Abstract

The amplified-spontaneous-emission spectrum of the light field in the nonlasing supermode of two coupled semiconductor lasers is analyzed with linearized Langevin equations. It is shown that the interference between the laser mode and the fluctuating light field in the nonlasing mode causes spatial hole burning. This effect introduces a phase-sensitive coupling between the laser field and the fluctuations of the nonlasing mode. For high laser fields this coupling splits the spectrum of the nonlasing mode into a triplet consisting of two relaxation oscillation sidebands that are in phase with the laser light and a centerline at the lasing frequency with a phase shift of ±π/2 relative to the laser light. As the laser intensity is increased close to threshold, the spectrum shows a continuous transition from the single amplified-spontaneous-emission line at the frequency of the nonlasing mode to the triplet structure. An analytical expression for this transition is derived, and typical features are discussed.

© 1999 Optical Society of America

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References

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  1. H. F. Hofmann and O. Hess, “Quantum noise and polarization fluctuations in vertical cavity surface emitting lasers,” Phys. Rev. A 56, 868 (1997).
    [CrossRef]
  2. A. K. J. van Doorn, M. P. van Exter, A. M. van der Lee, and J. P. Woerdman, “Coupled-mode description for the polarization state of a vertical-cavity semiconductor laser,” Phys. Rev. A 55, 1473 (1997).
    [CrossRef]
  3. H. van der Lem and D. Lenstra, “Saturation-induced frequency shift in the noise spectrum of a birefringent vertical-cavity surface emitting laser,” Opt. Lett. 22, 1698 (1997).
    [CrossRef]
  4. H. F. Hofmann and O. Hess, “The split density model: a unified description of polarization and array dynamics for vertical cavity surface emitting lasers,” Quantum Semiclassic. Opt. 9, 749 (1997).
    [CrossRef]
  5. H. G. Winful and S. S. Wang, “Stability of phase locking in coupled semiconductor laser arrays,” Appl. Phys. Lett. 53, 1894 (1988).
    [CrossRef]
  6. S. S. Wang and H. G. Winful, “Dynamics of phase-locked semiconductor laser arrays,” Appl. Phys. Lett. 52, 1774 (1988).
    [CrossRef]
  7. M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, and N. G. Stoffel, “Large two-dimensional arrays of phase-locked vertical cavity surface emitting lasers,” Appl. Phys. Lett. 60, 1535 (1992).
    [CrossRef]
  8. R. A. Morgan, K. Kojima, T. Mullally, G. D. Guth, M. W. Focht, R. E. Leibenguth, and M. Asom, “High-power coherently coupled 8×8 vertical cavity surface emitting laser array,” Appl. Phys. Lett. 61, 1160 (1992).
    [CrossRef]
  9. R. A. Morgan and K. Kojima, “Optical characteristics of two-dimensional coherently coupled vertical-cavity surface-emitting laser arrays,” Opt. Lett. 18, 352 (1993).
    [CrossRef] [PubMed]
  10. J. M. Catchmark, L. E. Rogers, R. A. Morgan, M. T. Asom, G. D. Guth, and D. N. Christodoulides, “Optical characteristics of multitransverse-mode two-dimensional vertical-cavity top surface-emitting laser arrays,” IEEE J. Quantum Electron. 32, 986 (1996).
    [CrossRef]
  11. H. F. Hofmann and O. Hess, “Spontaneous-emission spectrum of the nonlasing supermodes in semiconductor laser arrays,” Opt. Lett. 23, 391 (1998).
    [CrossRef]
  12. M. Münkel, F. Kaiser, and O. Hess, “Stabilization of spatiotemporally chaotic semiconductor laser arrays by means of delayed optical feedback,” Phys. Rev. E 56, 3868–3875 (1997).
    [CrossRef]

1998 (1)

1997 (5)

M. Münkel, F. Kaiser, and O. Hess, “Stabilization of spatiotemporally chaotic semiconductor laser arrays by means of delayed optical feedback,” Phys. Rev. E 56, 3868–3875 (1997).
[CrossRef]

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

A. K. J. van Doorn, M. P. van Exter, A. M. van der Lee, and J. P. Woerdman, “Coupled-mode description for the polarization state of a vertical-cavity semiconductor laser,” Phys. Rev. A 55, 1473 (1997).
[CrossRef]

H. van der Lem and D. Lenstra, “Saturation-induced frequency shift in the noise spectrum of a birefringent vertical-cavity surface emitting laser,” Opt. Lett. 22, 1698 (1997).
[CrossRef]

H. F. Hofmann and O. Hess, “The split density model: a unified description of polarization and array dynamics for vertical cavity surface emitting lasers,” Quantum Semiclassic. Opt. 9, 749 (1997).
[CrossRef]

1996 (1)

J. M. Catchmark, L. E. Rogers, R. A. Morgan, M. T. Asom, G. D. Guth, and D. N. Christodoulides, “Optical characteristics of multitransverse-mode two-dimensional vertical-cavity top surface-emitting laser arrays,” IEEE J. Quantum Electron. 32, 986 (1996).
[CrossRef]

1993 (1)

1992 (2)

M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, and N. G. Stoffel, “Large two-dimensional arrays of phase-locked vertical cavity surface emitting lasers,” Appl. Phys. Lett. 60, 1535 (1992).
[CrossRef]

R. A. Morgan, K. Kojima, T. Mullally, G. D. Guth, M. W. Focht, R. E. Leibenguth, and M. Asom, “High-power coherently coupled 8×8 vertical cavity surface emitting laser array,” Appl. Phys. Lett. 61, 1160 (1992).
[CrossRef]

1988 (2)

H. G. Winful and S. S. Wang, “Stability of phase locking in coupled semiconductor laser arrays,” Appl. Phys. Lett. 53, 1894 (1988).
[CrossRef]

S. S. Wang and H. G. Winful, “Dynamics of phase-locked semiconductor laser arrays,” Appl. Phys. Lett. 52, 1774 (1988).
[CrossRef]

Asom, M.

R. A. Morgan, K. Kojima, T. Mullally, G. D. Guth, M. W. Focht, R. E. Leibenguth, and M. Asom, “High-power coherently coupled 8×8 vertical cavity surface emitting laser array,” Appl. Phys. Lett. 61, 1160 (1992).
[CrossRef]

Asom, M. T.

J. M. Catchmark, L. E. Rogers, R. A. Morgan, M. T. Asom, G. D. Guth, and D. N. Christodoulides, “Optical characteristics of multitransverse-mode two-dimensional vertical-cavity top surface-emitting laser arrays,” IEEE J. Quantum Electron. 32, 986 (1996).
[CrossRef]

Catchmark, J. M.

J. M. Catchmark, L. E. Rogers, R. A. Morgan, M. T. Asom, G. D. Guth, and D. N. Christodoulides, “Optical characteristics of multitransverse-mode two-dimensional vertical-cavity top surface-emitting laser arrays,” IEEE J. Quantum Electron. 32, 986 (1996).
[CrossRef]

Christodoulides, D. N.

J. M. Catchmark, L. E. Rogers, R. A. Morgan, M. T. Asom, G. D. Guth, and D. N. Christodoulides, “Optical characteristics of multitransverse-mode two-dimensional vertical-cavity top surface-emitting laser arrays,” IEEE J. Quantum Electron. 32, 986 (1996).
[CrossRef]

Florez, L. T.

M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, and N. G. Stoffel, “Large two-dimensional arrays of phase-locked vertical cavity surface emitting lasers,” Appl. Phys. Lett. 60, 1535 (1992).
[CrossRef]

Focht, M. W.

R. A. Morgan, K. Kojima, T. Mullally, G. D. Guth, M. W. Focht, R. E. Leibenguth, and M. Asom, “High-power coherently coupled 8×8 vertical cavity surface emitting laser array,” Appl. Phys. Lett. 61, 1160 (1992).
[CrossRef]

Guth, G. D.

J. M. Catchmark, L. E. Rogers, R. A. Morgan, M. T. Asom, G. D. Guth, and D. N. Christodoulides, “Optical characteristics of multitransverse-mode two-dimensional vertical-cavity top surface-emitting laser arrays,” IEEE J. Quantum Electron. 32, 986 (1996).
[CrossRef]

R. A. Morgan, K. Kojima, T. Mullally, G. D. Guth, M. W. Focht, R. E. Leibenguth, and M. Asom, “High-power coherently coupled 8×8 vertical cavity surface emitting laser array,” Appl. Phys. Lett. 61, 1160 (1992).
[CrossRef]

Harbison, J. P.

M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, and N. G. Stoffel, “Large two-dimensional arrays of phase-locked vertical cavity surface emitting lasers,” Appl. Phys. Lett. 60, 1535 (1992).
[CrossRef]

Hess, O.

H. F. Hofmann and O. Hess, “Spontaneous-emission spectrum of the nonlasing supermodes in semiconductor laser arrays,” Opt. Lett. 23, 391 (1998).
[CrossRef]

H. F. Hofmann and O. Hess, “The split density model: a unified description of polarization and array dynamics for vertical cavity surface emitting lasers,” Quantum Semiclassic. Opt. 9, 749 (1997).
[CrossRef]

M. Münkel, F. Kaiser, and O. Hess, “Stabilization of spatiotemporally chaotic semiconductor laser arrays by means of delayed optical feedback,” Phys. Rev. E 56, 3868–3875 (1997).
[CrossRef]

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

Hofmann, H. F.

H. F. Hofmann and O. Hess, “Spontaneous-emission spectrum of the nonlasing supermodes in semiconductor laser arrays,” Opt. Lett. 23, 391 (1998).
[CrossRef]

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

H. F. Hofmann and O. Hess, “The split density model: a unified description of polarization and array dynamics for vertical cavity surface emitting lasers,” Quantum Semiclassic. Opt. 9, 749 (1997).
[CrossRef]

Kaiser, F.

M. Münkel, F. Kaiser, and O. Hess, “Stabilization of spatiotemporally chaotic semiconductor laser arrays by means of delayed optical feedback,” Phys. Rev. E 56, 3868–3875 (1997).
[CrossRef]

Kapon, E.

M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, and N. G. Stoffel, “Large two-dimensional arrays of phase-locked vertical cavity surface emitting lasers,” Appl. Phys. Lett. 60, 1535 (1992).
[CrossRef]

Kojima, K.

R. A. Morgan and K. Kojima, “Optical characteristics of two-dimensional coherently coupled vertical-cavity surface-emitting laser arrays,” Opt. Lett. 18, 352 (1993).
[CrossRef] [PubMed]

R. A. Morgan, K. Kojima, T. Mullally, G. D. Guth, M. W. Focht, R. E. Leibenguth, and M. Asom, “High-power coherently coupled 8×8 vertical cavity surface emitting laser array,” Appl. Phys. Lett. 61, 1160 (1992).
[CrossRef]

Leibenguth, R. E.

R. A. Morgan, K. Kojima, T. Mullally, G. D. Guth, M. W. Focht, R. E. Leibenguth, and M. Asom, “High-power coherently coupled 8×8 vertical cavity surface emitting laser array,” Appl. Phys. Lett. 61, 1160 (1992).
[CrossRef]

Lenstra, D.

Morgan, R. A.

J. M. Catchmark, L. E. Rogers, R. A. Morgan, M. T. Asom, G. D. Guth, and D. N. Christodoulides, “Optical characteristics of multitransverse-mode two-dimensional vertical-cavity top surface-emitting laser arrays,” IEEE J. Quantum Electron. 32, 986 (1996).
[CrossRef]

R. A. Morgan and K. Kojima, “Optical characteristics of two-dimensional coherently coupled vertical-cavity surface-emitting laser arrays,” Opt. Lett. 18, 352 (1993).
[CrossRef] [PubMed]

R. A. Morgan, K. Kojima, T. Mullally, G. D. Guth, M. W. Focht, R. E. Leibenguth, and M. Asom, “High-power coherently coupled 8×8 vertical cavity surface emitting laser array,” Appl. Phys. Lett. 61, 1160 (1992).
[CrossRef]

Mullally, T.

R. A. Morgan, K. Kojima, T. Mullally, G. D. Guth, M. W. Focht, R. E. Leibenguth, and M. Asom, “High-power coherently coupled 8×8 vertical cavity surface emitting laser array,” Appl. Phys. Lett. 61, 1160 (1992).
[CrossRef]

Münkel, M.

M. Münkel, F. Kaiser, and O. Hess, “Stabilization of spatiotemporally chaotic semiconductor laser arrays by means of delayed optical feedback,” Phys. Rev. E 56, 3868–3875 (1997).
[CrossRef]

Orenstein, M.

M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, and N. G. Stoffel, “Large two-dimensional arrays of phase-locked vertical cavity surface emitting lasers,” Appl. Phys. Lett. 60, 1535 (1992).
[CrossRef]

Rogers, L. E.

J. M. Catchmark, L. E. Rogers, R. A. Morgan, M. T. Asom, G. D. Guth, and D. N. Christodoulides, “Optical characteristics of multitransverse-mode two-dimensional vertical-cavity top surface-emitting laser arrays,” IEEE J. Quantum Electron. 32, 986 (1996).
[CrossRef]

Stoffel, N. G.

M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, and N. G. Stoffel, “Large two-dimensional arrays of phase-locked vertical cavity surface emitting lasers,” Appl. Phys. Lett. 60, 1535 (1992).
[CrossRef]

van der Lee, A. M.

A. K. J. van Doorn, M. P. van Exter, A. M. van der Lee, and J. P. Woerdman, “Coupled-mode description for the polarization state of a vertical-cavity semiconductor laser,” Phys. Rev. A 55, 1473 (1997).
[CrossRef]

van der Lem, H.

van Doorn, A. K. J.

A. K. J. van Doorn, M. P. van Exter, A. M. van der Lee, and J. P. Woerdman, “Coupled-mode description for the polarization state of a vertical-cavity semiconductor laser,” Phys. Rev. A 55, 1473 (1997).
[CrossRef]

van Exter, M. P.

A. K. J. van Doorn, M. P. van Exter, A. M. van der Lee, and J. P. Woerdman, “Coupled-mode description for the polarization state of a vertical-cavity semiconductor laser,” Phys. Rev. A 55, 1473 (1997).
[CrossRef]

Wang, S. S.

H. G. Winful and S. S. Wang, “Stability of phase locking in coupled semiconductor laser arrays,” Appl. Phys. Lett. 53, 1894 (1988).
[CrossRef]

S. S. Wang and H. G. Winful, “Dynamics of phase-locked semiconductor laser arrays,” Appl. Phys. Lett. 52, 1774 (1988).
[CrossRef]

Winful, H. G.

S. S. Wang and H. G. Winful, “Dynamics of phase-locked semiconductor laser arrays,” Appl. Phys. Lett. 52, 1774 (1988).
[CrossRef]

H. G. Winful and S. S. Wang, “Stability of phase locking in coupled semiconductor laser arrays,” Appl. Phys. Lett. 53, 1894 (1988).
[CrossRef]

Woerdman, J. P.

A. K. J. van Doorn, M. P. van Exter, A. M. van der Lee, and J. P. Woerdman, “Coupled-mode description for the polarization state of a vertical-cavity semiconductor laser,” Phys. Rev. A 55, 1473 (1997).
[CrossRef]

Appl. Phys. Lett. (4)

H. G. Winful and S. S. Wang, “Stability of phase locking in coupled semiconductor laser arrays,” Appl. Phys. Lett. 53, 1894 (1988).
[CrossRef]

S. S. Wang and H. G. Winful, “Dynamics of phase-locked semiconductor laser arrays,” Appl. Phys. Lett. 52, 1774 (1988).
[CrossRef]

M. Orenstein, E. Kapon, J. P. Harbison, L. T. Florez, and N. G. Stoffel, “Large two-dimensional arrays of phase-locked vertical cavity surface emitting lasers,” Appl. Phys. Lett. 60, 1535 (1992).
[CrossRef]

R. A. Morgan, K. Kojima, T. Mullally, G. D. Guth, M. W. Focht, R. E. Leibenguth, and M. Asom, “High-power coherently coupled 8×8 vertical cavity surface emitting laser array,” Appl. Phys. Lett. 61, 1160 (1992).
[CrossRef]

IEEE J. Quantum Electron. (1)

J. M. Catchmark, L. E. Rogers, R. A. Morgan, M. T. Asom, G. D. Guth, and D. N. Christodoulides, “Optical characteristics of multitransverse-mode two-dimensional vertical-cavity top surface-emitting laser arrays,” IEEE J. Quantum Electron. 32, 986 (1996).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. A (2)

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

A. K. J. van Doorn, M. P. van Exter, A. M. van der Lee, and J. P. Woerdman, “Coupled-mode description for the polarization state of a vertical-cavity semiconductor laser,” Phys. Rev. A 55, 1473 (1997).
[CrossRef]

Phys. Rev. E (1)

M. Münkel, F. Kaiser, and O. Hess, “Stabilization of spatiotemporally chaotic semiconductor laser arrays by means of delayed optical feedback,” Phys. Rev. E 56, 3868–3875 (1997).
[CrossRef]

Quantum Semiclassic. Opt. (1)

H. F. Hofmann and O. Hess, “The split density model: a unified description of polarization and array dynamics for vertical cavity surface emitting lasers,” Quantum Semiclassic. Opt. 9, 749 (1997).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Relaxation of f(t=0)=1 and f(t=0)=0 for κ-=400 GHz, s=3 GHz, Ω=1 GHz, γ+2Γ=10 GHz, α=3, and wI0=1 GHz. The oscillations effectively correspond to a mainly counterclockwise rotation.

Fig. 2
Fig. 2

Amplified-spontaneous-emission spectra for κ-=400 GHz, s=3 GHz, Ω=1 GHz, γ+2Γ=10 GHz, and α=3. (a) Contour plot of the spectrum as a function of laser intensity from wI0=0 to wI0=2 GHz; (b) spectra at wI0=0 (no offset), wI0=0.5 GHz (offset of 6/GHz), wI0=1.0 GHz (offset of 12/GHz), and wI0=1.5 GHz (offset of 18/GHz).

Fig. 3
Fig. 3

Amplified-spontaneous-emission spectra close to threshold for the same parameters as in Fig. 1. (a) Contour plot of the spectrum as a function of laser intensity from wI0=0 to wI0=0.3 GHz; (b) spectra at wI0=0 (no offset), wI0=0.1 GHz (offset of 3/GHz), wI0=0.2 GHz (offset of 6/GHz), and wI0=0.3 GHz (offset of 9/GHz).

Fig. 4
Fig. 4

Amplified-spontaneous-emission spectrum for κ-=400 GHz, s=3 GHz, Ω=1 GHz, γ+2Γ=200 GHz, and α=3. (a) Contour plot of the spectrum as a function of laser intensity from wI0=0 to wI0=2 GHz; (b) spectra at wI0=0 (no offset), wI0=0.5 GHz (offset of 6/GHz), wI0=1.0 GHz (offset of 12/GHz), and wI0=1.5 GHz (offset of 18/GHz).

Fig. 5
Fig. 5

Stability boundary for κ-=100 GHz, s=5 GHz, Ω=10 GHz, α=3, and variable carrier-recombination and -diffusion rates γ+2Γ. Diagonal line, the approximate boundary for sideband undamping given by γ+2Γ+wI0+s-αΩ=0. The horizontal line at γ+2Γ=15 GHz represents the choice of parameters in Subsection 4.C and in Fig. 6.

Fig. 6
Fig. 6

Amplified-spontaneous-emission spectrum for κ-=100 GHz, s=5 GHz, Ω=10 GHz, γ+2Γ=15 GHz, and α=3. (a) Contour plot of the spectrum as a function of laser intensity from wI0=0 to wI0=10 GHz; (b) spectra at wI0=3 GHz (no offset), wI0=4 GHz (offset of 5/GHz), wI0=5 GHz (offset of 10/GHz), wI0=6 GHz (offset of 15/GHz), and wI0=7 GHz (offset of 20/GHz).

Fig. 7
Fig. 7

Amplified-spontaneous-emission spectrum for κ-=100 GHz, s=5 GHz, Ω=10 GHz, γ+2Γ=50 GHz, and α=3. (a) Contour plot of the spectrum as a function of laser intensity from wI0=0 to wI0=10 GHz; (b) spectra at wI0=0 (no offset), wI0=2 GHz (offset of 0.2/GHz), wI0=4 GHz (offset of 0.4/GHz), wI0=6 GHz (offset of 0.6/GHz), wI0=8 GHz (offset of 0.8/GHz), and wI0=10 GHz (offset of 1.0/GHz).

Equations (41)

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ddtE1=w2N1(1-iα)E1-(κ¯+iω¯)E1-s2-iΩ2E2,
ddtN1=μ2-γN1-Γ(N1-N2)-2wE1*E1N1,
ddtE2=w2N2(1-iα)E2-(κ¯+iω¯)E2-s2-iΩ2E1,
ddtN2=μ2-γN2-Γ(N2-N1)-2wE2*E2N2,
Γ=4π2Ddiffr2.
ddtE+=w2N(1-iα)E+-(κ++iω+)E++w2Δ(1-iα)E-,
ddtE-=w2N(1-iα)E--(κ-+iω-)E-+w2Δ(1-iα)E+,
ddtN=μ-γN-w(E+*E++E-*E-)N-w(E+*E-+E-*E+)Δ,
ddtΔ=-(γ+2Γ)Δ-w(E+*E++E-*E-)Δ-w(E+*E-+E-*E+)N,
Δ=0,
N=2κ-/w,
E+=0,
E-=μ4κ--γ2w1/2 exp[-i(ω-+ακ-)t]=I0 exp[-i(ω-+ακ-)t],
ddtE+=-(κ+-κ-+iω++iακ-)E++w2Δ(1-iα)I0 exp[-i(ω-+ακ-)t],
ddtΔ=-(γ+2Γ+wI0)Δ-2κ-I0{exp[-i(ω-+ακ-)t]E+*+exp[+i(ω-+ακ-)t]E+}.
E+=(f-if)exp[-i(ω-+ακ-)t].
ddtffΔ=-s+ΩwI0/2-Ω-sαwI0/2-4κ-I00-γ-2Γ-wI0ffΔ+QQ0.
Q(t)Q(t+τ)=Q(t)Q(t+τ)=κ-δ(τ).
Δf0,
ff0 exp[-(s+αΩ)t].
ff0 cos(2κ-wI0t)×exp[-(γ+2Γ+wI0+s-αΩ)t/2],
fαf0 cos(2κ-wI0t)×exp[-(γ+2Γ+wI0+s-αΩ)t/2],
Δ-22κ-/wf0 sin(2κ-wI0t)×exp[-(γ+2Γ+wI0+s-αΩ)t/2].
f(t)=cos(2κ-wI0t)×exp[-(γ+2Γ+wI0+s-αΩ)t/2],
f(t)=α(cos{2κ-wI0t)exp[-(γ+2Γ+wI0+s-αΩ)t/2]-exp[-(s+αΩ)]}).
S=s-Ω-wI0/2Ωs-αwI0/24κ-I00γ+2Γ+wI0
det{S-iω}=0.
αΩ<s+γ+2Γ+wI0+sκ-wI0[Ω2+(s+γ+2Γ+wI0)2].
γ+2Γ+s>αΩ1+2sκ--2sκ-1+sκ-×1+1α21/2.
G(ω)=1(s+iω)2+Ω2+M(αΩ+s+iω)×s+iωΩ-Ω-αMs+iω+M,
M=2κ-wI0γ+2Γ+wI0+iω.
|f(ω)|2f*(ω)f(ω)f(ω)*f(ω)|f(ω)|2=κ-2πG(ω)G(ω).
I+(ω)=E+*(ω)E+(ω)=|f(ω)|2+|f(ω)|2+if*(ω)f(ω)-if*(ω)f(ω).
I+(ω)=κ-2π2[s2+(ω-Ω)2]+(1-iα)[s-i(ω-Ω)]M+(1+iα)[s+i(ω-Ω)]M*+(1+α2)M*M|(s+iω)2+Ω2+M(s+iω+αΩ)|2.
I+(ω)M=0=κ-π1s2+(ω+Ω)2.
I+(ω)M=κ-2π1+α2(s+αΩ)2+ω2.
I+(ω)κ-2π1+α2(s+αΩ)2+ω2+κ-2π1+α2+2[Ω+α(s+γ+2Γ+wI0)]/2κ-wI0(γ+2Γ+wI0+s-αΩ)2+4(ω+2κ-wI0)2+κ-2π1+α2-2[Ω+α(s+γ+2Γ+wI0)]/2κ-wI0(γ+2Γ+wI0+s-αΩ)2+4(ω-(2κ-wI0)2.
Icenterline=(1+α2)κ-2(s+αΩ),
Isidebands=(1+α2)κ-4(γ+2Γ+wI0+s-αΩ)×1±2Ω+2α(s+γ+2Γ+wI0)(1+α2)2κ-wI0.
r1µm240GHzΓ.
d(ωp)d(wI0)=ακ-(γ+2Γ).

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