Abstract
1. INTRODUCTION
A fascinating phenomenon called branched flow occurs when a wave passes through a weak disordered potential whose relative length exceeds its wavelength. The gradual variation in disorder produces focused, elongated filaments that branch out, forming tree-like structures rather than random speckle patterns. This branched flow appears to be irreversible, from the trunk to the branch, and originates from the ray deflections due to weakly correlated changes in potential, leading to the formation of caustics [1]. These caustics reflect the folding of the Lagrangian manifolds in phase space, which corresponds to the concentration of rays and high field strengths along specific lines on 2D or 3D surfaces [2]. When initially observed in electrons [3–7] and microwave cavities [8,9], branched flow is thought to occur across a broad spectrum of wavelengths. In fact, branched flows are believed to play a pivotal role in focusing ocean waves [10,11]. In the case of light, however, this phenomenon has only recently been subtly observed on soap bubbles [12].
An interesting but easily overlooked point is that the branched flow phenomenon has a time arrow. Because of the existence of the random potential field, the time inversion symmetry of the Helmholtz equation is broken, and the optical path loses reversibility. In the intuitive physical picture, when we consider the inversion of time, the branched waves within these filamentous structures refocus to create robust trunks [13]. This counterintuitive process seems to violate the principle of entropy increase, which dictates a natural tendency towards chaos. Yet on curved surfaces, specifically those with constant Gaussian curvature surfaces (CGCSs), it is possible to achieve this effect to some extent, thanks to the periodic properties of light transmission. The general theory of relativity offers rich potential for studying electromagnetic waves in curved spacetime, where two-dimensional surfaces are often used as laboratory simulations of the effects of gravitational fields [14]. In addition to advances in geometric optics, such as gravitational lensing [15], more interesting effects were discovered, such as the relativistic Hall effect [16,17], the Wolf effect [18,19], and chaotic [20] and wave redirection [21]. Among them, the electromagnetic wave equation on the curved surface established by Batz and Peschel [22] has become a key tool. They predicted that the coherent electromagnetic wave transmission on the CGCS has periodic properties, and in one period, the second half period can be regarded as the time inversion of the first half period, which can also be obtained by the matrix optical method [23,24]. This gives us a scheme to achieve the temporal inversion of branched flow phenomena.
In recent experiments with optical branching flows on soap bubbles [12], the bubbles they studied were large enough to consider flat, smooth thickness variations in the film acting as a correlated disordered potential. They verified that the distance from the launch point to the first branching point leads to a scaling law that depends upon the optical potential strength and its correlation length. In this paper, we propose that this relationship on the curved surface should be corrected by the curvature of the surface and can be described by the scintillation index () [25]. The is one important parameter for describing the appearance of branches, as they periodically become abnormally larger, which is caused by the optical sink flow of the surfaces.
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The magical sink flow on the curved surface is a phenomenon of reverse branched flow, which is a process from disorder to order. Following the concept of thermodynamics, we define a new index called “branched flow entropy,” and prove that the entropy increase principle is temporarily broken in the inverse branch flow on a curved surface. The degree of entropy reduction is closely related to the curvature of the surface and the parameters of the random potential field. Further, we explore that this is because the decoherence process is blocked by the curved surface, and the decreasing coherence of the light will rise periodically, which gives us inspiration for studying the influence of spatial structures on the conservation of information. In addition, since the CGCSs are often used to simulate curved spacetime with uniform mass densities, the Friedmann–Robertson–Walker (FRW) metric, our research can also be extended to the cosmology [26]. Considering that macroscopic surfaces can be approximately flat, the refocusing phenomenon of branched flow on curved surfaces will be more significant at microscopic levels, such as in droplets [27] or biological cells [28]. Therefore, microscopic particle size analysis may be one of the possible applications of this research. Additionally, this research could use extremely intense lasers or electron streams [29,30], which could help in the study of extreme radiation and field.
2. RESULTS
We begin our study of optical branched flow and the corresponding time reversal on a special type of 2D curved surface in 3D space, CGCS, by investigating the plane wave and the Gaussian beam propagation on the rough curved surfaces with random fluctuations. The CGCS has revealed various properties on which light is transmitted, such as periodic convergence and divergence of the light spot and oscillation of the transmission trajectory [26,31,32]. By convention, we define the coordinate system, as shown in Fig. 1(a), where is the equatorial, and is the arc length of the generating line (longitude and latitude). The light transporting on a two-dimensional curved surface can be described by the scalar Helmholtz equation [33],
Figure 1.CGCS images with positive Gaussian curvature. (a) Smooth and (b) rough. (c) Light beam transmits on the rough CGCS. (d) Local rough surface.
It has been known in previous discussions, namely [32], that the effect of the curvature on the CGCS in the paraxial approximation can be equivalent to a second-order potential field , and the Helmholtz equation with paraxial approximation on a CGCS can be written as
The generatrix equation of CGCS is , where the relationship between and determines the shape. In particular, is spherical, and we take the olive shape of as the light transport surface for our paper. In our simulation, as shown in Fig. 1(d), the branched flow of light also appears on the curved surface, but then, time inversion occurs, and other abnormal peaks in the appear. The is used to describe the normalized variance of the branched flow intensity,
Based on Eq. (1), in a random potential field with Gaussian correlation, the evolutions of light transmission on a flat surface and on a CGCS are demonstrated, as shown in Figs. 2(a)–2(d), by using the method of fast Fourier transform. In our simulation, according to Ref. [12], the beam size of the Gaussian light is set as 20 μm, and the wavelength of the light is set as 500 nm. The correlation length of 100–300 μm is performed in the calculation, which is approximately hundreds of times the wavelength and 10 times the size of the Gaussian spot. For convenience, we set all the calculation parameters to be dimensionless in the following. The white line shows the change of the of the plane waves with transmission. The highest point of the corresponds to the first branching position of the light beam, and it obeys the scaling law as the distance to the first caustic position when the light travels in flat space [1,9], as shown on the dark point of the black curve in Fig. 2(e). We also find that the first caustic position on the curved surface appears earlier than that on the flat surface. Furthermore, as the curvature increases or the amplitude of the random potential field increases, the branching position will be more advanced, as shown in Figs. 2(e) and 2(f). This advance is closely related to the focusing effect of the positive curvature surface on the light, which shortens the effective transmission distance of the light and speeds up the branching process. In Fig. 2(f), the fitted lines represent a proportional relationship between the curvature and the first focus position . For Gaussian beams, this branch position advance phenomenon also appears on the curved surface. But because the initial scintillation index of Gaussian light is large, it is difficult to identify the position of the first peak; here, we only show the changes with the transmission distance of the plane waves.
Figure 2.Beam is transmitting in a Gaussian correlated random potential. (a) and (b) are the plane waves propagating on a flat surface and a CGCS, respectively, and the white line is the evolution of the
In addition, in Fig. 2(e), when the curvature of the surface increases to a certain value, two or more peaks of will appear. The presence of the remaining peaks indicates that the light field has an anomalous flare after branching. We discover that the first peak is at the location where the branched flow is generated, while the second peak corresponds to a special convergence position of the light on the curved surface.
This similar phenomenon can happen to the Gaussian beam. Figure 3(a) presents the evolution of the Gaussian beam in the random field on the CGCS. The size of the main peak of the waveform will first expand and then converge. As shown in Figs. 3(b) and 3(c), a single-peak Gaussian light changes into multiple branches and then returns to one main peak with multiple secondary peaks, meanwhile the maximum light intensity decreases first and then increases, although the intensity of the main peak cannot return to the original value.
Figure 3.(a) Evolution of the Gaussian beams in random fields on the CGCS. (b) Change of the maximum intensity of light with respect to transmission. (c) Evolution of the transverse light intensity distribution.
During the propagation, the branched flow process of light is partially reversed, and the scattered branches of light flow converge again, becoming more orderly, which is known as sink flow. This interesting phenomenon indicates that when light travels on a curved surface, its disorder does not always increase, but it may cause anomalous changes at some positions. In order to better explain this phenomenon, we introduce a concept similar to thermodynamic entropy to describe the chaos degree of light rays. Referring to the concept of thermodynamics, we define the entropy of branched flow as the following expression to describe the chaos of the light beam:
Figure 4.(a) Evolution of the entropy of the optical branched flow when it transmits on a curved surface. (P1)–(P4) Light intensity probability distribution. For the different correlation lengths, the branched flow entropy of the beam varies with the transmission on (b) the flat surface and (c) the CGCS.
3. DISCUSSION
In flat space, when an electromagnetic wave travels in a random medium over a long distance, the light intensity is a strong fluctuating quantity, which is classically described by Rayleigh statistics. The distribution function is a negative exponential [37],
Before studying the curved surface, we can first demonstrate the influence of the random potential field on the coherence of the flat surface. By solving the Helmholtz equation numerically, the electric field and light intensity distribution of the optical branch flow are obtained. During the numerical simulation, 500 random fields with consistent parameters were generated, and the evolution of optical coherence was obtained on average. It is found that with the light transmission, the coherence can be viewed as decreasing exponentially, , and gradually tends to 0, as shown in Fig. 5(a). The coherence attenuation factor is related to the fluctuation and the correlation length of the random potential field, as shown in Figs. 5(b) and 5(c).
Figure 5.(a) Evolution of the coherence degree of the branched flow with transmission on a flat surface. (b) and (c) are the relationship between the attenuation factor
On the curved surface, as long as the light intensity moment is expanded (see Appendix A), the evolution of light intensity distribution can be naturally obtained. Then, the relationship between the intensity distribution of the partially coherent light and the coherence degree after multiple scattering by random mediums will take the following form [8]:
From Figs. 5(b) and 5(c), is proportional to the square of while the variation of with is a logarithmic Gaussian distribution. The coherence of the optical branched flow will decrease with the increasing , which leads to the acceleration of the entropy increase process. In addition, on a curved surface, an increase of will weaken the entropy reduction effect and even make it disappear. Furthermore, from Fig. 5(f), it is found that the position of the first peak in the curve of Fig. 5(e) is positive with the reciprocal of the curvature. Through the synthesis of the above numerical relations, we can deduce an analytic expression,
In particular, the CGCS can be regarded as a rotation in phase space composed of the momentum and position of the light. What we calculated above is the spatial coherence (transverse coherence). At the same time, the CGCS will also become such a transformation system, which periodically converts the temporal coherence (longitudinal coherence) into the spatial coherence, resulting in a brief reversal of the decoherence process. In fact, the reduction of temporal coherence has been reported in other literature, e.g., the so-called Wolf effect [19]. It shows that the curved spaces with larger positive curvature accelerate and enhance the spectral shifts (blue shifts or red shifts) of light during their propagation.
It is worth emphasizing that in this study we do not have a clear delineation of the material that constitutes the curved surface. The optofluidic system can be formed by the optical branched flow with liquid surfaces, such as soap bubbles mentioned in the literature [12], biological cells, and even the ocean. It can also be formed with solid structures such as nanostructure, fiber, or metasurface. In optofluidic systems, there are some examples that can be further considered, such as the optical control of the thermocapillary effects [40], stochastic solitons, or occurrence of turbulence at low Reynolds numbers. Applications to solid systems include particle manipulation [41], random scattering quasi-two-dimensional resonator [8], and so on. The universality of the discussion in this paper means that it is not related to the constituent matter but only to the curvature of the surface. This is reminiscent of the same “hairless” black hole, so we cannot help but ask whether the entropy reduction here has the same deeper roots as Bekenstein entropy [42], and how the entropy reduction relates to the thermodynamic effects of the event horizon.
4. CONCLUSION
In conclusion, we have investigated the phenomenon of optical branched flow on a CGCS. The CGCS is rough, with a random slight fluctuation potential field, which satisfies both the generation condition of the branched flow and the weak field approximation. In the case of paraxial transmission, the first branching position of the optical branched flow on the CGCS is earlier than that on the flat surface, which shows that the focusing characteristic of the positive curvature surface shortens the length of the wave caustics. On the curved surface, there is a phenomenon of anti-branched flow, called sink flow, which is impossible to appear on the flat surface. Through simulations, we found that the “branched flow entropy” near the sink flow point decreases temporarily with the transmission, indicating that the optical intensity distribution becomes ordered at this location.
In the discussion, we demonstrated that the entropy reduction is closely related to the coherence of the optical field, and the increase of the curvature of the surface will accelerate the conversion from the longitudinal coherence to the transverse to resist the decoherence of the light caused by the random field. Moreover, the effects of the random field and the surface parameters on the evolution of the optical coherence were studied. Based on this, we deduced a theoretical expression of entropy reduction, which agrees with the simulation results.
We obtained this time reversal of branched flow because of the unique focusing property of the CGCS, which is a non-local effect. In fact, if we consider the transmission of branched flow on surfaces with slowly changing curvature, this inversion property is also possible, but much worse than that of the CGCS. If the optical branches propagate on the CGCS with negative curvature for the location of the scintillation index and the first branching point of the branched flow, compared to the flat surface, then the negative curvature surface will have a hysteresis effect, which is the opposite of the CGCS with the positive curvature. Furthermore, our simulation showed that CGCS with negative curvature will lose the temporary entropy reduction effect of the optical branched flow, and the time inversion phenomenon will disappear simultaneously. The entropy reduction, which is monitored by the curvature of the surface and roughness parameters, can be used to enlighten multi-mode fiber amplifiers to improve pump absorption efficiency under complex backgrounds with nonuniform transverses [43,44]. The research can also be extended to other types of wave transport on curved surfaces, such as the transmission of ocean currents branching on the Earth’s surface [11]. In addition, according to Einstein, gravitational fields can be thought of as curved spacetime. The exploration of branched flow and entropy within the context of non-Euclidean geometry may give a new perspective in the study of conservation of information and chaos in cosmology.
APPENDIX A: PROOF OF THE RELATION BETWEEN INTENSITY PROBABILITY DISTRIBUTION AND COHERENCE DEGREE
Because of the completeness of the exponential family of distributions, we can write for different evolutionary processes or different coherence degrees as follows. This is physically equivalent to a superposition of a heat distribution with a certain temperature distribution width,
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