## Degree of phase-space separability of statistical pulses |

Optics Express, Vol. 20, Issue 3, pp. 2548-2555 (2012)

http://dx.doi.org/10.1364/OE.20.002548

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### 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

## 1. Introduction: Wigner function description of statistical pulses

16. B. H. Kolner and M. Nazarathy, “Temporal imaging with a time lens,” Opt. Lett. **14**, 630–632 (1989). [CrossRef] [PubMed]

*E*(

*t*)} and decompose the electric field

*E*(

*t*) into a slowly-varying envelope

*U*(

*t*) and a carrier wave [17] 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. The Wigner distribution (WD) of the ensemble is then defined as where we introduced the variables

*t*= (

*t*

_{1}+

*t*

_{2})/2 and

*τ*=

*t*

_{1}–

*t*

_{2}. The intensity

*I*(

*t*) and spectrum

*S*(

*ω*) of the pulse are determined by the appropriate marginals of the Wigner distribution viz., Many phase space characteristics of the pulses, such as the position of the pulse center (in time), central frequency of the envelope, rms temporal and spectral widths etc., can be determined as the corresponding moments of

*I*or

*S*. Notice also that it follows at once from Eqs. (2) and (3) that fully as well as partially temporarily and/or spectrally coherent pulses can be treated in the same way using the Wigner distribution, the fully coherent case being just the limiting situation when the cross-correlation function factorizes.

## 2. Quantifying phase-space separability of statistical pulses

*λ*} and eigenfunctions, {

_{n}*χ*} and {

_{n}*ϕ*}, are real due to reality of the WD. It is known [25] that the two sets of eigenfunctions obey the Fredholm integral equations which, in our case, take the form and The suitably normalized eigenfunctions form an orthonormal set in the sense that

_{n}*ν*(

_{n}*z*)’s such that

*ρ*(

*z*) ≤ 1, attaining its maximum if there is only the lowest-order eigenvalue present in the expansion (13). This corresponds to the ideal case of complete separability of the WD. Thus the proposed measure conforms to our intuitive perception of the degree of separability.

## 3. Degree of separability of chirped Gaussian Schell-model (CGSM) pulses

*t*is a characteristic pulse width,

_{p}*t*is a pulse coherence time,

_{c}*C*is a dimensionless chirp parameter, and Γ

_{00}is a normalization constant. Using the definition of WD (3), we can express the WD of a CGSM pulse as where It can be readily inferred from Eq. (19) that because of phase chirping, the WD of a CGSM source is not separable.

*σ*(

*z*) given by and the effective chirp parameter

*C*(

*z*) as

*t*→ ∞. The evolution scenario of

_{c}*σ*depends on the sign of

*Cβ*

_{2}. If

*Cβ*

_{2}≥ 0,

*σ*increases monotonously with the propagation distance. On the other hand, if

*Cβ*

_{2}< 0, the propagation factor attains a minimum, at the distance This behavior is qualitatively sketched in Fig. 1. We also note that at

*z*

_{*}the effective chirp is equal to zero–the accrued chirp on propagation in the medium unchirps the initial chirp of the opposite sign imposed by the time lens.

*ω*. We arrive, after straightforward algebra, at the expression It is clear from Eq. (26) that the propagation factor

*σ*governs the dynamics of the pulse width and coherence time.

*ζ*| ≤ 1. Next, we introduce the scaling factors

*a*(

*z*) and

*b*(

*z*) such that in the scaled variables

*T̃*=

*t/a*(

*z*) and

*ω*̃ =

*ω/b*(

*z*), the WD of a CGSM takes the form On comparing Eqs. (28) and (27), we conclude that the WD in the original variables can be expanded into a bi-orthogonal series (13) with the eigenvalues and the eigenfunctions and In Eqs. (30) and (31), the scaling factors are given by the expressions and where and positive roots are assumed in Eqs. (32) and (33). We note in passing that the bi-orthogonal decomposition (13) should not be confused with the more familiar coherent-mode expansion of a GSM source [28

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]

*χ*} and {

_{n}*ϕ*} belong to different domains which is formally reflected in quite different scaling of

_{n}*a*and

*b*with the propagation distance.

*ρ*as a function of the propagation distance qualitatively depends on the sign of the initial chirp. If

*C*≥ 0, the degree of phase separability monotonously decreases with the propagation distance. If, on the other hand,

*C*< 0,

*ρ*goes through a maximum attained at

*z*=

*z*

_{*}. These scenarios are exhibited in Fig. 2 where we present the behavior of

*ρ*as a function of dimensionless propagation distance in the coherent case using three values of the chirp,

*C*= 0 and

*C*= ±1 for illustration. The influence of pulse coherence time on the evolution of

*ρ*is illustrated in Fig. 3 for several values of

*t*and

_{c}*C*= −1. Interestingly, the WD in the CGSM case becomes separable precisely at the distance where the pulses are the most compressed and least coherent. Chirping GSM pulses with a time lens then provides a powerful tool to control pulse coherence and phase-space separability at the same time.

## References and links

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

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

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

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

5. | V. Torres-Company, H. Lajunen, and A. T. Friberg, “Cohrence theory of noise in ultrashort pulse trains,” J. Opt. Soc. Am. B |

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

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

8. | A. T. Friberg, H. Lajunen, and V. Torres-Company, “Spectral elementary-coherence-function representation for partially coherent light pulses,” Opt. Express |

9. | J. Ojeda-Castaneda, J. Lancis, C. M. Gomez-Sarabia, V. Torres-Company, and P. Andrés, “Ambuguity function analysis of pulse train propagation: applications to temporal Lau filtering,” J. Opt. Soc. Am. A |

10. | M. A. Alonso, “Wigner functions in optics: describing beams as ray bundles and pulses as particle ensembles,” Adv. Opt. Photon. |

11. | S. A. Ponomarenko, “Complex Gaussian representation of statistical pulses,” Opt. Express |

12. | G. P. Agrawal, |

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

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

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 |

16. | B. H. Kolner and M. Nazarathy, “Temporal imaging with a time lens,” Opt. Lett. |

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, |

19. | M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. |

20. | M. J. Bastiaans, “The Wigner function and its application to first-order optics,” J. Opt. Soc. Am. |

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

22. | J. Lancis, V. Torres-Company, E. Silvestre, and P. Andrés, “Space-time analogy for partially coherent plane-wave-type pulses,” Opt. Lett. |

23. | V. Torres-Company, J. Lancis, and P. Andrés, “Space-time analogies in optics,” Prog. Opt. |

24. | M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. |

25. | P. M. Morse and H. Feshbach, |

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

27. | M. Abramowitz and I. A. Stegan, |

28. | F. Gori, “Collett-Wolf sources and multimode lasers,” Opt. Commun. |

29. | A. Starikov and E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and their radiation fields,” J. Opt. Soc. Am. |

**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: December 13, 2011

Revised Manuscript: January 9, 2012

Manuscript Accepted: January 10, 2012

Published: January 19, 2012

**Citation**

Sergey A. Ponomarenko, "Degree of phase-space separability of statistical pulses," Opt. Express **20**, 2548-2555 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-3-2548

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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]
- 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. Express11, 1894–1899 (2003). [CrossRef] [PubMed]
- 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]
- V. Torres-Company, H. Lajunen, and A. T. Friberg, “Cohrence theory of noise in ultrashort pulse trains,” J. Opt. Soc. Am. B24, 1441–1450 (2007). [CrossRef]
- B. Davis, “Measurable coherence theory for statistically periodic fields,” Phys. Rev. A76, 043843 (2007). [CrossRef]
- P. Vahimaa and J. Turunen, “Independent-elementary-pulse representation for non-stationary fields,” Opt. Express14, 5007–5012 (2006). [CrossRef] [PubMed]
- A. T. Friberg, H. Lajunen, and V. Torres-Company, “Spectral elementary-coherence-function representation for partially coherent light pulses,” Opt. Express15, 5160–5165 (2007). [CrossRef] [PubMed]
- J. Ojeda-Castaneda, J. Lancis, C. M. Gomez-Sarabia, V. Torres-Company, and P. Andrés, “Ambuguity function analysis of pulse train propagation: applications to temporal Lau filtering,” J. Opt. Soc. Am. A242268–2273 (2007). [CrossRef]
- M. A. Alonso, “Wigner functions in optics: describing beams as ray bundles and pulses as particle ensembles,” Adv. Opt. Photon.3, 272–365 (2011). [CrossRef]
- S. A. Ponomarenko, “Complex Gaussian representation of statistical pulses,” Opt. Express19, 17086–17091 (2011). [CrossRef] [PubMed]
- G. P. Agrawal, Fiber-Optic Communication Systems, 3rd ed., (Wiley, New York, NY2002). [CrossRef]
- 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]
- 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. Express18, 14979–14991 (2011). [CrossRef]
- B. H. Kolner and M. Nazarathy, “Temporal imaging with a time lens,” Opt. Lett.14, 630–632 (1989). [CrossRef] [PubMed]
- 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).
- M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun.25, 26–30, (1978). [CrossRef]
- M. J. Bastiaans, “The Wigner function and its application to first-order optics,” J. Opt. Soc. Am.69, 1710–1716 (1979). [CrossRef]
- J. C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena, 2nd ed. (Academic Press, Amsterdam2006).
- 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]
- V. Torres-Company, J. Lancis, and P. Andrés, “Space-time analogies in optics,” Prog. Opt.56, 1–80 (2011), ed. E. Wolf. [CrossRef]
- 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]
- 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).
- F. Gori, “Collett-Wolf sources and multimode lasers,” Opt. Commun.34, 301–305 (1980). [CrossRef]
- 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]

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