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Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 1 — Jan. 14, 2013
  • pp: 190–195
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Non-uniformly correlated partially coherent pulses

Hanna Lajunen and Toni Saastamoinen  »View Author Affiliations


Optics Express, Vol. 21, Issue 1, pp. 190-195 (2013)
http://dx.doi.org/10.1364/OE.21.000190


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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

Stochastic fluctuations affect inevitably the behavior of many types of light fields. While the optical coherence theory has traditionally concentrated on studying such effects occurring in stationary fields [1

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

], similar phenomena concern also optical pulses that play a crucial role, for example, in optical telecommunications [2

2. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 1995).

]. The influence of the coherence properties on the evolution of pulses upon propagation has been studied in various media, including linear [3

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

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]

] and nonlinear [5

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]

] dispersive media, coherent linear absorbers [6

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]

] and scattering random media [7

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]

]. A typical conclusion found in linear dispersive media is that decreased coherence degrades the performance of optical systems by leading to faster temporal spreading of pulses.

In most of the studies concerning coherence properties of pulses, the basic Gaussian Schell-model (GSM) pulses [8

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

] have been chosen to represent the field. However, it is possible to have ensembles of pulses with correlation distributions that do not follow the simple Schell model where the degree of coherence between two instants of time is assumed to depend only on their temporal separation. For instance, it has been shown to be possible to introduce partially coherent pulses that propagate as dispersion-free localized waves [9

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

] or maintain their shape upon propagation in resonant linear absorbers [10

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]

]. The coherence distributions may also differ from the basic Gaussian Schell-model due to the noise properties of the pulsed source [11

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]

], or their shape may be distorted upon propagation in nonlinear media [5

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]

]. Besides, a scheme for experimental generation of pulses with predetermined coherence distributions has been suggested [12

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]

].

Recently we introduced a model for optical scalar beams with new type of non-uniform spatial correlations [13

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

]. The modified coherence distribution was shown to lead to self-focusing and lateral shifts of the beam maxima in free-space propagation. The propagation of such beams has been studied also in random media [14

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

] and the model has been extended for electromagnetic beams [15

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

]. Due to the space-time analogy between paraxial propagation of partially coherent beams and the evolution of partially coherent plane-wave pulses in linear second-order dispersive media [16

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]

], it can be expected that corresponding coherence-induced effects could be seen also in the temporal evolution of pulses.

In this paper we apply a model similar to that of Ref. [13

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

] to the correlation functions of partially coherent pulses. The basic theory for defining non-uniform correlation functions is discussed in Sect. 2. In Sects. 3 and 4 the model is introduced in time and frequency domains, respectively, and the propagation characteristics of the resulting pulses in linear dispersive media are studied. Finally, the conclusions are presented in Sect. 5.

2. Theory

Let us consider partially coherent plane-wave pulses propagating in the direction of the positive z-axis. In the space-time domain the coherence properties of the pulses can be defined by their mutual coherence function Γ(t1, t2) = 〉U*(t1)U(t2)〉, 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, γ(t1,t2)=Γ(t1,t2)/I(t1)I(t2). Its absolute value varies between 0 and 1, the extremes corresponding to fully incoherent or coherent fields, correspondingly.

In general, since the exact forms of all the pulse realizations are not known, the correlation properties of the pulse ensemble must be described by choosing a suitable mathematical form for the mutual coherence function. In order to be a proper correlation function, Γ(t1, t2) must correspond to a non-negative definite kernel [1

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

]. 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]

], a necessary and sufficient condition for the non-negative definiteness is that the mutual coherence function can be represented in the form
Γ(t1,t2)=p(v)V*(t1,v)V(t2,v)dv,
(1)
where 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.

The coherence properties of pulses can be studied also in the space-frequency domain through the cross-spectral density W(ω1, ω2) that measures the correlations between two angular frequencies ω1 and ω2. Here, we present all the frequencies with respect to the central frequency of the pulses ω0, i.e. the real frequencies are obtained by ω′ = ω +ω0. W is connected to Γ through the generalized Wiener-Khintchine theorem [1

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

],
W(ω1,ω2)=1(2π)2Γ(t1,t2)exp[i(ω1t1ω2t2)]dt1dt2,
(2)
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 μ(ω1,ω2)=W(ω1,ω2)/S(ω1)S(ω2). Inserting Eq. (1) into (2) and assuming that the kernel V(t, v) has inverse Fourier transform with respect to the variable t, Ṽ(ω, v), we readily get
W(ω1,ω2)=1(2π)2p(v)V˜*(ω1,v)V˜(ω2,v)dv.
(3)
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 Ṽ 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.

Equations (1) and (3) resemble the elementary-pulse representations introduced for partially coherent fields in temporal and spectral domains [18

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

, 19

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]

]. However, in the elementary-pulse representations the coherence functions are defined as an integrated continuum of shifted identical pulses, i.e., the kernel functions in the above equations are assumed to be of the form V(t, v) = V(tv) or Ṽ(ω, v) = Ṽ(ωv). 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.

Based on the theory presented above, various different coherence functions with non-uniform correlation distributions can be defined. In the following sections, we concentrate on a basic case where both the weight and kernel functions are Gaussian. Starting from such model first in the time domain and then in the frequency domain, we study the effects of varying coherence distributions on pulse propagation in second-order dispersive media.

3. Non-uniformly correlated temporally Gaussian pulses

First, we consider partially coherent pulses with non-uniform correlation function obtained assuming Gaussian weight and kernel functions in temporal domain. Inserting
p(v)=(πa)1exp(v2/a2)
(4)
and
V0(t,v)=exp(t22T02)exp[iκv(ttc)2],
(5)
where a and κ are positive real constants and tc is a real constant, into Eq. (1) we get mutual coherence function
Γ0(t1,t2)=exp(t12+t222T02)exp{[(t2tc)2(t1tc)2]2Tc4},
(6)
where Tc=2/κa. If the latter exponential term in Eq. (5) was replaced by exp(−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 tc. This leads to a situation where the correlations are highest around tc. 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).

Fig. 1 (a) The mutual coherence function Γ(t1, t2) and (b) the degree of coherence γ(t1, t2) given by Eq. (6) with T0 = 15 ps, Tc = 10 ps and tc = −10 ps. (c) The corresponding cross-spectral density W(ω1, ω2) and (d) degree of spectral coherence μ(ω1, ω2) calculated numerically by Eq.(2).

In the frequency domain we can solve the corresponding modes of Eq. (3) by taking the inverse Fourier transform of Eq. (5), and using a well-known integration formula (Eq. 3.323-2 in page 337 of Ref. [20

20. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products7th ed. (Academic Press, 2007).

]):
V˜0(ω,v)=[πA(v)]1/2exp[(ω+2κvtc)2/A(v)iκvtc2],
(7)
where A(v)=2/T02+4iκv. Even though these modes still are chirped and have Gaussian shapes, each of them is spectrally shifted depending both on the variable v and the parameter tc. 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(ω1, ω2) 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.

For studying the coherence-induced propagation effects, we assume now a medium characterized by the second-order dispersion coefficient β2[2

2. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 1995).

], and study the propagation of pulses whose coherence function at the starting plane z = 0 is the same as in Fig. 1. The basic methods for modeling the propagation of the coherence functions in dispersive media are described in several articles [3

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

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]

, 16

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]

]. Unfortunately, the propagated intensity distribution of non-uniformly correlated pulses cannot be expressed analytically in closed form. Our numerical simulation results for normally dispersive media are shown in Fig. 2. The intensity distribution of the pulses follows a skewed trajectory in the (I,z)-plane, which means that the maximum peak is accelerating and also focused and increased for a specific range of propagation distances. Analogously to the similar beams in the spatial domain [13

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

], the maximum temporal shift of the peak equals approximately the decentering parameter tc of the coherence function.

Fig. 2 (a) Evolution of the intensity distribution I(t) of pulses given at z = 0 by Eq. (6) with T0 = 15 ps, Tc = 10 ps and tc = −10 ps in a medium with β2 = 50 ps2/km. (b) Pulse shapes at selected propagation distances. The time coordinate is measured in the reference frame moving at the group velocity of the pulse.

If we assume anomalously dispersive media with negative β2, 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 tc in normally dispersive media. Likewise, a positive tc in anomalously dispersive media leads to similar accelerating as shown in Fig. 2. Thus, by choosing properly the parameter tc, we can obtain accelerating or decelerating behavior in both types of dispersive media.

4. Non-uniformly correlated spectrally Gaussian pulses

Fig. 3 (a) Evolution of the intensity distribution I(t) of pulses given at z = 0 by Eq. (9) with Ω0 = 0.3 THz*rad, Ωc = 0.3 THz*rad and ωc = −0.2 THz*rad in a medium with β2 = 50 ps2/km. (b) Pulse shapes at selected propagation distances. The time coordinate is measured in the reference frame moving at the group velocity of the pulse.

5. Conclusions

We have studied the propagation of pulses with non-uniform correlation distributions in second-order dispersive media. The modified coherence properties lead to various changes upon propagation, including possible self-focusing and acceleration or deceleration of the intensity maximum of the pulses. Similar effects can be obtained both in normally and anomalously dispersive media. Our results imply that tailoring the coherence distributions offers a novel way to manipulate the propagation characteristics of pulsed fields.

Acknowledgments

This work was supported by the Academy of Finland (grant 118329).

References and links

1.

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

2.

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 1995).

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]

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]

17.

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

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]

20.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products7th ed. (Academic Press, 2007).

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

  1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, UK, 1995).
  2. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 1995).
  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. Express18, 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. A80, 053804 (2009). [CrossRef]
  10. 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]
  11. 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]
  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. A29, 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]
  17. F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett.32, 3531–3533 (2007). [CrossRef] [PubMed]
  18. P. Vahimaa and J. Turunen, “Independent-elementary-pulse representation for non-stationary fields,” Opt. Express14, 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. Express15, 5160–5165 (2007). [CrossRef] [PubMed]
  20. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products7th ed. (Academic Press, 2007).

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