Abstract

We introduce the concept of phase-space separability degree of statistical pulses and show how it can be determined using a bi-orthogonal decomposition of the pulse Wigner distribution. We present explicit analytical results for the case of chirped Gaussian Schell-model pulses. We also demonstrate that chirping of the pulsed source serves as a powerful tool to control coherence and phase-space separability of statistical pulses.

© 2012 OSA

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  1. T. Brabec and F. Krausz, “Intense few-cycle laser fields: Frontiers of nonlinear optics,” Rev. Mod. Phys. 72, 545–591 (2000).
    [CrossRef]
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    [CrossRef]
  3. H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, “Spectral coherence properties of temporarily modulated stationary light sources,” Opt. Express 11, 1894–1899 (2003).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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  12. G. P. Agrawal, Fiber-Optic Communication Systems, 3rd ed., (Wiley, New York, NY2002).
    [CrossRef]
  13. Q. Lin, L. Wang, and S. Zhu, “Partially coherent light pulse and its propagation,” Opt. Commun. 219, 65–70 (2003).
    [CrossRef]
  14. M. Brunel and S. Coëtlemec, “Fractional-order Fourier formulation of the propagation of partially coherent light pulses,” Opt. Commun. 230, 1–5 (2004).
    [CrossRef]
  15. H. Lajunen, V. Torres-Company, J. Lancis, E. Silvestre, and P. Andrés, “Pulse-by-pulse method to characterize partially coherent pulse propagation in instanteneous nonlonear media,” Opt. Express 18, 14979–14991 (2011).
    [CrossRef]
  16. B. H. Kolner and M. Nazarathy, “Temporal imaging with a time lens,” Opt. Lett. 14, 630–632 (1989).
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  17. Although the slowly-varying envelope approximation breaks down for few-cycle long femtosecond pulses, the decomposition into the envelope and carrier wave makes sense even in this case, see Ref. [21] for details.
  18. G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic Press, Amsterdam, 2007).
  19. M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30, (1978).
    [CrossRef]
  20. M. J. Bastiaans, “The Wigner function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
    [CrossRef]
  21. J. C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena, 2nd ed. (Academic Press, Amsterdam2006).
  22. J. Lancis, V. Torres-Company, E. Silvestre, and P. Andrés, “Space-time analogy for partially coherent plane-wave-type pulses,” Opt. Lett. 30, 2973–2975 (2005).
    [CrossRef] [PubMed]
  23. V. Torres-Company, J. Lancis, and P. Andrés, “Space-time analogies in optics,” Prog. Opt. 56, 1–80 (2011), ed. E. Wolf.
    [CrossRef]
  24. M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
    [CrossRef] [PubMed]
  25. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Part I.
  26. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).
  27. M. Abramowitz and I. A. Stegan, Handbook of Mathematical Functions (Dover, New York, 1972).
  28. F. Gori, “Collett-Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
    [CrossRef]
  29. A. Starikov and E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and their radiation fields,” J. Opt. Soc. Am. 72, 923–928 (1982).
    [CrossRef]

2011 (4)

2007 (4)

2006 (1)

2005 (1)

2004 (2)

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. Coëtlemec, “Fractional-order Fourier formulation of the propagation of partially coherent light pulses,” Opt. Commun. 230, 1–5 (2004).
[CrossRef]

2003 (2)

2002 (1)

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53–58 (2002).
[CrossRef]

2000 (1)

T. Brabec and F. Krausz, “Intense few-cycle laser fields: Frontiers of nonlinear optics,” Rev. Mod. Phys. 72, 545–591 (2000).
[CrossRef]

1994 (1)

M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

1989 (1)

1982 (1)

1980 (1)

F. Gori, “Collett-Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

1979 (1)

1978 (1)

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30, (1978).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegan, Handbook of Mathematical Functions (Dover, New York, 1972).

Agrawal, G. P.

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]

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic Press, Amsterdam, 2007).

G. P. Agrawal, Fiber-Optic Communication Systems, 3rd ed., (Wiley, New York, NY2002).
[CrossRef]

Alonso, M. A.

Andrés, P.

Bastiaans, M. J.

M. J. Bastiaans, “The Wigner function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
[CrossRef]

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30, (1978).
[CrossRef]

Beck, M.

M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

Brabec, T.

T. Brabec and F. Krausz, “Intense few-cycle laser fields: Frontiers of nonlinear optics,” Rev. Mod. Phys. 72, 545–591 (2000).
[CrossRef]

Brunel, M.

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

Coëtlemec, S.

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

Davis, B.

B. Davis, “Measurable coherence theory for statistically periodic fields,” Phys. Rev. A 76, 043843 (2007).
[CrossRef]

Diels, J. C.

J. C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena, 2nd ed. (Academic Press, Amsterdam2006).

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Part I.

Friberg, A. T.

Gomez-Sarabia, C. M.

Gori, F.

F. Gori, “Collett-Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

Kolner, B. H.

Krausz, F.

T. Brabec and F. Krausz, “Intense few-cycle laser fields: Frontiers of nonlinear optics,” Rev. Mod. Phys. 72, 545–591 (2000).
[CrossRef]

Lajunen, H.

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]

Mandel, L.

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

McAlister, D. F.

M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Part I.

Nazarathy, M.

Ojeda-Castaneda, J.

Pääkkönen, P.

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53–58 (2002).
[CrossRef]

Ponomarenko, S. A.

Raymer, M. G.

M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

Rudolph, W.

J. C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena, 2nd ed. (Academic Press, Amsterdam2006).

Silvestre, E.

Starikov, A.

Stegan, I. A.

M. Abramowitz and I. A. Stegan, Handbook of Mathematical Functions (Dover, New York, 1972).

Tervo, J.

Torres-Company, V.

Turunen, J.

Vahimaa, P.

Wang, L.

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

Wolf, E.

Wyrowski, F.

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

P. Pääkkönen, 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]

Adv. Opt. Photon. (1)

J. Opt. Soc. Am. (2)

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

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

Opt. Commun. (5)

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. Coëtlemec, “Fractional-order Fourier formulation of the propagation of partially coherent light pulses,” Opt. Commun. 230, 1–5 (2004).
[CrossRef]

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53–58 (2002).
[CrossRef]

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30, (1978).
[CrossRef]

F. Gori, “Collett-Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

Opt. Express (5)

Opt. Lett. (3)

Phys. Rev. A (1)

B. Davis, “Measurable coherence theory for statistically periodic fields,” Phys. Rev. A 76, 043843 (2007).
[CrossRef]

Phys. Rev. Lett. (1)

M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

Prog. Opt. (1)

V. Torres-Company, J. Lancis, and P. Andrés, “Space-time analogies in optics,” Prog. Opt. 56, 1–80 (2011), ed. E. Wolf.
[CrossRef]

Rev. Mod. Phys. (1)

T. Brabec and F. Krausz, “Intense few-cycle laser fields: Frontiers of nonlinear optics,” Rev. Mod. Phys. 72, 545–591 (2000).
[CrossRef]

Other (7)

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Part I.

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

M. Abramowitz and I. A. Stegan, Handbook of Mathematical Functions (Dover, New York, 1972).

G. P. Agrawal, Fiber-Optic Communication Systems, 3rd ed., (Wiley, New York, NY2002).
[CrossRef]

J. C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena, 2nd ed. (Academic Press, Amsterdam2006).

Although the slowly-varying envelope approximation breaks down for few-cycle long femtosecond pulses, the decomposition into the envelope and carrier wave makes sense even in this case, see Ref. [21] for details.

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic Press, Amsterdam, 2007).

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Figures (3)

Fig. 1
Fig. 1

Sketching the behavior of the propagation factor of a CGSM pulse as a function of dimensionless propagation distance Z = β2z/tp.

Fig. 2
Fig. 2

Degree of phase-space separability of a fully coherent CGSM pulse as a function of dimensionless propagation distance Z = β2z/tp for three values of the initial chirp: C = 0 and C = ±1.

Fig. 3
Fig. 3

Degree of phase-space separability of a partially coherent CGSM pulse as a function of dimensionless propagation distance Z = β2z/tp; solid, tc = ∞, dotted, tc = tp and dashed t c = 2 t p / 3. The initial chirp is C = −1.

Equations (35)

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E ( t ) = U ( t ) e i ω c t ,
Γ ( t 1 , t 2 ) = U * ( t 1 ) U ( t 2 ) ,
𝒲 ( t , ω ) = d τ Γ ( t τ / 2 , t + τ / 2 ) e i ω τ ,
I ( t ) = d ω 𝒲 ( t , ω ) ; S ( ω ) = d t 𝒲 ( t , ω ) .
i z U 1 2 β 2 s s 2 U = 0 ,
i z Γ + 1 2 β 2 ( s 1 s 1 2 s 2 s 2 2 ) Γ = 0 .
z 𝒲 + β 2 T 𝒲 = 0 ,
T = s 1 + s 2 2 , and τ = s 1 s 2 ,
𝒲 ( T , ω ; z ) = 𝒲 0 ( T β 2 ω , ω ) .
𝒲 ( t , ω ) I ( t ) S ( ω ) .
Γ GSM ( t 1 , t 2 ) = I ( t 1 + t 2 2 ) g ( t 1 t 2 ) ,
Γ qs ( t 1 , t 2 ) I ( t 1 + t 2 2 ) γ ( t 1 t 2 ) ,
𝒲 ( T , ω ; z ) = n λ n ( z ) χ n ( T , z ) ϕ n ( ω , z ) ,
λ n ( z ) ϕ n ( ω , z ) = d T 𝒲 ( T , ω ; z ) ϕ n ( T , z ) ,
λ n ( z ) χ n ( T , z ) = d ω 𝒲 ( T , ω ; z ) χ n ( ω , z ) .
d x χ n ( x , z ) ϕ m ( x , z ) = δ n m , x = T , ω .
ρ ( z ) = 1 n = 0 ν n 2 ( z ) .
Γ CGSM ( t 1 , t 2 ) = Γ 00 exp [ ( 1 i C ) t 1 2 2 t p 2 ( 1 + i C ) t 2 2 2 t p 2 ( t 1 t 2 ) 2 2 t c 2 ] ,
𝒲 CGSM ( t , ω ) = 𝒲 00 exp [ t 2 t p 2 ( 1 + C 2 t eff 2 2 t p 2 ) ω 2 t eff 2 2 + C t eff 2 t p 2 ω t ] ,
1 t eff 2 = 1 t p 2 + 1 2 t c 2 .
𝒲 CGSM ( T , ω ; z ) = 𝒲 00 exp [ T 2 t p 2 ( 1 + C 2 t eff 2 2 t p 2 ) ω 2 σ 2 ( z ) t eff 2 2 + C ( z ) t eff 2 t p 2 ω T ] .
σ 2 ( z ) = ( 1 + C β 2 z t p 2 ) 2 + 2 β 2 2 z 2 t p 2 t eff 2 ,
C ( z ) = C + β 2 z t p 2 ( C 2 + 2 t p 2 t eff 2 ) .
σ min = 2 t p 2 / t eff 2 C 2 + 2 t p 2 / t eff 2 ,
z * = C t eff 2 / β 2 C 2 + 2 t p 2 / t eff 2 .
Γ CGSM ( s 1 , s 2 ; z ) = Γ 00 σ ( z ) exp [ ( s 1 2 + s 2 2 ) 2 t p 2 σ 2 ( z ) ( s 1 s 2 ) 2 2 t c 2 σ 2 ( z ) + i C ( z ) 2 t p 2 σ 2 ( z ) ( s 1 2 s 2 2 ) ] .
exp ( x 2 + y 2 2 x y ζ 1 ζ 2 ) = 1 + ζ 2 e x 2 y 2 n = 0 ζ n 2 n n ! H n ( x ) H n ( y ) ,
𝒲 CGSM ( T ˜ , ω ˜ ; z ) = 𝒲 00 exp ( T ˜ 2 + ω ˜ 2 2 T ˜ ω ˜ ζ ( z ) 1 ζ 2 ( z ) ) .
λ n ( z ) = 𝒲 00 [ 1 + ζ 2 ( z ) ] a ( z ) b ( z ) ζ 2 n ( z ) ,
χ n ( T , z ) = 1 2 n n ! π a ( z ) e T 2 / a 2 ( z ) H n [ T a ( z ) ] ,
ϕ n ( ω , z ) = 1 2 n n ! π b ( z ) e ω 2 / b 2 ( z ) H n [ ω b ( z ) ] .
a ( z ) = 2 t p 4 / t eff 2 ( C 2 + 2 t p 2 / t eff 2 ) [ 1 ζ 2 ( z ) ] ,
b ( z ) = 2 σ 2 ( z ) [ 1 ζ 2 ( z ) ] t eff 2 ,
ζ 2 ( z ) = C 2 ( z ) σ 2 ( z ) ( C 2 + 2 t p 2 / t eff 2 ) ,
ρ ( z ) = σ min 2 / σ 2 ( z ) .

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