Abstract

A broad class of partially coherent non-stationary fields can be expressed in terms of the recently proposed independent-elementary-pulse model. In this work we first introduce a corresponding dual representation in the frequency domain and then extend this concept by considering shifted and weighted elementary spectral coherence functions. We prove that this method, which closely describes practical optical systems, leads to properly defined correlation functions. As an example, we demonstrate that our new model characterizes, in a natural way, trains of ultra-short pulses, affected by noise and timing jitter, emitted by usual modulators employed in telecom applications.

© 2007 Optical Society of America

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References

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  1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, UK, 1995).
  2. G. P. Agrawal, Fiber-Optic Communication Systems, 3rd ed. (Wiley, New York, NY, 2002).
    [CrossRef]
  3. B. E. A. Saleh and M. I. Irshid, "Collet-Wolf equivalence theorem and propagation of a pulse in a single-mode optical fiber," Opt. Lett. 7, 342-343 (1982).
    [CrossRef] [PubMed]
  4. J. Lancis, V. Torres-Company, E. Silvestre, and P. Andres, "Space-time analogy for partially coherent planewave- type pulses," Opt. Lett. 30, 2973-2975 (2005).
    [CrossRef] [PubMed]
  5. H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, "Spectral coherence properties of temporally modulated stationary light sources," Opt. Express 11, 1894-1899 (2003).
    [CrossRef] [PubMed]
  6. M. J. Ablowitz, B. Ilan, and S. T. Cundiff, "Noise-induced linewidth in frequency combs," Opt. Lett. 31, 1875- 1877 (2006).
    [CrossRef] [PubMed]
  7. V. Torres-Company, H. Lajunen, and A. T. Friberg, "Effects of partial coherence on frequency combs," J. Eur. Opt. Soc. Rapid Publ. 2, 07007:1-4 (2007).
    [CrossRef]
  8. V. Torres-Company, H. Lajunen, and A. T. Friberg, "Coherence theory of noise in ultrashort-pulse trains," J. Opt. Soc. Am. B (in press).
  9. S. T. Cundiff and J. Ye, "Colloquium: Femtosecond optical frequency combs," Rev. Mod. Phys. 75, 325-342 (2003).
    [CrossRef]
  10. P. P¨a¨akk¨onen, J. Turunen, P. Vahimaa, A. T. Friberg, and F.Wyrowski, "Partially coherent Gaussian pulses," Opt. Commun. 204, 53-58 (2002).
    [CrossRef]
  11. P. Vahimaa and J. Turunen, "Independent-elementary-pulse representation for non-stationary fields," Opt. Express 14, 5007-5012 (2006).
    [CrossRef] [PubMed]
  12. P. Vahimaa and J. Turunen, "Finite-elementary-source model for partially coherent radiation," Opt. Express 14, 1376-1381 (2006).
    [CrossRef] [PubMed]
  13. J. Capmany "A tutorial on microwave photonic filters," J. Lightwave Technol. 24, 201-229 (2006).
    [CrossRef]
  14. S. A. Ponomarenko, G. P. Agrawal, and E. Wolf, "Energy spectrum of a nonstationary ensemble of pulses," Opt. Lett. 29, 394-396 (2004).
    [CrossRef] [PubMed]
  15. M. Bertolotti, L. Sereda, and A. Ferrari, "Application of the spectral representation of stochastic processes to the study of nonstationary light radiation: a tutorial," JEOS A: Pure Appl. Opt. 6, 153 (1997).
    [CrossRef]

2007 (1)

V. Torres-Company, H. Lajunen, and A. T. Friberg, "Effects of partial coherence on frequency combs," J. Eur. Opt. Soc. Rapid Publ. 2, 07007:1-4 (2007).
[CrossRef]

2006 (4)

2005 (1)

2004 (1)

2003 (2)

2002 (1)

P. P¨a¨akk¨onen, J. Turunen, P. Vahimaa, A. T. Friberg, and F.Wyrowski, "Partially coherent Gaussian pulses," Opt. Commun. 204, 53-58 (2002).
[CrossRef]

1997 (1)

M. Bertolotti, L. Sereda, and A. Ferrari, "Application of the spectral representation of stochastic processes to the study of nonstationary light radiation: a tutorial," JEOS A: Pure Appl. Opt. 6, 153 (1997).
[CrossRef]

1982 (1)

Ablowitz, M. J.

Agrawal, G. P.

Andres, P.

Bertolotti, M.

M. Bertolotti, L. Sereda, and A. Ferrari, "Application of the spectral representation of stochastic processes to the study of nonstationary light radiation: a tutorial," JEOS A: Pure Appl. Opt. 6, 153 (1997).
[CrossRef]

Capmany, J.

Cundiff, S. T.

M. J. Ablowitz, B. Ilan, and S. T. Cundiff, "Noise-induced linewidth in frequency combs," Opt. Lett. 31, 1875- 1877 (2006).
[CrossRef] [PubMed]

S. T. Cundiff and J. Ye, "Colloquium: Femtosecond optical frequency combs," Rev. Mod. Phys. 75, 325-342 (2003).
[CrossRef]

Ferrari, A.

M. Bertolotti, L. Sereda, and A. Ferrari, "Application of the spectral representation of stochastic processes to the study of nonstationary light radiation: a tutorial," JEOS A: Pure Appl. Opt. 6, 153 (1997).
[CrossRef]

Friberg, A. T.

V. Torres-Company, H. Lajunen, and A. T. Friberg, "Effects of partial coherence on frequency combs," J. Eur. Opt. Soc. Rapid Publ. 2, 07007:1-4 (2007).
[CrossRef]

V. Torres-Company, H. Lajunen, and A. T. Friberg, "Coherence theory of noise in ultrashort-pulse trains," J. Opt. Soc. Am. B (in press).

Ilan, B.

Irshid, M. I.

Lajunen, H.

V. Torres-Company, H. Lajunen, and A. T. Friberg, "Effects of partial coherence on frequency combs," J. Eur. Opt. Soc. Rapid Publ. 2, 07007:1-4 (2007).
[CrossRef]

H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, "Spectral coherence properties of temporally modulated stationary light sources," Opt. Express 11, 1894-1899 (2003).
[CrossRef] [PubMed]

V. Torres-Company, H. Lajunen, and A. T. Friberg, "Coherence theory of noise in ultrashort-pulse trains," J. Opt. Soc. Am. B (in press).

Lancis, J.

Ponomarenko, S. A.

Saleh, B. E. A.

Sereda, L.

M. Bertolotti, L. Sereda, and A. Ferrari, "Application of the spectral representation of stochastic processes to the study of nonstationary light radiation: a tutorial," JEOS A: Pure Appl. Opt. 6, 153 (1997).
[CrossRef]

Silvestre, E.

Tervo, J.

Torres-Company, V.

V. Torres-Company, H. Lajunen, and A. T. Friberg, "Effects of partial coherence on frequency combs," J. Eur. Opt. Soc. Rapid Publ. 2, 07007:1-4 (2007).
[CrossRef]

J. Lancis, V. Torres-Company, E. Silvestre, and P. Andres, "Space-time analogy for partially coherent planewave- type pulses," Opt. Lett. 30, 2973-2975 (2005).
[CrossRef] [PubMed]

V. Torres-Company, H. Lajunen, and A. T. Friberg, "Coherence theory of noise in ultrashort-pulse trains," J. Opt. Soc. Am. B (in press).

Turunen, J.

Vahimaa, P.

Wolf, E.

Wyrowski, F.

Ye, J.

S. T. Cundiff and J. Ye, "Colloquium: Femtosecond optical frequency combs," Rev. Mod. Phys. 75, 325-342 (2003).
[CrossRef]

J. Eur. Opt. Soc. Rapid Publ. (1)

V. Torres-Company, H. Lajunen, and A. T. Friberg, "Effects of partial coherence on frequency combs," J. Eur. Opt. Soc. Rapid Publ. 2, 07007:1-4 (2007).
[CrossRef]

J. Lightwave Technol. (1)

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

V. Torres-Company, H. Lajunen, and A. T. Friberg, "Coherence theory of noise in ultrashort-pulse trains," J. Opt. Soc. Am. B (in press).

JEOS A: Pure Appl. Opt. (1)

M. Bertolotti, L. Sereda, and A. Ferrari, "Application of the spectral representation of stochastic processes to the study of nonstationary light radiation: a tutorial," JEOS A: Pure Appl. Opt. 6, 153 (1997).
[CrossRef]

Opt. Commun. (1)

P. P¨a¨akk¨onen, J. Turunen, P. Vahimaa, A. T. Friberg, and F.Wyrowski, "Partially coherent Gaussian pulses," Opt. Commun. 204, 53-58 (2002).
[CrossRef]

Opt. Express (3)

Opt. Lett. (4)

Rev. Mod. Phys. (1)

S. T. Cundiff and J. Ye, "Colloquium: Femtosecond optical frequency combs," Rev. Mod. Phys. 75, 325-342 (2003).
[CrossRef]

Other (2)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, UK, 1995).

G. P. Agrawal, Fiber-Optic Communication Systems, 3rd ed. (Wiley, New York, NY, 2002).
[CrossRef]

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Equations (21)

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Γ ( t 1 , t 2 ) = U * ( t 1 ) U ( t 2 ) = g ( t ´ ) Γ e ( t 1 t ´ , t 2 t ´ ) dt ´ ,
W ( ω 1 , ω 2 ) = A * ( ω 1 ω 0 ) A ( ω 2 ω 0 ) G ( ω 2 ω 1 ) ,
W ( ω 1 , ω 2 ) = W N ( ω ´ ) W e ( ω 1 ω ´ , ω 2 ω ´ ) ´ .
S ( ω ) = W N ( ω ) U ˜ e ( ω ) 2 ;
μ ( ω 1 , ω 2 ) = W ( ω 1 , ω 2 ) S ( ω 1 ) S ( ω 2 ) ,
Γ ( t 1 , t 2 ) = W ( ω 1 , ω 2 ) exp [ i ( ω 1 t 1 ω 2 t 2 ) ] 1 2 .
Γ ( t 1 , t 2 ) = a * ( t 1 ) a ( t 2 ) exp [ i ω 0 ( t 2 t 1 ) ] Γ N ( t 2 t 1 ) = Γ e ( t 1 , t 2 ) Γ N ( t 2 t 1 ) ,
Γ N ( t 2 t 1 ) = W N ( ω ) exp [ ( t 2 t 1 ) ] .
γ ( t 1 , t 2 ) = Γ ( t 1 , t 2 ) I ( t 1 ) I ( t 2 ) ,
W pc ( ω 1 , ω 2 ) = U ˜ pc * ( ω 1 ) U ˜ pc ( ω 2 ) ,
W ( ω 1 , ω 2 ) = W N ( ω ´ ) W pc ( ω 1 ω ´ , ω 2 ω ´ ) ´ ,
Γ ( t 1 , t 2 ) = Γ N ( t 2 t 1 ) Γ pc ( t 1 , t 2 ) ,
Γ pc ( t 1 , t 2 ) = a pc * ( t 1 ) a pc ( t 2 ) exp [ i ω 0 ( t 2 t 1 ) ] .
U ( t ) = exp ( i ω 0 t ) N ( t ) M [ t Tj ( t ) ] ,
Γ ( t 1 , t 2 ) = Γ N ( t 2 t 1 ) [ M * ( t 1 ) M ( t 2 ) + T 2 M ˙ * ( t 1 ) M ˙ ( t 2 ) Γ 2 ( t 1 , t 2 ) ] exp [ 0 ( t 2 t 1 ) ] ,
Γ pc ( t 1 , t 2 ) = exp [ 0 ( t 2 t 1 ) ] [ M * ( t 1 ) M ( t 2 ) + T 2 M ˙ * ( t 1 ) M ˙ ( t 2 ) Γ j ( t 1 , t 2 ) ] .
W pc ( ω 1 , ω 2 ) = M ˜ * ( Ω 1 ) M ˜ ( Ω 2 ) + T 2 Ω 1 Ω 2 M ˜ * ( Ω 1 ) M ˜ ( Ω 2 ) 2 W j ( Ω 1 , Ω 2 ) ,
f * ( ω 1 ) f ( ω 2 ) W ( ω 1 , ω 2 ) 1 2 0 ,
f * ( ω 1 ) f ( ω 2 ) W pc ( ω 1 ω ´ , ω 2 ω′ ) W N ( ω′ ) 1 2 dω′
= f * ( ω 1 ) U ˜ pc * ( ω 1 ω ´ ) 1 f ( ω 2 ) U ˜ pc ( ω 2 ω ´ ) 2 W N ( ω ´ ) ´
= f ( ω ´ ) U ˜ pc ( ω ´ ) 2 W N ( ω ´ ) ´ 0 ,

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