In a space laser communication system with a few-mode fiber as the receiving terminal, the influence of random angular jitter on the communication performance cannot be ignored. Therefore, it is necessary to research the coupling characteristics of the space laser–FMF system with random angular jitter. A space laser–FMF coupling model with random angular jitter is proposed based on the Laguerre–Gaussian mode. The coupling efficiency and bit error rate (BER) of the communication system are optimized by the relative aperture of the coupling lens. When the relative aperture is 0.15, the standard deviation of random jitter is 3, the signal-to-noise ratio is 5, the coupling efficiency after optimization is improved by 29%, and the BER is reduced by 46%. In this case, the maximum BER reduction after optimization is 55%. The research shows that in the presence of random angular jitter, adjusting the relative aperture can significantly optimize the coupling efficiency and the BER of the space laser–FMF coupling system. The ability of the relative aperture to optimize the coupling efficiency increases with random jitter standard deviation, which exhibits an upper limit for the optimization of the BER.

## I. INTRODUCTION

Space laser communication has become a research hotspot due to its many advantages, such its large amount of data transfer, fast speed, and flexible form. In space laser communication systems using optical fibers as receivers, the prevalence of the few-mode fiber (FMF) is gradually increasing. The FMF provides more light-guiding modes than the single-mode fiber (SMF), and the coupling efficiency with space lasers is higher.^{1–4} The mode number of the FMF is less than that of the multi-mode fiber (MMF), and the laser is less affected by mode dispersion during the propagation of the FMF.^{5–8} In the space laser–FMF coupled system, random angular jitter is a factor that cannot be ignored. Random angular jitter refers to the variation in the deviation angle between the receiving optical axis and the incident beam.^{9,10} Its existence will reduce the coupling efficiency of the space laser and FMF and increase the bit error rate (BER) of the communication system. The efficient coupling between space laser and FMF is the focus of space laser communication systems.

Wang *et al.* proved experimentally that the coupling efficiency of the FMF and space laser is 12.1% higher than that of the SMF when the random angular jitter is 5 *µ*m.^{11} Therefore, the coupling efficiency can be improved by using the FMF as the receiver in the space laser communication system. Guang *et al.* analyzed the coupling efficiencies of ultrafast laser pulses with different wavelengths, durations, incident angles, positions, beam spot sizes, and curvature radii with different LP modes of FMFs.^{12,13} These studies can provide a reference for calculating the coupling efficiency of the FMF under different operating conditions. Fardoost *et al.* constructed a coupling model of the Laguerre–Gaussian mode space laser and the FMF.^{14} The simulation results show that the optimal normalized frequency under ideal conditions was 4.94 and the maximum total coupling efficiency exceeded 99%. The Laguerre–Gaussian coupling model was discussed in detail, but the loss of coupling efficiency caused by the coupling environment was not considered when calculating the coupling efficiency. Fan *et al.* researched a coupling efficiency model of a space laser–FMF system in the presence of atmospheric turbulence and random angular jitter.^{15} They used the ratio of the coupling lens radius to the fiber core radius as an optimization parameter, and its values were obtained based on the LP01 mode. Based on this method, when the normalized jitter value was 0.4, the maximum coupling efficiency simulation value was about 0.65. The method of obtaining the optimization parameter based on the LP01 mode can simplify the calculation, but it ignores the contribution of other modes in the FMF to the coupling efficiency. Therefore, further research is needed to improve the coupling model and communication performance of the space laser–FMF system with random angle jitter through new optimization methods.

The atmosphere has an absorption effect on space lasers, which is weak only at specific wavelengths, such as 800, 1060, and 1550 nm. Therefore, these communication wavelengths are often selected for space laser communication systems. The fiber core diameter cannot reflect all the characteristics of the FMF. The normalized frequency of the fiber is an important parameter to determine the usable mode number of the FMF. Based on the above-mentioned reasons, using the laser wavelength or fiber core diameter as the optimization parameter of space laser–FMF coupling systems is not very applicable in practical engineering. When designing such systems, the space laser wavelength and fiber parameters are usually determined first, and then the coupling lens is designed. Selecting the relative aperture of the coupling lens as the optimization parameter can leave more selectivity for the FMF and the space laser, which not only conforms to engineering practices but also makes the design of the coupling system more flexible. However, research on optimizing the space laser–FMF coupling efficiency based on the relative aperture of the coupling lens is still lacking. Therefore, it is necessary to deeply analyze the optimization effect of the relative aperture of the coupling lens on the coupling efficiency and BER of the space laser–FMF communication system.

This paper constructs a space laser–FMF coupling model to analyze the relationship between the space laser–FMF coupling efficiency and the relative aperture of the coupling lens in the presence of random angular jitter. The enhancement of the communication performance by the optimal relative aperture of the coupling lens is investigated. The study assumes that the space laser–FMF coupling is in a non-turbulent atmosphere, and the influence of atmospheric turbulence on coupling efficiency is not considered. The communication distance is between 20 and 100 km, and a space laser participating in coupling is regarded as a Gaussian beam. In the case of not using the mode-division multiplexing communication method, the study ignores the inter-mode crosstalk of the FMF. The FMF is assumed to be a weakly guiding step-index fiber to simplify the calculation. This research is helpful to improve the communication performance of the space laser–FMF system in the presence of random jitter, and the optimization results can be used as a reference for the design of the coupling lens.

## II. THEORETICAL FORMULATION

### A. Ideal coupling efficiency of few-mode fiber

The space laser–FMF communication system can be simplified to a coupling model consisting of a space laser, a coupling lens, and an FMF. When analyzing the influence of the coupling lens on the coupling efficiency, the coupling lens can be equivalent to a thin lens with a focal length of *f* and a diameter of *D*. The space laser–FMF coupling model is shown in Fig. 1.

As shown in Fig. 1, plane A is the entrance pupil plane of the coupling lens, plane B is the focal plane of the coupling lens, and the end face of the FMF is located at the focal plane of the coupling lens. The space laser incident on plane A is condensed to plane B through the coupling lens and coupled with the FMF.

The coupling efficiency of the FMF is defined as the ratio of the light energy coupled into the FMF to the available light energy. According to Parseval’s theorem, it is equivalent to calculate the coupling efficiency in either plane A or plane B.^{16,17} Instead of calculating the space laser–FMF coupling efficiency at plane A,^{15} we calculate it at plane B, where we only need to calculate the optical field distribution of the laser in the focal plane once, whereas plane A requires calculating the mode field distribution of all modes of the FMF at the entrance pupil plane. Thus, it is simpler to calculate the coupling efficiency at plane B.

After the space laser passes through the lens, its optical field distribution at focal plane B can be expressed as^{18}

where *λ* is the laser wavelength, *k* is the wavenumber *k* = 2*π*/*λ*, and $J1\u22c5$ is the Bessel function of the first kind of order 1. The phasor actor $expikf+r2/2f$ can be ignored when performing intensity calculations. The optical field distribution of the laser at plane B can be rewritten as

Ignoring the polarization mode coupling component, the mode field distribution of the weakly guided step-index FMF can be approximated by a Laguerre–Gaussian distribution, and the normalized expression of the linearly polarized LP mode field of the FMF can be expressed as^{19,20}

where *m* and *l* are the axial and angular coordinates of the mode field, respectively, $Cm,l\u22c5$ is the normalization constant for the light fields, $Lml\u22c5$ is a Laguerre polynomial, sin *lφ* and cos *lφ* correspond to the two forms of the simplex mode, and *ω* is the relative radius parameter, $\omega =r0/kN\u22c5A.$, where *r*_{0} is the core radius of the FMF and *N.A.* is the numerical aperture of the fiber.

The coupling efficiency of the *j*-th order mode in an FMF is the overlap integral of the mode field of this mode and the space laser, which can be expressed as^{21}

where $EB*r$ is the complex conjugate of the light field of the space laser in plane B, $Fjr$ is the normalized fiber mode field of the *j*-th order mode, and $\u222bBFjr2dr=1$. Equation (4) is rewritten as

In the case of ignoring the crosstalk between n modes, the coupling between space laser and each mode of the FMF is a linear process,^{22,23} and the total coupling efficiency of the FMF is equal to the sum of the coupling efficiencies of each mode,

### B. Space laser–FMF coupling efficiency with random angular jitter

In the space laser–FMF coupling system with random angular jitter, the angle between the space laser at the entrance pupil plane and the optical axis of the coupling lens has a random jitter angle *θ*, which can approximately satisfy the Rayleigh distribution, and its probability density function can be expressed as^{20}

where *σ*_{θ} is the standard deviation of the random deviation angle.

The incident laser has a random jitter angle *θ* with respect to the optical axis, and the space laser spot in plane B will have a random deviation displacement *Δr* with respect to the fiber. The relationship between the random deviation displacement *Δr* in plane B and the random jitter angle *θ* can be expressed as^{13}

Equation (8) shows that the random jitter displacement also satisfies the Rayleigh distribution, and the probability density of the random displacement jitter can be expressed as

where *σ*_{r} is the standard deviation of the random displacement jitter,

where *σ*_{θ} is the measured value and *σ*_{r} is the calculated value.

In the space laser–FMF coupling system, the coupling efficiency value calculated on plane A with the random jitter angle equals the coupling efficiency value calculated on plane B with the random deviation displacement.^{24,25} In plane B, the coupling effect of the random deviation displacement of the space laser facular relative to the FMF is equivalent to the random jitter displacement of the FMF relative to the space laser facular. After the fiber has displaced in the focal plane, the fiber mode field function concerning the parameters can be represented by a Nakagami–Rice distribution.^{14,18} Taking a quad-mode fiber as an example, with the random deviation displacement *Δr* on plane B according Eq. (3), LP_{01}, LP_{11}, LP_{02}, and LP_{21} are expressed as

where $I0\u22c5$ is a modified Bessel function of the first kind of order 0 and the other parameters have the same meaning as those in Eq. (3).

The coupling efficiency should be averaged over all states of the random deviation displacement *Δr*, which is the average coupling efficiency. In plane B, the coupling efficiency of the *j*-th mode in an FMF with random angular jitter is expressed as

As the Laguerre–Gaussian function is used to approximate the laser mode, Eq. (11) shows that the high-order mode of the FMF contains the sum of squares of absolute values. The coupling efficiency calculation can be simplified as

Substituting Eq. (13) into Eq. (12), the coupling efficiency of the *j*-th order mode can be expressed as

where *K* is a constant term in the integral and *α* and *β* are positive integer multiples of 1/2,

Substituting Eq. (15) into Eq. (14), the coupling efficiency of the *j*-th order mode can be expressed as

### C. Average BER of space laser–FMF coupling with random jitter

The average coupling efficiency of the space laser–FMF system with random jitter was calculated previously. The coupling efficiency of the FMF is converted into the level of the photodetector. The BER of an optical receiver for an intensity-modulation and direct-detection system with non-return-to-zero can be expressed as^{15,18}

where erfc(·) is the complementary error function and *Q* is the SNR ratio parameter.

The change in the average bit error rate of the communication system caused by the random angle jitter of the entrance pupil plane is equivalent to the change in the average BER of the communication system caused by the random displacement of the focal plane. Therefore, the BER of the space laser–FMF coupling structure with random angular jitter can be expressed as

## III. NUMERICAL RESULTS AND DISCUSSIONS

The space laser wavelength used in the numerical simulation is 1550 nm, the diameter of the quad-mode fiber is 15 *µ*m, and the numerical aperture is 0.15.

### A. Coupling efficiency of space laser and FMF without random jitter

FMFs have multiple modes, and it is necessary to calculate the coupling efficiency of each mode when determining the optimal coupling parameters according to the coupling efficiency. The coupling efficiency for each mode and the total coupling efficiency of a quad-mode fiber are plotted in Fig. 2.

Figure 2 shows the coupling efficiency for each mode of the quad-mode fiber as a function of the relative aperture without random jitter. Under the same coupling conditions, the coupling efficiency between the LP_{01} mode and the space laser is the highest, and it plays a major role in the coupling process between the quad-mode fiber and the space laser. The coupling efficiency of the LP_{02} mode is significantly lower than that of the LP_{01} mode. The coupling efficiency of the LP_{11} mode and LP_{21} mode is similar: 0. The optimal value of the coupling lens’ relative aperture corresponding to the quad-mode fiber’s total coupling efficiency is smaller than the coupling efficiency corresponding to the basic mode because under the same coupling conditions, the total coupling efficiency of the quad-mode fiber and the space laser is larger than that of the basic mode, and the coupling ability of the coupling lens is required to be stronger; thus, the corresponding relative aperture value becomes smaller.

The optimal value of the total coupling efficiency and the basic mode coupling efficiency corresponds to different relative aperture values. Although the coupling efficiency of the LP_{02} mode is small, its influence on the optimal value of the total coupling efficiency cannot be ignored. When using the coupling lens’ relative aperture to optimize the FMF’s coupling efficiency, the relative aperture value obtained based on the best value of the total coupling efficiency is better than that of the basic-mode coupling efficiency.

### B. Coupling efficiency of space laser–FMF with random jitter

As shown in Fig. 3, the total coupling efficiency of the quad-mode fiber with random jitter standard deviations is 1, 2, and 3. The random jitter standard deviation σ_{r} increases from 1 to 3, and the coupling efficiency greatly decreases from 0.89 to 0.34. The increase in the random jitter standard deviation leads to a fast decrease in the optimal value of the coupling efficiency of the quad-mode fiber. This result is predictable because the larger the random jitter standard deviation, the more the space laser spot on the focal plane deviates from the fiber core, and the more obvious the decrease in the coupling efficiency.

The optimal value of the relative aperture of the coupled lens also decreases with the increase in the standard deviation of the random jitter. When the σ_{r} value is 1, the optimal value of the relative aperture value is 0.21, σ_{r} increases to 3, and the optimal value of the relative aperture value decreases to 0.16. These results can be explained by the fact that the larger the standard deviation of random jitter, the worse the coupling effect of the space laser-quad-mode fiber. Reducing the lens’ relative aperture can improve its coupling ability to the space laser, increasing the coupling efficiency of the FMF. In the presence of random jitter, the optimal value obtained based on all modes of the FMF improves the maximum coupling efficiency by about 5% compared to that based on the LP_{01} mode^{15} in the FMF.

Figure 4 shows that as the standard deviation of random jitter increases, the optimization method based on the relative aperture of the coupling lens significant improves the coupling efficiency. When σ_{r} is 0, the improvement in the coupling efficiency is more than 0, which can be explained by the fact that the standard deviation of random jitter is 0 but the random jitter value is not 0, and there is still a random jitter resulting in reduced coupling efficiency. When σ_{r} is less than 1, the optimal value of the relative aperture improves the coupling efficiency by less than 5%. This result indicates that when the random jitter standard deviation is small, the improvement in the coupling efficiency by the optimal value of the relative aperture is not obvious because when the random jitter standard deviation is slight, the coupling efficiency is still close to that without random jitter. When σ_{r} is 5, the coupling efficiency after optimization is increased by 29% compared with that before optimization. The standard deviation of random jitter increases, and the growth rate of coupling efficiency also increases significantly after optimization, which indicates that the coupling ability of the coupling lens to the space laser increases synchronously with the increase in random jitter, and the value of coupling efficiency after optimization is significantly higher than that before optimization. Thus, in an environment with large random jitter, the method based on the optimal value of the relative aperture of the coupling lens can significantly improve the coupling efficiency and have a good optimization effect.

### C. Average BER with random jitter

Figure 5 shows the BER of the space laser–quad-mode fiber as a function of the relative aperture of the coupling lens when the coupling parameter *Q* = 5. In the space laser–FMF coupling system with random angular jitter, the BER corresponding to the optimal value of the relative aperture is the smallest. As the relative aperture value deviates from the optimum value, the BER increases rapidly, which shows that the numerical value of the BER is sensitive to the optimal value of the relative aperture value.

The coupling lens designed based on the optimal value of the relative aperture can significantly reduce the BER of the space laser–FMF communication system; compared with the optimization method based on the optimal parameter of the basic mode,^{15} the average BER is reduced by 10%. When the relative aperture of the coupling lens is at the optimal value, σ_{r} increases from 1 to 3, leading to the minimum value of the average BER increasing by a factor of 10^{4}. With the increase in random jitter standard deviation, the average BER of the coupling system increases significantly, and the reliability of communication performance is dramatically reduced. The drop rate of the average BER is plotted in Fig. 6 by optimizing the optimal value of the relative aperture when *Q* = 5.

Figure 6 shows the average BER reduction with different random jitter standard deviations after optimization of the relative aperture optimum for *Q* = 5. As the standard deviation of random jitter increases, there is a maximum value of the reduction rate of the BER by the relative aperture optimization. When σ_{r} is less than 2, the reduction rate of the BER optimized based on the relative aperture optimum value increases as the standard deviation increases. When σ_{r} is 2, the maximum reduction ratio of the average BER optimized by the optimal value of the relative aperture is 55%. This result can be explained by the fact that the greater the standard deviation of random jitter, the lower the value of the BER of the coupling system, and the more difficult it is to reduce the BER by the optimal value of the relative aperture. The method of optimizing the coupling system based on the optimal value of the relative aperture has an upper limit on the reduction rate of the BER.

As shown in Fig. 7, when the SNR parameter *Q* is 3, 6, and 9, the reduction ratio of the BER of the space laser–quad-mode fiber coupling system is a function of the standard deviation of random jitter. When *Q* increases from 3 to 9, the maximum value of the reduction ratio of the average BER after optimization by the relative aperture optimum value increases from 27% to 89%, which indicates that in the coupling system composed of the space laser–quad-mode fiber, the maximum value of the reduction ratio of the average BER relates to the SNR parameter *Q*. The larger the *Q* in the coupling system, the higher the optimal upper limit of the optimal value of the relative aperture to the average BER.

In the coupling environment of the quad-mode fiber and the space laser with three different SNR parameters, the reduction ratio of the BER reaches a maximum when σ = 2. When σ_{r} is greater than 2, the optimization effect by the relative aperture on the BER gradually decreases; with the increase in the standard deviation of random jitter, the optimization effect of the relative aperture optimum value on the BER decreases, independent of the SNR parameter *Q*.

## IV. CONCLUSION

Based on the Laguerre–Gaussian model, this paper established a coupling model of the space laser–FMF system with random angular jitter and simulated and analyzed it with a quad-mode fiber. Studies have shown that the coupling lens has an optimal relative aperture that results in the best coupling efficiency and BER. Under the influence of different random jitter standard deviations, the coupling system’s ability to resist random jitter is improved by optimizing the coupling lens’ relative aperture, which is reflected in the increase in the coupling efficiency and the decrease in the BER after optimization. With the increase in the standard deviation of random jitter, the optimization degree of the optimal value of the relative aperture to the coupling efficiency gradually increases. It is worth noting that, affected by random jitter, the optimal value of the relative aperture has an upper limit on the optimization ability of the average BER. The greater the SNR in the communication system, the greater the reduction rate of the optimized BER. Our future works will consider more complex cases, including optimizing the coupling efficiency in the presence of thermal effects.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose

### Author Contributions

**Zhaoyuan Zhang**: Formal analysis (lead); Software (lead); Writing – original draft (lead); Writing – review & editing (lead).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request