Random Phase Screen in Scattering Media with Multi-Parameter Coupling

Abstract

The modeling of light propagation in scattering media is an important topic that has attracted considerable attention in recent decades. Coupling microscopic parameters such as particle concentration, particle size, and the refractive index of the medium can broaden the applicability of the model and improve simulation accuracy. In this work, these parameters are used to regulate the anisotropy factor and the mean free path. They are then integrated into a random phase screen model constructed using the Monte Carlo and the Gerchberg–Saxton algorithm. An optical experimental setup was established, in which a Laguerre–Gaussian beam was employed as the incident light source and diffusers with mesh numbers of 1500, 600, and 220 were used as the scattering media. The model was validated through comparative analysis between simulated and experimental results. Correlation coefficients between the simulated and experimental beam profiles exceeded 0.73, and the maximum relative error in power-in-the-bucket was only 4.9%, confirming the model’s accuracy and reliability. Numerical simulations were performed based on the established model to investigate beam propagation behavior. The results indicate that increasing particle concentration and particle size both lead to enhanced beam centroid shift and beam broadening. This modeling method provides a useful tool for analyzing beam propagation in complex scattering media and holds potential applications in wavefront correction and structured beam recognition.

1. Introduction

In free-space optical (FSO) communication, the existence of scattering media introduces both technical challenges and potential prospects for research [1,2,3,4,5]. On the one hand, the presence of fog, smoke, and aerosol particles causes serious signal attenuation and wavefront distortion, which can significantly impair communication quality [6,7]. On the other hand, accurate modeling of scattering media facilitates wavefront correction and provides theoretical foundations for the development of novel modulation schemes [8,9]. In the modeling of scattering media, parameters such as particle size distribution, concentration, and medium thickness play a crucial role in determining the propagation characteristics of light beams. These parameters govern the distribution features of the scattered field, thereby influencing the phase evolution of the beam during propagation. Therefore, the proper incorporation of these multi-physical parameters into scattering models is essential for enhancing both the physical fidelity and applicability of the model.

Figure 1 illustrates several representative methods involved in modeling light propagation in scattering media: Mie theory can accurately analyze the angular distribution under single-scattering conditions, but its extension to multiple scattering becomes extremely complex and computationally intensive, making direct application impractical [10]. The Monte Carlo (MC) method is widely used, but obtaining accurate results requires tracing a very large number of photon paths, which often takes many hours of computation [11]. The transmission matrix (TM) approach can reconstruct optical transmission information through a medium, but it relies heavily on experimental measurements and is difficult to apply to novel scattering media without established databases [12,13]. In contrast, the random phase screen (RPS) model captures wavefront distortion by applying random phase perturbations. The RPS model is a simplification of complex optical systems into a point-source imaging framework. It has been widely used to simulate random media, such as atmospheric or oceanic turbulence [14,15,16,17]. However, the application of this model in scattering media remains relatively unexplored.

In 2015, Schott developed a random phase screen model for biological tissue, coupling the anisotropy factor of the scattering medium as a key parameter in the transmission model. They experimentally validated the reliability of the model using the optical memory effect [18]. In 2019, Liu [19] refined Schott’s model by providing a theoretical derivation of the phase screen model. The role of the ballistic light was under their consideration, making the model more consistent with real-world conditions. In 2020, Qiao et al. analyzed the frequency characteristics of the phase screen. They introduce the zero-frequency component in the spatial frequency domain and analyze the impact of screen spacing on simulation accuracy. These advancements contributed to the model’s enhanced realism [20]. Should parameters such as particle concentration, particle size, and refractive index of the scattering medium be integrated into the random phase screen model, the applicability of the model would be significantly extended, and its accuracy in simulating complex scattering phenomena would be greatly enhanced.

In this work, we use the parameters, including particle concentration, particle size, and refractive index of the scattering medium, to systematically regulate the anisotropy factor and the mean free path (MFP). Then, we couple these parameters into a comprehensive scattering model and devise a random phase screen simulation method based on the MC and Gerchberg–Saxton (GS) algorithms. The Laguerre–Gaussian (LG) beam is employed as the light source to verify the simulation method through both theoretical analysis and experimental measurements.

2. Method

In scattering environments, beam propagation is not governed by a single parameter but results from the combined influence of multiple physical factors. In this work, three microscopic parameters—particle concentration, particle size, and refractive index of scattering medium—are used to regulate the anisotropy factor and the mean free path (MFP). These parameters together modulate the wavefront distortion imposed on the beam by the scattering media.

2.1. Multi-Parameter Coupling

As illustrated in Figure 2a, the anisotropy factor, defined as $g=〈\cos \theta 〉$, represents the average cosine of the scattering angle. Its value ranges from −1 to 1. The value of g approaching 1 indicates a predominance of forward scattering, whereas the value closer to −1 corresponds to backward scattering [21,22], The value of g can be calculated as

$$g={\displaystyle \frac{4}{Q_s{x}^2}}\left[{\displaystyle \sum _{n=1}^{n_{\max }}\frac{n(n+2)}{n+1}}\mathrm{Re}\left(a_n{a}_{n+1}^{*}+b_n{b}_{n+1}^{*}\right)+{\displaystyle \sum _{n=1}^{n_{\max }}\frac{2n+1}{n+1}}\mathrm{Re}\left(a_n{b}_n^{*}\right)\right],$$

(1)

where $x=2\pi r/\lambda$ denotes the size parameters of the particle, $\lambda$ represents the wavelength of light. The Mie scattering coefficients $a_n$ and $b_n$, expressed via Bessel and Hankel functions, explicitly incorporates the particle radius r and the complex refractive index m of the scattering medium.

The MFP refers to the average distance between two successive scattering events. Numerically, it is equal to the reciprocal of the attenuation coefficient ${\alpha }_t$. The attenuation coefficient is defined as the sum of the scattering coefficient ${\alpha }_s$ and the absorption coefficient ${\alpha }_a$:

$${\alpha }_t={\alpha }_s+{\alpha }_a,$$

(2)

The scattering and absorption coefficients are given by

$${\alpha }_s={\sum }_{s}C,{\alpha }_a={\sum }_{a}C,$$

(3)

where C denotes the concentration of the scattering medium. And the ${\sum }_s$ and ${\sum }_a$ characterizes the scattering cross-section and the absorption cross-section, respectively. It can be expressed as

$${\sum }_s={\displaystyle \frac{2\pi }{k^2}}{\displaystyle \sum _{n=1}^{n_{\max }}\left(2n+1\right)\left({\left|a_n\right|}^2+{\left|b_n\right|}^2\right)},{\sum }_a={\displaystyle \frac{2\pi }{k^2}}{\displaystyle \sum _{n=1}^{n_{\max }}\left(2n+1\right)\mathrm{Re}\left(a_n+b_n\right)}.$$

(4)

Water, acrylic, diffuser, and polystyrene—having refractive indices of 1.33, 1.49, 1.53, and 1.59, respectively—are selected as the scattering media. The variation in the anisotropy factor with particle radius is calculated for each material. As shown in Figure 3a, the anisotropy factor increases gradually with particle radius and tends to stabilize when the radius reaches approximately 20 μm. Materials with higher refractive indices demonstrate lower stabilized values of the anisotropy factor, indicating stronger backscattering at the particle surface.

To clearly illustrate how MFP varies under different particle radii and concentrations, a base-10 logarithmic transformation of MFP is employed for visualization, as shown in Figure 3b. Overall, the MFP exhibits a decreasing trend with increasing particle size and concentration. Specifically, higher particle concentration leads to more scattering events per unit volume, causing a rapid reduction in MFP. While the increase in particle radius affects MFP indirectly by enlarging the scattering cross-section, resulting in a more moderate decline. Therefore, particle concentration serves as the primary controlling parameter for tuning MFP.

2.2. Construction of the Phase Screen

As demonstrated in Figure 2b, to build the forward scattering model with coupled parameters, we directly use the anisotropy factor g and the mean free path (MFP) in the Monte Carlo simulation. The MFP defines the random step length l of photon transmission in the scattering medium [23]. This length can be generated based on

$$l=-\ln \left(\xi \right)\cdot \mathrm{MFP}$$

(5)

where $\xi$ is a uniformly distributed random variable within the range [0, 1]. Figure 4 shows the boxplots of photon step-length distributions at particle concentrations of 1 × 10¹⁰ m⁻³, 5 × 10¹⁰ m⁻³, 1 × 10¹¹ m⁻³, and 5 × 10¹¹ m⁻³. The black pentagram indicates the mean value of l at each concentration. As the particle concentration increases, the overall photon step length decreases significantly. The mean value becomes smaller, and the distribution becomes more concentrated. This trend indicates that in high-concentration scattering media, the average photon travel distance is effectively reduced, corresponding to a lower MFP.

When the photon reaches its randomly assigned step length, a scattering event is triggered. The scattering angle is then determined based on the Henyey–Greenstein (HG) scattering phase function, which is used in this work to establish the relationship between the scattering angle and anisotropy, describing the angular distribution of a single scattering event. It is widely applied in radiative transfer calculations in astrophysics and ocean optics, and is often used to simulate Mie scattering [24]. The HG function can be given by

$$\mathrm{HG}\left(\theta ,g\right)={\displaystyle \frac{1}{4\pi }}{\displaystyle \frac{1-g^2}{{\left(1+g^2-2g\cos \theta \right)}^{3/2}}},$$

(6)

where $\theta$ represents the angle between the scattered light and the incident light. The HG function offers a good approximation of the angular scattering distribution, especially in multiple-scattering regimes where forward scattering dominates. Moreover, it significantly reduces computational overhead, making it well-suited for large-scale Monte Carlo simulations.

The receiving plane determines whether a photon reaches the detection region based on its coordinates. The position of received photons on the plane is recorded to obtain the angular distribution of scattered photons. Due to the inherent randomness of scattering, the distribution must be stochastically modulated to maintain statistical variability. Using the coordinate transformation, which can be written as

$$\theta =\arcsin \left(\lambda \sqrt{f_x^2+f_y^2}\right),\phi =\arctan \left(f_y/f_x\right),$$

(7)

where $\left(f_x,f_y\right)$ represents the spatial frequencies. the angular distribution ${\left|S\left(\theta ,\phi \right)\right|}^2$ is converted from the $\left(\theta ,\phi \right)$ domain to the $\left(f_x,f_y\right)$ domain, resulting in the spectral amplitude distribution in the frequency domain. To obtain the phase distribution of the phase screen, the Gerchberg–Saxton (GS) algorithm can be used to reconstruct the random phase profile through iterative phase retrieval [25,26,27].

As shown in Figure 5, the GS algorithm solves for the optimal phase solution through multiple iterations between the real plane and the Fourier plane. Here, $\left|F\left(u\right)\right|$ denotes the scattered light intensity distribution, $t_n$ and $T_n$ represent the wave functions in the source plane and Fourier plane, respectively. When the phase difference between two iterations is smaller than a given threshold, the optimal phase solution is obtained.

3. Experiment and Simulation

3.1. Simulation

Based on the above method, the phase screen models were established for 1500, 600, and 220 mesh diffusers, and their statistical distributions were obtained. In the Monte Carlo angular scattering simulation, a total of 10¹⁰ photons were emitted for statistical analysis.

Figure 6a–c correspond to phase screens generated by 1500, 600, and 220 mesh diffusers, respectively, along with their associated phase statistical distributions. It can be observed that as the mesh number decreases, the range of phase modulation broadens significantly. The calculated mean and standard deviation show that the phase screen produced by the 1500-mesh diffuser exhibits weaker modulation, with a more concentrated phase distribution and a smaller standard deviation. In contrast, the phase screens corresponding to the 600-mesh and 220-mesh diffusers demonstrate stronger modulation, characterized by larger phase fluctuation ranges and higher standard deviations.

This phenomenon occurs because a higher mesh number corresponds to smaller particle radii. Smaller particles not only lead to a larger mean free path (MFP) but also exhibit a broader scattering phase function, with stronger large-angle components. As a result, forward scattering is reduced, and less multiply scattered light reaches the detector. In the Fourier domain, a broader phase function corresponds to lower frequency perturbations, leaving high-frequency information largely unaffected. Consequently, the effective phase modulation on the beam is weaker for higher mesh numbers. Conversely, for lower mesh numbers with larger particles, the MFP decreases and forward scattering is more dominant, resulting in stronger phase modulation and a phase distribution approaching uniformity.

3.2. Experiment

To verify the validity and reliability of the proposed random phase screen model, an optical experiment was built using the LG beam as the light source and the diffusers as the scattering media. This system was used to analyze the impact of the scattering medium on beam propagation.

As shown in Figure 7, a 532 nm solid-state laser (LR-GSP-532/300 mW) was used to emit a Gaussian beam. A 4-f beam expansion system with lenses of focal lengths 25 mm and 125 mm was employed to expand and collimate the beam, ensuring its spatial quality during propagation. A diaphragm was used to shape the beam to a suitable size, improve beam uniformity, suppress stray light and background noise, and enhance the signal-to-noise ratio. A polarizer was placed in the path to produce a polarized beam.

To generate a stable Laguerre–Gaussian (LG) beam with a specific topological charge l = 2 and radial index p = 0, a spatial light modulator (SLM) was used. The scattering media were a series of GCL-2011 diffusers, each with a thickness of 2 mm. Three diffusers with mesh numbers 1500, 600, and 220 were selected. The corresponding particle radii were calculated to be 8.5 μm, 21 μm, and 57 μm, respectively. The associated particle concentrations were estimated as 3.2 × 10¹¹ m⁻³, 6.4 × 10¹⁰ m⁻³, and 8.7 × 10⁹ m⁻³. At the receiver, a CCD camera (STC-MCS231U3V, SENTECH Gesellschaft für Sensortechnik mbH, Berlin, Germany) was used to record the evolution of the beam after transmission through the scattering medium.

3.3. Discussion

Simulation and experimental studies were conducted on diffusers with mesh numbers of 1500, 600, and 220. The corresponding intensity distributions and correlation coefficients are shown together in Figure 8. The top row gives the simulation results, and the bottom row gives the experimental results. When using the 1500-mesh diffuser, the beam maintains a noticeable ring-shaped profile. As the mesh number decreases, the scattering becomes stronger, leading to reduced central intensity and a more uniform energy distribution. This is because low-mesh-number diffusers have larger particle sizes and a sparser distribution, which decreases the number of scattering particles per unit area and enhances the single-scattering effect of each particle. In contrast, high-mesh-number diffusers exhibit denser particle distributions and stronger overall directionality. The correlation coefficients under all three scattering conditions are greater than 0.73, indicating high similarity between simulation and experiment.

The comparison of PIB is shown in Figure 9. The maximum relative error between simulation and experiment is only 4.9%, which further demonstrates that the proposed model can accurately capture the energy distribution after scattering. These results confirm that the phase-screen model can effectively reproduce the optical phenomena observed in experiments and is applicable under different scattering conditions.

4. Analysis of Propagation Characteristics

Based on the previously constructed multi-parameter coupled phase screen model, numerical simulations were conducted to analyze the scattering propagation of Laguerre–Gaussian beam in complex scattering media. Particle radii corresponding to three types of diffusers (1500 mesh, 600 mesh, and 220 mesh) were selected. For each case, we investigated the effect of varying particle concentration on beam centroid wander and beam spreading [28,29].

As shown in Figure 10a,b, the three curves represent the simulated beam centroid wander and beam spreading for diffusers with particle radii of 8.7 μm, 21 μm, and 57 μm. the simulation results indicate that both centroid wander and beam spreading increase and gradually stabilize with higher particle concentrations. Under larger particle radii, the centroid wander and beam spreading become more pronounced, suggesting that stronger scattering leads to more severe beam distortion, more complex speckle patterns, and a more divergent spatial distribution.

In Figure 10a,b, the experimental data points scattered outside the simulation curves (marked with red triangles, yellow circles, and blue squares) correspond to measurements for the three types of scattering media. To further verify the accuracy of the model, centroid positions and beam widths were extracted from CCD-captured speckle images under experimental conditions for three types of scattering media. These values were overlaid on the simulation curves for comparison. The results show that the experimental data points align well with the simulated trends, indicating strong parameter correspondence. Based on the calculated relative errors, the centroid wander exhibited a maximum deviation of less than 35%, while the error of beam spreading remained within 20% across the three representative cases. These findings confirm the reliability and accuracy of the proposed model in simulating beam propagation through scattering media.

5. Conclusions

This work analyzes complex scattering media by coupling particle concentration, particle size, and refractive index of the medium to jointly regulate the anisotropy factor and the MFP. These parameters are integrated into the phase screen model. Phase screen models corresponding to diffusers with mesh numbers of 1500, 600, and 220 were theoretically constructed. An experimental platform was established using a Laguerre–Gaussian beam on the diffusers to validate the model. Comparison between experimental and theoretical results was conducted through the correlation coefficient and the relative error of PIB. The correlation coefficients between experimental and simulated results all exceeded 0.7, and the maximum relative error of PIB was only 3.4%, confirming the effectiveness of the proposed model in simulating realistic scattering media. Based on the phase screen model, numerical simulations of beam propagation through scattering media were performed. The results show that increases in particle concentration and particle size lead to enhanced centroid wander and beam spreading.

In summary, the proposed modeling method effectively characterizes the statistical propagation behavior of beams in scattering media. These findings provide theoretical support for beam control and stability optimization in complex scattering environments. In addition, in terms of wavefront correction, the model can numerically simulate phase distortions induced by scattering media, serving as input for adaptive optics systems to test and optimize wavefront sensing and correction strategies. In terms of mode recognition, the model is capable of reproducing modal crosstalk, thereby providing a virtual environment for the design and validation of modal detection and demultiplexing. Overall, the proposed model provides an effective tool for advanced optical research in complex scattering conditions.