Self-focusing and self-splitting properties of partially coherent temporal pulses propagating in dispersive media

Huilong Liu; Zhenhua Du; Yuzhao Li; Hong Chen; Yanfei Lü

doi:10.1364/OE.481555

1. Introduction

It is universally acknowledged that, for quasi-homogeneous wide-sense stationary sources, the spatial source correlation functions play important role in the spatial intensity evolution behaviors of statistically stationary fields [1]. So far, there are various stationary sources with different far field intensity profiles have been proposed, for instance, the Gaussian Schell-model sources due to their tractability and universality [1], the J₀-correlated sources having a Gaussian intensity profile [2,3], the multi-Gaussian Schell-model sources shaping flat-topped profiles [4], elliptical multi-Gaussian Schell-model sources with elliptical flat-topped profiles [5], rectangular multi-Gaussian Schell-model sources with rectangular flat-topped profiles in far fields [6], the Laguerre-Gaussian correlated Schell-model and Bessel–Gaussian Schell-model beams forming dark-hollow beam (the central dark region beam) in the far-field plane [7], the cosine-Gaussian Shell-model sources producing shape-invariant ring-shaped intensity distribution [8], rectangular cosine-Gaussian Shell-model displaying a four-beamlet array profile intensity in the far field [9], rectangular Laguerre-Gaussian correlated Schell-model exhibiting self-splitting and self-combining features during transmission [10], and rectangular Hermite-Gaussian correlated Schell-model sources evolving into two or four beam spots in the far field [11], the GSM array sources producing far fields with optical lattice average intensity patterns [12], the circular multi-sinc Schell-model sources showing multi-rings far field distribution and rectangular multi-sinc Schell-model sources generating lattice intensity patterns in the far field [13], the multi-cosine Laguerre Gaussian correlated Schell-model sources producing a ring-shaped optical array profile consisting of dark-hollow-shaped beamlets [14], the rectangular sinc-correlated sources have self-focusing features during propagation and rectangular correlated vortex sources are with a rotation of the spectral density around the axis upon propagation [15], the twisted sinc-correlation Schell-model sources are presented and theirs irradiance profile of light intensity always rotates to 90 degree [16], the self-focusing formation of a partially coherent beam with a non-uniform correlation structure in a non-linear medium is investigated [17], and non-uniformly correlated sources with self-focusing effects and laterally shifted intensity maxima upon propagation [18–24].

All the above sources have a limitation of the frame of a spatial domain to produce broad sense statistically stationary sources. Recently, intensive efforts have been devoted to extending the stationary optical fields into the non-stationary optical fields owing to their various applications in optical imaging, pulse shaping, fiber optics, and ghost imaging [25–32]. Researchers have proposed a series of pulse sources, of which correlated functions distribution complying with non-Gaussian distribution and the evolution behaviors propagating through dispersive media have been examined. The non-uniformly correlated pulses have been considered and shown accelerating or decelerating properties of the maximum peak of the intensity during propagation [33]. The multi-Gaussian Schell-model pulse sources will shape the pulse flat-topped intensity and the width of the flat center can be adjusted by source parameters [34]. The cosine-Gaussian Shell-model, Laguerre-Gaussian correlated Schell-model and Hermite-Gaussian correlated Schell-model pulses have time-domain self-splitting properties, and two or more peak pulses will occur at the temporal far field [35,36]. The sinc Schell-model pulses on propagation in a dispersive medium can produce flat-topped intensity profiles of controllable width and height [37]. The electromagnetic SSM pulses will transform into a double-layer flat-top distribution from the Gaussian distribution of the source plane during the propagation process [38]. The temporal pulse source with fractional multi-Gaussian correlated Schell-model function will form a strongly sharp pulse intensity pattern when it propagates into far fields [39]. Temporal analogues of the complex screen and phase screen methods are developed and the intensity evolution of the cosine-Gaussian correlated Schell-model pulse in a nonlinear dispersive medium is explored [40]. According to the above analysis, few sources have self-splitting properties or form a controlled number of subpulses in far fields.

In this article, our aim is to discuss a novel class of partially coherent pulse sources with temporal MCGCSM correlations. The evolution behaviors of the MCGCSM pulse beams on propagation through the dispersive media are investigated. The interesting phenomenon is that different from reported partially coherent pulse sources, the MCGCSM pulses are able to produce far fields with a flat-topped profile pulse or split from a single pulse of the source plane into multiple subpulses by adjusting the source parameters.

2. Theoretical formulation

Suppose a beam-like pulse field in the space-time domain, whose generation is from a planar pulse source in the plane z = 0, propagates into the half-space z > 0. Its coherence properties can be described by the TMCF as the following form

(1)$$\varGamma ({{t_1},{t_2}} )= \left\langle {{E^\ast }({{t_1}} )E({{t_2}} )} \right\rangle .$$

where the brackets denote an ensemble averaging, E(t) is the complex analysis signal realized by the pulse at time t, and the asterisk (∗) represents the complex conjugation. A variety of functions can be chosen as the TMCF, but it must meet the nonnegative definiteness constraint. The TMCF with the well-defined condition is given as a superposition integral as [1]

(2)$$\varGamma \left( {t_1,t_2} \right) = \int {p\left( v \right)} H^*\left( {t_1,v} \right)H\left( {t_2,v} \right)dv,$$

where H (t,v) is an arbitrary kernel with Fourier-transformable form leading to the Schell-model source class. Equation (2) can be considered as an incoherent superposition of kernel H (t,v) weighted by function p (v). The function p (v) is a suitable nonnegative function, whose choice decides the profile of the correlation function and the generation of the prescribed far-field intensity distributions.

The function H(t, v) determines the correlation class with Fourier-transformable form, and p (v) plays a dominant role in the distributions of the spectral density in far fields. To generate a temporal pulse source shaping far fields with a flat top profile or multiple subpulses, we set the functions H (t,v) and p (v) as the following expression

(3)$${H}({t,v} )= \exp \left[ { - \frac{{{t^2}}}{{4T_0^2}}} \right]\exp [{ - ivt} ],$$

(4)$$p(v )= \frac{{\sqrt \pi {T_c}}}{{\sqrt 2 N}}\sum\limits_{n ={-} {{({N - 1} )} / 2}}^{{{({N - 1} )} / 2}} {\{{\exp [{ - 2{\pi^2}{{({{T_c}v + n{T_s}} )}^2}} ]+ \exp [{ - 2{\pi^2}{{({{T_c}v - n{T_s}} )}^2}} ]} \}} ,$$

where T_c is the r.m.s. source correlation width determining the TDOCs of the pulse. T₀ is the pulse duration. T_s is a positive real constant and N takes a positive integer. By some simple calculations, Eq. (4) can also be written as

(5)$$p(v )= \frac{{\sqrt {2\pi } {T_c}}}{N}\exp ({ - 2{\pi^2}{T_c}v} )\sum\limits_{n ={-} {{({N - 1} )} / 2}}^{{{({N - 1} )} / 2}} {\cosh ({ - 4{\pi^2}n{T_c}{T_s}v} )\exp ({ - 2{\pi^2}{n^2}T_s^2} )} ,$$

where cosh (t) denotes the hyperbolic cosine function. Since cosh (t) is larger than zero for any t, the weight p(v) is a non-negative function.

After substituting from Eqs. (5) and (3) into Eq. (2), the TMCF of the MCGCSM pulse can be given by

(6)$$\varGamma ({{t_1},{t_2}} )= \exp \left[ { - \frac{{t_1^2 + t_2^2}}{{4T_0^2}}} \right]\mu ({{t_1},{t_2}} ),$$

with

(7)$$\mu ({{t_1},{t_2}} )= \frac{1}{N}\exp \left[ { - \frac{{{{({{t_1} - {t_2}} )}^2}}}{{2T_c^2}}} \right]\sum\limits_{n ={-} {{({N - 1} )} / 2}}^{{{({N - 1} )} / 2}} {\cos \left[ {\frac{{2\pi n{T_s}({{t_1} - {t_2}} )}}{{{T_c}}}} \right]} .$$

Equation (7) shows the TDOC of the MCGCSM pulse field at a pair of time t₁ and t₂ in the source plane. Figure 1 shows the profiles of the TDOC of the MCGCSM pulse in the source plane for different values of parameter N. The Gaussian term in Eq. (7) is an action of envelope modulation, i.e., it modulates the whole profiles of the TDOC into Gaussian distributions. For the odd N, the number of the main peak of the TDOC is 9. As N increases, a sub-peak occurs between the adjacent main peak and its number is (N-1)/2-1. For the even N, the curves oscillate more significantly in the negative range of the TDOC.

Fig. 1. The temporally degree of coherence of the MCGCSM pulse with T_c= 1 ps and T_s= 1.5 in the source plane.

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With help of the generalized Collins formula in the temporal domain, the evolution dynamics of the MCGCSM pulse propagating through an optical ABCD system could be studied. We can obtain the propagation expression of the TMCF in the dispersive media which can be treated by the following integral formulas [41,42]

(8)$$\varGamma ({{\tau_1},{\tau_2},z} )= \frac{{{\omega _0}}}{{2\pi B}}\int\!\!\!\int {{\varGamma ^{(0 )}}({{t_1},{t_2}} )\exp \left\{ {\frac{{i{\omega_0}}}{{2B}}[{A({t_1^2 - t_2^2} )+ D({\tau_1^2 - \tau_2^2} )- 2({{t_1}{\tau_1} - {t_2}{\tau_2}} )} ]} \right\}} d{t_1}d{t_2},$$

where A, B and D are the elements of the transfer matrix for the temporal optical system. ω₀ is the carrier frequency of pulse. An assumption that the measurement of time coordinate is in a reference frame with the group velocity of the pulse is taken.

On substituting from Eq. (6) into Eq. (8) and performing tedious but straightforward integrations, the analytical expression of the TMCF of the MCGCSM pulse beams in the dispersive media yields the following expression

(9)$$\begin{aligned} \varGamma ({{\tau_1},{\tau_2}} )&= \frac{{{\omega _0}}}{{4BN}}\sqrt {\frac{{2T_0^2}}{\Pi }} \exp \left[ {\frac{{i{\omega_0}D}}{{2B}}({\tau_1^2 - \tau_2^2} )} \right]\\ &\quad \times \exp \left[ { - \frac{{\omega_0^2T_0^2}}{{2{B^2}}}{{({{\tau_1} - {\tau_2}} )}^2}} \right]\sum\limits_{n ={-} {{({N - 1} )} / 2}}^{{{({N - 1} )} / 2}} {\left[ {\exp \left( {\frac{{\varGamma_ +^2}}{\Pi }} \right) + \exp \left( {\frac{{\varGamma_ -^2}}{\Pi }} \right)} \right]} , \end{aligned}$$

(10)$$\frac{1}{{{\Omega ^2}}} = \frac{1}{{4T_0^2}} + \frac{1}{{T_c^2}},$$

(11)$$\Pi = \frac{1}{2}\left( {\frac{{\omega_0^2{A^2}T_0^2}}{{{B^2}}} + \frac{1}{{{\Omega ^2}}}} \right),$$

(12)$${\varGamma _ \pm } = \frac{{i{\omega _0}}}{{4B}}({{\tau_1} + {\tau_2}} )- \frac{{A\omega _0^2T_0^2}}{{4{B^2}}}({{\tau_1} - {\tau_2}} )\pm \frac{{i\pi n{T_s}}}{{{T_c}}}.$$

Now, the TMCF of the MCGCSM pulse beams spreading through an optical ABCD system at the output screen z is obtained. As a case of the pulse beam in a linearly dispersive medium, the temporally optical ABCD matrix is given by

(13)$$\left( {\begin{array}{cc} A&B\\ C&D \end{array}} \right) = \left( {\begin{array}{cc} {1 + s{\beta_2}z}&{{\omega_0}{\beta_2}z}\\ {{s / {{\omega_0}}}}&1 \end{array}} \right),$$

where β₂ and s are the dispersion and the chirp coefficient, respectively. By insetting Eq. (13) into Eq. (9) and letting τ₁=τ₂=τ, the average intensity distribution expression for the MCGCSM pulse beams propagation on the dispersive media has the form as

(14)$$\begin{aligned} \varGamma ({\tau ,z} )&= \frac{1}{{2N}}\sum\limits_{n ={-} {{({N - 1} )} / 2}}^{{{({N - 1} )} / 2}} {\frac{1}{{T(z )}}} \\ &\quad \times \left\{ {\exp \left[ { - \frac{1}{{T_0^2{T^2}(z )}}{{\left( {\frac{\tau }{2} + \frac{{\pi n{T_s}{\beta_2}z}}{{{T_c}}}} \right)}^2}} \right] + \exp \left[ { - \frac{1}{{T_0^2{T^2}(z )}}{{\left( {\frac{\tau }{2} - \frac{{\pi n{T_s}{\beta_2}z}}{{{T_c}}}} \right)}^2}} \right]} \right\}, \end{aligned}$$

where

(15)$${T^2}(z )= {({1 + s{\beta_2}z} )^2} + \frac{{\beta _2^2{z^2}}}{{{\Omega ^2}T_0^2}}.$$

For the propagation of the MCGCSM pulse beams in dispersive media, the evolution dynamics of the TDOC can be calculated by the following formula

(16)$$\mu ({{\tau_1},{\tau_2},z} )= \frac{{\varGamma ({{\tau_1},{\tau_2},z} )}}{{\sqrt {\varGamma ({{\tau_1},{\tau_1},z} )} \sqrt {\varGamma ({{\tau_2},{\tau_2},z} )} }}.$$

3. Numerical results and analysis

In this section, applying the formulae obtained in the last section, the numerical calculations are applied to reveal the temporal propagation dynamics of the MCGCSM pulse beams spreading in dispersive media with a chirper or without a chirper. The parameters of the pulse and medium are chosen as s = -2 × 10⁻² ps^-2, β₂= 20 ps²/km, T₀= 10 ps, T_c= 5 ps, and T_s= 1.5, these values will be used as calculation parameters unless otherwise stated.

Figure 2 shows the evolution of the average intensity of the MCGCSM pulse beams propagating in dispersive media in τ-z plane for several values N when the chirp coefficient s = 0. One can see from Fig. 2 that the MCGCSM pulse beams exhibit self-splitting properties during propagation and evolve from a single pulse of the source plane into N subpulses in the far field. For the odd N, there is a subpulse on the optical axis, while for the even N, no subpulse is formed along the optical axis. To show more self-splitting characteristics, the influences of the source correlation width T_c, the pulse duration T₀, the dispersion coefficient β₂, and the real constant T_s on the average intensity of the MCGCSM pulse beams at propagation length z = 3 km are shown in Fig. 3. One can find that the average intensity maximum of subpulse becomes larger with the increment of T₀ and T_c or reduction of β₂. The interval between adjacent subpulse is invariant for the change of T₀, but it will broaden as T_c decreases or β₂ increases. In Fig. 3(b), the variation of T_s will not affect the peak value of the subpulses, but it will lead to the change of the whole intensity profile, i.e., when T_s= 1.5, each subpulse is completely separated, and but with the decrease in T_s, the intensity between adjacent subpulse is non-zero and the whole intensity distribution will be narrower.

Fig. 2. The evolution of the average intensity of the MCGCSM pulse beams propagating through dispersive media with s = 0 in τ-z plane for several values of N (a) N = 2, (b) N = 3, (c) N = 4, (d) N = 5.

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Fig. 3. Transverse average intensity distributions of the MCGCSM pulse beams atpropagation length z = 3 km (a) for different the pulse duration T₀, (b) for different real constant T_s, (c) for different the source correlation width T_c, and (d) for different the dispersion coefficient β₂.

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In Fig. 4 we show the propagation of the MCGCSM pulse beams with T_c= 1ps, T_s= 0.25, and N = 4. It can be found that unlike the propagation of the MCGCSM pulse beams in Fig. 2, no self-splitting phenomena can be observed in Fig. 4 due to the reduction of T_s. From Fig. 4(b) we find that the pulse beams can generate flat intensity profiles as the propagation distance grows. The longer the propagation distance is, the smaller the height of intensity and the flatter the pattern of intensity becomes. The influences of the parameter N on the propagation of the MCGCSM pulse beams generating far fields with flat-topped distributions are shown in Fig. 5. It can be seen that the average intensity distributions of the MCGCSM pulse beams for any N are the same Gaussian profile in the source plane. With the growing propagation distance, the smaller the height of intensity becomes. At sufficiently long distances, the average intensity distributions of the MCGCSM pulse beams exhibit a flatter pattern. The larger the N is, the lower the height of on-axis intensity and the flatter the profile of intensity is, and the shorter the propagation distance is for the generation of a flat top pattern.

Fig. 4. Propagation of the MCGCSM pulse beams with T_c= 1 ps, T_s= 0.25 and N = 4 (a) The evolution of the average intensity as a function of propagation distance z and time τ. (b) Transverse average intensity distributions at several propagation distances

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Fig. 5. Transverse average intensity distributions of the MCGCSM pulse beams at several propagation distances (a) 0 km, (b) 0.5 km, (c) 1.5 km, (d) 3 km. Several curves correspond to different values of N: N = 4 black curves, N = 5 red curves, N = 6 blue curves, N = 7 green curves. Other parameters are shown in Fig. 4.

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Figure 6 illustrates the evolution properties of average intensity of the MCGCSM pulse beams in the presence of dispersive media with a chirp coefficient s = -2 × 10⁻² ps^-2 as a function of distance z and time τ for several values of N. It clearly shows that the MCGCSM pulse beams propagating through dispersive media with a chirp coefficient s = -2 × 10⁻² ps^-2 display the self-focusing phenomenon and two self-focusing processes can be observed, one of them occurs near the source plane, and the other is on the subpulses. The focus spot intensity of the former is larger than that of the latter, so we in turn call them the main maximum intensity and secondary maximum intensity. As the value of N increases, the secondary maximum intensity will weaken.

Fig. 6. The evolution of average intensity of the MCGCSM pulse beams in dispersive media with a chirp coefficient s = -2 × 10⁻² ps^-2 for several values of N (a) N = 3, (b) N = 4, (c) N = 5, (d) N = 6.

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Figure 7 shows the evolution of average intensity of the MCGCSM pulse beams with N = 4 propagating through dispersive media as a function of distance z and time τ for several values of chirp coefficient s. Figure 8 shows the transverse average intensity profile in Fig. 7 at propagation distances of 0.5 km and 3 km. As can be seen in Fig. 7(a), the two self-focusing processes can be clearly observed for s = -2 × 10⁻² ps^-2. The self-focusing properties are unobvious for s = -1 × 10⁻² ps^-2. In Fig. 7(c) and (d), no self-focusing phenomenon can be observed and the self-splitting properties also disappear. From Fig. 8(a) we can see clearly that the main maximum intensity goes through from existence to nonexistence when s changes from -2 × 10⁻² ps^-2 to 2 × 10⁻² ps^-2. In Fig. 8(b) the chirp coefficient s plays an important role in the distributions of the average intensity of the MCGCSM pulse beams in the far field. The profile of average intensity evolves from a flat top pattern to Gaussian-like distribution when s changes from 1 × 10⁻² ps^-2 to 2 × 10⁻² ps^-2. Combining Fig. 7 and Fig. 2(c), we can come to the conclusion that the self-splitting phenomenon occurs and is reduced to disappear when s changes from -2 × 10⁻² ps^-2 to 0.

Fig. 7. The evolution of average intensity of the MCGCSM pulse beams with N = 4 propagating through dispersive media with a chirper as a function of distance z and time τ for several values of chirp coefficient s (a) s = -2 × 10⁻² ps^-2, (b) s = -1 × 10⁻² ps^-2, (c) s = 1 × 10⁻² ps^-2, (d) s = 2 × 10⁻² ps^-2.

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Fig. 8. Transverse average intensity distributions in Fig. 7 at propagation distance (a) 0.5 km and (b) 3 km.

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Figure 9 shows the influences of the pulse parameters and medium parameters on the evolution of the average intensity in the optical axis of the MCGCSM pulse beams versus the propagation distance z. In Fig. 2 we obtain that a subpulse will be evolved from the pulse in the source plane only for the odd N and its transmission characteristics perfectly reflect properties of other subpulses, so we chose N = 5 in Fig. 9. In Fig. 9(a) we also plot the evolution of on-axis average intensity of the MCGCSM pulse beams with N = 1, one can see that only a strong self-focusing process occurs during the propagation. The physical fact is that the MCGCSM pulse beams with N = 1 will reduce to GSM pulse beam which will not split on its own. It should be noted that the two self-focusing processes of MCGCSM pulse beams with N > 1 caused by the self-splitting properties, so the position of secondary maximum intensity is equal to the focal position of the MCGCSM pulse beams with N = 1. Of course, we can also determine the self-splitting properties of the MCGCSM pulse beams in accordance with the transmission distance z_p corresponding to the average intensity minima between two self-focusing processes, i.e., the self-splitting of the MCGCSM pulse beams has completed when it propagates to z_p. The MCGCSM pulse beams for any N > 1 has the same z_p (see Fig. 9(a)). In Fig. 9(b) the alteration of T_s is independent of the value and position of secondary maximum intensity. From Fig. 9(c) and (d) one finds that as the T₀ or T_c decreases, the value of secondary maximum intensity is reduced, and its focal position is closer to the source plane. The influence of the dispersion coefficient β₂ on the evolution of the on-axis average intensity of the MCGCSM pulse beams is shown in Fig. 9(e). It can be seen that the variation of dispersion coefficient β₂ is unrelated with the values of the main maximum intensity and secondary maximum intensity, but with addition of β₂, their focal positions shift toward the source plane. In Fig. 9(f) the occurrence of self-focusing states is on the condition of s < 0, otherwise no self-focusing can be observed for s ≥ 0.

Fig. 9. On-axis average intensity of the MCGCSM pulse beams versus the propagation distance z for different values of (a) N, (b) T_s, (c) T₀, (d) T_c, (e) β₂, (f) s.

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Next, we will give a physical interpretation to explain the self-focusing process of the MCGCSM pulse beams during propagation. Setting T²(z) = 0 in Eq. (15) which is a quadratic equation related with propagation distance z and after solving for the derivative of T(z) over z, we can obtain

(17)$${z_m} ={-} \frac{{{\Omega ^2}T_0^2s}}{{{\beta _2}({{\Omega ^2}T_0^2{s^2} + 1} )}}.$$

z_m is the focal position of secondary maximum intensity. Inserting the parameters of Fig. 9 (a) into Eq. (17), we can obtain z_m= 2.293 km which is consistent with the focal position of secondary maximum intensity in Fig. 9 (a). It is clear from Eq. (17) that z_m is dependent of T₀, T_c, β₂ and s, but is independent of N and T_s. According to Eq. (17), some results in Fig. 9 are easy to understand. To clearly explain how the chirp coefficient s affects the self-focusing characteristics, the temporal ABCD matrix from Eq. (13) can be rewritten as

(18)$$\left( {\begin{array}{cc} {1 + s{\beta_2}z}&{{\omega_0}{\beta_2}z}\\ {{s / {{\omega_0}}}}&1 \end{array}} \right) = {M_1} \times {M_2} = \left( {\begin{array}{cc} 1&{{\omega_0}{\beta_2}z}\\ 0&1 \end{array}} \right) \times \left( {\begin{array}{cc} 1&0\\ {{s / {{\omega_0}}}}&1 \end{array}} \right).$$

where M₁ demotes beam transfer matrix of after a propagation length of z'=ω₀βz in free space. The matrix M₂ is transfer matrix of a lens, and its focal length is l, ${{{{ - 1} / l} = s} / {{\omega _0}}}$. When the chirp coefficient s < 0, the matrix M₂ represents the focusing lens. That is to say, the Eq. (18) represents the beam passes through a focusing lens with focal length –ω₀/s, and then propagates a distance of z’ in free space. When the chirp coefficient s > 0, M₂ represents obviously a negative lens. According to the interpretation given above, it is easy to understand the self-focusing phenomena of the MCGCSM pulse beams spreading through dispersive media, shown in Fig. 7, Fig. 8 and Fig. 9(f).

Figure 10 and 11 show the TDOCs of the MCGCSM pulse beams propagating through dispersive media in Fig. 2 and that in Fig. 4, respectively. We can see from Fig. 10 that the curves of the TDOCs in the source plane are closely related with N, and as the propagation length grows, the oscillations of the curves gradually weaken. The evolution process of the TDOCs of the pulse beams with even N is inconsistent with that of the pulse beams with odd N. At the same propagation distance in the far field, the latter will ultimately transform into Gaussian-like distribution, but the former still has oscillations. In Fig. 11 the profiles of the TDOCs of the MCGCSM pulse beams with different values of N in the source plane are similar to that of the MGSM pulse beams with different values of M in Ref. [34]. The MGSM pulse beams can generate far fields with a flat center of pulse intensity profile and the larger the value of M is, the flatter the pulse profile becomes. As expected, the MCGCSM pulse beams also can produce optical fields with flat top profiles, and the larger the value of N is, the flatter the pulse profile will be. In Fig. 11(b) with growing the propagation distance, the TDOCs of the MCGCSM pulse beams turn into Gaussian-like profiles in the far field, which is in agreement with the evolution of the spectral DOCs of the statistically stationary MGSM beams [4].

Fig. 10. The TDOCs of the MCGCSM pulse beams propagating through dispersive media for several propagation distances and values of N. The other parameters are shown in Fig. 2.

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Fig. 11. The TDOCs of the MCGCSM pulse beams propagating through dispersive media for several propagation distances and values of N. The other parameters are shown in Fig. 4.

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Figure 12 shows the TDOCs of the MCGCSM pulse beams propagating through dispersive media for several propagation distances and values of chirp coefficient s for N = 4. One can find that the distributions of TDOCs at the source plane are the same for different values of the chirp coefficient s. This is because the pulse beam has not entered the dispersive media, the chirp coefficient s will not affect TDOCs distributions. The distribution profile of TDOCs will increase with increasing s from negative to positive. This can be explained by Eq. (18), the dispersive medium can be regarded as a focusing lens when s < 0, and a divergent lens when s > 0.

Fig. 12. The TDOCs of the MCGCSM pulse beams propagating through dispersive media for several propagation distances and values of chirp coefficient s for N = 4. The other parameters are shown in Fig. 2.

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4. Conclusion

A new class of pulsed sources is proposed in this paper, named as MCGCSM pulse sources, producing far fields with flat-topped profile of average intensity or splitting into multiple subpulses in the far field by adjusting source parameters. The analytical expression for the TMCF of the MCGCSM pulse beams in dispersive media is derived and then the evolution of the average intensity and TDOCs in the dispersive media is analyzed. The obtained results show that the MCGCSM pulse beams will exhibit robust self-splitting properties during propagation and evolve from a pulse of the source plane into N subpulses in the far field. For the odd N, there is a subpulse on the optical axis, while for the even N, no subpulse is formed along the optical axis. By adjusting the source parameters, the MCGCSM pulse beams also can produce optical fields with flat top profiles, and the larger the value of N is, the flatter the pulse profile becomes. The MCGCSM pulse beams propagating in dispersive media with chirp coefficient s < 0 will exhibit two self-focusing processes. Our analysis came to the conclusion that when the chirp coefficient s < 0, the dispersive media are equivalent to a focusing lens which can lead to the focusing effects, and the self-splitting properties of the MCGCSM pulse beams result in the occurrence of two self-focusing processes. The robust self-splitting properties of the MCGCSM pulse beams have potential applications in optical communications, since they may provide a possible way to encrypt data by decomposing themselves. On the other hand, the robust flat-topped intensity pulses obtained by the MCGCSM pulse beams can be used for material surface processing in the case that the unavoidable presence of the media is from the source to the object. The experimental production of the MCGCSM pulse beam is a simple setup including diffraction gratings and a fluctuating spectral phase diffuser realized by a spatial light modulator [43].

Funding

National Natural Science Foundation of China (61965016, 62175209, 62241506); Yunnan Provincial Science and Technology Department (106220615033).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

Data availability

Data underlying the results presented in this paper are available from the corresponding authors upon reasonable request.

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Self-focusing and self-splitting properties of partially coherent temporal pulses propagating in dispersive media

Abstract

1. Introduction

2. Theoretical formulation

3. Numerical results and analysis

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (12)

Equations (18)

Optics Express