Photonics Technol. Lett. 8, 752-755 (1996).
[CrossRef]

Yariv, A.

R. J. Lang and A. Yariv, “An exact formulation of coupled-mode theory for coupled-cavity lasers,” IEEE J. Quantum Electron. 24, 66-72 (1988).
[CrossRef]

Zhang, Y.

J. Sun, Y. Dai, Y. Zhang, X. Chen, and S. Xie, “Dual-wavelength DFB fiber laser based on unequalized phase shifts,” IEEE Photonics Technol. Lett. 18, 2493-2495 (2006).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

L. A. Coldren, B. I. Miller, K. Iga, and J. A. Rentschler, “Monolithic two-section GaInAsP/InP active-optical-resonator devices formed by reactive-ion-etching,” Appl. Phys. Lett. 38, 315-317 (1981).
[CrossRef]

Electron. Lett. (1)

J. J. O'Reilly, P. M. Lane, R. Heidemann, and R. Hofstetter, “Optical generation of very narrow linewidth millimeter wave signals,” Electron. Lett. 28, 2309-2310 (1992).

IEEE J. Quantum Electron. (6)

L. A. Coldren and T. L. Koch, “Analysis and design of coupled-cavity lasers--Part I: Threshold gain analysis and design guidelines,” IEEE J. Quantum Electron. 20, 659-670 (1984).
[CrossRef]

R. J. Lang and A. Yariv, “An exact formulation of coupled-mode theory for coupled-cavity lasers,” IEEE J. Quantum Electron. 24, 66-72 (1988).
[CrossRef]

D. Marcuse, “Computer simulation of laser photon fluctuations: theory of single-cavity laser,” IEEE J. Quantum Electron. 20, 1139-1148 (1984).
[CrossRef]

D. Marcuse, “Computer simulation of laser photon fluctuations: single-cavity laser results,” IEEE J. Quantum Electron. 20, 1148-1155 (1984).
[CrossRef]

D. Marcuse, “Computer simulation of laser photon fluctuations: coupled-cavity lasers,” IEEE J. Quantum Electron. 21, 154-161 (1985).
[CrossRef]

D. Marcuse and T. P. Lee, “Rate equation model of a coupled-cavity laser,” IEEE J. Quantum Electron. 20, 166-176 (1984).
[CrossRef]

IEEE Photonics Technol. Lett. (4)

X. Liu, “A novel dual-wavelength DFB fiber laser based on symmetrical FBG structure,” IEEE Photonics Technol. Lett. 19, 632-634 (2007).
[CrossRef]

J. Sun, Y. Dai, Y. Zhang, X. Chen, and S. Xie, “Dual-wavelength DFB fiber laser based on unequalized phase shifts,” IEEE Photonics Technol. Lett. 18, 2493-2495 (2006).
[CrossRef]

F. Pozzi, R. M. DeLa Rue, and M. Sorel, “Dual-wavelength InAlGaAs-InP laterally coupled distributed feedback laser,” IEEE Photonics Technol. Lett. 18, 2563-2565 (2006).
[CrossRef]

J. J. Huang, C. C. Yang, and D. W. Huang, “Carrier capture competition between two different quantum wells in dual-wavelength semiconductor lasers,” IEEE Photonics Technol. Lett. 8, 752-755 (1996).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

David Wake, Claudio R. Lima, and Phillip A. Davies, “Optical generation of millimeter--wave signals for fiber-radio systems using a dual-mode DFB semiconductor laser,” IEEE Trans. Microwave Theory Tech. 43, 2270-2276 (1995).
[CrossRef]

J. Appl. Phys. (1)

T. L. Koch and L. A. Coldren, “Optimum coupling junction and cavity lengths for coupled-cavity semiconductor lasers,” J. Appl. Phys. 57, 740-754 (1985).
[CrossRef]

J. Lightwave Technol. (1)

G. Bjork and O. Nilsson, “A new exact and efficient numerical matrix theory of complicated laser structures: properties of asymmetric phase-shifted DFB lasers,” J. Lightwave Technol. 5, 140-146 (1987).
[CrossRef]

Phys. Lett. A (1)

A. Gearba and G. Cone, “Numerical analysis of laser mode competition and stability,” Phys. Lett. A 269, 112-119 (2000).
[CrossRef]

Other (4)

Y. Suematsu and A. R. Adams, Handbook of Semiconductor Lasers and Photonic Integrated Circuits (Chapman and Hall, 1994).

W. T. Tsang, “The cleaved-coupled-cavity (C3) laser,” in Semiconductors and Semimetals, Vol. 22, part B, R. K. Willardson, A. C. Beer, and W. T. Tsang, eds. (Academic, 1985), pp. 258-369.

K. E. Razavi and P. A. Davies, “Semiconductor laser sources for the generation of millimeter-wave signals,” in Proceedings of IEEE Conference on Optoelectronics (IEEE, 1998), pp. 159-163.
[CrossRef]

G. P. Agrawal and N. K. Dutta, “Coupled-cavity semiconductor lasers,” in Long-Wavelength Semiconductor Lasers. (Van Nostrand Reinhold, 1986), pp. 333-371.

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Figures (10)

Fig. 1
Fig. 1

Schematic diagram of the coupled-cavity dual-wavelength laser.

Fig. 2
Fig. 2

Reflectivity spectra of the two etalon filters of length 19.6 μ m (dash-dotted curve) and 61 μ m (dashed curve), respectively, both separated by a 1.25 μ m wide trench with a dielectric filling (refractive index 1.55) from the active cavity. The solid curve represents the combined filtering spectra of the two etalon filters.

Fig. 3
Fig. 3

Small signal gain spectra of the coupled-cavity laser with active cavity length L 1 = L 2 = 214.1     μ m and air gap size of 5 4 λ : (a) without the passive etalon filters; (b) with the passive etalon filters.

Fig. 4
Fig. 4

Frequency interval of the doublet modes as a function of the active cavity gain coefficient difference | g 1 g 2 | while keeping the sum of the two gain coefficients constant at the threshold condition for the cases (a) L 1 = L 2 = 428.5     μ m , g 1 + g 2 = 32.44 cm 1 and (b) L 1 = L 2 = 214.1     μ m , g 1 + g 2 = 59.91 cm 1 .

Fig. 5
Fig. 5

Normalized saturation and suppression coefficients as functions of the active cavity gain coefficient difference | g 1 g 2 | while keeping the sum of the two gain coefficients constant for the case L 1 = L 2 = 214.1     μ m , g 1 + g 2 = 59.91 cm 1 .

Fig. 6
Fig. 6

Relationship between the pumping current density J 1 and J 2 of the two active cavities for different values of photon density S and for different values of gain coefficient difference g 1 g 2 .

Fig. 7
Fig. 7

Time evolution of photon densities of the two competing modes. The two active cavities ( L 1 = L 2 = 214.1     μ m , g 1 + g 2 = 59.91 cm 1 ) are pumped in different conditions: (a) J 1 = J 2 = 3.54 J th , g 1 = g 2 = 29.95     cm 1 , β = 0.705 ε , θ = 0.663 ε ; and (b) J 1 = 6.71 J th , J 2 = 0.63 J th , g 1 = 54.95 cm 1 , g 2 = 4.95 cm 1 , β = 0.718 ε , θ = 0.711 ε . The threshold gains of the two modes are equal in both cases. Langevin noise sources are not included. The photon density curves for both modes are nearly overlapped, and cases (a) and (b) are not distinguishable in the figure.

Fig. 8
Fig. 8

Similar to Fig. 7, but with Langevin noise sources included.

Fig. 9
Fig. 9

Similar to Fig. 8b, taking into account the electronic intraband relaxation. An excess suppression of 2% is assumed. The saturation and suppression coefficients now become β = 0.704 ε , θ = 0.711 ε .

Fig. 10
Fig. 10

Time evolution of (a) the photon densities of the two competing modes and (b) the threshold gain difference adjustment between the two competing modes. The parameters are the same as in the case of Fig. 9.

Tables (1)

Tables Icon

Table 1 Parameter Values Used in the Numerical Example

Equations (44)

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

A 11 = { [ ( n + n 0 ) 2 e i ( k 1 L 1 + k 2 L 2 ) + ( n n 0 ) 2 R 1 R 2 e i ( k 1 L 1 + k 2 L 2 ) ( n 2 n 0 2 ) R 2 e i ( k 1 L 1 k 2 L 2 ) ( n 2 n 0 2 ) R 1 e i ( k 1 L 1 k 2 L 2 ) ] e i k 0 L g [ ( n n 0 ) 2 e i ( k 1 L 1 + k 2 L 2 ) + ( n + n 0 ) 2 R 1 R 2 e i ( k 1 L 1 + k 2 L 2 ) ( n 2 n 0 2 ) R 2 e i ( k 1 L 1 k 2 L 2 ) ( n 2 n 0 2 ) R 1 e i ( k 1 L 1 k 2 L 2 ) ] e i k 0 L g } 1 4 n 0 n ( 1 R 1 ) ( 1 + R 2 ) .
e ( g th α ) L = R 1 R 2 e ( g th α ) L .
g th = 1 4 L 1 n ( 1 R 1 R 2 ) + α .
A 11 = [ ( n + n 0 ) 2 ( R 1 R 2 ) 1 2 2 i sin ( 2 β L + k 0 L g ) ( n n 0 ) 2 ( R 1 R 2 ) 1 2 2 i sin ( 2 β L k 0 L g ) ( n 2 n 0 2 ) R 2 e ( g 1 g 2 ) L 2 2 i sin ( k 0 L g ) ( n 2 n 0 2 ) R 1 e ( g 1 g 2 ) L 2 2 i sin ( k 0 L g ) ] 1 4 n 0 n ( 1 R 1 ) ( 1 + R 2 ) .
sin ( 2 β L ) = 0 ,
2 n 2 + n 0 2 n 2 n 0 2 cos ( 2 β L ) = ( R 2 R 1 ) 1 4 e ( g 1 g 2 ) L 2 + ( R 1 R 2 ) 1 4 e ( g 1 g 2 ) L 2 .
Δ β = 1 2 L cos 1 [ 1 2 ( n 2 n 0 2 n 2 + n 0 2 ) ( ( R 2 R 1 ) 1 4 e ( g 1 g 2 ) L 2 + ( R 1 R 2 ) 1 4 e ( g 1 g 2 ) L 2 ) ] .
Δ β max = 1 2 L cos 1 ( n 2 n 0 2 n 2 + n 0 2 ) .
d S 1 d t = Γ v g g 1 S 1 S 1 τ p 1 + Γ β sp B N 2 ,
d S 2 d t = Γ v g g 2 S 2 S 2 τ p 2 + Γ β sp B N 2 ,
g j = h ν act g ( z ) S j ( z ) d z act W j ( z ) d z ,
g ( z ) = d g d N ( N N tr ) [ 1 ε S ( z ) ] ,
S ( z ) S 1 ( z ) + S 2 ( z ) .
g 1 = h ν d g d N ( N N tr ) act S 1 ( z ) d z ε act S 1 ( z ) S 1 ( z ) d z ε act S 1 ( z ) S 2 ( z ) d z act W 1 ( z ) d z ,
g 2 = h ν d g d N ( N N tr ) act S 2 ( z ) d z ε act S 2 ( z ) S 2 ( z ) d z ε act S 2 ( z ) S 1 ( z ) d z act W 2 ( z ) d z .
E j = E 0 j [ e j + ( z ) + e j ( z ) ] ,
S j ( z ) = E 0 j 2 | e j + ( z ) + e j ( z ) | 2 v g S 0 j | e j + ( z ) + e j ( z ) | 2 ,
W j ( z ) = h ν E 0 j 2 ( | e j + ( z ) | 2 + | e j ( z ) | 2 ) v g h ν S 0 j ( | e j + ( z ) | 2 + | e j ( z ) | 2 ) .
g 1 = d g d N ( N N tr ) ( χ β 1 S 01 θ 12 S 02 ) ,
g 2 = d g d N ( N N tr ) ( χ β 2 S 02 θ 21 S 01 ) ,
χ = act | e j + ( z ) + e j ( z ) | 2 d z act [ | e j + ( z ) | 2 + | e j ( z ) | 2 ] d z ,
β j = ε act | e j + ( z ) + e j ( z ) | 4 d z act [ | e j + ( z ) | 2 + | e j ( z ) | 2 ] d z ,
θ 12 = ε act | e 1 + ( z ) + e 1 ( z ) | 2 | e 2 + ( z ) + e 2 ( z ) | 2 d z act [ | e 1 + ( z ) | 2 + | e 1 ( z ) | 2 ] d z ,
θ 21 = ε act | e 1 + ( z ) + e 1 ( z ) | 2 | e 2 + ( z ) + e 2 ( z ) | 2 d z act [ | e 2 + ( z ) | 2 + | e 2 ( z ) | 2 ] d z ,
d S 1 d t = Γ v g d g d N ( N N tr ) ( χ β S 1 θ S 2 ) S 1 S 1 τ p 1 + Γ β sp B N 2 ,
d S 2 d t = Γ v g d g d N ( N N tr ) ( χ β S 2 θ S 1 ) S 2 S 2 τ p 2 + Γ β sp B N 2 .
d N i d t = η i I i e V i A N i B N i 2 C N i 3 j = 1 2 v g g i j S j + F N i ( t ) , i = 1 , 2
d S j d t = Γ v g 2 i = 1 2 g i j S j S j τ p j + i = 1 2 Γ β sp B N i 2 + F Sj ( t ) , j = 1 , 2
g i j = d g d N ( N i N tr ) ( χ k = 1 2 ε j k S k ) , i = 1 , 2 , j = 1 , 2
ε j k = { β , j = k θ , j k . .
a = d g d N ( χ β S θ S ) .
N i = g i a + N tr .
[ Γ v g ( g 2 + Δ 2 Γ ) 1 τ p ] S + Γ β sp B [ ( Γ g 2 + Δ Γ a + N tr ) 2 + ( g 2 a + N tr ) 2 ] = 0.
η i I i e V i = A N i + B N i 2 + C N i 3 + 2 v g g i S .
F N 1 ( t ) F N 2 ( t ) = 0 ,
F S 1 ( t ) F S 2 ( t ) = 0 ,
F N i ( t ) F N i ( t ) = V e i 2 δ ( t t ) , i = 1 , 2
F S j F S j = V ν j 2 δ j j δ ( t t ) , j = 1 , 2 ; j = 1 , 2
F N i ( t ) F S j ( t ) = r i j V e i V ν j δ ( t t ) , i = 1 , 2 ; j = 1 , 2.
V e i 2 = { η i I i e + ( A N i + B N i 2 + C N i 3 ) V i + j = 1 2 v g g i j ( + ) S j V i } V i 2 , i = 1 , 2
V ν j 2 = { 1 2 Γ v g i = 1 2 g i j ( + ) S j V P + S j V P τ p j + i = 1 2 Γ β sp B N i 2 V P } V P 2 , j = 1 , 2
r i j V e i V ν j = { Γ β sp B N i 2 V P + 1 2 Γ v g g i j ( + ) S j V P } V i V P ,
i = 1 , 2 , j = 1 , 2
g i j ( + ) = d g d N ( N i + N tr ) ( 1 k = 1 2 ε j k S k ) , i = 1 , 2 , j = 1 , 2

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