Abstract

A new class of partially coherent pulses of Schell type with cosine-Gaussian temporal degree of coherence is introduced. Such waves are termed the Cosine-Gaussian Schell-model (CGSM) pulses. The analytic expression for the temporal mutual coherence function of the CGSM pulse in dispersive media is derived and used to study the evolution of its intensity distribution and its temporal degree of coherence. Further, the numerical calculations are performed in order to show the dependence of the intensity profile and the temporal degree of coherence of the CGSM pulse on the incident pulse duration, the initial temporal coherence length, the order-parameter n and the dispersion of the medium. The most important feature of the novel pulsed wave is its ability to split into two pulses on passage in a dispersive medium at some critical propagation distance. Such critical distance and the subsequent evolution of the split pulses are shown to depend on the source parameters and on the properties of the medium in which the pulse travels.

© 2014 Optical Society of America

1. Introduction

It is a well-known fact that the states of spatial and temporal coherence of a partially coherent light field in the source plane are intimately related with its far-field spatial and temporal intensity patterns via the Fourier-type reciprocity relations [1, 2]. Such relations can be readily established in spatial domain for the Continuous Waves [CW] of Schell-model type [2]. For such waves the complex degree of coherence at two spatial arguments is assumed to be only a function of the separation between the points, and, hence, the state of coherence is uniform across the source plane [2]. Apart from the well-established Gaussian Schell-model source [2] other Schell-type model sources have been recently introduced, namely, the J0-correlated Schell-model sources [3, 4], the Multi-Gaussian Schell-model sources [57], the Bessel-Gaussian and Laguerre-Gaussian Schell-model sources [8], and the Cosine-Gaussian Schell-model sources [9, 10]. A class of CW sources with non-Schell-model correlation functions and beams they generate has also been recently introduced and analyzed [11,12].

Unlike in case of deterministic beams, the beams generated by partially coherent sources can exhibit very interesting features on propagation. For instance, the intensity profiles of beams originated by J0-correlated Schell-model sources have properties analogous to those of the Bessel-Gaussian beams but their degree of coherence does not keep the J0(x) profile nor shift-invariance [4]; the Multi-Gaussian Schell-model sources can generate far fields with tunable flat profiles, whether circular [5,6] or rectangular [7]; the Bessel-Gaussian and the Laguerre-Gaussian Schell-model sources are capable of producing far fields with ring-shaped intensity distributions [8]; the far-field spectral density produced by a cosine-Gaussian Schell-model beams takes on the dark-hollow profile [9]. In addition, the beams generated by non-uniformly correlated light sources can self-focus and possess lateral shifts of the beam intensity maxima in free-space propagation [11]. Such shifts are shown to be suppressed by natural media [12].

On the other hand the non-stationary light fields, also called stochastic optical pulses exhibit partial coherence spectrally or temporally. They represent a wide class of partially coherent fields that find numerous applications in optical telecommunications, optical imaging, fiber optics, etc [13]. In recent years, the influence of the temporal coherence properties on the evolution of pulses upon propagation has been studied extensively [1424]. The results of these investigations imply that temporal coherence of optical pulses play a crucial role in the evolution of the pulse. However, almost all the previously introduced models of temporally coherent optical pulses are also limited to classical Gaussian Schell-type correlations, with the exception of Ref [25]. by Lajunen and Saastamoinen who introduced the non-uniformly correlated partially coherent pulses with non-Gaussian Schell-model correlations distribution of temporal coherence, and found some interesting characteristics in dispersive-media propagation. Non-stationary non-uniformly-correlated pulses were also shown to have interesting features in regards to their polarimetric properties [26].

Taking into account the possibility of split in the CW (stationary) cosine-Gaussian correlated Schell-model beams on propagation, an interesting question comes up: can it also occur in temporal domain? The purpose of this paper is to explore the evolution of the CGSM pulses in dispersive media, and to examine the influence of pulse parameters and the dispersive properties of the medium on the temporal pulse intensity and the temporal degree of coherence.

2. Theoretical formulation

In the space-time domain the coherence properties of the pulses can be characterized by their mutual coherence function Γ(t1,t2) = <E*(t1)E (t2)>, where E(t) represents the complex analytic signal of pulse realizations at time t, and the angular brackets denote the ensemble average. In general, for a mutual coherence function to be genuine, i.e., physically realizable, Γ(t1,t2) must correspond to a non-negative definite kernel [2]. As has been shown for correlation functions in the spatial domain [27], a sufficient condition for the non-negative definiteness is that the mutual coherence function must be expressed as a superposition integral of the form

Γ(t1,t2)=p(v)H(t1,v)H(t2,v)dv,
where p(v) is a non-negative, Fourier-transformable function and H is an arbitrary kernel. By choosing suitable p(v) and H, one can define a wide variety of temporal coherence functions for a partially coherent pulse.

In order to introduce a CGSM pulse, we choose p(v) and H as follows:

p(v)=Tc2πcosh(n2πTcv)exp(Tc2v2+2n2π2),
H(t,v)=exp(t24T02)exp(ivt),
where Tc is the r.m.s source correlation determining the temporal degree of coherence of a pulse, n is a positive real constant, not necessarily an integer, cosh(x) is the hyperbolic cosine function, T0 represents the typical pulse duration. Substituting Eqs. (2) and (3) into Eq. (1), we obtain the mutual coherence function of a CGSM pulse in the form
Γ(t1,t2)=exp(t12+t224T02)γ(t1,t2),
where

γ(t1,t2)=cos[n2π(t2t1)Tc]exp[(t2t1)22Tc2].

Equation (5) denotes the temporal degree of coherence of the CGSM pulse at a pair of time instants, t1 and t2 in the source plane. If we set n = 0, γ (t1, t2) can be expressed as

γ(t1,t2)=exp[(t2t1)22Tc2].

Thus, we can see that the temporal degree of coherence of the CGSM pulse reduce to that of the conventional Gaussian correlated Schell-model pulse for n = 0, while for n ≠ 0 the coherence is modulated by the cosine function.

We will now study the propagation of a CGSM pulse in the second-order dispersive media. Propagation of the mutual coherence function in such media can be studied by the generalized Collins formula in the temporal domain [15, 28]

Γ(t1,t2,z)=12πβ2zΓ(t10,t20)exp{i2β2z[(t102t202)+(t12t22)2(t10t1-t20t2)]}dt10dt20,
where β2 represents the group velocity dispersion coefficient. Here we have assumed that the time coordinate is measured in the reference frame moving at the group velocity of the pulse.

On substituting from Eq. (4) into Eq. (7), after tedious integral calculations we obtain the following analytic formula of the propagating mutual coherence function of the CGSM pulses.

Γ(t1,t2,z)=T022β2zΔ(z)exp[(t1t2)2R(z)+i2β2z(t12t22)][exp(η+2Δ2(z))+exp(η2Δ2(z))],
where

1R(z)=T022β22z2,
Δ2(z)=1R(z)+12ξ2,
1ξ2=14T02+1Tc2,
η±=1R(z)(t1t2)i4β2z(t1+t2)±in2π2Tc.

From the mutual coherence function (8), the average intensity I(t, z) and the temporal degree of coherence γ (t1, t2, z) of the CGSM pulse in the second-order dispersive media can be calculated by the expressions [19]

I(t,z)=Γ(t,t,z),
γ(t1,t2,z)=Γ(t1,t2,z)Γ(t1,t1,z)Γ(t2,t2,z).

3. Numerical calculations

In this section the results of numerical calculation are given to present the temporal evolution of the CGSM pulse in dispersive media. Here the time coordinate is measured in the reference frame moving at the group velocity of the pulse.

Figure 1 gives the normalized intensity distribution I(t, z) of the CGSM pulse as a function of propagation distance z and time t for different values of pulse durations (a) T0 = 15ps, (b) T0 = 25ps, and (c) T0 = 35ps. Figures 1(d), 1(e) and 1(f) are the color-coded plots corresponding to Figs. 1(a), 1(b) and 1(c), respectively. The other parameters are Tc = 10ps, n = 2, β2 = 20ps2km−1. As shown in Fig. 1, on dispersive-media propagation at critical distance of zc = 2km a CGSM pulse begins to split into two pulses. With the increasing pulse duration of the incident pulse (see Figs. 1(b), 1(e), 1(c) and 1(f)), the critical propagation distance zc of the two resulting pulses moves farer away from the incident pulse and the CGSM pulse broadens upon propagation.

 

Fig. 1 Normalized intensity distribution I(t, z) of the CGSM pulse as a function of propagation distance z and time t for different values of pulse durations (a) T0 = 15ps, (b) T0 = 25ps, (c) T0 = 35ps and (d), (e), (f) the color-coded plot corresponding to (a), (b), (c), respectively. The other parameters are Tc = 10ps, n = 2, β2 = 20ps2km−1.

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Figure 2 gives the normalized intensity distribution I(t, z) of the CGSM pulse as a function of propagation distance z and time t for different values of temporal coherence lengths (a) Tc = 10ps, (b) Tc = 20ps, (c) Tc = 30ps. Figures 2(d), 2(e) and 2(f) are the color-coded plots corresponding to Figs. 2(a), 2(b) and 2(c), respectively. The other parameters are T0 = 15ps, n = 2, β2 = 20ps2km−1. It is shown that, with the increasing temporal coherence length of the incident pulse (see Figs. 2(b), 2(e), 2(c) and 2(f)), the critical propagation distance zc, at which the pulse begins to split, moves far away from the incident pulse and the two split pulses become closer.

 

Fig. 2 Normalized intensity distribution I(t, z) of the CGSM pulse as a function of propagation distance z and time t for different values of temporal coherence lengths (a) Tc = 10ps, (b) Tc = 20ps, (c) Tc = 30ps and (d), (e), (f) the color-coded plot corresponding to (a), (b), (c), respectively. The other parameters are T0 = 15ps, n = 2, β2 = 20ps2km−1.

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Figure 3 gives the normalized intensity distribution I(t, z) of the CGSM pulse as a function of propagation distance z and time t for different values of parameters (a) n = 2, (b) n = 4, (c) n = 6. Figures 3(d), 3(e) and 3(f) are the color-coded plot corresponding to Figs. 3(a), 3(b) and 3(c), respectively. The other parameters are T0 = 15ps, Tc = 10ps, β2 = 20ps2km−1. As shown by Fig. 3, with increasing parameters n(see Figs. 3(b), 3(e), 3(c) and 3(f)), the critical propagation distance zc, where the pulse begins to split, moves towards the incident pulse and the two split pulses become far away from each other.

 

Fig. 3 Normalized intensity distribution I(t, z) of the CGSM pulse as a function of propagation distance z and time t for different values of parameters (a) n = 2, (b) n = 4, (c) n = 6 and (d), (e), (f) the color-coded plot corresponding to (a), (b), (c), respectively. The other parameters are T0 = 15ps, Tc = 10ps, β2 = 20ps2km−1.

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Figure 4 gives the normalized intensity distribution I(t, z) of CGSM pulse as a function of propagation distance z and time t for different values of second-order dispersion coefficients (a) β2 = 20ps2km−1, (b) β2 = 35ps2km−1, (c) β2 = 50ps2km−1. Figures 4(d), 4(e) and 4(f) are the color-coded plot corresponding to Figs. 4(a), 4(b) and 4(c), respectively. The other parameters are T0 = 15ps, Tc = 10ps, n = 2. It is seen that the medium dispersion affects the intensity distribution I(t, z) of CGSM pulse. And with the increasing second-order dispersion coefficient β2 (see Figs. 4(b), 4(e), 4(c) and 4(f)), the critical propagation distance zc, at which the pulse begins to split, moves towards the incident pulse and the two split pulses become far away from each other.

 

Fig. 4 Normalized intensity distribution I(t, z) of the CGSM pulse as a function of propagation distance z and time t for different values of second-order dispersion coefficients (a) β2 = 20ps2km−1, (b) β2 = 35ps2km−1, (c) β2 = 50ps2km−1 and (d), (e), (f) the color-coded plot corresponding to (a), (b), (c), respectively. The other parameters are T0 = 15ps, Tc = 10ps, n = 2.

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Figure 5 shows the temporal degree of coherence γ (t1, t2, z) of the CGSM pulse as a function of separation between two time points td = t1-t2 at some propagation distances z. The calculation parameters are (a) n = 0, (b) n = 2, (c) n = 4, and the other parameters are T0 = 15ps, Tc = 10ps and β2 = 20ps2km−1. As shown in Fig. 5, the temporal degree of coherence γ (t1, t2, z) of the CGSM pulse has Gaussian form for n = 0 and starts to oscillate like a sinc-function as n increases for relatively short propagation distance z. We can also see that as the propagation distance increases, the oscillations in the temporal degree of coherence profiles gradually weaken and it eventually degenerates into a Gaussian shape when the propagation distance is large enough.

 

Fig. 5 The temporal degree of coherence γ (t1, t2, z) of the CGSM pulse as a function of separation between two time points td at some propagation distances z. The calculation parameters are (a) n = 0, (b) n = 2, (c) n = 4, and the other parameters are T0 = 15ps, Tc = 10ps and β2 = 20ps2km−1.

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Figure 6(a) shows the temporal degree of coherence γ (t1, t2, z) of the CGSM pulse as a function of separation between two time instants td for different values of pulse duration T0 = 15ps, T0 = 25ps, T0 = 35ps, respectively, and Tc = 10ps. As can be seen from Fig. 6(a) the oscillations in the temporal degree of coherence profiles gradually intensify with increasing pulse duration of the incident pulse. Figure 6(b) shows the temporal degree of coherence γ (t1, t2, z) as a function of separation between two time instants td for different values of temporal coherence length Tc = 10ps, Tc = 20ps, Tc = 30ps, respectively, and T0 = 15ps. The other parameters are n = 2 and β2 = 20ps2km−1. As shown in Fig. 6(b), the profiles of the temporal degree of coherence move towards the large value td with increasing temporal coherence length of the incident pulse.

 

Fig. 6 (a) The temporal degree of coherence γ (t1, t2, z) of the CGSM pulse as a function of separation between two time points td for different values of pulse durations T0 = 15ps, T0 = 25ps, T0 = 35ps, respectively, and Tc = 10ps. (b) The temporal degree of coherence γ (t1, t2, z) as a function of separation between two time points td for different values of temporal coherence lengths (b) Tc = 10ps, Tc = 20ps, Tc = 30ps, respectively, and T0 = 15ps. The other parameters are n = 2 and β2 = 20ps2km−1.

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Figure 7 shows the temporal degree of coherence γ (t1, t2, z) of the CGSM pulse as a function of separation between two time instants td for different values of the second-order dispersion coefficients β2 = 20ps2km−1, β2 = 35ps2km−1, β2 = 50ps2km−1, β2 = 123ps2km−1, respectively. The other parameters are T0 = 15ps, Tc = 10ps, n = 2. It is shown that the medium dispersion plays the important role on the shape in the temporal degree of coherence, that gradually weaken and eventually degenerate into a Gaussian shape with increasing second-order dispersion coefficient β2.

 

Fig. 7 The temporal degree of coherence γ (t1, t2, z) of the CGSM pulse as a function of separation between two time points td for different values of the second-order dispersion coefficients β2 = 20ps2km−1, β2 = 35ps2km−1, β2 = 50ps2km−1, β2 = 123ps2km−1, respectively. The other parameters are T0 = 15ps, Tc = 10ps, n = 2.

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4. Physical interpretation

The main result that one CGSM pulse is able to split into two parts on the dispersive-media propagation can be interpreted as follows.

Equation (13) can be rewritten as

I(t,z)=T02T02+(14T02+1Tc2)β22z2{exp[-12(tn2πTcβ2z)2T02+(14T02+1Tc2)β22z2]+exp[-12(t+n2πTcβ2z)2T02+(14T02+1Tc2)β22z2]},
which is the superposition of two Gaussian functions, i.e. I (t, z) = I+(t, z) + I-(t, z), in which
I±(t,z)=T02T02+(14T02+1Tc2)β22z2exp[-12(t±n2πTcβ2z)2T02+(14T02+1Tc2)β22z2],
where the functions I+(t, z) and I-(t, z) indicate two split pulses, respectively. And the function I(t, z) indicates the superposition of two split pulses. Figure 8 gives the functions I+(t, z), I-(t, z) and I (t, z) versus the time t for different values of (a,d) z = 4km, (b,e) z = 2km and (c,f) z = 0, respectively. The other parameters are T0 = 15ps, Tc = 10ps, n = 2, β2 = 20ps2km−1. As shown in Fig. 8, as the propagation distance z = 4km, there is very short common region for two split pulses. With decreasing z, two split pulses move towards each other, and therefore, there is a large common region for two split pulses as z = 2km. The two split pulses occur at the same position as z = 0. Thus, only one pulse, which is the superposition of two split pulses, can be seen. That is why one CGSM pulse is able to split into two parts in dispersive-media propagation.

 

Fig. 8 The functions I+(t, z), I-(t, z) and I (t, z) versus the time t for different values of (a,d) z = 4km, (b,e) z = 2km and (c,f) z = 0, respectively. The other parameters are T0 = 15ps, Tc = 10ps, n = 2, β2 = 20ps2km−1.

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5. Conclusion

In conclusion, we have introduced a new class of partially coherent pulses of Schell type with cosine-Gaussian temporal degree of coherence, i.e. the cosine-Gaussian correlated Schell-model (CGSM) pulses, which are physically realizable using the similar method to the conventional Gaussian correlated Schell-model pulses [21]. We have derived the analytical expression of the mutual coherence function of the CGSM pulses in the dispersive-media propagation. According to the expression obtained, we have studied and analyzed the evolution of the intensity and the temporal degree of coherence of the CGSM pulse, where the incident pulse parameters (T0, Tc, n) and the propagation medium coefficient β2 play the important role for the evolution. We have found that at some critical propagation distances zc, one CGSM pulse begins to split into two parts in dispersive-media propagation. With increasing pulse duration or temporal coherence length of the incident pulse, the critical propagation distance zc, where the pulse begins to split, moves far away from the incident pulse. By contrast, with the increasing values of parameter n or the second-order dispersion coefficient β2, the critical propagation distance zc, where the pulse begins to split, moves toward the incident pulse. The temporal degree of coherence of the CGSM pulse has Gaussian form for n = 0 and exhibit oscillations like a sinc-function as n increases for relatively short propagation distances z. As the propagation distance increases or the second-order dispersion coefficient β2 increases, the oscillations of the temporal degree of coherence profiles gradually weaken and eventually degenerate into a Gaussian shape. However, the oscillations of the temporal degree of coherence profiles gradually intensify with the increasing pulse duration of the incident pulse. The results obtained can offer some new methods to manipulate the propagation characteristics of pulsed fields in order to the application of partially coherent pulse in dispersive media [13].

Acknowledgments

Chaoliang Ding, Yongtao Zhang and Liuzhan Pan’s research is supported by the National Natural Science Foundation of China under Grant Nos. 61275150, 61078077 and 61108090, the Education Department of Henan Province Project 13A140797, the Program for Science & Technology Innovation Talents in Universities of Henan Province (13HASTIT048) and the Program for Innovative Research Team (in Science and Technology) in University of Henna Province (Grant No. 13IRTSTHN020). O. Korotkova’s research is supported by US AFOSR (FA9550-12-1-0449) and US ONR (N0018913P1226).

References and links

1. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

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

3. C. Palma, R. Borghi, and G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125(1-3), 113–121 (1996). [CrossRef]  

4. F. Gori, M. Santarsiero, and R. Borghi, “Modal expansion for J0-correlated electromagnetic sources,” Opt. Lett. 33(16), 1857–1859 (2008). [CrossRef]   [PubMed]  

5. S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012). [CrossRef]   [PubMed]  

6. O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012). [CrossRef]   [PubMed]  

7. O. Korotkova, “Random sources for rectangular far fields,” Opt. Lett. 39(1), 64–67 (2014). [CrossRef]   [PubMed]  

8. Z. R. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013). [CrossRef]   [PubMed]  

9. Z. R. Mei and O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013). [CrossRef]   [PubMed]  

10. Z. R. Mei, E. Shchepakina, and O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express 21(15), 17512–17519 (2013). [CrossRef]   [PubMed]  

11. H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011). [CrossRef]   [PubMed]  

12. Z. S. Tong and O. Korotkova, “Nonuniformly correlated light beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012). [CrossRef]   [PubMed]  

13. G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).

14. P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204(1-6), 53–58 (2002). [CrossRef]  

15. Q. Lin, L. G. Wang, and S. Y. Zhu, “Partially coherent light pulse and its propagation,” Opt. Commun. 219(1-6), 65–70 (2003). [CrossRef]  

16. L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 056613 (2003). [CrossRef]   [PubMed]  

17. H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, “Spectral coherence properties of temporally modulated stationary light sources,” Opt. Express 11(16), 1894–1899 (2003). [CrossRef]   [PubMed]  

18. 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(11), 2117–2123 (2004). [CrossRef]   [PubMed]  

19. H. Lajunen, P. Vahimaa, and J. Tervo, “Theory of spatially and spectrally partially coherent pulses,” J. Opt. Soc. Am. A 22(8), 1536–1545 (2005). [CrossRef]   [PubMed]  

20. J. Lancis, V. Torres-Company, E. Silvestre, and P. Andrés, “Space-time analogy for partially coherent plane-wave-type pulses,” Opt. Lett. 30(22), 2973–2975 (2005). [CrossRef]   [PubMed]  

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

22. V. Torres-Company, H. Lajunen, J. Lancis, and A. T. Friberg, “Ghost interference with classical partially coherent light pulses,” Phys. Rev. A 77(4), 043811 (2008). [CrossRef]  

23. C. L. Ding, Y. J. Cai, O. Korotkova, Y. T. Zhang, and L. Z. Pan, “Scattering-induced changes in the temporal coherence length and the pulse duration of a partially coherent plane-wave pulse,” Opt. Lett. 36(4), 517–519 (2011). [CrossRef]   [PubMed]  

24. L. Mokhtarpour and S. A. Ponomarenko, “Complex area correlation theorem for statistical pulses in coherent linear absorbers,” Opt. Lett. 37(17), 3498–3500 (2012). [CrossRef]   [PubMed]  

25. H. Lajunen and T. Saastamoinen, “Non-uniformly correlated partially coherent pulses,” Opt. Express 21(1), 190–195 (2013). [CrossRef]   [PubMed]  

26. C. L. Ding, Y. J. Cai, Y. T. Zhang, H. X. Wang, Z. G. Zhao, and L. Z. Pan, “Stochastic electromagnetic plane-wave pulse with non-uniform correlation distribution,” Phys. Lett. A 377(25-27), 1563–1565 (2013). [CrossRef]  

27. F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007). [CrossRef]   [PubMed]  

28. S. P. Dijaili, A. Dienes, and J. S. Smith, “ABCD matrices for dispersive pulse propagation,” IEEE J. Quantum Electron. 26(6), 1158–1164 (1990). [CrossRef]  

References

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  1. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).
  2. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  3. C. Palma, R. Borghi, and G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125(1-3), 113–121 (1996).
    [Crossref]
  4. F. Gori, M. Santarsiero, and R. Borghi, “Modal expansion for J0-correlated electromagnetic sources,” Opt. Lett. 33(16), 1857–1859 (2008).
    [Crossref] [PubMed]
  5. S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
    [Crossref] [PubMed]
  6. O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012).
    [Crossref] [PubMed]
  7. O. Korotkova, “Random sources for rectangular far fields,” Opt. Lett. 39(1), 64–67 (2014).
    [Crossref] [PubMed]
  8. Z. R. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
    [Crossref] [PubMed]
  9. Z. R. Mei and O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013).
    [Crossref] [PubMed]
  10. Z. R. Mei, E. Shchepakina, and O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express 21(15), 17512–17519 (2013).
    [Crossref] [PubMed]
  11. H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011).
    [Crossref] [PubMed]
  12. Z. S. Tong and O. Korotkova, “Nonuniformly correlated light beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012).
    [Crossref] [PubMed]
  13. G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).
  14. P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204(1-6), 53–58 (2002).
    [Crossref]
  15. Q. Lin, L. G. Wang, and S. Y. Zhu, “Partially coherent light pulse and its propagation,” Opt. Commun. 219(1-6), 65–70 (2003).
    [Crossref]
  16. L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 056613 (2003).
    [Crossref] [PubMed]
  17. H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, “Spectral coherence properties of temporally modulated stationary light sources,” Opt. Express 11(16), 1894–1899 (2003).
    [Crossref] [PubMed]
  18. 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(11), 2117–2123 (2004).
    [Crossref] [PubMed]
  19. H. Lajunen, P. Vahimaa, and J. Tervo, “Theory of spatially and spectrally partially coherent pulses,” J. Opt. Soc. Am. A 22(8), 1536–1545 (2005).
    [Crossref] [PubMed]
  20. J. Lancis, V. Torres-Company, E. Silvestre, and P. Andrés, “Space-time analogy for partially coherent plane-wave-type pulses,” Opt. Lett. 30(22), 2973–2975 (2005).
    [Crossref] [PubMed]
  21. V. Torres-Company, G. Mínguez-Vega, J. Lancis, and A. T. Friberg, “Controllable generation of partially coherent light pulses with direct space-to-time pulse shaper,” Opt. Lett. 32(12), 1608–1610 (2007).
    [Crossref] [PubMed]
  22. V. Torres-Company, H. Lajunen, J. Lancis, and A. T. Friberg, “Ghost interference with classical partially coherent light pulses,” Phys. Rev. A 77(4), 043811 (2008).
    [Crossref]
  23. C. L. Ding, Y. J. Cai, O. Korotkova, Y. T. Zhang, and L. Z. Pan, “Scattering-induced changes in the temporal coherence length and the pulse duration of a partially coherent plane-wave pulse,” Opt. Lett. 36(4), 517–519 (2011).
    [Crossref] [PubMed]
  24. L. Mokhtarpour and S. A. Ponomarenko, “Complex area correlation theorem for statistical pulses in coherent linear absorbers,” Opt. Lett. 37(17), 3498–3500 (2012).
    [Crossref] [PubMed]
  25. H. Lajunen and T. Saastamoinen, “Non-uniformly correlated partially coherent pulses,” Opt. Express 21(1), 190–195 (2013).
    [Crossref] [PubMed]
  26. C. L. Ding, Y. J. Cai, Y. T. Zhang, H. X. Wang, Z. G. Zhao, and L. Z. Pan, “Stochastic electromagnetic plane-wave pulse with non-uniform correlation distribution,” Phys. Lett. A 377(25-27), 1563–1565 (2013).
    [Crossref]
  27. F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
    [Crossref] [PubMed]
  28. S. P. Dijaili, A. Dienes, and J. S. Smith, “ABCD matrices for dispersive pulse propagation,” IEEE J. Quantum Electron. 26(6), 1158–1164 (1990).
    [Crossref]

2014 (1)

2013 (5)

2012 (4)

2011 (2)

2008 (2)

V. Torres-Company, H. Lajunen, J. Lancis, and A. T. Friberg, “Ghost interference with classical partially coherent light pulses,” Phys. Rev. A 77(4), 043811 (2008).
[Crossref]

F. Gori, M. Santarsiero, and R. Borghi, “Modal expansion for J0-correlated electromagnetic sources,” Opt. Lett. 33(16), 1857–1859 (2008).
[Crossref] [PubMed]

2007 (2)

2005 (2)

2004 (1)

2003 (3)

Q. Lin, L. G. Wang, and S. Y. Zhu, “Partially coherent light pulse and its propagation,” Opt. Commun. 219(1-6), 65–70 (2003).
[Crossref]

L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 056613 (2003).
[Crossref] [PubMed]

H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, “Spectral coherence properties of temporally modulated stationary light sources,” Opt. Express 11(16), 1894–1899 (2003).
[Crossref] [PubMed]

2002 (1)

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204(1-6), 53–58 (2002).
[Crossref]

1996 (1)

C. Palma, R. Borghi, and G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125(1-3), 113–121 (1996).
[Crossref]

1990 (1)

S. P. Dijaili, A. Dienes, and J. S. Smith, “ABCD matrices for dispersive pulse propagation,” IEEE J. Quantum Electron. 26(6), 1158–1164 (1990).
[Crossref]

Andrés, P.

Borghi, R.

F. Gori, M. Santarsiero, and R. Borghi, “Modal expansion for J0-correlated electromagnetic sources,” Opt. Lett. 33(16), 1857–1859 (2008).
[Crossref] [PubMed]

C. Palma, R. Borghi, and G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125(1-3), 113–121 (1996).
[Crossref]

Cai, Y. J.

C. L. Ding, Y. J. Cai, Y. T. Zhang, H. X. Wang, Z. G. Zhao, and L. Z. Pan, “Stochastic electromagnetic plane-wave pulse with non-uniform correlation distribution,” Phys. Lett. A 377(25-27), 1563–1565 (2013).
[Crossref]

C. L. Ding, Y. J. Cai, O. Korotkova, Y. T. Zhang, and L. Z. Pan, “Scattering-induced changes in the temporal coherence length and the pulse duration of a partially coherent plane-wave pulse,” Opt. Lett. 36(4), 517–519 (2011).
[Crossref] [PubMed]

Chen, H.

L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 056613 (2003).
[Crossref] [PubMed]

Cincotti, G.

C. Palma, R. Borghi, and G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125(1-3), 113–121 (1996).
[Crossref]

Dienes, A.

S. P. Dijaili, A. Dienes, and J. S. Smith, “ABCD matrices for dispersive pulse propagation,” IEEE J. Quantum Electron. 26(6), 1158–1164 (1990).
[Crossref]

Dijaili, S. P.

S. P. Dijaili, A. Dienes, and J. S. Smith, “ABCD matrices for dispersive pulse propagation,” IEEE J. Quantum Electron. 26(6), 1158–1164 (1990).
[Crossref]

Ding, C. L.

C. L. Ding, Y. J. Cai, Y. T. Zhang, H. X. Wang, Z. G. Zhao, and L. Z. Pan, “Stochastic electromagnetic plane-wave pulse with non-uniform correlation distribution,” Phys. Lett. A 377(25-27), 1563–1565 (2013).
[Crossref]

C. L. Ding, Y. J. Cai, O. Korotkova, Y. T. Zhang, and L. Z. Pan, “Scattering-induced changes in the temporal coherence length and the pulse duration of a partially coherent plane-wave pulse,” Opt. Lett. 36(4), 517–519 (2011).
[Crossref] [PubMed]

Friberg, A. T.

V. Torres-Company, H. Lajunen, J. Lancis, and A. T. Friberg, “Ghost interference with classical partially coherent light pulses,” Phys. Rev. A 77(4), 043811 (2008).
[Crossref]

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

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204(1-6), 53–58 (2002).
[Crossref]

Gori, F.

Korotkova, O.

Lajunen, H.

Lancis, J.

Lin, Q.

L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 056613 (2003).
[Crossref] [PubMed]

Q. Lin, L. G. Wang, and S. Y. Zhu, “Partially coherent light pulse and its propagation,” Opt. Commun. 219(1-6), 65–70 (2003).
[Crossref]

Mei, Z. R.

Mínguez-Vega, G.

Mokhtarpour, L.

Pääkkönen, P.

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204(1-6), 53–58 (2002).
[Crossref]

Palma, C.

C. Palma, R. Borghi, and G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125(1-3), 113–121 (1996).
[Crossref]

Pan, L. Z.

C. L. Ding, Y. J. Cai, Y. T. Zhang, H. X. Wang, Z. G. Zhao, and L. Z. Pan, “Stochastic electromagnetic plane-wave pulse with non-uniform correlation distribution,” Phys. Lett. A 377(25-27), 1563–1565 (2013).
[Crossref]

C. L. Ding, Y. J. Cai, O. Korotkova, Y. T. Zhang, and L. Z. Pan, “Scattering-induced changes in the temporal coherence length and the pulse duration of a partially coherent plane-wave pulse,” Opt. Lett. 36(4), 517–519 (2011).
[Crossref] [PubMed]

Ponomarenko, S. A.

Saastamoinen, T.

Sahin, S.

Santarsiero, M.

Shchepakina, E.

Silvestre, E.

Smith, J. S.

S. P. Dijaili, A. Dienes, and J. S. Smith, “ABCD matrices for dispersive pulse propagation,” IEEE J. Quantum Electron. 26(6), 1158–1164 (1990).
[Crossref]

Tervo, J.

Tong, Z. S.

Torres-Company, V.

Turunen, J.

H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, “Spectral coherence properties of temporally modulated stationary light sources,” Opt. Express 11(16), 1894–1899 (2003).
[Crossref] [PubMed]

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204(1-6), 53–58 (2002).
[Crossref]

Vahimaa, P.

Wang, H. X.

C. L. Ding, Y. J. Cai, Y. T. Zhang, H. X. Wang, Z. G. Zhao, and L. Z. Pan, “Stochastic electromagnetic plane-wave pulse with non-uniform correlation distribution,” Phys. Lett. A 377(25-27), 1563–1565 (2013).
[Crossref]

Wang, L. G.

L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 056613 (2003).
[Crossref] [PubMed]

Q. Lin, L. G. Wang, and S. Y. Zhu, “Partially coherent light pulse and its propagation,” Opt. Commun. 219(1-6), 65–70 (2003).
[Crossref]

Wyrowski, F.

H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, “Spectral coherence properties of temporally modulated stationary light sources,” Opt. Express 11(16), 1894–1899 (2003).
[Crossref] [PubMed]

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204(1-6), 53–58 (2002).
[Crossref]

Zhang, Y. T.

C. L. Ding, Y. J. Cai, Y. T. Zhang, H. X. Wang, Z. G. Zhao, and L. Z. Pan, “Stochastic electromagnetic plane-wave pulse with non-uniform correlation distribution,” Phys. Lett. A 377(25-27), 1563–1565 (2013).
[Crossref]

C. L. Ding, Y. J. Cai, O. Korotkova, Y. T. Zhang, and L. Z. Pan, “Scattering-induced changes in the temporal coherence length and the pulse duration of a partially coherent plane-wave pulse,” Opt. Lett. 36(4), 517–519 (2011).
[Crossref] [PubMed]

Zhao, Z. G.

C. L. Ding, Y. J. Cai, Y. T. Zhang, H. X. Wang, Z. G. Zhao, and L. Z. Pan, “Stochastic electromagnetic plane-wave pulse with non-uniform correlation distribution,” Phys. Lett. A 377(25-27), 1563–1565 (2013).
[Crossref]

Zhu, S. Y.

Q. Lin, L. G. Wang, and S. Y. Zhu, “Partially coherent light pulse and its propagation,” Opt. Commun. 219(1-6), 65–70 (2003).
[Crossref]

L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 056613 (2003).
[Crossref] [PubMed]

IEEE J. Quantum Electron. (1)

S. P. Dijaili, A. Dienes, and J. S. Smith, “ABCD matrices for dispersive pulse propagation,” IEEE J. Quantum Electron. 26(6), 1158–1164 (1990).
[Crossref]

J. Opt. Soc. Am. A (3)

Opt. Commun. (3)

C. Palma, R. Borghi, and G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125(1-3), 113–121 (1996).
[Crossref]

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204(1-6), 53–58 (2002).
[Crossref]

Q. Lin, L. G. Wang, and S. Y. Zhu, “Partially coherent light pulse and its propagation,” Opt. Commun. 219(1-6), 65–70 (2003).
[Crossref]

Opt. Express (3)

Opt. Lett. (12)

H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011).
[Crossref] [PubMed]

Z. S. Tong and O. Korotkova, “Nonuniformly correlated light beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012).
[Crossref] [PubMed]

J. Lancis, V. Torres-Company, E. Silvestre, and P. Andrés, “Space-time analogy for partially coherent plane-wave-type pulses,” Opt. Lett. 30(22), 2973–2975 (2005).
[Crossref] [PubMed]

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

F. Gori, M. Santarsiero, and R. Borghi, “Modal expansion for J0-correlated electromagnetic sources,” Opt. Lett. 33(16), 1857–1859 (2008).
[Crossref] [PubMed]

S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
[Crossref] [PubMed]

O. Korotkova, “Random sources for rectangular far fields,” Opt. Lett. 39(1), 64–67 (2014).
[Crossref] [PubMed]

Z. R. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
[Crossref] [PubMed]

Z. R. Mei and O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013).
[Crossref] [PubMed]

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
[Crossref] [PubMed]

C. L. Ding, Y. J. Cai, O. Korotkova, Y. T. Zhang, and L. Z. Pan, “Scattering-induced changes in the temporal coherence length and the pulse duration of a partially coherent plane-wave pulse,” Opt. Lett. 36(4), 517–519 (2011).
[Crossref] [PubMed]

L. Mokhtarpour and S. A. Ponomarenko, “Complex area correlation theorem for statistical pulses in coherent linear absorbers,” Opt. Lett. 37(17), 3498–3500 (2012).
[Crossref] [PubMed]

Phys. Lett. A (1)

C. L. Ding, Y. J. Cai, Y. T. Zhang, H. X. Wang, Z. G. Zhao, and L. Z. Pan, “Stochastic electromagnetic plane-wave pulse with non-uniform correlation distribution,” Phys. Lett. A 377(25-27), 1563–1565 (2013).
[Crossref]

Phys. Rev. A (1)

V. Torres-Company, H. Lajunen, J. Lancis, and A. T. Friberg, “Ghost interference with classical partially coherent light pulses,” Phys. Rev. A 77(4), 043811 (2008).
[Crossref]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 056613 (2003).
[Crossref] [PubMed]

Other (3)

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

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

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Figures (8)

Fig. 1
Fig. 1 Normalized intensity distribution I(t, z) of the CGSM pulse as a function of propagation distance z and time t for different values of pulse durations (a) T0 = 15ps, (b) T0 = 25ps, (c) T0 = 35ps and (d), (e), (f) the color-coded plot corresponding to (a), (b), (c), respectively. The other parameters are Tc = 10ps, n = 2, β2 = 20ps2km−1.
Fig. 2
Fig. 2 Normalized intensity distribution I(t, z) of the CGSM pulse as a function of propagation distance z and time t for different values of temporal coherence lengths (a) Tc = 10ps, (b) Tc = 20ps, (c) Tc = 30ps and (d), (e), (f) the color-coded plot corresponding to (a), (b), (c), respectively. The other parameters are T0 = 15ps, n = 2, β2 = 20ps2km−1.
Fig. 3
Fig. 3 Normalized intensity distribution I(t, z) of the CGSM pulse as a function of propagation distance z and time t for different values of parameters (a) n = 2, (b) n = 4, (c) n = 6 and (d), (e), (f) the color-coded plot corresponding to (a), (b), (c), respectively. The other parameters are T0 = 15ps, Tc = 10ps, β2 = 20ps2km−1.
Fig. 4
Fig. 4 Normalized intensity distribution I(t, z) of the CGSM pulse as a function of propagation distance z and time t for different values of second-order dispersion coefficients (a) β2 = 20ps2km−1, (b) β2 = 35ps2km−1, (c) β2 = 50ps2km−1 and (d), (e), (f) the color-coded plot corresponding to (a), (b), (c), respectively. The other parameters are T0 = 15ps, Tc = 10ps, n = 2.
Fig. 5
Fig. 5 The temporal degree of coherence γ (t1, t2, z) of the CGSM pulse as a function of separation between two time points td at some propagation distances z. The calculation parameters are (a) n = 0, (b) n = 2, (c) n = 4, and the other parameters are T0 = 15ps, Tc = 10ps and β2 = 20ps2km−1.
Fig. 6
Fig. 6 (a) The temporal degree of coherence γ (t1, t2, z) of the CGSM pulse as a function of separation between two time points td for different values of pulse durations T0 = 15ps, T0 = 25ps, T0 = 35ps, respectively, and Tc = 10ps. (b) The temporal degree of coherence γ (t1, t2, z) as a function of separation between two time points td for different values of temporal coherence lengths (b) Tc = 10ps, Tc = 20ps, Tc = 30ps, respectively, and T0 = 15ps. The other parameters are n = 2 and β2 = 20ps2km−1.
Fig. 7
Fig. 7 The temporal degree of coherence γ (t1, t2, z) of the CGSM pulse as a function of separation between two time points td for different values of the second-order dispersion coefficients β2 = 20ps2km−1, β2 = 35ps2km−1, β2 = 50ps2km−1, β2 = 123ps2km−1, respectively. The other parameters are T0 = 15ps, Tc = 10ps, n = 2.
Fig. 8
Fig. 8 The functions I+(t, z), I-(t, z) and I (t, z) versus the time t for different values of (a,d) z = 4km, (b,e) z = 2km and (c,f) z = 0, respectively. The other parameters are T0 = 15ps, Tc = 10ps, n = 2, β2 = 20ps2km−1.

Equations (16)

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Γ( t 1 , t 2 )= p(v) H ( t 1 ,v)H( t 2 ,v)dv,
p(v)= T c 2π cosh( n 2π T c v )exp( T c 2 v 2 +2 n 2 π 2 ),
H(t,v)=exp( t 2 4 T 0 2 )exp( ivt ),
Γ( t 1 , t 2 )=exp( t 1 2 + t 2 2 4 T 0 2 )γ( t 1 , t 2 ),
γ( t 1 , t 2 )=cos[ n 2π ( t 2 t 1 ) T c ]exp[ ( t 2 t 1 ) 2 2 T c 2 ].
γ( t 1 , t 2 )=exp[ ( t 2 t 1 ) 2 2 T c 2 ].
Γ( t 1 , t 2 ,z )= 1 2π β 2 z Γ( t 10 , t 20 ) exp{ i 2 β 2 z [ ( t 10 2 t 20 2 )+( t 1 2 t 2 2 )2( t 10 t 1 - t 20 t 2 ) ] }d t 10 d t 20 ,
Γ( t 1 , t 2 ,z )= T 0 2 2 β 2 zΔ(z) exp[ ( t 1 t 2 ) 2 R(z) + i 2 β 2 z ( t 1 2 t 2 2 ) ][ exp( η + 2 Δ 2 (z) )+exp( η 2 Δ 2 (z) ) ],
1 R(z) = T 0 2 2 β 2 2 z 2 ,
Δ 2 (z)= 1 R(z) + 1 2 ξ 2 ,
1 ξ 2 = 1 4 T 0 2 + 1 T c 2 ,
η ± = 1 R(z) ( t 1 t 2 ) i 4 β 2 z ( t 1 + t 2 )± in 2π 2 T c .
I( t,z )=Γ( t,t,z ),
γ( t 1 , t 2 ,z )= Γ( t 1 , t 2 ,z ) Γ( t 1 , t 1 ,z ) Γ( t 2 , t 2 ,z ) .
I ( t , z ) = T 0 2 T 0 2 + ( 1 4 T 0 2 + 1 T c 2 ) β 2 2 z 2 { e x p [ - 1 2 ( t n 2 π T c β 2 z ) 2 T 0 2 + ( 1 4 T 0 2 + 1 T c 2 ) β 2 2 z 2 ] + e x p [ - 1 2 ( t + n 2 π T c β 2 z ) 2 T 0 2 + ( 1 4 T 0 2 + 1 T c 2 ) β 2 2 z 2 ] } ,
I ± ( t , z ) = T 0 2 T 0 2 + ( 1 4 T 0 2 + 1 T c 2 ) β 2 2 z 2 e x p [ - 1 2 ( t ± n 2 π T c β 2 z ) 2 T 0 2 + ( 1 4 T 0 2 + 1 T c 2 ) β 2 2 z 2 ] ,

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