Abstract

We analyze the entire power spectrum of pulse trains generated by a continuously operating actively mode-locked laser in the presence of noise. We consider the effect of amplitude, pulse-shape, and timing-jitter fluctuations that are characterized by stationary processes. Effects of correlations between different parameters of these fluctuations are studied also. The nonstationary timing-jitter fluctuations of passively mode-locked lasers and their influence on the power spectrum is discussed as well.

© 1996 Optical Society of America

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References

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  1. D. von der Linde, “Characterization of the noise in continuously operating mode-locked lasers,” Appl. Phys. B 39, 201–217 (1986).
    [CrossRef]
  2. M. L. Lambsdorff and J. Kuhl, “Low noise hybrid mode-locking of a femtosecond continuous wave dye laser with Gires–Tournois interferometers,” J. Opt. Soc. Am. B 5, 2311–2314 (1988).
    [CrossRef]
  3. M. J. W. Rodwell, D. M. Bloom, and K. J. Weingarten, “Subpicosecond laser timing stabilization,” IEEE J. Quantum Electron. 25, 817–827 (1989).
    [CrossRef]
  4. A. Finch, X. Zhu, P. N. Kean, and W. Sibbett, “Noise characterization of mode-locked color-center laser sources,” IEEE J. Quantum Electron. 26, 1115–1123 (1990).
    [CrossRef]
  5. P. A. Harten, S. G. Lee, J. P. Sokoloff, J. R. Salcedo, and N. Peyghambarian, “Noise in a dual dye jet hybridly mode-locked near infrared femtosecond laser,” Opt. Commun. 91, 465–473 (1992).
    [CrossRef]
  6. J. Son, J. V. Rudd, and J. F. Whitaker, “Noise characterization of a self-mode-locked Ti:sapphire laser,” Opt. Lett. 17, 733–735 (1992).
    [CrossRef] [PubMed]
  7. D. Henderson and A. G. Roddie, “A comparison of spectral and temporal techniques for the measurement of timing jitter and their application in a modelocked argon ion and dye laser system,” Opt. Commun. 100, 456–460 (1993).
    [CrossRef]
  8. A. Yariv, Optical Electronics, 4th ed. (Saunders, Philadelphia, Pa., 1991), p. 360.
  9. H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. 29, 983–996 (1993).
    [CrossRef]
  10. D. Eliyahu, R. A. Salvatore, and A. Yariv, “The effect of noise on the power spectrum of passively mode-locked lasers,” submitted to J. Opt. Soc. Am. B.
  11. I. G. Fuss, “An interpretation of the spectral measurement of optical pulse train noise,” IEEE J. Quantum Electron. 30, 2707–2710 (1994).
    [CrossRef]

1994 (1)

I. G. Fuss, “An interpretation of the spectral measurement of optical pulse train noise,” IEEE J. Quantum Electron. 30, 2707–2710 (1994).
[CrossRef]

1993 (2)

D. Henderson and A. G. Roddie, “A comparison of spectral and temporal techniques for the measurement of timing jitter and their application in a modelocked argon ion and dye laser system,” Opt. Commun. 100, 456–460 (1993).
[CrossRef]

H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. 29, 983–996 (1993).
[CrossRef]

1992 (2)

P. A. Harten, S. G. Lee, J. P. Sokoloff, J. R. Salcedo, and N. Peyghambarian, “Noise in a dual dye jet hybridly mode-locked near infrared femtosecond laser,” Opt. Commun. 91, 465–473 (1992).
[CrossRef]

J. Son, J. V. Rudd, and J. F. Whitaker, “Noise characterization of a self-mode-locked Ti:sapphire laser,” Opt. Lett. 17, 733–735 (1992).
[CrossRef] [PubMed]

1990 (1)

A. Finch, X. Zhu, P. N. Kean, and W. Sibbett, “Noise characterization of mode-locked color-center laser sources,” IEEE J. Quantum Electron. 26, 1115–1123 (1990).
[CrossRef]

1989 (1)

M. J. W. Rodwell, D. M. Bloom, and K. J. Weingarten, “Subpicosecond laser timing stabilization,” IEEE J. Quantum Electron. 25, 817–827 (1989).
[CrossRef]

1988 (1)

1986 (1)

D. von der Linde, “Characterization of the noise in continuously operating mode-locked lasers,” Appl. Phys. B 39, 201–217 (1986).
[CrossRef]

Bloom, D. M.

M. J. W. Rodwell, D. M. Bloom, and K. J. Weingarten, “Subpicosecond laser timing stabilization,” IEEE J. Quantum Electron. 25, 817–827 (1989).
[CrossRef]

Eliyahu, D.

D. Eliyahu, R. A. Salvatore, and A. Yariv, “The effect of noise on the power spectrum of passively mode-locked lasers,” submitted to J. Opt. Soc. Am. B.

Finch, A.

A. Finch, X. Zhu, P. N. Kean, and W. Sibbett, “Noise characterization of mode-locked color-center laser sources,” IEEE J. Quantum Electron. 26, 1115–1123 (1990).
[CrossRef]

Fuss, I. G.

I. G. Fuss, “An interpretation of the spectral measurement of optical pulse train noise,” IEEE J. Quantum Electron. 30, 2707–2710 (1994).
[CrossRef]

Harten, P. A.

P. A. Harten, S. G. Lee, J. P. Sokoloff, J. R. Salcedo, and N. Peyghambarian, “Noise in a dual dye jet hybridly mode-locked near infrared femtosecond laser,” Opt. Commun. 91, 465–473 (1992).
[CrossRef]

Haus, H. A.

H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. 29, 983–996 (1993).
[CrossRef]

Henderson, D.

D. Henderson and A. G. Roddie, “A comparison of spectral and temporal techniques for the measurement of timing jitter and their application in a modelocked argon ion and dye laser system,” Opt. Commun. 100, 456–460 (1993).
[CrossRef]

Kean, P. N.

A. Finch, X. Zhu, P. N. Kean, and W. Sibbett, “Noise characterization of mode-locked color-center laser sources,” IEEE J. Quantum Electron. 26, 1115–1123 (1990).
[CrossRef]

Kuhl, J.

Lambsdorff, M. L.

Lee, S. G.

P. A. Harten, S. G. Lee, J. P. Sokoloff, J. R. Salcedo, and N. Peyghambarian, “Noise in a dual dye jet hybridly mode-locked near infrared femtosecond laser,” Opt. Commun. 91, 465–473 (1992).
[CrossRef]

Mecozzi, A.

H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. 29, 983–996 (1993).
[CrossRef]

Peyghambarian, N.

P. A. Harten, S. G. Lee, J. P. Sokoloff, J. R. Salcedo, and N. Peyghambarian, “Noise in a dual dye jet hybridly mode-locked near infrared femtosecond laser,” Opt. Commun. 91, 465–473 (1992).
[CrossRef]

Roddie, A. G.

D. Henderson and A. G. Roddie, “A comparison of spectral and temporal techniques for the measurement of timing jitter and their application in a modelocked argon ion and dye laser system,” Opt. Commun. 100, 456–460 (1993).
[CrossRef]

Rodwell, M. J. W.

M. J. W. Rodwell, D. M. Bloom, and K. J. Weingarten, “Subpicosecond laser timing stabilization,” IEEE J. Quantum Electron. 25, 817–827 (1989).
[CrossRef]

Rudd, J. V.

Salcedo, J. R.

P. A. Harten, S. G. Lee, J. P. Sokoloff, J. R. Salcedo, and N. Peyghambarian, “Noise in a dual dye jet hybridly mode-locked near infrared femtosecond laser,” Opt. Commun. 91, 465–473 (1992).
[CrossRef]

Salvatore, R. A.

D. Eliyahu, R. A. Salvatore, and A. Yariv, “The effect of noise on the power spectrum of passively mode-locked lasers,” submitted to J. Opt. Soc. Am. B.

Sibbett, W.

A. Finch, X. Zhu, P. N. Kean, and W. Sibbett, “Noise characterization of mode-locked color-center laser sources,” IEEE J. Quantum Electron. 26, 1115–1123 (1990).
[CrossRef]

Sokoloff, J. P.

P. A. Harten, S. G. Lee, J. P. Sokoloff, J. R. Salcedo, and N. Peyghambarian, “Noise in a dual dye jet hybridly mode-locked near infrared femtosecond laser,” Opt. Commun. 91, 465–473 (1992).
[CrossRef]

Son, J.

von der Linde, D.

D. von der Linde, “Characterization of the noise in continuously operating mode-locked lasers,” Appl. Phys. B 39, 201–217 (1986).
[CrossRef]

Weingarten, K. J.

M. J. W. Rodwell, D. M. Bloom, and K. J. Weingarten, “Subpicosecond laser timing stabilization,” IEEE J. Quantum Electron. 25, 817–827 (1989).
[CrossRef]

Whitaker, J. F.

Yariv, A.

D. Eliyahu, R. A. Salvatore, and A. Yariv, “The effect of noise on the power spectrum of passively mode-locked lasers,” submitted to J. Opt. Soc. Am. B.

A. Yariv, Optical Electronics, 4th ed. (Saunders, Philadelphia, Pa., 1991), p. 360.

Zhu, X.

A. Finch, X. Zhu, P. N. Kean, and W. Sibbett, “Noise characterization of mode-locked color-center laser sources,” IEEE J. Quantum Electron. 26, 1115–1123 (1990).
[CrossRef]

Appl. Phys. B (1)

D. von der Linde, “Characterization of the noise in continuously operating mode-locked lasers,” Appl. Phys. B 39, 201–217 (1986).
[CrossRef]

IEEE J. Quantum Electron. (4)

M. J. W. Rodwell, D. M. Bloom, and K. J. Weingarten, “Subpicosecond laser timing stabilization,” IEEE J. Quantum Electron. 25, 817–827 (1989).
[CrossRef]

A. Finch, X. Zhu, P. N. Kean, and W. Sibbett, “Noise characterization of mode-locked color-center laser sources,” IEEE J. Quantum Electron. 26, 1115–1123 (1990).
[CrossRef]

H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. 29, 983–996 (1993).
[CrossRef]

I. G. Fuss, “An interpretation of the spectral measurement of optical pulse train noise,” IEEE J. Quantum Electron. 30, 2707–2710 (1994).
[CrossRef]

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

Opt. Commun. (2)

D. Henderson and A. G. Roddie, “A comparison of spectral and temporal techniques for the measurement of timing jitter and their application in a modelocked argon ion and dye laser system,” Opt. Commun. 100, 456–460 (1993).
[CrossRef]

P. A. Harten, S. G. Lee, J. P. Sokoloff, J. R. Salcedo, and N. Peyghambarian, “Noise in a dual dye jet hybridly mode-locked near infrared femtosecond laser,” Opt. Commun. 91, 465–473 (1992).
[CrossRef]

Opt. Lett. (1)

Other (2)

A. Yariv, Optical Electronics, 4th ed. (Saunders, Philadelphia, Pa., 1991), p. 360.

D. Eliyahu, R. A. Salvatore, and A. Yariv, “The effect of noise on the power spectrum of passively mode-locked lasers,” submitted to J. Opt. Soc. Am. B.

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Figures (5)

Fig. 1
Fig. 1

PI(ω)/|F(ω)|2 as a function of ωT/2π for the stationary white-noise timing-jitter case. Here GT(k) = GT(0)δ0,k and GT(0)/T2 = 10−4.

Fig. 2
Fig. 2

PI(ω)/|F(ω)|2 − exp[−ω2GT(0)]H(ωT) versus ωT/2π, where GT(k) = GT(0)exp(−αk) and α = 1.5. The dashed curve represents the peak values of the power spectrum for low-frequency approximation [ω2PT(2π/T)].

Fig. 3
Fig. 3

(a) Same as in Fig. 2, but α = 0.5; (b) the power-spectrum sidebands for the first 20 harmonics as a function of the frequency deviation Δω.

Fig. 4
Fig. 4

Power spectrum for the nonstationary timing-jitter fluctuations [Eq. (23)]. The first ten sidebands are plotted as a function of the frequency deviation Δω in a log-log scale. The value of D/2T = 10−4 was assumed.

Fig. 5
Fig. 5

Normalized timing-jitter autocorrelation GT(k)/GT(0) (solid curves) and the normalized amplitude autocorrelation GA(k)/GA(0) (dotted curves) as a function of k = t/T, the ratio between the measured time to the repetition period. The calculations are based on Eqs. (35)(37) and on the first and the tenth sidebands of the restored power spectrum in Figs. 7(a) and 7(b) of Ref. 4. In (a) the integration boundaries are 25 Hz to 500 Hz, and in (b) and (c) these boundaries are 50 Hz to 500 Hz and 75 Hz to 50 Hz, respectively.

Equations (49)

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I Δ T ( t ) = n = N N f n ( t T n ) ,
P I ( ω ) = + G I ( τ ) exp ( i ω τ ) d τ ,
G I ( τ ) = lim Δ T 1 Δ T Δ T / 2 Δ T / 2 I Δ T ( t ) I Δ T ( t + τ ) d t .
P I ( ω ) = lim Δ T 1 Δ T I Δ T ( ω ) I Δ T * ( ω ) ,
I Δ T ( ω ) = Δ T / 2 Δ T / 2 I Δ T ( t ) exp ( i ω t ) d t ,
I Δ T ( ω ) = n = N N F n ( ω ) exp ( i ω T n ) ,
F n ( ω ) = f n ( t ) exp ( i ω t ) d t .
P I ( ω ) = lim N 1 2 N + 1 n , m = N N F n ( ω ) F m * ( ω ) × exp [ i ω ( T n T m ) ] .
P I ( ω ) = | F ( ω ) | 2 lim N H N ( ω T ) ,
H N ( ω T ) = 1 2 N + 1 | n = N N exp ( i ω n T ) | 2 = 1 2 N + 1 { sin [ ( 2 N + 1 ) ω T / 2 ] sin ( ω T / 2 ) } 2 .
H ( ω T ) = lim N H N ( ω T ) = 2 π T n = δ ( ω 2 π n / T ) .
δ A n δ A m = δ A 0 δ A | n m |
P I ( ω ) = | S ( ω ) | 2 [ H ( ω T ) + 2 P A ( ω T ) ] ,
P A ( ω ) = G A ( 0 ) 2 + k = 1 G A ( k ) cos ( k ω T ) ,
G A ( k ) = δ A 0 δ A k .
P I ( ω ) = | F ( ω ) | 2 lim N 1 2 N + 1 n , m = N N exp [ i ω T ( n m ) ] × exp [ i ω ( δ T n δ T m ) ] ,
exp [ i ω ( δ T n δ T m ) ] = exp { ω 2 [ G T ( 0 ) G T ( | n m | ) ] } ,
G T ( k ) = δ T 0 δ T k .
P I ( ω ) = | F ( ω ) | 2 [ 1 + exp [ ω 2 G T ( 0 ) ] ( 2 k = 1 { exp [ ω 2 G T ( k ) ] 1 } cos ( k ω T ) + H ( ω T ) 1 ) ] .
P I ( ω ) = | F ( ω ) | 2 { [ 1 ω 2 G T ( 0 ) ] H ( ω T ) + 2 ω 2 P T ( ω ) } ,
P T ( ω ) = G T ( 0 ) 2 + k = 1 G T ( k ) cos ( k ω T ) .
exp [ i ω ( δ T n δ T m ) ] = exp ( ω 2 2 D T | n m | ) ,
P I ( ω ) = | F ( ω ) | 2 sinh ( ω 2 D T / 2 ) cosh ( ω 2 D T / 2 ) cos ( ω T ) .
F n ( ω ) F m * ( ω ) exp [ i ω ( T n T m ) ] = | S ( ω ) | 2 exp [ i ω T ( n m ) ] × ( A + δ A n ) ( A + δ A m ) × exp [ i ω ( δ T n δ T m ) ] .
P I ( ω ) / | S ( ω ) | 2 = A 2 + A 2 exp [ ω 2 G T ( 0 ) ] × ( 2 k = 1 { exp [ ω 2 G T ( k ) ] 1 } × cos ( k ω T ) + H ( ω T ) 1 ) + G A ( 0 ) + exp [ ω 2 G T ( 0 ) ] × { 2 k = 1 G A ( k ) exp [ ω 2 G T ( k ) ] cos ( k ω T ) } .
P I ( ω ) = 2 | S ( ω ) | 2 { [ 1 ω 2 G T ( 0 ) ] [ A 2 H ( ω T ) / 2 + P A ( ω ) ] + A 2 ω 2 P T ( ω ) + ω 2 P A ( ω ) P T ( ω ) } ,
P A ( ω ) P T ( ω ) = T π π / T π / T P A ( ω ) P T ( ω ω ) d ω = G A ( 0 ) G T ( 0 ) 2 + k = 1 G A ( k ) G T ( k ) cos ( k ω T ) .
( A + δ A n ) ( A + δ A m ) exp [ i ω ( δ T n δ T m ) ] .
i ω A ( δ A n + δ A m ) ( δ T n δ T m ) = 0 ,
( A + δ A n ) ( A + δ A m ) exp [ i ω ( δ T n δ T m ) ] = { A 2 + G A ( | n m | ) + ω 2 [ G A T ( 0 ) G A T ( | n m | ) ] 2 } × exp { ω 2 [ G T ( 0 ) G T ( | n m | ) ] } ,
P I ( ω ) / | S ( ω ) | 2 = A 2 + A 2 exp [ ω 2 G T ( 0 ) ] × ( 2 k = 1 { exp [ ω 2 G T ( k ) ] 1 } × cos ( k ω T ) + H ( ω T ) 1 ) + G A ( 0 ) + exp [ ω 2 G T ( 0 ) ] × ( 2 k = 1 { G A ( k ) + ω 2 [ G A T ( 0 ) G A T ( k ) ] 2 } × exp [ ω 2 G T ( k ) ] cos ( k ω T ) ) .
ω 2 exp [ ω 2 G T ( 0 ) ] [ G A T ( 0 ) ] 2 ( 2 k = 1 { exp [ ω 2 G T ( k ) ] 1 } × cos ( k ω T ) + H ( ω T ) 1 ) .
1 | S ( 2 π n / T ) | 2 T 2 π 2 π ( n 1 / 2 ) T 2 π ( n + 1 / 2 ) T P I ( ω ) d ω = A 2 + G A ( 0 ) ,
T π ω L ω H P I ( ω ) d ω G A ( 0 ) + ( 2 π n T ) 2 [ A 2 G T ( 0 ) G A T 2 ( 0 ) ] ,
b n ( k ) = T π ω L ω H P I ( ω ) cos ( k ω T ) d ω G A ( k ) + A 2 ( 2 π n T ) 2 G T ( k ) .
G T ( k ) / G T ( 0 ) = b n ( k ) b m ( k ) b n ( 0 ) b m ( 0 ) ,
G A ( k ) / G A ( 0 ) = m 2 b n ( k ) n 2 b m ( k ) m 2 b n ( 0 ) n 2 b m ( 0 ) ,
P I C ( ω ) = lim Δ T 1 Δ T I Δ T ( ω ) I Δ T * ( ω ) C ,
lim Δ T 1 Δ T I Δ T ( ω ) I Δ T * ( ω ) = | S ( ω ) | 2 { A 2 + [ ω G A T ( 0 ) ] 2 } × H ( ω T ) exp [ ω 2 G T ( 0 ) ] .
F n ( ω ) F m * ( ω ) = { | F n ( ω ) | 2 n = m | F n ( ω ) | 2 n m ,
P I ( ω ) = | F n ( ω ) | 2 + | F n ( ω ) | 2 [ H ( ω T ) 1 ] .
f n ( t ) = exp [ t 2 2 ( τ + δ τ n ) 2 ] ,
F n ( ω ) = 2 π ( τ + δ τ n ) exp [ ω 2 ( τ + δ τ n ) 2 2 ] .
P I ( ω ) τ 2 { 1 ω 2 [ τ 2 + 3 G τ ( 0 ) ] } × H ( ω T ) + 2 P τ ( ω ) ( 1 3 τ 2 ω 2 ) ,
f n ( t ) = 1 τ + δ τ n exp [ t 2 2 ( τ + δ τ n ) 2 ] ,
F n ( ω ) = 2 π exp [ ω 2 ( τ + δ τ n ) 2 2 ] .
| F n ( ω ) | 2 = 2 π 1 + ω 2 G τ ( 0 ) exp [ τ 2 ω 2 1 + ω 2 G τ ( 0 ) ] ,
| F n ( ω ) | 2 = 2 π 1 + 2 ω 2 G τ ( 0 ) exp [ τ 2 ω 2 1 + 2 ω 2 G τ ( 0 ) ] ,
P I ( ω ) 2 π ω 4 τ 2 G τ 2 ( 0 ) + 2 π { 1 ω 2 [ τ 2 + G τ ( 0 ) ] } H ( ω T ) .

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