Abstract

By employing the standard frequency-domain eigenfunction expansion technique, we present an analysis on the residual timing jitter of actively harmonic mode-locked lasers incorporated with high-finesse Fabry–Perot filters. Simple formulas that directly relate the pulse train residual timing jitter performance to several experimentally measurable qualities, such as the Fabry–Perot filter finesse and the residual phase-modulation noise knee frequency, are given and discussed.

© 2006 Optical Society of America

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  1. R. A. Barry, V. Chan, K. Hall, E. S. Kintzer, J. D. Moores, K. A. Rauschenbach, E. A. Swanson, L. E. Adams, C. R. Doerr, S. G. Finn, H. A. Haus, E. P. Ippen, W. S. Wong, and M. Haner, "All-optical network consortium--ultrafast TDM networks," IEEE J. Sel. Areas Commun. 14, 999-1013 (1996).
    [CrossRef]
  2. P. W. Juodawlkis, J. C. Twichell, G. E. Betts, J. J. Hargreaves, R. D. Younger, J. L. Wasserman, F. J. O'Donnell, K. G. Ray, and R. C. Williamson, "Optically sampled analog-to-digital converters," IEEE Trans. Microwave Theory Tech. 49, 1840-1853 (2001).
    [CrossRef]
  3. D. von der Linde, "Characterization of noise in continuously operating mode-locking lasers," Appl. Phys. B 39, 201-217 (1986).
    [CrossRef]
  4. C. M. DePriest, T. Yilmaz, P. J. Delfyett, S. Etemad, A. Braun, and J. Abeles, "Ultralow noise and supermode suppression in an actively mode-locked external-cavity semiconductor diode ring laser," Opt. Lett. 27, 719-721 (2002).
    [CrossRef]
  5. T. Yilmaz, C. M. DePriest, and P. J. Delfyett, "Complete noise characterization of external cavity semiconductor laser hybrid mode-locked at 10 GHz," Electron. Lett. 37, 1338-1339 (2001).
    [CrossRef]
  6. F. Rana, H. L. T. Lee, R. J. Ram, M. E. Grein, L. A. Jiang, E. P. Ippen, and H. A. Haus, "Characterization of the noise and correlations in harmonically mode-locked lasers," J. Opt. Soc. Am. B 19, 2609-2621 (2002).
    [CrossRef]
  7. C. T. Harvey and L. F. Mollenauer, "Harmonically mode-locked fiber laser with an internal Fabry-Perot stabilizer for soliton transmission," Opt. Lett. 18, 107-109 (1993).
    [CrossRef] [PubMed]
  8. P. T. Ho, "Phase and amplitude fluctuations in a mode-locked laser," IEEE J. Quantum Electron. 21, 1806-1803 (1985).
    [CrossRef]
  9. D. R. Hjelme and A. R. Mickelson, "Theory of timing jitter in actively mode-locked lasers," IEEE J. Quantum Electron. 28, 1594-1605 (1992).
    [CrossRef]
  10. H. A. Haus, "Mode-locking of lasers," IEEE J. Sel. Top. Quantum Electron. 8, 1173-1185 (2000).
  11. T. Yilmaz, C. M. DePriest, P. J. Delfyett, S. Etemad, A. Braun, and J. H. Abeles, "Supermode suppression to below −130 dBc/Hz in 10 GHz harmonically mode-locked external sigma cavity semiconductor laser," Opt. Express 11, 1090-1095 (2003).
    [CrossRef] [PubMed]
  12. H. A. Haus and W. S. Wong, "Solitons in optical communications," Rev. Mod. Phys. 68, 423-444 (1996).
    [CrossRef]

2003 (1)

2002 (2)

2001 (2)

T. Yilmaz, C. M. DePriest, and P. J. Delfyett, "Complete noise characterization of external cavity semiconductor laser hybrid mode-locked at 10 GHz," Electron. Lett. 37, 1338-1339 (2001).
[CrossRef]

P. W. Juodawlkis, J. C. Twichell, G. E. Betts, J. J. Hargreaves, R. D. Younger, J. L. Wasserman, F. J. O'Donnell, K. G. Ray, and R. C. Williamson, "Optically sampled analog-to-digital converters," IEEE Trans. Microwave Theory Tech. 49, 1840-1853 (2001).
[CrossRef]

2000 (1)

H. A. Haus, "Mode-locking of lasers," IEEE J. Sel. Top. Quantum Electron. 8, 1173-1185 (2000).

1996 (2)

H. A. Haus and W. S. Wong, "Solitons in optical communications," Rev. Mod. Phys. 68, 423-444 (1996).
[CrossRef]

R. A. Barry, V. Chan, K. Hall, E. S. Kintzer, J. D. Moores, K. A. Rauschenbach, E. A. Swanson, L. E. Adams, C. R. Doerr, S. G. Finn, H. A. Haus, E. P. Ippen, W. S. Wong, and M. Haner, "All-optical network consortium--ultrafast TDM networks," IEEE J. Sel. Areas Commun. 14, 999-1013 (1996).
[CrossRef]

1993 (1)

1992 (1)

D. R. Hjelme and A. R. Mickelson, "Theory of timing jitter in actively mode-locked lasers," IEEE J. Quantum Electron. 28, 1594-1605 (1992).
[CrossRef]

1986 (1)

D. von der Linde, "Characterization of noise in continuously operating mode-locking lasers," Appl. Phys. B 39, 201-217 (1986).
[CrossRef]

1985 (1)

P. T. Ho, "Phase and amplitude fluctuations in a mode-locked laser," IEEE J. Quantum Electron. 21, 1806-1803 (1985).
[CrossRef]

Abeles, J.

Abeles, J. H.

Adams, L. E.

R. A. Barry, V. Chan, K. Hall, E. S. Kintzer, J. D. Moores, K. A. Rauschenbach, E. A. Swanson, L. E. Adams, C. R. Doerr, S. G. Finn, H. A. Haus, E. P. Ippen, W. S. Wong, and M. Haner, "All-optical network consortium--ultrafast TDM networks," IEEE J. Sel. Areas Commun. 14, 999-1013 (1996).
[CrossRef]

Barry, R. A.

R. A. Barry, V. Chan, K. Hall, E. S. Kintzer, J. D. Moores, K. A. Rauschenbach, E. A. Swanson, L. E. Adams, C. R. Doerr, S. G. Finn, H. A. Haus, E. P. Ippen, W. S. Wong, and M. Haner, "All-optical network consortium--ultrafast TDM networks," IEEE J. Sel. Areas Commun. 14, 999-1013 (1996).
[CrossRef]

Betts, G. E.

P. W. Juodawlkis, J. C. Twichell, G. E. Betts, J. J. Hargreaves, R. D. Younger, J. L. Wasserman, F. J. O'Donnell, K. G. Ray, and R. C. Williamson, "Optically sampled analog-to-digital converters," IEEE Trans. Microwave Theory Tech. 49, 1840-1853 (2001).
[CrossRef]

Braun, A.

Chan, V.

R. A. Barry, V. Chan, K. Hall, E. S. Kintzer, J. D. Moores, K. A. Rauschenbach, E. A. Swanson, L. E. Adams, C. R. Doerr, S. G. Finn, H. A. Haus, E. P. Ippen, W. S. Wong, and M. Haner, "All-optical network consortium--ultrafast TDM networks," IEEE J. Sel. Areas Commun. 14, 999-1013 (1996).
[CrossRef]

Delfyett, P. J.

DePriest, C. M.

Doerr, C. R.

R. A. Barry, V. Chan, K. Hall, E. S. Kintzer, J. D. Moores, K. A. Rauschenbach, E. A. Swanson, L. E. Adams, C. R. Doerr, S. G. Finn, H. A. Haus, E. P. Ippen, W. S. Wong, and M. Haner, "All-optical network consortium--ultrafast TDM networks," IEEE J. Sel. Areas Commun. 14, 999-1013 (1996).
[CrossRef]

Etemad, S.

Finn, S. G.

R. A. Barry, V. Chan, K. Hall, E. S. Kintzer, J. D. Moores, K. A. Rauschenbach, E. A. Swanson, L. E. Adams, C. R. Doerr, S. G. Finn, H. A. Haus, E. P. Ippen, W. S. Wong, and M. Haner, "All-optical network consortium--ultrafast TDM networks," IEEE J. Sel. Areas Commun. 14, 999-1013 (1996).
[CrossRef]

Grein, M. E.

Hall, K.

R. A. Barry, V. Chan, K. Hall, E. S. Kintzer, J. D. Moores, K. A. Rauschenbach, E. A. Swanson, L. E. Adams, C. R. Doerr, S. G. Finn, H. A. Haus, E. P. Ippen, W. S. Wong, and M. Haner, "All-optical network consortium--ultrafast TDM networks," IEEE J. Sel. Areas Commun. 14, 999-1013 (1996).
[CrossRef]

Haner, M.

R. A. Barry, V. Chan, K. Hall, E. S. Kintzer, J. D. Moores, K. A. Rauschenbach, E. A. Swanson, L. E. Adams, C. R. Doerr, S. G. Finn, H. A. Haus, E. P. Ippen, W. S. Wong, and M. Haner, "All-optical network consortium--ultrafast TDM networks," IEEE J. Sel. Areas Commun. 14, 999-1013 (1996).
[CrossRef]

Hargreaves, J. J.

P. W. Juodawlkis, J. C. Twichell, G. E. Betts, J. J. Hargreaves, R. D. Younger, J. L. Wasserman, F. J. O'Donnell, K. G. Ray, and R. C. Williamson, "Optically sampled analog-to-digital converters," IEEE Trans. Microwave Theory Tech. 49, 1840-1853 (2001).
[CrossRef]

Harvey, C. T.

Haus, H. A.

F. Rana, H. L. T. Lee, R. J. Ram, M. E. Grein, L. A. Jiang, E. P. Ippen, and H. A. Haus, "Characterization of the noise and correlations in harmonically mode-locked lasers," J. Opt. Soc. Am. B 19, 2609-2621 (2002).
[CrossRef]

H. A. Haus, "Mode-locking of lasers," IEEE J. Sel. Top. Quantum Electron. 8, 1173-1185 (2000).

H. A. Haus and W. S. Wong, "Solitons in optical communications," Rev. Mod. Phys. 68, 423-444 (1996).
[CrossRef]

R. A. Barry, V. Chan, K. Hall, E. S. Kintzer, J. D. Moores, K. A. Rauschenbach, E. A. Swanson, L. E. Adams, C. R. Doerr, S. G. Finn, H. A. Haus, E. P. Ippen, W. S. Wong, and M. Haner, "All-optical network consortium--ultrafast TDM networks," IEEE J. Sel. Areas Commun. 14, 999-1013 (1996).
[CrossRef]

Hjelme, D. R.

D. R. Hjelme and A. R. Mickelson, "Theory of timing jitter in actively mode-locked lasers," IEEE J. Quantum Electron. 28, 1594-1605 (1992).
[CrossRef]

Ho, P. T.

P. T. Ho, "Phase and amplitude fluctuations in a mode-locked laser," IEEE J. Quantum Electron. 21, 1806-1803 (1985).
[CrossRef]

Ippen, E. P.

F. Rana, H. L. T. Lee, R. J. Ram, M. E. Grein, L. A. Jiang, E. P. Ippen, and H. A. Haus, "Characterization of the noise and correlations in harmonically mode-locked lasers," J. Opt. Soc. Am. B 19, 2609-2621 (2002).
[CrossRef]

R. A. Barry, V. Chan, K. Hall, E. S. Kintzer, J. D. Moores, K. A. Rauschenbach, E. A. Swanson, L. E. Adams, C. R. Doerr, S. G. Finn, H. A. Haus, E. P. Ippen, W. S. Wong, and M. Haner, "All-optical network consortium--ultrafast TDM networks," IEEE J. Sel. Areas Commun. 14, 999-1013 (1996).
[CrossRef]

Jiang, L. A.

Juodawlkis, P. W.

P. W. Juodawlkis, J. C. Twichell, G. E. Betts, J. J. Hargreaves, R. D. Younger, J. L. Wasserman, F. J. O'Donnell, K. G. Ray, and R. C. Williamson, "Optically sampled analog-to-digital converters," IEEE Trans. Microwave Theory Tech. 49, 1840-1853 (2001).
[CrossRef]

Kintzer, E. S.

R. A. Barry, V. Chan, K. Hall, E. S. Kintzer, J. D. Moores, K. A. Rauschenbach, E. A. Swanson, L. E. Adams, C. R. Doerr, S. G. Finn, H. A. Haus, E. P. Ippen, W. S. Wong, and M. Haner, "All-optical network consortium--ultrafast TDM networks," IEEE J. Sel. Areas Commun. 14, 999-1013 (1996).
[CrossRef]

Lee, H. L. T.

Mickelson, A. R.

D. R. Hjelme and A. R. Mickelson, "Theory of timing jitter in actively mode-locked lasers," IEEE J. Quantum Electron. 28, 1594-1605 (1992).
[CrossRef]

Mollenauer, L. F.

Moores, J. D.

R. A. Barry, V. Chan, K. Hall, E. S. Kintzer, J. D. Moores, K. A. Rauschenbach, E. A. Swanson, L. E. Adams, C. R. Doerr, S. G. Finn, H. A. Haus, E. P. Ippen, W. S. Wong, and M. Haner, "All-optical network consortium--ultrafast TDM networks," IEEE J. Sel. Areas Commun. 14, 999-1013 (1996).
[CrossRef]

O'Donnell, F. J.

P. W. Juodawlkis, J. C. Twichell, G. E. Betts, J. J. Hargreaves, R. D. Younger, J. L. Wasserman, F. J. O'Donnell, K. G. Ray, and R. C. Williamson, "Optically sampled analog-to-digital converters," IEEE Trans. Microwave Theory Tech. 49, 1840-1853 (2001).
[CrossRef]

Ram, R. J.

Rana, F.

Rauschenbach, K. A.

R. A. Barry, V. Chan, K. Hall, E. S. Kintzer, J. D. Moores, K. A. Rauschenbach, E. A. Swanson, L. E. Adams, C. R. Doerr, S. G. Finn, H. A. Haus, E. P. Ippen, W. S. Wong, and M. Haner, "All-optical network consortium--ultrafast TDM networks," IEEE J. Sel. Areas Commun. 14, 999-1013 (1996).
[CrossRef]

Ray, K. G.

P. W. Juodawlkis, J. C. Twichell, G. E. Betts, J. J. Hargreaves, R. D. Younger, J. L. Wasserman, F. J. O'Donnell, K. G. Ray, and R. C. Williamson, "Optically sampled analog-to-digital converters," IEEE Trans. Microwave Theory Tech. 49, 1840-1853 (2001).
[CrossRef]

Swanson, E. A.

R. A. Barry, V. Chan, K. Hall, E. S. Kintzer, J. D. Moores, K. A. Rauschenbach, E. A. Swanson, L. E. Adams, C. R. Doerr, S. G. Finn, H. A. Haus, E. P. Ippen, W. S. Wong, and M. Haner, "All-optical network consortium--ultrafast TDM networks," IEEE J. Sel. Areas Commun. 14, 999-1013 (1996).
[CrossRef]

Twichell, J. C.

P. W. Juodawlkis, J. C. Twichell, G. E. Betts, J. J. Hargreaves, R. D. Younger, J. L. Wasserman, F. J. O'Donnell, K. G. Ray, and R. C. Williamson, "Optically sampled analog-to-digital converters," IEEE Trans. Microwave Theory Tech. 49, 1840-1853 (2001).
[CrossRef]

von der Linde, D.

D. von der Linde, "Characterization of noise in continuously operating mode-locking lasers," Appl. Phys. B 39, 201-217 (1986).
[CrossRef]

Wasserman, J. L.

P. W. Juodawlkis, J. C. Twichell, G. E. Betts, J. J. Hargreaves, R. D. Younger, J. L. Wasserman, F. J. O'Donnell, K. G. Ray, and R. C. Williamson, "Optically sampled analog-to-digital converters," IEEE Trans. Microwave Theory Tech. 49, 1840-1853 (2001).
[CrossRef]

Williamson, R. C.

P. W. Juodawlkis, J. C. Twichell, G. E. Betts, J. J. Hargreaves, R. D. Younger, J. L. Wasserman, F. J. O'Donnell, K. G. Ray, and R. C. Williamson, "Optically sampled analog-to-digital converters," IEEE Trans. Microwave Theory Tech. 49, 1840-1853 (2001).
[CrossRef]

Wong, W. S.

R. A. Barry, V. Chan, K. Hall, E. S. Kintzer, J. D. Moores, K. A. Rauschenbach, E. A. Swanson, L. E. Adams, C. R. Doerr, S. G. Finn, H. A. Haus, E. P. Ippen, W. S. Wong, and M. Haner, "All-optical network consortium--ultrafast TDM networks," IEEE J. Sel. Areas Commun. 14, 999-1013 (1996).
[CrossRef]

H. A. Haus and W. S. Wong, "Solitons in optical communications," Rev. Mod. Phys. 68, 423-444 (1996).
[CrossRef]

Yilmaz, T.

Younger, R. D.

P. W. Juodawlkis, J. C. Twichell, G. E. Betts, J. J. Hargreaves, R. D. Younger, J. L. Wasserman, F. J. O'Donnell, K. G. Ray, and R. C. Williamson, "Optically sampled analog-to-digital converters," IEEE Trans. Microwave Theory Tech. 49, 1840-1853 (2001).
[CrossRef]

Appl. Phys. B (1)

D. von der Linde, "Characterization of noise in continuously operating mode-locking lasers," Appl. Phys. B 39, 201-217 (1986).
[CrossRef]

Electron. Lett. (1)

T. Yilmaz, C. M. DePriest, and P. J. Delfyett, "Complete noise characterization of external cavity semiconductor laser hybrid mode-locked at 10 GHz," Electron. Lett. 37, 1338-1339 (2001).
[CrossRef]

IEEE J. Quantum Electron. (2)

P. T. Ho, "Phase and amplitude fluctuations in a mode-locked laser," IEEE J. Quantum Electron. 21, 1806-1803 (1985).
[CrossRef]

D. R. Hjelme and A. R. Mickelson, "Theory of timing jitter in actively mode-locked lasers," IEEE J. Quantum Electron. 28, 1594-1605 (1992).
[CrossRef]

IEEE J. Sel. Areas Commun. (1)

R. A. Barry, V. Chan, K. Hall, E. S. Kintzer, J. D. Moores, K. A. Rauschenbach, E. A. Swanson, L. E. Adams, C. R. Doerr, S. G. Finn, H. A. Haus, E. P. Ippen, W. S. Wong, and M. Haner, "All-optical network consortium--ultrafast TDM networks," IEEE J. Sel. Areas Commun. 14, 999-1013 (1996).
[CrossRef]

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

H. A. Haus, "Mode-locking of lasers," IEEE J. Sel. Top. Quantum Electron. 8, 1173-1185 (2000).

IEEE Trans. Microwave Theory Tech. (1)

P. W. Juodawlkis, J. C. Twichell, G. E. Betts, J. J. Hargreaves, R. D. Younger, J. L. Wasserman, F. J. O'Donnell, K. G. Ray, and R. C. Williamson, "Optically sampled analog-to-digital converters," IEEE Trans. Microwave Theory Tech. 49, 1840-1853 (2001).
[CrossRef]

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

Opt. Express (1)

Opt. Lett. (2)

Rev. Mod. Phys. (1)

H. A. Haus and W. S. Wong, "Solitons in optical communications," Rev. Mod. Phys. 68, 423-444 (1996).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Illustration of the optical spectrum of a 2 N + 1 order actively harmonic mode-locked laser of which there exists one set of strong oscillating main modes (solid lines) and 2 N sets of weak oscillating background supermodes (dashed lines). Note that for the purpose of simplicity, only one set of weak oscillating supermodes is depicted. The topmost comblike solid line represents the transmission function of the intracavity high-finesse Fabry–Perot filter. ω 0 and ω 1 are the carrier reference frequency, ω c is the cavity fundamental frequency, ω m = ( 2 N + 1 ) ω c is the active loss modulation frequency, and Γ denotes the extra loss experienced by the misaligned weak oscillating supermode mode sets.

Fig. 2
Fig. 2

Calculated supermode spur suppression [Eq. (14)] as a function of filter finesse. The knee frequency used in the calculation are f knee = 20 kHz and f knee = 200 kHz , respectively. Other parameters used are f m = 10 GHz and f c 10 MHz .

Fig. 3
Fig. 3

Calculated total timing jitter suppression [Eq. (15)] as a function of filter finesse. The knee frequency used in the calculation are f knee = 20 kHz and f knee = 200 kHz , respectively. Other parameters used are f m = 10 GHz and f c 10 MHz .

Fig. 4
Fig. 4

Calculated total timing jitter suppression [Eq. (15)] as a function of the order of harmonic mode locking. The pulse width used in the calculation are τ = 3 ps and τ = 10 ps , respectively. Other parameters used are m = 0.5 , f m = 10 GHz , and F = 500 .

Equations (26)

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

A s , n t = { ( g r s ) g n 2 ω m 2 Ω f 2 } A s , n + 1 2 M { A s , n + 1 + A s , n 1 2 A s , n } + S s , n ,
A s ( ω , t ) t = { ( g r s ) g ω 2 Ω f 2 } A s ( ω , t ) + 1 2 M ω m 2 2 A s ( ω , t ) ω 2 + S s ( ω , t ) .
{ 1 2 M ω m 2 2 ω 2 g ω 2 Ω f 2 } φ l ( ω τ ) = ( l + 1 2 ) M ω m 2 τ 2 φ l ( ω τ ) ,
A s ( ω , t ) = l = 0 c s , l ( t ) φ l ( ω τ ) .
c s , l ( t ) t = Δ s , l c s , l ( t ) + s s , l ( t ) ,
Δ s , l = r s g + ( l + 1 2 ) M ω m 2 τ 2 ,
s s , l ( t ) = φ l ( ω τ ) S s ( ω , t ) d ω τ .
E ( t ) = exp ( i ω 0 t ) ω m τ s = N N l = 0 p s , l ( t ) ,
Δ J = s = N N 2 c s , 1 c 0 , 0 τ exp ( i s ω c t ) .
Δ J 2 ( ω ) = 2 τ 2 s = N N 2 Δ 0 , 0 [ ( ω s ω c ) 2 + Δ s , 1 2 ] .
Δ J 2 = + Δ J 2 ( ω ) d f = 2 τ 2 s = N N Δ 0 , 0 Δ s , 1 .
T FP ( ω ) = 1 1 + ( ω ω m 2 F ) 2 ,
r s = r 0 + f c 2 ln [ 1 + 4 F 2 s 2 ( 2 N + 1 ) 2 ] ,
Δ 0 , 0 = r 0 g + 1 2 M ω m 2 τ 2 ,
Δ s , l M ω m 2 τ 2 + f c 2 ln [ 1 + 4 F 2 s 2 ( 2 N + 1 ) 2 ] .
Δ J 2 ( ω ) = 2 ( G 1 ) n sp ω 0 f c 2 τ 2 P avg s = N N 1 ( ω s ω c ) 2 + ( m f c ω m 2 τ 2 + f c 2 ln [ 1 + 4 F 2 s 2 ( 2 N + 1 ) 2 ] ) 2 ,
Δ J 2 = ( G 1 ) n sp N ph 1 m ω m 2 1 2 N + 1 s = N N 1 1 + 1 2 m ω m 2 τ 2 ln [ 1 + 4 F 2 s 2 ( 2 N + 1 ) 2 ] .
r spur = Δ J 2 ( ω c ) Δ J 2 ( 0 ) { 1 + f c 4 π f knee ln [ 1 + 4 F 2 ( 2 N + 1 ) 2 ] } 2 ,
r jitter = Δ J 2 Δ J 2 F = 0 = 1 2 N + 1 s = N N { 1 + f c 4 π f knee ln [ 1 + 4 F 2 s 2 ( 2 N + 1 ) 2 ] } 1 ,
Δ J 2 2 N + 1 = ( G 1 ) n sp τ 2 N ph F F F d x 2 m ω m 2 τ 2 + ln ( 1 + x 2 ) .
Δ J 2 2 N + 1 = ( G 1 ) n sp τ 2 N ph 1 ln ( F ) .
S s ( ω , t ) S s 1 * ( ω 1 , t 1 ) = ( G 1 ) n sp ω 0 f c 2 ω m δ ( t t 1 ) δ ( ω ω 1 ) δ s , s 1 .
s s , l ( t ) s s 1 , l 1 * ( t 1 ) = ( G 1 ) n sp ω 0 f c 2 ω m τ δ ( t t 1 ) δ s , s 1 δ l , l 1 ,
c s , l ( t ) c s 1 , l 1 * ( t ) = ( G 1 ) n sp ω 0 f c 2 ω m τ 1 2 Δ s , l δ s , s 1 δ l , l 1 .
P avg E 0 ( t ) 2 c 0 , 0 2 φ 0 ( ω τ ) 2 d ω τ ω m τ .
P avg = c 1 , 0 2 1 ω m τ = ( G 1 ) n sp ω 0 f c 2 1 2 Δ 0 , 0 .

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