Abstract

The polarization competition in vertical-cavity surface-emitting lasers with two nearly degenerate modes is analyzed with the aim of studying their noise properties and, in particular, amplitude fluctuations and spectral linewidth. The coupling between the two modes is attributed to carriers, different spin populations, and structural anisotropies, including the consequences of the electro-optic effect that results from current injection. The model is based on quantum equations of motion with Langevin noise sources. Results for the noise spectra in the regions where mode competition is stronger are reported and discussed.

© 1999 Optical Society of America

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References

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  1. M. San Miguel, Q. Feng, and J. V. Maloney, “Light-polarization dynamics in surface-emitting semiconductor lasers,” Phys. Rev. A 52, 1728–1739 (1995).
    [CrossRef] [PubMed]
  2. M. Travagnin, M. P. van Exter, A. K. Jansen van Doorn, and P. Woerdman, “Role of optical anisotropies in the polarization properties of surface-emitting semiconductor lasers,” Phys. Rev. A 54, 1647–1660 (1996).
    [CrossRef] [PubMed]
  3. J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33, 765–783 (1997).
    [CrossRef]
  4. M. Travagnin, “Linear anisotropies and polarization properties of vertical cavity surface emitting semiconductor lasers,” Phys. Rev. A 56, 4094–5005 (1997).
    [CrossRef]
  5. M. P. van Exter, A. K. Jansen van Doorn, and J. P. Woerdman, “Electro-optic effect and birefringence in semiconductor vertical-cavity lasers,” Phys. Rev. A 56, 845–853 (1997).
    [CrossRef]
  6. K. D. Choquette, D. A. Richie, and R. E. Leibenguth, “Temperature dependence of gain-guided vertical-cavity surface emitting laser polarization,” Appl. Phys. Lett. 64, 2062–2064 (1994).
    [CrossRef]
  7. H. F. Hofmann and O. Hess, “Quantum noise and polarization fluctuations in vertical-cavity surface-emitting lasers,” Phys. Rev. A 56, 868–876 (1997).
    [CrossRef]
  8. Y. Yamamoto, S. Machida, and O. Nilsson, “Amplitude squeezing in a pump-noise suppressed laser oscillator,” Phys. Rev. A 34, 4025–4042 (1986).
    [CrossRef] [PubMed]
  9. Y. Yamamoto, S. Machida, and G. Björk, “Microcavity semiconductor laser with enhanced spontaneous emission,” Phys. Rev. A 44, 657–668 (1991).
    [CrossRef] [PubMed]
  10. D. Marcuse, “Computer simulation of laser photon fluctuations: theory of single cavity laser,” IEEE J. Quantum Electron. QE-20, 1139–1148 (1984).
    [CrossRef]
  11. M. Yamada, “Variation of intensity noise and frequency noise with spontaneous emission factor in semiconductor lasers,” IEEE J. Quantum Electron. 30, 1511–1519 (1994).
    [CrossRef]
  12. J. L. Vey, K. Auen, and W. Elsaesser, “Quantum noise properties of vertical cavity surface emitting lasers: theory and experiment,” Phys. Status Solidi B 206, 427–436 (1998).
    [CrossRef]
  13. M. P. van Exter, M. B. Willemsen, and J. P. Woerdman, “Polarization fluctuations in vertical-cavity semiconductor lasers,” Phys. Rev. A 58, 4191–4205 (1998).
    [CrossRef]
  14. T. C. Damen, L. Vina, J. E. Cunningham, J. Shah, and L. J. Sham, “Subpicosecond spin relaxation dynamics of excitons and free carriers in GaAs quantum wells,” Phys. Rev. Lett. 67, 3432–3435 (1991).
    [CrossRef] [PubMed]
  15. G. P. Bava and P. Debernardi, “Spontaneous emission in semiconductor microcavity post lasers,” IEE Proc. Optoelectron. 145, 37–42 (1998).
    [CrossRef]
  16. M. Asada and Y. Suematsu, “Density-matrix theory of semiconductors lasers with relaxation model. Gain and gain-suppression in semiconductor lasers,” IEEE J. Quantum Electron. QE-21, 434–442 (1985).
    [CrossRef]
  17. W. Chow, S. Koch, and M. Sargent, Semiconductor Laser Physics (Springer-Verlag, Berlin, 1994).
  18. H. Haug, “Quantum mechanical theory of fluctuations and relaxation in semiconductor lasers,” Z. Phys. 200, 57–68 (1967).
    [CrossRef]
  19. Y. Yamamoto and N. Imoto, “Internal and external field fluctuations of laser oscillator. I. Quantum mechanical Langevin treatment,” IEEE J. Quantum Electron. QE-22, 2032–2042 (1986).
    [CrossRef]

1998 (3)

J. L. Vey, K. Auen, and W. Elsaesser, “Quantum noise properties of vertical cavity surface emitting lasers: theory and experiment,” Phys. Status Solidi B 206, 427–436 (1998).
[CrossRef]

M. P. van Exter, M. B. Willemsen, and J. P. Woerdman, “Polarization fluctuations in vertical-cavity semiconductor lasers,” Phys. Rev. A 58, 4191–4205 (1998).
[CrossRef]

G. P. Bava and P. Debernardi, “Spontaneous emission in semiconductor microcavity post lasers,” IEE Proc. Optoelectron. 145, 37–42 (1998).
[CrossRef]

1997 (4)

J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33, 765–783 (1997).
[CrossRef]

M. Travagnin, “Linear anisotropies and polarization properties of vertical cavity surface emitting semiconductor lasers,” Phys. Rev. A 56, 4094–5005 (1997).
[CrossRef]

M. P. van Exter, A. K. Jansen van Doorn, and J. P. Woerdman, “Electro-optic effect and birefringence in semiconductor vertical-cavity lasers,” Phys. Rev. A 56, 845–853 (1997).
[CrossRef]

H. F. Hofmann and O. Hess, “Quantum noise and polarization fluctuations in vertical-cavity surface-emitting lasers,” Phys. Rev. A 56, 868–876 (1997).
[CrossRef]

1996 (1)

M. Travagnin, M. P. van Exter, A. K. Jansen van Doorn, and P. Woerdman, “Role of optical anisotropies in the polarization properties of surface-emitting semiconductor lasers,” Phys. Rev. A 54, 1647–1660 (1996).
[CrossRef] [PubMed]

1995 (1)

M. San Miguel, Q. Feng, and J. V. Maloney, “Light-polarization dynamics in surface-emitting semiconductor lasers,” Phys. Rev. A 52, 1728–1739 (1995).
[CrossRef] [PubMed]

1994 (2)

K. D. Choquette, D. A. Richie, and R. E. Leibenguth, “Temperature dependence of gain-guided vertical-cavity surface emitting laser polarization,” Appl. Phys. Lett. 64, 2062–2064 (1994).
[CrossRef]

M. Yamada, “Variation of intensity noise and frequency noise with spontaneous emission factor in semiconductor lasers,” IEEE J. Quantum Electron. 30, 1511–1519 (1994).
[CrossRef]

1991 (2)

T. C. Damen, L. Vina, J. E. Cunningham, J. Shah, and L. J. Sham, “Subpicosecond spin relaxation dynamics of excitons and free carriers in GaAs quantum wells,” Phys. Rev. Lett. 67, 3432–3435 (1991).
[CrossRef] [PubMed]

Y. Yamamoto, S. Machida, and G. Björk, “Microcavity semiconductor laser with enhanced spontaneous emission,” Phys. Rev. A 44, 657–668 (1991).
[CrossRef] [PubMed]

1986 (2)

Y. Yamamoto and N. Imoto, “Internal and external field fluctuations of laser oscillator. I. Quantum mechanical Langevin treatment,” IEEE J. Quantum Electron. QE-22, 2032–2042 (1986).
[CrossRef]

Y. Yamamoto, S. Machida, and O. Nilsson, “Amplitude squeezing in a pump-noise suppressed laser oscillator,” Phys. Rev. A 34, 4025–4042 (1986).
[CrossRef] [PubMed]

1985 (1)

M. Asada and Y. Suematsu, “Density-matrix theory of semiconductors lasers with relaxation model. Gain and gain-suppression in semiconductor lasers,” IEEE J. Quantum Electron. QE-21, 434–442 (1985).
[CrossRef]

1984 (1)

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

1967 (1)

H. Haug, “Quantum mechanical theory of fluctuations and relaxation in semiconductor lasers,” Z. Phys. 200, 57–68 (1967).
[CrossRef]

Abraham, N. B.

J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33, 765–783 (1997).
[CrossRef]

Asada, M.

M. Asada and Y. Suematsu, “Density-matrix theory of semiconductors lasers with relaxation model. Gain and gain-suppression in semiconductor lasers,” IEEE J. Quantum Electron. QE-21, 434–442 (1985).
[CrossRef]

Auen, K.

J. L. Vey, K. Auen, and W. Elsaesser, “Quantum noise properties of vertical cavity surface emitting lasers: theory and experiment,” Phys. Status Solidi B 206, 427–436 (1998).
[CrossRef]

Bava, G. P.

G. P. Bava and P. Debernardi, “Spontaneous emission in semiconductor microcavity post lasers,” IEE Proc. Optoelectron. 145, 37–42 (1998).
[CrossRef]

Björk, G.

Y. Yamamoto, S. Machida, and G. Björk, “Microcavity semiconductor laser with enhanced spontaneous emission,” Phys. Rev. A 44, 657–668 (1991).
[CrossRef] [PubMed]

Choquette, K. D.

K. D. Choquette, D. A. Richie, and R. E. Leibenguth, “Temperature dependence of gain-guided vertical-cavity surface emitting laser polarization,” Appl. Phys. Lett. 64, 2062–2064 (1994).
[CrossRef]

Cunningham, J. E.

T. C. Damen, L. Vina, J. E. Cunningham, J. Shah, and L. J. Sham, “Subpicosecond spin relaxation dynamics of excitons and free carriers in GaAs quantum wells,” Phys. Rev. Lett. 67, 3432–3435 (1991).
[CrossRef] [PubMed]

Damen, T. C.

T. C. Damen, L. Vina, J. E. Cunningham, J. Shah, and L. J. Sham, “Subpicosecond spin relaxation dynamics of excitons and free carriers in GaAs quantum wells,” Phys. Rev. Lett. 67, 3432–3435 (1991).
[CrossRef] [PubMed]

Debernardi, P.

G. P. Bava and P. Debernardi, “Spontaneous emission in semiconductor microcavity post lasers,” IEE Proc. Optoelectron. 145, 37–42 (1998).
[CrossRef]

Elsaesser, W.

J. L. Vey, K. Auen, and W. Elsaesser, “Quantum noise properties of vertical cavity surface emitting lasers: theory and experiment,” Phys. Status Solidi B 206, 427–436 (1998).
[CrossRef]

Feng, Q.

M. San Miguel, Q. Feng, and J. V. Maloney, “Light-polarization dynamics in surface-emitting semiconductor lasers,” Phys. Rev. A 52, 1728–1739 (1995).
[CrossRef] [PubMed]

Haug, H.

H. Haug, “Quantum mechanical theory of fluctuations and relaxation in semiconductor lasers,” Z. Phys. 200, 57–68 (1967).
[CrossRef]

Hess, O.

H. F. Hofmann and O. Hess, “Quantum noise and polarization fluctuations in vertical-cavity surface-emitting lasers,” Phys. Rev. A 56, 868–876 (1997).
[CrossRef]

Hofmann, H. F.

H. F. Hofmann and O. Hess, “Quantum noise and polarization fluctuations in vertical-cavity surface-emitting lasers,” Phys. Rev. A 56, 868–876 (1997).
[CrossRef]

Imoto, N.

Y. Yamamoto and N. Imoto, “Internal and external field fluctuations of laser oscillator. I. Quantum mechanical Langevin treatment,” IEEE J. Quantum Electron. QE-22, 2032–2042 (1986).
[CrossRef]

Leibenguth, R. E.

K. D. Choquette, D. A. Richie, and R. E. Leibenguth, “Temperature dependence of gain-guided vertical-cavity surface emitting laser polarization,” Appl. Phys. Lett. 64, 2062–2064 (1994).
[CrossRef]

Machida, S.

Y. Yamamoto, S. Machida, and G. Björk, “Microcavity semiconductor laser with enhanced spontaneous emission,” Phys. Rev. A 44, 657–668 (1991).
[CrossRef] [PubMed]

Y. Yamamoto, S. Machida, and O. Nilsson, “Amplitude squeezing in a pump-noise suppressed laser oscillator,” Phys. Rev. A 34, 4025–4042 (1986).
[CrossRef] [PubMed]

Maloney, J. V.

M. San Miguel, Q. Feng, and J. V. Maloney, “Light-polarization dynamics in surface-emitting semiconductor lasers,” Phys. Rev. A 52, 1728–1739 (1995).
[CrossRef] [PubMed]

Marcuse, D.

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

Martin-Regalado, J.

J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33, 765–783 (1997).
[CrossRef]

Nilsson, O.

Y. Yamamoto, S. Machida, and O. Nilsson, “Amplitude squeezing in a pump-noise suppressed laser oscillator,” Phys. Rev. A 34, 4025–4042 (1986).
[CrossRef] [PubMed]

Prati, F.

J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33, 765–783 (1997).
[CrossRef]

Richie, D. A.

K. D. Choquette, D. A. Richie, and R. E. Leibenguth, “Temperature dependence of gain-guided vertical-cavity surface emitting laser polarization,” Appl. Phys. Lett. 64, 2062–2064 (1994).
[CrossRef]

San Miguel, M.

J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33, 765–783 (1997).
[CrossRef]

M. San Miguel, Q. Feng, and J. V. Maloney, “Light-polarization dynamics in surface-emitting semiconductor lasers,” Phys. Rev. A 52, 1728–1739 (1995).
[CrossRef] [PubMed]

Shah, J.

T. C. Damen, L. Vina, J. E. Cunningham, J. Shah, and L. J. Sham, “Subpicosecond spin relaxation dynamics of excitons and free carriers in GaAs quantum wells,” Phys. Rev. Lett. 67, 3432–3435 (1991).
[CrossRef] [PubMed]

Sham, L. J.

T. C. Damen, L. Vina, J. E. Cunningham, J. Shah, and L. J. Sham, “Subpicosecond spin relaxation dynamics of excitons and free carriers in GaAs quantum wells,” Phys. Rev. Lett. 67, 3432–3435 (1991).
[CrossRef] [PubMed]

Suematsu, Y.

M. Asada and Y. Suematsu, “Density-matrix theory of semiconductors lasers with relaxation model. Gain and gain-suppression in semiconductor lasers,” IEEE J. Quantum Electron. QE-21, 434–442 (1985).
[CrossRef]

Travagnin, M.

M. Travagnin, “Linear anisotropies and polarization properties of vertical cavity surface emitting semiconductor lasers,” Phys. Rev. A 56, 4094–5005 (1997).
[CrossRef]

M. Travagnin, M. P. van Exter, A. K. Jansen van Doorn, and P. Woerdman, “Role of optical anisotropies in the polarization properties of surface-emitting semiconductor lasers,” Phys. Rev. A 54, 1647–1660 (1996).
[CrossRef] [PubMed]

van Doorn, A. K. Jansen

M. P. van Exter, A. K. Jansen van Doorn, and J. P. Woerdman, “Electro-optic effect and birefringence in semiconductor vertical-cavity lasers,” Phys. Rev. A 56, 845–853 (1997).
[CrossRef]

M. Travagnin, M. P. van Exter, A. K. Jansen van Doorn, and P. Woerdman, “Role of optical anisotropies in the polarization properties of surface-emitting semiconductor lasers,” Phys. Rev. A 54, 1647–1660 (1996).
[CrossRef] [PubMed]

van Exter, M. P.

M. P. van Exter, M. B. Willemsen, and J. P. Woerdman, “Polarization fluctuations in vertical-cavity semiconductor lasers,” Phys. Rev. A 58, 4191–4205 (1998).
[CrossRef]

M. P. van Exter, A. K. Jansen van Doorn, and J. P. Woerdman, “Electro-optic effect and birefringence in semiconductor vertical-cavity lasers,” Phys. Rev. A 56, 845–853 (1997).
[CrossRef]

M. Travagnin, M. P. van Exter, A. K. Jansen van Doorn, and P. Woerdman, “Role of optical anisotropies in the polarization properties of surface-emitting semiconductor lasers,” Phys. Rev. A 54, 1647–1660 (1996).
[CrossRef] [PubMed]

Vey, J. L.

J. L. Vey, K. Auen, and W. Elsaesser, “Quantum noise properties of vertical cavity surface emitting lasers: theory and experiment,” Phys. Status Solidi B 206, 427–436 (1998).
[CrossRef]

Vina, L.

T. C. Damen, L. Vina, J. E. Cunningham, J. Shah, and L. J. Sham, “Subpicosecond spin relaxation dynamics of excitons and free carriers in GaAs quantum wells,” Phys. Rev. Lett. 67, 3432–3435 (1991).
[CrossRef] [PubMed]

Willemsen, M. B.

M. P. van Exter, M. B. Willemsen, and J. P. Woerdman, “Polarization fluctuations in vertical-cavity semiconductor lasers,” Phys. Rev. A 58, 4191–4205 (1998).
[CrossRef]

Woerdman, J. P.

M. P. van Exter, M. B. Willemsen, and J. P. Woerdman, “Polarization fluctuations in vertical-cavity semiconductor lasers,” Phys. Rev. A 58, 4191–4205 (1998).
[CrossRef]

M. P. van Exter, A. K. Jansen van Doorn, and J. P. Woerdman, “Electro-optic effect and birefringence in semiconductor vertical-cavity lasers,” Phys. Rev. A 56, 845–853 (1997).
[CrossRef]

Woerdman, P.

M. Travagnin, M. P. van Exter, A. K. Jansen van Doorn, and P. Woerdman, “Role of optical anisotropies in the polarization properties of surface-emitting semiconductor lasers,” Phys. Rev. A 54, 1647–1660 (1996).
[CrossRef] [PubMed]

Yamada, M.

M. Yamada, “Variation of intensity noise and frequency noise with spontaneous emission factor in semiconductor lasers,” IEEE J. Quantum Electron. 30, 1511–1519 (1994).
[CrossRef]

Yamamoto, Y.

Y. Yamamoto, S. Machida, and G. Björk, “Microcavity semiconductor laser with enhanced spontaneous emission,” Phys. Rev. A 44, 657–668 (1991).
[CrossRef] [PubMed]

Y. Yamamoto, S. Machida, and O. Nilsson, “Amplitude squeezing in a pump-noise suppressed laser oscillator,” Phys. Rev. A 34, 4025–4042 (1986).
[CrossRef] [PubMed]

Y. Yamamoto and N. Imoto, “Internal and external field fluctuations of laser oscillator. I. Quantum mechanical Langevin treatment,” IEEE J. Quantum Electron. QE-22, 2032–2042 (1986).
[CrossRef]

Appl. Phys. Lett. (1)

K. D. Choquette, D. A. Richie, and R. E. Leibenguth, “Temperature dependence of gain-guided vertical-cavity surface emitting laser polarization,” Appl. Phys. Lett. 64, 2062–2064 (1994).
[CrossRef]

IEE Proc. Optoelectron. (1)

G. P. Bava and P. Debernardi, “Spontaneous emission in semiconductor microcavity post lasers,” IEE Proc. Optoelectron. 145, 37–42 (1998).
[CrossRef]

IEEE J. Quantum Electron. (5)

M. Asada and Y. Suematsu, “Density-matrix theory of semiconductors lasers with relaxation model. Gain and gain-suppression in semiconductor lasers,” IEEE J. Quantum Electron. QE-21, 434–442 (1985).
[CrossRef]

Y. Yamamoto and N. Imoto, “Internal and external field fluctuations of laser oscillator. I. Quantum mechanical Langevin treatment,” IEEE J. Quantum Electron. QE-22, 2032–2042 (1986).
[CrossRef]

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

M. Yamada, “Variation of intensity noise and frequency noise with spontaneous emission factor in semiconductor lasers,” IEEE J. Quantum Electron. 30, 1511–1519 (1994).
[CrossRef]

J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33, 765–783 (1997).
[CrossRef]

Phys. Rev. A (8)

M. Travagnin, “Linear anisotropies and polarization properties of vertical cavity surface emitting semiconductor lasers,” Phys. Rev. A 56, 4094–5005 (1997).
[CrossRef]

M. P. van Exter, A. K. Jansen van Doorn, and J. P. Woerdman, “Electro-optic effect and birefringence in semiconductor vertical-cavity lasers,” Phys. Rev. A 56, 845–853 (1997).
[CrossRef]

M. San Miguel, Q. Feng, and J. V. Maloney, “Light-polarization dynamics in surface-emitting semiconductor lasers,” Phys. Rev. A 52, 1728–1739 (1995).
[CrossRef] [PubMed]

M. Travagnin, M. P. van Exter, A. K. Jansen van Doorn, and P. Woerdman, “Role of optical anisotropies in the polarization properties of surface-emitting semiconductor lasers,” Phys. Rev. A 54, 1647–1660 (1996).
[CrossRef] [PubMed]

H. F. Hofmann and O. Hess, “Quantum noise and polarization fluctuations in vertical-cavity surface-emitting lasers,” Phys. Rev. A 56, 868–876 (1997).
[CrossRef]

Y. Yamamoto, S. Machida, and O. Nilsson, “Amplitude squeezing in a pump-noise suppressed laser oscillator,” Phys. Rev. A 34, 4025–4042 (1986).
[CrossRef] [PubMed]

Y. Yamamoto, S. Machida, and G. Björk, “Microcavity semiconductor laser with enhanced spontaneous emission,” Phys. Rev. A 44, 657–668 (1991).
[CrossRef] [PubMed]

M. P. van Exter, M. B. Willemsen, and J. P. Woerdman, “Polarization fluctuations in vertical-cavity semiconductor lasers,” Phys. Rev. A 58, 4191–4205 (1998).
[CrossRef]

Phys. Rev. Lett. (1)

T. C. Damen, L. Vina, J. E. Cunningham, J. Shah, and L. J. Sham, “Subpicosecond spin relaxation dynamics of excitons and free carriers in GaAs quantum wells,” Phys. Rev. Lett. 67, 3432–3435 (1991).
[CrossRef] [PubMed]

Phys. Status Solidi B (1)

J. L. Vey, K. Auen, and W. Elsaesser, “Quantum noise properties of vertical cavity surface emitting lasers: theory and experiment,” Phys. Status Solidi B 206, 427–436 (1998).
[CrossRef]

Z. Phys. (1)

H. Haug, “Quantum mechanical theory of fluctuations and relaxation in semiconductor lasers,” Z. Phys. 200, 57–68 (1967).
[CrossRef]

Other (1)

W. Chow, S. Koch, and M. Sargent, Semiconductor Laser Physics (Springer-Verlag, Berlin, 1994).

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Figures (6)

Fig. 1
Fig. 1

Stability regions of two-mode operation in the F0P plane. White region, mode 1; black areas, mode 2; hatched areas, both modes stable. At the borders of the hatched regions the small shaded areas correspond to modes with higher ellipticity.

Fig. 2
Fig. 2

Characteristics of the elliptical polarization (tilting angle and axis ratio) in the smooth transition region between modes 1 and 2 versus P for three values of F0: continuous curves; F0=0.08 GHz; dotted curves, F0=0.3 GHz; dashed curves, F0=0.5 GHz.

Fig. 3
Fig. 3

Same as Fig. 1, but here the electro-optic effect has been neglected.

Fig. 4
Fig. 4

NPFS versus P for three values of F0; the values of F0 and the types of curve are the same as in Fig. 2.

Fig. 5
Fig. 5

NPFS versus F0 for two values of P. Continuous curves, γr is due completely to the back mirror; dashed curves, γr is due completely to the output mirror.

Fig. 6
Fig. 6

Spectral linewidth ΔfPout versus P for three values of F0; the values of F0 and the types of curve are the same as in Fig. 2.

Equations (113)

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

glη=(ωcv2/4ωl)1/2El·dη,
daldt=-iωlal-(i/) Va dV k,η glη*pη+-½γllal-½γlmam+Lal,
dpηdt=iωcvpη-(i/) l al+glη*(ncη-nvη)-γppη+Lpη,
dncηdt=-(i/) l(glηpηal-glη*al+pη+)+Λc-Rcη+γc(fcη-ncη)+γcs(ncη-ncη)+Lncη,
dnvηdt=+(i/) l(glηpηal-glη*al+pη+)+Λv-Rvη+γv(fvη-nvη)+γvs(nvη-nvη)+Lnvη.
nv=nv=nv.
ncη=nc+sηn dn¯cdN¯,
nc=½(nc+nc)
N=Va dV k(nc+nc)
n=Va dV k(nc-nc)
L(t)L+(s)=2Dδ(t-s),
2Dam+al=γml*nth,
2Damal+=γml(1+nth),
2Dpηpη+=2γpn¯cη(1-n¯v)=X+sηn¯x,
2Dpη+pη=2γpn¯v(1-n¯cη)=Y+sηn¯y,
2Dnc,vηnc,vη=Λc,v+Rc,vη,
2Dnc,vηnc,vη=γc,vs ηn¯c,vη(1-n¯c,vη),
X=2γp(1-fv)fc,
Y=2γp(1-fc)fv,
x=2γp(1-fv) dfcdN¯,
y=-2γpfv dfcdN¯.
am=Am exp(-iωt),
pη=Pη exp(iωt)
Pη=1m gmη*Dk(ncη-nv)Am++i LpηDkexp(-iωt),
Dk=(ωcv-ω)+iγp.
Va dVgmη*glη
=Mml2 Va dV(Emx-sηiEmy)(Elx+sηiEly)
=Mml2(Cml+sηiC¯ml),
Mml2=Rk2ωcv24εaωmωlRk2ωcv24εaω=M2,
Cml=Va εa(EmxElx+EmyEly)dV,
C¯ml=Va εa(EmxEly-EmyElx)dV.
dA1dt=[i(ω-ω1)+½(G-γ11)]A1-½(γ12-ign)A2+LA1,
dA2dt=[i(ω-ω2)+½(G-γ22)]A2-½(γ21+ign)A1+LA2,
dNdt=Iq-R-Gr(A1+A1+A2+A2)-igrn(A1+A2-A2+A1)+LN,
dndt=-γnn-grn(A1+A1+A2+A2)-iGr(A1+A2-A2+A1)+Ln,
G=Gr+iGi=-4i k M2C2Dk*(fc-fv)
g=gr+igi=-4i k M2C2Dk*dfcdN¯.
A=A1A2=exp(iϕ1+iΔϕˆ1)00exp(iϕ2+iΔϕˆ2)×M1+ΔMˆ1M2+ΔMˆ2=exp[i(ϕD+ΔϕˆD)](M+ΔMˆ),
α=α0+Δα.
dα0dt=L(α0)=0,
L(α0)=(L+δ)M1-[(γr+½gin¯)cos φ+(γi-½grn¯)sin φ]M2(L-δ)M2-[(γr-½gin¯)cos φ+(γi-½grn¯)sin φ]M1Σ+[(M2/M1)+(M1/M2)](γi-½grn¯)cos φ+{[(M2/M1)+(M1/M2)]γr+[(M2/M1)-(M1/M2)]½gin¯}sin φIq-R-Gr(M12+M22)+2grn¯M1M2 sin φ-γnn¯-grn¯(M12+M22)+2GrM1M2 sin φ,
F=Σ2π=F0-0.067P[GHz]
δ=(3.5+1.2P)10-3[ns-1],
H=i(A+KA0-A0+K*TA),
KK*T=γu=γ11γ21γ12γ22u,
σ=σ1σ2=K*TA-R0fe,
d(Δα)dt=JΔα+f,
LMm=½[LAm exp(-iϕm)+exp(+iϕm)LAm+],
Lϕm=(1/2iMm)[LAm exp(-iϕm)-exp(+iϕm)LAm+],
Lφ=Lϕ2-Lϕ1
LA=LA+Kfe,
ΘT=12exp(-iφ1)×100exp(-iφ)i 1M1-i exp(-iφ)M20000.
Δα(Ω)=Λ(Ω)f(Ω),
Λ=(iΩD-J)-1;
f(Ω)f+(Ω)=2π2Dδ(Ω-Ω),
fe(Ω)fe+(Ω)=2πIδ(Ω-Ω),
fe+(Ω)fe(Ω)=fe(Ω)fe(Ω)=fe+(Ω)fe+(Ω)=0,
Aexp(iϕD)M+exp(iϕD)ΔM+i exp(iϕD)MDΔϕ,
ΔΦ=MTΓΔM+ΔM+ΓM+iMT(ΓΔϕ-Δϕ+Γ)M-R0[MT exp(-iϕD)Kfe+feTK*T exp(iϕD)M],
MTΓΔM+ΔM+ΓM=2MT Re(Γ)ΔM,
MT(ΓΔϕ-Δϕ+Γ)M=M1M2(Γ12-Γ21)Δφ,
ΔΦ=RΔm-R0(S*Tfe+fe+S),
2πδ(Ω-Ω)SΔΦ(Ω)=ΔΦ(Ω)ΔΦ+(Ω),
SΔΦ(Ω)=RΛ2DΛ*TRT+R0MTΓM-2R0R Re[ΛΘ exp(iϕD)Γ]M.
NPFS=SΔΦ(0)Φ0.
I(Ω)= dt exp(-iΩt)Ab(t)Ab+(0),
I(Ω)= dt exp(-iΩt)i,j (σb(t)σb+(0))i,j,
I(Ω)=i,j MCij(Ω)M,
Cij(Ω)= dt exp(-iΩt)exp{i[ϕi(t)-ϕj(0)]}.
Cij(Ω)= dt exp(-iΩt)exp{-½[ϕi(t)-ϕj(0)]2}.
[ϕi(t)-ϕj(0)]2
=12π[ϕi(ω)ϕi*(ω)+ϕj(ω)ϕj*(ω)
-2 Re(ϕi(ω)ϕj*(ω)exp(iωt))]dω
αijt,
ϕi(ω)ϕi*(ω)=(Λϕ2DϕΛϕ)ij,
LAm=Lam exp(iωt)-1VadV kη gmη*Dk*Lpη+ exp(iωt),
LN=Va dV k(Lnc+Lnc),
Ln=Va dV k(Lnc-Lnc),
Lncη=Lncη+1Lpη exp(-iωt)m gmηDkAm+Lpη+ exp(iωt)m gmη*Dk*Am+
D=DM1M1DM1M2*DM1φ*DM1N*DM1n*DM1M2DM2M2DM2φ*DM2N*DM2N*DM1φDM2φDφφDφN*Dφn*DM1NDM2NDφNDNNDNn*DM1nDM2nDφnDNnDnn.
DMmMl=¼[DAmAl+ exp(iφlm)+DAm+Al exp(-iφlm)],
DMmϕl=14iMl[-DAmAl+ exp(iφlm)+DAm+Al exp(-iφlm)],
DMmφ=DMmφ2-DMmφ1,
Dφφ=DM1M1M12+DM2M2M22-DM1M2M1M2-DM2M1M1M2,
2DAm+Al=2Dam+a1+Va dV kη gmηglη*2|Dk|22Dpηpη+
=2Dam+al+kη M22|Dk|2[(XCml-inxC¯ml)],
2DAmAl+=2Damal++Va dV kη gmη*glη2|Dk|22Dpη+pη
=2Damal++kη M22|Dk|2[(YCml+inyC¯ml)].
S=kη 2γpM2C2|Dk|2(1-fv)fc
s=kη 2γpM2C2|Dk|2(1-fv) dfcdN,
kη M2C2|Dk|2(X+Y)=2S-Gr,
kη M2C2|Dk|2(X-Y)=Gr,
kη M2C2|Dk|2(x+y)=2s-gr,
kη M2C2|Dk|2(x-y)=gr.
2DMmMl=¼{γml exp(iφlm)+cos φlm[Cml(2S-Gr)-ingrC¯ml]+i sin φlm[-GrCml+in(2s-gr)C¯ml]},
2DMmϕl=14iMl{-γml exp(iφlm)+cos φlm[CmlGr-in(2s-gr)C¯ml]+i sin φlm[-(2S-Gr)Cml+ingrC¯ml]}.
[LAm exp(-iϕm)±exp(iϕm)LAm+](Lnc±Lnc),
2DAmncη exp(-iϕm)
=-l gmη*glηMl exp(iφlm)k 2Dpη+pη2|Dk|2,
2DAm+ncη exp(iϕm)
=-l gmηglη*Ml exp(-iφlm)k 2Dpηpη+2|Dk|2.
2DMlN=-½{CMl(2S-Gr)-nMmC¯lm[igr cos φml+(2s-gr)sin φml]},
2DMln=-½{-C¯lmMm[(2S-Gr)sin φml+iGr cos φml]+nCMl(2s-gr)},
2DϕlN=-(1/2iMl){-GrCMl+nC¯lmMm[gr sin φml+i(2s-gr)cos φml]},
2Dϕln=-(1/2iMl){C¯lmMm[Gr sin φml+i(2S-Gr)cos φml]-nCMlgr}.
DNN=Va dV k(Dncnc+Dncnc+2Dncnc,
Dnn=Va dV k(Dncnc+Dncnc-2Dncnc),
DNn=Va dV k(Dncnc-Dncnc).
Dncηncη=Dncηncη+(Dpη+pη+Dpηpη+)×mlgmηglη*2|Dk|2MmMl exp(iφml),
Dncηncη=Dncηncη,
2DNN=R+C(2S-Gr)(M12+M22)-2nCM1M2(2s-gr)sin φ+2γcsn2 kdfcdN¯2,
2Dnn=R+C(2S-Gr)(M12+M22)-2nCM1M2(2s-gr)sin φ+2γcs k(2fc-fc2),
2DNn=dRdNn+nC(2s-gr)(M12+M22)-2CM1M2(2S-Gr)sin φ.

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