Abstract

We introduce a class of temporally partially coherent non-stationary wave fields, which can be represented with the aid of a packet of mutually independent, fully temporally coherent pulses of finite length. The spectral coherence properties of these pulses are determined. Fully coherent pulses and stationary fields are obtained from the model at the appropriate limits. Gaussian and exponential elementary pulses are considered as examples. Together with a Gaussian weight function they admit closed-form algebraic expressions for both temporal and spectral correlation functions and lead to Gaussian and Lorentzian spectra, respectively.

© 2006 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, 1995).
  2. J. H. Eberly and K. Wódkiewich, "The time-dependent physical spectrum of light," J. Opt. Soc. Am. 67,1252-1261 (1977).
    [CrossRef]
  3. R. Gase and M. Schubert, "On the determination of spectral coherence properties of non-stationary radiation," Opt. Acta 29,1331-1347 (1982).
    [CrossRef]
  4. 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]
  5. M. Bertolotti, A. Ferrari, and L. Sereda, "Coherence properties of nonstationary polychromatic light sources," J. Opt. Soc. Am. B 12,341-347 (1995).
    [CrossRef]
  6. M. Bertolotti, L. Sereda, and A. Ferrari, "Application of the spectral representation of stochastic processes to the study of nonstationary light radiation: a tutorial," Pure Appl. Opt. 6,153-171 (1997).
    [CrossRef]
  7. L. Sereda, M. Bertolotti, and A. Ferrari, "Coherence properties of nonstationary light wave fields," J. Opt. Soc. Am. A 15,695-705 (1998).
    [CrossRef]
  8. P. Paakkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, "Partially coherent Gaussian pulses," Opt. Commun. 204,53-58 (2002).
    [CrossRef]
  9. 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]
  10. Q. Lin, L. Wang, and S. Zhu, "Partially coherent light pulse and its propagation," Opt. Commun. 219,65-70 (2003).
    [CrossRef]
  11. M. Brunel and S. Cöetlemec, "Fractional-order Fourier formulation of the propagation of partially coherent light pulses," Opt. Commun. 230,1-5 (2004).
    [CrossRef]
  12. H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, "Spectrally partially coherent pulse trains in dispersive media," Opt. Commun. 255,12-22 (2005).
    [CrossRef]
  13. J. Lancis, V. Torres-Company, E. Silvestre, and P. Andr´es, "Space-time analogy for partially coherent plane-wave pulses," Opt. Lett. 30,2973-2975 (2005).
    [CrossRef] [PubMed]
  14. P. Vahimaa and J. Turunen, "Finite-elementary-source model for partially coherent radiation," Opt. Express 14,1376-1381 (2006).
    [CrossRef] [PubMed]
  15. We do not assume a(t) to be a ’slow’ function of t; the elementary field is written in the form of Eq. (1) for notational convenience only.
  16. This requirement follows from the positive definiteness of Γ.
  17. In view of the normalization (3), this limit should be understood as g(t_) = 1/T_ over an interval −T’/2 <t’ < T’/2, where T’ →∞.

2006

2005

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

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, "Spectrally partially coherent pulse trains in dispersive media," Opt. Commun. 255,12-22 (2005).
[CrossRef]

2004

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]

M. Brunel and S. Cöetlemec, "Fractional-order Fourier formulation of the propagation of partially coherent light pulses," Opt. Commun. 230,1-5 (2004).
[CrossRef]

2003

2002

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

1998

1997

M. Bertolotti, L. Sereda, and A. Ferrari, "Application of the spectral representation of stochastic processes to the study of nonstationary light radiation: a tutorial," Pure Appl. Opt. 6,153-171 (1997).
[CrossRef]

1995

1982

R. Gase and M. Schubert, "On the determination of spectral coherence properties of non-stationary radiation," Opt. Acta 29,1331-1347 (1982).
[CrossRef]

1977

Agrawal, G. P.

Andr´es, P.

Bertolotti, M.

Brunel, M.

M. Brunel and S. Cöetlemec, "Fractional-order Fourier formulation of the propagation of partially coherent light pulses," Opt. Commun. 230,1-5 (2004).
[CrossRef]

Cöetlemec, S.

M. Brunel and S. Cöetlemec, "Fractional-order Fourier formulation of the propagation of partially coherent light pulses," Opt. Commun. 230,1-5 (2004).
[CrossRef]

Eberly, J. H.

Ferrari, A.

Friberg, A. T.

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

Gase, R.

R. Gase and M. Schubert, "On the determination of spectral coherence properties of non-stationary radiation," Opt. Acta 29,1331-1347 (1982).
[CrossRef]

Lajunen, H.

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, "Spectrally partially coherent pulse trains in dispersive media," Opt. Commun. 255,12-22 (2005).
[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]

Lancis, J.

Lin, Q.

Q. Lin, L. Wang, and S. Zhu, "Partially coherent light pulse and its propagation," Opt. Commun. 219,65-70 (2003).
[CrossRef]

Paakkönen, P.

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

Ponomarenko, S. A.

Schubert, M.

R. Gase and M. Schubert, "On the determination of spectral coherence properties of non-stationary radiation," Opt. Acta 29,1331-1347 (1982).
[CrossRef]

Sereda, L.

Silvestre, E.

Tervo, J.

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, "Spectrally partially coherent pulse trains in dispersive media," Opt. Commun. 255,12-22 (2005).
[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]

Torres-Company, V.

Turunen, J.

P. Vahimaa and J. Turunen, "Finite-elementary-source model for partially coherent radiation," Opt. Express 14,1376-1381 (2006).
[CrossRef] [PubMed]

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, "Spectrally partially coherent pulse trains in dispersive media," Opt. Commun. 255,12-22 (2005).
[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]

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

Vahimaa, P.

P. Vahimaa and J. Turunen, "Finite-elementary-source model for partially coherent radiation," Opt. Express 14,1376-1381 (2006).
[CrossRef] [PubMed]

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, "Spectrally partially coherent pulse trains in dispersive media," Opt. Commun. 255,12-22 (2005).
[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]

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

Wang, L.

Q. Lin, L. Wang, and S. Zhu, "Partially coherent light pulse and its propagation," Opt. Commun. 219,65-70 (2003).
[CrossRef]

Wódkiewich, K.

Wolf, E.

Wyrowski, F.

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, "Spectrally partially coherent pulse trains in dispersive media," Opt. Commun. 255,12-22 (2005).
[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]

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

Zhu, S.

Q. Lin, L. Wang, and S. Zhu, "Partially coherent light pulse and its propagation," Opt. Commun. 219,65-70 (2003).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Acta

R. Gase and M. Schubert, "On the determination of spectral coherence properties of non-stationary radiation," Opt. Acta 29,1331-1347 (1982).
[CrossRef]

Opt. Commun.

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

Q. Lin, L. Wang, and S. Zhu, "Partially coherent light pulse and its propagation," Opt. Commun. 219,65-70 (2003).
[CrossRef]

M. Brunel and S. Cöetlemec, "Fractional-order Fourier formulation of the propagation of partially coherent light pulses," Opt. Commun. 230,1-5 (2004).
[CrossRef]

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, "Spectrally partially coherent pulse trains in dispersive media," Opt. Commun. 255,12-22 (2005).
[CrossRef]

Opt. Express

Opt. Lett.

Pure Appl. Opt.

M. Bertolotti, L. Sereda, and A. Ferrari, "Application of the spectral representation of stochastic processes to the study of nonstationary light radiation: a tutorial," Pure Appl. Opt. 6,153-171 (1997).
[CrossRef]

Other

We do not assume a(t) to be a ’slow’ function of t; the elementary field is written in the form of Eq. (1) for notational convenience only.

This requirement follows from the positive definiteness of Γ.

In view of the normalization (3), this limit should be understood as g(t_) = 1/T_ over an interval −T’/2 <t’ < T’/2, where T’ →∞.

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

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Equations (32)

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U e ( t ) = a ( t ) exp ( i ω 0 t ) .
Γ e ( t 1 , t 2 ) = U e * ( t 1 ) U e ( t 2 ) = a * ( t 1 ) a ( t 2 ) exp [ i ω 0 ( t 1 t 2 ) ] .
g ( t ' ) d t ' = 1 ,
Γ ( t 1 , t 2 ) = g ( t ' ) Γ e ( t 1 t ' , t 2 t ' ) d t '
= exp [ i ω 0 ( t 1 t 2 ) ] g ( t ' ) a * ( t 1 t ' ) a ( t 2 t ' ) d t ' .
I ( t ) = Γ ( t , t ) = g ( t ' ) a ( t t ' ) 2 d t '
γ ( t 1 , t 2 ) = Γ ( t 1 , t 2 ) I ( t 1 ) I ( t 2 ) ,
Γ ( t 1 , t 2 ) = U * ( t 1 ) U ( t 2 ) ,
U ( t ) = p ( t ' ) U e ( t t ' ) d t '
p * ( t ' ) p ( t " ) = g ( t ' ) δ ( t ' t " )
W ( ω 1 , ω 2 ) = Γ ( t 1 , t 2 ) exp [ i ( ω 1 t 1 ω 2 t 2 ) ] d t 1 d t 2 ,
S ( ω ) = W ( ω , ω )
μ ( ω 1 , ω 2 ) = W ( ω 1 , ω 2 ) S ( ω 1 ) S ( ω 2 ) ,
W ( ω 1 , ω 2 ) = A * ( ω 1 ω 0 ) A ( ω 2 ω 0 ) G ( ω 2 ω 1 ) ,
S ( ω ) = A ( ω ω 0 ) 2
μ ( ω 1 , ω 2 ) = G ( ω 2 ω 1 ) exp [ i Φ ( ω 1 , ω 2 ) ] ,
a ( t ) = a 0 exp ( t 2 T e 2 ) ,
g ( t ) = g 0 exp ( 2 t 2 T g 2 ) ,
I ( t ) = I 0 exp ( 2 t 2 T 2 )
T = T e [ 1 + ( T g T e ) 2 ] 1 2
γ ( t 1 , t 2 ) = exp [ ( t 1 t 2 ) 2 2 T c 2 ] exp [ i ω 0 ( t 1 t 2 ) ] ,
T c = T e [ 1 + ( T e T g ) 2 ] 1 2 .
S ( ω ) = S 0 exp [ 2 Ω 2 ( ω ω 0 ) 2 ] ,
Ω = 2 T e
μ ( ω 1 , ω 2 ) = exp [ ( ω 1 ω 2 ) 2 2 Ω c 2 ] ,
Ω c = 2 T g
a ( t ) = { a 0 exp ( t T e ) t 0 0 t < 0 ,
Φ ( ω 1 , ω 2 ) = arctan [ T e ( ω 1 ω 0 ) ] arctan [ T e ( ω 2 ω 0 ) ]
S ( ω ) = S 0 K K 2 + ( ω ω 0 ) 2 ,
K = 1 T e
I ( t ) = I 0 exp ( 2 t T e ) erfc ( 2 t T g + T g 2 T e )
γ ( t 1 , t 2 ) = exp [ i ω 0 ( t 1 t 2 ) ] [ erfc ( 2 t 1 T g + T g 2 T e ) erfc ( 2 t 2 T g + T g 2 T e ) ] 1 2 ,

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