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Enhancing fiber-coupling efficiency of beam-to-fiber links in turbulence by spatial non-uniform coherence engineering

Zhiwen Yan, Ying Xu, Shuqin Lin, Hao Chang, Xinlei Zhu, Yangjian Cai, and Jiayi Yu

Open Access Open Access

Abstract

We present a general formula for the fiber-coupling efficiency of various types of non-uniformly correlated beams propagating in a turbulent atmosphere. With it, we calculate the fiber-coupling efficiency of a specific type of non-uniformly correlated beams, Laguerre non-uniformly correlated (LNUC) beams, to investigate how the non-uniform correlation structure plays a role in enhancing the fiber-coupling efficiency. Compared with conventional Gaussian Schell-model beams, the LNUC beams possess better coupling behavior, and the initial coherence length and beam order of such beams can be adjusted to further improve the fiber-coupling efficiency in turbulence. Our results demonstrate how non-uniformly correlated beams can be used for fiber-coupling applications, and demonstrate their intriguing potential for free-space optical communications.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Free-space optical communication is a new generation technology of laser communication, which uses free space as an information transmission channel and light beams as an information carrier to transmit and exchange information between different platforms. One part of this technology is to couple spatial light beams into fibers [1], however, before this, the light beams need to propagate through a turbulent atmosphere. The fluctuations of turbulent refractive index will cause random fluctuations in the beam wavefront, resulting in negative effects such as beam spot deformation, beam wander, and scintillation, which makes it difficult to receive optical signals and seriously affects the quality of optical communications [2]. Therefore, it is particularly important to study how to reduce or overcome the negative impact of turbulence on light beams, thereby improving beam quality and fiber-coupling efficiency in turbulence.

In order to solve the above challenge, people mainly adopt two types of methods: designing and optimizing the fiber coupling system [3–5]; correcting and compensating the distorted beams by some algorithms [6,7]. For the former, studies have shown that using optimized inverted nanocones [3] and attaching a small pure silica sphere on the surface of single-mode fiber [4] can improve fiber-coupling efficiency. Adjusting the fiber parameters to adapt the beam width can maximize fiber-coupling efficiency of Laguerre-Gaussian beams and vortex beams [5]. For the latter, such as an adaptive optical compensation system of the incident signal optical field induced by a turbulent atmosphere was used to obtain high fiber-coupling efficiency in satellite to ground laser communication [6]. The hybrid technology of adaptive optics and coherent fiber array can further improve the fiber-coupling efficiency of light beams in turbulence [7].

Light beams whose spatial characteristics are modulated to produce novel physics features, known as structured light beams, have been extensively studied in recent years [8]. It provides a new idea for solving the above challenge. Coherence is one of the important natural characteristics of light beams, and numerous studies have shown that light beams with reduced spatial coherence, called partially coherent structured beams [9,10], have a wide range of applications [11–13]. The beams exhibit better resistance to turbulence than coherent beams [14,15], which has been physically explained from the perspective of coherent mode representation [16,17]. Therefore, the fiber-coupling efficiency of different kinds of partially coherent structured beams in turbulence has been extensively studied. L. Tan et al. derived the fiber-coupling efficiency equation for Gaussian Shell-model (GSM) beams in turbulence and discussed the effect of spatial coherence on fiber-coupling efficiency [18]. The effects of partial coherence and partial polarization of random electromagnetic beams on fiber-coupling efficiency are discussed in detail [19]. X. Zhu et al. analyzed the effect of joint modulation of polarization, phase, and coherence of partially coherent radially polarized vortex beams on the fiber-coupling efficiency [20]. Recently, Y. Yuan et al. found that the fiber-coupling efficiency of a partially coherent flat-topped beam is higher than that of Gaussian beam and GSM beam, and the coupling efficiency can be improved by adjusting the beam order of the beam [21].

On the other hand, existing studies have shown that the turbulence resistance of partially coherent structured beams is better than that of coherent beams, and it is enhanced as the spatial coherence decreases [14,15]. However, numerous studies including the above studies have shown that reducing the coherence of partially coherent structured beams will reduce the fiber coupling efficiency of the beams in turbulence, which is a contradiction and challenge in the field of fiber coupling of partially coherent structured beams. Therefore, it is urgent to find a special type of partially coherent structured beams, which can simultaneously improve the turbulence resistance and fiber coupling efficiency by simple parameter adjustment such as reducing coherence.

Correlation structure is a unique second-order statistical characteristic of partially coherent structured beams, which can be modulated to generate non-uniformly correlated beams, and has been extensively studied in recent years [22–32]. The non-uniform correlation structure enables light beams to exhibit self-focusing propagation features, which plays a crucial role in many applications, such as optical trapping and manipulation of particles [33], measuring the refractive index of crystal [34], and so on. Our previous paper showed that non-uniformly correlated beams not only exhibit self-focusing propagation characteristics in turbulence, but also have excellent resistance to turbulence, furthermore, reducing the spatial coherence can simultaneously enhance their self-focusing and turbulence resistance [24], which will certainly play an important role in enhancing the fiber-coupling efficiency.

In this paper, we focus on the effect of non-uniform correlation structures on the fiber-coupling efficiency of beams to single-mode fiber in turbulence. A general formula for calculating the fiber-coupling efficiency of various types of non-uniformly correlated beams in a turbulent atmosphere is derived. With it, we calculate the fiber-coupling efficiency of a specific type of non-uniformly correlated beams, Laguerre non-uniformly correlated (LNUC) beams, and discuss the effect of different initial beam parameters on the fiber-coupling efficiency. Furthermore, we introduce a strategy on how to adjust the coherence length and beam order to enhance the fiber-coupling efficiency, which will help the design of free-space optical communication systems.

2. Formulations

In free-space optical communications, the cooperation of space light-fiber coupling technology is required for signal transmission and exchange. Figure 1 is a schematic diagram of spatial light-fiber coupling in a turbulent atmosphere. The light beam propagates a distance $z$ in the turbulence before being incident on the surface of a coupling lens. The lens then converges the light beam distorted by turbulence and couples it into a fiber. The bit error rate of optical signals is affected by the fiber-coupling efficiency $\eta _c$, which is defined as the ratio of the average power $\left \langle P_c\right \rangle$ coupled into the fiber to the average optical power $\left \langle P_a\right \rangle$ incident on the receiving plane of the optical system [35], i.e., $\eta _c=\left \langle P_c\right \rangle /\left \langle P_a\right \rangle$. Here, we assume that the incident light beam is coupled into a single-mode fiber through a coupling lens with an aperture $D$. In practice, the lens is a hard aperture. To simplify the integration, we approximate this diameter by a Gaussian aperture of radius $w$ related to the aperture diameter, $w^2=D^2/8$. For partially coherent structured beams, they need to be characterized by the cross spectral density, so we have [18,21,36]

$$\eta_c=\frac{\left\langle{P_c}\right\rangle}{\left\langle{P_a}\right\rangle}=\frac{\iint W\left({\boldsymbol{\rho}}_1,{\boldsymbol{\rho}}_2,z\right)F^*\left({\boldsymbol{\rho}}_1\right)F\left({\boldsymbol{\rho}}_2\right)\exp\left[-\left({\boldsymbol{\rho}}_1^2+{\boldsymbol{\rho}}_2^2\right)/w^2\right]d^2{\boldsymbol{\rho}}_1 d^2{\boldsymbol{\rho}}_2}{\int W\left({\boldsymbol{\rho}},{\boldsymbol{\rho}},z\right)\exp\left({-}2{\boldsymbol{\rho}}^2/w^2\right)d^2{\boldsymbol{\rho}}},$$
where ${\boldsymbol {\rho }}\equiv (\rho _x, \rho _y)$ is the position vectors transverse to the direction of propagation in the receiving plane; the asterisk denotes the complex conjugation; $W({\boldsymbol {\rho }}_1,{\boldsymbol {\rho }}_2,z)$ is the cross spectral density of partially coherent structured beams in the receiving plane; $F({\boldsymbol {\rho }})$ describes the mold field distribution in the receiving/aperture plane, and is given by
$$F\left({\boldsymbol{\rho}}\right)=\sqrt{\frac{2}{\pi w_a^2}}\exp\left(-\frac{{\boldsymbol{\rho}}^2}{w_a^2}\right),$$
where $w_a=\lambda f/(\pi w_j)$ with $\lambda$ is the wavelength, $f$ is the focal length of the coupling lens, and $w_j$ is the fiber-mode field radius.

 figure: Fig. 1.

Fig. 1. Schematic diagram of spatial light-fiber coupling in a turbulent atmosphere.

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From the definition of fiber-coupling efficiency, it can be seen that the average power of the light beam entering the fiber determines the coupling efficiency. For the case where the beam energy is concentrated in a small area and irradiated on the tangent plane of the fiber, the fiber-coupling efficiency in this case must be higher than the case where the beam energy is divergent or no beam is irradiated on the tangent plane of the fiber. Therefore, the light intensity distribution on the receiving plane will inevitably affect the fiber-coupling efficiency. It is well known that the intensity evolution of partially coherent structured beams on propagation can be controlled by manipulating their correlation structures [9–11], thus the correlation structure becomes a degree of freedom to enhance the fiber-coupling efficiency. Heretofore, numerous studies have shown that partially coherent structured beams with non-uniform correlation structures exhibit self-focusing features in both free space and turbulence, and the control of the self-focusing feature can be realized by adjusting the initial beam parameters of the correlation structure, including the spatial position and intensity of the focal point [23,24,31]. Therefore, non-uniformly correlated beams are of great significance for enhancing the fiber-coupling efficiency in turbulence.

Next, we will derive a general formula for calculating the fiber-coupling efficiency of various types of non-uniformly correlated beams propagating through a turbulent atmosphere. Based on Eq. (1), it is necessary to obtain the cross spectral density of the considered beams at the receiving plane. We assume paraxial propagation along the $z$-axis, then, the cross spectral density of non-uniformly correlated beams in turbulence can be described by the extended Huygens-Fresnel integral [2]

$$\begin{aligned}W\left({\boldsymbol{\rho}}_1,{\boldsymbol{\rho}}_2,z\right)&=\frac{1}{\lambda^2 z^2}\iint W\left({\mathbf{r}}_1,{\mathbf{r}}_2\right)\exp\left[-\frac{ik}{2z}\left({\mathbf{r}}_1-{\boldsymbol{\rho}}_1\right)^2+\frac{ik}{2z}\left({\mathbf{r}}_2-{\boldsymbol{\rho}}_2\right)^2\right] \\ &\times\left\langle\exp\left[\Psi^*\left({\mathbf{r}}_1,{\boldsymbol{\rho}}_1\right)+\Psi\left({\mathbf{r}}_2,{\boldsymbol{\rho}}_2\right)\right]\right\rangle d^2{\mathbf{r}}_1d^2{\mathbf{r}}_2, \end{aligned}$$
where $k=2\pi /\lambda$ is the wavenumber. ${\mathbf {r}}\equiv (x, y)$ is the position vectors in the source plane and $W({\mathbf {r}}_1,{\mathbf {r}}_2)$ denotes the cross spectral density of the source. $\Psi ({\mathbf {r}}, {\boldsymbol {\rho }})$ is the complex phase perturbation caused by the refractive index fluctuations of the turbulent medium between ${\mathbf {r}}$ and ${\boldsymbol {\rho }}$. The ensemble average over the turbulence, last term in above equation, can be expressed as [2]
$$\begin{aligned}&\left\langle\exp\left[\Psi^*\left({\mathbf{r}}_1,{\boldsymbol{\rho}}_1\right)+\Psi\left({\mathbf{r}}_2,{\boldsymbol{\rho}}_2\right)\right]\right\rangle \\ &=\exp\left\{{-}4\pi^2 k^2 z\int_0^1 \int_0^\infty\boldsymbol\kappa\Phi_n\left(\boldsymbol\kappa\right)\cdot\left\{ 1-J_0\left[\left|\left(1-\xi\right){\boldsymbol{\rho}}_d+\xi{\mathbf{r}}_d\right|\boldsymbol\kappa\right]\right\}d^2\boldsymbol\kappa d\xi\right\}, \end{aligned}$$
where ${\boldsymbol {\rho }}_d={\boldsymbol {\rho }}_1-{\boldsymbol {\rho }}_2$, ${\mathbf {r}}_d={\mathbf {r}}_1-{\mathbf {r}}_2$, $J_0$ is the zero-order Bessel function that can be approximated as [2]
$$J_0\left[\left|\left(1-\xi\right){\boldsymbol{\rho}}_d+\xi{\mathbf{r}}_d\right|\boldsymbol\kappa\right]\sim 1-\frac{1}{4}\left[\left(1-\xi\right){\boldsymbol{\rho}}_d+\xi{\mathbf{r}}_d\right]^2\boldsymbol\kappa^2.$$

Substituting the relation Eq. (5) into Eq. (4) to obtain the ensemble average term

$$\begin{aligned}&\left\langle\exp\left[\Psi^*\left({\mathbf{r}}_1,{\boldsymbol{\rho}}_1\right)+\Psi\left({\mathbf{r}}_2,{\boldsymbol{\rho}}_2\right)\right]\right\rangle \\ &=\exp\left\{{-}k\Omega Tz^{{-}1}\left[\left({\boldsymbol{\rho}}_1-{\boldsymbol{\rho}}_2\right)^2+\left({\boldsymbol{\rho}}_1-{\boldsymbol{\rho}}_2\right)\cdot\left({\mathbf{r}}_1-{\mathbf{r}}_2\right)+\left({\mathbf{r}}_1-{\mathbf{r}}_2\right)^2\right]\right\}, \end{aligned}$$
with $\Omega =k\pi ^2z^2/3$ and $T=\int _0^\infty \boldsymbol \kappa ^3\Phi \left (\boldsymbol \kappa \right )d^2\boldsymbol \kappa$ with $\Phi (\boldsymbol \kappa )$ represents spatial power spectrum of the refractive-index fluctuations of the turbulence.

For partially coherent structured sources, their cross spectral density must correspond to a non-negative definite kernel, which is fulfilled if the function can be written as [37]

$$W\left({\mathbf{r}}_1,{\mathbf{r}}_2\right)=\int p\left(\mathbf{v} \right)H^*\left({\mathbf{r}}_1,\mathbf{v} \right)H\left({\mathbf{r}}_2,\mathbf{v} \right)d^2\mathbf{v},$$
where $H({\mathbf {r}},\mathbf {v})$ is an arbitrary integrable kernel, and $p(\mathbf {v})$ is a non-negative weight function with $\mathbf {v}\equiv (v_x,v_y)$ is a two-dimensional vector in Fourier space. The discretized Eq. (7) can be viewed as the cross spectral density of partially coherent structured sources as a linear superposition of mutually uncorrelated, fully coherent, and suitably weighted scalar fields. Therefore, $H({\mathbf {r}},\mathbf {v})$ can be viewed as pseudo-modes with $p(\mathbf {v})$ is their corresponding weights. To generate partially coherent structured sources with self-focusing unique propagation feature, we consider the pseudo-modes as a scalar field with a quadratic phase factor as
$$H\left({\mathbf{r}},\mathbf{v}\right)=\tau\left({\mathbf{r}}\right)\exp\left[{-}ik\left(v_x+v_y\right){\mathbf{r}}^2\right],$$
where $\tau ({\mathbf {r}})=\exp (-{\mathbf {r}}^2/w_0^2)$ is the deterministic complex amplitude with $w_0$ is the beam width. Based on the above setting, we can get the expression of the non-uniformly correlated sources as
$$W\left({\mathbf{r}}_1,{\mathbf{r}}_2\right)\propto\tau^*\left({\mathbf{r}}_1\right)\tau\left({\mathbf{r}}_2\right)\tilde p\left({\mathbf{r}}_1^2-{\mathbf{r}}_2^2\right),$$
where $\tilde p\left ({\mathbf {r}}_1^2-{\mathbf {r}}_2^2\right )$ is the Fourier transform of the weight function $p\left (\mathbf {v}\right )$. Therefore, with different choices of weight function $p(\mathbf {v})$, we can generate a variety of non-uniformly correlated beams based on Eq. (9). And we may derive the spectral intensity and the fiber-coupling efficiency for various types of non-uniformly correlated beams in turbulence using the above equations. After a complex and lengthy integral calculation (using the exchanged integration order in the process), we obtained a general expression of the spectral intensity $S({\boldsymbol {\rho }},z)$ and the fiber-coupling efficiency $\eta _c$ of variety of non-uniformly correlated beams in turbulence
$$S({\boldsymbol{\rho}},z)=\int \frac{w_0^2}{2w_z^2}p(\mathbf{v})\exp\left(\frac{{\boldsymbol{\rho}}^2}{w_z^2}\right)d^2\mathbf{v},$$
and
$$\eta_c=2\frac{\int p\left(\mathbf{v}\right)\left(w_a w_z\sqrt{C_z}\right)^{{-}2}d^2\mathbf{v}}{\int p\left(\mathbf{v}\right)\left(1+2w_z^2/{w^2}\right)^{{-}1}d^2\mathbf{v}},$$
where
$$w_z^2=\frac{w_0^2}{2}\left[1-2\left(v_x+v_y\right)z\right]^2+\frac{2z^2}{w_0^2 k^2}+\frac{4\Omega Tz}{k},$$
$$C_z=\left(A_z+\frac{B_z^2}{w_z^2}+\frac{1}{4w_z^2}+\frac{1}{w_a^2}+\frac{1}{w^2}\right)^2-\left(A_z+\frac{B_z^2}{w_z^2}-\frac{1}{4w_z^2}\right)^2+\left(\frac{k}{2z}\right)^2,$$
$$A_z=\frac{w_0^2 k^2}{8z^2}+\Omega T;\quad B_z=\frac{kw_0^2}{4z}\left[1-2\left(v_x+v_y\right)z\right]-\Omega T.$$

Therefore, users can calculate the spectral intensity evolution and fiber-coupling efficiency in turbulence by simply substituting the weight function $p(\mathbf {v})$ corresponding to the considered non-uniformly correlated beams into Eq. (10) and Eq. (13). Comparing the numerical results to find out which beam has excellent coupling performance.

Therefore, the user only needs to substitute the weight function $p(\mathbf {v})$ corresponding to the considered inhomogeneously correlated beam into Eq. (10) and Eq (11) to calculate the fiber coupling efficiency of different types of inhomogeneously correlated beams in turbulent flow and compare the numerical results , to find out which beam has excellent coupling properties.

In order to demonstrate the excellent fiber-coupling efficiency of partially coherent beams with prescribed non-uniform correlation structure, we now focus on the particular example of partially coherent structured beams whose weight function $p(\mathbf {v})$ has the form as

$$p\left(\mathbf{v} \right)=\left(\alpha^{n+1}/\pi n!\right)\mathbf{v}^{2n}\exp\left(-\alpha\mathbf{v}^2\right),$$
with $\alpha$ is a positive real constant and $n$ is beam order. Then, according to Eq. (9), one obtains the cross spectral density of the proposed beams as
$$W\left({\mathbf{r}}_1,{\mathbf{r}}_2\right)=\exp\left(-\frac{{\mathbf{r}}_1^2+{\mathbf{r}}_2^2}{w_0^2}\right)\exp\left[-\frac{\left({\mathbf{r}}_1^2-{\mathbf{r}}_2^2 \right)^2}{r_c^4}\right]L_n^0\left[\frac{\left({\mathbf{r}}_1^2-{\mathbf{r}}_2^2\right)^2}{r_c^4}\right],$$
with $r_c=(2\alpha /k^2)^{1/4}$ is the spatial coherence length, and $L_n^0$ represents the Laguerre polynomial of mode orders $n$ and 0. We note that the correlation function of the proposed beams is an inhomogeneous function of Laguerre form. Therefore, we label such partially coherent structured beams as LNUC beams. We infer from earlier work that LNUC beams will exhibit self-focusing features, enabling improved fiber-coupling efficiency. This is due to the fact that the pseudo-modes $H({\mathbf {r}},\mathbf {v})$ of the beams possess a quadratic phase factor, which causes them to converge. So far, based on the above analysis, we can obtain the spectral intensity evolution and the fiber-coupling efficiency of LNUC beams in a turbulent atmosphere using Eqs. (10), (11) and (15).

3. Numerical results

In this section, we use the expressions derived above to study the intensity evolution and the fiber-coupling efficiency of the LNUC beams in a turbulent atmosphere by detailed numerical calculations. In the following numerical calculations, we assume that the turbulence obeys Kolmogorov statistics, and use the von Karman spectrum as the power spectrum of the index-of-refraction fluctuations of the atmospheric turbulence, expressed as [38]

$$\Phi\left(\boldsymbol\kappa\right)=0.033C_n^2\left(\boldsymbol\kappa^2+\kappa_0^2\right)^{{-}11/6}\exp\left(-\boldsymbol\kappa^2/\kappa_m^2\right),$$
where $\kappa _0=2\pi /L_0$ and $\kappa _m=5.92/l_0$ with $L_0$ and $l_0$ being the outer and inner scale of turbulence, respectively. $C^2_n$ is a generalized refractive-index structure parameter, and its value represents the strength of turbulence. The initial parameters of the beams, the turbulence and the fiber coupling system are initially set as follows unless otherwise stated: $\lambda =632.8\rm {nm}$, $w_0=4\rm {cm}$, $r_c=3\rm {cm}$; $C^2_n$=$5\times 10^{-15}\rm {m}^{-2/3}$, $l_0=1\rm {mm}$, $L_0=1\rm {m}$; $w_j=5.15\,{\mathrm{\mu} \mathrm{m}}$, $D=10\rm {cm}$, and $f=20\rm {cm}$. For convenience, the wavelength was taken to be that of common HeNe-type laser diodes; the results presented here are qualitatively similar for optical communications wavelengths.

Figure 2 shows the evolution of the intensity distribution of LNUC beams in the turbulent atmosphere. We find from Fig. 2(a) that the size of the beam spot decreases with increasing propagation distance over short ranges, while at long ranges, the spot size increases. Therefore, if we make full use of the converging process of such beams, which will facilitate the beams to be coupled into the fiber. Figure 2(b) and Fig. 2(c) are the evolution of the normalized intensity on-axis of the LNUC beams with different coherence lengths and beam orders in the turbulence, respectively. The intensity is normalized by setting the on-axis intensity of the source to unity. We also confirm from Fig. 2(b) that the LNUC beams exhibit self-focusing propagation characteristics, the intensity on-axis reach a peak at a short propagation distance, and the energy convergence helps to improve the fiber-coupling efficiency. Then, as the propagation distance further increases, the intensity on-axis gradually decreases. Furthermore, we find that as the coherence length decreases, the self-focusing characteristic will become more drastic, which implies that the fiber-coupling efficiency can be improved by adjusting the coherence length. Figure 2(c) that the LNUC beam with a larger beam order has more obvious self-focusing properties. Comparing the dotted line and solid line, we confirm the self-focusing properties induced by the non-uniform correlation structure, while the conventional GSM beam does not have such feature on propagation, since the coherent modes of the synthesized conventional GSM beam do not carry a quadratic phase factor. Therefore, based on the intensity evolution of such beams in a turbulent atmosphere, we can preliminarily judge that adjusting the initial coherence length and beam order of the beams can effectively enhance the fiber-coupling efficiency of the beam passing through the turbulent atmosphere.

 figure: Fig. 2.

Fig. 2. (a) Density plot of the normalized intensity and (b), (c) normalized intensity on-axis of the LNUC beams upon propagation in turbulence for different coherence lengths with $n=1$ and different beam orders with $r_c=3.0\rm {cm}$.

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Next, we discuss in detail the effects of the initial coherence length and beam order of the LNUC beams on their fiber-coupling efficiency in turbulence. Figure 3 shows the evolution of the fiber-coupling efficiency after beams of different coherence lengths pass through turbulent atmosphere, and clarifies the fiber-coupling efficiency of the LNUC beams in turbulence first increases and then decreases with the increase of propagation distance. The reason for this phenomenon is that such non-uniformly correlated beams possess self-focusing features during propagation (as shown in Fig. 2), and the spot size reduction and energy convergence caused by the self-focusing characteristics improve the fiber-coupling efficiency of the beam after propagation. Comparing the fiber-coupling efficiency of LNUC beams with different coherence lengths, one confirms that with the decrease of the initial coherence length, the peak value of the fiber-coupling efficiency first increases and then decreases, that is, the maximum value of the fiber-coupling efficiency will correspond to an optimal value of the coherence length. As shown in Fig. 3, when beam order $n=1$, the maximum fiber-coupling efficiency $\eta _c=13.65{\%}$ of the LNUC beams in the turbulent atmosphere occurs approximately when the coherence length $r_c=2.5\rm {cm}$. Here we temporarily define $r_c<2.5\rm {cm}$ and $r_c>2.5\rm {cm}$ is the increasing and decreasing regions. Notably, comparing Fig. 2(b) with Fig. 3, we find that the LNUC beam with $n=1$ and $r_c=1.5\rm {cm}$ has high intensity on-axis but exhibits low fiber-coupling efficiency. This is because such beam with low coherence exhibits drastic self-focusing properties, and the divergence angle of the beam after self-focusing is too large to converge by the coupling lens, resulting in a low fiber-coupling efficiency.

 figure: Fig. 3.

Fig. 3. The fiber-coupling efficiency of the LNUC beams in turbulence for different coherence lengths with $n=1$.

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Next, we focus on discussing the effect of the initial beam order on the fiber-coupling efficiency of the LNUC beams in turbulence when their coherence length $r_c=2.0\rm {cm}$ (in increasing region) and $r_c=3.0\rm {cm}$ (in decreasing region). It can be seen from Fig. 4 that no matter which region the coherence length is in, we can adjust their beam order to enhance the value of the fiber-coupling efficiency. It should be noted that it is not that the larger the beam order is, the higher the maximum fiber-coupling efficiency is, and the maximum fiber-coupling efficiency still corresponds to the optimal value of beam order. Furthermore, when the coherence length is in the decreasing region, adjusting the beam order can further increase the maximum fiber-coupling efficiency, as shown in Fig. 4(b), when $n=5$, the maximum fiber-coupling efficiency can reach $\eta _c=14.76{\%}$. Therefore, the beam order can be used as another adjustable initial beam parameter to enhance the fiber-coupling efficiency. Moreover, the black solid line in Fig. 4 represents the fiber-coupling efficiency of GSM beams with the same coherence length ($r_c=3\rm {cm}$) under the same turbulent conditions. Comparing the fiber-coupling efficiency of GSM beams with LNUC beams (i.e., the black line and other dotted lines), it can be found that the fiber-coupling efficiency of GSM beams decreases with the increase of propagation distance in turbulence. This conclusion is consistent with the general knowledge. However, in the short distance range, it shows that the fiber-coupling efficiency of the LNUC beams is low, and is even smaller than that of GSM beams. This is because in the case of short distances, due to the self-focusing characteristics of the non-uniformly correlated beams and focusing effect of the lens, the actual focus of the spot has a focus shift, that is, the actual focus is located in front of the geometric focus of the lens, resulting in the divergence of the intensity at the tangent plane of the fiber, reducing the fiber-coupling efficiency. Therefore, the fiber-coupling efficiency of GSM beams is better than that of LNUC beams over short distances.

 figure: Fig. 4.

Fig. 4. The fiber-coupling efficiency of the LNUC beams and a GSM beam in turbulence for different beam orders with (a) $r_c=2.0\rm {cm}$; (b) $r_c=3.0\rm {cm}$.

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Figure 5 shows the fiber-coupling efficiency of the LNUC beams versus coherence length for different beam orders in turbulence at a certain propagation distance. We find that the fiber-coupling efficiency first increases and then decreases with the increase of the beam coherence length, which is consistent with the results obtained in Fig. 2. It can also be found from Fig. 5 that at a certain propagation distance, the maximum fiber-coupling efficiency of the LNUC beams with different beam orders corresponds to different beam coherence lengths. As the beam order increases, the maximum fiber-coupling efficiency increases, the LNUC beam at this case requires a corresponding high coherence. Comparing Fig. 5(a) with Fig. 5(b), one confirms that at different propagation distances, the maximum fiber-coupling efficiency corresponds to different beam orders and different coherence lengths.

 figure: Fig. 5.

Fig. 5. The fiber-coupling efficiency of the LNUC beams in turbulence versus coherence length for different beam order at distance (a) $z=1 \rm {km}$; (b) $z=2 \rm {km}$.

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The above conclusions show us that the fiber-coupling efficiency of the LNUC beams in turbulence is related to the initial coherence length and beam order of such beams, and there is a set of optimal values corresponding to the maximum fiber-coupling efficiency. In practice, in order to facilitate the user to directly call the optimal value of the initial coherence length and beam order of the beams according to the actual situation, we present in Fig. 6 that the maximum fiber-coupling efficiency and the optimal value of the corresponding coherence length for different beam orders on propagation in turbulence. One confirms from Fig. 6 that the maximum fiber-coupling efficiency increases gradually over the near field to the far field, after which it decreases gently, which indicates that the maximum fiber-coupling efficiency occurs at a defined distance. The maximum coupling efficiency under different distances corresponds to different coherence lengths, and the optimal value of the corresponding coherence length increases gradually first and then increases rapidly, resulting in such partially coherent structured beam approximating a fully coherent beam, thereby losing the self-focusing propagation features. Thus, Fig. 6 only shows the maximum fiber-coupling efficiency and the corresponding optimal values of coherence length and beam order in the distance range of $0\sim 2.8\rm {km}$. Here, we present four results for $n=0,1,2,3$, users can use the results to determine for themselves which coherence lengths and beam orders provide optimal results for their application.

 figure: Fig. 6.

Fig. 6. Maximum fiber-coupling efficiency and the optimal value of the corresponding coherence length on propagation in turbulence for $n=0,1,2,3$.

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4. Discussion and conclusion

It is worth noting that the fiber-coupling efficiency of the LNUC beams in turbulence is closely related to the values of coherence length and beam order. This is because the position of the self-focusing focus exhibited by such beam is closely related to the coherence length and beam order. The smaller coherence length and larger beam order, the shorter the distance required for the self-focusing focus to appear; the larger coherence length and smaller beam order, the longer the distance required for the self-focusing focus to appear. Therefore, by adjusting the coherence length and beam order, the position of the self-focusing focus in turbulence can be manipulated to adapt to the spatial position of the fiber coupling system, thereby achieving the optimal coupling efficiency. On the other hand, we can optimize the fiber-coupling efficiency by improving the beam model. From the discrete necessary and sufficient condition, one knows that a partially coherent structured source can be regarded as a superposition over the coherent modes according to a prescribed weight. Therefore, we can construct partially coherent structured beams with novel propagation properties by modifying the coherent modes and weight distributions, thereby further improving the fiber-coupling efficiency.

We first derive a general formula for calculating the fiber-coupling efficiency of various types of non-uniformly correlated beams propagating through a turbulent atmosphere, and then theoretically investigate the intensity evolution and fiber-coupling efficiency of the LNUC beam in turbulence. Theoretical demonstrations show that the fiber-coupling efficiency of the LNUC beams is affected by the initial coherence length and beam order, and the fiber-coupling efficiency can be enhanced by adjusting these two beam parameters. Our results of the general formula will be useful for follow-up studies, and the non-uniformly correlated beams show intriguing potential in free-space optical communications.

Funding

National Key Research and Development Program of China (2019YFA0705000, 2022YFA1404800); National Natural Science Foundation of China (11974218, 12004218, 12192254, 92250304); China Postdoctoral Science Foundation (2020M680093, 2022M721992); Natural Science Foundation of Shandong Province (ZR2020QA067); The Open Fund of the Guangdong Provincial Key Laboratory of Optical Fiber Sensing and Communications (2020GDSGXCG08).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

1. V. W. S. Chan, “Free-Space Optical Communications,” J. Lightwave Technol. 24(12), 4750–4762 (2006). [CrossRef]  

2. L. C. Andrews and R. L. Philips, Laser Beam Propagation through Random Media (SPIE, 2005).

3. C. G. Torun, P. I. Schneider, M. Hammerschmidt, S. Burger, J. H. D. Munns, and T. Schröder, “Optimized diamond inverted nanocones for enhanced color center to fiber coupling,” Appl. Phys. Lett. 118(23), 234002 (2021). [CrossRef]  

4. X. Chen, X. Hu, L. Liao, Y. Xing, G. Chen, L. Yang, J. Peng, H. Li, N. Dai, and J. Li, “High coupling efficiency technology of large core hollow-core fiber with single mode fiber,” Opt. Express 27(23), 33135–33142 (2019). [CrossRef]  

5. S. Rojas-Rojas, G. Ca nas, G. Saavedra, E. S. Gómez, S. P. Walborn, and G. Lima, “Evaluating the coupling efficiency of OAM beams into ring-core optical fibers,” Opt. Express 29(15), 23381–23392 (2021). [CrossRef]  

6. D. Rui, C. Liu, M. Chen, B. Lan, and H. Xian, “Probability enhancement of fiber coupling efficiency under turbulence with adaptive optics compensation,” Opt. Fiber Technol. 60, 102343 (2020). [CrossRef]  

7. H. Wu, H. Yan, and X. Li, “Modal correction for fiber-coupling efficiency in free-space optical communication systems through atmospheric turbulence,” Optik 121(19), 1789–1793 (2010). [CrossRef]  

8. A. Forbes, M. de Oliveira, and M. R. Dennis, “Structured light,” Nat. Photonics 15(4), 253–262 (2021). [CrossRef]  

9. G. Gbur and T. D. Visser, “The structure of partially coherent fields,” Prog. Opt. 55, 285–341 (2010). [CrossRef]  

10. Y. Chen, F. Wang, and Y. Cai, “Partially coherent light beam shaping via complex spatial coherence structure engineering,” Adv. Phys.: X 7(1), 2009742 (2022). [CrossRef]  

11. O. Korotkova and G. Gbur, “Applications of optical coherence theory,” Prog. Opt. 65, 43–104 (2020). [CrossRef]  

12. D. Peng, Z. Huang, Y. Liu, Y. Chen, F. Wang, S. A. Ponomarenko, and Y. Cai, “Optical coherence encryption with structured random light,” PhotoniX 2(1), 6–15 (2021). [CrossRef]  

13. Y. Liu, Y. Chen, F. Wang, Y. Cai, C. Liang, and O. Korotkova, “Robust far-field imaging by spatial coherence engineering,” Opto-Electron. Adv. 4(12), 210027 (2021). [CrossRef]  

14. G. Gbur, “Partially coherent beam propagation in atmospheric turbulence (invited),” J. Opt. Soc. Am. A 31(9), 2038–2045 (2014). [CrossRef]  

15. F. Wang, X. Liu, and Y. Cai, “Propagation of partially coherent beam in turbulent atmosphere: a review (Invited review),” Prog. Electromagnetics Res. 150, 123–143 (2015). [CrossRef]  

16. G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19(8), 1592–1598 (2002). [CrossRef]  

17. T. Shirai, A. Dogariu, and E. Wolf, “Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 20(6), 1094–1102 (2003). [CrossRef]  

18. L. Tan, M. Li, Q. Yang, and J. Ma, “Fiber-coupling efficiency of Gaussian Schell model for optical communication through atmospheric turbulence,” Appl. Opt. 54(9), 2318–2325 (2015). [CrossRef]  

19. M. Salem and G. P. Agrawal, “Effects of coherence and polarization on the coupling of stochastic electromagnetic beams into optical fibers,” J. Opt. Soc. Am. A 26(11), 2452–2458 (2009). [CrossRef]  

20. X. Zhu, K. Wang, F. Wang, C. Zhao, and Y. Cai, “Coupling efficiency of a partially coherent radially polarized vortex beam into a single-mode fiber,” Appl. Sci. 8(8), 1313 (2018). [CrossRef]  

21. Y. Yuan, J. Zhang, J. Dang, W. Zheng, G. Zheng, P. Fu, J. Qu, B. J. Hoenders, Y. Zhao, and Y. Cai, “Enhanced fiber-coupling efficiency via high-order partially coherent flat-topped beams for free-space optical communications,” Opt. Express 30(4), 5634–5643 (2022). [CrossRef]  

22. S. A. Ponomarenko, “A class of partially coherent beams carrying optical vortices,” J. Opt. Soc. Am. A 18(1), 150–156 (2001). [CrossRef]  

23. H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011). [CrossRef]  

24. J. Yu, F. Wang, L. Liu, Y. Cai, and G. Gbur, “Propagation properties of Hermite non-uniformly correlated beams in turbulence,” Opt. Express 26(13), 16333–16343 (2018). [CrossRef]  

25. Z. Mei, “Hyperbolic sine-correlated beams,” Opt. Express 27(5), 7491–7497 (2019). [CrossRef]  

26. J. Yu, X. Zhu, S. Lin, F. Wang, G. Gbur, and Y. Cai, “Vector partially coherent beams with prescribed non-uniform correlation structure,” Opt. Lett. 45(13), 3824–3827 (2020). [CrossRef]  

27. X. Zhu, J. Yu, Y. Chen, F. Wang, O. Korotkova, and Y. Cai, “Experimental synthesis of random light sources with circular coherence by digital micro-mirror device,” Appl. Phys. Lett. 117(12), 121102 (2020). [CrossRef]  

28. S. Zhu, P. Li, Z. Li, Y. Cai, and W. He, “Generating non-uniformly correlated twisted sources,” Opt. Lett. 46(20), 5100–5103 (2021). [CrossRef]  

29. X. Zhu, J. Yu, F. Wang, Y. Chen, Y. Cai, and O. Korotkova, “Synthesis of vector nonuniformly correlated light beams by a single digital mirror device,” Opt. Lett. 46(12), 2996–2999 (2021). [CrossRef]  

30. M. Luo, M. Koivurova, M. Ornigotti, and C. Ding, “Turbulence-resistant self-focusing vortex beams,” New J. Phys. 24(9), 093036 (2022). [CrossRef]  

31. S. Lin, J. Wu, Y. Xu, X. Zhu, G. Gbur, Y. Cai, and J. Yu, “Analysis and experimental demonstration of propagation features of radially polarized specific non-uniformly correlated beams,” Opt. Lett. 47(2), 305–308 (2022). [CrossRef]  

32. Y. Xu, Y. Guan, Y. Liu, S. Lin, X. Zhu, Y. Cai, J. Yu, and G. Gbur, “Generating multi-focus beams with a spatial non-uniform coherence structure,” Opt. Lett. 48(10), 2631–2634 (2023). [CrossRef]  

33. J. Yu, Y. Xu, S. Lin, X. Zhu, G. Gbur, and Y. Cai, “Longitudinal optical trapping and manipulating Rayleigh particles by spatial nonuniform coherence engineering,” Phys. Rev. A 106(3), 033511 (2022). [CrossRef]  

34. R. Lin, M. Chen, Y. Liu, H. Zhang, G. Gbur, Y. Cai, and J. Yu, “Measuring refractive indices of a uniaxial crystal by structured light with non-uniform correlation,” Opt. Lett. 46(10), 2268–2271 (2021). [CrossRef]  

35. P. J. Winzer and W. R. Leeb, “Fiber coupling efficiency for random light and its applications to lidar,” Opt. Lett. 23(13), 986–988 (1998). [CrossRef]  

36. C. Zhai, L. Tan, S. Yu, and J. Ma, “Fiber coupling efficiency for a Gaussian-beam wave propagating through non-Kolmogorov turbulence,” Opt. Express 23(12), 15242–15255 (2015). [CrossRef]  

37. F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007). [CrossRef]  

38. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008). [CrossRef]  

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (6)
Fig. 1.
Fig. 1. Schematic diagram of spatial light-fiber coupling in a turbulent atmosphere.

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Fig. 2.
Fig. 2. (a) Density plot of the normalized intensity and (b), (c) normalized intensity on-axis of the LNUC beams upon propagation in turbulence for different coherence lengths with $n=1$ and different beam orders with $r_c=3.0\rm {cm}$.

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Fig. 3.
Fig. 3. The fiber-coupling efficiency of the LNUC beams in turbulence for different coherence lengths with $n=1$.

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Fig. 4.
Fig. 4. The fiber-coupling efficiency of the LNUC beams and a GSM beam in turbulence for different beam orders with (a) $r_c=2.0\rm {cm}$; (b) $r_c=3.0\rm {cm}$.

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Fig. 5.
Fig. 5. The fiber-coupling efficiency of the LNUC beams in turbulence versus coherence length for different beam order at distance (a) $z=1 \rm {km}$; (b) $z=2 \rm {km}$.

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Fig. 6.
Fig. 6. Maximum fiber-coupling efficiency and the optimal value of the corresponding coherence length on propagation in turbulence for $n=0,1,2,3$.

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Equations (17)

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η c = ⟨ P c ⟩ ⟨ P a ⟩ = ∬ W ( ρ 1 , ρ 2 , z ) F ∗ ( ρ 1 ) F ( ρ 2 ) exp ⁡ [ − ( ρ 1 2 + ρ 2 2 ) / w 2 ] d 2 ρ 1 d 2 ρ 2 ∫ W ( ρ , ρ , z ) exp ⁡ ( − 2 ρ 2 / w 2 ) d 2 ρ ,
F ( ρ ) = 2 π w a 2 exp ⁡ ( − ρ 2 w a 2 ) ,
W ( ρ 1 , ρ 2 , z ) = 1 λ 2 z 2 ∬ W ( r 1 , r 2 ) exp ⁡ [ − i k 2 z ( r 1 − ρ 1 ) 2 + i k 2 z ( r 2 − ρ 2 ) 2 ] × ⟨ exp ⁡ [ Ψ ∗ ( r 1 , ρ 1 ) + Ψ ( r 2 , ρ 2 ) ] ⟩ d 2 r 1 d 2 r 2 ,
⟨ exp ⁡ [ Ψ ∗ ( r 1 , ρ 1 ) + Ψ ( r 2 , ρ 2 ) ] ⟩ = exp ⁡ { − 4 π 2 k 2 z ∫ 0 1 ∫ 0 ∞ κ Φ n ( κ ) ⋅ { 1 − J 0 [ | ( 1 − ξ ) ρ d + ξ r d | κ ] } d 2 κ d ξ } ,
J 0 [ | ( 1 − ξ ) ρ d + ξ r d | κ ] ∼ 1 − 1 4 [ ( 1 − ξ ) ρ d + ξ r d ] 2 κ 2 .
⟨ exp ⁡ [ Ψ ∗ ( r 1 , ρ 1 ) + Ψ ( r 2 , ρ 2 ) ] ⟩ = exp ⁡ { − k Ω T z − 1 [ ( ρ 1 − ρ 2 ) 2 + ( ρ 1 − ρ 2 ) ⋅ ( r 1 − r 2 ) + ( r 1 − r 2 ) 2 ] } ,
W ( r 1 , r 2 ) = ∫ p ( v ) H ∗ ( r 1 , v ) H ( r 2 , v ) d 2 v ,
H ( r , v ) = τ ( r ) exp ⁡ [ − i k ( v x + v y ) r 2 ] ,
W ( r 1 , r 2 ) ∝ τ ∗ ( r 1 ) τ ( r 2 ) p ~ ( r 1 2 − r 2 2 ) ,
S ( ρ , z ) = ∫ w 0 2 2 w z 2 p ( v ) exp ⁡ ( ρ 2 w z 2 ) d 2 v ,
η c = 2 ∫ p ( v ) ( w a w z C z ) − 2 d 2 v ∫ p ( v ) ( 1 + 2 w z 2 / w 2 ) − 1 d 2 v ,
w z 2 = w 0 2 2 [ 1 − 2 ( v x + v y ) z ] 2 + 2 z 2 w 0 2 k 2 + 4 Ω T z k ,
C z = ( A z + B z 2 w z 2 + 1 4 w z 2 + 1 w a 2 + 1 w 2 ) 2 − ( A z + B z 2 w z 2 − 1 4 w z 2 ) 2 + ( k 2 z ) 2 ,
A z = w 0 2 k 2 8 z 2 + Ω T ; B z = k w 0 2 4 z [ 1 − 2 ( v x + v y ) z ] − Ω T .
p ( v ) = ( α n + 1 / π n ! ) v 2 n exp ⁡ ( − α v 2 ) ,
W ( r 1 , r 2 ) = exp ⁡ ( − r 1 2 + r 2 2 w 0 2 ) exp ⁡ [ − ( r 1 2 − r 2 2 ) 2 r c 4 ] L n 0 [ ( r 1 2 − r 2 2 ) 2 r c 4 ] ,
Φ ( κ ) = 0.033 C n 2 ( κ 2 + κ 0 2 ) − 11 / 6 exp ⁡ ( − κ 2 / κ m 2 ) ,
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