Numerical Study of Non-Schell Model Pulses in Nonlinear Dispersive Media with the Monte Carlo-Based Pseudo-Mode Superposition Method

Abstract

Recently, we introduced random complex and phase screen methods as powerful tools for numerically investigating the evolution of partially coherent pulses (PCPs) in nonlinear dispersive media. However, these methods are restricted to the Schell model type. Non-Schell model light has attracted growing attention in recent years for its distinctive characteristics, such as self-focusing, self-shifting, and non-diffraction properties as well as its critical applications in areas such as particle trapping and information encryption. In this study, we incorporate the Monte Carlo method into the pseudo-mode superposition method to derive the random electric field of any PCPs, including non-Schell model pulses (nSMPs). By solving the nonlinear Schrödinger equations through numerical simulations, we systematically explore the propagation dynamics of nSMPs in nonlinear dispersive media. By leveraging the nonlinearity and optical coherence, this approach allows for effective control over the focal length, peak power, and full width at half the maximum of the pulses. We believe this method offers valuable insights into the behavior of coherence-related phenomena in nonlinear dispersive media, applicable to both temporal and spatial domains.

1. Introduction

Optical coherence, a fundamental property of light, describes the correlation of the optical field at different points in space and time [1]. Partially coherent light effectively suppresses light interference and exhibits strong resilience to environmental disturbances, making it invaluable for applications such as speckle-free optical imaging, super-resolution imaging, optical communication, and photonic computing [2,3,4,5,6]. The degree of coherence, defined as the normalized correlation function of the electric field, offers a distinctive degree of freedom for partially coherent light, driving extensive research from fundamental physics to practical applications. Studies have explored the interactions between the degree of coherence and other optical parameters, enabling the customization of optical fields [7,8,9,10,11,12,13] and advanced applications in diverse fields such as optical imaging, prime number factorization, angular velocity measurement, particle trapping, information encryption, and free-space optical communication [14,15,16,17,18,19,20]. Partially coherent light is categorized based on whether its degree of coherence depends on position: Schell model light (or uniformly correlated partially coherent light) and non-Schell model light (or non-uniformly correlated partially coherent light). The former features a position-independent degree of coherence, while the latter does not. Past research has primarily focused on Schell model light due to its accessibility in laboratories [1]. Common techniques, such as manipulating spontaneous emission light sources, using rotating ground glass disks, and employing holographic methods like random complex screens have been widely utilized [8,9]. These techniques are typically based on the van Cittert–Zernike theorem, which specifically applies to Schell model light customization.

In contrast, non-Schell model light exhibits greater diversity and more versatile generation mechanisms. Its degree of coherence possesses higher dimensionality, encapsulating richer information. Inspired by the pioneering work of Lajunnen [21], extensive studies have been conducted on non-Schell model beams and their variants, including electromagnetic, higher order, and other versions [22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40]. These beams exhibit peculiar propagation characteristics in free space and turbulent atmospheres, including traditional phenomena such as self-shifting and self-focusing, as well as novel characteristics like diffraction-free behavior [22,23,24,25,26,27,28,29]. Additionally, they demonstrate strong self-healing capabilities, resilience against orbital angular momentum spectrum degradation, and significantly enhanced resistance to environmental disturbances [30,31,32]. These unique properties make them highly applicable in particle trapping, optical communication, and information encryption [33,34]. Techniques such as pseudo-mode superposition, random phase screen methods, and optical coordinate transformations have been developed to enable the customization of non-Schell model light [35,36,37,38,39,40].

Attributed to the well-known space–time analogy, the coherence-induced phenomena observed in spatially partially coherent beams in free space are also expected to occur in PCPs propagating through linear second-order dispersive media [41]. Numerous PCPs with prescribed degrees of coherence have been explored [42,43,44,45,46,47,48], demonstrating the capability to shape these pulses on demand in the far zone by leveraging the degree of coherence in the source. While the evolution behavior of PCPs in linear dispersive media can be effectively analyzed using integral formulas [49], studying their behavior in nonlinear dispersive media typically requires solving coupled nonlinear Schrödinger equations, posing considerable challenges. To address this, the typical solution is to derive the random electric field of the PCPs. The Karhunen–Loève expansion method has been proposed [1]. Although applicable to any PCPs, the method has largely been limited to very simple models, such as Gaussian Schell model pulses, due to the mathematical challenges associated with solving homogeneous Fredholm integral equations [1]. Recently, we extended the complex screen method from the spatial domain to the temporal domain [50]. It significantly simplifies the representation of PCPs and makes it easy to derive their random electric fields. Using this approach, we analyzed the propagation properties of complex PCPs in nonlinear dispersive media, demonstrating that reducing optical coherence effectively mitigates nonlinear effects and enables the on-demand customization of far-field pulse properties through manipulation of source pulse properties [51]. However, this protocol is restricted to Schell model pulses. The nSMPs were first introduced a decade ago [52], but their study has remained confined to linear dispersive media [53,54]. Research has shown that, similar to their spatial counterparts, nSMPs exhibit self-focusing behavior during propagation, leading to pulse self-compression. The space–time analogy highlights their significant potential for applications in pulse shaping, material processing, and information encryption. However, to date, the propagation properties of nSMPs in nonlinear dispersive media remain unexplored.

In this Letter, we introduce the Monte Carlo method into the pseudo-mode superposition method [30] to generate the random electric fields (or modes) of any PCPs, including nSMPs. By applying the split-step Fourier method to solve the nonlinear Schrödinger equation, we systematically investigate the dynamic behavior of nSMPs during propagation in nonlinear dispersive media.

2. Theoretical Model

This section is divided into three parts: an introduction to the Monte Carlo-based pseudo-mode superposition method, a description of the specific non-Schell model light source, and an overview of the pulse propagation model.

2.1. Introduction to the Monte Carlo-Based Pseudo-Mode Superposition Method

Now, we consider a statistically nonstationary optical pulse propagating along the z-direction. In the time domain, its second-order statistical properties can be characterized by the mutual coherence function [1], represented by

$$\Gamma \left(t_1,t_2\right)=〈U^{\ast }\left(t_1\right)U\left(t_2\right)〉={\displaystyle \int p\left(f\right)H^{\ast }\left(t_1,f\right)}H\left(t_2,f\right)df,$$

(1)

where $U\left(t\right)$ denotes the random electric field of the PCPs, $p\left(f\right)$ is a nonnegative function, and $H\left(t,f\right)$ is an any kernel function. Here, t and f characterize the time and frequency, respectively. By appropriately selecting $p\left(f\right)$ and $H\left(t,f\right)$, we can construct the PCPs on demand, including the Schell model and non-Schell model types.

To obtain the corresponding random electric field of the PCPs, we first utilize the pseudo-mode superposition principle [30] and rewrite Equation (1) as a discretized summation in the following form:

$$\Gamma \left(t_1,t_2\right)\approx {\displaystyle \sum _m^{M}p\left(f_m\right)H^{\ast }\left(t_1,f_m\right)H\left(t_2,f_m\right)\Delta f_m},$$

(2)

where $f_m$ is evaluated at separate locations labeled by the index m along the frequency axis. $\Delta f_m$ represents the sampling interval, M denotes the number of samples, and M should be sufficiently large to ensure the validation of the above equation. In practical simulations, we typically use a finite number of samples to save time. For the choice of M, it is required that as M increases, the right side of Equation (2) essentially becomes unchanged. The value of M at this point is considered the sought-after value. Here, $p\left(f_m\right)$ and $H\left(t,f_m\right)$ are treated as the mode weight and pseudo-mode, respectively. From the perspective of the pseudo-mode superposition method, M also characterizes the number of pseudo-modes.

Next, we express the function $p$ in Equation (2) as follows:

$$p\left(f_m\right)=\sqrt{p\left(f_{m1}\right)}\sqrt{p\left(f_{m2}\right)}\delta \left(f_{m1}-f_{m2}\right),$$

(3)

where $\delta$ denotes the Dirac delta function, which takes 1 when $f_{m1}=f_{m2}$ and 0 otherwise. To further develop our method, we represent $\delta$ function by

$$\delta \left(f_{m1}-f_{m2}\right)=〈C_{m1}^{\ast }{C}_{m2}〉.$$

(4)

Here, the angular bracket stands for the ensemble average. $C_m$ denotes the random complex number, which obeys the circularly symmetric standard complex normal distribution.

By applying Equations (1)–(4), we can obtain the random electric field of the PCPs, expressed as

$$U\left(t\right)\approx {\displaystyle \sum _m^M\sqrt{p\left(f_m\right)}{C}_{m}H\left(t,f_m\right)\sqrt{\Delta f}}.$$

(5)

It is important to note that we have integrated the Monte Carlo method into the pseudo-mode superposition approach, allowing us to derive the random electric field of any PCPs due to the universality of Equation (1).

Compared with the traditional pseudo-mode superposition method [38,39,40], the advantage of our proposed proposal is it can effectively study the dynamic evolution of nSMPs in nonlinear dispersive media. For traditional techniques, the propagation behavior of nSMPs is analyzed by superimposing all individual modes after their independent propagation to the receiver plane. Each mode carries only a small portion of the total energy (determined by its mode weight), leading to incorrect nonlinear effects. In our method, we perform an incoherent superposition of all modes to ensure that all modes experience statistically identical nonlinear effects.

2.2. Description of the Specific nSMPs

To explore the propagation evolution of nSMPs, we first construct a specific source. Here, we define the expressions of $p\left(f\right)$ and $H\left(t,f\right)$ as follows [27]:

$$p\left(f\right)={\left(\pi b^2\right)}^{-1/2}\exp \left(-{\displaystyle \frac{f^2}{b^2}}\right),$$

(6)

and

$$H\left(t,f\right)=\sqrt{P_0}\exp \left(-{\displaystyle \frac{t^2}{2t_p^2}}\right)\exp \left(-i{k}_{0}f{t}^n\right),$$

(7)

where $k_0=2/b{t}_c^n$, b is a constant in the dimension of s⁻¹, n serves as the mode order, used to control the self-focusing capacity [27], $P_0$ represents the peak power, and $t_c$ and $t_p$ denote the coherence time and pulse width, respectively. By substituting Equations (6) and (7) into Equation (1), we derive the analytical expression of the nSMP as follows

$$\Gamma \left(t_1,t_2\right)=\exp \left(-{\displaystyle \frac{t_1^2+t_2^2}{2t_p^2}}\right)\exp \left[-{\displaystyle \frac{{\left(t_2^n-t_1^n\right)}^2}{t_c^{2n}}}\right].$$

(8)

Meanwhile, by substituting Equations (6) and (7) into Equation (5), we can attain the random electric field of such an nSMP. To validate the feasibility of our proposed method, we numerically simulate the intensity and degree of coherence of the nSMPs with Equations (5)–(7). The intensity and the degree of coherence are calculated by

$$\overline{I}\left(t\right)\approx {\displaystyle \frac{1}{K}}{\displaystyle \sum _{\alpha }^K{I}_{\alpha }\left(t\right)}={\displaystyle \frac{1}{K}}{\displaystyle \sum _{\alpha }^K{\left|U_{\alpha }\left(t\right)\right|}^2},$$

(9)

and

$$\mu \left(t_1,t_2\right)\approx {\displaystyle \frac{{\displaystyle \sum _{\alpha }^K{U}_{\alpha }^{\ast }\left(t_1\right)U_{\alpha }\left(t_2\right)}}{\sqrt{{\displaystyle \sum _{\alpha }^K{\left|U_{\alpha }\left(t_1\right)\right|}^2}}\sqrt{{\displaystyle \sum _{\alpha }^K{\left|U_{\alpha }\left(t_2\right)\right|}^2}}}},$$

(10)

respectively. $I_{\alpha }\left(t\right)={\left|U_{\alpha }\left(t\right)\right|}^2$ characterizes the random intensity of individual realization. Here, we use the subscript $\mathsf{\alpha }$ for the electric field $U$ to denote the $\mathsf{\alpha }$-th realization. Typically, K needs to be sufficiently large to ensure that the left-hand side and the right-hand side of the above two equations are approximately equal. The numerical simulation and analytical results for the intensity and degree of coherence of the nSMPs are presented in Figure 1. The random intensity curves of five realizations are shown in Figure 1a. Due to the randomness of $\left\{C_m\right\}$, the shape and peak power of the modes vary significantly. However, due to the even nature of the mode order n, the random intensity of all realizations exhibits even functions of t. When the mode order n is odd, these realizations are randomly distributed and do not exhibit the even function property. To save space, we do not include the results for the case when n is odd. The averaged intensity profile of the nSMP, obtained from K = 10,000 realizations, is depicted in Figure 1b (blue dots). The analytical intensity result, obtained by Equation (8), is represented by the red curve. It is evident that the two perfectly match. Furthermore, the numerical simulation for the degree of coherence and its difference with the analytical results (derived using Equation (8)) are shown in Figure 1c,d, respectively. The difference values presented in Figure 1d are very small, which demonstrates the feasibility of our method.

2.3. Introduction to the Pulse Propagation Model

The propagation of the pulse through a nonlinear dispersive medium is described by the nonlinear Schrödinger equation, expressed as

$$i{\displaystyle \frac{\partial U}{\partial z}}-{\displaystyle \frac{{\beta }_2}{2}}{\displaystyle \frac{{\partial }^{2}U}{\partial t^2}}=-\gamma {\left|U\right|}^{2}U,$$

(11)

where $U\left(t\right)$ represents the slowly varying pulse envelope, and $\gamma$ and ${\beta }_2$ are the Kerr nonlinearity coefficient and group velocity dispersion, respectively. The formation of optical solitons is only supported when the group velocity dispersion is negative. The time coordinate t is assumed to be defined relative to a reference frame moving at the pulse’s group velocity.

The random electric fields $U_n\left(t\right)$ of nSMPs are determined by Equations (5)–(7), which is used as the input of Equation (11). We solve the nonlinear Schrödinger equation using the split-step Fourier method and obtain the electric field $U_n\left(t,z\right)$ at the output. By refreshing the random number $C_m$ to update the input $U_n\left(t\right)$ and repeating the above process, a new output electric field $U_n\left(t,z\right)$ can be obtained. Through extensive iterative processes, a substantial dataset of output electric fields $\left\{U_n\left(t,z\right)\right\}$ can be generated. It allows for a detailed study of the pulse properties of the nSMPs propagating in the nonlinear dispersive media, also utilizing Equations (9) and (10).

3. Results and Analysis

In this section, we aim to explore the intensity evolution of the nSMP during propagation in nonlinear dispersive media using the theoretical model described above. The relevant parameters are set to $t_p=15\mathrm{ps}$, $\gamma =0.1{\mathrm{W}}^{-1}{\mathrm{km}}^{-1}$, $b=0.0817{\mathrm{ps}}^{-1}$, M = 3000, and ${\beta }_2=-50{\mathrm{ps}}^2/\mathrm{km}$. First, we present the density map of the intensity evolution of the nSMPs for varying mode order n and their on-axis intensity curves during propagation in Figure 2. To highlight the impact of nonlinear effects on pulse behavior, the top row of Figure 2 illustrates the propagation evolution of nSMPs in linear dispersive media. The second and third rows correspond to nonlinear cases with soliton parameters N = 2 and 4, respectively, where the soliton parameter N is defined by $N=\sqrt{t_p^2\gamma P_0/\left|{\beta }_2\right|}$. Typically, we change the value of the peak power $P_0$ to vary the soliton parameter N. A larger N signifies stronger nonlinear effects. In the bottom row of Figure 2, we present the on-axis intensity curves. To clearly display the self-focusing properties, the pulse intensity is normalized by the source peak power. Influenced by the source degree of coherence [54], the pulse undergoes self-focusing during propagation, followed by a gradual decrease in intensity (e.g., see Figure 2(a1)). From left to right, as mode order n increases (from 2 to 6), the pulse evolutions in the linear dispersive media slightly change (see the top row). With stronger nonlinear effects (increasing N from 2 to 4), the pulse becomes more compressed, its peak power rises, and it exhibits stable propagation characteristics [e.g., Figure 2(b1,c1)]. The on-axis intensity curves (bottom row) confirm that the pulse intensity initially increases with the propagation distance, then decreases, and eventually stabilizes. Additionally, as nonlinear effects intensify, the pulse peak power increases further, indicating greater compression of the pulse width. In

Table 1. Full width at half maximum (FWHM) and the focal length $z_f$ (defined as the distance between the source plane and the position where the pulse power reaches its maximum) of the nSMPs for various values of the parameters n and N.

Exponent Values Soliton Parameter	n = 2 N = 0 N = 2 N = 4	n = 4 N = 0 N = 2 N = 4	n = 6 N = 0 N = 2 N = 4
FWHM (ps)	6.3 3.1 1.7	5.8 2.8 1.5	5.3 2.5 1.3
Zf (km)	1.22 1.01 0.54	1.04 0.99 0.63	1.06 0.99 0.68

, we summarize the full width at half maximum (FWHM) and focal length (the distance from the source plane to the position where the pulse power reaches its maximum) of the pulse for varying values of n and N. As n and N increase, the pulse width decreases, leading to higher peak power. Overall, pulse compression induced by nonlinear effects is more pronounced than the influence of the degree of coherence under moderate optical coherence ($t_c=t_p$). Notably, the focal length decreases with the nonlinear parameter N. In linear or weakly nonlinear dispersive media, higher mode order reduces the focal length. However, under strong nonlinear effects, the focal length increases with mode order. This phenomenon occurs because both nonlinearity and the degree of coherence of the nSMPs contribute to pulse compression. Following compression, the degree of coherence causes the pulse to diverge. Thus, the pulse’s focusing behavior is dictated by mutual assistance and competition between these two factors.

Next, we investigate the effect of the source optical coherence (coherence width $t_c$) on the evolution of the nSMPs during propagation in the linear and nonlinear dispersive media. In the linear dispersive regime, the propagation characteristics of the nSMPs are primarily dictated by their source intensity and degree of coherence. Under conditions of low coherence, the degree of coherence predominantly governs their propagation dynamics. It is well established that nSMPs exhibit self-focusing behavior during propagation due to the source degree of coherence. Consequently, as optical coherence decreases from high to low, the self-focusing ability of the pulses intensifies, as illustrated in the top row of Figure 3. However, when the optical coherence is further reduced, the self-focusing effect gradually saturates. For high optical coherence (see Figure 3, first column), enhanced optical nonlinearity leads to pronounced pulse compression, causing the pulse to evolve into a stable structure with increasing propagation distance. As shown in Figure 3(d1), the on-axis intensity initially increases, then decreases, and eventually stabilizes at a certain value, consistent with the trends discussed earlier. As optical coherence decreases, the intensity evolution and corresponding on-axis intensity profiles of the nSMPs exhibit similar patterns. However, with further reductions in optical coherence, differences arising from varying nonlinear effects diminish. As highlighted in our recent work [53], nonlinear effects can be mitigated by reducing the source optical coherence of the pulse. Under low coherence conditions (see Figure 3, last column), the pulse becomes unstable, with the intensity along the axis initially increasing and then decreasing. To further elucidate these dynamics, we calculate the FWHM and focal length of nSMPs under varying levels of source optical coherence, with the results summarized in

Table 2. FWHM and focal length $z_f$ of the nSMPs for various values of the parameters $t_c$ and N.

Coherent Time (ps) Soliton Parameter	tc = 25 N = 0 N = 2 N = 4	tc = 15 N = 0 N = 2 N = 4	tc = 5 N = 0 N = 2 N = 4
FWHM (ps)	7.1 3.3 4.1	6.3 3.1 1.6	2.6 2.1 1.3
Zf (km)	2.57 1.35 0.68	1.22 1.01 0.54	0.14 0.16 0.17

. As discussed above, lower optical coherence results in higher peak pulse power, narrower pulse width, and shorter focal length. Conversely, under high coherence conditions, stronger nonlinearity similarly enhances peak power, narrows pulse width, and shortens focal length. A notable distinction under low coherence, however, is that the focal length increases. This phenomenon, as hypothesized earlier, arises due to the competition between optical coherence and nonlinearity. These findings underscore the ability to control self-focusing behavior through careful manipulation of the source degree of coherence and nonlinear effects.

4. Conclusions

In this manuscript, we integrate the Monte Carlo method into the pseudo-mode superposition method, enabling the calculation of the random electric field for any PCPs, including nSMPs. By taking a statistically meaningful dataset of random electric fields, we demonstrate that the numerically simulated intensity and degree of coherence align perfectly with their analytical counterparts, thereby validating the feasibility and accuracy of our approach. Using the split-step Fourier method to solve the nonlinear Schrödinger equation, we investigate the propagation characteristics of nSMPs in nonlinear dispersive media. Our results reveal that both optical nonlinearity and the source degree of coherence significantly influence pulse compression during propagation. In cases of high or moderate optical coherence, propagation dynamics are predominantly governed by nonlinearity. Conversely, under low optical coherence, the pulse exhibits an enhanced ability to mitigate nonlinear effects, with the beam’s focusing behavior primarily dictated by the degree of coherence. For the experimental realization, we suggest utilizing spectral control techniques. The optical system consists of two gratings, two lenses, and a spatial light modulator, as detailed in Refs. [48,55]. To generate the on-demand random electric fields of PCPs (as described in Equation (5)), we follow a complex amplitude modulation encoding algorithm to customize the gratings [56], which are then loaded onto the spatial light modulator in the spectral plane. Then, we refresh the gratings to update the random electric fields. After the incoherent superposition in the output, we are able to construct the PCPs, including nSMPs, in the laboratory. These findings achieved in our work provide valuable insights into applications such as pulse shaping and material processing, particularly in nonlinear environments.