Abstract

We study theoretically the effect of an intracavity etalon on actively mode-locked fiber lasers by solving the master equation for the laser when nonlinearity in the laser is negligible. The first-order dispersion of the material inside the etalon can increase the pulse duration by a factor of 10. The minimum pulse duration is obtained when the relative frequency offset between the free spectral range of the etalon and the modulation frequency of the active mode locking is of the order of 102. The group-velocity dispersion of the material inside the etalon as well as the finesse of the etalon affect the total cavity dispersion. The etalon helps to suppress both a simultaneous lasing in several supermodes and lasing in higher-order pulse modes of the master equation. The etalon also helps lock the central wavelength of the laser to the etalon comb.

© 2007 Optical Society of America

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  1. T. R. Clark, T. F. Carruthers, P. J. Matthews, and I. N. Duling III, "Phase noise measurements of ultrafast 10 GHz harmonically modelocked fibre laser," Electron. Lett. 35, 720-721 (1999).
    [CrossRef]
  2. M. E. Grein, H. A. Haus, Y. Chen, and E. P. Ippen, "Quantum-limited timing jitter in actively modelocked lasers," IEEE J. Quantum Electron. 40, 1458-1470 (2004).
    [CrossRef]
  3. M. Horowitz, C. R. Menyuk, T. F. Carruthers, and I. N. Dulling III, "Theoretical and experimental study modelocked fiber laser for optical communication systems," J. Lightwave Technol. 18, 1565-1574 (2000).
    [CrossRef]
  4. G. T. Harvey and L. F. Mollenauer, "Harmonically mode-locked fiber ring laser with an internal Fabry-Perot stabilizer for soliton transmission," Opt. Lett. 18, 107-109 (1993).
    [CrossRef] [PubMed]
  5. J. E. Malowicki, M. L. Fanto, M. J. Hayduk, and P. J. Delfyett, Jr., "Harmonically mode-locked glass waveguide laser with 21-fs timing jitter," IEEE Photon. Technol. Lett. 17, 40-42 (2005).
    [CrossRef]
  6. S. Gee, F. Quinlan, S. Ozharar, and P. J. Delfyett, Jr., "Simultaneous optical comb frequency stabilization and super-mode noise suppression of harmonically mode-locked semiconductor ring laser using an intracavity etalon," IEEE Photon. Technol. Lett. 17, 199-201 (2005).
    [CrossRef]
  7. J. S. Way, J. Goldhar, and G. L. Burdge, "Active harmonic modelocking of an erbium fiber laser with intracavity Fabry-Perot filters," J. Lightwave Technol. 15, 1171-1180 (1997).
    [CrossRef]
  8. D. J. Kuizenga and A. E. Siegman, "FM and AM mode locking of the homogeneous laser--Part I: theory," IEEE J. Quantum Electron. QE-6, 694-708 (1970).
    [CrossRef]
  9. R. J. Jones and Jean-Claude Diels, "Stabilization of femtosecond lasers for optical frequency metrology and direct optical to radio frequency," Phys. Rev. Lett. 86, 3288-3291 (2001).
    [CrossRef] [PubMed]
  10. H. A. Haus, "Mode-locking of lasers," IEEE J. Sel. Top. Quantum Electron. 6, 1173-1184 (2000).
    [CrossRef]
  11. Ch. Fabry and A. Perot, "Thérie et applications d'une nouvelle méthode interférentielle," Ann. Chim. Phys. 16, 115-144 (1899).
  12. M.Bass, ed., Handbook of Optics (McGraw-Hill, 1995), Vol. 1.
  13. S. Choi, M. Yoshida, and M. Nakazawa, "Measurements of longitudinal linewidths of 10 GHz, picosecond mode-locked erbium-doped fiber lasers using a heterodyne detection method," Trans. Inst. Electron. Commun. Eng. Jpn., Part C J86-C, 1054-1062 (2003).
  14. F. K. Fatemi, J. W. Lou, and T. F. Carruthers, "Frequency comb linewidth of an actively mode-locked fiber laser," Opt. Lett. 29, 944-946 (2004).
    [CrossRef] [PubMed]
  15. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 1995), Chap. 1.

2005 (2)

J. E. Malowicki, M. L. Fanto, M. J. Hayduk, and P. J. Delfyett, Jr., "Harmonically mode-locked glass waveguide laser with 21-fs timing jitter," IEEE Photon. Technol. Lett. 17, 40-42 (2005).
[CrossRef]

S. Gee, F. Quinlan, S. Ozharar, and P. J. Delfyett, Jr., "Simultaneous optical comb frequency stabilization and super-mode noise suppression of harmonically mode-locked semiconductor ring laser using an intracavity etalon," IEEE Photon. Technol. Lett. 17, 199-201 (2005).
[CrossRef]

2004 (2)

M. E. Grein, H. A. Haus, Y. Chen, and E. P. Ippen, "Quantum-limited timing jitter in actively modelocked lasers," IEEE J. Quantum Electron. 40, 1458-1470 (2004).
[CrossRef]

F. K. Fatemi, J. W. Lou, and T. F. Carruthers, "Frequency comb linewidth of an actively mode-locked fiber laser," Opt. Lett. 29, 944-946 (2004).
[CrossRef] [PubMed]

2003 (1)

S. Choi, M. Yoshida, and M. Nakazawa, "Measurements of longitudinal linewidths of 10 GHz, picosecond mode-locked erbium-doped fiber lasers using a heterodyne detection method," Trans. Inst. Electron. Commun. Eng. Jpn., Part C J86-C, 1054-1062 (2003).

2001 (1)

R. J. Jones and Jean-Claude Diels, "Stabilization of femtosecond lasers for optical frequency metrology and direct optical to radio frequency," Phys. Rev. Lett. 86, 3288-3291 (2001).
[CrossRef] [PubMed]

2000 (2)

1999 (1)

T. R. Clark, T. F. Carruthers, P. J. Matthews, and I. N. Duling III, "Phase noise measurements of ultrafast 10 GHz harmonically modelocked fibre laser," Electron. Lett. 35, 720-721 (1999).
[CrossRef]

1997 (1)

J. S. Way, J. Goldhar, and G. L. Burdge, "Active harmonic modelocking of an erbium fiber laser with intracavity Fabry-Perot filters," J. Lightwave Technol. 15, 1171-1180 (1997).
[CrossRef]

1993 (1)

1970 (1)

D. J. Kuizenga and A. E. Siegman, "FM and AM mode locking of the homogeneous laser--Part I: theory," IEEE J. Quantum Electron. QE-6, 694-708 (1970).
[CrossRef]

1899 (1)

Ch. Fabry and A. Perot, "Thérie et applications d'une nouvelle méthode interférentielle," Ann. Chim. Phys. 16, 115-144 (1899).

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 1995), Chap. 1.

Burdge, G. L.

J. S. Way, J. Goldhar, and G. L. Burdge, "Active harmonic modelocking of an erbium fiber laser with intracavity Fabry-Perot filters," J. Lightwave Technol. 15, 1171-1180 (1997).
[CrossRef]

Carruthers, T. F.

Chen, Y.

M. E. Grein, H. A. Haus, Y. Chen, and E. P. Ippen, "Quantum-limited timing jitter in actively modelocked lasers," IEEE J. Quantum Electron. 40, 1458-1470 (2004).
[CrossRef]

Choi, S.

S. Choi, M. Yoshida, and M. Nakazawa, "Measurements of longitudinal linewidths of 10 GHz, picosecond mode-locked erbium-doped fiber lasers using a heterodyne detection method," Trans. Inst. Electron. Commun. Eng. Jpn., Part C J86-C, 1054-1062 (2003).

Clark, T. R.

T. R. Clark, T. F. Carruthers, P. J. Matthews, and I. N. Duling III, "Phase noise measurements of ultrafast 10 GHz harmonically modelocked fibre laser," Electron. Lett. 35, 720-721 (1999).
[CrossRef]

Delfyett, P. J.

J. E. Malowicki, M. L. Fanto, M. J. Hayduk, and P. J. Delfyett, Jr., "Harmonically mode-locked glass waveguide laser with 21-fs timing jitter," IEEE Photon. Technol. Lett. 17, 40-42 (2005).
[CrossRef]

S. Gee, F. Quinlan, S. Ozharar, and P. J. Delfyett, Jr., "Simultaneous optical comb frequency stabilization and super-mode noise suppression of harmonically mode-locked semiconductor ring laser using an intracavity etalon," IEEE Photon. Technol. Lett. 17, 199-201 (2005).
[CrossRef]

Diels, Jean-Claude

R. J. Jones and Jean-Claude Diels, "Stabilization of femtosecond lasers for optical frequency metrology and direct optical to radio frequency," Phys. Rev. Lett. 86, 3288-3291 (2001).
[CrossRef] [PubMed]

Duling, I. N.

T. R. Clark, T. F. Carruthers, P. J. Matthews, and I. N. Duling III, "Phase noise measurements of ultrafast 10 GHz harmonically modelocked fibre laser," Electron. Lett. 35, 720-721 (1999).
[CrossRef]

Dulling, I. N.

Fabry, Ch.

Ch. Fabry and A. Perot, "Thérie et applications d'une nouvelle méthode interférentielle," Ann. Chim. Phys. 16, 115-144 (1899).

Fanto, M. L.

J. E. Malowicki, M. L. Fanto, M. J. Hayduk, and P. J. Delfyett, Jr., "Harmonically mode-locked glass waveguide laser with 21-fs timing jitter," IEEE Photon. Technol. Lett. 17, 40-42 (2005).
[CrossRef]

Fatemi, F. K.

Gee, S.

S. Gee, F. Quinlan, S. Ozharar, and P. J. Delfyett, Jr., "Simultaneous optical comb frequency stabilization and super-mode noise suppression of harmonically mode-locked semiconductor ring laser using an intracavity etalon," IEEE Photon. Technol. Lett. 17, 199-201 (2005).
[CrossRef]

Goldhar, J.

J. S. Way, J. Goldhar, and G. L. Burdge, "Active harmonic modelocking of an erbium fiber laser with intracavity Fabry-Perot filters," J. Lightwave Technol. 15, 1171-1180 (1997).
[CrossRef]

Grein, M. E.

M. E. Grein, H. A. Haus, Y. Chen, and E. P. Ippen, "Quantum-limited timing jitter in actively modelocked lasers," IEEE J. Quantum Electron. 40, 1458-1470 (2004).
[CrossRef]

Harvey, G. T.

Haus, H. A.

M. E. Grein, H. A. Haus, Y. Chen, and E. P. Ippen, "Quantum-limited timing jitter in actively modelocked lasers," IEEE J. Quantum Electron. 40, 1458-1470 (2004).
[CrossRef]

H. A. Haus, "Mode-locking of lasers," IEEE J. Sel. Top. Quantum Electron. 6, 1173-1184 (2000).
[CrossRef]

Hayduk, M. J.

J. E. Malowicki, M. L. Fanto, M. J. Hayduk, and P. J. Delfyett, Jr., "Harmonically mode-locked glass waveguide laser with 21-fs timing jitter," IEEE Photon. Technol. Lett. 17, 40-42 (2005).
[CrossRef]

Horowitz, M.

Ippen, E. P.

M. E. Grein, H. A. Haus, Y. Chen, and E. P. Ippen, "Quantum-limited timing jitter in actively modelocked lasers," IEEE J. Quantum Electron. 40, 1458-1470 (2004).
[CrossRef]

Jones, R. J.

R. J. Jones and Jean-Claude Diels, "Stabilization of femtosecond lasers for optical frequency metrology and direct optical to radio frequency," Phys. Rev. Lett. 86, 3288-3291 (2001).
[CrossRef] [PubMed]

Kuizenga, D. J.

D. J. Kuizenga and A. E. Siegman, "FM and AM mode locking of the homogeneous laser--Part I: theory," IEEE J. Quantum Electron. QE-6, 694-708 (1970).
[CrossRef]

Lou, J. W.

Malowicki, J. E.

J. E. Malowicki, M. L. Fanto, M. J. Hayduk, and P. J. Delfyett, Jr., "Harmonically mode-locked glass waveguide laser with 21-fs timing jitter," IEEE Photon. Technol. Lett. 17, 40-42 (2005).
[CrossRef]

Matthews, P. J.

T. R. Clark, T. F. Carruthers, P. J. Matthews, and I. N. Duling III, "Phase noise measurements of ultrafast 10 GHz harmonically modelocked fibre laser," Electron. Lett. 35, 720-721 (1999).
[CrossRef]

Menyuk, C. R.

Mollenauer, L. F.

Nakazawa, M.

S. Choi, M. Yoshida, and M. Nakazawa, "Measurements of longitudinal linewidths of 10 GHz, picosecond mode-locked erbium-doped fiber lasers using a heterodyne detection method," Trans. Inst. Electron. Commun. Eng. Jpn., Part C J86-C, 1054-1062 (2003).

Ozharar, S.

S. Gee, F. Quinlan, S. Ozharar, and P. J. Delfyett, Jr., "Simultaneous optical comb frequency stabilization and super-mode noise suppression of harmonically mode-locked semiconductor ring laser using an intracavity etalon," IEEE Photon. Technol. Lett. 17, 199-201 (2005).
[CrossRef]

Perot, A.

Ch. Fabry and A. Perot, "Thérie et applications d'une nouvelle méthode interférentielle," Ann. Chim. Phys. 16, 115-144 (1899).

Quinlan, F.

S. Gee, F. Quinlan, S. Ozharar, and P. J. Delfyett, Jr., "Simultaneous optical comb frequency stabilization and super-mode noise suppression of harmonically mode-locked semiconductor ring laser using an intracavity etalon," IEEE Photon. Technol. Lett. 17, 199-201 (2005).
[CrossRef]

Siegman, A. E.

D. J. Kuizenga and A. E. Siegman, "FM and AM mode locking of the homogeneous laser--Part I: theory," IEEE J. Quantum Electron. QE-6, 694-708 (1970).
[CrossRef]

Way, J. S.

J. S. Way, J. Goldhar, and G. L. Burdge, "Active harmonic modelocking of an erbium fiber laser with intracavity Fabry-Perot filters," J. Lightwave Technol. 15, 1171-1180 (1997).
[CrossRef]

Yoshida, M.

S. Choi, M. Yoshida, and M. Nakazawa, "Measurements of longitudinal linewidths of 10 GHz, picosecond mode-locked erbium-doped fiber lasers using a heterodyne detection method," Trans. Inst. Electron. Commun. Eng. Jpn., Part C J86-C, 1054-1062 (2003).

Ann. Chim. Phys. (1)

Ch. Fabry and A. Perot, "Thérie et applications d'une nouvelle méthode interférentielle," Ann. Chim. Phys. 16, 115-144 (1899).

Electron. Lett. (1)

T. R. Clark, T. F. Carruthers, P. J. Matthews, and I. N. Duling III, "Phase noise measurements of ultrafast 10 GHz harmonically modelocked fibre laser," Electron. Lett. 35, 720-721 (1999).
[CrossRef]

IEEE J. Quantum Electron. (2)

M. E. Grein, H. A. Haus, Y. Chen, and E. P. Ippen, "Quantum-limited timing jitter in actively modelocked lasers," IEEE J. Quantum Electron. 40, 1458-1470 (2004).
[CrossRef]

D. J. Kuizenga and A. E. Siegman, "FM and AM mode locking of the homogeneous laser--Part I: theory," IEEE J. Quantum Electron. QE-6, 694-708 (1970).
[CrossRef]

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

H. A. Haus, "Mode-locking of lasers," IEEE J. Sel. Top. Quantum Electron. 6, 1173-1184 (2000).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

J. E. Malowicki, M. L. Fanto, M. J. Hayduk, and P. J. Delfyett, Jr., "Harmonically mode-locked glass waveguide laser with 21-fs timing jitter," IEEE Photon. Technol. Lett. 17, 40-42 (2005).
[CrossRef]

S. Gee, F. Quinlan, S. Ozharar, and P. J. Delfyett, Jr., "Simultaneous optical comb frequency stabilization and super-mode noise suppression of harmonically mode-locked semiconductor ring laser using an intracavity etalon," IEEE Photon. Technol. Lett. 17, 199-201 (2005).
[CrossRef]

J. Lightwave Technol. (2)

J. S. Way, J. Goldhar, and G. L. Burdge, "Active harmonic modelocking of an erbium fiber laser with intracavity Fabry-Perot filters," J. Lightwave Technol. 15, 1171-1180 (1997).
[CrossRef]

M. Horowitz, C. R. Menyuk, T. F. Carruthers, and I. N. Dulling III, "Theoretical and experimental study modelocked fiber laser for optical communication systems," J. Lightwave Technol. 18, 1565-1574 (2000).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. Lett. (1)

R. J. Jones and Jean-Claude Diels, "Stabilization of femtosecond lasers for optical frequency metrology and direct optical to radio frequency," Phys. Rev. Lett. 86, 3288-3291 (2001).
[CrossRef] [PubMed]

Trans. Inst. Electron. Commun. Eng. Jpn., Part C (1)

S. Choi, M. Yoshida, and M. Nakazawa, "Measurements of longitudinal linewidths of 10 GHz, picosecond mode-locked erbium-doped fiber lasers using a heterodyne detection method," Trans. Inst. Electron. Commun. Eng. Jpn., Part C J86-C, 1054-1062 (2003).

Other (2)

M.Bass, ed., Handbook of Optics (McGraw-Hill, 1995), Vol. 1.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 1995), Chap. 1.

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Figures (9)

Fig. 1
Fig. 1

Schematic of the laser cavity that is analyzed in this paper. EDFA, an erbium-doped fiber amplifier.

Fig. 2
Fig. 2

Schematic of the etalon modes (dotted curve) and the modes of the laser (solid line) when (a) δ ω = 0 , (b) δ ω 0 . The parameter N is the mode number. The dashed–dotted curve represents the etalon filtering.

Fig. 3
Fig. 3

(a) Pulse duration 2 T 0 , normalized to the Kuizenga–Siegman limit 2 τ s and (b) the chirp parameter C, as a function of the normalized frequency detuning between the etalon and the laser modes, Δ Ω Ω M , for a laser with a modulation frequency Ω M 2 π = 10 GHz , a modulation depth M = 0.2 , a total cavity dispersion D = 0.13 ps 2 , a constant loss δ = 0.1 , and an intracavity etalon with a finesse F = 100 and with material dispersion coefficients γ 1 ω 0 = 0.0186 , γ 2 Ω M ω 0 = 1.22 × 10 6 .

Fig. 4
Fig. 4

Dependence of (a) the minimum pulse duration normalized to the Kuizenga–Siegman limit and (b) the chirp parameter on the etalon finesse calculated using the parameters in Fig. 3 and a normalized detuning, Δ Ω Ω M = 0.012858 .

Fig. 5
Fig. 5

Dependence of (a) the minimum pulse duration normalized to the Kuizenga–Siegman limit and (b) the chirp parameter on the etalon finesse calculated for a total cavity dispersion D = 0.13 ps 2 and a normalized detuning, Δ Ω Ω M = 0.012858 . The other laser parameters are as used in Fig. 3.

Fig. 6
Fig. 6

(a) Dependence of threshold gain, g m , for different pulse modes: m = 0 (solid line), 1 (dashed line), 2 (dashed-dotted line), 3 (dotted line). (b) A close-up view near the point where the minimum pulse duration is obtained. The parameters of the laser are the same as used in Fig. 3.

Fig. 7
Fig. 7

Dependence of the threshold gain, g 0 , on the frequency offset δ ω , normalized to half of the FWHM of the etalon modes, δ ω 0 , for an etalon with a finesse of F = 200 with a frequency detuning parameter given by h Ω M = 2 × 10 4 (dotted curve), 10 4 (dashed-dotted curve), 0.5 × 10 4 (dashed curve), 10 5 (solid curve).

Fig. 8
Fig. 8

Dependence of the threshold gain, g 0 , on the normalized frequency offset δ ω δ ω 100 , where δ ω 100 is half of the FWHM of an etalon with a finesse F = 100 . The values of the etalon finesse are F = 100 (solid curve), 200 (dashed curve), 400 (dashed-dotted curve). The detuning parameter is given by h Ω M = 0.5 × 10 4 .

Fig. 9
Fig. 9

Dependence of the central frequency offset Re ( B ) , normalized to the modulation frequency Ω M , as a function of the frequency offset δ ω , normalized to half of the FWHM of an etalon mode with a finesse F = 200 , for h Ω M = 2 × 10 4 (solid line) 10 4 (dashed line), 5 × 10 5 (dashed–dotted line), and 10 5 (dashed–double-dotted line).

Equations (43)

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T F ( ω tot ) = ( 1 R ) exp ( i φ 2 ) 1 R exp ( i φ ) ,
ω tot = ω 0 + δ ω + N Ω e + N Δ Ω ,
ω tot = ϵ 1 ω 0 + N Ω e + ϵ ( δ ω + N Δ Ω ) ,
n ( ω ) = n 0 + ϵ 2 γ 1 [ N Ω e + ϵ ( δ ω + N Δ Ω ) ] + ϵ 3 γ 2 [ N Ω e + ϵ ( δ ω + N Δ Ω ) ] 2 ,
φ = ϵ 1 φ 1 + φ 0 + ϵ φ 1 + ϵ 2 φ 2 ,
φ 1 = 2 l c n 0 ω 0 = 2 π m ,
φ 0 = 2 l c n 0 N Ω e = 2 π N ,
φ 1 = 2 l c n 0 ( δ ω + N h ) ,
φ 2 = 2 l c [ γ 1 ω 0 ( δ ω + N Δ Ω ) + ( γ 1 + γ 2 ω 0 ) ( N Ω e ) 2 ] ,
h = Δ Ω + Ω e γ 1 ω 0 n 0 ,
T E ( φ ) = 1 + P φ φ = 0 ( ϵ φ 1 + ϵ 2 φ 2 ) + 1 2 2 P φ 2 φ = 0 ( ϵ φ 1 + ϵ 2 φ 2 ) 2
1 + i J R 2 ( ϵ φ 1 + ϵ 2 φ 2 ) ϵ 2 J ( 1 + R ) φ 1 2 8 + O ( ϵ 3 ) ,
T E ( ω tot ) = P ¯ 0 + P ¯ 1 ϵ + P ¯ 2 ϵ 2 ,
P ¯ 0 = 1 ,
P ¯ 1 = [ i C 2 ( 2 l n 0 c ) δ ω ] + [ i C 2 ( 2 l n 0 c ) h ] N ,
P ¯ 2 = [ i C 2 ( 2 l n 0 c ) ( γ 1 n 0 ) ω 0 Δ ω + C 1 ( 2 l n 0 c ) 2 ( δ ω ) 2 ] + [ i C 2 ( 2 l n 0 c ) ( γ 1 n 0 ) ω 0 δ Ω + C 1 ( 2 l n 0 c ) 2 2 h δ ω ] N + [ i C 2 ( 2 l n 0 c ) [ ( γ 1 + ω 0 γ 2 ) n 0 ] Ω e 2 + C 1 ( 2 l n 0 c ) 2 h 2 ] N 2 ,
T E ( N ) = P ̃ 0 + P ̃ 1 N + P ̃ 2 N 2 ,
P ̃ 0 = 1 J ( 1 + R ) 8 ( 2 l n 0 c ) 2 δ ω 2 + i J R 2 2 l c ( ω 0 γ 1 δ ω + n 0 δ ω ) ,
P ̃ 1 = J ( 1 + R ) 8 ( 2 l n 0 c ) 2 2 h δ ω + i J R 2 2 l c ( h n 0 + ω 0 γ 1 Δ Ω ) ,
P ̃ 2 = J ( 1 + R ) 8 ( 2 l n 0 c ) 2 h 2 + i J R 2 2 l c ( γ 1 + ω 0 γ 2 ) Ω e 2 .
E ( T , t ) = m a m ( T , t m T M ) ,
T R a m ( T , t ) T = ( g δ ) a m ( T , t ) + ( i D + g Ω g 2 + K Ω K 2 ) 2 a m ( T , t ) t 2 M [ 1 cos ( Ω M t ) ] a m ( T , t ) + w m ( T , t ) ,
w m ( T , t ) = ( 1 R ) j = 0 R j a m j [ T , t j ( T M T e ) ] a m [ T , t ] ,
w ( T , t ) = ( 1 R ) j = 0 R j a [ T , t j ( T M T e ) ] a [ T , t ] .
T E ( ω ) = P 0 + P 1 ω + P 2 ω 2 ,
w ( T , ω ) = a ̃ ( T , ω ) [ T E ( ω ) 1 ] ,
P i = P ̃ i ( Ω M ) i ( i = 0 , , 2 ) .
a ̃ ( T , ω ) = d t exp ( i ω t ) a ( T , t ) .
T R a ̃ ( T , ω ) T = M 2 Ω M 2 2 a ̃ ( T , ω ) ω 2 ( i D + g Ω g 2 + K Ω K 2 P 2 ) ω 2 a ̃ ( T , ω ) + [ g δ + Re ( P 0 1 ) ] a ̃ ( T , ω ) + ω Re ( P 1 ) a ̃ ( T , ω ) .
a ̃ ( T , ω ) = m = 0 c m exp ( ω 2 2 τ m 2 ) H m ( ω τ m ) exp ( T T R Λ m ) ,
τ m 4 = W s ,
Λ m = g m δ ( 2 m + 1 ) W s ,
s = M 2 Ω M 2 , W = g m Ω g 2 + K Ω K 2 + i D P 2 .
a ( T , t ) = m = 0 c m exp ( t 2 2 τ m 2 ) H m ( t τ m ) exp ( T T R Λ m ) .
τ 0 = { 1 2 Ω g 2 + [ ( 1 2 Ω g 2 ) 2 + P ̂ i D ̂ s ] 1 2 } 1 2 ,
P ̂ = δ Ω g 2 + K Ω K 2 J ( 1 + R ) 8 ( 2 l n 0 c ) 2 h 2 1 Ω M 2 ,
D ̂ = D + J R l c ( γ 1 Ω e 2 + ω 0 γ 2 Ω e 2 ) 1 Ω M 2 Im ( Λ 0 ) Ω g 2 .
a ( t ) = a 0 exp [ ( 1 + i C ) 2 t 2 T 0 2 ] ,
g m = δ + ( 2 m + 1 ) Re [ M Ω M 2 2 ( g m Ω g 2 + K Ω K 2 + i D P 2 ) ] 1 2 .
a ̃ ( T , ω ) = A 0 exp [ ( ω + B ) 2 2 τ 2 ] exp ( T T R Λ ) .
B = δ ω h Ω M Re ( P 2 ) W ,
s τ 4 = W .
g = δ Re [ ( δ ω h Ω M ) 2 H Re ( P 2 ) W s W ] ,

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