Abstract

We develop a general representation for ensembles of non-stationary random pulses in terms of statistically uncorrelated, time-delayed, frequency-shifted Gaussian pulses which are classical counterparts of coherent states of a quantum harmonic oscillator. We show that the two-time correlation function describing second-order statistics of the pulses can be expanded in terms of the complex Gaussian pulses. We also demonstrate how the novel formalism can be applied to describe recently introduced Gaussian Schell-model pulses and pulse trains generated by typical mode-locked lasers.

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References

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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]
  2. E. Wolf, Introduction to the Theory of Coherence and Polarization (Cambridge University Press, 2007).
  3. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).
  4. M. Bertolotti, A. Ferrari, and L. Sereda, “Coherence properties of nonstationary polychromatic light sources,” J. Opt. Soc. Am. B 12, 341–347 (1995).
    [CrossRef]
  5. L. Sereda, M. Bertolotti, and A. Ferrari, “Coherence properties of nonstationary light wave fields,” J. Opt. Soc. Am. B 15, 695–705 (1998).
    [CrossRef]
  6. 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]
  7. B. Davis, “Measurable coherence theory for statistically periodic fields,” Phys. Rev. A 76, 043843 (2007).
    [CrossRef]
  8. P. Pääkkönen, 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 temporarily modulated stationary light sources,” Opt. Express 11, 1894–1899 (2003).
    [CrossRef] [PubMed]
  10. P. Vahimaa and J. Turunen, “Independent-elementary-pulse representation for non-stationary fields,” Opt. Express 14, 5007–5012 (2006).
    [CrossRef] [PubMed]
  11. Q. Lin, L. Wang, and S. Zhu, “Partially coherent light pulse and its propagation,” Opt. Commun. 219, 65–70 (2003).
    [CrossRef]
  12. M. Brunel and S. Coëtlemec, “Fractional-order Fourier formulation of the propagation of partially coherent light pulses,” Opt. Commun. 230, 1–5 (2004).
    [CrossRef]
  13. R. W. Schoonover, B. J. Davis, and P. S. Carney, “The generalized Wolf shift for cyclostationary fields,” Opt. Express 17, 4705–4711 (2009).
    [CrossRef] [PubMed]
  14. G. Arfken, Mathematical Methods for Physicists (Academic Press, 1970).
  15. 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. [19] for details.
  16. S. A. Ponomarenko, J. J. Greffet, and E. Wolf, “Diffusion of partially coherent beams in turbulent media,” Opt. Commun. 208, 1–8 (2002).
    [CrossRef]
  17. E. Wolf, “New theory of partial coherence in space-frequency domain. Part I: Spectra and cross-spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982).
    [CrossRef]
  18. P. W. Milonni and J. H. Eberly, Lasers (Wiley, 1988).
  19. J. C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena, 2nd ed. (Academic Press, 2006).

2009

2007

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

2006

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

2003

2002

S. A. Ponomarenko, J. J. Greffet, and E. Wolf, “Diffusion of partially coherent beams in turbulent media,” Opt. Commun. 208, 1–8 (2002).
[CrossRef]

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

2000

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

1998

L. Sereda, M. Bertolotti, and A. Ferrari, “Coherence properties of nonstationary light wave fields,” J. Opt. Soc. Am. B 15, 695–705 (1998).
[CrossRef]

1995

1982

Agrawal, G. P.

Arfken, G.

G. Arfken, Mathematical Methods for Physicists (Academic Press, 1970).

Bertolotti, M.

L. Sereda, M. Bertolotti, and A. Ferrari, “Coherence properties of nonstationary light wave fields,” J. Opt. Soc. Am. B 15, 695–705 (1998).
[CrossRef]

M. Bertolotti, A. Ferrari, and L. Sereda, “Coherence properties of nonstationary polychromatic light sources,” J. Opt. Soc. Am. B 12, 341–347 (1995).
[CrossRef]

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]

Carney, P. S.

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]

Davis, B. J.

Diels, J. C.

J. C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena, 2nd ed. (Academic Press, 2006).

Eberly, J. H.

P. W. Milonni and J. H. Eberly, Lasers (Wiley, 1988).

Ferrari, A.

L. Sereda, M. Bertolotti, and A. Ferrari, “Coherence properties of nonstationary light wave fields,” J. Opt. Soc. Am. B 15, 695–705 (1998).
[CrossRef]

M. Bertolotti, A. Ferrari, and L. Sereda, “Coherence properties of nonstationary polychromatic light sources,” J. Opt. Soc. Am. B 12, 341–347 (1995).
[CrossRef]

Friberg, A. T.

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

Greffet, J. J.

S. A. Ponomarenko, J. J. Greffet, and E. Wolf, “Diffusion of partially coherent beams in turbulent media,” Opt. Commun. 208, 1–8 (2002).
[CrossRef]

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.

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).

Milonni, P. W.

P. W. Milonni and J. H. Eberly, Lasers (Wiley, 1988).

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.

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]

S. A. Ponomarenko, J. J. Greffet, and E. Wolf, “Diffusion of partially coherent beams in turbulent media,” Opt. Commun. 208, 1–8 (2002).
[CrossRef]

Rudolph, W.

J. C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena, 2nd ed. (Academic Press, 2006).

Schoonover, R. W.

Sereda, L.

L. Sereda, M. Bertolotti, and A. Ferrari, “Coherence properties of nonstationary light wave fields,” J. Opt. Soc. Am. B 15, 695–705 (1998).
[CrossRef]

M. Bertolotti, A. Ferrari, and L. Sereda, “Coherence properties of nonstationary polychromatic light sources,” J. Opt. Soc. Am. B 12, 341–347 (1995).
[CrossRef]

Tervo, J.

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.

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]

S. A. Ponomarenko, J. J. Greffet, and E. Wolf, “Diffusion of partially coherent beams in turbulent media,” Opt. Commun. 208, 1–8 (2002).
[CrossRef]

E. Wolf, “New theory of partial coherence in space-frequency domain. Part I: Spectra and cross-spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982).
[CrossRef]

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

E. Wolf, Introduction to the Theory of Coherence and Polarization (Cambridge University Press, 2007).

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]

J. Opt. Soc. Am.

J. Opt. Soc. Am. B

M. Bertolotti, A. Ferrari, and L. Sereda, “Coherence properties of nonstationary polychromatic light sources,” J. Opt. Soc. Am. B 12, 341–347 (1995).
[CrossRef]

L. Sereda, M. Bertolotti, and A. Ferrari, “Coherence properties of nonstationary light wave fields,” J. Opt. Soc. Am. B 15, 695–705 (1998).
[CrossRef]

Opt. Commun.

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]

S. A. Ponomarenko, J. J. Greffet, and E. Wolf, “Diffusion of partially coherent beams in turbulent media,” Opt. Commun. 208, 1–8 (2002).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

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

Rev. Mod. Phys.

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

Other

E. Wolf, Introduction to the Theory of Coherence and Polarization (Cambridge University Press, 2007).

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

G. Arfken, Mathematical Methods for Physicists (Academic Press, 1970).

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. [19] for details.

P. W. Milonni and J. H. Eberly, Lasers (Wiley, 1988).

J. C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena, 2nd ed. (Academic Press, 2006).

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Equations (34)

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ψ ( t ; t s , ω s ) = A exp [ ( t t s ) 2 2 t * 2 ] e i ω s t ,
ψ α ( T ) = e ( Im α ) 2 π 1 / 4 exp [ ( T 2 α ) 2 2 ] .
α = 1 2 ( T s + i Ω s ) .
d T | ψ α ( T ) | 2 = 1 .
| α = 𝒜 n = 0 α n n ! | n ,
ψ α ( x ) = x | α = 𝒜 n = 0 α n n ! x | n ,
x | n = 1 π 1 / 2 2 n n ! H n ( x ) e x 2 / 2 .
e 2 s x s 2 = n = 0 s n n ! H n ( x ) ,
ψ α ( x ) = e ( Im α ) 2 π 1 / 4 exp [ ( x 2 α ) 2 2 ] .
d 2 α | α α | = 1 .
d 2 α ψ α ( T ) ψ α * ( T ) = δ ( T T ) ,
E ( T ) = U ( T ) e i Ω c T ,
Γ ( T 1 , T 2 ) = U * ( T 1 ) U ( T 2 ) ,
Γ ( T 1 , T 2 ) T 2 | Γ ^ | T 1 .
Γ ^ = d 2 α 𝒫 ( α ) | α α | .
T 2 | Γ ^ | T 1 = d 2 α 𝒫 ( α ) T 2 | α α | T 1 ,
Γ ( T 1 , T 2 ) = d 2 α 𝒫 ( α ) ψ α * ( T 1 ) ψ α ( T 2 ) .
𝒫 ( α ) = 1 π 2 d 2 β e | α | 2 + | β | 2 β | Γ | β e β * α α * β ,
𝒫 ( α ) = e | α | 2 π 2 d 2 β e | β | 2 e β * α α * β × d T 1 d T 2 Γ ( T 1 , T 2 ) ψ β * ( T 1 ) ψ β ( T 2 ) .
U ( T ) = d 2 α c ( α ) ψ α ( T ) .
c * ( α ) c ( α ) = 𝒫 ( α ) δ ( α α ) .
Γ ( T 1 , T 2 ) = n λ n ϕ n * ( T 1 ) ϕ n ( T 2 ) ,
d T ϕ m * ( T ) ϕ n ( T ) = δ n m .
d T 1 Γ ( T 1 , T 2 ) ϕ n ( T 1 ) = λ n ϕ n ( T 2 ) .
U ( T ) = n a n ϕ n ( T ) ,
a m * a n = λ n δ n m .
Γ ( T 1 , T 2 ) = I 0 exp [ T 1 2 + T 2 2 2 σ p 2 ] exp [ ( T 1 T 2 ) 2 2 σ c 2 ] ,
𝒫 ( T s , Ω s ) = 2 I 0 π 2 ( 1 1 / σ p 2 ) ( 2 / σ c 2 + 1 / σ p 2 1 ) × exp [ T s 2 σ p 2 ( 1 1 / σ p 2 ) Ω s 2 2 / σ c 2 + 1 / σ p 2 1 ] .
𝒫 ( T s , Ω s ) = I 0 σ c π δ ( T s ) e σ c 2 Ω s 2 / 2 .
𝒫 ( α ) = n w n δ ( α α n ) ,
Γ ( t 1 , t 2 ) = n w n ψ α n * ( t 1 ) ψ α n ( t 2 ) .
I ( t ) = n = N N w n | ψ α n ( t ) | 2 = 1 π t * n = N N w n exp [ ( t n t 0 ) 2 t * 2 ] .
c ( α ) = n c n δ ( α α n ) ,
U ( t ) = 1 π 1 / 2 t * n = N N c n e i n ω 0 t * exp [ ( t n t 0 ) 2 2 t * 2 ] .

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