## Non-uniformly correlated partially coherent pulses |

Optics Express, Vol. 21, Issue 1, pp. 190-195 (2013)

http://dx.doi.org/10.1364/OE.21.000190

Acrobat PDF (791 KB)

### Abstract

We consider partially coherent plane-wave pulses with non-uniform correlation distributions and study their propagation in linear second-order dispersive media. Particular models for coherence functions are introduced both in time and frequency domains. It is shown that the maximum peak of the pulse energy can be accelerating or decelerating and also self-focusing effects are possible due to coherence-induced propagation effects.

© 2013 OSA

## 1. Introduction

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

4. H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, “Spectrally partially coherent pulse trains in dispersive media,” Opt. Commun. **255**, 12–22 (2005). [CrossRef]

5. H. Lajunen, V. Torres-Company, J. Lancis, E. Silvestre, and P. Andres, “Pulse-by-pulse method to characterize partially coherent pulse propagation in instantaneous nonlinear media,” Opt. Express **18**, 14979–14991 (2010). [CrossRef] [PubMed]

6. L. Mokhtarpour and S. A. Ponomarenko, “Complex area correlation theorem for statistical pulses in coherent linear absorbers,” Opt. Lett. **37**, 3498–3500 (2012). [CrossRef] [PubMed]

7. C. Ding, Y. Cai, O. Korotkova, Y. Zhang, and L. Pan, “Scattering-induced changes in the temporal coherence length and the pulse duration of a partially coherent plane-wave pulse,” Opt. Lett. **36**, 517–519 (2011). [CrossRef] [PubMed]

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. K. Saastamoinen, J. Turunen, P. Vahimaa, and A. T. Friberg, “Spectrally partially coherent propagation-invariant fields,” Phys. Rev. A **80**, 053804 (2009). [CrossRef]

10. L. Mokhtarpour, G. H. Akter, and S. A. Ponomarenko, “Partially coherent self-similar pulses in resonant linear absorbers,” Opt. Express **20**, 17816–17822 (2012). [CrossRef] [PubMed]

11. V. Torres-Company, H. Lajunen, and A. T. Friberg, “Coherence theory of noise in ultrashort-pulse trains,” J. Opt. Soc. Am. B **24**, 1441–1450 (2007). [CrossRef]

5. H. Lajunen, V. Torres-Company, J. Lancis, E. Silvestre, and P. Andres, “Pulse-by-pulse method to characterize partially coherent pulse propagation in instantaneous nonlinear media,” Opt. Express **18**, 14979–14991 (2010). [CrossRef] [PubMed]

12. V. Torres-Company, G. Minguez-Vega, J. Lancis, and A. T. Friberg, “Controllable generation of partially coherent light pulses with direct space-to-time pulse shaper,” Opt. Lett. **32**, 1608–1610 (2007). [CrossRef] [PubMed]

13. H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. **36**, 4104–4106 (2011). [CrossRef] [PubMed]

14. Z. Tong and O. Korotkova, “Nonuniformly correlated light beams in uniformly correlated media,” Opt. Lett. **37**, 3240–3242 (2012). [CrossRef] [PubMed]

15. Z. Tong and O. Korotkova, “Electromagnetic nonuniformly correlated beams,” J. Opt. Soc. Am. A **29**, 2154–2158 (2012). [CrossRef]

16. J. Lancis, V. Torres-Company, E. Silvestre, and P. Andres, “Space-time analogy for partially coherent plane-wave-type pulses,” Opt. Lett. **30**, 2973–2975 (2005). [CrossRef] [PubMed]

13. H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. **36**, 4104–4106 (2011). [CrossRef] [PubMed]

## 2. Theory

*z*-axis. In the space-time domain the coherence properties of the pulses can be defined by their mutual coherence function Γ(

*t*

_{1},

*t*

_{2}) = 〉

*U*

^{*}(

*t*

_{1})

*U*(

*t*

_{2})〉, where

*U*(

*t*) represents the complex analytic signal of pulse realizations at time

*t*, and the angle brackets denote ensemble average. The average intensity of the partially coherent pulses is then obtained as

*I*(

*t*) = Γ(

*t,t*). The degree of coherence of the pulses is defined as the normalized form of the mutual coherence function,

*t*

_{1},

*t*

_{2}) must correspond to a non-negative definite kernel [1]. As has been shown for correlation functions in the spatial domain [17

17. F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. **32**, 3531–3533 (2007). [CrossRef] [PubMed]

*V*is an arbitrary kernel and

*p*is an arbitrary non-negative function. Starting from this condition, a wide variety of different temporal correlation distributions can be defined for partially coherent pulses.

*W*(

*ω*

_{1},

*ω*

_{2}) that measures the correlations between two angular frequencies

*ω*

_{1}and

*ω*

_{2}. Here, we present all the frequencies with respect to the central frequency of the pulses

*ω*

_{0}, i.e. the real frequencies are obtained by

*ω*′ =

*ω*+

*ω*

_{0}.

*W*is connected to Γ through the generalized Wiener-Khintchine theorem [1], and to the average spectrum of the pulses as

*S*(

*ω*) =

*W*(

*ω*,

*ω*). Further, the degree of spectral coherence is given analogously to the corresponding time-domain function by

*V*(

*t*,

*v*) has inverse Fourier transform with respect to the variable

*t*,

*V*̃(

*ω*,

*v*), we readily get This proves that for partially coherent fields defined by Eq. (1) the non-negative definiteness condition is naturally fulfilled also in the frequency domain. On the other hand, it is equally possible to first define the correlation function in the space-frequency domain by choosing kernels

*V*̃ and weight function

*p*in Eq. (3). It should be noted that even though Eqs. (1) and (3) are formally analogous, they do not generally result in a pair of corresponding coherence functions that have similar forms in the time and frequency domain.

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

19. A. T. Friberg, H. Lajunen, and V. Torres-Company, “Spectral elementary-coherence-function representation for partially coherent light pulses,” Opt. Express **15**, 5160–5165 (2007). [CrossRef] [PubMed]

*V*(

*t*,

*v*) =

*V*(

*t*−

*v*) or

*V*̃(

*ω*,

*v*) =

*V*̃(

*ω*−

*v*). Allowing the elementary fields to change as a function of

*v*we can define more versatile coherence functions that do not necessarily follow the basic Schell-model.

## 3. Non-uniformly correlated temporally Gaussian pulses

*a*and

*κ*are positive real constants and

*t*

_{c}is a real constant, into Eq. (1) we get mutual coherence function where

*ivt*), we would get conventional GSM pulses. Based on the definition of the kernel function, Eq. (5), the correlation function is obtained as a weighted superposition of linearly chirped Gaussian modes for which the amount and sign of chirp changes as a function of the variable

*v*. In addition, the chirping is decentered compared to maximum intensity of the modes by the time constant

*t*

_{c}. This leads to a situation where the correlations are highest around

*t*

_{c}. An example of the shape of the mutual coherence function and the corresponding degree of coherence are illustrated in Fig. 1. The intensity of the defined partially coherent pulses has Gaussian shape, as can easily be seen by definition from Eq. (6).

*v*and the parameter

*t*

_{c}. Further, because of the more complicated dependence on

*v*, we cannot get an analytical closed form solution for the integral of Eq. (3). However,

*W*(

*ω*

_{1},

*ω*

_{2}) can be easily calculated numerically using Eq. (2). The resulting cross-spectral density and the degree of spectral coherence of the partially coherent pulses are also illustrated in Fig. 1, which shows a clear difference to the corresponding coherence functions in the time domain. Unlike the intensity, the spectrum is not Gaussian but it has a slightly sharper line shape.

*β*

_{2}, the trajectory of the intensity distribution is reversed and the pulse maximum is thus decelerating. Similar change is observed if we choose a positive value for the parameter

*t*

_{c}in normally dispersive media. Likewise, a positive

*t*

_{c}in anomalously dispersive media leads to similar accelerating as shown in Fig. 2. Thus, by choosing properly the parameter

*t*

_{c}, we can obtain accelerating or decelerating behavior in both types of dispersive media.

## 4. Non-uniformly correlated spectrally Gaussian pulses

13. H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. **36**, 4104–4106 (2011). [CrossRef] [PubMed]

*ω*

_{c}. Identically to the previous section, we get where

*ω*

_{c}affects to the angle at which the peak intensity is directed. However, when the pulses spread upon further propagation, their maximum finally returns to the center. Similar considerations as above about changing the sign of parameter

*ω*

_{c}or the type of dispersion for controlling which way the maximum is shifted are valid also in this case.

## 5. Conclusions

## Acknowledgments

## References and links

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

2. | G. P. Agrawal, |

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

4. | H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, “Spectrally partially coherent pulse trains in dispersive media,” Opt. Commun. |

5. | H. Lajunen, V. Torres-Company, J. Lancis, E. Silvestre, and P. Andres, “Pulse-by-pulse method to characterize partially coherent pulse propagation in instantaneous nonlinear media,” Opt. Express |

6. | L. Mokhtarpour and S. A. Ponomarenko, “Complex area correlation theorem for statistical pulses in coherent linear absorbers,” Opt. Lett. |

7. | C. Ding, Y. Cai, O. Korotkova, Y. Zhang, and L. Pan, “Scattering-induced changes in the temporal coherence length and the pulse duration of a partially coherent plane-wave pulse,” Opt. Lett. |

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

9. | K. Saastamoinen, J. Turunen, P. Vahimaa, and A. T. Friberg, “Spectrally partially coherent propagation-invariant fields,” Phys. Rev. A |

10. | L. Mokhtarpour, G. H. Akter, and S. A. Ponomarenko, “Partially coherent self-similar pulses in resonant linear absorbers,” Opt. Express |

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

12. | V. Torres-Company, G. Minguez-Vega, J. Lancis, and A. T. Friberg, “Controllable generation of partially coherent light pulses with direct space-to-time pulse shaper,” Opt. Lett. |

13. | H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. |

14. | Z. Tong and O. Korotkova, “Nonuniformly correlated light beams in uniformly correlated media,” Opt. Lett. |

15. | Z. Tong and O. Korotkova, “Electromagnetic nonuniformly correlated beams,” J. Opt. Soc. Am. A |

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

17. | F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. |

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

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

20. | I. S. Gradshteyn and I. M. Ryzhik, |

**OCIS Codes**

(030.1640) Coherence and statistical optics : Coherence

(030.6600) Coherence and statistical optics : Statistical optics

(320.5550) Ultrafast optics : Pulses

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: October 26, 2012

Revised Manuscript: November 26, 2012

Manuscript Accepted: December 1, 2012

Published: January 3, 2013

**Citation**

Hanna Lajunen and Toni Saastamoinen, "Non-uniformly correlated partially coherent pulses," Opt. Express **21**, 190-195 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-1-190

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### References

- L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, UK, 1995).
- G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 1995).
- Q. Lin, L. G. Wang, and S. Y. Zhu, “Partially coherent light pulse and its propagation,” Opt. Commun.219, 65–70 (2003). [CrossRef]
- H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, “Spectrally partially coherent pulse trains in dispersive media,” Opt. Commun.255, 12–22 (2005). [CrossRef]
- H. Lajunen, V. Torres-Company, J. Lancis, E. Silvestre, and P. Andres, “Pulse-by-pulse method to characterize partially coherent pulse propagation in instantaneous nonlinear media,” Opt. Express18, 14979–14991 (2010). [CrossRef] [PubMed]
- L. Mokhtarpour and S. A. Ponomarenko, “Complex area correlation theorem for statistical pulses in coherent linear absorbers,” Opt. Lett.37, 3498–3500 (2012). [CrossRef] [PubMed]
- C. Ding, Y. Cai, O. Korotkova, Y. Zhang, and L. Pan, “Scattering-induced changes in the temporal coherence length and the pulse duration of a partially coherent plane-wave pulse,” Opt. Lett.36, 517–519 (2011). [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]
- K. Saastamoinen, J. Turunen, P. Vahimaa, and A. T. Friberg, “Spectrally partially coherent propagation-invariant fields,” Phys. Rev. A80, 053804 (2009). [CrossRef]
- L. Mokhtarpour, G. H. Akter, and S. A. Ponomarenko, “Partially coherent self-similar pulses in resonant linear absorbers,” Opt. Express20, 17816–17822 (2012). [CrossRef] [PubMed]
- V. Torres-Company, H. Lajunen, and A. T. Friberg, “Coherence theory of noise in ultrashort-pulse trains,” J. Opt. Soc. Am. B24, 1441–1450 (2007). [CrossRef]
- V. Torres-Company, G. Minguez-Vega, J. Lancis, and A. T. Friberg, “Controllable generation of partially coherent light pulses with direct space-to-time pulse shaper,” Opt. Lett.32, 1608–1610 (2007). [CrossRef] [PubMed]
- H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett.36, 4104–4106 (2011). [CrossRef] [PubMed]
- Z. Tong and O. Korotkova, “Nonuniformly correlated light beams in uniformly correlated media,” Opt. Lett.37, 3240–3242 (2012). [CrossRef] [PubMed]
- Z. Tong and O. Korotkova, “Electromagnetic nonuniformly correlated beams,” J. Opt. Soc. Am. A29, 2154–2158 (2012). [CrossRef]
- J. Lancis, V. Torres-Company, E. Silvestre, and P. Andres, “Space-time analogy for partially coherent plane-wave-type pulses,” Opt. Lett.30, 2973–2975 (2005). [CrossRef] [PubMed]
- F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett.32, 3531–3533 (2007). [CrossRef] [PubMed]
- 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]
- I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products7th ed. (Academic Press, 2007).

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