Branched flows have been observed in different types of waves 3–8, an important phenomenon involving wave propagation through weakly disordered potential, but the phenomenon for light waves has not been perceived experimentally until recently1. Briefly, this experiment indicated that the branched flow of light can be produced when a laser beam interacts with a soap film, and simulations were conducted according to the paraxial wave equation to support the experimental outcomes. Different from the simulation methods in the previous work1, 9–13, a new quasiparticle concept was proposed in our previous work to explain the branched flow phenomenon of light, which can be successfully applied to generate branched flow of light with the classical geometric optical theory14. Here, one issue may inspire us: *how can we manually generate or control of branched flow patterns of light in reality?*

In this paper, using the quasiparticle concept, we developed an experiment to efficiently realize manually branched flow of light by exploring laser light on tiny glass spheres in pure water. Furthermore, two simulation methods were employed to interpret and evaluate the experimental findings, and the computational results indicated that the observed branched flow of light can be attributed to the new concept of the main branched flow of light, while the sub-branched flow of light that cannot be detected in the experiment may be overwhelmed by the strong background scattered light. Thus, this work not only supports our previous work on the quasiparticle concept in ref. 2, but can more significantly provide guidance for obtaining controllable branched flow patterns of light through material microstructure design.

The experiment is based on the concept of quasiparticles (~ 10− 4m) 2, e.g., an ensemble of disordered quasiparticles, which may have different refractive indices (\({n}_{i}\) for the *i*th quasiparticle), can be embedded in a liquid background with constant refractive index of \({n}_{0}\) to generate the branched flow of light. Besides, \({n}_{0}\) should be smaller than \({n}_{i}\), and a more detailed explanation can be found in ref. 2. Here, we acquired the branched flow of light based on the simplified model experimentally, and the corresponding experimental setup is shown in Fig. 1(a). Briefly, the quasiparticles in the experiment were tiny glass pellets with the identical diameter of \(0.6 mm\) and the same refractive index \({n}_{i}\) of 1.55, and hundreds to thousands of glass spheres were placed in a glass beaker filled with ultrapure water (\({n}_{0}=1.33\)). Next, by emitting a laser beam (\(\ge 1.5mW\), \(632.8nm\)) on the glass balls through a transparent beaker, the incident light trajectories can be explored with the glass pellets in different configurations.

To better understand the experimental results, the classical geometric ray theory was first implemented based on a JAVA code to simulate different configurations of glass spheres, e.g., two rays should follow the optical paths shown in Fig. 1(b). If the incidence angle of one ray is \({\theta }_{0}\) and the angle of refraction is \({\theta }_{i}\), then the two angles of the ray should obey the following equations

$$sin{\theta }_{i}=\left(\frac{{n}_{0}}{{n}_{i}}\right)\sqrt{1-{\left(cos{\theta }_{0}\right)}^{2}}$$

$$\overrightarrow{b}=\overrightarrow{a}+2cos{\theta }_{0}\overrightarrow{t}$$

$$\overrightarrow{c}=\left(\frac{{n}_{0}}{{n}_{i}}\right)\overrightarrow{a}+(\frac{{n}_{0}}{{n}_{i}}cos{\theta }_{0}-cos{\theta }_{i})\overrightarrow{k}$$

$${R}_{s}={\left|\frac{{n}_{0}cos{\theta }_{0}-{n}_{i}cos{\theta }_{i}}{{n}_{0}cos{\theta }_{0}+{n}_{i}cos{\theta }_{i}}\right|}^{2}$$

$${R}_{p}={\left|\frac{{n}_{0}cos{\theta }_{i}-{n}_{i}cos{\theta }_{0}}{{n}_{0}cos{\theta }_{i}+{n}_{i}cos{\theta }_{0}}\right|}^{2}$$

Where \(\overrightarrow{a}\), \(\overrightarrow{b}\), and \(\overrightarrow{c}\) are the normalized direction vectors of the incidence, reflected, and refracted rays, respectively, and \(\overrightarrow{t}\) is the normalized plane normal vector.\({R}_{s}\) and \({R}_{p}\) correspond to the reflectance of s-polarized light and p-polarized light. \({T}_{s}\) is the transmitted power of s-polarized light and \({T}_{p}\) is the transmitted power of p-polarized light. In a word, any specific ray should follow Snell’s law, reflection law, energy conservation with Ts + Rs = 1, where transmission and reflection coefficient obey Fresnel’s equations, etc.

To make the simulation results more convincing, geometrical rays can be explored from the electrical field \(\overrightarrow{E}\) of the electromagnetic (EM) field, which can be written as

$$\overrightarrow{E}=A{e}^{i\phi }=A{e}^{i(\overrightarrow{k}\bullet \overrightarrow{r}-\omega t+{\phi }_{0})}$$

Where \(\overrightarrow{k}=\frac{\partial \phi }{\partial \overrightarrow{r}}\), which is the EM wavevector, \(\omega =-\frac{\partial \phi }{\partial t}\) is the angular frequency and \({\phi }_{0}\) is the initial phase. Thus, the basic equation of the geometrical ray should have the coupled differential equations as15

\(\frac{d\overrightarrow{k}}{dt}=-\frac{\partial \omega }{\partial \overrightarrow{r}}\) , \(\frac{d\overrightarrow{r}}{dt}=-\frac{\partial \omega }{\partial \overrightarrow{k}}\)

This can be analogous to the classical Hamiltonian *H* equations with the generalized momentum \(\overrightarrow{p}\) of a moving particle

$$\frac{d\overrightarrow{p}}{dt}=-\frac{\partial H}{\partial \overrightarrow{r}},\frac{d\overrightarrow{r}}{dt}=-\frac{\partial H}{\partial \overrightarrow{p}}$$

Optical trajectories can be calculated by the finite-element method (FEM), e.g., the differential equations of \(\overrightarrow{k}\) and \(\overrightarrow{r}\) components need to be resolved mathematically. The discontinuity between pure water and glass particles should be similarly based on the law of reflection, Snell’s law, and Fresnel’s equations. The experimental and simulation findings will be discussed in the next section.

Figures 2(a), (b) and (c) indicate optical flow images from three different glass pellet configurations in a beaker, both of which demonstrate that when the light leaves the intersection of the laser and the glass balls, it branches off into the finer streams, as anticipated from the quasiparticle model calculations based on the geometrical ray theory in ref. 2. However, rather than observing the dozens of light flow branches in the predicted quasiparticle model, only two or three rivulets of light flow were evidently detected as the fitted light flow pattern for the white lines at the bottom of the images. A previous experimental investigation of branched flow of light was the interaction between a laser beam and a soap film1, whereas here we designed a hybrid system of glass spheres and an aqueous background to propagate light in place of the soap film. This is to say, the branched flow of light in the experiment we obtained can be identified as a manual design, and the branching pattern of light is closely related to the distribution of the glass spheres. Moreover, many tributaries may have been so thin that they were obscured by scattered light and experimentally difficult to observe, e.g., applying the above experimental procedure to glass balls of smaller size (d = 0.3 mm): the branched flow of light, although it can also be observable, was experimentally blurred by scattered light from smaller glass balls.

The experimental results support the proposed model of quasiparticles that can be applied to generate branched flow of light, but we did not perceive the desired branching number of light flow. To gain a more complete picture of this unexpected outcome, two different simulation methods described above were used to calculate the ray trajectories of a disordered two-dimensional glass spherical lens. It should be noted that our simulation model is primarily according to the previously proposed quasiparticle model, and the critical point of the model is that the *random* potential (quasiparticle) landscape requires to be incorporated and the relevant physical parameters should be replicated as much as possible. In other words, the site of a specific individual glass particle in this experiment is not vital, what is important is that the overall glass pellet position is disordered. Consequently, here we consider two different 2D distributions of the spherical lens with an average spherical size of about 0.6 mm and an index of refraction of 1.55, and the lens positions are random and embedded in a background with a refractive index of 1.33, as Figs. 3 (a) and (b) show. From the simulation results, both approaches suggest that we should in principle observe more light branching in the random quasiparticle distribution, although the light branching patterns are completely dissimilar due to the difference in particle spreading. However, the simulation findings indicate that when the ultrafine branched rivulets happen to converge in space, thicker branches can form and their brightness will exceed the background scattered light, which is possible to observe in our experiments.

To further validate this explanation, we propose two original concepts to elucidate the branching phenomena observed in the experiments, namely the main branched flow of light and sub-branched flow of light. Moreover, a 2D periodic structure was designed to illustrate the two new notations. Briefly, Fig. 3(a) is a periodic glass pellet structure, and the spherical lens are in contact with each other without any gaps, and Fig. 3(b) has a similar periodic structure, but the difference is that the gap between the lens is 0.1 mm. Also, we can set the refractive index of the spherical lens to 1.33, while the refractive index of the background should be 1.55. Here, the incidence angle of light beam is 38°. As can be seen Figs. 4(a) and (b), the two different approaches indicate very good agreement, leading to the conclusion that the branched flow of light is essentially a classical phenomenon and could be manually designed. Moreover, the overall branching pattern may be divided into two regions: The top three branches can be seen as the main branched flows of light, which can be perceived effortlessly in Fig. 2(b). Other ultrafine rivulets can be identified as the sub-branched flows of light, which are difficult to detect in the designed experiment. Correspondingly, it can be found that the glass spheres used in our experiments cannot have the ideal transparency comparable to the simulated glass lens, and the strong scattered light may cover the sub-branched flows of light. Compared with the periodic microstructures in Figs. 4(a) and (b), the positions of the glass spheres in the experiments are more random and larger quasiparticles are easily formed. The computation findings in Fig. 4(c) may be closer to the attained experimental results: perhaps the specific glass sphere position is possibly no longer significant, and the most important point is the randomness of the quasiparticles. From the perspective of the physical mechanism behind branched flow of light, as a natural phenomenon, the occurrence of light branched flow requires random potentials in the experiment, and more detailed researches are required in the future.

In conclusion, our findings outline a new approach to experimentally manipulate branched flow of light through the interplay between light and glass spheres emerged in pure water, and also provide the evidence for the quasiparticle notation in the previous study. Moreover, the main branched flow of light and sub-branched flow of light, which may be important for understanding the experimental results, are proposed in this work. To the end, what we primarily explored here is the branched flow of light based on the two-dimensional microstructures of spherical lens. In future experiments, we will explore more three-dimensional quasiparticle structure configurations, but further requires to reduce the scattered light of glass pellets and improve the processing technology of microstructures. In other aspects, it is also possibly to sense the light branching phenomenon in a structure of biological colloids. Thus, our work can offer guidance for future studies to achieve controllable branched flow pattern of light by material structure design.