Abstract

We show by means of analytical and numerical calculations that saturation, in combination with the linewidth-enhancement factor and the finite rate of spin–flip processes, causes a redshift of the spontaneous-emission peak with respect to the off-lasing-mode frequency in the optical spectrum of a quantum-well vertical-cavity surface emitting laser.

© 1997 Optical Society of America

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References

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  1. A. K. Jansen van Doorn, M. P. van Exter, and J. P. Woerdman, Appl. Phys. Lett. 69, 1041 (1996).
    [CrossRef]
  2. M. P. van Exter, A. K. Jansen van Doorn, and J. P. Woerdman, Phys. Rev. A 56, 845 (1997).
    [CrossRef]
  3. R. F. M. Hendriks, M. P. van Exter, and J. P. Woerdman, Appl. Phys. Lett. 69, 869 (1996).
    [CrossRef]
  4. J. Martin-Regalado, M. San Miguel, N. B. Abraham, and F. Prati, Opt. Lett. 21, 351 (1996).
    [CrossRef]
  5. J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, IEEE J. Quantum Electron. 33, 765 (1997).
    [CrossRef]
  6. M. San Miguel, Q. Feng, and J. V. Moloney, Phys. Rev. A 52, 1728 (1995).
    [CrossRef] [PubMed]
  7. A. K. Jansen van Doorn, M. P. van Exter, A. M. van der Lee, and J. P. Woerdman, Phys. Rev. A 55, 1473 (1997).
    [CrossRef]
  8. We recently heard of a related theoretical study by M. P. van Exter, R. F. M. Hendriks, and J. P. Woerdman, [Phys. Rev. A 56, 845 (1997)] on the calculation of the eigenvalues that correspond to the linearly polarized solutions by use of the same nonlinear model that we use here.
    [CrossRef]

1997 (4)

M. P. van Exter, A. K. Jansen van Doorn, and J. P. Woerdman, Phys. Rev. A 56, 845 (1997).
[CrossRef]

J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, IEEE J. Quantum Electron. 33, 765 (1997).
[CrossRef]

A. K. Jansen van Doorn, M. P. van Exter, A. M. van der Lee, and J. P. Woerdman, Phys. Rev. A 55, 1473 (1997).
[CrossRef]

We recently heard of a related theoretical study by M. P. van Exter, R. F. M. Hendriks, and J. P. Woerdman, [Phys. Rev. A 56, 845 (1997)] on the calculation of the eigenvalues that correspond to the linearly polarized solutions by use of the same nonlinear model that we use here.
[CrossRef]

1996 (3)

J. Martin-Regalado, M. San Miguel, N. B. Abraham, and F. Prati, Opt. Lett. 21, 351 (1996).
[CrossRef]

A. K. Jansen van Doorn, M. P. van Exter, and J. P. Woerdman, Appl. Phys. Lett. 69, 1041 (1996).
[CrossRef]

R. F. M. Hendriks, M. P. van Exter, and J. P. Woerdman, Appl. Phys. Lett. 69, 869 (1996).
[CrossRef]

1995 (1)

M. San Miguel, Q. Feng, and J. V. Moloney, Phys. Rev. A 52, 1728 (1995).
[CrossRef] [PubMed]

Abraham, N. B.

J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, IEEE J. Quantum Electron. 33, 765 (1997).
[CrossRef]

J. Martin-Regalado, M. San Miguel, N. B. Abraham, and F. Prati, Opt. Lett. 21, 351 (1996).
[CrossRef]

Feng, Q.

M. San Miguel, Q. Feng, and J. V. Moloney, Phys. Rev. A 52, 1728 (1995).
[CrossRef] [PubMed]

Hendriks, R. F. M.

We recently heard of a related theoretical study by M. P. van Exter, R. F. M. Hendriks, and J. P. Woerdman, [Phys. Rev. A 56, 845 (1997)] on the calculation of the eigenvalues that correspond to the linearly polarized solutions by use of the same nonlinear model that we use here.
[CrossRef]

R. F. M. Hendriks, M. P. van Exter, and J. P. Woerdman, Appl. Phys. Lett. 69, 869 (1996).
[CrossRef]

Jansen van Doorn, A. K.

M. P. van Exter, A. K. Jansen van Doorn, and J. P. Woerdman, Phys. Rev. A 56, 845 (1997).
[CrossRef]

A. K. Jansen van Doorn, M. P. van Exter, A. M. van der Lee, and J. P. Woerdman, Phys. Rev. A 55, 1473 (1997).
[CrossRef]

A. K. Jansen van Doorn, M. P. van Exter, and J. P. Woerdman, Appl. Phys. Lett. 69, 1041 (1996).
[CrossRef]

Martin-Regalado, J.

J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, IEEE J. Quantum Electron. 33, 765 (1997).
[CrossRef]

J. Martin-Regalado, M. San Miguel, N. B. Abraham, and F. Prati, Opt. Lett. 21, 351 (1996).
[CrossRef]

Moloney, J. V.

M. San Miguel, Q. Feng, and J. V. Moloney, Phys. Rev. A 52, 1728 (1995).
[CrossRef] [PubMed]

Prati, F.

J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, IEEE J. Quantum Electron. 33, 765 (1997).
[CrossRef]

J. Martin-Regalado, M. San Miguel, N. B. Abraham, and F. Prati, Opt. Lett. 21, 351 (1996).
[CrossRef]

San Miguel, M.

J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, IEEE J. Quantum Electron. 33, 765 (1997).
[CrossRef]

J. Martin-Regalado, M. San Miguel, N. B. Abraham, and F. Prati, Opt. Lett. 21, 351 (1996).
[CrossRef]

M. San Miguel, Q. Feng, and J. V. Moloney, Phys. Rev. A 52, 1728 (1995).
[CrossRef] [PubMed]

van der Lee, A. M.

A. K. Jansen van Doorn, M. P. van Exter, A. M. van der Lee, and J. P. Woerdman, Phys. Rev. A 55, 1473 (1997).
[CrossRef]

van Exter, M. P.

A. K. Jansen van Doorn, M. P. van Exter, A. M. van der Lee, and J. P. Woerdman, Phys. Rev. A 55, 1473 (1997).
[CrossRef]

M. P. van Exter, A. K. Jansen van Doorn, and J. P. Woerdman, Phys. Rev. A 56, 845 (1997).
[CrossRef]

We recently heard of a related theoretical study by M. P. van Exter, R. F. M. Hendriks, and J. P. Woerdman, [Phys. Rev. A 56, 845 (1997)] on the calculation of the eigenvalues that correspond to the linearly polarized solutions by use of the same nonlinear model that we use here.
[CrossRef]

R. F. M. Hendriks, M. P. van Exter, and J. P. Woerdman, Appl. Phys. Lett. 69, 869 (1996).
[CrossRef]

A. K. Jansen van Doorn, M. P. van Exter, and J. P. Woerdman, Appl. Phys. Lett. 69, 1041 (1996).
[CrossRef]

Woerdman, J. P.

M. P. van Exter, A. K. Jansen van Doorn, and J. P. Woerdman, Phys. Rev. A 56, 845 (1997).
[CrossRef]

A. K. Jansen van Doorn, M. P. van Exter, A. M. van der Lee, and J. P. Woerdman, Phys. Rev. A 55, 1473 (1997).
[CrossRef]

We recently heard of a related theoretical study by M. P. van Exter, R. F. M. Hendriks, and J. P. Woerdman, [Phys. Rev. A 56, 845 (1997)] on the calculation of the eigenvalues that correspond to the linearly polarized solutions by use of the same nonlinear model that we use here.
[CrossRef]

A. K. Jansen van Doorn, M. P. van Exter, and J. P. Woerdman, Appl. Phys. Lett. 69, 1041 (1996).
[CrossRef]

R. F. M. Hendriks, M. P. van Exter, and J. P. Woerdman, Appl. Phys. Lett. 69, 869 (1996).
[CrossRef]

Appl. Phys. Lett. (2)

A. K. Jansen van Doorn, M. P. van Exter, and J. P. Woerdman, Appl. Phys. Lett. 69, 1041 (1996).
[CrossRef]

R. F. M. Hendriks, M. P. van Exter, and J. P. Woerdman, Appl. Phys. Lett. 69, 869 (1996).
[CrossRef]

IEEE J. Quantum Electron. (1)

J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, IEEE J. Quantum Electron. 33, 765 (1997).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (4)

M. P. van Exter, A. K. Jansen van Doorn, and J. P. Woerdman, Phys. Rev. A 56, 845 (1997).
[CrossRef]

M. San Miguel, Q. Feng, and J. V. Moloney, Phys. Rev. A 52, 1728 (1995).
[CrossRef] [PubMed]

A. K. Jansen van Doorn, M. P. van Exter, A. M. van der Lee, and J. P. Woerdman, Phys. Rev. A 55, 1473 (1997).
[CrossRef]

We recently heard of a related theoretical study by M. P. van Exter, R. F. M. Hendriks, and J. P. Woerdman, [Phys. Rev. A 56, 845 (1997)] on the calculation of the eigenvalues that correspond to the linearly polarized solutions by use of the same nonlinear model that we use here.
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

Simulated spectra of a VCSEL for different values of γs. The narrow peak, which corresponds to the y-polarized lasing field, is attenuated by a factor of 3000. Parameter values: γp=2 ns-1, γ=1 ns-1, κ=300 ns-1, μ=1.4, α=4, β=5×10-4.

Fig. 2
Fig. 2

Comparison of the approximated expression [Eq.  (13)] with the numerically calculated value of Δω. μ=1.4, α=4, γs=100 ns-1, κ=300 ns-1, γ=1 ns-1.

Equations (14)

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

E˙±=κ1+iαN±n-1E±-iγpE+βN±n1/2χ±,
N˙=-γN-μ-γNE+2+E-2-γnE+2-E-2,
n˙=-γsn-γnE+2+E-2+γNE+2-E-2,
E+=E-=Q2exp-iγpt,
E+=-E-=iQ2expiγpt,
N=1, n=0, Q2=μ-1.
E±±iQ2+δq±expiγpt+δϕ±,
N1+δN, nδn.
λ3+γs+γQ2λ2+2κQ2γ+4γp2λ+4καQ2γγp+4γs+γQ2γp2=0.
κμ-1γ/γsγpγs,
λ2,3=-κQ2γ/γs±i2γp+καQ2γ/γs.
ΔωFWHM=2κμ-1γ/γs,
ωsp=-γp-καμ-1γ/γs,
ωconj=3γp+καμ-1γ/γs.

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