Information Entropy and Its Periodic Features in Hermite–Gaussian Correlated Schell-Model Beams in a Gradient-Index Fiber

Abstract

This paper investigates the evolution of information entropy (IE) in Hermite–Gaussian correlated Schell-model (HGcSM) beams propagating through a gradient-index (GRIN) fiber using Shannon information theory. Our results reveal that the IE of such beams evolves periodically, with the beam order significantly influencing its initial distribution. Compared with traditional Gaussian Schell-model beams, HGcSM beams exhibit more complex IE dynamics, characterized by periodically emerging low-entropy regions whose IE decreases with increasing beam order. Furthermore, the fiber’s central refractive index and core radius strongly affect the evolution period and fluctuation amplitude of IE. These findings provide a theoretical basis for optimizing partially coherent beams in optical fiber applications.

1. Introduction

Since the inception of Shannon information theory, information entropy (IE) has served as a quantitative measure of disorder in information sources, providing a standardized approach to assessing randomness [1]. In the field of optics, IE has garnered significant attention and is widely used as a tool to quantify the amount of information and uncertainty within light fields. In optical communication, for instance, in-depth analysis of the IE of optical signals has led to optimized coding and modulation strategies, thereby enhancing the efficiency and reliability of data transmission [2]. In optical imaging, IE helps evaluate the information acquisition capacity of imaging systems, guiding the improvement of imaging algorithms and system designs, thus enhancing overall image quality [3]. Furthermore, in fields like optical encryption, IE is utilized to assess the security and confidentiality of encrypted information [4,5]. These applications demonstrate the indispensable role that optical IE plays in evaluating and optimizing optical systems.

Optical spatial coherence is another crucial, controllable aspect of light fields [6,7]. Light beams with reduced spatial coherence, known as partially coherent (PC) beams, have become an important class of beams in contemporary optics [8,9]. Traditional fully coherent light beam faces limitations in practical applications, including sensitivity to environmental factors and prone to speckle [10]. The advent of PC beams offers a solution to these issues [8,9]. These beams exhibit greater resilience to external disturbances during propagation, suppress speckle effects more effectively [11,12], and demonstrate improved stability in environments such as atmospheric turbulence and complex media [13,14,15,16,17] and have important applications in many fields [18,19,20,21,22,23,24,25]. By manipulating the two-dimensional spatial coherence structure of PC beams, the propagation features of these beams, such as self-shaping [26], self-focusing [16], and self-splitting [17], can be realized. These novel properties have been widely applied in optical imaging [27,28], communication [29,30], and optical tweezers [31,32,33]. However, research on the IE of PC beams remains limited [34,35,36]. Exploring it can provide deeper insights into their characteristics and propagation behaviors from an information-theoretic perspective, thereby offering valuable theoretical support for optimizing the use of PC beams in various applications.

Gradient-index (GRIN) fibers, as a key optical transmission medium [37], have demonstrated significant potential in fields such as optical communication, fiber sensing, and optical imaging. The unique refractive index distribution in GRIN fibers enables light beams to self-focus during propagation, reducing intermodal dispersion and enhancing signal bandwidth and transmission stability. Studying PC beams in GRIN fibers is of great importance [38,39,40,41]. Due to their robust anti-interference properties and ability to suppress speckles, PC beams offer high transmission stability in complex environments. When coupled with GRIN fibers, these beams can theoretically optimize light propagation characteristics and broaden the application scope of optical fibers. Recent studies have explored the propagation properties of partially coherent Gaussian vortex beams [42] and Laguerre–Gaussian correlated Schell-model beams [43] in GRIN fibers. Findings indicate that the interaction between the light field and the fiber medium induces periodic propagation behaviors in PC beams. However, many aspects of the propagation characteristics of PC beams in GRIN fibers remain underexplored, particularly with regard to their IE. Conducting research in this area could provide a deeper understanding of the transmission dynamics of PC beams in this unique medium and offer valuable insights for the development of novel optical devices and applications based on GRIN fibers.

2. Statistical Properties of Hermite–Gaussian Correlated Schell-Model Beams in a Gradient-Index Fiber

As a prominent class of PC beams, Hermite–Gaussian correlated Schell-model (HGcSM) beams have been widely and deeply discussed in previous research. In this paper, we focus on the evolution of IE in these beams as they propagate through a GRIN fiber. The cross-spectral density function of the HGcSM beam provides an accurate representation of the beam’s spatial characteristics. In a rectangular coordinate system, the expression is given by [6,7]

$$W\left({\mathbf{r}}_1,{\mathbf{r}}_2\right)=\tau \left({\mathbf{r}}_1\right){\tau }^{*}\left({\mathbf{r}}_2\right)\mu \left({\mathbf{r}}_1,{\mathbf{r}}_2\right),$$

(1)

where $\tau \left(\mathbf{r}\right)=exp\left(-{\mathbf{r}}^2/4w^2\right)$ represents the amplitude, which is critical for the beam’s energy distribution; w is the beam width, determining the spot size; and $\mathbf{r}=\left(x,y\right)$ is the spatial coordinate vector of the source plane. The spatial coherence structure of the HGcSM source is expressed as [17]

$$\mu \left({\mathbf{r}}_1,{\mathbf{r}}_2\right)=\prod _{\xi =x,y}\left\{G_0{H}_{2m}\left({\displaystyle \frac{{\xi }_1-{\xi }_2}{\sqrt{2}{r}_{c\xi }}}\right)exp\left[-{\displaystyle \frac{{\left({\xi }_1-{\xi }_2\right)}^2}{2r_{c\xi }^2}}\right]\right\},$$

(2)

here, $\mathsf{\Pi }$ denotes the product of terms, the constant $G_0=1/H_{2m}\left(0\right)$, and $H_{2m}$ represents the $2m$ order Hermite polynomial, which shapes the beam’s intensity distribution and, consequently, its spot geometry. The zero-order HGcSM beam reduces to the well-known Gaussian Schell-model beam. $r_{c\xi }$ represents the spatial coherence length along the $\xi$-direction, which is crucial for describing the beam’s coherence in a given direction and is directly linked to its interference and diffraction properties during propagation. In the following analysis, we assume the coherence lengths in the x- and y-directions are equal and set to $r_x=r_y=r_c$ for simplicity.

Under the paraxial approximation, the generalized Collins integral formula is a vital tool for analyzing the propagation characteristics of PC beams. The formula is expressed as [44]

$$\begin{array}{cc}\hfill W\left({\mathit{\rho }}_1,{\mathit{\rho }}_2\right)& ={\displaystyle \frac{1}{{\left(\lambda B\right)}^2}}exp\left[-{\displaystyle \frac{ikD}{2B}}\left({\mathit{\rho }}_1^2-{\mathit{\rho }}_2^2\right)\right]\hfill \\ \hfill & \times \underset{-\infty }{\overset{\infty }{\int \int }}W\left({\mathbf{r}}_1,{\mathbf{r}}_2\right)exp\left[-{\displaystyle \frac{ikA}{2B}}\left({\mathbf{r}}_1^2-{\mathbf{r}}_2^2\right)\right]exp\left[{\displaystyle \frac{ik}{B}}\left({\mathbf{r}}_1·{\mathit{\rho }}_1-{\mathbf{r}}_2·{\mathit{\rho }}_2\right)\right]d^2{\mathbf{r}}_1{d}^2{\mathbf{r}}_2,\hfill \end{array}$$

(3)

where $\lambda$ is the wavelength, k is the wavenumber, A, B, C, and D are the matrix elements of the optical system’s transmission matrix, which describe the system’s effect on beam propagation. $W({\mathbf{r}}_1,{\mathbf{r}}_2)$ and $W({\mathit{\rho }}_1,{\mathit{\rho }}_2)$ are the cross-spectral density functions of the light source and target planes, respectively, with $\mathit{\rho }=({\mathit{\rho }}_x,{\mathit{\rho }}_y)$ representing the spatial position vector on the target plane. Substituting the cross-spectral density function of the HGcSM source into the generalized Collins integral formula allows for the precise derivation of the cross-spectral density function $W({\mathit{\rho }}_1,{\mathit{\rho }}_2)$ at the target plane. To further investigate the statistical properties of this beam in the GRIN fiber medium, it is essential to identify the matrix elements of the $ABCD$ transmission matrix for the fiber.

In the weak guidance approximation with a radial distribution, the refractive index of the GRIN fiber exhibits a parabolic profile, given by [42,43]

$$n^2=\left\{\begin{array}{c}{n}_0^2\left(1-{\alpha }^2{\rho }^2\right),{\rho }^2<R_0^2\hfill \\ n_0^2\left(1-{\alpha }^2{R}_0^2\right),{\rho }^2\ge R_0^2\hfill \end{array}\right.,$$

(4)

where $R_0$ is the core radius, defining the fiber’s effective propagation area; $\rho =\left|\mathit{\rho }\right|$ is the radial distance; $n_0$ is the central refractive index of the fiber; and $\alpha$ is the radial gradient parameter of the refractive index

$${\alpha }^2={\displaystyle \frac{1}{R_0^2}}\left(1-{\displaystyle \frac{n_1^2}{n_0^2}}\right),$$

(5)

where $n_1$ is the refractive index at the fiber boundary. This unique refractive index distribution causes the light beam to experience refractive index variations along the radial direction, leading to a self-focusing effect during propagation. Specifically, as the beam enters the fiber, the refractive index difference at different radial positions causes the light to refract progressively toward the axis, eventually achieving stable transmission. The matrix element of the $ABCD$ transmission matrix for a GRIN fiber is

$${\left[\begin{array}{cc}A& B\\ C& D\end{array}\right]}_z=\left[\begin{array}{cc}cos\left(\alpha z\right)& {\displaystyle \frac{sin\left(\alpha z\right)}{n_0\alpha }}\\ -n_0\alpha sin\left(\alpha z\right)& cos\left(\alpha z\right)\end{array}\right],$$

(6)

where z is the distance from the target plane to the light source plane. After complex calculations, the cross-spectral density function of the HGcSM beam at the target plane is obtained as

$$\begin{array}{cc}\hfill W\left({\mathit{\rho }}_1,{\mathit{\rho }}_2\right)& ={\displaystyle \frac{G_0^2{w}^2{k}^2{r}_c^2}{a{B}^2}}\left(1-{\displaystyle \frac{1}{a}}\right)exp\left[-{\displaystyle \frac{ikD}{2B}}\left({\mathit{\rho }}_1^2-{\mathit{\rho }}_2^2\right)-{\displaystyle \frac{k^2{w}^2}{2B^2}}{\left({\mathit{\rho }}_1-{\mathit{\rho }}_2\right)}^2\right]\hfill \\ \hfill & \prod _{\xi =x,y}{H}_{2m}\left[-{\displaystyle \frac{b_{\xi }}{2a}}{\left(1-{\displaystyle \frac{1}{a}}\right)}^{-1/2}\right]exp\left({\displaystyle \frac{b_{\xi }^2}{4a}}\right),\hfill \end{array}$$

(7)

with

$$a=1+{\displaystyle \frac{r_c^2}{4w^2}}+{\displaystyle \frac{A^2{k}^2{w}^2{r}_c^2}{B^2}};b_{\xi }={\displaystyle \frac{ik{r}_c}{\sqrt{2}B}}\left({\rho }_{1\xi }+{\rho }_{2\xi }\right)+{\displaystyle \frac{\sqrt{2}A{k}^2{w}^2{r}_c}{B^2}}\left({\rho }_{1\xi }-{\rho }_{2\xi }\right).$$

(8)

By combining the cross-spectral density function of the propagated HGcSM beam with the $ABCD$ matrix elements for the GRIN fiber, we can investigate the statistical properties of the beam in such fiber.

3. Information Entropy of Hermite–Gaussian Correlated Schell-Model Beams in a Gradient-Index Fiber

IE serves as a standardized measure for quantifying the uncertainty associated with random variables. Intuitively, the greater the disorder of an event, the higher its IE value, and vice versa. For a specific random event, the IE of its probability distribution is expressed as [45]

$$S=-\sum _{n}P\left(n\right)log\left[P\left(n\right)\right],$$

(9)

where $P\left(n\right)$ represents the probability of the occurrence of event n.

For the continuous electric field $E\left(\mathbf{r}\right)$, discretization is required, after which the IE can be approximated using the above formula. When the discretization accuracy is sufficiently small, the IE of the continuous electric field can be expressed as

$$S=-\int P\left[E\left(\mathbf{r}\right)\right]log\left[P_{\delta }\left[E\left(\mathbf{r}\right)\right]\right]dE\left(\mathbf{r}\right),$$

(10)

where $P_{\delta }\left[E\left(\mathbf{r}\right)\right]={\delta }^{4}P\left[E\left(\mathbf{r}\right)\right]$, $P\left[E\left(\mathbf{r}\right)\right]$ is the probability density function of the electric field at point $\mathbf{r}$ in space, and $\delta$ is the discretization accuracy.

However, for scalar PC light fields, the electric field alone cannot fully characterize the randomness at specific spatial points. The cross-spectral density function $W({\mathbf{r}}_1,{\mathbf{r}}_2)$ is used for a more complete description. For a random vector ${[E\left({\mathbf{r}}_1\right),E\left({\mathbf{r}}_2\right)]}^T$, the superscript T stands for transpose, composed of electric fields $E\left({\mathbf{r}}_1\right)$ and $E\left({\mathbf{r}}_2\right)$ at two points in space with Gaussian statistical properties, the IE is given by

$$S=log\left\{{\pi }^2{e}^2{\delta }^{-4}det\left[\mathsf{\Phi }\left({\mathbf{r}}_1,{\mathbf{r}}_2\right)\right]\right\},$$

(11)

where e is Euler’s number and det denotes the matrix determinant,

$$\mathsf{\Phi }\left({\mathbf{r}}_1,{\mathbf{r}}_2\right)=\left[\begin{array}{cc}W\left({\mathbf{r}}_1,{\mathbf{r}}_1\right)& W\left({\mathbf{r}}_1,{\mathbf{r}}_2\right)\\ W\left({\mathbf{r}}_2,{\mathbf{r}}_1\right)& W\left({\mathbf{r}}_2,{\mathbf{r}}_2\right)\end{array}\right].$$

(12)

Using Equations (11) and (12), we derive the IE expression for the PC beams,

$$S=2log\left(\pi e\right)+log\left[{\delta }^{-2}I\left({\mathbf{r}}_1\right)\right]+log\left[{\delta }^{-2}I\left({\mathbf{r}}_2\right)\right]+log\left[1-{\left|\mu \left({\mathbf{r}}_1,{\mathbf{r}}_2\right)\right|}^2\right],$$

(13)

where $I\left(\mathbf{r}\right)=W\left(\mathbf{r},\mathbf{r}\right)$ is the spectral intensity at position $\mathbf{r}$ in space. When the discretization accuracy is small relative to the spectral intensity $I\left(\mathbf{r}\right)$, the change in IE due to spatial variations in spectral intensity can be neglected. In this case, the final term in the above expression becomes the primary factor determining the distribution and evolution of the beam’s IE, defined as the IE difference,

$$I{E}_d=log\left[1-{\left|\mu \left({\mathbf{r}}_1,{\mathbf{r}}_2\right)\right|}^2\right].$$

(14)

The IE difference is solely dependent on the coherence function, meaning that once the coherence function at the target plane is known, the IE difference for the beam can be computed. By normalizing Equation (7) to obtain the coherence function of the HGcSM beam at the target plane, and substituting it into Equation (14), the evolution features of the IE difference for this beam in a GRIN fiber can be calculated.

4. Numerical Calculation Results

In this section, we examine the evolution of IE in HGcSM beams propagating through a GRIN fiber. Our analysis is based on derived expressions and detailed numerical simulations. The HGcSM beam is a uniformly coherent structured beam whose degree of coherence depends on the spatial position difference, specifically $\mu \left({\mathbf{r}}_1,{\mathbf{r}}_2\right)=\mu \left({\mathbf{r}}_1-{\mathbf{r}}_2\right)$. For clarity in the subsequent discussion, we define the position intervals as $\Delta x=x_1-x_2$ and $\Delta y=y_1-y_2$, which serve as spatial coordinate variables. The initial beam parameters are set as follows: wavelength $\lambda =632.8\mathrm{nm}$, beam waist w = 10 $\mathsf{\mu }\mathrm{m}$, and coherence length $r_c$ = 2.888 $\mathsf{\mu }\mathrm{m}$. For convenience, we used the wavelength typical of common HeNe-type laser diodes. The results presented are qualitatively similar to those for other wavelengths. As a typical GRIN fiber whose cladding is made of pure ${\mathrm{SiO}}_2$ and a core center made of silica glass doped with $7.9\%$ ${\mathrm{GeO}}_2$, the boundary refractive index $n_1=1.45702$, the central refractive index $n_0=1.46977$ [42], and we set the core radius $R_0$ = 60 $\mathsf{\mu }\mathrm{m}$. Unless otherwise specified, these parameters remain constant throughout the numerical calculations.

Figure 1 illustrates the distribution of the IE difference for the HGcSM source. Notably, at the center point $(\Delta x=\Delta y=0)$, where the degree of coherence $\mu (\Delta x=\Delta y=0)=1$, the IE difference approaches negative infinity. To enhance clarity, the distribution area that tends toward infinity and has minimal research significance (i.e., the white area in the figure) is truncated in the density distribution diagram. Figure 1a–c present the IE difference distribution under different beam orders. When $m=0$, the HGcSM beam reduces to a Gaussian Schell-model beam, and its IE difference displays a circularly symmetric distribution. This symmetry indicates that the information uncertainty in all directions is consistent. However, as the beam order increases to $m=1$, the IE difference distribution becomes more complex, showing a rectangularly symmetric array with enhanced entropy regions in specific directions. This suggests that as the beam order increases, the distribution of the IE difference becomes more complex, with varying degrees of information disorder in different directions. At $m=2$, this pattern is further pronounced, with additional enhanced regions of IE difference. These observations confirm that the beam order plays a significant role in determining the complexity of the IE difference distribution. To quantitatively demonstrate the side lobes of the IE difference, Figure 1d displays the corresponding crossline, where the sidelobes of the IE difference become more pronounced with increasing beam order. This highlights the ability to control the IE difference sidelobe by adjusting the beam order, thereby reducing disorder and enhancing spatial information correlation.

Figure 2 demonstrates the evolution of the IE difference for both a Gaussian Schell-model beam $(m=0)$ and a HGcSM beam $(m=1)$ in a GRIN fiber. Figure 2a shows the IE difference distribution for the former at multiple propagation distances. Despite maintaining a circular symmetry, the amplitude of the distribution fluctuates periodically (indicated by the changing size of the white area). Figure 2b presents the evolution of the IE difference for the latter. Unlike the Gaussian Schell-model beam, the IE difference distribution of the HGcSM beam exhibits a complex rectangularly symmetric array. Although complex structural changes occur during propagation, the IE difference continues to exhibit periodic behavior. To better illustrate the periodic evolution of the IE difference in the GRIN fiber, the density distribution of the IE difference is plotted on the x–z plane for both beam types.

Figure 3 presents the evolution of IE difference for Gaussian Schell-model beams and HGcSM beams of different orders on the x–z plane in the GRIN fiber. Figure 3a shows the evolution of the IE difference for a Gaussian Schell-model beam, which displays a relatively simple, uniform periodic change without significant structural complexity. Figure 3b,c show the evolution of the HGcSM beams of orders $m=1$ and $m=2$. These figures reveal that as the beam order increases, the complexity of the IE difference distribution also increases. In particular, the distribution exhibits alternating peaks and troughs along the x-direction, with the amplitude of the IE difference fluctuating more intensively. This demonstrates that higher-order PC beams lead to more complex distributions and greater fluctuations in IE. The periodic blue regions in these figures correspond to lower IE difference areas, where the light field information correlation is higher, reflecting improved beam quality. Figure 3d shows the IE difference crossline at a specific reference point $\left(\Delta x=5\mathsf{\mu }\mathrm{m},\Delta y=0\right)$, where the periodic fluctuations are evident in all three curves. The amplitude of fluctuation increases with beam order, confirming that while the beam order influences the amplitude of the IE difference fluctuation, it does not alter the evolution period.

From Figure 1, we confirm by adjusting the initial coherence length, we can control the spatial positioning of the IE difference sidelobes, thus managing the region of reduced disorder. Therefore, we need to jointly control the beam order and coherence length to ensure that the valley value of the IE difference sidelobe is kept at the selected reference point $\left(\Delta x=5\mathsf{\mu }\mathrm{m},\Delta y=0\right)$ for quantitative analysis. Figure 4 shows the minimum IE difference and the corresponding coherence length under different beam orders, aiming to provide readers with a basis for selecting the appropriate beam order and coherence length according to actual needs. We find that as the beam order increases, the coherence of the beam needs to increase accordingly, and the minimum value of the IE difference at this time will gradually decrease, thereby effectively reducing the disorder of the beam.

Next, we discuss the influence of GRIN fiber parameters on the periodic features of the IE difference in PC beams. Figure 5 explores the influence of central refractive index $n_0$ on the periodic features of the IE difference. Figure 5a–c illustrate the IE difference evolution for Gaussian Schell-model beams and HGcSM beams under different central refractive indices. The results show that the central refractive index significantly affects both the period and fluctuation amplitude of the IE difference evolution. As $n_0$ increases, the periodic changes and fluctuation amplitude become more pronounced. This is because a higher central refractive index increases the refractive index difference during beam propagation, causing the light to deflect more strongly toward the axis and reducing the required axial distance for the periodic change, thus shortening the period. Notably, the evolution behavior of the Gaussian Schell-model beam in Figure 5a differs from that of the HGcSM beams in Figure 5b,c. When the central refractive index is small, diffraction effects dominate the early stages of propagation, leading to opposite trends in the IE difference evolution compared with cases with higher central refractive indices. Regardless, the IE difference remains periodic in all cases.

Figure 6 examines the impact of core radius on the periodic features of the IE difference. Figure 6a–c correspond to core radii of 55 $\mathsf{\mu }\mathrm{m}$, 60 $\mathsf{\mu }\mathrm{m}$, and 65 $\mathsf{\mu }\mathrm{m}$, respectively, showing how the IE difference evolves with propagation distance under different core radii. Figure 6d zooms in on the IE difference evolution of the Gaussian Schell-model beam in Figure 6c. The comparison of Figure 6a–c reveals that the core radius significantly affects both the periodicity and fluctuation amplitude of the IE difference. As the core radius increases, the evolution period becomes longer, and the fluctuation amplitude decreases. This indicates that the core radius is a key factor influencing the periodic behavior of the IE difference in optical fibers.

5. Summary and Discussion

In this study, we explore the IE distribution of HGcSM beams, a special class of PC beams. Our findings reveal that the beam order significantly influences the IE distribution on the source plane, with different beam orders leading to distinct entropy trends. This result is crucial for understanding the initial information characteristics of these beams. Additionally, we observe that the evolution of IE in a GRIN fiber exhibits periodic behavior. Compared with traditional Gaussian Schell-model beams, the entropy evolution of HGcSM beams is more complex, with periodic low-entropy regions appearing during propagation. As the beam order increases, these low-entropy regions shrink, suggesting improved control over information uncertainty and enhanced orderliness in higher-order beams.

We also examine the impact of GRIN fiber parameters—specifically the central refractive index and core radius—on the IE evolution. Our results indicate that fibers with higher central refractive indices and smaller core radii produce larger cycles and more pronounced fluctuations in IE, highlighting the intrinsic connection between fiber parameters and beam IE evolution. This finding offers valuable insights for optimizing optical fiber transmission.

While this study does not address non-uniform coherent structured beams, their unique coherence properties suggest that they may exhibit novel IE evolution behaviors during propagation, warranting further investigation.