Abstract

We analyze the influence of thermal effects on the polarization-resolved light-current (LI) characteristics of vertical-cavity surface-emitting lasers (VCSELs). We use a model that is an extension of the spin-flip model incorporating material gain that is frequency and temperature dependent, and a rate equation for the temperature of the active region, which takes into account decay to a fixed substrate temperature, Joule heating and nonradiative recombination heating. The model also incorporates the red shift for increasing temperature of the gain curve and of the cavity resonance. The temperature sensitivity of the lasing threshold current is found to be in good qualitative agreement with observations and with previous reports based on detailed microscopic models. The temperature dependence of the polarization switching point, when the dominant polarization turn off and the orthogonal polarization emerges, is characterized in terms of various model parameters, such as the room-temperature gain-cavity offset, the subtracte temperature, and the size of the active region.

© 2008 Optical Society of America

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  1. C. Lei, and J. K. Guenter, eds., Vertical-Cavity Surface-Emitting Lasers X II, Proc. SPIE 6908, 2008.
  2. Y. Suematsu and K. Iga, "Semiconductor lasers in photonics," J. Lightwave Technol. 26, 1132-1144 (2008).
    [CrossRef]
  3. S. Mogg, N. Chitica, U. Christiansson, R. Schatz, P. Sundgren, C. Asplund, and M. Hammar, "Temperature sensitivity of the threshold current of long-wavelength InGaAs-GaAsVCSELs with large gain-cavity detuning," IEEE J. Quantum Electron. 40, 453-462 (2004).
    [CrossRef]
  4. E. S. Bjorlin, J. Geske, M. Mehta, J. Piprek, and J. E. Bowers, "Temperature dependence of the relaxation resonance frequency of long-wavelength vertical-cavity lasers," IEEE Photon. Technol. Lett. 17, 944-946 (2005).
    [CrossRef]
  5. A. Mircea, A. Caliman, V. Iakovlev, A. Mereuta, G. Suruceanu, C. A. Berseth, P. Royo, A. Syrbu, and E. Kapon, "Cavity mode-gain peak tradeoff for 1320-nm wafer-fused VCSELs with 3-mW single-mode emission power and 10-Gb/s modulation speed up to 70 degrees C," IEEE Photon. Technol. Lett. 19, 121-123 (2007).
    [CrossRef]
  6. C. J. Chang-Hasnain, J. P. Harbison, G. Hasnain, A. C. Vonlehmen, L. T. Florez and N. G. Stoffel, "Dynamic, polarization and transverse-mode characteristics of vertical cavity surface emitting lasers," IEEE J. Quantum Electron. 27, 1402-1409 (1991).
    [CrossRef]
  7. K. D. Choquette, R. P. Schneider, K. L. Lear and R. E. Leibenguth, "Gain-dependent polarization properties of vertical-cavity lasers," IEEE J. Sel. Top. Quantum Electron. 1, 661-666 (1995).
    [CrossRef]
  8. M. B. Willemsen, M. U. F. Khalid, M. P. van Exter, and J. P. Woerdman, "Polarization switching of a vertical-cavity semiconductor laser as a Kramers hopping problem," Phys. Rev. Lett. 82, 4815-4818 (1999).
    [CrossRef]
  9. J. Kaiser, C. Degen, and W. Elsasser, "Polarization-switching influence on the intensity noise of vertical-cavity surface-emitting lasers," J. Opt. Soc. Am. B 19, 672-677 (2002).
    [CrossRef]
  10. J. Paul, C. Masoller, Y. Hong, P. S. Spencer, and K. A. Shore, "Experimental study of polarization switching of vertical-cavity surface-emitting lasers as a dynamical bifurcation,"Opt. Lett. 31, 748-750 (2006).
    [CrossRef] [PubMed]
  11. 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]
  12. S. Balle, E. Tolkachova, M. San Miguel, J. R. Tredicce, J. Martin-Regalado, and A. Gahl, "Mechanisms of polarization switching in single-transverse-mode vertical-cavity surface-emitting lasers: thermal shift and nonlinear semiconductor dynamics," Opt. Lett. 24, 1121-1123 (1999).
    [CrossRef]
  13. M. Sondermann, M. Weinkath, T. Ackemann, J. Mulet, and S. Balle, "Two-frequency emission and polarization dynamics at lasing threshold in vertical-cavity surface-emitting lasers," Phys. Rev. A. 68, 033822 (2003).
    [CrossRef]
  14. C. Z. Ning and J. V. Moloney, "Thermal effects on the threshold of vertical-cavity surface-emitting lasers: first- and second-order phase transitions," Opt. Lett. 20, 1151-1153 (1995).
    [CrossRef] [PubMed]
  15. T. Rossler, R. A. Indik, G. K. Harkness, J. V. Moloney, and C. Z. Ning, "Modeling the interplay of thermal effects and transverse mode behavior in native-oxide-confined vertical-cavity surfaceemitting lasers," Phys. Rev. A. 58, 3279-3292 (1998).
    [CrossRef]
  16. S. F. Yu, W. N. Wong, P. Shum, and E. H. Li, "Theoretical analysis of modulation response and second-order harmonic distortion in vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 32, 2139-2147 (1996).
    [CrossRef]
  17. P. V. Mena, J. J. Morikuni, S.-M. Kang, A. V. Harton, and K. W. Wyat, "A comprehensive circuit-level model of vertical-cavity surface-emitting lasers," J. Lightwave Technol. 17, 2612-2632 (1999).
    [CrossRef]
  18. H. C. Schneider, A. J. Fischer, W. W. Chow, and J. F. Klem, "Temperature dependence of laser threshold in an InGaAsN vertical-cavity surface-emitting laser," Appl. Phys. Lett. 78, 3391-3393 (2001).
    [CrossRef]
  19. C. Degen, I. Fischer, W. Elsasser, L. Fratta, P. Debernardi, G. P. Bava, M. Brunner, R. Hovel, M. Moser, and K. Gulden, "Transverse modes in thermally detuned oxide-confined vertical-cavity surface-emitting lasers," Phys. Rev. A 63, 023817 (2001).
    [CrossRef]
  20. Y. Liu, W. C. Ng, K. D. Choquette, and K. Hess, "Numerical investigation of self-heating effects of oxide-confined vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 41, 15-25 (2005).
    [CrossRef]
  21. C. Chen, P. O. Leisher, A. A. Allerman, K. M. Geib, and K. D. Choquette, "Temperature analysis of threshold current in infrared vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 42, 1078-1083 (2006).
    [CrossRef]
  22. C. Degen, I. Fischer, and W. Elsasser, "Thermally induced local gain suppression in vertical-cavity surface-emitting lasers," Appl. Phys. Lett. 76, 3352-3354 (2000).
    [CrossRef]
  23. J. S. Gustavsson, J. A. Vukusic, J. Bengtsson, and A. Larsson, "A comprehensive model for the modal dynamics of vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 38, 203-212 (2002).
    [CrossRef]
  24. P. V. Paulau, A. J. Scroggie, A. Naumenko, T. Ackemann, N. A. Loiko, and W. J. Firth, "Localized traveling waves in vertical-cavity surface-emitting lasers with frequency-selective optical feedback," Phys. Rev. E 75, 056208 (2007).
    [CrossRef]
  25. L. Spinelli, G. Tissoni, L. A. Lugiato, and M. Brambilla, "Thermal effects and transverse structures in semiconductor microcavities with population inversion," Phys. Rev. A 66, 023817 (2002).
    [CrossRef]
  26. C. Masoller, T. Sorrentino, M. Chevrollier, and M. Oria, "Bistability in semiconductor lasers with polarization-rotated frequency-dependent optical feedback," IEEE J. Quantum Electron. 43, 261-268 (2007).
    [CrossRef]
  27. F. Prati, P. Caccia and F. Castelli, "Effects of gain saturation on polarization switching in verticalcavity surface-emitting lasers," Phys. Rev. A 66, 063811 (2002).
    [CrossRef]
  28. S. Barland, P. Spinicelli, G. Giacomelli, and F. Marin, "Measurement of the working parameters of an air-post vertical-cavity surface-emitting laser," IEEE J. Quantum Electron. 41, 1235-1243 (2005).
    [CrossRef]
  29. E. L. Blansett, M. G. Raymer, G. Khitrova, H. M. Gibbs, D. K. Serkland, A. A. Allerman, and K. M. Geib, "Ultrafast polarization dynamics and noise in pulsed vertical-cavity surface-emitting lasers," Opt. Express 9, 312-318 (2001).
    [CrossRef] [PubMed]
  30. G. Van der Sande, M. Peeters, I. Veretennicoff, J. Danckaert, G. Verschaffelt, and S. Balle, "The effects of stress, temperature, and spin flips on polarization switching in vertical-cavity surfaceemitting lasers," IEEE J. Quantum Electron. 42, 898-906 (2006).
    [CrossRef]
  31. A. Homayounfar and M. J. Adams, "Analysis of SFM dynamics in solitary and optically-injected VCSELs," Opt. Express 15, 10504-10519 (2007).
    [CrossRef] [PubMed]
  32. C. Carlsson, H. Martinsson, R. Schatz, J. Halonen, and A. Larsson, "Analog modulation properties of oxide confined VCSELs at microwave frequencies," J. Lightwave Technol. 20, 1740-1749 (2002).
    [CrossRef]
  33. G. Verschaffelt, J. Albert, B. Nagler, M. Peeters, J. Danckaert, S. Barbay, G. Giacomelli, and F. Marin, "Frequency response of polarization switching in vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 39, 1177-1186 (2003).
    [CrossRef]
  34. C. Degen, I. Fischer and W. Elsasser, "Transverse modes in oxide confined VCSELs: Influence of pump profile, spatial hole burning, and thermal effects," Opt. Express 5, 38-47 (1999)
    [CrossRef] [PubMed]
  35. D. M. Grasso and K. D. Choquette, "Temperature-dependent polarization characteristics of composite-resonator vertical-cavity lasers," IEEE J. Quantum Electron. 41, 127-131 (2005).
    [CrossRef]
  36. J. Rudolph, S. Dohrmann, D. Hagele, M. Oestreich, and W. Stolz, "Room-temperature threshold reduction in vertical-cavity surface-emitting lasers by injection of spin-polarized electrons," Appl. Phys. Lett. 87, 241117 (2005).
    [CrossRef]
  37. M. Holub, J. Shin, D. Saha, and P. Bhattacharya, "Electrical spin injection and threshold reduction in a semiconductor laser," Phys. Rev. Lett. 98, 146603 (2007).
    [CrossRef] [PubMed]
  38. G. P. Agrawal and N. K. Dutta, Semiconductor Lasers, 2nd ed. (Amsterdam, The Netherlands: Kluwer, 1993).

2008 (1)

2007 (5)

A. Homayounfar and M. J. Adams, "Analysis of SFM dynamics in solitary and optically-injected VCSELs," Opt. Express 15, 10504-10519 (2007).
[CrossRef] [PubMed]

A. Mircea, A. Caliman, V. Iakovlev, A. Mereuta, G. Suruceanu, C. A. Berseth, P. Royo, A. Syrbu, and E. Kapon, "Cavity mode-gain peak tradeoff for 1320-nm wafer-fused VCSELs with 3-mW single-mode emission power and 10-Gb/s modulation speed up to 70 degrees C," IEEE Photon. Technol. Lett. 19, 121-123 (2007).
[CrossRef]

P. V. Paulau, A. J. Scroggie, A. Naumenko, T. Ackemann, N. A. Loiko, and W. J. Firth, "Localized traveling waves in vertical-cavity surface-emitting lasers with frequency-selective optical feedback," Phys. Rev. E 75, 056208 (2007).
[CrossRef]

C. Masoller, T. Sorrentino, M. Chevrollier, and M. Oria, "Bistability in semiconductor lasers with polarization-rotated frequency-dependent optical feedback," IEEE J. Quantum Electron. 43, 261-268 (2007).
[CrossRef]

M. Holub, J. Shin, D. Saha, and P. Bhattacharya, "Electrical spin injection and threshold reduction in a semiconductor laser," Phys. Rev. Lett. 98, 146603 (2007).
[CrossRef] [PubMed]

2006 (3)

G. Van der Sande, M. Peeters, I. Veretennicoff, J. Danckaert, G. Verschaffelt, and S. Balle, "The effects of stress, temperature, and spin flips on polarization switching in vertical-cavity surfaceemitting lasers," IEEE J. Quantum Electron. 42, 898-906 (2006).
[CrossRef]

C. Chen, P. O. Leisher, A. A. Allerman, K. M. Geib, and K. D. Choquette, "Temperature analysis of threshold current in infrared vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 42, 1078-1083 (2006).
[CrossRef]

J. Paul, C. Masoller, Y. Hong, P. S. Spencer, and K. A. Shore, "Experimental study of polarization switching of vertical-cavity surface-emitting lasers as a dynamical bifurcation,"Opt. Lett. 31, 748-750 (2006).
[CrossRef] [PubMed]

2005 (5)

Y. Liu, W. C. Ng, K. D. Choquette, and K. Hess, "Numerical investigation of self-heating effects of oxide-confined vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 41, 15-25 (2005).
[CrossRef]

E. S. Bjorlin, J. Geske, M. Mehta, J. Piprek, and J. E. Bowers, "Temperature dependence of the relaxation resonance frequency of long-wavelength vertical-cavity lasers," IEEE Photon. Technol. Lett. 17, 944-946 (2005).
[CrossRef]

D. M. Grasso and K. D. Choquette, "Temperature-dependent polarization characteristics of composite-resonator vertical-cavity lasers," IEEE J. Quantum Electron. 41, 127-131 (2005).
[CrossRef]

J. Rudolph, S. Dohrmann, D. Hagele, M. Oestreich, and W. Stolz, "Room-temperature threshold reduction in vertical-cavity surface-emitting lasers by injection of spin-polarized electrons," Appl. Phys. Lett. 87, 241117 (2005).
[CrossRef]

S. Barland, P. Spinicelli, G. Giacomelli, and F. Marin, "Measurement of the working parameters of an air-post vertical-cavity surface-emitting laser," IEEE J. Quantum Electron. 41, 1235-1243 (2005).
[CrossRef]

2004 (1)

S. Mogg, N. Chitica, U. Christiansson, R. Schatz, P. Sundgren, C. Asplund, and M. Hammar, "Temperature sensitivity of the threshold current of long-wavelength InGaAs-GaAsVCSELs with large gain-cavity detuning," IEEE J. Quantum Electron. 40, 453-462 (2004).
[CrossRef]

2003 (2)

M. Sondermann, M. Weinkath, T. Ackemann, J. Mulet, and S. Balle, "Two-frequency emission and polarization dynamics at lasing threshold in vertical-cavity surface-emitting lasers," Phys. Rev. A. 68, 033822 (2003).
[CrossRef]

G. Verschaffelt, J. Albert, B. Nagler, M. Peeters, J. Danckaert, S. Barbay, G. Giacomelli, and F. Marin, "Frequency response of polarization switching in vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 39, 1177-1186 (2003).
[CrossRef]

2002 (5)

J. Kaiser, C. Degen, and W. Elsasser, "Polarization-switching influence on the intensity noise of vertical-cavity surface-emitting lasers," J. Opt. Soc. Am. B 19, 672-677 (2002).
[CrossRef]

C. Carlsson, H. Martinsson, R. Schatz, J. Halonen, and A. Larsson, "Analog modulation properties of oxide confined VCSELs at microwave frequencies," J. Lightwave Technol. 20, 1740-1749 (2002).
[CrossRef]

J. S. Gustavsson, J. A. Vukusic, J. Bengtsson, and A. Larsson, "A comprehensive model for the modal dynamics of vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 38, 203-212 (2002).
[CrossRef]

F. Prati, P. Caccia and F. Castelli, "Effects of gain saturation on polarization switching in verticalcavity surface-emitting lasers," Phys. Rev. A 66, 063811 (2002).
[CrossRef]

L. Spinelli, G. Tissoni, L. A. Lugiato, and M. Brambilla, "Thermal effects and transverse structures in semiconductor microcavities with population inversion," Phys. Rev. A 66, 023817 (2002).
[CrossRef]

2001 (3)

H. C. Schneider, A. J. Fischer, W. W. Chow, and J. F. Klem, "Temperature dependence of laser threshold in an InGaAsN vertical-cavity surface-emitting laser," Appl. Phys. Lett. 78, 3391-3393 (2001).
[CrossRef]

C. Degen, I. Fischer, W. Elsasser, L. Fratta, P. Debernardi, G. P. Bava, M. Brunner, R. Hovel, M. Moser, and K. Gulden, "Transverse modes in thermally detuned oxide-confined vertical-cavity surface-emitting lasers," Phys. Rev. A 63, 023817 (2001).
[CrossRef]

E. L. Blansett, M. G. Raymer, G. Khitrova, H. M. Gibbs, D. K. Serkland, A. A. Allerman, and K. M. Geib, "Ultrafast polarization dynamics and noise in pulsed vertical-cavity surface-emitting lasers," Opt. Express 9, 312-318 (2001).
[CrossRef] [PubMed]

2000 (1)

C. Degen, I. Fischer, and W. Elsasser, "Thermally induced local gain suppression in vertical-cavity surface-emitting lasers," Appl. Phys. Lett. 76, 3352-3354 (2000).
[CrossRef]

1999 (4)

1998 (1)

T. Rossler, R. A. Indik, G. K. Harkness, J. V. Moloney, and C. Z. Ning, "Modeling the interplay of thermal effects and transverse mode behavior in native-oxide-confined vertical-cavity surfaceemitting lasers," Phys. Rev. A. 58, 3279-3292 (1998).
[CrossRef]

1997 (1)

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]

1996 (1)

S. F. Yu, W. N. Wong, P. Shum, and E. H. Li, "Theoretical analysis of modulation response and second-order harmonic distortion in vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 32, 2139-2147 (1996).
[CrossRef]

1995 (2)

C. Z. Ning and J. V. Moloney, "Thermal effects on the threshold of vertical-cavity surface-emitting lasers: first- and second-order phase transitions," Opt. Lett. 20, 1151-1153 (1995).
[CrossRef] [PubMed]

K. D. Choquette, R. P. Schneider, K. L. Lear and R. E. Leibenguth, "Gain-dependent polarization properties of vertical-cavity lasers," IEEE J. Sel. Top. Quantum Electron. 1, 661-666 (1995).
[CrossRef]

1991 (1)

C. J. Chang-Hasnain, J. P. Harbison, G. Hasnain, A. C. Vonlehmen, L. T. Florez and N. G. Stoffel, "Dynamic, polarization and transverse-mode characteristics of vertical cavity surface emitting lasers," IEEE J. Quantum Electron. 27, 1402-1409 (1991).
[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]

Ackemann, T.

P. V. Paulau, A. J. Scroggie, A. Naumenko, T. Ackemann, N. A. Loiko, and W. J. Firth, "Localized traveling waves in vertical-cavity surface-emitting lasers with frequency-selective optical feedback," Phys. Rev. E 75, 056208 (2007).
[CrossRef]

M. Sondermann, M. Weinkath, T. Ackemann, J. Mulet, and S. Balle, "Two-frequency emission and polarization dynamics at lasing threshold in vertical-cavity surface-emitting lasers," Phys. Rev. A. 68, 033822 (2003).
[CrossRef]

Adams, M. J.

Albert, J.

G. Verschaffelt, J. Albert, B. Nagler, M. Peeters, J. Danckaert, S. Barbay, G. Giacomelli, and F. Marin, "Frequency response of polarization switching in vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 39, 1177-1186 (2003).
[CrossRef]

Allerman, A. A.

C. Chen, P. O. Leisher, A. A. Allerman, K. M. Geib, and K. D. Choquette, "Temperature analysis of threshold current in infrared vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 42, 1078-1083 (2006).
[CrossRef]

E. L. Blansett, M. G. Raymer, G. Khitrova, H. M. Gibbs, D. K. Serkland, A. A. Allerman, and K. M. Geib, "Ultrafast polarization dynamics and noise in pulsed vertical-cavity surface-emitting lasers," Opt. Express 9, 312-318 (2001).
[CrossRef] [PubMed]

Asplund, C.

S. Mogg, N. Chitica, U. Christiansson, R. Schatz, P. Sundgren, C. Asplund, and M. Hammar, "Temperature sensitivity of the threshold current of long-wavelength InGaAs-GaAsVCSELs with large gain-cavity detuning," IEEE J. Quantum Electron. 40, 453-462 (2004).
[CrossRef]

Balle, S.

G. Van der Sande, M. Peeters, I. Veretennicoff, J. Danckaert, G. Verschaffelt, and S. Balle, "The effects of stress, temperature, and spin flips on polarization switching in vertical-cavity surfaceemitting lasers," IEEE J. Quantum Electron. 42, 898-906 (2006).
[CrossRef]

M. Sondermann, M. Weinkath, T. Ackemann, J. Mulet, and S. Balle, "Two-frequency emission and polarization dynamics at lasing threshold in vertical-cavity surface-emitting lasers," Phys. Rev. A. 68, 033822 (2003).
[CrossRef]

S. Balle, E. Tolkachova, M. San Miguel, J. R. Tredicce, J. Martin-Regalado, and A. Gahl, "Mechanisms of polarization switching in single-transverse-mode vertical-cavity surface-emitting lasers: thermal shift and nonlinear semiconductor dynamics," Opt. Lett. 24, 1121-1123 (1999).
[CrossRef]

Barbay, S.

G. Verschaffelt, J. Albert, B. Nagler, M. Peeters, J. Danckaert, S. Barbay, G. Giacomelli, and F. Marin, "Frequency response of polarization switching in vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 39, 1177-1186 (2003).
[CrossRef]

Barland, S.

S. Barland, P. Spinicelli, G. Giacomelli, and F. Marin, "Measurement of the working parameters of an air-post vertical-cavity surface-emitting laser," IEEE J. Quantum Electron. 41, 1235-1243 (2005).
[CrossRef]

Bava, G. P.

C. Degen, I. Fischer, W. Elsasser, L. Fratta, P. Debernardi, G. P. Bava, M. Brunner, R. Hovel, M. Moser, and K. Gulden, "Transverse modes in thermally detuned oxide-confined vertical-cavity surface-emitting lasers," Phys. Rev. A 63, 023817 (2001).
[CrossRef]

Bengtsson, J.

J. S. Gustavsson, J. A. Vukusic, J. Bengtsson, and A. Larsson, "A comprehensive model for the modal dynamics of vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 38, 203-212 (2002).
[CrossRef]

Berseth, C. A.

A. Mircea, A. Caliman, V. Iakovlev, A. Mereuta, G. Suruceanu, C. A. Berseth, P. Royo, A. Syrbu, and E. Kapon, "Cavity mode-gain peak tradeoff for 1320-nm wafer-fused VCSELs with 3-mW single-mode emission power and 10-Gb/s modulation speed up to 70 degrees C," IEEE Photon. Technol. Lett. 19, 121-123 (2007).
[CrossRef]

Bhattacharya, P.

M. Holub, J. Shin, D. Saha, and P. Bhattacharya, "Electrical spin injection and threshold reduction in a semiconductor laser," Phys. Rev. Lett. 98, 146603 (2007).
[CrossRef] [PubMed]

Bjorlin, E. S.

E. S. Bjorlin, J. Geske, M. Mehta, J. Piprek, and J. E. Bowers, "Temperature dependence of the relaxation resonance frequency of long-wavelength vertical-cavity lasers," IEEE Photon. Technol. Lett. 17, 944-946 (2005).
[CrossRef]

Blansett, E. L.

Bowers, J. E.

E. S. Bjorlin, J. Geske, M. Mehta, J. Piprek, and J. E. Bowers, "Temperature dependence of the relaxation resonance frequency of long-wavelength vertical-cavity lasers," IEEE Photon. Technol. Lett. 17, 944-946 (2005).
[CrossRef]

Brambilla, M.

L. Spinelli, G. Tissoni, L. A. Lugiato, and M. Brambilla, "Thermal effects and transverse structures in semiconductor microcavities with population inversion," Phys. Rev. A 66, 023817 (2002).
[CrossRef]

Brunner, M.

C. Degen, I. Fischer, W. Elsasser, L. Fratta, P. Debernardi, G. P. Bava, M. Brunner, R. Hovel, M. Moser, and K. Gulden, "Transverse modes in thermally detuned oxide-confined vertical-cavity surface-emitting lasers," Phys. Rev. A 63, 023817 (2001).
[CrossRef]

Caccia, P.

F. Prati, P. Caccia and F. Castelli, "Effects of gain saturation on polarization switching in verticalcavity surface-emitting lasers," Phys. Rev. A 66, 063811 (2002).
[CrossRef]

Caliman, A.

A. Mircea, A. Caliman, V. Iakovlev, A. Mereuta, G. Suruceanu, C. A. Berseth, P. Royo, A. Syrbu, and E. Kapon, "Cavity mode-gain peak tradeoff for 1320-nm wafer-fused VCSELs with 3-mW single-mode emission power and 10-Gb/s modulation speed up to 70 degrees C," IEEE Photon. Technol. Lett. 19, 121-123 (2007).
[CrossRef]

Carlsson, C.

Castelli, F.

F. Prati, P. Caccia and F. Castelli, "Effects of gain saturation on polarization switching in verticalcavity surface-emitting lasers," Phys. Rev. A 66, 063811 (2002).
[CrossRef]

Chang-Hasnain, C. J.

C. J. Chang-Hasnain, J. P. Harbison, G. Hasnain, A. C. Vonlehmen, L. T. Florez and N. G. Stoffel, "Dynamic, polarization and transverse-mode characteristics of vertical cavity surface emitting lasers," IEEE J. Quantum Electron. 27, 1402-1409 (1991).
[CrossRef]

Chen, C.

C. Chen, P. O. Leisher, A. A. Allerman, K. M. Geib, and K. D. Choquette, "Temperature analysis of threshold current in infrared vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 42, 1078-1083 (2006).
[CrossRef]

Chevrollier, M.

C. Masoller, T. Sorrentino, M. Chevrollier, and M. Oria, "Bistability in semiconductor lasers with polarization-rotated frequency-dependent optical feedback," IEEE J. Quantum Electron. 43, 261-268 (2007).
[CrossRef]

Chitica, N.

S. Mogg, N. Chitica, U. Christiansson, R. Schatz, P. Sundgren, C. Asplund, and M. Hammar, "Temperature sensitivity of the threshold current of long-wavelength InGaAs-GaAsVCSELs with large gain-cavity detuning," IEEE J. Quantum Electron. 40, 453-462 (2004).
[CrossRef]

Choquette, K. D.

C. Chen, P. O. Leisher, A. A. Allerman, K. M. Geib, and K. D. Choquette, "Temperature analysis of threshold current in infrared vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 42, 1078-1083 (2006).
[CrossRef]

D. M. Grasso and K. D. Choquette, "Temperature-dependent polarization characteristics of composite-resonator vertical-cavity lasers," IEEE J. Quantum Electron. 41, 127-131 (2005).
[CrossRef]

Y. Liu, W. C. Ng, K. D. Choquette, and K. Hess, "Numerical investigation of self-heating effects of oxide-confined vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 41, 15-25 (2005).
[CrossRef]

K. D. Choquette, R. P. Schneider, K. L. Lear and R. E. Leibenguth, "Gain-dependent polarization properties of vertical-cavity lasers," IEEE J. Sel. Top. Quantum Electron. 1, 661-666 (1995).
[CrossRef]

Chow, W. W.

H. C. Schneider, A. J. Fischer, W. W. Chow, and J. F. Klem, "Temperature dependence of laser threshold in an InGaAsN vertical-cavity surface-emitting laser," Appl. Phys. Lett. 78, 3391-3393 (2001).
[CrossRef]

Christiansson, U.

S. Mogg, N. Chitica, U. Christiansson, R. Schatz, P. Sundgren, C. Asplund, and M. Hammar, "Temperature sensitivity of the threshold current of long-wavelength InGaAs-GaAsVCSELs with large gain-cavity detuning," IEEE J. Quantum Electron. 40, 453-462 (2004).
[CrossRef]

Danckaert, J.

G. Van der Sande, M. Peeters, I. Veretennicoff, J. Danckaert, G. Verschaffelt, and S. Balle, "The effects of stress, temperature, and spin flips on polarization switching in vertical-cavity surfaceemitting lasers," IEEE J. Quantum Electron. 42, 898-906 (2006).
[CrossRef]

G. Verschaffelt, J. Albert, B. Nagler, M. Peeters, J. Danckaert, S. Barbay, G. Giacomelli, and F. Marin, "Frequency response of polarization switching in vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 39, 1177-1186 (2003).
[CrossRef]

Debernardi, P.

C. Degen, I. Fischer, W. Elsasser, L. Fratta, P. Debernardi, G. P. Bava, M. Brunner, R. Hovel, M. Moser, and K. Gulden, "Transverse modes in thermally detuned oxide-confined vertical-cavity surface-emitting lasers," Phys. Rev. A 63, 023817 (2001).
[CrossRef]

Degen, C.

J. Kaiser, C. Degen, and W. Elsasser, "Polarization-switching influence on the intensity noise of vertical-cavity surface-emitting lasers," J. Opt. Soc. Am. B 19, 672-677 (2002).
[CrossRef]

C. Degen, I. Fischer, W. Elsasser, L. Fratta, P. Debernardi, G. P. Bava, M. Brunner, R. Hovel, M. Moser, and K. Gulden, "Transverse modes in thermally detuned oxide-confined vertical-cavity surface-emitting lasers," Phys. Rev. A 63, 023817 (2001).
[CrossRef]

C. Degen, I. Fischer, and W. Elsasser, "Thermally induced local gain suppression in vertical-cavity surface-emitting lasers," Appl. Phys. Lett. 76, 3352-3354 (2000).
[CrossRef]

C. Degen, I. Fischer and W. Elsasser, "Transverse modes in oxide confined VCSELs: Influence of pump profile, spatial hole burning, and thermal effects," Opt. Express 5, 38-47 (1999)
[CrossRef] [PubMed]

Dohrmann, S.

J. Rudolph, S. Dohrmann, D. Hagele, M. Oestreich, and W. Stolz, "Room-temperature threshold reduction in vertical-cavity surface-emitting lasers by injection of spin-polarized electrons," Appl. Phys. Lett. 87, 241117 (2005).
[CrossRef]

Elsasser, W.

Fischer, A. J.

H. C. Schneider, A. J. Fischer, W. W. Chow, and J. F. Klem, "Temperature dependence of laser threshold in an InGaAsN vertical-cavity surface-emitting laser," Appl. Phys. Lett. 78, 3391-3393 (2001).
[CrossRef]

Fischer, I.

C. Degen, I. Fischer, W. Elsasser, L. Fratta, P. Debernardi, G. P. Bava, M. Brunner, R. Hovel, M. Moser, and K. Gulden, "Transverse modes in thermally detuned oxide-confined vertical-cavity surface-emitting lasers," Phys. Rev. A 63, 023817 (2001).
[CrossRef]

C. Degen, I. Fischer, and W. Elsasser, "Thermally induced local gain suppression in vertical-cavity surface-emitting lasers," Appl. Phys. Lett. 76, 3352-3354 (2000).
[CrossRef]

C. Degen, I. Fischer and W. Elsasser, "Transverse modes in oxide confined VCSELs: Influence of pump profile, spatial hole burning, and thermal effects," Opt. Express 5, 38-47 (1999)
[CrossRef] [PubMed]

Florez, L. T.

C. J. Chang-Hasnain, J. P. Harbison, G. Hasnain, A. C. Vonlehmen, L. T. Florez and N. G. Stoffel, "Dynamic, polarization and transverse-mode characteristics of vertical cavity surface emitting lasers," IEEE J. Quantum Electron. 27, 1402-1409 (1991).
[CrossRef]

Fratta, L.

C. Degen, I. Fischer, W. Elsasser, L. Fratta, P. Debernardi, G. P. Bava, M. Brunner, R. Hovel, M. Moser, and K. Gulden, "Transverse modes in thermally detuned oxide-confined vertical-cavity surface-emitting lasers," Phys. Rev. A 63, 023817 (2001).
[CrossRef]

Gahl, A.

Geib, K. M.

C. Chen, P. O. Leisher, A. A. Allerman, K. M. Geib, and K. D. Choquette, "Temperature analysis of threshold current in infrared vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 42, 1078-1083 (2006).
[CrossRef]

E. L. Blansett, M. G. Raymer, G. Khitrova, H. M. Gibbs, D. K. Serkland, A. A. Allerman, and K. M. Geib, "Ultrafast polarization dynamics and noise in pulsed vertical-cavity surface-emitting lasers," Opt. Express 9, 312-318 (2001).
[CrossRef] [PubMed]

Geske, J.

E. S. Bjorlin, J. Geske, M. Mehta, J. Piprek, and J. E. Bowers, "Temperature dependence of the relaxation resonance frequency of long-wavelength vertical-cavity lasers," IEEE Photon. Technol. Lett. 17, 944-946 (2005).
[CrossRef]

Giacomelli, G.

S. Barland, P. Spinicelli, G. Giacomelli, and F. Marin, "Measurement of the working parameters of an air-post vertical-cavity surface-emitting laser," IEEE J. Quantum Electron. 41, 1235-1243 (2005).
[CrossRef]

G. Verschaffelt, J. Albert, B. Nagler, M. Peeters, J. Danckaert, S. Barbay, G. Giacomelli, and F. Marin, "Frequency response of polarization switching in vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 39, 1177-1186 (2003).
[CrossRef]

Gibbs, H. M.

Grasso, D. M.

D. M. Grasso and K. D. Choquette, "Temperature-dependent polarization characteristics of composite-resonator vertical-cavity lasers," IEEE J. Quantum Electron. 41, 127-131 (2005).
[CrossRef]

Gulden, K.

C. Degen, I. Fischer, W. Elsasser, L. Fratta, P. Debernardi, G. P. Bava, M. Brunner, R. Hovel, M. Moser, and K. Gulden, "Transverse modes in thermally detuned oxide-confined vertical-cavity surface-emitting lasers," Phys. Rev. A 63, 023817 (2001).
[CrossRef]

Gustavsson, J. S.

J. S. Gustavsson, J. A. Vukusic, J. Bengtsson, and A. Larsson, "A comprehensive model for the modal dynamics of vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 38, 203-212 (2002).
[CrossRef]

Hagele, D.

J. Rudolph, S. Dohrmann, D. Hagele, M. Oestreich, and W. Stolz, "Room-temperature threshold reduction in vertical-cavity surface-emitting lasers by injection of spin-polarized electrons," Appl. Phys. Lett. 87, 241117 (2005).
[CrossRef]

Halonen, J.

Hammar, M.

S. Mogg, N. Chitica, U. Christiansson, R. Schatz, P. Sundgren, C. Asplund, and M. Hammar, "Temperature sensitivity of the threshold current of long-wavelength InGaAs-GaAsVCSELs with large gain-cavity detuning," IEEE J. Quantum Electron. 40, 453-462 (2004).
[CrossRef]

Harbison, J. P.

C. J. Chang-Hasnain, J. P. Harbison, G. Hasnain, A. C. Vonlehmen, L. T. Florez and N. G. Stoffel, "Dynamic, polarization and transverse-mode characteristics of vertical cavity surface emitting lasers," IEEE J. Quantum Electron. 27, 1402-1409 (1991).
[CrossRef]

Harkness, G. K.

T. Rossler, R. A. Indik, G. K. Harkness, J. V. Moloney, and C. Z. Ning, "Modeling the interplay of thermal effects and transverse mode behavior in native-oxide-confined vertical-cavity surfaceemitting lasers," Phys. Rev. A. 58, 3279-3292 (1998).
[CrossRef]

Harton, A. V.

Hasnain, G.

C. J. Chang-Hasnain, J. P. Harbison, G. Hasnain, A. C. Vonlehmen, L. T. Florez and N. G. Stoffel, "Dynamic, polarization and transverse-mode characteristics of vertical cavity surface emitting lasers," IEEE J. Quantum Electron. 27, 1402-1409 (1991).
[CrossRef]

Hess, K.

Y. Liu, W. C. Ng, K. D. Choquette, and K. Hess, "Numerical investigation of self-heating effects of oxide-confined vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 41, 15-25 (2005).
[CrossRef]

Holub, M.

M. Holub, J. Shin, D. Saha, and P. Bhattacharya, "Electrical spin injection and threshold reduction in a semiconductor laser," Phys. Rev. Lett. 98, 146603 (2007).
[CrossRef] [PubMed]

Homayounfar, A.

Hong, Y.

Hovel, R.

C. Degen, I. Fischer, W. Elsasser, L. Fratta, P. Debernardi, G. P. Bava, M. Brunner, R. Hovel, M. Moser, and K. Gulden, "Transverse modes in thermally detuned oxide-confined vertical-cavity surface-emitting lasers," Phys. Rev. A 63, 023817 (2001).
[CrossRef]

Iakovlev, V.

A. Mircea, A. Caliman, V. Iakovlev, A. Mereuta, G. Suruceanu, C. A. Berseth, P. Royo, A. Syrbu, and E. Kapon, "Cavity mode-gain peak tradeoff for 1320-nm wafer-fused VCSELs with 3-mW single-mode emission power and 10-Gb/s modulation speed up to 70 degrees C," IEEE Photon. Technol. Lett. 19, 121-123 (2007).
[CrossRef]

Iga, K.

Indik, R. A.

T. Rossler, R. A. Indik, G. K. Harkness, J. V. Moloney, and C. Z. Ning, "Modeling the interplay of thermal effects and transverse mode behavior in native-oxide-confined vertical-cavity surfaceemitting lasers," Phys. Rev. A. 58, 3279-3292 (1998).
[CrossRef]

Kaiser, J.

Kang, S.-M.

Kapon, E.

A. Mircea, A. Caliman, V. Iakovlev, A. Mereuta, G. Suruceanu, C. A. Berseth, P. Royo, A. Syrbu, and E. Kapon, "Cavity mode-gain peak tradeoff for 1320-nm wafer-fused VCSELs with 3-mW single-mode emission power and 10-Gb/s modulation speed up to 70 degrees C," IEEE Photon. Technol. Lett. 19, 121-123 (2007).
[CrossRef]

Khalid, M. U. F.

M. B. Willemsen, M. U. F. Khalid, M. P. van Exter, and J. P. Woerdman, "Polarization switching of a vertical-cavity semiconductor laser as a Kramers hopping problem," Phys. Rev. Lett. 82, 4815-4818 (1999).
[CrossRef]

Khitrova, G.

Klem, J. F.

H. C. Schneider, A. J. Fischer, W. W. Chow, and J. F. Klem, "Temperature dependence of laser threshold in an InGaAsN vertical-cavity surface-emitting laser," Appl. Phys. Lett. 78, 3391-3393 (2001).
[CrossRef]

Larsson, A.

C. Carlsson, H. Martinsson, R. Schatz, J. Halonen, and A. Larsson, "Analog modulation properties of oxide confined VCSELs at microwave frequencies," J. Lightwave Technol. 20, 1740-1749 (2002).
[CrossRef]

J. S. Gustavsson, J. A. Vukusic, J. Bengtsson, and A. Larsson, "A comprehensive model for the modal dynamics of vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 38, 203-212 (2002).
[CrossRef]

Lear, K. L.

K. D. Choquette, R. P. Schneider, K. L. Lear and R. E. Leibenguth, "Gain-dependent polarization properties of vertical-cavity lasers," IEEE J. Sel. Top. Quantum Electron. 1, 661-666 (1995).
[CrossRef]

Leibenguth, R. E.

K. D. Choquette, R. P. Schneider, K. L. Lear and R. E. Leibenguth, "Gain-dependent polarization properties of vertical-cavity lasers," IEEE J. Sel. Top. Quantum Electron. 1, 661-666 (1995).
[CrossRef]

Leisher, P. O.

C. Chen, P. O. Leisher, A. A. Allerman, K. M. Geib, and K. D. Choquette, "Temperature analysis of threshold current in infrared vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 42, 1078-1083 (2006).
[CrossRef]

Li, E. H.

S. F. Yu, W. N. Wong, P. Shum, and E. H. Li, "Theoretical analysis of modulation response and second-order harmonic distortion in vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 32, 2139-2147 (1996).
[CrossRef]

Liu, Y.

Y. Liu, W. C. Ng, K. D. Choquette, and K. Hess, "Numerical investigation of self-heating effects of oxide-confined vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 41, 15-25 (2005).
[CrossRef]

Loiko, N. A.

P. V. Paulau, A. J. Scroggie, A. Naumenko, T. Ackemann, N. A. Loiko, and W. J. Firth, "Localized traveling waves in vertical-cavity surface-emitting lasers with frequency-selective optical feedback," Phys. Rev. E 75, 056208 (2007).
[CrossRef]

Lugiato, L. A.

L. Spinelli, G. Tissoni, L. A. Lugiato, and M. Brambilla, "Thermal effects and transverse structures in semiconductor microcavities with population inversion," Phys. Rev. A 66, 023817 (2002).
[CrossRef]

Marin, F.

S. Barland, P. Spinicelli, G. Giacomelli, and F. Marin, "Measurement of the working parameters of an air-post vertical-cavity surface-emitting laser," IEEE J. Quantum Electron. 41, 1235-1243 (2005).
[CrossRef]

G. Verschaffelt, J. Albert, B. Nagler, M. Peeters, J. Danckaert, S. Barbay, G. Giacomelli, and F. Marin, "Frequency response of polarization switching in vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 39, 1177-1186 (2003).
[CrossRef]

Martin-Regalado, J.

Martinsson, H.

Masoller, C.

C. Masoller, T. Sorrentino, M. Chevrollier, and M. Oria, "Bistability in semiconductor lasers with polarization-rotated frequency-dependent optical feedback," IEEE J. Quantum Electron. 43, 261-268 (2007).
[CrossRef]

J. Paul, C. Masoller, Y. Hong, P. S. Spencer, and K. A. Shore, "Experimental study of polarization switching of vertical-cavity surface-emitting lasers as a dynamical bifurcation,"Opt. Lett. 31, 748-750 (2006).
[CrossRef] [PubMed]

Mehta, M.

E. S. Bjorlin, J. Geske, M. Mehta, J. Piprek, and J. E. Bowers, "Temperature dependence of the relaxation resonance frequency of long-wavelength vertical-cavity lasers," IEEE Photon. Technol. Lett. 17, 944-946 (2005).
[CrossRef]

Mena, P. V.

Mereuta, A.

A. Mircea, A. Caliman, V. Iakovlev, A. Mereuta, G. Suruceanu, C. A. Berseth, P. Royo, A. Syrbu, and E. Kapon, "Cavity mode-gain peak tradeoff for 1320-nm wafer-fused VCSELs with 3-mW single-mode emission power and 10-Gb/s modulation speed up to 70 degrees C," IEEE Photon. Technol. Lett. 19, 121-123 (2007).
[CrossRef]

Mircea, A.

A. Mircea, A. Caliman, V. Iakovlev, A. Mereuta, G. Suruceanu, C. A. Berseth, P. Royo, A. Syrbu, and E. Kapon, "Cavity mode-gain peak tradeoff for 1320-nm wafer-fused VCSELs with 3-mW single-mode emission power and 10-Gb/s modulation speed up to 70 degrees C," IEEE Photon. Technol. Lett. 19, 121-123 (2007).
[CrossRef]

Mogg, S.

S. Mogg, N. Chitica, U. Christiansson, R. Schatz, P. Sundgren, C. Asplund, and M. Hammar, "Temperature sensitivity of the threshold current of long-wavelength InGaAs-GaAsVCSELs with large gain-cavity detuning," IEEE J. Quantum Electron. 40, 453-462 (2004).
[CrossRef]

Moloney, J. V.

T. Rossler, R. A. Indik, G. K. Harkness, J. V. Moloney, and C. Z. Ning, "Modeling the interplay of thermal effects and transverse mode behavior in native-oxide-confined vertical-cavity surfaceemitting lasers," Phys. Rev. A. 58, 3279-3292 (1998).
[CrossRef]

C. Z. Ning and J. V. Moloney, "Thermal effects on the threshold of vertical-cavity surface-emitting lasers: first- and second-order phase transitions," Opt. Lett. 20, 1151-1153 (1995).
[CrossRef] [PubMed]

Morikuni, J. J.

Moser, M.

C. Degen, I. Fischer, W. Elsasser, L. Fratta, P. Debernardi, G. P. Bava, M. Brunner, R. Hovel, M. Moser, and K. Gulden, "Transverse modes in thermally detuned oxide-confined vertical-cavity surface-emitting lasers," Phys. Rev. A 63, 023817 (2001).
[CrossRef]

Mulet, J.

M. Sondermann, M. Weinkath, T. Ackemann, J. Mulet, and S. Balle, "Two-frequency emission and polarization dynamics at lasing threshold in vertical-cavity surface-emitting lasers," Phys. Rev. A. 68, 033822 (2003).
[CrossRef]

Nagler, B.

G. Verschaffelt, J. Albert, B. Nagler, M. Peeters, J. Danckaert, S. Barbay, G. Giacomelli, and F. Marin, "Frequency response of polarization switching in vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 39, 1177-1186 (2003).
[CrossRef]

Naumenko, A.

P. V. Paulau, A. J. Scroggie, A. Naumenko, T. Ackemann, N. A. Loiko, and W. J. Firth, "Localized traveling waves in vertical-cavity surface-emitting lasers with frequency-selective optical feedback," Phys. Rev. E 75, 056208 (2007).
[CrossRef]

Ng, W. C.

Y. Liu, W. C. Ng, K. D. Choquette, and K. Hess, "Numerical investigation of self-heating effects of oxide-confined vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 41, 15-25 (2005).
[CrossRef]

Ning, C. Z.

T. Rossler, R. A. Indik, G. K. Harkness, J. V. Moloney, and C. Z. Ning, "Modeling the interplay of thermal effects and transverse mode behavior in native-oxide-confined vertical-cavity surfaceemitting lasers," Phys. Rev. A. 58, 3279-3292 (1998).
[CrossRef]

C. Z. Ning and J. V. Moloney, "Thermal effects on the threshold of vertical-cavity surface-emitting lasers: first- and second-order phase transitions," Opt. Lett. 20, 1151-1153 (1995).
[CrossRef] [PubMed]

Oestreich, M.

J. Rudolph, S. Dohrmann, D. Hagele, M. Oestreich, and W. Stolz, "Room-temperature threshold reduction in vertical-cavity surface-emitting lasers by injection of spin-polarized electrons," Appl. Phys. Lett. 87, 241117 (2005).
[CrossRef]

Oria, M.

C. Masoller, T. Sorrentino, M. Chevrollier, and M. Oria, "Bistability in semiconductor lasers with polarization-rotated frequency-dependent optical feedback," IEEE J. Quantum Electron. 43, 261-268 (2007).
[CrossRef]

Paul, J.

Paulau, P. V.

P. V. Paulau, A. J. Scroggie, A. Naumenko, T. Ackemann, N. A. Loiko, and W. J. Firth, "Localized traveling waves in vertical-cavity surface-emitting lasers with frequency-selective optical feedback," Phys. Rev. E 75, 056208 (2007).
[CrossRef]

Peeters, M.

G. Van der Sande, M. Peeters, I. Veretennicoff, J. Danckaert, G. Verschaffelt, and S. Balle, "The effects of stress, temperature, and spin flips on polarization switching in vertical-cavity surfaceemitting lasers," IEEE J. Quantum Electron. 42, 898-906 (2006).
[CrossRef]

G. Verschaffelt, J. Albert, B. Nagler, M. Peeters, J. Danckaert, S. Barbay, G. Giacomelli, and F. Marin, "Frequency response of polarization switching in vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 39, 1177-1186 (2003).
[CrossRef]

Piprek, J.

E. S. Bjorlin, J. Geske, M. Mehta, J. Piprek, and J. E. Bowers, "Temperature dependence of the relaxation resonance frequency of long-wavelength vertical-cavity lasers," IEEE Photon. Technol. Lett. 17, 944-946 (2005).
[CrossRef]

Prati, F.

F. Prati, P. Caccia and F. Castelli, "Effects of gain saturation on polarization switching in verticalcavity surface-emitting lasers," Phys. Rev. A 66, 063811 (2002).
[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]

Raymer, M. G.

Rossler, T.

T. Rossler, R. A. Indik, G. K. Harkness, J. V. Moloney, and C. Z. Ning, "Modeling the interplay of thermal effects and transverse mode behavior in native-oxide-confined vertical-cavity surfaceemitting lasers," Phys. Rev. A. 58, 3279-3292 (1998).
[CrossRef]

Royo, P.

A. Mircea, A. Caliman, V. Iakovlev, A. Mereuta, G. Suruceanu, C. A. Berseth, P. Royo, A. Syrbu, and E. Kapon, "Cavity mode-gain peak tradeoff for 1320-nm wafer-fused VCSELs with 3-mW single-mode emission power and 10-Gb/s modulation speed up to 70 degrees C," IEEE Photon. Technol. Lett. 19, 121-123 (2007).
[CrossRef]

Rudolph, J.

J. Rudolph, S. Dohrmann, D. Hagele, M. Oestreich, and W. Stolz, "Room-temperature threshold reduction in vertical-cavity surface-emitting lasers by injection of spin-polarized electrons," Appl. Phys. Lett. 87, 241117 (2005).
[CrossRef]

Saha, D.

M. Holub, J. Shin, D. Saha, and P. Bhattacharya, "Electrical spin injection and threshold reduction in a semiconductor laser," Phys. Rev. Lett. 98, 146603 (2007).
[CrossRef] [PubMed]

San Miguel, M.

Schatz, R.

S. Mogg, N. Chitica, U. Christiansson, R. Schatz, P. Sundgren, C. Asplund, and M. Hammar, "Temperature sensitivity of the threshold current of long-wavelength InGaAs-GaAsVCSELs with large gain-cavity detuning," IEEE J. Quantum Electron. 40, 453-462 (2004).
[CrossRef]

C. Carlsson, H. Martinsson, R. Schatz, J. Halonen, and A. Larsson, "Analog modulation properties of oxide confined VCSELs at microwave frequencies," J. Lightwave Technol. 20, 1740-1749 (2002).
[CrossRef]

Schneider, H. C.

H. C. Schneider, A. J. Fischer, W. W. Chow, and J. F. Klem, "Temperature dependence of laser threshold in an InGaAsN vertical-cavity surface-emitting laser," Appl. Phys. Lett. 78, 3391-3393 (2001).
[CrossRef]

Schneider, R. P.

K. D. Choquette, R. P. Schneider, K. L. Lear and R. E. Leibenguth, "Gain-dependent polarization properties of vertical-cavity lasers," IEEE J. Sel. Top. Quantum Electron. 1, 661-666 (1995).
[CrossRef]

Scroggie, A. J.

P. V. Paulau, A. J. Scroggie, A. Naumenko, T. Ackemann, N. A. Loiko, and W. J. Firth, "Localized traveling waves in vertical-cavity surface-emitting lasers with frequency-selective optical feedback," Phys. Rev. E 75, 056208 (2007).
[CrossRef]

Serkland, D. K.

Shin, J.

M. Holub, J. Shin, D. Saha, and P. Bhattacharya, "Electrical spin injection and threshold reduction in a semiconductor laser," Phys. Rev. Lett. 98, 146603 (2007).
[CrossRef] [PubMed]

Shore, K. A.

Shum, P.

S. F. Yu, W. N. Wong, P. Shum, and E. H. Li, "Theoretical analysis of modulation response and second-order harmonic distortion in vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 32, 2139-2147 (1996).
[CrossRef]

Sondermann, M.

M. Sondermann, M. Weinkath, T. Ackemann, J. Mulet, and S. Balle, "Two-frequency emission and polarization dynamics at lasing threshold in vertical-cavity surface-emitting lasers," Phys. Rev. A. 68, 033822 (2003).
[CrossRef]

Sorrentino, T.

C. Masoller, T. Sorrentino, M. Chevrollier, and M. Oria, "Bistability in semiconductor lasers with polarization-rotated frequency-dependent optical feedback," IEEE J. Quantum Electron. 43, 261-268 (2007).
[CrossRef]

Spencer, P. S.

Spinelli, L.

L. Spinelli, G. Tissoni, L. A. Lugiato, and M. Brambilla, "Thermal effects and transverse structures in semiconductor microcavities with population inversion," Phys. Rev. A 66, 023817 (2002).
[CrossRef]

Spinicelli, P.

S. Barland, P. Spinicelli, G. Giacomelli, and F. Marin, "Measurement of the working parameters of an air-post vertical-cavity surface-emitting laser," IEEE J. Quantum Electron. 41, 1235-1243 (2005).
[CrossRef]

Stoffel, N. G.

C. J. Chang-Hasnain, J. P. Harbison, G. Hasnain, A. C. Vonlehmen, L. T. Florez and N. G. Stoffel, "Dynamic, polarization and transverse-mode characteristics of vertical cavity surface emitting lasers," IEEE J. Quantum Electron. 27, 1402-1409 (1991).
[CrossRef]

Stolz, W.

J. Rudolph, S. Dohrmann, D. Hagele, M. Oestreich, and W. Stolz, "Room-temperature threshold reduction in vertical-cavity surface-emitting lasers by injection of spin-polarized electrons," Appl. Phys. Lett. 87, 241117 (2005).
[CrossRef]

Suematsu, Y.

Sundgren, P.

S. Mogg, N. Chitica, U. Christiansson, R. Schatz, P. Sundgren, C. Asplund, and M. Hammar, "Temperature sensitivity of the threshold current of long-wavelength InGaAs-GaAsVCSELs with large gain-cavity detuning," IEEE J. Quantum Electron. 40, 453-462 (2004).
[CrossRef]

Suruceanu, G.

A. Mircea, A. Caliman, V. Iakovlev, A. Mereuta, G. Suruceanu, C. A. Berseth, P. Royo, A. Syrbu, and E. Kapon, "Cavity mode-gain peak tradeoff for 1320-nm wafer-fused VCSELs with 3-mW single-mode emission power and 10-Gb/s modulation speed up to 70 degrees C," IEEE Photon. Technol. Lett. 19, 121-123 (2007).
[CrossRef]

Syrbu, A.

A. Mircea, A. Caliman, V. Iakovlev, A. Mereuta, G. Suruceanu, C. A. Berseth, P. Royo, A. Syrbu, and E. Kapon, "Cavity mode-gain peak tradeoff for 1320-nm wafer-fused VCSELs with 3-mW single-mode emission power and 10-Gb/s modulation speed up to 70 degrees C," IEEE Photon. Technol. Lett. 19, 121-123 (2007).
[CrossRef]

Tissoni, G.

L. Spinelli, G. Tissoni, L. A. Lugiato, and M. Brambilla, "Thermal effects and transverse structures in semiconductor microcavities with population inversion," Phys. Rev. A 66, 023817 (2002).
[CrossRef]

Tolkachova, E.

Tredicce, J. R.

Van der Sande, G.

G. Van der Sande, M. Peeters, I. Veretennicoff, J. Danckaert, G. Verschaffelt, and S. Balle, "The effects of stress, temperature, and spin flips on polarization switching in vertical-cavity surfaceemitting lasers," IEEE J. Quantum Electron. 42, 898-906 (2006).
[CrossRef]

van Exter, M. P.

M. B. Willemsen, M. U. F. Khalid, M. P. van Exter, and J. P. Woerdman, "Polarization switching of a vertical-cavity semiconductor laser as a Kramers hopping problem," Phys. Rev. Lett. 82, 4815-4818 (1999).
[CrossRef]

Veretennicoff, I.

G. Van der Sande, M. Peeters, I. Veretennicoff, J. Danckaert, G. Verschaffelt, and S. Balle, "The effects of stress, temperature, and spin flips on polarization switching in vertical-cavity surfaceemitting lasers," IEEE J. Quantum Electron. 42, 898-906 (2006).
[CrossRef]

Verschaffelt, G.

G. Van der Sande, M. Peeters, I. Veretennicoff, J. Danckaert, G. Verschaffelt, and S. Balle, "The effects of stress, temperature, and spin flips on polarization switching in vertical-cavity surfaceemitting lasers," IEEE J. Quantum Electron. 42, 898-906 (2006).
[CrossRef]

G. Verschaffelt, J. Albert, B. Nagler, M. Peeters, J. Danckaert, S. Barbay, G. Giacomelli, and F. Marin, "Frequency response of polarization switching in vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 39, 1177-1186 (2003).
[CrossRef]

Vonlehmen, A. C.

C. J. Chang-Hasnain, J. P. Harbison, G. Hasnain, A. C. Vonlehmen, L. T. Florez and N. G. Stoffel, "Dynamic, polarization and transverse-mode characteristics of vertical cavity surface emitting lasers," IEEE J. Quantum Electron. 27, 1402-1409 (1991).
[CrossRef]

Vukusic, J. A.

J. S. Gustavsson, J. A. Vukusic, J. Bengtsson, and A. Larsson, "A comprehensive model for the modal dynamics of vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 38, 203-212 (2002).
[CrossRef]

Weinkath, M.

M. Sondermann, M. Weinkath, T. Ackemann, J. Mulet, and S. Balle, "Two-frequency emission and polarization dynamics at lasing threshold in vertical-cavity surface-emitting lasers," Phys. Rev. A. 68, 033822 (2003).
[CrossRef]

Willemsen, M. B.

M. B. Willemsen, M. U. F. Khalid, M. P. van Exter, and J. P. Woerdman, "Polarization switching of a vertical-cavity semiconductor laser as a Kramers hopping problem," Phys. Rev. Lett. 82, 4815-4818 (1999).
[CrossRef]

Woerdman, J. P.

M. B. Willemsen, M. U. F. Khalid, M. P. van Exter, and J. P. Woerdman, "Polarization switching of a vertical-cavity semiconductor laser as a Kramers hopping problem," Phys. Rev. Lett. 82, 4815-4818 (1999).
[CrossRef]

Wong, W. N.

S. F. Yu, W. N. Wong, P. Shum, and E. H. Li, "Theoretical analysis of modulation response and second-order harmonic distortion in vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 32, 2139-2147 (1996).
[CrossRef]

Wyat, K. W.

Yu, S. F.

S. F. Yu, W. N. Wong, P. Shum, and E. H. Li, "Theoretical analysis of modulation response and second-order harmonic distortion in vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 32, 2139-2147 (1996).
[CrossRef]

Appl. Phys. Lett. (3)

H. C. Schneider, A. J. Fischer, W. W. Chow, and J. F. Klem, "Temperature dependence of laser threshold in an InGaAsN vertical-cavity surface-emitting laser," Appl. Phys. Lett. 78, 3391-3393 (2001).
[CrossRef]

C. Degen, I. Fischer, and W. Elsasser, "Thermally induced local gain suppression in vertical-cavity surface-emitting lasers," Appl. Phys. Lett. 76, 3352-3354 (2000).
[CrossRef]

J. Rudolph, S. Dohrmann, D. Hagele, M. Oestreich, and W. Stolz, "Room-temperature threshold reduction in vertical-cavity surface-emitting lasers by injection of spin-polarized electrons," Appl. Phys. Lett. 87, 241117 (2005).
[CrossRef]

IEEE J. Quantum Electron. (12)

C. Masoller, T. Sorrentino, M. Chevrollier, and M. Oria, "Bistability in semiconductor lasers with polarization-rotated frequency-dependent optical feedback," IEEE J. Quantum Electron. 43, 261-268 (2007).
[CrossRef]

S. Barland, P. Spinicelli, G. Giacomelli, and F. Marin, "Measurement of the working parameters of an air-post vertical-cavity surface-emitting laser," IEEE J. Quantum Electron. 41, 1235-1243 (2005).
[CrossRef]

G. Van der Sande, M. Peeters, I. Veretennicoff, J. Danckaert, G. Verschaffelt, and S. Balle, "The effects of stress, temperature, and spin flips on polarization switching in vertical-cavity surfaceemitting lasers," IEEE J. Quantum Electron. 42, 898-906 (2006).
[CrossRef]

G. Verschaffelt, J. Albert, B. Nagler, M. Peeters, J. Danckaert, S. Barbay, G. Giacomelli, and F. Marin, "Frequency response of polarization switching in vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 39, 1177-1186 (2003).
[CrossRef]

D. M. Grasso and K. D. Choquette, "Temperature-dependent polarization characteristics of composite-resonator vertical-cavity lasers," IEEE J. Quantum Electron. 41, 127-131 (2005).
[CrossRef]

J. S. Gustavsson, J. A. Vukusic, J. Bengtsson, and A. Larsson, "A comprehensive model for the modal dynamics of vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 38, 203-212 (2002).
[CrossRef]

S. F. Yu, W. N. Wong, P. Shum, and E. H. Li, "Theoretical analysis of modulation response and second-order harmonic distortion in vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 32, 2139-2147 (1996).
[CrossRef]

Y. Liu, W. C. Ng, K. D. Choquette, and K. Hess, "Numerical investigation of self-heating effects of oxide-confined vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 41, 15-25 (2005).
[CrossRef]

C. Chen, P. O. Leisher, A. A. Allerman, K. M. Geib, and K. D. Choquette, "Temperature analysis of threshold current in infrared vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 42, 1078-1083 (2006).
[CrossRef]

C. J. Chang-Hasnain, J. P. Harbison, G. Hasnain, A. C. Vonlehmen, L. T. Florez and N. G. Stoffel, "Dynamic, polarization and transverse-mode characteristics of vertical cavity surface emitting lasers," IEEE J. Quantum Electron. 27, 1402-1409 (1991).
[CrossRef]

S. Mogg, N. Chitica, U. Christiansson, R. Schatz, P. Sundgren, C. Asplund, and M. Hammar, "Temperature sensitivity of the threshold current of long-wavelength InGaAs-GaAsVCSELs with large gain-cavity detuning," IEEE J. Quantum Electron. 40, 453-462 (2004).
[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]

IEEE J. Sel. Top. Quantum Electron. (1)

K. D. Choquette, R. P. Schneider, K. L. Lear and R. E. Leibenguth, "Gain-dependent polarization properties of vertical-cavity lasers," IEEE J. Sel. Top. Quantum Electron. 1, 661-666 (1995).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

E. S. Bjorlin, J. Geske, M. Mehta, J. Piprek, and J. E. Bowers, "Temperature dependence of the relaxation resonance frequency of long-wavelength vertical-cavity lasers," IEEE Photon. Technol. Lett. 17, 944-946 (2005).
[CrossRef]

A. Mircea, A. Caliman, V. Iakovlev, A. Mereuta, G. Suruceanu, C. A. Berseth, P. Royo, A. Syrbu, and E. Kapon, "Cavity mode-gain peak tradeoff for 1320-nm wafer-fused VCSELs with 3-mW single-mode emission power and 10-Gb/s modulation speed up to 70 degrees C," IEEE Photon. Technol. Lett. 19, 121-123 (2007).
[CrossRef]

J. Lightwave Technol. (3)

J. Opt. Soc. Am. B (1)

Opt. Express (3)

Opt. Lett. (3)

Phys. Rev. A (3)

F. Prati, P. Caccia and F. Castelli, "Effects of gain saturation on polarization switching in verticalcavity surface-emitting lasers," Phys. Rev. A 66, 063811 (2002).
[CrossRef]

C. Degen, I. Fischer, W. Elsasser, L. Fratta, P. Debernardi, G. P. Bava, M. Brunner, R. Hovel, M. Moser, and K. Gulden, "Transverse modes in thermally detuned oxide-confined vertical-cavity surface-emitting lasers," Phys. Rev. A 63, 023817 (2001).
[CrossRef]

L. Spinelli, G. Tissoni, L. A. Lugiato, and M. Brambilla, "Thermal effects and transverse structures in semiconductor microcavities with population inversion," Phys. Rev. A 66, 023817 (2002).
[CrossRef]

Phys. Rev. A. (2)

M. Sondermann, M. Weinkath, T. Ackemann, J. Mulet, and S. Balle, "Two-frequency emission and polarization dynamics at lasing threshold in vertical-cavity surface-emitting lasers," Phys. Rev. A. 68, 033822 (2003).
[CrossRef]

T. Rossler, R. A. Indik, G. K. Harkness, J. V. Moloney, and C. Z. Ning, "Modeling the interplay of thermal effects and transverse mode behavior in native-oxide-confined vertical-cavity surfaceemitting lasers," Phys. Rev. A. 58, 3279-3292 (1998).
[CrossRef]

Phys. Rev. E (1)

P. V. Paulau, A. J. Scroggie, A. Naumenko, T. Ackemann, N. A. Loiko, and W. J. Firth, "Localized traveling waves in vertical-cavity surface-emitting lasers with frequency-selective optical feedback," Phys. Rev. E 75, 056208 (2007).
[CrossRef]

Phys. Rev. Lett. (2)

M. B. Willemsen, M. U. F. Khalid, M. P. van Exter, and J. P. Woerdman, "Polarization switching of a vertical-cavity semiconductor laser as a Kramers hopping problem," Phys. Rev. Lett. 82, 4815-4818 (1999).
[CrossRef]

M. Holub, J. Shin, D. Saha, and P. Bhattacharya, "Electrical spin injection and threshold reduction in a semiconductor laser," Phys. Rev. Lett. 98, 146603 (2007).
[CrossRef] [PubMed]

Other (2)

G. P. Agrawal and N. K. Dutta, Semiconductor Lasers, 2nd ed. (Amsterdam, The Netherlands: Kluwer, 1993).

C. Lei, and J. K. Guenter, eds., Vertical-Cavity Surface-Emitting Lasers X II, Proc. SPIE 6908, 2008.

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Figures (7)

Fig. 1.
Fig. 1.

Left: Gain coefficient vs. wavelength for active region temperature in the range 240–3400 K. The circles indicate the location of the cavity mode. The gain and the cavity mode are aligned at room temperature (δ 0=0 nm). Right: Gain coefficient at the cavity mode vs. temperature of the active region, for three values of the gain-cavity offset at room temperature.

Fig. 2.
Fig. 2.

Left: Variation of the lasing threshold current, I th , with the radio of active region, R a . We compare measurements of [32] (open circles) with results of simulations (dots). The value of the device resistance, R, is indicated in the right vertical axis (triangles). Right: Thermal resistance, R th TI, vs. the active region ratio, R a (squares). The value of the temperature decay rate, γT , is indicated in the right vertical axis (triangles).

Fig. 3.
Fig. 3.

(a), (b) Dependence of the shape of the LI curve on the active region size. The aperture diameter is 6 µm (a) and 10 µm (b). (c) Variation of the shape of the LI curve with the substrate temperature. The RT gain-cavity offset is δ 0=-3 nm. The inset shows a detail of the lasing threshold.

Fig. 4.
Fig. 4.

Polarization-resolved LI curve (x polarization: red; y polarization: blue). The left (right) column is done with parameters corresponding to type I (type II) PS. It can be seen how the substrate temperature, T s , affects the PS points: for intermediate T s , a second PS appears near the thermally induced power rollover point; at high enough T s , both PSs abruptly disappear.

Fig. 5.
Fig. 5.

Threshold current (a) and polarization-switching current (b) versus the substrate temperature for γT =0.01 ns-1 and various values of the RT gain-cavity offset, δ 0 (in nm). Threshold current (c) and polarization-switching current (d) versus the substrate temperature for δ 0=-3 nm and various values of the temperature decay rate, γT (in ns-1).

Fig. 6.
Fig. 6.

Influence of the birefringence parameter, γ p . Polarization-switching current for type I PS (a) and for type II PS (b) versus the substrate temperature for fixed RT gain-cavity offset, δ 0=-3 nm, and various values of γp (in rad/ns). The dashed lines show the threshold current, that does not dependent on γp .

Fig. 7.
Fig. 7.

Dependence of the polarization-resolved LI curve on the duration of the current ramp. The injection current increases linearly in 1000 ns (a), (d) 5000 ns (b), (e) and 50000 ns (c), (f). The left (right) column is done with parameters corresponding to type I (type II) PS. The insets show a detail of the LI curve. The gain-cavity offset is δ 0=-3 nm, and the substrate temperature, T s =250C.

Tables (1)

Tables Icon

Table 1. Parameters used in the simulations

Equations (22)

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

d E ± dt = k ( 1 + i α ) ( N ± 1 ) E ± ( γ a + i γ p ) E + β sp ξ ± ,
d N ± dt = γ N ( N ± μ + 2 N ± E ± 2 ) γ j ( N ± N ) .
d E x , y dt = k ( 1 + i α ) [ ( N 1 ) E x , y ± in E y , x ] ( γ a + i γ p ) E x , y + β sp ξ x , y ,
d E ± dt = k ( 1 + i α ) [ g ( ω ± , T ) N ± 1 ] E ± ( γ a + i γ p ) E + β sp ξ ± ( t ) ,
d N ± dt = γ N ( N ± μ + 2 g ( ω ± , T ) N ± E ± 2 ) γ j ( N ± N ) .
g ( ω , T ) = T 0 T 1 + [ δ ( T ) ω ] 2 Δ ω g 2 ( T ) ,
h ¯ ω g ( T ) = ε g 0 α T 2 ( T + β ) ,
ω c ( T ) = ( 2 π c λ 0 ) [ 1 ( 1 η 0 ) ( d η dT ) ( T T 0 ) ] ,
dT dt = γ T ( T T s ) + γ N h ¯ ω c c q N + R S 2 c q V t J 2 ,
N = K ( N N 0 1 ) ,
N = K ( J J 0 1 ) ,
dT dt = γ T ( T T s ) + Z ( N K + 1 ) + P ( μ K + 1 ) 2 ,
dE dt = k ( 1 + i α ) ( N 1 ) E ,
dN dt = γ N ( N μ + N E 2 ) .
d dt = ( 1 2 ) ( 1 + i α ) [ v g Γ a ( N N 0 ) 1 τ p ] ,
d N dt = J ( e L a ) γ N N v g Γ a ( N N 0 ) 2 ,
N = τ p v g Γ a ( N N 0 ) .
v g Γ a ( N th N 0 ) = 1 τ p ,
N = ( N N 0 ) ( N th N 0 ) ,
μ = τ p v g Γ a [ J ( e L a γ N ) N 0 ] .
N = K ( N N 0 1 )
μ = K ( J J 0 1 ) .

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