## Complex Gaussian representation of statistical pulses |

Optics Express, Vol. 19, Issue 18, pp. 17086-17091 (2011)

http://dx.doi.org/10.1364/OE.19.017086

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### 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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## 1. P-representation of statistical pulses: introduction and preliminaries

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

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]

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]

*t*Gaussian pulse with the carrier frequency shifted to

_{s}*ω*; the pulse has the temporal profile where

_{s}*A*and

*t*are a real amplitude and width of the pulse. Transforming to dimensionless variables,

_{*}*T*=

*t/t*,

_{*}*T*=

_{s}*t*/

_{s}*t*, and Ω

_{*}*=*

_{s}*ω*

_{s}t_{*}–which we are going to use hereafter unless we indicate otherwise–we obtain, after elementary algebra, the following expression Here the complex displacement conveniently combines time delay and frequency shift viz., In Eq. (2) we chose the amplitude

*A*such that the pulse profile function is normalized to unity:

*n*〉 can be expressed in the coordinate representation as Performing summation over

*n*on the r.h.s of Eq. (6) with the aid of Eq. (7) and the generating function for Hermite polynomials

*H*(

_{n}*x*) in the form [14] we arrive at the final expression for the normalized coherent state wave function in the coordinate representation as Equation (9) is identical with Eq. (2) apart from the notation. Thus, the (normalized) complex Gaussian pulses have the same profiles as the coherent states.

*T*| and |

*T*〉, we can re-write the completeness relation explicitly in the temporal representation as where we denoted

*ψ*(

_{α}*T*) = 〈

*T*|

*α*〉.

## 2. P-representation of statistical pulses: general formalism

*E*(

*T*)}. Hereafter, we find it convenient to decompose the electric field

*E*(

*T*) into a (usually) slowly-varying envelope

*U*(

*T*) and a carrier wave [15] such that where Ω

*is a deterministic carrier frequency of the pulse. The second-order statistical properties of the ensemble {*

_{c}*U*(

*T*)} are specified by the cross-correlation function where the angle brackets denote ensemble averaging. Similarly to the spatial case [16

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]

*𝒫*- representation of the quantum density operator [3]; it can be written in the matrix form as implying in more “classical” notations that Notice that one can formally invert Eq. (17) to determine the classical P-distribution in terms of the two-time correlation function. With this purpose, we can again borrow the known expression for

*𝒫*(

*α*) in the matrix form [3] and expand the rhs in terms of complete sets of time ket-vectors to yield In principle, Eqs. (17) and (19) solve the problem of finding the appropriate complex Gaussian representation for any statistical pulse. In practice, of course, the integrals in Eq. (19) can fail to converge in the space of ordinary functions, making the P-representation cumbersome in the case.

10. P. Vahimaa and J. Turunen, “Independent-elementary-pulse representation for non-stationary fields,” Opt. Express **14**, 5007–5012 (2006). [CrossRef] [PubMed]

## 3. Examples and discussion

*c*(

*α*)}. In particular, considering with

*α*=

_{n}*n*(

*t*

_{0}/

*t*+

_{*}*iω*

_{0}

*t*)/ 2, we obtain a train of coherent Gaussian pulses with the field profile Equation (34) represents an ideal train of identical coherent Gaussian pulses provided that

_{*}*c*=

_{n}*c*

_{0}=

*const*,

*ω*

_{0}= 2

*π*/

*t*

_{0}and

*t*=

_{*}*t*

_{0}/

*N*.

*P*-representation of quantum optics–of the cross-correlation function of any statistical pulse in terms of complex Gaussian pulses with the appropriately distributed emission times and carrier frequencies. We showed how the complex Gaussian representation can describe statistical features of Gaussian Schell-model pulses and the output of realistic mode-locked lasers. The new representation is anticipated to find applications in ultrafast optics and temporal imaging with ultrashort pulses.

## References and links

1. | T. Brabec and F. Krausz, “Intense few-cycle laser fields: Frontiers of nonlinear optics,” Rev. Mod. Phys. |

2. | E. Wolf, |

3. | L. Mandel and E. Wolf, |

4. | M. Bertolotti, A. Ferrari, and L. Sereda, “Coherence properties of nonstationary polychromatic light sources,” J. Opt. Soc. Am. B |

5. | L. Sereda, M. Bertolotti, and A. Ferrari, “Coherence properties of nonstationary light wave fields,” J. Opt. Soc. Am. B |

6. | S. A. Ponomarenko, G. P. Agrawal, and E. Wolf, “Energy spectrum of a nonstationary ensemble of pulses,” Opt. Lett. |

7. | B. Davis, “Measurable coherence theory for statistically periodic fields,” Phys. Rev. A |

8. | P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. |

9. | H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, “Spectral coherence properties of temporarily modulated stationary light sources,” Opt. Express |

10. | P. Vahimaa and J. Turunen, “Independent-elementary-pulse representation for non-stationary fields,” Opt. Express |

11. | Q. Lin, L. Wang, and S. Zhu, “Partially coherent light pulse and its propagation,” Opt. Commun. |

12. | M. Brunel and S. Coëtlemec, “Fractional-order Fourier formulation of the propagation of partially coherent light pulses,” Opt. Commun. |

13. | R. W. Schoonover, B. J. Davis, and P. S. Carney, “The generalized Wolf shift for cyclostationary fields,” Opt. Express |

14. | G. Arfken, |

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. |

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. |

18. | P. W. Milonni and J. H. Eberly, |

19. | J. C. Diels and W. Rudolph, |

**OCIS Codes**

(030.0030) Coherence and statistical optics : Coherence and statistical optics

(320.0320) Ultrafast optics : Ultrafast optics

(320.5550) Ultrafast optics : Pulses

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: July 8, 2011

Revised Manuscript: July 29, 2011

Manuscript Accepted: July 29, 2011

Published: August 16, 2011

**Citation**

Sergey A. Ponomarenko, "Complex Gaussian representation of statistical pulses," Opt. Express **19**, 17086-17091 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-18-17086

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### References

- T. Brabec and F. Krausz, “Intense few-cycle laser fields: Frontiers of nonlinear optics,” Rev. Mod. Phys. 72, 545–591 (2000). [CrossRef]
- 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).
- 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]
- 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]
- B. Davis, “Measurable coherence theory for statistically periodic fields,” Phys. Rev. A 76, 043843 (2007). [CrossRef]
- P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53–58 (2002). [CrossRef]
- 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. Vahimaa and J. Turunen, “Independent-elementary-pulse representation for non-stationary fields,” Opt. Express 14, 5007–5012 (2006). [CrossRef] [PubMed]
- 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]
- 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]
- 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.
- 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]
- 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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