## Partially coherent self-similar pulses in resonant linear absorbers |

Optics Express, Vol. 20, Issue 16, pp. 17816-17822 (2012)

http://dx.doi.org/10.1364/OE.20.017816

Acrobat PDF (891 KB)

### Abstract

We theoretically describe several classes of ultrashort partially coherent pulses that maintain their shape on propagation in coherent linear absorbers near optical resonance.

© 2012 OSA

## 1. Introduction

1. G. P. Agrawal, *Fiber-Optic Communication Systems*, 3rd ed. (Wiley, 2002). [CrossRef]

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

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

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

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

5. 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]

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

10. S. A. Ponomarenko, “Complex Gaussian representation of statistical pulses,” Opt. Express **19**, 17086–17091 (2011). [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]

14. S. A. Ponomarenko, “Degree of phase-space separability of statistical pulses,” Opt. Express **20**, 2548–2555 (2012). [CrossRef] [PubMed]

15. H. Lajunen, V. Torres-Company, J. Lancis, E. Silvestre, and P. Andrés, “Pulse-by-pulse method to characterize partially coherent pulse propagation in instantaneous nonlinear media,” Opt. Express **18**, 14979–14991 (2011). [CrossRef]

16. S. Haghgoo and S. A. Ponomarenko, “Self-similar pulses in coherent linear amplifiers,” Opt. Express **19**, 9750–9758 (2011). [CrossRef] [PubMed]

17. S. Haghgoo and S. A. Ponomarenko, “Shape-invariant pulses in resonant linear absorbers,” Opt. Lett. **37**, 1328–1330 (2012). [CrossRef] [PubMed]

19. H. Lajunen, J. Tervo, and P. Vahimaa, “Overall coherence and coherent-mode expansion of spectrally partially coherent plane-wave pulses,” J. Opt. Soc. Am. A **21**, 2117–2123 (2004). [CrossRef]

## 2. Partially coherent self-similar pulses in coherent linear absorbers

*N*∼ 10

^{11}cm

^{−3}at room temperature at the pressure of

*P*= 0.1 Torr, say. One can then estimate a characteristic spectral width due to collision broadening as

*δν*∼ 10

_{c}^{3}MHz [20]. Assuming further that dipole relaxation is mainly due to collisions, we can estimate a typical dipole relaxation time,

*L*=

_{A}*α*

^{−1}≃ 2 mm, where

*α*=

*NT*

_{⊥}

*e*

^{2}/2

*ε*

_{0}

*mc*is a small-signal absorption coefficient. Thus, the proposed self-similar pulses can be realized with nanosecond small-area input pulses in a few-centimeter long cell filled with the homogeneously broadened dilute vapor.

17. S. Haghgoo and S. A. Ponomarenko, “Shape-invariant pulses in resonant linear absorbers,” Opt. Lett. **37**, 1328–1330 (2012). [CrossRef] [PubMed]

*α*is a small-signal inverse absorption length,

*T*

_{⊥}is an individual dipole relaxation time, and

*ζ*

_{0}determines the spectral mode profile at the source. Introducing dimensionless variables Ω =

*ωT*

_{⊥}and

*Z*=

*αζ*and restricting ourselves to the integer index modes (

*s*=

*n*), we can rewrite the spectral field amplitude of each shape-invariant mode in the form

*λ*≥ 0 and the angle brackets indicate ensemble averaging. The second-order statistical properties of pulses in the spectral domain are described by the cross-spectral density distribution defined as It follows from Eqs. (4) and (5) that the cross-spectral density can then be expressed as a Mercer-type series in the form [18] Although a multitude of partially coherent shape-invariant pulses can be represented by the expansion (6), closed-form results can only be obtained for a few classes of pulses. In the following sections we consider two such cases.

_{n}## 3. Pulses with the power-law distribution of modal weights

*λ*}’s have a power distribution: where

_{n}*𝒜*> 0 is a normalization constant specifying the overall intensity of a partially coherent pulse, and

*λ*≥ 0. It follows at once from Eqs. (2), (6), and (7) that the spectrum of the pulse, defined as

*S*(Ω,

*Z*) ≡

*W*(Ω, Ω,

*Z*) [6

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]

*λ*< 2/

*Z*

_{0}is imposed. The latter is rather stringent as it stipulates that

*λ*be fairly small for sufficiently large

*Z*

_{0}. Notice also that the spectrum (8) has the shape reminiscent of the zero-index mode spectrum |ℰ̃

_{0}(Ω,

*Z*)|

^{2}of Ref. [17

17. S. Haghgoo and S. A. Ponomarenko, “Shape-invariant pulses in resonant linear absorbers,” Opt. Lett. **37**, 1328–1330 (2012). [CrossRef] [PubMed]

*λ*and

*Z*

_{0}. In particular, the pulse spectrum at the source can have either a hole, or a dip, or else a peak at the center, depending on

*λ*and

*Z*

_{0}. The situation is illustrated in Fig. 1 where the pulse spectrum evolution is displayed for three sets of parameters: (a)

*λ*= 0.1,

*Z*

_{0}= 5; (b)

*λ*= 0.3,

*Z*

_{0}= 3, and (c)

*λ*= 10,

*Z*

_{0}= 0.1. As is seen in Fig. 1(a), the spectrum with a hole at the center propagates in a self-similar fashion from the outset. Whenever there is only a dip at the center of the incident pulse–as is shown in Fig. 1(b)–the dip deepens on propagation until a hole is burnt at the center and the pulse enters its self-similar evolution stage. If, on the other hand, the spectrum has a central peak (see Fig. 1(c)), the latter eventually transforms into a hole with the subsequent self-similar pulse evolution.

*λ*= 0.1,

*Z*

_{0}= 5 and (b)

*λ*= 10,

*Z*

_{0}= 0.1. As is seen in the figure, the spectral coherence properties of the pulses are nonuniform and dependent on the values of parameters. To illustrate these points, we display in Fig. 3 |

*μ*| as a function of Ω

_{1}, say, for fixed Ω

_{2}. It is seen in the figure that the degree of coherence can have a local maximum or minimum at the center, Ω

_{1}= 0, depending on the position of the other spectral point. The magnitudes of the maxima and minima depend on the values of the other parameters.

## 4. Pulses with modal weight distributions decaying faster than the power-law

*ℬ*being a positive normalization constant. It can be inferred from Eqs. (2), (6), and (10) using the representation for the zero-order modified Bessel function [21], that the partially coherent pulse spectrum in the case takes the form We conclude by examining Eq. (12) that provided

*λ*< 1, the spectral hole or dip presence at the pulse center in the source plane depends on the values of

*λ*and

*Z*

_{0}. However, unlike in the previously considered case, if

*λ*> 1, there can be no spectral hole at the source.

*λ*= 0.9,

*Z*

_{0}= 0.1; (b)

*λ*= 0.9,

*Z*

_{0}= 15; (c)

*λ*= 2,

*Z*

_{0}= 0.1, and (d)

*λ*= 2,

*Z*

_{0}= 15. It is seen by comparing Figs. 4(a) and 4(c) that for sufficiently small

*Z*

_{0}= 0.1, there is a peak at the center in the source plane in both figures. At the same time, as

*Z*

_{0}increases to 15, say, a hole is formed at the center for

*λ*= 0.9 < 1 as is seen in Fig. 4(b). Yet, the pulse spectrum has a peak at the center for

*λ*= 2 > 1 as shown in Fig. 4(d). Our numerical simulations indicate that no matter how close the value of

*λ*approaches unity from the above, only a dip but not a hole can be formed at the center of the pulse spectrum.

*Z*

_{0}= 15 with

*λ*= 0.9 (left) and

*λ*= 2 (right). Note that there is no constraint on the magnitude of

*λ*in Eq. (12) which makes this class of pulses wider than the previously considered one. On comparing Fig. 2 and Fig. 5, we observe that first, the spectral degree of coherence is rotationally symmetric in the former figure while the rotational symmetry is broken in the latter. Second, we notice that the quantitative dependence of |

*μ*| on the parameters is stronger for the second class of pulses than is for the first. To bring these points home, we exhibit in Fig. 6 the cross-sectional plot of |

*μ*| for a couple fixed values of one of the frequencies, Ω

_{2}= −15 in Fig. 6(a) and Ω

_{2}= 0, in Fig. 6(b). On comparing Figs. 3 and 6, we can see that not only does the qualitative behavior of |

*μ*| richer in Fig. 6, but pulse coherence properties can be tuned within a wider range for the second class of pulses than for the first one.

## References and links

1. | G. P. Agrawal, |

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

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

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

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

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

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

10. | S. A. Ponomarenko, “Complex Gaussian representation of statistical pulses,” 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. | E. Wolf, |

14. | S. A. Ponomarenko, “Degree of phase-space separability of statistical pulses,” Opt. Express |

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

16. | S. Haghgoo and S. A. Ponomarenko, “Self-similar pulses in coherent linear amplifiers,” Opt. Express |

17. | S. Haghgoo and S. A. Ponomarenko, “Shape-invariant pulses in resonant linear absorbers,” Opt. Lett. |

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

19. | H. Lajunen, J. Tervo, and P. Vahimaa, “Overall coherence and coherent-mode expansion of spectrally partially coherent plane-wave pulses,” J. Opt. Soc. Am. A |

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

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

**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: May 14, 2012

Revised Manuscript: July 11, 2012

Manuscript Accepted: July 14, 2012

Published: July 20, 2012

**Citation**

L. Mokhtarpour, G. H. Akter, and S. A. Ponomarenko, "Partially coherent self-similar pulses in resonant linear absorbers," Opt. Express **20**, 17816-17822 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-16-17816

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### References

- G. P. Agrawal, Fiber-Optic Communication Systems, 3rd ed. (Wiley, 2002). [CrossRef]
- M. Bertolotti, A. Ferrari, and L. Sereda, “Coherence properties of nonstationary polychromatic light sources,” J. Opt. Soc. Am. B12, 341–347 (1995). [CrossRef]
- L. Sereda, M. Bertolotti, and A. Ferrari, “Coherence properties of nonstationary light wave fields,” J. Opt. Soc. Am. B15, 695–705 (1998). [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]
- 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. Lahunen, and V. Torres-Company, “Spectral elementary-coherence-function representation for partially coherent light pulses,” Opt. Express15, 5160–5165 (2007). [CrossRef] [PubMed]
- S. A. Ponomarenko, “Complex Gaussian representation of statistical pulses,” Opt. Express19, 17086–17091 (2011). [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]
- E. Wolf, Introduction to the Theory of Coherence and Polarization (Cambridge University Press, 2007).
- S. A. Ponomarenko, “Degree of phase-space separability of statistical pulses,” Opt. Express20, 2548–2555 (2012). [CrossRef] [PubMed]
- H. Lajunen, V. Torres-Company, J. Lancis, E. Silvestre, and P. Andrés, “Pulse-by-pulse method to characterize partially coherent pulse propagation in instantaneous nonlinear media,” Opt. Express18, 14979–14991 (2011). [CrossRef]
- S. Haghgoo and S. A. Ponomarenko, “Self-similar pulses in coherent linear amplifiers,” Opt. Express19, 9750–9758 (2011). [CrossRef] [PubMed]
- S. Haghgoo and S. A. Ponomarenko, “Shape-invariant pulses in resonant linear absorbers,” Opt. Lett.37, 1328–1330 (2012). [CrossRef] [PubMed]
- L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).
- H. Lajunen, J. Tervo, and P. Vahimaa, “Overall coherence and coherent-mode expansion of spectrally partially coherent plane-wave pulses,” J. Opt. Soc. Am. A21, 2117–2123 (2004). [CrossRef]
- P. W. Milonni and J. H. Eberly, Lasers (Wiley, 1985).
- M. Abramowitz and I. A. Stegan, Handbook of Mathematical Functions (Dover, 1972).

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