1. Introduction

The increasing amount of satellite data and its acquisition demand urgently requires improving the satellite-ground data downlink capability to meet the high-speed transmission [1,2]. Free-space optical communication (FSOC) using fiber-based optical receivers is an excellent solution by using lasers as information carriers enabling communication between satellites and ground stations with high data rates, ultra-bandwidth, and low power consumption [3–5]. However, atmospheric turbulence in the free space propagation induces refractive index disturbances, causing laser signals to experience intensity scintillation, beam drift, beam spread, and angle-of-arrival fluctuations, which reduces the coupling efficiency of optical fibers and affects the performance of practical laser transmission systems [6,7]. To combat these issues, mode diversity technology has been investigated as an effective means of resisting atmospheric turbulence and improving the reception performance of FSOC systems [8–10]. This technique refers to the collection of spatial light using few-mode fibers (FMFs) or multimode fibers (MMFs). By using mode demultiplexers, the turbulent light is effectively decomposed into an array of single-mode fibers (SMFs). Subsequently, different combining methods, including selection combining (SC), equal gain combining (EGC), and maximal ratio combining (MRC), are adopted to combine the outputs of the single-mode end of demultiplexers to improve the signal quality [10]. The likelihood of simultaneous deep fading occurring in all copies carrying the same information in the SMF array is low. Therefore, mode diversity technology is effective in improving the quality of the received signal, which can be applied for satellite-to-ground optical communication systems.

For different FSOC scenarios, various studies have been conducted on mode diversity receivers (MDR) by various mode demultiplexers and combining technologies to improve systems [11–13]. The MDR with FMF coupling for high-speed FSOC could improve the power by 4-5 dB compared to SMF-coupled receivers [13]. In fiber-based MDR systems, mode demultiplexers play a crucial role in decomposing spatial light induced by turbulence into SMFs. Two common types of mode demultiplexers in fiber-based MDR systems are multi-plane light converters (MPLCs) and photonic lanterns (PLs). In [14–16], researchers used MPLC to decompose the turbulence-corrupted optical beam into N spatially separated Gaussian modes, to improve the coupling efficiency between the perturbed beam and fibers. It is confirmed that FSOC receivers based on MPLCs could collect more spatial modes due to their large area and numerical aperture (NA) compared to SMF receivers and this feature enables a strong resilience to wavefront distortions. In [15], researchers verified the coupling efficiency improvement of MPLCs with 10, 15, and 20 output modes for various turbulence strengths in a horizontal link of 2 km. Another research further demonstrated that MPLC-based FSOC receivers showed much less power penalty and were much more robust to atmospheric turbulence compared to SMF receivers in a realistic FSOC link from a GEO satellite to an optical ground station. For example, the average coupling loss of MPLC with 10 and 15 output fibers is reduced by 4.8 and 6.5 dB compared to SMF coupling. However, MPLCs typically have complex structures and require high-precision fabrication, which leads to higher fabrication costs and implementation difficulties. Moreover, increasing the number of output fibers may introduce additional problems, such as low mode coupling efficiency, large mode crosstalk, and complex design of coherent receivers. PLs are popular all-fiber type mode demultiplexers used in horizontal link or satellite-to-ground laser communication receiving systems. Depending on the characteristics of the transmitted electromagnetic wave modes in SMFs, they can be divided into mode-selective PLs (MSPLs) and non-mode-selective PLs (NSPLs) [17]. Both MSPLs and NSPLs are involved in the currently reported MDR systems, which can effectively improve the performance of receiving systems [18,12]. Most of the reported MDR systems use 3-mode or 6-mode PLs with FMF. In [12], researchers demonstrated diversity receivers using 3-mode PLs reduced the transmit power by 6 dB compared to receivers using SMF. Other researchers have compared the receiving performance of MDR systems using different combining techniques [19,20]. Researchers verified that under specific turbulence conditions, the combination of NSPLs and the EGC method provides the best receiving performance [20].

The above-mentioned research using FMF-based or multimode receivers aimed to improve the power, coupling efficiency, and signal-to-noise ratio of the receiving system by various solutions. However, few studies have analyzed the effect of fiber position error (FPE) on receiving performance. Although FMF-receivers using traditional PLs have the advantages of better coupling efficiency, less transmitting power, and a certain tolerance of position deviation compared to SMF, their performance is still vulnerable to fiber position errors, which are inevitable degradation factors in practical optical receiving systems, due to their small NA and core diameters [21–23]. MMF receivers have been proposed to increase the receiving area of light and reduce the effect of fiber position errors on spatial light-to-fiber coupling [20,24]. However, satisfying the mode-number matching condition requires hundreds of SMF ports for MMF-based PLs, which significantly increases the complexity of the mode diversity receiver by requiring too many coherent receivers for combining the signals output from SMFs.

In this study, we propose a mode-mismatching multimode photonic lantern (MM-PL) with only a few SMFs at the single-mode end and a relatively high NA and core diameter at the multimode end. We aim to increase the tolerance of PLs to position deviation without increasing the complexity of receivers. We analyze the effects of optimal focal length deviation and lateral offsets on the coupling of spatial light and MM-PLs in satellite-to-ground downlink scenarios and compared it with the case of FM-PLs with the same number SMFs channels. The results confirmed that our proposed MM-PLs can tolerate larger position errors under weak-to-strong turbulence. The optimal focal length tolerance for a 1 dB drop in average coupled power is 5-6 m and 2-3 m for MM-PL and FM-PL, respectively. Additionally, the lateral offset tolerance corresponding to a 0.5 dB decrease in average coupling power for MM-PL exceeds 12 µm, and there is a gentle reduction in the mean SNR of 0.67-1.16 dB with a 12 µm offset under weak turbulence. However, FM-PLs show a high sensitivity to lateral offsets with a decrease in the mean SNR of 6.50-8.49 dB at an offset of 12 µm. These results demonstrate the advantages of MM-PLs over FM-PLs for the satellite-ground downlink considering fiber position errors. In Section 2, we present the satellite-to-ground downlink model. Section 3 discusses the design of PLs, and Section 4 presents simulation analysis and comparison results. In Section 5, we summarize this paper.

2. Mode diversity using MM-PL for satellite-to-ground optical communication downlink model

The laser transmission from a satellite to the ground involves a long-distance vacuum transmission section and a near-ground section affected by atmospheric turbulence. The optical signal will undergo severe distortion after satellite-to-ground downlink, due to the effects of long-distance transmission and atmospheric turbulence. To improve the quality of the optical signal, mode diversity technology can be utilized, which exploits the uncorrelated characteristics between multiple copies transmitted through different mode channels, thereby reducing signal fading. FMFs or MMFs that support multiple orthogonal modes can be utilized to receive turbulent light, resulting in higher receiving power compared to SMFs. The distorted wavefront can be seen as a superposition of fundamental and higher-order modes, as many higher-order modes as possible can be received and decoupled into a multichannel fundamental mode signal. Finally, the signal can be combined to compensate for atmospheric turbulence.

Figure 1 depicts a simplified schematic diagram of the satellite-to-ground downlink. The optical link has a total distance of 400 km, which includes a high-altitude transmission segment of 380 km with a fixed geometric loss, and a near-ground segment of 20 km that is susceptible to atmospheric turbulence. The light source emitted by the satellite is a Gaussian beam with a wavelength of 1550 nm, a beam waist radius of 0.075 m, and a power of 0dBm. Modeling the propagation of beams in the high-altitude segment through the fundamental fluctuation equation and the Huygens-Fresnel principle. Non-equally spaced 20 layers of phase screens are utilized to simulate atmospheric turbulence in the near-ground 20 km. The transmission of Gaussian beam through turbulence is simulated by stepwise Fresnel diffraction transmission and phase modulation. Thereby the spatial light, which has been disturbed by atmospheric turbulence, is obtained and then focused onto fibers using a lens with a diameter of 0.6 m. Following this, the spatial light is demultiplexed into multiple single modes by PLs, and the single mode fiber array undergoes coherent reception, merging, and signal processing to recover the transmitted information. The position error of the coupling fiber affects the reception of the spatial light. The position error includes lateral displacement and longitudinal distance, which are set as Δx and Δz, respectively, as shown in Fig. 1.

 

Fig. 1. The schematic diagram of satellite-to-ground downlink transmission.

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In this paper, the Von-Karman power spectrum inversion method [25] is used to simulate the phase screens. To facilitate the description of the atmospheric turbulence intensity variation, the atmospheric structural refractive index constant $C_n^2$ is used to represent the turbulence intensity. The height distribution profile model of $C_n^2(h)$ is the Hufnagel-Valley 5/7 model [26]. The relationship between $C_n^2$ and h is shown in Eq. (1).

In Eq. (1), the value of $C_n^2(0)$ approximately corresponds to the atmospheric refractive index structure constant A for near-ground, which is generally taken as 1.7 × 10⁻¹⁴m^-2/3, while the root-mean-square value of wind speed w is 21 m/s. To simulate strong, moderate, and weak turbulence intensity, we respectively set A to 1 × 10⁻¹³, 2 × 10⁻¹⁴, and 1 × 10⁻¹⁵. The turbulence-induced optical distortion inside the lens aperture as well as the spatial light after the lens focusing for different turbulence strengths are detailed in Fig. 1. The greater the turbulence intensity, the more severe the distortion of the received spatial light.

3. Design and simulation of PL

PL is fabricated by placing multiple SMFs into a low refractive index capillary, which is then adiabatically tapered to form an MMF. The cladding of the SMFs and the capillary gradually form a new MMF core and cladding, respectively. PL can be classified into mode-selective and non-mode-selective types according to whether the SMFs are completely identical. For the former, the sizes of the SMFs are different, and they transmit different electromagnetic wave modes, resulting in the adiabatic coupling of the high-order modes to the specified SMF port. In contrast, the latter uses identical SMF that transmit the same electromagnetic wave mode, and the modes of the multimode end are coupled to each SMF port after tapering. For conventional few-mode PLs (FM-PLs), the number of SMFs at the single-mode end and the number of modes supported by the few-mode end are equal. This is done to satisfy the mode-matching condition and ensure lossless transmission of all guided modes. However, the proposed MM-PL in this paper tapers the SMFs into an MMF with a relatively high NA and core diameter, while only having a few SMFs at its single-mode end. The first few modes of the multimode end are converted into the fundamental mode, while the other higher-order modes evolve into cladding modes and gradually disappear during the tapering process.

To compare the performance of different PLs in receiving spatial light distorted by different strengths of turbulence, 8 PLs were designed, including a few-mode 3-mode photonic lantern (FM-3PL), a multimode 3-mode photonic lantern (MM-3PL), a few-mode 6-mode photonic lantern (FM-6PL), a multimode 6-mode photonic lantern (MM-6PL), a non-mode-selective few-mode 3-mode photonic lantern (NSFM-3PL), a non-mode-selective multimode 3-mode photonic lantern (NSMM-3PL), a non-mode-selective few-mode 6-mode photonic lantern (NSFM-6PL), and a non-mode-selective multimode 6-mode photonic lantern (NSMM-6PL). The MM-6PL is used as an example to illustrate its design principle. Figure 2(a) illustrates three-dimensional (3D) schematic diagrams of the MM-6PL. 6 SMFs in the capillary gradually become smaller during the adiabatic taper process, eventually forming a multimode end with the SMF cladding as the core layer and the capillary as the cladding. The finite element method (FEM) is utilized to simulate the mode evolution process of the single-mode end of PLs under different taper ratios, and the mode field evolution diagrams are shown in Fig. 2(b). In Fig. 2(b), the label “TR” represents the taper ratio, while the term “Mode” denotes fiber modes. The taper ratio should be chosen at the position where the single-mode fiber's eigenmode completely transforms into the multimode fiber mode, considering the optimization of mode field distribution and taper length. The final taper ratio is determined to be 0.179, aligning with the characteristics of the multimode fiber. The mode-evolution process demonstrates that the 6 single modes are well converted to the first 6 higher-order modes at the multimode end of the MM-6PL at the selected taper ratio. Additionally, Fig. 2(c) shows the effective index variation curves during the tapering process, indicating that the effective refractive indices of the modes can be separated at a tapering ratio of 0.179.

 

Fig. 2. (a) 3D schematic diagram (b) mode field variation, (c) effective index at various taper ratios of MM-6PL.

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Based on the same design method, the above 8 types of PLs were designed, and the parameters of each PL are shown in Table 1. In the header of Table 1, the labels “D_C”, “RI_C”, “D_SMF”, “NA_MMF/FMF”, “RI_MMF/FMF”, “NA_SMFs”, “RI_SMFs”, and “TR” respectively represent the inner and outer diameter of the capillary, the refractive index of the capillary, the diameter of the SMF, NA of MMF or FMF, the core refractive index of the MMF or FMF, and the core refractive index of SMF. Besides, the capillary is composed of F-doped + SiO2 material. After parameter optimization, it is found that the insertion loss from the multimode end to the single-mode end of PLs ranges from 0.1 to 0.2 dB.

Table 1. Parameter of PLs.

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4. Analysis and discussion

4.1 Analysis of optimal focal length

The focal length of the receiving aperture affects the distorted spatial light-to-fiber coupling, so selecting a suitable focal length is crucial. The focal length corresponding to the maximum coupling efficiency of the fiber under turbulence-free conditions is regarded as the optimal focal length in the absence of turbulence, as shown in Eq. (2), where D is the diameter of lens and ${w_0}$ is the mode-field radius of the fiber [11]. Besides, $\lambda$ represents the wavelength, and all simulations in this paper are conducted at 1550 nm. The optimal focal length for different receiving fibers under turbulence is determined by calculating the average coupling power for different focal lengths.

The average coupling power of different receiving fibers at different focal lengths was calculated under strong, moderate, and weak turbulence conditions. Figure 3 shows the variation of the average coupling power with the focal length for 3-mode few-mode fiber (FMF-3modes) and 3-mode multimode fiber (MMF-3modes), while Fig. 4 shows the variation of the average coupling power with the focal length for 6-mode few-mode fiber (FMF-6modes) and 6-mode multimode fiber (MMF-6modes), respectively.

 

Fig. 3. Average coupling power of FMF-3modes for (a) strong, (b) moderate, and (c) weak turbulence and MMF-3modes for (d) strong, (e) moderate, and (f) weak turbulence.

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Fig. 4. Average coupling power of FMF-6modes for (a) strong, (b) moderate, and (c) weak turbulence and MMF-6modes for (d) strong, (e) moderate, and (f) weak turbulence.

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Regardless of turbulence intensity, as the focal length gradually increases, the average coupling power first increases and then decreases, and there exists an optimal focal length corresponding to the maximum coupling power. As the turbulence intensity increases, the optimal focal length becomes relatively smaller. Furthermore, the optimal focal length increases with increasing lens diameter. In the case of a fixed focal length, a larger lens diameter results in a higher average coupled power. As illustrated in Fig. 3 and Fig. 4, the variation of the coupling power with the focal length for 6-mode fiber follows a similar pattern as 3-mode fiber. Regardless of strong or weak turbulence intensity, the change of the average coupling power of MMF with focal length is relatively gentle compared with FMF. This indicates that the variation of the reception coupling power of MMF is not significant within a certain deviation from the optimal focal length. For example, when D = 0.6 m, the optimal focal lengths of FMF-3modes and MMF-3modes are 2.5 m and 7 m for moderate turbulence density, respectively. When the focal length deviates by 2 m from the optimal focal length, the average coupling power of FMF-3modes and MMF-3modes decreases by 1.76 dB and 0.43 dB respectively from the optimal coupling power. Under the above conditions of turbulence and deviation from optimal focal length, the average coupling power of FMF-6modes and MMF-6modes decreased by 2.93 dB and 0.66 dB, respectively.

We selected a coupling lens diameter of 0.6 m based on the model presented in Section 2. The optimum focal lengths for the different fibers under weak-to-strong turbulence intensity are determined based on the results of Fig. 3 and Fig. 4. Additionally, we define the focal length tolerance as the sum of the upper and lower focal deviations from the optimal focal range within an average coupling power drop of approximately 1 dB, which is noted Δf. Table 1 presents a comparison of the optimal focal length and the focal length tolerance for each receiving fiber under both turbulent and non-turbulent conditions. AT-f and NAT-f in Table 2 represent the optimal focus lengths in the presence and absence of turbulence, respectively. The optimal focal length with turbulence is smaller than the optimal focal length without turbulence and the difference between the optimal focal length with and without turbulence for MMF is greater than that for FMF. Moreover, the focal length tolerance of MMF is greater than that of FMF under strong-to-weak turbulence. The optimal focal length tolerance to achieve a 1 dB reduction in average coupled power is 2-3 m and 5-6 m for FMF and MMF, respectively. Although significant focal length deviations are unlikely in practical engineering, MMF receivers are still well-suited for satellite-to-ground reception under varying focal lengths and turbulence intensities, even without zoom capability in the receiving system.

Table 2. Optimal focal lengths for different fiber receivers

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4.2 Analysis of MS-PLs and NS-PLs under strong-to-weak turbulence

To compare the coupling power of different PLs under weak-to-strong turbulence conditions, the appropriate focal lengths of different fibers and their corresponding received photonic lanterns were selected based on the analysis in Section 4.1. Specifically, the appropriate focal lengths for FM-3PL, MM-3PL, FM-6PL, and MM-6PL under weak-to-strong turbulence conditions were determined to be 2.5 m, 7.5 m, 2.5 m, and 6.5 m, respectively. 100 tests were conducted under a reasonable focal length to obtain the focused disturbed spatial light that would be coupled to various fibers. Then the spatial light was demultiplexed by the above PLs and the optical power of the SMF array was obtained.

Mode-selective PLs (MSPLs) are characterized by the use of SMFs with unique core diameters, allowing high-order modes at the multi-mode end to be output with high coupling efficiency to the fundamental mode through the specific SMF. Conversely, non-mode-selective PLs (NSPLs) employ SMFs with identical core diameters, and the modes at the multi-mode end are coupled respectively to each SMF.

Taking MM-3PL as an example, we calculated and compared the cumulative probability curves of the power at the single-mode ports of MM-3PL and NSMM-3PL under strong-to-weak turbulence, as shown in Fig. 5. SMF1, SMF2, SMF3, and SUM represent the power and summed power of three SMFs at the single-mode end of the MM-3PL, depicted by solid lines. SMF1-NS, SMF2-NS, SMF3-NS, and SUM-NS denote the power and summed power of three SMFs at the single-mode end of the NSMM-3PL, symbolized by dotted lines. The spatial beam passing through strong turbulence is seriously distorted with more high-frequency components, which can excite more high-order modes in the MMF. There is not obvious difference in the proportion of the first few modes of MMF. For example, when the cumulative probability reaches 0.2, the powers of the three SMFs of MM-3PL are -41.05 dBm, -40.56 dBm, and -42.48 dBm, while for NSMM-3PL, they are -41.44 dBm, -42.48 dBm, and -42.93 dBm, respectively, as shown in Fig. 5(a). Regardless of whether the PL has mode-selective characteristics or not, the output power distribution of the corresponding SMF port is similar under strong turbulence conditions. While the spatial light induced by weak turbulence mostly drifts without significant distortion, making it easier to couple to the LP₀₁ mode. For MSPLs, most of the power of the turbulent light coupled to LP₀₁ is concentrated in a specific SMF port, whereas for NSPLs, the turbulent light at the multimode port is uniformly coupled to each fiber at single-mode ports. As depicted in Fig. 5(c), the cumulative distribution curves of the power at the single-mode end of MM-3PL and NSMM-3PL distinctly diverge. When the cumulative probability reaches 0.2, the powers of three SMFs of MM-3PL are -23.79 dBm, -32.77 dBm, and -32.19 dBm, while they are -28.02 dBm, -29.24 dBm, and -30.29 dBm for the NSMM-3PL. In terms of the total coupled power, the mode-selective characteristics have little effect on the total coupling power of PLs for weak-to-strong turbulence, which is clearly depicted in Fig. 5 by the overlapping of the solid red line and the dashed red line. However, the coupling between the spatial light and fibers varies with different turbulence intensities, resulting in distinct output power distributions of SMFs after mode evolution through PLs.

 

Fig. 5. The coupling power distribution of MM-3PL and NSMM-3PL for (a) strong, (b) moderate, and (c)weak turbulence.

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In addition, we calculated the SNR distributions of MS-PLs and NS-PLs for different turbulence strengths. The signal-to-noise ($SNR$) under the shot noise limit conditions of a coherent detection system can be expressed as Eq. (3). The MRC method was employed to merge the SNR of the output from the SMFs of the PL.

In the Eq. (3), $\overline {{P_s}}$ represents the average optical power of each SMF; R denotes the responsivity of the photodetector; q denotes the electronic charge; $\eta$ is the quantum efficiency, assumed to be 100%; $h$ is Planck's constant (≈6.626E-34); v is the optical frequency (≈1.9355E14 at 1550 nm), and $\Delta F$ is the bandwidth, which is set to 100 GHz. Using MM-6PL as examples, their SNR distributions are shown in Fig. 6. Strong turbulence causes severe scattering of the spatial light, resulting in relatively scattered power distribution at the output from the SMF end of PLs. As the turbulence intensity decreases, the power distribution becomes more concentrated, which affects the SNR distribution of each SMF. The difference of SNR between the MS-PLs and NS-PLs is small and their SNR distributions are similar for both moderate and strong turbulence intensity. However, for the influence of weak turbulence, there is a gap between the SNR distributions of the MS-PLs and the NS-PLs. This is because most of the power of the turbulent light is coupled to the LP₀₁ mode and then output from the specific SMF by MS-PLs. While for the NS-PLs, the spatial light from the multimode ports couples uniformly to each fiber of the single-mode ports. The power and corresponding weight distribution at the single-mode end of NS-PLs exhibit a greater degree of uniformity compared to MS-PLs. Because signals with higher output power have larger weights in the MRC method, the combined SNR calculated when the power is concentrated and output to one of the N output ports is greater than the combined SNR calculated when the power is uniformly distributed and output through N paths. Therefore, under equivalent noise levels, the combined SNR of the single-mode end in MS-PLs outperforms that of NS-PLs. Consequently, we utilized MS-PLs for mode diversity reception in the satellite-to-ground downlink.

 

Fig. 6. SNRs distribution of MM-6PL for the influence of (a) strong, (b) moderate, and (c) weak turbulence.

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4.3 Analysis of lateral offsets

Due to pointing errors at the end of fibers, the turbulence-distorted spatial light may experience lateral displacement after lens focusing. The lateral displacement is defined as offset, which is set to 0-12µm. Based on the focal length parameters selected in Section 4.2, the input turbulent spots of different MS-PLs are obtained separately, and their coupling power and SNR are calculated with different lateral offsets. The lateral offset corresponding to a decrease of approximately 0.5 dB in the total average coupling power is defined as the tolerance of the lateral offset, which is denoted as Δoffset. The Δoffset of FM-3PL and FM-6PL is 3-6 µm and 6-9 µm under moderate-to-weak and strong turbulence conditions. However, the Δoffset of MM-3PL and MM-6PL is greater than 12 µm regardless of strong or weak turbulence. Figure 7 shows the cumulative probability distribution curves of SNR for different MS-PLs at various lateral offsets. Here, the cumulative probability refers to the probability that the SNR is less than a certain value. For example, the cumulative probabilities of SNR less than a certain value are 0.1 and 0.2, which can be interpreted as the probabilities of SNR greater than that value are 90% and 80%. The SNRs corresponding to a cumulative probability of 0.1 and 0.2 at offsets of 0 and 12 µm are labeled for different PLs in Fig. 7. Additionally, the difference between the SNR at offsets of 0 and 12 µm is defined as ΔSNR₁ and ΔSNR₂ when the cumulative probability is 0.1 and 0.2, respectively. ΔSNR₁ and ΔSNR₂ of different PLs for strong-to-weak turbulence are shown in Table 3.

 

Fig. 7. SNR of different PLs at different offsets for (a) strong, (b)moderate and (c) weak turbulence.

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Table 3. ΔSNR1 and ΔSNR2 of different PLs.

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Moreover, the mean SNR of each PL at various offsets is calculated, as illustrated in Fig. 8. The mean SNR of each PL decreases with increasing turbulence intensity and lateral offsets. As shown in Fig. 8 (a), as the offset increases, the mean SNR of MM-PLs remains stable within the offset range of 0-9 µm, with a decrease of 0.11 dB and 0.13 dB at an offset of 12 µm compared to that of no offset input for MM-3PL and MM-6PL. However, the average SNR of FM-PLs shows a decreasing trend as the input offset increases. For instance, the mean SNR of FM-3PL and FM-6PL at an offset of 12 µm decreases by 0.96 dB and 1.45 dB compared to that of the no-offset input. Under moderate turbulence conditions, the mean SNR of FM-PLs gradually decreases with increasing offset, and the degree of change is significantly greater than that of MM-PL, as shown in Fig. 8(b). Figure 8(c) illustrates the mean SNR of FM-PLs decreases dramatically with increasing offset, while the mean SNR of MM-PLs does not decrease significantly at large offsets for the influence of weak turbulence. For example, the mean SNR of FM-3PL and FM-6PL is 8.49 dB and 6.50 dB lower than that of no offset at an offset of 12 µm, while for the MM-3PLand MM-6PL, the mean SNR decreases by 1.16 dB and 0.67 dB respectively. Therefore, regardless of whether under strong or weak turbulence conditions, MM-PLs have better resistance to lateral deviation compared to FM-PLs.

 

Fig. 8. The mean SNR of PLs at different offsets for (a) strong, (b) moderate and (c) weak turbulence.

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The main reason is that the multimode end of MM-PLs exhibits a larger core diameter and numerical aperture (NA) compared to FM-PLs. For distorted spatial light under various levels of turbulent influence, MM-PLs possess greater receiving power in the presence of offsets, increasing the likelihood of high coupling efficiency between distorted light and fiber. Consequently, for aberrant spatial light that deviates from the fiber position, MM-PLs experience smooth fluctuations in the receiving power at the single mode ports. The fluctuations in the output power of PLs at the single mode end directly affect the combined SNR at the single-mode end. Due to the lower output power fluctuations of MM-PLs compared to FM-PLs, MM-PLs exhibit smoother SNR fluctuations by the MRC method under different lateral offset inputs.

5. Conclusion

In conclusion, we have proposed novel mode-mismatching multimode photon lanterns (MM-PLs) for mode diversity reception in satellite-to-ground downlinks. We have demonstrated photonic lanterns with mode selectivity are superior to non-mode-selective photonic lanterns under weak turbulent conditions. Besides, the comparative simulation results verify the advantages of MM-PLs over FMF-based photonic lanterns in terms of turbulence resistance in the satellite-ground downlink. The proposed MM-PLs exhibited better stability and could tolerate greater position error in terms of optimal focus length deviation and lateral offsets, which demonstrates the practical value of MM-PLs. Furthermore, we will fabricate MM-PLs and evaluate their receiving performance under strong-to-weak turbulence in a desktop experiment simulating satellite-to-ground downlink transmission in the future.