Enhanced fiber-coupling efficiency via high-order partially coherent flat-topped beams for free-space optical communications

Yangsheng Yuan; Yangsheng Yuan; Jiqian Zhang; Junjie Dang; Wenjie Zheng; Guochen Zheng; Peng Fu; Jun Qu; Bernhard J. Hoenders; Yuefeng Zhao; Yangjian Cai; Yangjian Cai; Yangjian Cai

doi:10.1364/OE.450737

1. Introduction

In free-space optical (FSO) communications, integrated photonic waveguides, lidar systems and other optics applications, one of the main challenges is to achieve highly stable and efficient fiber-coupling of the signal beam coupled from space and/or other optical elements into a fiber [1–6]. The outage probability of an optical link is significantly affected by the fiber-coupling efficiency [7]. Therefore, fiber-coupling efficiency has been widely studied both theoretically and experimentally. In an FSO communication system, the beam carried signal is degraded significantly by the turbulent atmosphere because the propagation properties of the signal beam are distorted [8]. The fiber-coupling efficiency is mainly influenced by the intensity distribution, beam wander, intensity fluctuation, random jitter and the degree of coherence and polarization of the signal beam induced by the turbulent atmosphere [9–16]. Fan et al. studied fiber-coupling efficiency of a scale-adapted set of Laguerre-Gaussian modes used in FSO communications and found that the coupling efficiency was influenced by various modes, atmospheric turbulence, or random jitter [9]. The mean-coupling efficiency of a plane wave in a turbulent atmosphere influenced by the random angular jitter was analyzed for different optical fibers [10]. The fiber-coupling efficiencies of the plane, spherical, and Gaussian beam waves are disturbed by the non-Kolmogorov turbulent atmosphere [11,12]. Further, it has also been observed that the plane and spherical waves have similar fiber-coupling efficiency results. The fiber-coupling efficiency of Gaussian-beam waves can be increased by optimizing the parameters of the beam in a weak or strong non-Kolmogorov turbulent atmosphere. In addition, many studies have shown that fiber-coupling efficiencies are affected by the degree of coherence and polarization of the beam when a partially coherent radially polarized vortex, the Gaussian Schell-model (GSM), and/or electromagnetic GSM beams are considered in random media, such as a turbulent atmosphere [13–16].

Recently, many methods have been applied to enhance fiber-coupling efficiency [17–26]. In one of these studies, the enhanced fiber-coupling efficiency was investigated using optimized inverted nanocone designs [17]. By attaching a small pure silica ball at the surface of a single-mode fiber, the coupling efficiency could be increased from a hollow-core fiber with a large core diameter to a single-mode fiber [18]. The maximization of the fiber-coupling efficiency of the Laguerre-Gaussian beam and perfect vortex beam was obtained by correctly adjusting the fiber parameters to the beamwidth [19]. Single-mode fiber-coupling efficiency was enhanced by an adaptive optical system based on a modal version of the stochastic parallel gradient descent algorithm. The results show that approximately 3.7 dB of the single-mode fiber-coupling efficiency was improved, and the fluctuation of the beam disturbed by the strong atmospheric turbulence was significantly suppressed in an 8 km urban terrestrial free-space channel [20]. 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 [21]. Based on the method of analyzing the impact of the FSO communication systems before and after aberrations correction on fiber efficiency, a holographic modal wavefront sensor with fast detection rates and insensitivity to scintillations was used to increase the fiber-coupling efficiency of the signal beam used in FSO communications [22]. The hybrid technique of adaptive optics and coherent fiber array was used to enhance the coupling efficiency of the partially coherent beam through atmospheric turbulence. The results show that the hybrid technique is superior to any independent technique [23]. A few-mode fiber was used to replace the single-mode fiber to increase the fiber-coupling efficiency for a beam in free space coupled to a fiber. The axial alignment tolerance of the few-mode fibers was better than that of the single-mode fibers. Unwinding the front section of the helical core fibers in a hydrogen flame was proposed to increase the fiber-coupling efficiency between the helical core fibers and single-mode fibers [24,25]. The fiber-coupling efficiency can be significantly increased when a coherent fiber array is used to reduce the perturbation due to the turbulent atmosphere [26]. The partially coherent beams that resisted the turbulent atmosphere perturbation were better than the completely coherent beam [27–29]. When partially coherent beams carrying bump on incoherent background propagate through the turbulent atmosphere, the intensity distribution of these beams become the asymptotic Gaussian shape. The beam property is not influenced by the degree of coherence and the strength of turbulence [30]. The above indicate that the coupling efficiency of a partially coherent beam through atmospheric turbulence has been studied widely [13–16]. In this letter, we theoretically studied the optical fiber-coupling efficiency of partially coherent flat-topped beams disturbed by atmospheric turbulence. The fiber-coupling efficiency equation of partially coherent flat-topped beam propagation through a turbulent atmosphere is derived using the Tatarski spectrum and cross-spectral density (CSD) function of the beam. Our results show that partially coherent flat-topped beams with a larger order are advantageous over lower order beams for mitigating the effects of atmospheric turbulence.

2. Formulation

2.1 Cross-spectral density function

The CSD is used to describe the intensity and turbulence-induced beam spreading with the second-order moment of the beam electric field. The CSD function $W({\rho _1},{\rho _2},0)$ of the partially coherent flat-topped beam at the source plane is expressed as follows [31–33]:

(1)$$W({{\rho_1},{\rho_2},0} )= \sum\limits_{n = 1}^N {\sum\limits_{m = 1}^N {\frac{{{{({ - 1} )}^{n + m}}}}{{{N^2}}}\left( {\begin{array}{c} N\\ n \end{array}} \right)\left( {\begin{array}{c} N\\ m \end{array}} \right)} } \left\{ {\exp ( - \frac{{n\rho_1^2}}{{w_0^2}} - \frac{{m\rho_2^2}}{{w_0^2}})\exp [ - \frac{{{{({\rho_1} - {\rho_2})}^2}}}{{2\sigma_0^2}})]} \right\}$$

where ${w_0}$ denotes the beam width, ${\rho _1}$ and ${\rho _2}$ are the arbitrary position vectors at the source plane, respectively. The parameter N is the beam order of the partially coherent flat-topped beams; Eq. (1) describes the GSM beam when N =1. ${\sigma _0}$ shows the transverse spatial coherence width, and $({_n^N} )$ is the binomial coefficient. Based on the generalized Huygens-Fresnel integral, the CSD of the partially coherent flat-topped beams at the receiver plane can be obtained as follows:

(2)$$\begin{aligned} W({{r_1},{r_2},z} )&= {(\frac{k}{{2\pi z}})^2}\int\!\!\!\int {{d^2}} {\rho _1}\int\!\!\!\int {{d^2}} {\rho _2}W({{\rho_1},{\rho_2},0} )\\ &\times \exp \left[ { - ik\frac{{{{({r_1} - {\rho_1})}^2} - {{({r_2} - {\rho_2})}^2}}}{{2z}}} \right]\\ &\times \left\langle {\exp [{{\psi^ \ast }({r_1},{\rho_1},z) + \psi ({r_2},{\rho_2},z)} ]} \right\rangle \end{aligned}, $$

where ${r_1}$ and ${r_2}$ are the arbitrary position vectors at the receiver plane, respectively. $k = {{2\pi } / \lambda }$ is the wave number with the wavelength $\lambda$. The symbol < > represents the ensemble average of the complex phase disturbed by the turbulent atmosphere. Based on the complex phase perturbations of the Rytov approximation, the last term of Eq. (2) can be obtained as follows [34]:

(3)$$\begin{aligned} &\left\langle {\exp [{{\psi^ \ast }({{\rho_1},{r_1},z} )+ \psi ({{\rho_2},{r_2},z} )} ]} \right\rangle \\ &\textrm{ } \cong \exp \{{ - M[{{{({\rho_1} - {\rho_2})}^2} + ({\rho_1} - {\rho_2})({r_1} - {r_2}) + {{({r_1} - {r_2})}^2}} ]} \}\end{aligned}, $$

The parameter M refers to the strength of the atmospheric turbulence;

(4)$$M = \textrm{0}\textrm{.33}{\pi ^\textrm{2}}{k^2}z\int_0^\infty {{\kappa ^3}{\phi _n}(\kappa )} d\kappa, $$

and $\kappa$ denotes the spatial wave number. The Tatarski spectrum for the index-of-refraction fluctuations of the turbulent atmosphere is used to represent the power spectrum of the refractive index fluctuation, as follows [8,35]:

(5)$${\phi _n}(\kappa )= 0.033C_n^2{\kappa ^{ - 11/3}}\exp \left( { - \frac{{{\kappa^2}}}{{\kappa_m^2}}} \right), $$

where ${\kappa _m} = {{5.92} / {{l_0}}}$, and ${l_0}$ is the inner scale with the microscale of the smaller eddies. $\kappa \gg {1 / {{L_0}}}$, and ${L_0}$ is the outer scale with the macroscale of the larger eddies. The strong or weak turbulence have the smaller or larger inner scales, respectively. $C_n^2$ is the structural constant of a turbulent atmosphere. The weak to moderate turbulence was described by the values of the structural constant $C_n^2$ from 0 to ${10^{ - 13}}{\textrm{m}^{ - 2/3}}$. The value of the structural constant $C_n^2$ larger than ${10^{ - 13}}{\textrm{m}^{ - 2/3}}$ describes the strong turbulence. By substituting Eq. (1) and Eq. (3) into Eq. (2), we obtain (the imaginary part is omitted):

(6)$$\begin{aligned} W({{r_1},{r_2},z} )&= {\left( {\frac{k}{{2\pi z}}} \right)^2}\sum\limits_{n = 1}^N {\sum\limits_{m = 1}^N {\frac{{{{({ - 1} )}^{n + m}}}}{{{N^2}}}\left( {\begin{array}{c} N\\ n \end{array}} \right)\left( {\begin{array}{c} N\\ m \end{array}} \right)} } \\ &\textrm{ } \times \left( {\frac{{4{\pi^2}C{z^2} + 8ik{\pi^2}Bz}}{{{C^2}{z^2} + 4{B^2}{k^2}}}} \right)\exp \{{ - H{{({{r_1} - {r_2}} )}^2}} \}\\ &\textrm{ } \times \exp ({F{{({{r_1} + {r_2}} )}^2}} )\exp [{G({{r_1}^2 - {r_2}^2} )} ]\end{aligned}, $$

where

(7)$$\begin{aligned} A &= \frac{{n + m}}{{{w_0}^2}},\textrm{ }B = \frac{{n - m}}{{{w_0}^2}},\textrm{ }C = {A^2} + \frac{{2A({\zeta _s} - 1)}}{{w_0^2}} + 4AM - {B^2} + \frac{{{k^2}}}{{{z^2}}}\\ D &= \frac{A}{4} + \frac{{{\zeta _s} - 1}}{{2w_0^2}} + M,\textrm{ }E = {C^2}{z^4} + 4{k^2}{B^2}{z^2}\\ F &= \frac{{ - {k^2}}}{{16{z^2}D}} - \frac{{({{B^2}C{z^2} - C{k^2} - 4{k^2}{B^2}} ){k^2}}}{{16DE}}\\ G &= \frac{{{k^2}}}{{2DE}}({BCD{z^2} - 2{k^2}BD + BCM{z^2} + {z^2}{B^3}M - {k^2}BM} )\\ H &= M - \frac{{{M^2}}}{{4D}} + \frac{{CD{k^2}{z^2}}}{E} + \frac{{CM{k^2}{z^2}}}{E} + \frac{{{k^2}C{M^2}{z^2}}}{{4DE}}\\ &\textrm{ } + \frac{{2{k^2}{B^2}M{z^2}}}{E} + \frac{{{k^2}{B^2}{M^2}{z^2}}}{{DE}} - \frac{{{B^2}C{M^2}{z^4}}}{{4DE}} \end{aligned}, $$

We used the following integral formula to obtain the above derivation:

(8)$$\int_{ - \infty }^\infty {\exp ({ - {s^2}{x^2} \pm qx} )dx} = \frac{{\sqrt \pi }}{s}\exp \left( {\frac{{{q^2}}}{{4{s^2}}}} \right), $$

where ${\zeta _s} = 1 + w_0^2/\sigma _0^2 \ge 1$ describes the source coherent parameter of the partially coherent beam at the source plane, and ${\zeta _s} = 1$ denotes the wholly coherent beam.

2.2 Fiber-coupling efficiency

In FSO communications, the laser beam must be coupled to the fiber to filter crosstalk and analyze the signal carried by the laser beam at the receiver plane. Figure 1 schematically shows the optical signals carried by the laser beam coupled to the single-mode optical fibers when the signal beam propagates through atmospheric turbulence. The bit error rate (BER) of the signals is influenced by the fiber-coupling efficiency. The value of the fiber-coupling efficiency $\left\langle {{\eta_c}} \right\rangle $ is determined by the average power $\left\langle {{P_c}} \right\rangle $ coupled to the fiber and the average power $\left\langle {{P_a}} \right\rangle $ sent to the receiver plane with the aperture, as follows [14–16]:

(9)$$\left\langle {{\eta_c}} \right\rangle = \frac{{\left\langle {{P_c}} \right\rangle }}{{\left\langle {{P_a}} \right\rangle }} = \frac{{\int\!\!\!\int_A {F_A^\ast ({{r_1}} ){F_A}({{r_2}} )W({{r_1},{r_2},z} )} d{r_1}d{r_2}}}{{\left\langle {{{\int\!\!\!\int_A {|{{E_A}(r )} |} }^2}dr} \right\rangle }}, $$

where ${E_A}(r)$ is the field at the receiver plane, $W({{r_1},{r_2},z} )$ is the CSD function of a partially coherent beam with the arbitrary position vectors $({r_1},{r_2})$, z is the propagating distance, the asterisk denotes the complex conjugation, and ${F_A}(r )$ describes the normalized mode field distribution of the optical fiber, as follows [14–16]:

(10)$${F_A}(r )= \frac{{\sqrt {2\pi } {w_a}}}{{\lambda f}}\exp \left[ { - {{\left( {\frac{{r\pi {w_a}}}{{\lambda f}}} \right)}^2}} \right] = \sqrt {\frac{2}{{\pi w_m^2}}} \exp \left( { - \frac{{{r^2}}}{{w_m^2}}} \right), $$

where ${w_m} = \lambda f/\pi {w_a}$ and ${w_a}$ describes the mode field radius at the fiber end surface, and f is the focal length of the lens. To analyze Eq. (9), we assumed that the mode field distribution of the fiber is known. The fiber-coupling efficiency can then be expressed as follows:

(11)$${\eta _c} = \frac{1}{{{A_R}}}\int {\int_A {F_A^\ast ({{r_1}} ){F_A}({{r_2}} )\mu ({{r_1},{r_2}} )d{r_1}d{r_2}} }, $$

where ${A_R} = {{\pi {d^2}} / 4}$ denotes the aperture area at the receiver plane with receiver aperture diameter d. The function $\mu ({{r_1},{r_2}} )$ describes the complex of coherence for the incident beam at the receiver plane, which can be calculated using the CSD, as follows:

(12)$$\mu ({r_1},{r_2},z) = \frac{{W({r_1},{r_2},z)}}{{\sqrt {W({r_1},{r_1},z)} \sqrt {W({r_2},{r_2},z)} }}, $$

By substituting Eqs. (6), (10), and (12) into Eq. (11), the fiber-coupling efficiency of partially coherent flat-topped beams in atmospheric turbulence can be obtained as follows:

(13)$$\begin{aligned} {\eta _c} &= 8{\beta ^2}\int_0^1 {\int_0^1 {\exp [{ - {\beta^2}({{x_1}^2 + {x_2}^2} )} ]} } \\ &\times \left\{ {\sum\limits_{n = 1}^N {\sum\limits_{m = 1}^N {\frac{{{{({ - 1} )}^{n + m}}}}{{{N^2}}}\left( {\begin{array}{c} N\\ n \end{array}} \right)\left( {\begin{array}{c} N\\ m \end{array}} \right)\left( {\frac{{4{\pi^2}C{z^2}}}{{{C^2}{z^2} + 4{B^2}{k^2}}}} \right)} } {I_0}\left[ {\frac{{({H + F} ){D^2}}}{2}{x_1}{x_2}} \right]} \right.\\ &\times \left. {\exp \left[ { - \frac{{({H - F} ){D^2}}}{4}({x_1^2 + x_2^2} )} \right]\exp \left[ {\frac{{G{D^2}}}{4}({x_1^2 - x_2^2} )} \right]} \right\}\\ &\times {\left[ {\sum\limits_{n = 1}^N {\sum\limits_{m = 1}^N {\frac{{{{({ - 1} )}^{n + m}}}}{{{N^2}}}\left( {\begin{array}{c} N\\ n \end{array}} \right)\left( {\begin{array}{c} N\\ m \end{array}} \right)\left( {\frac{{4{\pi^2}C{z^2}}}{{{C^2}{z^2} + 4{B^2}{k^2}}}} \right)} } \exp ({F{D^2}x_1^2} )} \right]^{ - 1/2}}\\ &\times {\left[ {\sum\limits_{n = 1}^N {\sum\limits_{m = 1}^N {\frac{{{{({ - 1} )}^{n + m}}}}{{{N^2}}}\left( {\begin{array}{c} N\\ n \end{array}} \right)\left( {\begin{array}{c} N\\ m \end{array}} \right)\left( {\frac{{4{\pi^2}C{z^2}}}{{{C^2}{z^2} + 4{B^2}{k^2}}}} \right)} } \exp ({F{D^2}x_2^2} )} \right]^{ - 1/2}}{x_1}{x_2}d{x_1}d{x_2} \end{aligned}, $$

where ${I_0}({\cdot} )$ denotes the modified Bessel function of the first kind and zeroth order, and the design parameter $\beta = d/2{w_m}$ denotes the ratio of the receiver lens diameter to the mode field radius. The above derivation was obtained with the help of Eq. (14) below, the integral of Eq. (15), and variables in Eq. (16).

(14)$${({{{\boldsymbol r}_1} - {{\boldsymbol r}_2}} )^2} = {r_1}^2 + {r_2}^2 - 2{r_1}{r_2}\cos ({{\varphi_1} - {\varphi_2}} ), $$

(15)$$\int_0^{2\pi } {\int_0^{2\pi } {\exp } } [{2({H + F} ){r_1}{r_2}\cos ({\varphi_1} - {\varphi_2})} ]d{\varphi _1}d{\varphi _2} = 4{\pi ^2}{I_0}[{2({H + F} ){r_1}{r_2}} ], $$

(16)$${x_1} = {{2{r_1}} / d},{x_2} = {{2{r_2}} / d}, $$

Equation (13) is the main analytical expression used in this study. Our results agree well with the Ref.14 when the beam order is N = 1. The numerical results provide a convenient way to demonstrate the fiber-coupling efficiency of partially coherent flat-topped beam propagation in a turbulent atmosphere.

Fig. 1. Illustration of coupling optical signals to the single-mode optical fiber in atmospheric turbulence.

Download Full Size | PDF

3. Numerical results

In this section, we analyze the fiber-coupling efficiency of partially coherent flat-topped beams induced by atmospheric turbulence. The parameters are chosen as wavelength $\lambda = 1550\textrm{nm}$, beam width ${w_0} = 0.04\textrm{m}$ and inner scale ${l_0} = 5\textrm{mm}$.

Figure 2 shows the fiber-coupling efficiency of the GSM beam and partially coherent flat-topped beams influenced by the diameter of the receiver lens, beam order, and design parameter with a turbulence strength of $C_n^2\textrm{ = 1} \times {10^{ - 15}}{\textrm{m}^{ - 2/3}}$, beam propagation distance of z = 5 km, and source coherent parameter ${\zeta _s} = 2$. From Fig. 2, there is a maximal value of the fiber-coupling efficiency for a certain value of the parameter β. This result indicates that the main power of the beam can be coupled to the optical fiber in the range around this value. The fiber-coupling efficiency decreases quickly when the parameter β is far from the optimal value. The maximal value of the fiber-coupling efficiency is due to the collineation between the incident beam and axial position of the SMF. The fiber-coupling efficiency is also affected by the order of the partially coherent flat-topped beam. The fiber-coupling efficiency is higher if the order of the beam is larger. The lowest fiber-coupling efficiency was observed for the GSM beams (see Fig. 2(d)). Figures 2(a)–(c) shows that the fiber-coupling efficiency increases when the aperturediameter of the receiver decreases. The maximal value of the fiber-coupling efficiency is affected by the variation in the order of the beams and the aperture diameter of the receiver. This phenomenon can be explained by the factor that the distribution of the intensities of the beams at the cross section becomes increasing uniform when the order of the partially coherent flat-topped beams increases. The misalignment between the incident beam and axial positioning of the SMF is small when the aperture diameter of the receiver is small. Therefore, the high fiber-coupling efficiency indicates that the fluctuation of the angle of arrival of the partially coherent flat-topped beam induced by the turbulent atmosphere is very small when the beams have a large order and the aperture diameter of the receiver is small.

Fig. 2. Fiber-coupling efficiency of partially coherent flat-topped beams in the presence of atmospheric turbulence versus the design parameter β.

Download Full Size | PDF

Figure 3 shows the changes in the fiber-coupling efficiency of partially coherent flat-topped beams with beam order, atmospheric turbulence, and aperture diameter of the receiver for a beam propagation distance of z = 5 km, the source coherent parameter ${\zeta _s} = 2$, and design parameter of $\beta = 1.12$. As shown in Fig. 3, the fiber-coupling efficiency of the beam increased when the aperture diameter of the receiver decreased, and the GSM beam had the smallest fiber-coupling efficiency. In free space, that is, $C_n^2\textrm{ = }0$, the fiber-coupling efficiency increases with decreasing receiver aperture diameter and increasing beam order. If there was no fluctuation in the angle of arrival, the fiber-coupling efficiency slightly decreased with the order of the beam when the aperture diameter of the receiver increased (see Fig. 3(a)). Under the same conditions, the fiber-coupling efficiency of the partially coherent flat-topped beam propagating through a turbulent atmosphere was smaller than that of the beam in free space (see Fig. 3(a), 3(b), and 3(c)). When the turbulent strength of the atmosphere increased, the fiber-coupling efficiency of the partially coherent flat-topped beam increased slightly as the beam order increased (see Fig. 3(d)).

Fig. 3. Fiber-coupling efficiency of partially coherent flat-topped beams for different atmospheric turbulence strengths versus the aperture diameter of the receiver d.

Download Full Size | PDF

Figure 4 shows the changes in the fiber-coupling efficiency of partially coherent flat-topped beams propagating through atmospheric turbulence with propagation distance, beam order, and source coherence parameter with the parameters of β=1.12, d = 0.1 m and $C_n^2\textrm{ = 5} \times {10^{ - 16}}{\textrm{m}^{ - 2/3}}$. The large source coherence parameter ${\zeta _\textrm{s}}$ describes the degree of coherence of the flat-topped beam reduction. From Fig. 4, the fiber-coupling efficiency of partially coherent flat-topped beams propagating through atmospheric turbulence and affected by the source coherence parameter is large in the near field with small ${\zeta _\textrm{s}}$. The opposite conclusion can be obtained when the source coherence parameter ${\zeta _\textrm{s}}$ is large in the far field. The fiber-coupling efficiency of the beam decreased as the source coherence parameter increased. When the source coherence parameter increased further, the fiber-coupling efficiency remained nearly constant. This phenomenon can be interpreted as follows: it the far field, the intensity of the partially coherent flat-topped beam induced by the turbulent atmosphere tends to a uniform distribution when the source coherence parameter ${\zeta _\textrm{s}}$ is large, if the receiver aperture diameter d is small.

Fig. 4. Fiber-coupling efficiency of partially coherent flat-topped beams propagating through atmospheric turbulence at different propagation distances versus source coherence parameter${\zeta _\textrm{s}}$.

Download Full Size | PDF

Figure 5 shows the effect of the propagation distance on the fiber-coupling efficiency of partially coherent flat-topped beams propagating through atmospheric turbulence with the parameters β=1.12, d = 0.1 m and $C_n^2\textrm{ = 5} \times {10^{ - 16}}{\textrm{m}^{ - 2/3}}$. As can be seen from Fig. 5, the fiber-coupling efficiency of the partially coherent flat-topped beams increases when the propagation distance z increases up to a certain value. When the propagation distance z increases further, the fiber-coupling efficiency starts to decrease because the cumulative effect of the atmospheric turbulence increasingly disturbs the partially coherent flat-topped beam. Further, it is observed that the fiber-coupling efficiency of a partially coherent flat-topped beam with large beam order is larger than that of the partially coherent flat-topped beam with small beam order. Under the same conditions, the fiber-coupling efficiency of the partially coherent flat-topped beam is larger than that of the Gaussian beam (black line) or the GSM beam (red line) in the far field. These results show that the influence of turbulence is weaker on partially coherent flat-topped beams than on the Gaussian beam or the GSM beam.

Fig. 5. Fiber-coupling efficiency of partially coherent flat-topped beams through atmospheric turbulence for different beam orders and source coherence parameter ${\zeta _\textrm{s}}$ versus the propagation distance z.

Download Full Size | PDF

4. Conclusion

In conclusion, we theoretically investigated the fiber-coupling efficiency of partially coherent flat-topped beams propagating through atmospheric turbulence. The theoretical demonstration showed that the fiber-coupling efficiency of partially coherent flat-topped beams was influenced by the beam order, the design parameter, the aperture diameter of the receiver, source coherence parameter, the propagation distance and the strength of the atmospheric turbulence. When the turbulence strength of the atmosphere and propagation distance were fixed, the fiber-coupling efficiency of the partially coherent flat-topped beams could be enhanced by optimizing the parameters of the beam and the fiber. These results will be useful for FSO communications.

Funding

National Key Research and Development Program of China (2019YFA0705000); National Natural Science Foundation of China (11974218, 11974219, 12074005, 12174227, 12192254, 9175020); Innovation Group of Jinan (2018GXRC010); Natural Science Foundation of Shandong Province (ZR2019MA028, ZR2020MA082); Local Science and Technology Development Project of the Central Government (YDZX20203700001766); Shandong Provincial Key Laboratory of Optics and Photonic Devices (K202010); National College Students Innovation and Entrepreneurship Training Program (202010445503, S202010445172); Undergraduate Research Foundation of Shandong Normal University (BKJJ2020041).

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. 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]

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

3. D. R. Gozzard, S. W. Schediwy, B. Stone, M. Messineo, and M. Tobar, “Stabilized free-space optical frequency transfer,” Phys. Rev. Appl. 10(2), 024046 (2018). [CrossRef]

4. X. Wen, K. Xu, and Q. Song, “Design of a barcode-like waveguide nanostructure for efficient chip–fiber coupling,” Photonics Res. 4(6), 209–213 (2016). [CrossRef]

5. N. Liu, J. Zhang, Z. Zhu, W. Xu, and K. Liu, “Efficient coupling between an integrated photonic waveguide and an optical fiber,” Opt. Express 29(17), 27396–27403 (2021). [CrossRef]

6. A. Noriki, I. Tamai, Y. Ibusuki, A. Ukita, S. Suda, D. Shimura, Y. Onawa, H. Yaegashi, and T. Amano, “Mirror-Based Broadband Silicon-Photonics Vertical I/O With Coupling Efficiency Enhancement for Standard Single-Mode Fiber,” J. Lightwave Technol. 38(12), 3147–3155 (2020). [CrossRef]

7. Y. Chang, Z. Liu, X. Ni, H. Yao, S. M. Gao, S. Y. Gao, and X. Tong, “Performance analysis of an all-optical double-hop free-space optical communication system considering fiber coupling,” Appl. Opt. 60(19), 5629–5637 (2021). [CrossRef]

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

9. X. Fan, D. Wang, J. Cheng, J. Yang, and J. Ma, “Few-mode fiber coupling efficiency for free-space optical communication,” J. Lightwave Technol. 39(6), 1823–1829 (2021). [CrossRef]

10. Y. Chen, L. Tan, L. Zhao, D. Kang, and J. Ma, “Coupling efficiency of a plane wave to few-mode fiber in the presence of beam random angular jitter,” J. Opt. Soc. Am. B 38(12), F8–F15 (2021). [CrossRef]

11. L. Tan, C. Zhai, S. Yu, Y. Cao, and J. Ma, “Fiber-coupling efficiency for optical wave propagating through non-Kolmogorov turbulence,” Opt. Commun. 331, 291–296 (2014). [CrossRef]

12. 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]

13. L. Yu and Y. Zhang, “Free-space to fiber coupling of electromagnetic Gaussian Schell-model beams in turbulent marine atmospheric channel,” IEEE Photonics J. 10(6), 7907909 (2018). [CrossRef]

14. 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]

15. B. Hu, Y. Zhang, and Y. Zhu, “New model of the fiber coupling efficiency of a partially coherent Gaussian beam in an ocean to fiber link,” Opt. Express 26(19), 25111–25119 (2018). [CrossRef]

16. 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]

17. 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]

18. 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]

19. S. Rojas-Rojas, G. Cañas, 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]

20. K. Yang, M. Abulizi, Y. Li, B. Zhang, S. Li, W. Liu, J. Yin, Y. Cao, J. Ren, and C. Peng, “Single-mode fiber coupling with a M-SPGD algorithm for long-range quantum communications,” Opt. Express 28(24), 36600–36610 (2020). [CrossRef]

21. 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]

22. W. Liu, W. Shi, K. Yao, J. Cao, P. Wu, and X. Chi, “Fiber coupling efficiency analysis of free space optical communication systems with holographic modal wave-front sensor,” Opt. Laser Technol. 60, 116–123 (2014). [CrossRef]

23. 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]

24. A. Fardoost, H. Wen, H. Liu, F. G. Vanani, and G. Li, “Optimizing free space to few-mode fiber coupling efficiency,” Appl. Opt. 58(13), D34–D38 (2019). [CrossRef]

25. G. Statkiewicz-Barabach, M. Napiorkowski, M. Bernas, L. Czyzewska, P. Mergo, and W. Urbanczyk, “Method for increasing coupling efficiency between helical-core and standard single-mode fibers,” Opt. Express 29(4), 5343–5357 (2021). [CrossRef]

26. Y. Dikmelik and F. M. Davidson, “Fiber-coupling efficiency for free-space optical communication through atmospheric turbulence,” Appl. Opt. 44(23), 4946–4952 (2005). [CrossRef]

27. 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 (2021). [CrossRef]

28. 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]

29. M. Dong, C. Zhao, Y. Cai, and Y. Yang, “Partially coherent vortex beams: Fundamentals and applications,” Sci. China Phys. Mech. Astron. 64(2), 224201 (2021). [CrossRef]

30. X. Li, S. A. Ponomarenko, Z. Xu, F. Wang, Y. Cai, and C. Liang, “Universal self-similar asymptotic behavior of optical bump spreading in random medium atop incoherent background,” Opt. Lett. 45(3), 698–701 (2020). [CrossRef]

31. Y. Dan and B. Zhang, “Beam propagation factor of partially coherent flat-topped beams in a turbulent atmosphere,” Opt. Express 16(20), 15563–15575 (2008). [CrossRef]

32. F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107(5-6), 335–341 (1994). [CrossRef]

33. Y. Li, “Light beam with flat-topped profiles,” Opt. Lett. 27(12), 1007–1009 (2002). [CrossRef]

34. O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012). [CrossRef]

35. Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboglu, Y. Baykal, and O. Korotkova, “M²-factor of coherent and partially coherent dark hollow beams propagating in turbulent atmosphere,” Opt. Express 17(20), 17344–17356 (2009). [CrossRef]

Enhanced fiber-coupling efficiency via high-order partially coherent flat-topped beams for free-space optical communications

Abstract

1. Introduction

2. Formulation

2.1 Cross-spectral density function

2.2 Fiber-coupling efficiency

3. Numerical results

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (5)

Equations (16)

Optics Express