Abstract

We investigate the threshold behavior of vertical-cavity surface-emitting lasers (VCSEL’s) by solving the set of coupled equations for carrier density, field amplitude, and plasma and lattice temperatures. We show that, because of plasma heating and lattice heating, the threshold of VCSEL’s can be continuous or discontinuous with bistability between lasing and nonlasing states (correspondingly second-or first-order phase transition), depending on the relative position of the cavity frequency to the gain bandwidth and on the ambient (substrate) temperature of the device.

© 1995 Optical Society of America

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  1. P. L. Gourley, S. K. Lyo, T. M. Brennan, B. E. Hammons, C. F. Schaus, S. Sun, Appl. Phys. Lett. 55, 2698 (1989).
    [CrossRef]
  2. J. W. Scott, S. W. Corzine, D. B. Young, L. A. Coldren, Appl. Phys. Lett. 62, 1050 (1993); IEEE J. Quantum Electron. 29, 1295 (1993).
    [CrossRef]
  3. B. Lu. P. Zhou, J. Cheng, K. J. Malloy, Appl. Phys. Lett. 65, 1337 (1994); Proc. Soc. Photo-Opt. Instrum. Eng. 2147, 12 (1994).
    [CrossRef]
  4. W. Nakwaski, M. Osinski, IEEE J. Quantum Electron. 27, 1391 (1991); Proc. Soc. Photo-Opt. Instrum. Eng. 2146, 365 (1994).
    [CrossRef]
  5. F. Jahnke, S. W. Koch, Opt. Lett. 18, 1438 (1993); F. Jahnke, K. Henneberger, W. Schäfer, S. W. Koch, J. Opt. Soc. Am. B 10, 2396 (1993).
    [CrossRef] [PubMed]
  6. C. Z. Ning, R. A. Indik, J. V. Moloney, “Selfconsistent approach to thermal effects in vertical-cavity surface-emitting lasers,” submitted toJ. Opt. Soc. Am. B.
  7. C. Z. Ning, J. V. Moloney, Appl. Phys. Lett. 66, 559 (1995).
    [CrossRef]
  8. H. Haken, Synergetics, An Introduction, 3rd ed. (Springer-Verlag, Berlin, 1983), Chap 8, p. 225.
  9. L. Lugiato, in Progress in Optics, E. Wolf, ed. (North- Holland, Amsterdam, 1984), Vol. XXI, p. 69.
    [CrossRef]
  10. B. Lu, Center for High Technological Materials, University of New Mexico, Albuquerque, N.M. 87131 (personal communication, 1994).

1995 (1)

C. Z. Ning, J. V. Moloney, Appl. Phys. Lett. 66, 559 (1995).
[CrossRef]

1994 (1)

B. Lu. P. Zhou, J. Cheng, K. J. Malloy, Appl. Phys. Lett. 65, 1337 (1994); Proc. Soc. Photo-Opt. Instrum. Eng. 2147, 12 (1994).
[CrossRef]

1993 (2)

J. W. Scott, S. W. Corzine, D. B. Young, L. A. Coldren, Appl. Phys. Lett. 62, 1050 (1993); IEEE J. Quantum Electron. 29, 1295 (1993).
[CrossRef]

F. Jahnke, S. W. Koch, Opt. Lett. 18, 1438 (1993); F. Jahnke, K. Henneberger, W. Schäfer, S. W. Koch, J. Opt. Soc. Am. B 10, 2396 (1993).
[CrossRef] [PubMed]

1991 (1)

W. Nakwaski, M. Osinski, IEEE J. Quantum Electron. 27, 1391 (1991); Proc. Soc. Photo-Opt. Instrum. Eng. 2146, 365 (1994).
[CrossRef]

1989 (1)

P. L. Gourley, S. K. Lyo, T. M. Brennan, B. E. Hammons, C. F. Schaus, S. Sun, Appl. Phys. Lett. 55, 2698 (1989).
[CrossRef]

Brennan, T. M.

P. L. Gourley, S. K. Lyo, T. M. Brennan, B. E. Hammons, C. F. Schaus, S. Sun, Appl. Phys. Lett. 55, 2698 (1989).
[CrossRef]

Cheng, J.

B. Lu. P. Zhou, J. Cheng, K. J. Malloy, Appl. Phys. Lett. 65, 1337 (1994); Proc. Soc. Photo-Opt. Instrum. Eng. 2147, 12 (1994).
[CrossRef]

Coldren, L. A.

J. W. Scott, S. W. Corzine, D. B. Young, L. A. Coldren, Appl. Phys. Lett. 62, 1050 (1993); IEEE J. Quantum Electron. 29, 1295 (1993).
[CrossRef]

Corzine, S. W.

J. W. Scott, S. W. Corzine, D. B. Young, L. A. Coldren, Appl. Phys. Lett. 62, 1050 (1993); IEEE J. Quantum Electron. 29, 1295 (1993).
[CrossRef]

Gourley, P. L.

P. L. Gourley, S. K. Lyo, T. M. Brennan, B. E. Hammons, C. F. Schaus, S. Sun, Appl. Phys. Lett. 55, 2698 (1989).
[CrossRef]

Haken, H.

H. Haken, Synergetics, An Introduction, 3rd ed. (Springer-Verlag, Berlin, 1983), Chap 8, p. 225.

Hammons, B. E.

P. L. Gourley, S. K. Lyo, T. M. Brennan, B. E. Hammons, C. F. Schaus, S. Sun, Appl. Phys. Lett. 55, 2698 (1989).
[CrossRef]

Indik, R. A.

C. Z. Ning, R. A. Indik, J. V. Moloney, “Selfconsistent approach to thermal effects in vertical-cavity surface-emitting lasers,” submitted toJ. Opt. Soc. Am. B.

Jahnke, F.

Koch, S. W.

Lu, B.

B. Lu, Center for High Technological Materials, University of New Mexico, Albuquerque, N.M. 87131 (personal communication, 1994).

Lugiato, L.

L. Lugiato, in Progress in Optics, E. Wolf, ed. (North- Holland, Amsterdam, 1984), Vol. XXI, p. 69.
[CrossRef]

Lyo, S. K.

P. L. Gourley, S. K. Lyo, T. M. Brennan, B. E. Hammons, C. F. Schaus, S. Sun, Appl. Phys. Lett. 55, 2698 (1989).
[CrossRef]

Malloy, K. J.

B. Lu. P. Zhou, J. Cheng, K. J. Malloy, Appl. Phys. Lett. 65, 1337 (1994); Proc. Soc. Photo-Opt. Instrum. Eng. 2147, 12 (1994).
[CrossRef]

Moloney, J. V.

C. Z. Ning, J. V. Moloney, Appl. Phys. Lett. 66, 559 (1995).
[CrossRef]

C. Z. Ning, R. A. Indik, J. V. Moloney, “Selfconsistent approach to thermal effects in vertical-cavity surface-emitting lasers,” submitted toJ. Opt. Soc. Am. B.

Nakwaski, W.

W. Nakwaski, M. Osinski, IEEE J. Quantum Electron. 27, 1391 (1991); Proc. Soc. Photo-Opt. Instrum. Eng. 2146, 365 (1994).
[CrossRef]

Ning, C. Z.

C. Z. Ning, J. V. Moloney, Appl. Phys. Lett. 66, 559 (1995).
[CrossRef]

C. Z. Ning, R. A. Indik, J. V. Moloney, “Selfconsistent approach to thermal effects in vertical-cavity surface-emitting lasers,” submitted toJ. Opt. Soc. Am. B.

Osinski, M.

W. Nakwaski, M. Osinski, IEEE J. Quantum Electron. 27, 1391 (1991); Proc. Soc. Photo-Opt. Instrum. Eng. 2146, 365 (1994).
[CrossRef]

Schaus, C. F.

P. L. Gourley, S. K. Lyo, T. M. Brennan, B. E. Hammons, C. F. Schaus, S. Sun, Appl. Phys. Lett. 55, 2698 (1989).
[CrossRef]

Scott, J. W.

J. W. Scott, S. W. Corzine, D. B. Young, L. A. Coldren, Appl. Phys. Lett. 62, 1050 (1993); IEEE J. Quantum Electron. 29, 1295 (1993).
[CrossRef]

Sun, S.

P. L. Gourley, S. K. Lyo, T. M. Brennan, B. E. Hammons, C. F. Schaus, S. Sun, Appl. Phys. Lett. 55, 2698 (1989).
[CrossRef]

Young, D. B.

J. W. Scott, S. W. Corzine, D. B. Young, L. A. Coldren, Appl. Phys. Lett. 62, 1050 (1993); IEEE J. Quantum Electron. 29, 1295 (1993).
[CrossRef]

Zhou, B. Lu. P.

B. Lu. P. Zhou, J. Cheng, K. J. Malloy, Appl. Phys. Lett. 65, 1337 (1994); Proc. Soc. Photo-Opt. Instrum. Eng. 2147, 12 (1994).
[CrossRef]

Appl. Phys. Lett. (4)

P. L. Gourley, S. K. Lyo, T. M. Brennan, B. E. Hammons, C. F. Schaus, S. Sun, Appl. Phys. Lett. 55, 2698 (1989).
[CrossRef]

J. W. Scott, S. W. Corzine, D. B. Young, L. A. Coldren, Appl. Phys. Lett. 62, 1050 (1993); IEEE J. Quantum Electron. 29, 1295 (1993).
[CrossRef]

B. Lu. P. Zhou, J. Cheng, K. J. Malloy, Appl. Phys. Lett. 65, 1337 (1994); Proc. Soc. Photo-Opt. Instrum. Eng. 2147, 12 (1994).
[CrossRef]

C. Z. Ning, J. V. Moloney, Appl. Phys. Lett. 66, 559 (1995).
[CrossRef]

IEEE J. Quantum Electron. (1)

W. Nakwaski, M. Osinski, IEEE J. Quantum Electron. 27, 1391 (1991); Proc. Soc. Photo-Opt. Instrum. Eng. 2146, 365 (1994).
[CrossRef]

Opt. Lett. (1)

Other (4)

C. Z. Ning, R. A. Indik, J. V. Moloney, “Selfconsistent approach to thermal effects in vertical-cavity surface-emitting lasers,” submitted toJ. Opt. Soc. Am. B.

H. Haken, Synergetics, An Introduction, 3rd ed. (Springer-Verlag, Berlin, 1983), Chap 8, p. 225.

L. Lugiato, in Progress in Optics, E. Wolf, ed. (North- Holland, Amsterdam, 1984), Vol. XXI, p. 69.
[CrossRef]

B. Lu, Center for High Technological Materials, University of New Mexico, Albuquerque, N.M. 87131 (personal communication, 1994).

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Figures (3)

Fig. 1
Fig. 1

Cw solution of Eqs. (1)(4), in which the solid (dotted) curves represent stable (unstable) solutions. Pth, Np, u, and Np, d represent the minimum output power and upward transition and downward transition pumping values, respectively. Parameters are γT/γ|| = 103, γa/γ|| = 102, Ta = 100 K, ħω0 = 1.419 eV, nb = 3.564, L = 7λ0/nb, Lm = 0.045L, γ|| = 109 s−1, γ = 1013 s−1, dipole size 0.5 nm, me = 0.0665m0, mh = 0.45m0, cq = 1.862 × 106 J/Km3, S = 100 μm2, R = 1000 Ω, and γnr = γ||. The reflectivity of the emitting end of the VCSEL is 0.998. The other end is assumed to be totally reflecting.

Fig. 2
Fig. 2

Output power versus pumping, with ħω0 = 1.42 eV. The other parameters are the same as in Fig. 1, except the ambient temperature, which is marked at each curve.

Fig. 3
Fig. 3

Threshold pumping (dashed and solid curves) and output power (dotted curve) at the downward transition threshold versus Ta, with ħω0 = 1.4 eV. The other parameters are the same as in Fig. 1, except that Ta is changing.

Equations (4)

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A ˙ = i ω 0 L m 2 n b 2 L β χ A - κ A ,
N ˙ = γ | | ( N 0 - N ) - 1 2 β G N A 2 ,
T ˙ p = γ | | [ J W ( W 0 - W ) - J N ( N 0 - N ) ] - 1 2 β G T p A 2 - γ T ( T p - T l ) ,
T ˙ l = - γ a ( T l - T a ) + γ T ( T p - T l ) + γ nr ω 0 c q N + S 2 R c q V t j 2 ,

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