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]
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  9. H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, F. Wyrowski, "Spectrally partially coherent pulse trains in dispersive media," Opt. Commun. 255, 12-22 (2005).
    [CrossRef]
  10. H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, F. Wyrowski, "Spectral coherence properties of temporally modulated stationary light sources," Opt. Express 11, 1894-1899 (2003).
    [CrossRef] [PubMed]
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    [CrossRef]
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2005

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

2004

2003

2002

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

2001

1997

B. Lacaze, "Calcul des spectres de puissance des processus caract`ere cyclostationnaire," Traitement du Signal,  14, 63-71 (1997).

Deparis, O.

Friberg, A.

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

Kiyan, R.

Lacaze, B.

B. Lacaze, "Calcul des spectres de puissance des processus caract`ere cyclostationnaire," Traitement du Signal,  14, 63-71 (1997).

Lajunen, H.

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

Lin, Q.

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

Linne, M.

Paakkonen, P.

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

Paciaroni, M.

Pottiez, O.

Tervo, J.

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

Turunen, J.

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

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

Vahimaa, P.

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

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

Wang, L.

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

Wyrowski, F.

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

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

Zhu, S.

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

Appl. Opt.

Opt. Commun.

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

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

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

Opt. Express

Opt. Lett.

Traitement du Signal

B. Lacaze, "Calcul des spectres de puissance des processus caract`ere cyclostationnaire," Traitement du Signal,  14, 63-71 (1997).

Other

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

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

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

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

H. Cramer, 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 al’eatoires pour communicatons num eriques, (Hermes, Paris, 2000).

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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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