Spatial-Temporal Self-Focusing of Partially Coherent Pulsed Beams in Dispersive Medium

Featured Application: The results obtained in this paper will provide a scientiﬁc reference for the applications of ultra-short pulses in laser micromachining. Abstract: Partially coherent pulsed beams have many applications in pulse shaping, ﬁber optics, ghost imaging, etc. In this paper, a novel class of partially coherent pulsed (PCP) sources with circular spatial coherence distribution and sinc temporal coherence distribution is introduced. The analytic formula for the spatial-temporal intensity of pulsed beams generated by this kind of source in dispersive media is derived. The evolution behavior of spatial-temporal intensity of the pulsed beams in water and air is investigated, respectively. It is found that the pulsed beams exhibit spatial-temporal self-focusing behavior upon propagation. Furthermore, a physical interpretation of the spatial-temporal self-focusing phenomenon is given. This is a phenomenon of optical nonlinearity, which may have potential application in laser micromachining and laser ﬁlamentation.


Introduction
As we all know, the coherence properties of a light source have a marked impact on the characteristics of the propagated field, especially on the intensity profile. Hence, it is a possible way to modulate the beam shape by regulating the coherence of the source [1][2][3][4][5][6][7]. In the past few decades, extensive work have been done to control the coherence distribution of light sources in order to obtain desired far-field intensity profile or extraordinary propagation behaviors , such as self-splitting, self-focusing, self-steering, and flat-topping, ring-shaped, rectangular, gridded, dark-hollow, or cusped intensity profiles, which have many applications in remote sensing [33], imaging [34], optical trapping and optical communications [35].
In 2017, a new class of partially coherent light sources with circular coherence was introduced [36] and synthesized through a time multiplexing approach [37]. Then, Hyde IV et al. proposed another alternative approach for synthesizing this class of sources [38]. The self-focusing propagation phenomenon of this class of sources was also founded, whose propagation properties through oceanic turbulence is also being explored [39]. All the above-mentioned references are confined to stationary optical fields or beams.
In this paper, we present a novel class of spatially and temporally PCP source with circular spatial coherence distribution and sinc temporal coherence distribution. The propagation properties of pulsed beams generated by this source in two common dispersive media, i.e., water and air are investigated. It is found that, the pulsed beams exhibit spatial-temporal self-focusing behavior upon propagation in water or air. The condition that the self-focusing takes place at spatial domain and temporal domain simultaneously is acquired. In Section 2, based on the optical coherence theory of non-stationary field, we obtain the spatial-temporal intensity distribution of the proposed PCP beams. In Section 3, we give numerical calculations, and present the spatial-temporal self-focusing behavior of the pulsed beams in water and air, respectively. In addition, a physical interpretation of the spatial-temporal self-focusing phenomenon is presented. In Section 4, we summarize the results.

Theory
Consider a statistically PCP source located in the plane z = 0, radiating a beam-like field that propagates into the positive half-space z > 0. The statistical properties of the 2D source at points ρ 1 = (x 1 , y 1 ) and ρ 2 = (x 2 , y 2 ), at different instant time t 10 and t 20 , can be characterized by the mutual coherent function (MCF), which is defined as a two-point two-time correlation function without spatiotemporal coupling: where where sincx = sinπx/πx and ω 0 is the carrier frequency of pulsed beams. w 0 and σ denotes beam width and spatial coherent parameter, respectively. T 0 and T c represent pulse duration and temporal coherent length, respectively. The beams generated by this kind of pulsed source expressed by Equation (1) are regarded as the spatially and temporally PCP beams with circular spatial and sinc temporal coherence distribution. The spatial part of mutual coherent function Equation (1) is the same as that in reference [39], which has a circular coherence distribution. Equation (1) can be also written as: where where p R (v 1 ) and p T (v 2 ) are rect weight functions of H(·). In the plane z > 0, by means of the extended Huygens-Fresnel integral with paraxial approximation, the propagation properties of PCP beams with circular spatial and sinc temporal coherence distribution in dispersive medium in spatiotemporal domain can be characterized by the following integral formula [1,62]: where k = n(ω)/c denotes the wave number, in which n(ω) is refractive index of medium and c is the speed of light in vacuum. In this paper, we assume that the medium is a linear dispersive medium whose refractive index is given by n(ω) = n a ω + n b , where n a and n b are constants. And a = ω 0 β 2 z, with β 2 representing the group velocity dispersion, which is related to n a by β 2 = n a /c. And n b = 2β 2 ω 0 c − c/v g , where v g is the group velocity of the pulse [63]. Here, the time coordinate is a retarded time, which is calculated in a frame moving with group velocity of the pulses. On substituting Equations (4) and (7) into (10), by interchanging the orders of the integrals, we can derive the formula: where and and By inserting Equations (8) and (9) into (13) and (14), respectively, after a tedious integral calculation, one can obtain: and where If we let r 1 = r 2 = r and t 1 = t 2 = t in Equations (15) and (16), respectively, we can obtain: According to Equation (11), we can get spatial-temporal intensity as: Equation (21) is the main formula derived in this paper, which is used to investigate the spatial-temporal intensity evolution of the PCP beams in dispersive medium.
If we choose the Gaussian weight function to replace Equations (5) and (6), respectively, i.e., and considering Equations (4), (7)-(9), we can obtain: where σ c = σ/ √ π and T c = T c / √ π. As can be seen from Equations (25) and (26), the spatial part and temporal parts of HCFs have non-uniform coherence distribution. And spatial and temporal parts of MCFs are the same as those in Ref. [64] and Ref. [56], respectively.

Spatial-Temporal Self-Focusing of PCP Beams in Dispersive Medium
In this section, the propagation properties of the proposed PCP beams in dispersive medium are investigated. Water and air are chosen as two common examples of dispersive medium. The evolution behavior of the proposed PCP beams upon propagation is explored by detailed numerical simulations. In the following calculation, the pulses and medium parameters are chosen to be w 0 = 1 cm, σ = 1 cm, T 0 = 6 ps, T c = 4 ps, ω 0 = 2.355 rad/fs and β 2 = 24.88 ps 2 km −1 unless different values are specified.  Figure 1 is the case of rect weight function case, which corresponds to the pulses with circular spatial and sinc temporal coherence distribution. Figure 2 is the case of Gaussian weight function, which corresponds to pulses with non-uniform coherence distribution. We choose water as the dispersive medium with refractive index n g = c/v g = 1.3425 (water, 20 • C). As can be seen, there is an intensity maximum upon propagation, i.e., the spatial-temporal self-focusing takes place upon the water medium propagation. In addition, detection time t has an important effect on the intensity distribution of pulsed beams. With increasing time t, the value of the peak intensity becomes small and the focal spot shifts slightly far from the source plane. Physically, the spatial-temporal intensities of the pulsed beams depend on the spatial positions and detection time t, respectively. As can be seen from Equation (20), detection time t has a close relationship with pulse duration T(z). Furthermore, as shown in Equation (18), T(z) is dependent of z. That is why the detection time affects intensity peak and position of peak intensity.

Water Case
Figures 1a-c and 2a-c give the evolution of normalized spatial-temporal intensity of PCP beams with different values of time t in the x-z plane, while the corresponding on-axis profile are given in Figures 1d and 2d, respectively. Figure 1 is the case of rect weight function case, which corresponds to the pulses with circular spatial and sinc temporal coherence distribution. Figure 2 is the case of Gaussian weight function, which corresponds to pulses with non-uniform coherence distribution. We choose water as the dispersive medium with refractive index ng = c/vg = 1.3425 (water, 20 °C). As can be seen, there is an intensity maximum upon propagation, i.e., the spatial-temporal self-focusing takes place upon the water medium propagation. In addition, detection time t has an important effect on the intensity distribution of pulsed beams. With increasing time t, the value of the peak intensity becomes small and the focal spot shifts slightly far from the source plane. Physically, the spatialtemporal intensities of the pulsed beams depend on the spatial positions and detection time t, respectively. As can be seen from Equation (20), detection time t has a close relationship with pulse duration T(z). Furthermore, as shown in Equation (18), T(z) is dependent of z. That is why the detection time affects intensity peak and position of peak intensity.   Figures 3d and 4d, respectively. Figure 3 is the case of rect weight function, while Figure 4 is the case of Gaussian weight function. As can be seen, the spatial-temporal self-focusing occurs not only at on-axis (x = 0) but also the off-axis (x ≠ 0). And, with increasing x, the focal spot shifts slightly  3a-c and 4a-c give the evolution of normalized spatial-temporal intensity of PCP beams with different values of x in the t-z plane, while the corresponding on-axis and off-axis profile are given in Figures 3d and 4d, respectively. Figure 3 is the case of rect weight function, while Figure 4 is the case of Gaussian weight function. As can be seen, the spatial-temporal self-focusing occurs not only at on-axis (x = 0) but also the off-axis (x 0). And, with increasing x, the focal spot shifts slightly towards the source plane. Physically, the spatial-temporal intensities of the pulsed beams depend on the spatial positions and detection time t, respectively. When detection time t is fixed, the intensity distribution has a marked impact on position coordinates, which can be seen easily from Equations (17) and (19). Figures 3a-c and 4a-c give the evolution of normalized spatial-temporal intensity of PCP beams with different values of x in the t-z plane, while the corresponding on-axis and off-axis profile are given in Figures 3d and 4d, respectively. Figure 3 is the case of rect weight function, while Figure 4 is the case of Gaussian weight function. As can be seen, the spatial-temporal self-focusing occurs not only at on-axis (x = 0) but also the off-axis (x ≠ 0). And, with increasing x, the focal spot shifts slightly towards the source plane. Physically, the spatial-temporal intensities of the pulsed beams depend on the spatial positions and detection time t, respectively. When detection time t is fixed, the intensity distribution has a marked impact on position coordinates, which can be seen easily from Equations (17) and (19).   (d) Normalized spatial-temporal intensity versus propagation distance z with t = 0. Figures 5 and 6 give the contour graph of normalized spatial-temporal intensity distribution of PCP beams at the z plane with different values of propagation distance z. Figure 5 is the case of rect weight function, while Figure 6 is the case of Gaussian weight function. It is shown that at the z = 0 plane, the pulse duration and beam spatial width is quite large. With increasing z, the pulses focus  Figures 5 and 6 give the contour graph of normalized spatial-temporal intensity distribution of PCP beams at the z plane with different values of propagation distance z. Figure 5 is the case of rect weight function, while Figure 6 is the case of Gaussian weight function. It is shown that at the z = 0 plane, the pulse duration and beam spatial width is quite large. With increasing z, the pulses focus first on the temporal dimension, then on the spatial dimension. However, at some critical distance, the pulses focus on temporal dimension and spatial dimension simultaneously (see Figure 5c   (d) Normalized spatial-temporal intensity versus propagation distance z with t = 0. Figures 5 and 6 give the contour graph of normalized spatial-temporal intensity distribution of PCP beams at the z plane with different values of propagation distance z. Figure 5 is the case of rect weight function, while Figure 6 is the case of Gaussian weight function. It is shown that at the z = 0 plane, the pulse duration and beam spatial width is quite large. With increasing z, the pulses focus first on the temporal dimension, then on the spatial dimension. However, at some critical distance, the pulses focus on temporal dimension and spatial dimension simultaneously (see Figure 5c    The spatial-temporal self-focusing behavior shown in Figures 5 and 6 can be interpreted as follows. There is some kind of relationship between spatial parameters and temporal parameters, which can be easily seen from the beam spatial width in Equation (17) and pulse duration in Equation (18) of elementary modes of PCP beams; this is because Equations (17) and (18) imply that there are The spatial-temporal self-focusing behavior shown in Figures 5 and 6 can be interpreted as follows. There is some kind of relationship between spatial parameters and temporal parameters, which can be easily seen from the beam spatial width in Equation (17) and pulse duration in Equation (18) of elementary modes of PCP beams; this is because Equations (17) and (18) imply that there are minima for the beam width and pulse duration, respectively. Hence, there are intensity maximum during propagation, when beam widths or pulse durations arrive minima at the same time. More specifically, Equations (17) and (18) can be expanded and re-written as: From Equations (27) and (28) we see that w 2 (z) and T 2 (z) are quadratic functions of z, and will reach minima when z 1 = −b 1 /2a 1 and z 2 = −b 2 /2a 2 , respectively, i.e., the intensity maximum appears at distances: where z 1,min depends on the v 1 and incident beam width w 0 . z 1,min has an internal relationship with spatial coherence length σ by the rect weight function p R (v 1 ) shown in Equation (5), while z 2,min depends on the v 2 and the pulse duration. And, z 2,min is in connection with temporal coherence length T c by the rect weight function p T (v 2 ) shown in Equation (6). Hence, when the self-focusing takes place at spatial and temporal dimensions simultaneously, the relationship between spatial parameters and temporal parameters can be obtained as follows: After a comparison between rect weight function and Gaussian weight function, it was seen that the intensity profiles of PCP beams with rect weight function are similar to the case of Gaussian weight function. However, specifically, the spatial-temporal self-focusing effect of the latter is more noticeable and the focal spot is much small-the value of peak intensity can reach 17.4 for the Gaussian weight function case. Nevertheless, the value is 9.3 for the rect weight function. In addition, the positions of focal spots where self-focusing in the spatial and temporal dimensions takes place simultaneously are different from each other. The positions of focal spot are z = 0.2 km and z = 0.04 km for the rect weight function and Gaussian weight function, respectively. That is to say, the spatial-temporal self-focusing effect of the latter takes place more early.

Air Case
In the following calculation, we present the evolution behavior of PCP beams in air medium. Because the refractive index n g = 1.00028 of air is much closer to the 1, and the second-order dispersion coefficient β 2 = 0.021233 ps 2 km −1 is very small. Hence, we choose more short pulse duration and temporal coherence length, namely, T 0 = 200 fs and T c = 100 fs in order to present the self-focusing more clearly. The other parameters are the same as the case of water. Figures 7 and 8 give the contour graph of normalized spatial-temporal intensity distribution I(x, t, z) of PCP beams at z plane with different values of z. Figure 7 is the case of rect weight function, while Figure 8 is the case of Gaussian weight function. From Figures 7 and 8, we can see that the similar spatial-temporal self-focusing phenomenon takes place, i.e., the pulses focus first on the temporal dimension, then focus on the spatial dimension with increasing propagation distance z. At some critical distance, the spatial-temporal self-focusing phenomenon takes place (see Figure 7c z = 0.13 km and Figure 8c z = 0.04 km). Compared with rect weight function, for the case of Gaussian weight function, the spatial-temporal self-focusing behavior appears more early and noticeable. Physically, Gaussian weight function compared with rect weight function has better energy focusability.   Figure 9 gives the on-axis and off-axis normalized intensity profile of PCP beams in air for the rect weight function (a,b) and Gaussian weight function (c,d), respectively. In general, the similar self-focusing phenomenon can be found for the rect weight function and Gaussian weight function of PCP beams. What is more, the self-focusing effect of PCP beams with Gaussian weight function is   Figure 9 gives the on-axis and off-axis normalized intensity profile of PCP beams in air for the rect weight function (a,b) and Gaussian weight function (c,d), respectively. In general, the similar self-focusing phenomenon can be found for the rect weight function and Gaussian weight function of PCP beams. What is more, the self-focusing effect of PCP beams with Gaussian weight function is  Figure 9 gives the on-axis and off-axis normalized intensity profile of PCP beams in air for the rect weight function (a,b) and Gaussian weight function (c,d), respectively. In general, the similar self-focusing phenomenon can be found for the rect weight function and Gaussian weight function of PCP beams. What is more, the self-focusing effect of PCP beams with Gaussian weight function is more noticeable than the case of rect weight function. Mathematically, there are high degrees of resemblance between rect function and Gaussian function. Physically, Gaussian weight function has better energy focusability than rect weight function.

Conclusions
In the present work, a novel class of spatially and temporally PCP source with circular spatial and sinc temporal coherence distribution is introduced. The evolution of pulsed beams generated by this kind of source in dispersive media is investigated. Water and air are chosen as two typical examples of dispersive medium. It is found that the pulsed beams exhibit spatial-temporal selffocusing behavior upon water/air propagation. The relationship between spatial parameters and temporal parameters of pulsed beams is acquired, where self-focusing takes place at spatial and temporal dimension simultaneously. A detailed comparison between the pulsed beams and the normal non-uniformly correlated PCP beams is performed. The results show that the spatialtemporal self-focusing phenomenon of non-uniformly correlated PCP beams with Gaussian weight function is more noticeable, and takes place more early. The results obtained can have potential applications in laser micromachining and laser filamentation [65]. For example, in laser micromachining, a proposed pulsed source might be generated in order to obtain a self-focusing spot upon propagation, where a suitable peak intensity is adopted to ablate the proposed material. In the process of micromachining, the profile of the focal spot can be modulated to improve fabrication resolution. In addition, in the field of laser filamentation, the intensity inside the filament can be enhanced by manipulating the mutual coherent function (MCF) of proposed pulsed source using a Spatial Light Modulater (SLM).
Author Contributions: Z.Z. Data Curation, writing-original draft, writing-review and editing; C.D. formal analysis, methodology; Y.Z. formal analysis; L.P. project administration. All authors read and approved the final manuscript.
Funding: This research was funded by the National Natural Science Foundation of China, grant numbers 61575091 and 61675094.

Conflicts of Interest:
The authors declare no conflict of interest.

Conclusions
In the present work, a novel class of spatially and temporally PCP source with circular spatial and sinc temporal coherence distribution is introduced. The evolution of pulsed beams generated by this kind of source in dispersive media is investigated. Water and air are chosen as two typical examples of dispersive medium. It is found that the pulsed beams exhibit spatial-temporal self-focusing behavior upon water/air propagation. The relationship between spatial parameters and temporal parameters of pulsed beams is acquired, where self-focusing takes place at spatial and temporal dimension simultaneously. A detailed comparison between the pulsed beams and the normal non-uniformly correlated PCP beams is performed. The results show that the spatial-temporal self-focusing phenomenon of non-uniformly correlated PCP beams with Gaussian weight function is more noticeable, and takes place more early. The results obtained can have potential applications in laser micromachining and laser filamentation [65]. For example, in laser micromachining, a proposed pulsed source might be generated in order to obtain a self-focusing spot upon propagation, where a suitable peak intensity is adopted to ablate the proposed material. In the process of micromachining, the profile of the focal spot can be modulated to improve fabrication resolution. In addition, in the field of laser filamentation, the intensity inside the filament can be enhanced by manipulating the mutual coherent function (MCF) of proposed pulsed source using a Spatial Light Modulater (SLM).