## 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 J_{0}-correlated Schell-model sources [3, 4], the Multi-Gaussian Schell-model sources [5–7], 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 J_{0}-correlated Schell-model sources have properties analogous to those of the Bessel-Gaussian beams but their degree of coherence does not keep the J_{0}(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 [14–24]. 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*(*t*_{1},*t*_{2}) = <*E**(*t*_{1})*E* (*t*_{2})>, 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*(*t*_{1},*t*_{2}) 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

*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:

*T*is the r.m.s source correlation determining the temporal degree of coherence of a pulse,

_{c}*n*is a positive real constant, not necessarily an integer, cosh(

*x*) is the hyperbolic cosine function,

*T*

_{0}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

Equation (5) denotes the temporal degree of coherence of the CGSM pulse at a pair of time instants, *t*_{1} and *t*_{2} in the source plane. If we set *n* = 0, *γ* (*t*_{1}, *t*_{2}) can be expressed as

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]

*β*

_{2}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.

From the mutual coherence function (8), the average intensity *I*(*t*, *z*) and the temporal degree of coherence *γ* (*t*_{1}, *t*_{2}, *z*) of the CGSM pulse in the second-order dispersive media can be calculated by the expressions [19]

## 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) *T*_{0} = 15*ps*, (b)
*T*_{0} = 25*ps*, and (c) *T*_{0}
= 35*ps*. 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 *T*_{c} = 10*ps*,
*n* = 2, *β*_{2} =
20ps^{2}km^{−1}. As shown in Fig.
1, on dispersive-media propagation at critical distance of
*z*_{c} = 2*km* 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 *z*_{c} of the two resulting pulses moves farer away
from the incident pulse and the CGSM pulse broadens upon propagation.

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) *T*_{c} = 10*ps*,
(b) *T*_{c} = 20*ps*, (c) *T*_{c} =
30*ps*. 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 *T*_{0} = 15*ps*,
*n* = 2, *β*_{2} =
20ps^{2}km^{−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
*z*_{c}, at which the pulse begins to split, moves far away from the
incident pulse and the two split pulses become closer.

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 *T*_{0} = 15*ps*,
*T*_{c} = 10*ps*, *β*_{2} =
20ps^{2}km^{−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 *z*_{c}, where the pulse begins to split, moves
towards the incident pulse and the two split pulses become far away from each other.

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) *β*_{2} =
20*ps*^{2}*km*^{−1}, (b)
*β*_{2} =
35*ps*^{2}*km*^{−1}, (c)
*β*_{2} =
50*ps*^{2}*km*^{−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
*T*_{0} = 15*ps*, *T*_{c} =
10*ps*, *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
*β*_{2} (see Figs. 4(b),
4(e), 4(c) and
4(f)), the critical propagation distance
*z*_{c}, at which the pulse begins to split, moves towards the incident
pulse and the two split pulses become far away from each other.

Figure 5 shows the temporal degree of coherence
*γ* (*t*_{1}, *t*_{2},
*z*) of the CGSM pulse as a function of separation between two time points
*t*_{d} = *t*_{1}-*t*_{2}
at some propagation distances *z*. The calculation parameters are (a)
*n* = 0, (b) *n* = 2, (c) *n* = 4, and the other
parameters are *T*_{0} = 15*ps*,
*T*_{c} = 10*ps* and
*β*_{2} = 20ps^{2}km^{−1}. As shown in
Fig. 5, the temporal degree of coherence
*γ* (*t*_{1}, *t*_{2},
*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.

Figure 6(a) shows the temporal degree of coherence
*γ* (*t*_{1}, *t*_{2},
*z*) of the CGSM pulse as a function of separation between two time instants
*t*_{d} for different values of pulse duration
*T*_{0} = 15*ps*, *T*_{0} =
25*ps*, *T*_{0} = 35*ps*, respectively,
and *T*_{c} = 10*ps*. 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* (*t*_{1}, *t*_{2},
*z*) as a function of separation between two time instants
*t*_{d} for different values of temporal coherence length
*T*_{c} = 10*ps*, *T*_{c} =
20*ps*, *T*_{c} = 30*ps*, respectively,
and *T*_{0} = 15*ps*. The other parameters are
*n* = 2 and *β*_{2} =
20ps^{2}km^{−1}. As shown in Fig.
6(b), the profiles of the temporal degree of coherence move towards the large value
*t*_{d} with increasing temporal coherence length of the incident
pulse.

Figure 7 shows the temporal degree of coherence
*γ* (*t*_{1}, *t*_{2},
*z*) of the CGSM pulse as a function of separation between two time instants
*t*_{d} for different values of the second-order dispersion coefficients
*β*_{2} =
20*ps*^{2}*km*^{−1},
*β*_{2} =
35*ps*^{2}*km*^{−1},
*β*_{2} =
50*ps*^{2}*km*^{−1},
*β*_{2} =
123*ps*^{2}*km*^{−1}, respectively. The
other parameters are *T*_{0} = 15*ps*,
*T*_{c} = 10*ps*, *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 *β*_{2}.

## 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*) =

*I*

_{+}(

*t*,

*z*) +

*I*

_{-}(

*t*,

*z*), in which

*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*= 4

*km*, (b,e)

*z*= 2

*km*and (c,f)

*z*= 0, respectively. The other parameters are

*T*

_{0}= 15

*ps*,

*T*

_{c}= 10

*ps*,

*n*= 2,

*β*

_{2}= 20ps

^{2}km

^{−1}. As shown in Fig. 8, as the propagation distance

*z*= 4

*km*, 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.

## 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 (*T*_{0}, *T*_{c}, *n*) and the propagation medium coefficient *β*_{2} play the important role for the evolution. We have found that at some critical propagation distances *z*_{c}, 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 *z*_{c}, 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 *β*_{2}, the critical propagation distance *z*_{c}, 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 *β*_{2} 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).

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