Abstract

Frequency combs generated by trains of pulses emitted from mode-locked lasers are analyzed when the center time and phase of the pulses undergo noise-induced random walk, which broadens the comb lines. Asymptotic analysis and computation reveal that, when the standard deviation of the center-time jitter of the nth pulse scales as np2, where p is a jitter exponent, the linewidth of the kth comb line scales as k2p. The linear-dispersionless (p=1) and pure-soliton (p=3) dynamics in lasers are derived as special cases of this time-frequency duality relation. In addition, the linewidth induced by phase jitter decreases with power Pout, as (Pout)1p.

© 2006 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
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2005 (3)

S.T.Cundiff and J.Ye, eds. Femtosecond Optical Frequency Comb Technology (Springer, 2005).

A. Marian, M. C. Stowe, D. Felinto, and J. Ye, Phys. Rev. Lett. 95, 023001 (2005).
[CrossRef] [PubMed]

Q. Quraishi, S. T. Cundiff, B. Ilan, and M. J. Ablowitz, Phys. Rev. Lett. 94, 243904 (2005).
[CrossRef]

2004 (2)

Y. Takushima, H. Sotobayashi, M. E. Grein, E. P. Ippen, and H. A. Haus, in Proc. SPIE 5595, 213 (2004).
[CrossRef]

F. X. Kärtner, U. Morgner, T. Schibli, R. Ell, H. A. Haus, J. G. Fujimoto, and E. P. Ippen, Vol. 95 of Topics in Applied Physics XIV (Springer-Verlag, 2004), p. 73.

2002 (1)

Th. Udem, R. Holzwarth, and T. W. Häncsh, Nature 416, 233 (2002).
[CrossRef] [PubMed]

2001 (1)

A. Papoulis and S. Unnikrishna Pillai, Probability, Random Variables and Stochastic Processes (McGraw-Hill, 2001).

1999 (1)

1993 (1)

H. A. Haus and A. Mecozzi, IEEE J. Quantum Electron. 29, 983 (1993).
[CrossRef]

1990 (1)

1986 (1)

1958 (1)

A. L. Schawlow and C. H. Townes, Phys. Rev. 112, 1940 (1958).
[CrossRef]

Ablowitz, M. J.

Q. Quraishi, S. T. Cundiff, B. Ilan, and M. J. Ablowitz, Phys. Rev. Lett. 94, 243904 (2005).
[CrossRef]

Chen, Y.

Cho, S. H.

Cundiff, S. T.

Q. Quraishi, S. T. Cundiff, B. Ilan, and M. J. Ablowitz, Phys. Rev. Lett. 94, 243904 (2005).
[CrossRef]

Ell, R.

F. X. Kärtner, U. Morgner, T. Schibli, R. Ell, H. A. Haus, J. G. Fujimoto, and E. P. Ippen, Vol. 95 of Topics in Applied Physics XIV (Springer-Verlag, 2004), p. 73.

Felinto, D.

A. Marian, M. C. Stowe, D. Felinto, and J. Ye, Phys. Rev. Lett. 95, 023001 (2005).
[CrossRef] [PubMed]

Fujimoto, J. G.

F. X. Kärtner, U. Morgner, T. Schibli, R. Ell, H. A. Haus, J. G. Fujimoto, and E. P. Ippen, Vol. 95 of Topics in Applied Physics XIV (Springer-Verlag, 2004), p. 73.

Y. Chen, F. X. Kärtner, U. Morgner, S. H. Cho, H. A. Haus, E. P. Ippen, and J. G. Fujimoto, J. Opt. Soc. Am. B 16, 1999 (1999).
[CrossRef]

Gordon, J. P.

Grein, M. E.

Y. Takushima, H. Sotobayashi, M. E. Grein, E. P. Ippen, and H. A. Haus, in Proc. SPIE 5595, 213 (2004).
[CrossRef]

Häncsh, T. W.

Th. Udem, R. Holzwarth, and T. W. Häncsh, Nature 416, 233 (2002).
[CrossRef] [PubMed]

Haus, H. A.

F. X. Kärtner, U. Morgner, T. Schibli, R. Ell, H. A. Haus, J. G. Fujimoto, and E. P. Ippen, Vol. 95 of Topics in Applied Physics XIV (Springer-Verlag, 2004), p. 73.

Y. Takushima, H. Sotobayashi, M. E. Grein, E. P. Ippen, and H. A. Haus, in Proc. SPIE 5595, 213 (2004).
[CrossRef]

Y. Chen, F. X. Kärtner, U. Morgner, S. H. Cho, H. A. Haus, E. P. Ippen, and J. G. Fujimoto, J. Opt. Soc. Am. B 16, 1999 (1999).
[CrossRef]

H. A. Haus and A. Mecozzi, IEEE J. Quantum Electron. 29, 983 (1993).
[CrossRef]

J. P. Gordon and H. A. Haus, Opt. Lett. 11, 665 (1986).
[CrossRef] [PubMed]

Holzwarth, R.

Th. Udem, R. Holzwarth, and T. W. Häncsh, Nature 416, 233 (2002).
[CrossRef] [PubMed]

Ilan, B.

Q. Quraishi, S. T. Cundiff, B. Ilan, and M. J. Ablowitz, Phys. Rev. Lett. 94, 243904 (2005).
[CrossRef]

Ippen, E. P.

F. X. Kärtner, U. Morgner, T. Schibli, R. Ell, H. A. Haus, J. G. Fujimoto, and E. P. Ippen, Vol. 95 of Topics in Applied Physics XIV (Springer-Verlag, 2004), p. 73.

Y. Takushima, H. Sotobayashi, M. E. Grein, E. P. Ippen, and H. A. Haus, in Proc. SPIE 5595, 213 (2004).
[CrossRef]

Y. Chen, F. X. Kärtner, U. Morgner, S. H. Cho, H. A. Haus, E. P. Ippen, and J. G. Fujimoto, J. Opt. Soc. Am. B 16, 1999 (1999).
[CrossRef]

Kärtner, F. X.

F. X. Kärtner, U. Morgner, T. Schibli, R. Ell, H. A. Haus, J. G. Fujimoto, and E. P. Ippen, Vol. 95 of Topics in Applied Physics XIV (Springer-Verlag, 2004), p. 73.

Y. Chen, F. X. Kärtner, U. Morgner, S. H. Cho, H. A. Haus, E. P. Ippen, and J. G. Fujimoto, J. Opt. Soc. Am. B 16, 1999 (1999).
[CrossRef]

Marian, A.

A. Marian, M. C. Stowe, D. Felinto, and J. Ye, Phys. Rev. Lett. 95, 023001 (2005).
[CrossRef] [PubMed]

Mecozzi, A.

H. A. Haus and A. Mecozzi, IEEE J. Quantum Electron. 29, 983 (1993).
[CrossRef]

Mollenauer, L. F.

Morgner, U.

F. X. Kärtner, U. Morgner, T. Schibli, R. Ell, H. A. Haus, J. G. Fujimoto, and E. P. Ippen, Vol. 95 of Topics in Applied Physics XIV (Springer-Verlag, 2004), p. 73.

Y. Chen, F. X. Kärtner, U. Morgner, S. H. Cho, H. A. Haus, E. P. Ippen, and J. G. Fujimoto, J. Opt. Soc. Am. B 16, 1999 (1999).
[CrossRef]

Papoulis, A.

A. Papoulis and S. Unnikrishna Pillai, Probability, Random Variables and Stochastic Processes (McGraw-Hill, 2001).

Quraishi, Q.

Q. Quraishi, S. T. Cundiff, B. Ilan, and M. J. Ablowitz, Phys. Rev. Lett. 94, 243904 (2005).
[CrossRef]

Schawlow, A. L.

A. L. Schawlow and C. H. Townes, Phys. Rev. 112, 1940 (1958).
[CrossRef]

Schibli, T.

F. X. Kärtner, U. Morgner, T. Schibli, R. Ell, H. A. Haus, J. G. Fujimoto, and E. P. Ippen, Vol. 95 of Topics in Applied Physics XIV (Springer-Verlag, 2004), p. 73.

Sotobayashi, H.

Y. Takushima, H. Sotobayashi, M. E. Grein, E. P. Ippen, and H. A. Haus, in Proc. SPIE 5595, 213 (2004).
[CrossRef]

Stowe, M. C.

A. Marian, M. C. Stowe, D. Felinto, and J. Ye, Phys. Rev. Lett. 95, 023001 (2005).
[CrossRef] [PubMed]

Takushima, Y.

Y. Takushima, H. Sotobayashi, M. E. Grein, E. P. Ippen, and H. A. Haus, in Proc. SPIE 5595, 213 (2004).
[CrossRef]

Townes, C. H.

A. L. Schawlow and C. H. Townes, Phys. Rev. 112, 1940 (1958).
[CrossRef]

Udem, Th.

Th. Udem, R. Holzwarth, and T. W. Häncsh, Nature 416, 233 (2002).
[CrossRef] [PubMed]

Unnikrishna Pillai, S.

A. Papoulis and S. Unnikrishna Pillai, Probability, Random Variables and Stochastic Processes (McGraw-Hill, 2001).

Ye, J.

A. Marian, M. C. Stowe, D. Felinto, and J. Ye, Phys. Rev. Lett. 95, 023001 (2005).
[CrossRef] [PubMed]

IEEE J. Quantum Electron. (1)

H. A. Haus and A. Mecozzi, IEEE J. Quantum Electron. 29, 983 (1993).
[CrossRef]

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

Nature (1)

Th. Udem, R. Holzwarth, and T. W. Häncsh, Nature 416, 233 (2002).
[CrossRef] [PubMed]

Opt. Lett. (2)

Phys. Rev. (1)

A. L. Schawlow and C. H. Townes, Phys. Rev. 112, 1940 (1958).
[CrossRef]

Phys. Rev. Lett. (2)

Q. Quraishi, S. T. Cundiff, B. Ilan, and M. J. Ablowitz, Phys. Rev. Lett. 94, 243904 (2005).
[CrossRef]

A. Marian, M. C. Stowe, D. Felinto, and J. Ye, Phys. Rev. Lett. 95, 023001 (2005).
[CrossRef] [PubMed]

Proc. SPIE (1)

Y. Takushima, H. Sotobayashi, M. E. Grein, E. P. Ippen, and H. A. Haus, in Proc. SPIE 5595, 213 (2004).
[CrossRef]

Other (3)

F. X. Kärtner, U. Morgner, T. Schibli, R. Ell, H. A. Haus, J. G. Fujimoto, and E. P. Ippen, Vol. 95 of Topics in Applied Physics XIV (Springer-Verlag, 2004), p. 73.

S.T.Cundiff and J.Ye, eds. Femtosecond Optical Frequency Comb Technology (Springer, 2005).

A. Papoulis and S. Unnikrishna Pillai, Probability, Random Variables and Stochastic Processes (McGraw-Hill, 2001).

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Figures (2)

Fig. 1
Fig. 1

Schematic of a pulse train (left) and its spectrum (right). In the absence of noise, the pulse’s spectrum determines the bandwidth, while the repetition time, T rep = T n + 1 T n , and overall phase change, Δ ϕ = ϕ n + 1 ϕ n , determine the comb function [Eqs. (1, 2, 3)]. The frequency of the kth comb line (enumerated around ω c ) is ω ̃ k = k ω rep + ω ̃ o , where ω rep and ω ̃ o are the repetition and offset frequencies. Noise induces a random jitter in the center time and phase, T n and ϕ n , which broadens the comb lines. The linewidth ( ω 1 2 in the inset) is the FWHM of the comb function [Eq. (2)] around a comb frequency [Eq. (7)].

Fig. 2
Fig. 2

Relative linewidth [i.e., Eq. (7)] induced solely by center-time jitter [i.e., Eq. (6) with c x y = c y = 0 and c x = 1 ] with ω rep = 2 π [ GHz ] , ω ̃ o = 0 , pulse width τ = 100 fs , N = 10 4 pulses, and noise strength σ = 1 . A, Log-log plot, where k = 0 corresponds to ω c . Within the asymptotic regime [Eq. (8)], the linewidths computed for the linear dispersionless ( p = 1 , dashes) and pure-soliton ( p = 3 , solid) dynamics obey the scaling law [Eq. (9)]; i.e., they fit the power laws δ ω 1 2 linear k 2.0 and δ ω 1 2 soliton k 0.67 , respectively. Near the center frequency for p = 1 , the TML linewidth scales as 1 N . B, Same as A using normalized absolute frequency. The linear-dispersionless and soliton linewidths increase as ( ω ω c ) 2 and ( ω ω c ) 2 3 , respectively. For reference, a single-pulse spectrum with a dimensional FWHM of 1 ( 10 τ ) is depicted.

Equations (9)

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F { n = 1 N E ( t T n ) e i ϕ n e i ω c t + c.c. } = E ̂ ( ω ̃ ) S ̂ ( ω ̃ ) + c.c. ,
S ̂ ( ω ̃ ) = n = 1 N e i ω ̃ T n + i ϕ n .
ω ̃ k = k ω rep + ω ̃ o , ω ̃ o = Δ ϕ 2 π ω rep ,
g n ( x , y ) = A e [ 1 2 ( 1 r 2 ) ] [ ( x σ x ) 2 ( 2 r 2 x y C x , y ) + ( y σ y ) 2 ] ,
σ x 2 = c x ( σ τ ) 2 n p , C x , y = c x y σ τ n p , σ y 2 = c y σ 2 n p ,
S ¯ ( ω ̃ k + 1 2 ω rep δ ω 1 2 ) 2 = 1 2 S ¯ ( ω ̃ k ) 2 .
1 σ τ N p 2 ω ω c 1 σ τ .
σ x σ τ n p 2 δ ω 1 2 ( σ τ ω rep ) 2 p k 2 p ,
σ y 2 P th P out n p δ ω 1 2 ( P th P out ) 1 p .

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