Abstract

The Gaussian-Schell (G-S) pulse model describes variations in Gaussian shaped light pulses. Recent studies proposed random functions fitted to G-S pulses in a non-stationary context. More specifically, this paper provides a cyclo-stationary model. The associated power spectrum is derived. A correlation between pulses is shown to remove the power spectrum Gaussian character.

© 2007 Optical Society of America

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References

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  1. S. B. Cavalcanti, “Theory of incoherent self-phase modulation of non-stationary pulses,” New J. Phys. 419.1–19.11 (2002).
    [Crossref]
  2. L. W. Couch II, Digital and Analog Communications,(MacMillan, 1990).
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  8. B. Lacaze, “Calcul des spectres de puissance des processus à caractére cyclostationnaire,” Traitement du Signal, 1463–71 (1997).
  9. H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, “Spectrally partially coherent pulse trains in dispersive media,” Opt. Commun. 25512–22 (2005).
    [Crossref]
  10. H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, “Spectral coherence properties of temporally modulated stationary light sources,” Opt. Express 111894–1899 (2003).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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2005 (1)

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, “Spectrally partially coherent pulse trains in dispersive media,” Opt. Commun. 25512–22 (2005).
[Crossref]

2004 (1)

2003 (2)

2002 (2)

P. Paakkonen, J. Turunen, P. Vahimaa, A. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Com-mun. 20453–58 (2002).
[Crossref]

S. B. Cavalcanti, “Theory of incoherent self-phase modulation of non-stationary pulses,” New J. Phys. 419.1–19.11 (2002).
[Crossref]

2001 (1)

2000 (1)

B. Lacaze, Processus alèatoires pour communicatons numèriques, (Hermes, Paris, 2000).

1997 (1)

B. Lacaze, “Calcul des spectres de puissance des processus à caractére cyclostationnaire,” Traitement du Signal, 1463–71 (1997).

Blondel, M.

Cavalcanti, S. B.

S. B. Cavalcanti, “Theory of incoherent self-phase modulation of non-stationary pulses,” New J. Phys. 419.1–19.11 (2002).
[Crossref]

Couch II, L. W.

L. W. Couch II, Digital and Analog Communications,(MacMillan, 1990).

Cramer, H.

H. Cramer and M. R. Leadbetter, Stationary and Related Stochastic Processes, (Wiley, 1966).

Deparis, O.

Feller, W.

W. Feller, An Introduction to Probability Theory and its Applications, (Wiley, 1966).

Friberg, A.

P. Paakkonen, J. Turunen, P. Vahimaa, A. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Com-mun. 20453–58 (2002).
[Crossref]

Gardner, A.

A. Gardner, Introduction to Random processes with Applications to Signals and Systems, (Mc Graw Hill, 1990).

Kay, S. M.

S. M. Kay, Modern Spectral Estimation, (Prentice Hall, 1988).

Kiyan, R.

Lacaze, B.

B. Lacaze, Processus alèatoires pour communicatons numèriques, (Hermes, Paris, 2000).

B. Lacaze, “Calcul des spectres de puissance des processus à caractére cyclostationnaire,” Traitement du Signal, 1463–71 (1997).

Lajunen, H.

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, “Spectrally partially coherent pulse trains in dispersive media,” Opt. Commun. 25512–22 (2005).
[Crossref]

H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, “Spectral coherence properties of temporally modulated stationary light sources,” Opt. Express 111894–1899 (2003).
[Crossref] [PubMed]

Leadbetter, M. R.

H. Cramer and M. R. Leadbetter, Stationary and Related Stochastic Processes, (Wiley, 1966).

Lin, Q.

Q. Lin, L. Wang, and S. Zhu, “Partially coherent light pulse and its propagation,” Opt. Commun. 21965–70 (2003).
[Crossref]

Linne, M.

Mègret, P.

Paakkonen, P.

P. Paakkonen, J. Turunen, P. Vahimaa, A. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Com-mun. 20453–58 (2002).
[Crossref]

Paciaroni, M.

Papoulis, A.

A. Papoulis, Probability, Random Variables and Stochastic Processes, (McGraw-Hill, 1965).

Pottiez, O.

Proakis, J. G.

J. G. Proakis, Digital Communications, (Mc Graw Hill, 1995).

Tervo, J.

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, “Spectrally partially coherent pulse trains in dispersive media,” Opt. Commun. 25512–22 (2005).
[Crossref]

H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, “Spectral coherence properties of temporally modulated stationary light sources,” Opt. Express 111894–1899 (2003).
[Crossref] [PubMed]

Turunen, J.

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, “Spectrally partially coherent pulse trains in dispersive media,” Opt. Commun. 25512–22 (2005).
[Crossref]

H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, “Spectral coherence properties of temporally modulated stationary light sources,” Opt. Express 111894–1899 (2003).
[Crossref] [PubMed]

P. Paakkonen, J. Turunen, P. Vahimaa, A. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Com-mun. 20453–58 (2002).
[Crossref]

Vahimaa, P.

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, “Spectrally partially coherent pulse trains in dispersive media,” Opt. Commun. 25512–22 (2005).
[Crossref]

H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, “Spectral coherence properties of temporally modulated stationary light sources,” Opt. Express 111894–1899 (2003).
[Crossref] [PubMed]

P. Paakkonen, J. Turunen, P. Vahimaa, A. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Com-mun. 20453–58 (2002).
[Crossref]

Wang, L.

Q. Lin, L. Wang, and S. Zhu, “Partially coherent light pulse and its propagation,” Opt. Commun. 21965–70 (2003).
[Crossref]

Wyrowski, F.

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, “Spectrally partially coherent pulse trains in dispersive media,” Opt. Commun. 25512–22 (2005).
[Crossref]

H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, “Spectral coherence properties of temporally modulated stationary light sources,” Opt. Express 111894–1899 (2003).
[Crossref] [PubMed]

P. Paakkonen, J. Turunen, P. Vahimaa, A. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Com-mun. 20453–58 (2002).
[Crossref]

Zhu, S.

Q. Lin, L. Wang, and S. Zhu, “Partially coherent light pulse and its propagation,” Opt. Commun. 21965–70 (2003).
[Crossref]

Appl. Opt. (1)

New J. Phys. (1)

S. B. Cavalcanti, “Theory of incoherent self-phase modulation of non-stationary pulses,” New J. Phys. 419.1–19.11 (2002).
[Crossref]

Opt. Com-mun. (1)

P. Paakkonen, J. Turunen, P. Vahimaa, A. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Com-mun. 20453–58 (2002).
[Crossref]

Opt. Commun. (2)

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, “Spectrally partially coherent pulse trains in dispersive media,” Opt. Commun. 25512–22 (2005).
[Crossref]

Q. Lin, L. Wang, and S. Zhu, “Partially coherent light pulse and its propagation,” Opt. Commun. 21965–70 (2003).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Traitement du Signal (1)

B. Lacaze, “Calcul des spectres de puissance des processus à caractére cyclostationnaire,” Traitement du Signal, 1463–71 (1997).

Other (8)

L. W. Couch II, Digital and Analog Communications,(MacMillan, 1990).

H. Cramer and M. R. Leadbetter, Stationary and Related Stochastic Processes, (Wiley, 1966).

W. Feller, An Introduction to Probability Theory and its Applications, (Wiley, 1966).

A. Gardner, Introduction to Random processes with Applications to Signals and Systems, (Mc Graw Hill, 1990).

S. M. Kay, Modern Spectral Estimation, (Prentice Hall, 1988).

B. Lacaze, Processus alèatoires pour communicatons numèriques, (Hermes, Paris, 2000).

J. G. Proakis, Digital Communications, (Mc Graw Hill, 1995).

A. Papoulis, Probability, Random Variables and Stochastic Processes, (McGraw-Hill, 1965).

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Equations (37)

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E [ A ( t ) A * ( t ' ) ] = a 2 π σ ' 2 exp [ i ω 0 ( t t ' ) ( t m ) 2 2 σ ' 2 ( t ' m ) 2 2 σ ' 2 ( t t ' ) 2 2 ρ ' 2 ]
A ( t ) = B ( t ) u ( t )
{ u ( t ) = 1 σ ' 2 π exp [ i ω 0 t ( t m ) 2 2 σ ' 2 ] E [ B ( t ) B * ( t ' ) ] = a exp [ ( t t ' ) 2 2 ρ ' 2 ]
[ K A ] ( ω , ω ' ) = 2 E [ A ( t ) A * ( t ' ) ] e iωt ' t ' dtdt '
[ K A ] ( ω , ω ' ) = σ ' exp [ im ( ω ω ' ) σ 2 2 ( ( ω ω 0 ) 2 + ( ω ' ω 0 ) 2 ) ρ 2 2 ( ω ω ' ) 2 ]
{ σ ' = σ 2 + 2 ρ 2 ρ ' = σ ρ σ 2 + 2 ρ 2 σ = σ ' ρ ' 2 σ ' 2 + ρ ' 2 ρ = σ ' 2 2 σ ' 2 + ρ ' 2
Z ( t ) = n = A n ( t n T ) .
Z ( t ) = A t ¯ ( t ¯ ) ,             t = t ¯ T + t ¯ ,             t ¯ , 0 t ¯ < T
{ A n ( t ) = B n ( t ) u ( t ) u ( t ) = 1 σ ' 2 π exp [ i ω 0 t ( t m ) 2 2 σ ' 2 ] E [ B n ( t ) B n * ( t ' ) ] = a exp [ ( t t ' ) 2 2 ρ ' 2 ]
{ P 1 : A n ( t ) = 0 , t ( 0 , 1 ) P 2 : E [ A n ( t ) ] = 0 , t∈ ( 0 , 1 ) P 3 : E [ A m ( t ) A m n * ( t ' ) ] = α n ( t , t ' ) is independent of m P 4 : lim n α n ( t , t ' ) = 0 with uniform convergence in ( t , t ' )
{ P 3 ' : E [ B m ( t ) B m n * ( t ' ) ] = β n ( t , t ' ) is a quantity independent of m P 4 ' : lim n β n ( t , t ' ) = 0 with uniform convergence in ( t , t ' )
E [ Z ( t ) Z * ( t ' ) ] = E [ Z ( t + 1 ) Z * ( t ' + 1 ) ]
s Z ( ω ) = 1 2 π n = [ ( 0 , 1 ) 2 α n ( t , t ' ) e i ω ( t ' t ) dtdt ' ] e i
α 0 ( t , t ' ) = E [ A n ( t ) A n * ( t ' ) ] = a u ( t ) u * ( t ' ) exp [ ( t t ' ) 2 2 ρ ' 2 ]
s Z ( ω ) = 2 πσ ' exp [ σ 2 ( ω ω 0 ) 2 ]
s Z ( ω ) = 1 a = 2 σ ' π .
{ s Z ( ω ) = 2 πσ ' T exp [ σ 2 ( ω ω 0 ) 2 ] s Z ( ω ) = 1 a = 2 σ ' T π
B n ( t ) = C ( t + n ) , t [ 0 , 1 [ , n
{ E [ C n ( t ) C n * ( t ' ) ] = a 2 exp [ ( t t ' ) 2 2 ρ ' 2 ] B n ( t ) = C n ( t ) + C n 1 ( t )
{ β 0 t t = a exp [ ( t t ) 2 2 ρ ' 2 ] β ± 1 t t = a 2 exp [ ( t t ' ) 2 2 ρ 2 ] β n t t = 0 , n 0 , ± 1 .
s Z ( ω ) = 2 πσ ( 1 + cos ω ) exp [ σ 2 ( ω ω 0 ) 2 ]
B n ( t ) = k b k C n k ( t )
{ s Z ( ω ) = P ( ω ) exp [ σ 2 ( ω ω 0 ) 2 ] P ( ω ) = 4 π σ k b k e ikω 2
n det [ Ω ] ( 2 π ) n 2 exp [ 1 2 ( ω ω 0 ) t Ω ( ω ω 0 ) + i u t ω ] = exp [ i u t ω 0 1 2 u t Ω 1 u ]
{ σ 2 ( ( ω ω 0 ) 2 + ( ω ω 0 ) 2 ) + ρ 2 ( ω ω ) 2 1 σ 2 ( ( t m ) 2 + ( t m ) 2 ) + 1 ρ 2 ( t t ) 2
Ω = [ σ 2 + ρ 2 ρ 2 ρ 2 σ 2 + ρ 2 ] ; Ω 1 = 1 σ 2 ( σ 2 + 2 ρ 2 ) [ σ 2 + ρ 2 ρ 2 ρ 2 σ 2 + ρ 2 ]
ω t = [ ω ω ] , ω 0 t = [ ω 0 ω 0 ] , u t = [ t + m t m ]
E [ Z ( t ) Z * ( t τ ) ] = E [ A t ¯ ( t ) A t τ ¯ * ( t τ ¯ ) ] = α t ¯ t τ ¯ t t τ ¯
U ( t ) = Z ( t Φ )
E [ U ( t ) U * ( t τ ) ] = 0 1 α t ϕ ¯ t τ ϕ ¯ t ϕ ¯ , t τ ϕ ¯ d ϕ
E [ U ( t ) U * ( t τ ) ] = 0 1 α u τ ¯ u u τ ¯ du
s Z ( ω ) = 1 2 π [ 0 1 α u τ ¯ u uτ ¯ du ] e iωτ
s Z ( ω ) = 1 2 π n = e inω [ ( 0,1 ) 2 α n t t ' e ( t t ) dtdt ' ]
α n t t = au ( t ) u * ( t ) exp [ 1 2 ρ 2 ( t t + n ) 2 ]
s Z ( ω ) = 2 π σ ' e σ 2 ( ω ω 0 ) 2 [ 1 + 2 n = 1 exp [ n 2 2 ( 2 σ 2 + ρ 2 ) ] cos [ 2 n ( ω ω 0 ) 2 + ( ρ σ ) 2 ] ]
β n t t = [ k b k b k n * ] E [ C 0 ( t ) C 0 * ( t ) ]
n , k b k b k n * e inω = k b k e ikω 2

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