Self-focusing of partially coherent beams based on complex screen and split-step Fourier transform methods

Fanglun Yang; Guowen Zhang; Xiaoqi Zhang; Yanli Zhang; Ruifeng Wang; Jianqiang Zhu

doi:10.3788/COL202321.071901

1. Introduction

High-power laser drivers play an important role in high-energy, high-density physics, such as inertial confinement fusion, laser plasma physics, and laboratory astrophysics. When a high-power laser propagates in a nonlinear medium, a self-focusing phenomenon occurs owing to nonlinear effects, which is a major factor that limits the laser output power and damages the beam quality. The Bespalov–Talanov (B-T) theory was the first to provide a simple and clear explanation of self-focusing filaments of coherent light^[1,2], which was experimentally verified by Bliss in the 1960s^[3,4]. Small-scale self-focusing is the main factor leading to the deterioration of beam quality and limits the load of the laser driver^[5–7]. Partially coherent beams (PCBs) have been developed as laser drivers because they suppress nonlinear effects and improve the uniformity of the optical field^[8–18]. To date, much work has been conducted on the propagation of fully coherent beams in nonlinear media. However, few studies have focused on PCBs^[19–21].

In optical coherence theory^[22], the calculation of the inevitable four-dimensional integrals during the propagation of PCBs increases the analytical complexity. Coherent modal representation (CMR)^[23,24], pseudo-modal representation (PMR)^[25], and random modal representation (RMR) were introduced to reduce computational difficulty^[26].

RMR primarily refers to the complex screen (CS) method used to construct PCBs, which can flexibly and reliably represent various beam distributions of Schell light sources. The random CS method has great potential for solving the problem of propagating of PCBs in nonlinear media. In 2014, David et al. proposed the Gaussian Schell-model (GSM) beam representation using the CS method and demonstrated its effectiveness by comparing it with the GSM constructed by the theoretical method^[27]. In 2015, this method was used to flexibly and conveniently simulate PCB fields with various far-field distributions^[28]. In 2017, Wang et al. used the CS method to calculate and experimentally verify the propagation of partially coherent crescent-like optical beams in free space and turbulent atmosphere^[29]. In 2022, Wang et al. extended the CS approach to the time-frequency domain of Schell-model beams and simulated the computational Schell-model transport in a nonlinear medium^[30] by comparing it with the pulse-by-phase method^[31] to verify the correctness of their method. In addition to the CS method, an analytical method for the steady-state self-focusing of the GSM in a nonlinear media was studied by Wang et al. in 2019^[32] and extended it to the quasi-steady-state case in 2020^[33]. This theory yields important propagation characteristics of the GSM in nonlinear media under the approximation that the beams maintain the GSM after passing through it. In 2022, Lu et al. studied the self-focusing property of PCBs with nonuniform correlation structures in nonlinear media and demonstrated the feasibility of controlling the self-focusing length by controlling the initial coherence radius^[34].

In this study, we propose a method for numerically solving the nonlinear Schrödinger (NLS) equation using the random CS and split-step Fourier methods, which can be used to calculate the self-focusing of PCBs in nonlinear media. To demonstrate the validity of the method, we compare the beam intensity evolution under the overall self-focusing of the GSM in the proposed and analytical formula methods^[32] and compare the small-scale self-focusing phenomenon of the GSM and a coherent Gaussian beam in this study. This method allows the numerical analysis of the degree of self-focusing of the Schell-model beams, extends the range of sources for which self-focusing can be calculated, and is important for assessing the load capability of new high-power laser drivers using spatially PCBs.

2. Theoretical Model

2.1. Random CS method

The Schell-model beams are expressed in terms of the cross-spectral density functions $W (r_{1}, r_{2}, z)$ , which can be expressed as $W (r_{1}, r_{2}, z) = 〈 T (r_{1}, z) T^{*} (r_{2}, z) 〉 = E (r_{1}) E^{*} (r_{2}) µ (r_{1}, r_{2}) .$ (1)

In the CS method^[26,27], the PCB is represented by an incoherent superposition of spatially random complex fields. The instantaneous scalar field (single realization) $T_{n} (r)$ can be expressed as the product of the coherent electric field $E (r)$ and random CS transmittance function $ψ_{n} (r)$ , $T_{n} (r) = E (r) ψ_{n} (r) .$ (2)Substituting Eq. (2) into Eq. (1), we obtain Eq. (3). $W (r_{1}, r_{2}, z) = E (r_{1}) E^{*} (r_{2}) 〈 ψ_{n} (r_{1}) ψ_{n}^{*} (r_{2}) 〉 .$ (3)According to Refs. [35,36], the coherence function $µ (r_{1}, r_{2})$ can be expressed as $µ (r_{1}, r_{2}) = \int p (v) \exp [- i (r_{1} - r_{2}) v] d v,$ (4)where $p (v)$ is a non-negative function. $〈 C_{n} (υ_{1}) C_{n}^{*} (υ_{2}) 〉 = δ (υ_{1} - υ_{2}),$ (5)where $δ (υ_{1} - υ_{2})$ is the Dirac delta function, and $C_{n} (υ)$ is a white noise electric field with the properties of Eq. (5). Expressing $ψ_{n} (r)$ in the form of Eq. (6) and substituting Eq. (4), Eq. (5), and Eq. (6) into Eq. (3), we obtain Eq. (1). $ψ_{n} (r) = \int \sqrt{p (v)} C_{n} (υ) \exp (- i 2 π r \cdot v) d v .$ (6)

When the number of complex screens $N$ is sufficiently large, $W (r_{1}, r_{2}, z)$ can be expressed as $W (r_{1}, r_{2}, z) \approx \frac{1}{N} \sum_{n = 1}^{N} T_{n} (r_{1}) T_{n}^{*} (r_{2}) .$ (7)

The cross-spectral density function of the GSM is expressed as $W (r_{1}, r_{2}, z) = I_{0} \exp (- \frac{r_{1}^{2} + r_{2}^{2}}{ω_{0}^{2}}) \exp [- \frac{{(r_{1} - r_{2})}^{2}}{2 ρ_{0}^{2}}],$ (8)where $I_{0}$ denotes the initial intensity. Using the CS method to represent the GSM, the coherent electric field, coherence function, and intensity distribution are given by Eqs. (9), (10), and (11), respectively. $E (r) = E_{0} \exp (- \frac{r^{2}}{ω_{0}^{2}}),$ (9) $µ (r_{1}, r_{2}) = \frac{W (r_{1}, r_{2}, z)}{\sqrt{W (r_{1}, r_{1}, z) W (r_{2}, r_{2}, z)}} = \exp [- \frac{{(r_{1} - r_{2})}^{2}}{2 ρ_{0}^{2}}],$ (10) $I (r) = 〈 {| T (r) |}^{2} 〉 \approx \frac{1}{N} \sum_{1}^{N} {| T_{n} (r) |}^{2} .$ (11)

2.2. Propagation of PCBs in a nonlinear Kerr medium

Under the steady-state and standard paraxial approximations, the NLS equation for PCBs in nonlinear Kerr medium is $[2 i k \frac{\partial}{\partial z} + (\nabla_{⊥ 1}^{2} - \nabla_{⊥ 2}^{2})] W (r_{1}, r_{2}, z) + 2 k^{2} \frac{n_{2} [W (r_{1}, r_{1}, z) - W (r_{2}, r_{2}, z)]}{n_{0}} W (r_{1}, r_{2}, z) = 0,$ (12)where $\nabla_{⊥}^{2} = \nabla_{x}^{2} + \nabla_{y}^{2}$ is the transverse Laplace operator and $n_{2}$ is the nonlinear refractive index. Usually the equation has no analytical solution and can be solved numerically by the split-step Fourier method^[37]. The key to solving the above equation using the split-step Fourier method is two parts: the diffraction part and the self-focusing effect part.

The diffraction part can be expressed as $2 i k \frac{\partial}{\partial z} W (r_{1}, r_{2}, z) + (\nabla_{⊥ 1}^{2} - \nabla_{⊥ 2}^{2}) W (r_{1}, r_{2}, z) = 0 .$ (13)

This equation represents the diffraction transport of the PCBs. The CS method perfectly represents the process and has been applied to the transport of PCBs in free space under atmospheric turbulence^[26,27].

The nonlinear part can be expressed as $\frac{\partial}{\partial z} W (r_{1}, r_{2}, z) = W (r_{1}, r_{2}, z) \times i k \frac{n_{2} [W (r_{1}, r_{1}, z) - W (r_{2}, r_{2}, z)]}{n_{0}} .$ (14)

For a small $Δ z$ , Eq. (14) can be expressed as $W (r_{1}, r_{2}, z + Δ z) = W (r_{1}, r_{2}, z) \times \exp [i k \frac{n_{2} [W (r_{1}, r_{1}, z) - W (r_{2}, r_{2}, z)]}{n_{0}} \times Δ z] .$ (15)

The CS rule is calculated as follows: $\frac{\partial}{\partial z} T_{n} (r, z) = i k \frac{n_{2} 〈 {| T (r) |}^{2} 〉}{n_{0}} T_{n} (r, z) .$ (16)

For a small $Δ z$ , Eq. (16) can be expressed as $T_{n} (r, z + Δ z) = \exp [i k \frac{〈 n_{2} {| T (r) |}^{2} 〉}{n_{0}} \times Δ z] \times T_{n} (r, z) .$ (17)

By substituting $T_{n}$ at $r_{1}, r_{2}$ into Eq. (2) according to Eq. (17), we obtain Eq. (15).

In addition, to demonstrate the small-scale self-focusing phenomenon of PCBs, we let the beam first pass through screen $t (x, y)$ of periodic transmittance as a small-scale modulation with the following transmittance before solving the NLS equation^[4,38]: $t (x, y) = {[1 + a \times \sin (2 π f_{x} x)]}^{1 / 2},$ (18)where $a$ and $f_{x}$ denote the initial modulation depth of the intensity and modulation period, respectively. The degree of self-focusing is characterized by the local modulation degree $(M)$ , where $I_{\max}$ and $I_{\min}$ denote the maximum and minimum intensities in the local area, respectively. $M = \frac{I_{\max} - I_{\min}}{I_{\max} + I_{\min}} .$ (19)

3. Numerical Results and Analysis

In this section, the above theoretical model is used to simulate the overall and small-scale self-focusing processes of the GSM. The nonlinear medium was neodymium glass, which is now extensively used in high-power laser systems. It has a refractive index ( $n_{0}$ ) of 1.54 and a nonlinear refractive index ( $n_{2}$ ) of $1.18 \times 10^{- 13} esu$ . The beam radius $(w_{0})$ is 4 mm, and $C = w_{0} / ρ_{0}$ denotes the ratio of the beam width $w_{0}$ to the coherence length $ρ_{0}$ .

First, we calculated the variation in the maximum intensity for the overall self-focusing of the GSM. Three cases of coherent beams, $C = 2$ , and $C = 3$ , with an initial intensity of $0.2 GW / {cm}^{2}$ and a length of 4.5 m for the nonlinear medium, were assigned.

The evolution of the maximum intensity of each beam under overall self-focusing, as calculated using the analytical formula and CS methods, is shown in Fig. 1. The coherent beam (NSL) is the numerical solution of the NLS equation using a coherent Gaussian beam. AF is the analytical formula in Ref. [32]. CS is the simulation result obtained using the random complex screen method. As shown in Fig. 1, the AF and CS methods fit well in all three cases, proving the reliability of the methods.

Figure 1.Comparison of the analytical formula and the CS methods for overall self-focusing.

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Next, the propagation of the coherent Gaussian beam and the GSM in a nonlinear medium at a certain intensity after the perturbation were simulated. The simulation method for the small-scale self-focusing of coherent beams has been well established and proven to be correct. Therefore, we included the simulation of a coherent beam as a reference for comparison with the PCBs. The length of the nonlinear medium was 400 mm. In the transmittance screen function $t (x, y), a = 0.1$ , and $f_{x} = 11.9 / cm$ . $C = 2$ , 5, 10, and 20 were set for the GSM, corresponding to the coherence lengths $ρ_{0}$ of 2, 0.8, 0.4, and 0.2 mm, respectively. The $w_{0}$ values are all 4 mm. The number of complex screens N is 2000, at which point the coherence function for each case can be better represented.

Figure 2 represents the intensity distribution of each beam with an initial intensity of $2.6 GW / {cm}^{2}$ after amplitude modulation and propagation through 400-mm Nd glass medium, where Figs. 2(a)–2(e) represent the coherent beam, and $C = 2$ , 5, 10, and 20, respectively, and Fig. 2(f) represents the one-dimensional distribution of each beam at $y = 0$ .

Figure 2.Beam distribution with the initial intensity of 2.6 GW/cm² after 400-mm transmission. (a), (b), (c), (d), and (e) correspond to the coherent beam, and C = 2, 5, 10, and 20, respectively. (f) The one-dimensional distribution corresponding to each light field at y = 0.

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Figure 3 shows the parameter variation of each beam on the transmission path for an initial intensity of $2.6 GW / {cm}^{2}$ . Figure 3(a) shows the variation in the degree of modulation on the transmission path, while Fig. 3(b) shows the variation in the maximum intensity on the transmission path. Figures 4 and 5 are similar to Figs. 2 and 3, except that Figs. 4 and 5 correspond to an initial intensity of $4 GW / {cm}^{2}$ . The coherence shown in Figs. 2 –5 is the numerical solution to the NLS equation using a coherent Gaussian beam. In the two cases of the combined initial intensity of $2.6 GW / {cm}^{2}$ and $4 GW / {cm}^{2}$ , the evolution of the beam distribution, modulation degree, and maximum intensity of $C = 2$ and the coherent beam can coincide more perfectly. This demonstrates the reliability of the method of solving the NLS equation to some extent.

Figure 3.Variation of the beam in (a) the modulation system and (b) the maximum intensity at an initial intensity of 2.6 GW/cm².

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Figure 4.Beam distribution with the initial intensity of 4 GW/cm² after 400-mm transmission. (a), (b), (c), (d), and (e) correspond to the coherent beam, and C = 2, 5, 10, and 20, respectively. (f) The one-dimensional distribution corresponding to each light field at y = 0.

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Figure 5.Variation of the beam in (a) the modulation system and (b) the maximum intensity at an initial intensity of 4 GW/cm².

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According to Figs. 3 and 5, the modulation degree $M$ and the maximum intensity increase with the diffraction distance when the initial intensity is the same. The change trends of both are essentially the same and are close to exponential growth, which is typical of the small-scale self-focusing effect. This phenomenon is also observed in PCBs. However, the modulation growth trend differs for each coherent GSM length.

Table 1 summarizes the final modulation degree at various coherence and initial intensities. Stronger initial intensities yield higher final modulation degrees. The higher coherence corresponds to the higher final modulation degrees. Therefore, reducing the initial intensity and coherence can slow the growth of the modulation degree and maximum intensity and suppress the self-focusing effect.

Table 1. Modulation Degree M after Propagating through 400-mm Nd Glass

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Table 1. Modulation Degree M after Propagating through 400-mm Nd Glass

Coherent C = 2 C = 5 C = 10 C = 20
2.6 GW/cm² 0.37 0.35 0.33 0.19 0.09
4 GW/cm² 0.72 0.70 0.66 0.46 0.19

At an initial intensity of $4 GW / {cm}^{2}$ , the final modulation degree of 0.19 for the $C = 20$ Gaussian Schell-mode was 73.6% lower than the 0.72 for the coherent beam. This is also significantly lower than those of the other GSMs. When the initial intensity is $2.6 GW / {cm}^{2}$ , the modulation degree even decreases from 0.1 to the final 0.09. In this case, the diffraction effect surpasses self-focusing, and small-scale perturbations appear as spatial diffusion.

4. Conclusion

This study presents a method for calculating the transmission of spatially PCBs in a nonlinear media using the CS method combined with the split-step Fourier method, which solves the problem of the transmission of PCBs in a nonlinear Kerr media. The evolution of the overall self-focused maximum intensity of the GSM was compared using the CS and the analytical formula methods, which verified the effectiveness of the method. Simulations of the nonlinear propagation of the GSM with modulation demonstrated the effects of factors such as spatial coherence, intensity, and transmission distance on the modulation degree of self-focusing. The simulation results showed that small-scale self-focusing also occurred in PCBs. However, low spatial coherence can suppress the degree of self-focusing. When the GSM coherence was high, the small-scale self-focusing behavior was very similar to that of the coherent beam, which laterally reflected the effectiveness of the CS method.

This method facilitates a quantitative analysis of the overall and small-scale self-focusing degree of the Schell-model beams and is expected to establish a theory applicable to the modulation instability of PCBs in nonlinear media, which is of great significance for designing the seed source of high-power PCB laser devices and evaluating the load capacity.

Category: Nonlinear Optics

Received: Feb. 14, 2023

Accepted: Apr. 14, 2023

Published Online: Jul. 14, 2023

The Author Email: Guowen Zhang (guowenzhang@siom.ac.cn), Jianqiang Zhu (jqzhu@siom.ac.cn)

DOI:10.3788/COL202321.071901

Table 1. Modulation Degree M after Propagating through 400-mm Nd Glass

Table 1. Modulation Degree M after Propagating through 400-mm Nd Glass