### Low-loss, high-purity, and ultrabroadband all-fiber LP_{41} mode converter employing a mode-selective photonic lantern [Invited]

_{41}mode converter is proposed and fabricated by tapering a nine-core single-mode fiber bundle. Experimental results indicate that this all-fiber LP

_{41}mode converter is low-loss, high-purity, and ultrabroadband. The insertion loss is less than 0.4 dB. The purity of odd LP

_{41}at 1310 nm is 95.09%, and the operating bandwidth exceeds 280 nm.

## 1. Introduction

High-order modes have potential applications in optical communication^{[1,2]}, particle manipulation^{[3]}, and optical fiber sensing^{[4]}. Methods to generate higher-order modes have been researched for several decades both in spatial^{[5,6]} and all-fiber systems^{[7–10]}. Compared with the spatial mode converters, all-fiber devices have the advantages of small volume and good compatibility. All-fiber high-order mode converters can be achieved through long-period fiber gratings (LPFGs)^{[7,8]}, mode selective couplers (MSCs)^{[9]}, and photonic lanterns (PLs)^{[10]}. A mode converter employing LPFGs can achieve mode conversion up to ${\mathrm{LP}}_{41}$^{[11,12]}. The disadvantage of LPFGs is their narrow working bandwidth. Even worse, as the mode order increases, the difficulty of production and additional insertion losses (ILs) also increase. For example, the 90% conversion efficiency bandwidth of the all-fiber fourth-order mode converter, realized by Chang *et al.* by employing an LPFG written by preset twist, is only 33 nm, and the IL of the LPFG is measured to be as large as 2.4 dB^{[1]}. MSCs usually have a wider operating bandwidth than LPFGs^{[13,14]}. Cui *et al.* fabricated MSCs for ${\mathrm{LP}}_{11}$, ${\mathrm{LP}}_{21}$, and ${\mathrm{LP}}_{31}$ modes with minimum losses of 2.5, 1.9, and 1.6 dB, respectively, by the side-polishing method^{[13]}. By optimizing the polishing process and precise control of phase-matching conditions, MSCs with six-mode groups (${\mathrm{LP}}_{01}$, ${\mathrm{LP}}_{11}$, ${\mathrm{LP}}_{21}$, ${\mathrm{LP}}_{02}$, ${\mathrm{LP}}_{12}$, and ${\mathrm{LP}}_{31}$) exhibit losses below 1.8 dB, and purity higher than 88.0% in the 1530–1600 nm wavelength range has been achieved by Zhang *et al.*^{[9]}. Recently, ultralow-loss five-LP (${\mathrm{LP}}_{01}$, ${\mathrm{LP}}_{11}$, ${\mathrm{LP}}_{21}$, ${\mathrm{LP}}_{02}$, and ${\mathrm{LP}}_{12}$) MSCs with a minimum IL of 0.02 dB (for ${\mathrm{LP}}_{11}$ mode) and a maximum 90% conversion bandwidth exceeding 174.31 nm (for ${\mathrm{LP}}_{21}$ mode) were achieved by Guo *et al.* by use of the fused biconical taper method^{[14]}. However, for the conversion of higher-order modes, the MSCs face an adiabatic transmission dilemma and high cross talk caused by the small difference in effective mode refractive index among high-order modes. Moreover, the precision taper control of large differences in core size between the two optical fibers for higher-order MSCs is also a great challenge.

The mode-group or mode-selective PL is a passive all-fiber device with the characteristics of ultralow loss, large bandwidth, and high capacity, playing the role of mode conversion and mode multiplexing/demultiplexing at the same time. Eznaveh *et al.* experimentally realized the mode selective excitation of ${\mathrm{LP}}_{11\mathrm{a}}$, ${\mathrm{LP}}_{11\text{\hspace{0.17em}}\mathrm{b}}$, ${\mathrm{LP}}_{21\mathrm{a}}$, and ${\mathrm{LP}}_{21\text{\hspace{0.17em}}\mathrm{b}}$ through a five-core PL, of which the IL was less than 3 dB^{[10]}. However, when the angular order is higher, modes with different radial orders will degenerate into one mode group due to the close propagation constants. It is difficult to achieve selective excitation of a single high-order mode. For example, Velázquez-Benítez *et al.* experimentally fabricated the 15-core PL multiplexing five LP mode groups, including radial high-order modes through microstructured templates^{[15]}. ${\mathrm{LP}}_{31}$ and ${\mathrm{LP}}_{12}$ degenerated into one mode group, and ${\mathrm{LP}}_{41}$, ${\mathrm{LP}}_{22}$, and ${\mathrm{LP}}_{03}$ degenerated into another mode group. The IL of the mode group including ${\mathrm{LP}}_{41}$, ${\mathrm{LP}}_{22}$, and ${\mathrm{LP}}_{03}$ is also as large as 1–2 dB.

Here, we propose and fabricate a fluorine-doped-fiber-assisted low-loss, high-purity, and ultrabroadband all-fiber ${\mathrm{LP}}_{41}$ mode converter. Selective excitation of ${\mathrm{LP}}_{41}^{\text{even}}$ and ${\mathrm{LP}}_{41}^{\text{odd}}$ is achieved by tapering a nine-core single-mode fiber bundle with an outer low refractive index capillary. The fluorine-doped fiber at the center of the fiber bundle is used to confine radial high-order modes at the few-mode end of the PL. The operating wavelength of the all-fiber ${\mathrm{LP}}_{41}$ mode converter is measured to cover 1260–1360 nm and 1460–1640 nm using a tunable laser source with limited spectral range. Through numerical mode decomposition of the collected intensity distribution, the purity of ${\mathrm{LP}}_{41}^{\text{odd}}$ at 1310 and 1565 nm is estimated to be about 95.09% and 93.37%, respectively. The IL of ${\mathrm{LP}}_{41}^{\text{odd}}$ is less than 0.2 dB at 1260–1360 nm and 1460–1640 nm. And the IL of ${\mathrm{LP}}_{41}^{\text{even}}$ is less than 0.4 dB at 1260–1360 nm and 1460–1640 nm.

## 2. Principle

The schematic of the proposed PL is shown in Fig. 1(a). A cross section of the single-mode end of the PL is shown in Fig. 1(b). A customized fluorine-doped fiber and five different types of single-mode fibers are used to construct the fiber bundle. The customized fluorine-doped fiber consists of an inner fluorine-doped layer and an outer pure silica layer. The numerical aperture of the inner fluorine-doped layer is 0.22, which means the refractive index of the inner fluorine-doped layer is much lower than that of ${\mathrm{SiO}}_{2}$. Therefore, at the few-mode end of the PL, radial high-order modes are not allowed. The diameter ${d}_{1}$ of the fluorine-doped layer is 170 µm, and the diameter ${d}_{2}$ of the pure silica layer is 240 µm. The numerical aperture of the low refractive index capillary is 0.22. The inner diameter ${d}_{3}$ and the outer diameter ${d}_{4}$ of the pretapered low refractive index capillary are 490 and 550 µm, respectively. Fiber ${\mathrm{F}}_{1}$ is placed at the right, and the others are symmetrically placed as shown in Fig. 1(b). The same color represents the same fiber. When fundamental mode is injected at the single-mode end of the PL, it will evolve into a corresponding mode at the few-mode end of the PL after passing through the tapered region with length $L$, as shown in Fig. 1(a). It is worth noting that Fig. 1 is not drawn to scale. The refractive index distributions of five kinds of single-mode fibers are shown in Fig. 2. Except for fiber ${\mathrm{F}}_{1}$, all other single-mode fibers have low refractive index regions around their cores.

Using the finite-element method and based on the experimental data, the geometric modeling of the PL is established, and the effective refractive index (ERI) of the eigenmodes is obtained, as shown in Fig. 3(a). When the fundamental mode is injected at the single-mode end of the PL, the energy will be confined inside the cores of the single-mode fibers when the propagation distance $z$ does not exceed 0.0500 m. There are several crossings in the ERI curves between the propagation distance $z=0$ and 0.0500 m, as shown in Fig. 3(a). The reason is that there are low doping regions around the core of single-mode fibers ${\mathrm{F}}_{2}\u2013{\mathrm{F}}_{5}$, which leads to a faster decrease in the ERI when energy diffuses to the cladding area, resulting in crossing. However, there is no coupling between these modes because they do not overlap spatially when the propagation distance $z$ does not exceed 0.0500 m. As the core size further decreases, the ERI of the mode also further decreases, as shown in Fig. 3(a). Until the fiber core is unable to confine light, energy diffuses to the cladding area. Then, spatial overlap and coupling are possible for modes injected into different cores. When coupling begins, the ERI of ${\mathrm{LP}}_{41}^{\text{even}}$ and ${\mathrm{LP}}_{41}^{\text{odd}}$ takes the eighth and ninth position. And the intrinsic energy exchanges along the taper between supermodes will make sure that supermodes corresponding to ${\mathrm{LP}}_{41}^{\text{even}}$ and ${\mathrm{LP}}_{41}^{\text{odd}}$ will always take the eighth and ninth position. The ERI of mode 1 and mode 3 is very close when the propagation distance $z=0.0771\text{\hspace{0.17em}}\mathrm{m}$, and the mode field distributions have a large spatial overlap, as shown by the red mark “D” in Fig. 3(a), which leads to strong coupling between ${\mathrm{LP}}_{01}$ and ${\mathrm{LP}}_{11}$ mode groups. The ERI of modes 2–5 is very close when the propagation distance $z=0.0566\text{\hspace{0.17em}}\mathrm{m}$, and mode fields of modes 2–5 almost completely overlap in space as shown by the red mark “B” in Fig. 3(a), which leads to strong coupling between ${\mathrm{LP}}_{11}$ and ${\mathrm{LP}}_{21}$ mode groups. And when the propagation distance $z=0.0663\text{\hspace{0.17em}}\mathrm{m}$, the ERI of mode 5 and mode 7 is close, and a large spatial overlap happens between them, as shown by the red mark “C” in Fig. 3(a), which leads to strong coupling between ${\mathrm{LP}}_{21}$ and ${\mathrm{LP}}_{31}$ mode groups. However, when the propagation distance $z=0.0552\text{\hspace{0.17em}}\mathrm{m}$, although the ERIs of modes 6, 7 and ${\mathrm{LP}}_{41}$ mode group are close, the spatial overlap is small as shown by the red mark “A” in Fig. 3(a). Therefore, the coupling between ${\mathrm{LP}}_{41}$ mode group and modes 6, 7 is weak. In summary, only the ${\mathrm{LP}}_{41}$ mode group is possible to be excited with high purity. The eigenmodes corresponding to the ${\mathrm{LP}}_{41}$ mode group at different propagation distances $z$ are shown in Fig. 3(b). It can be seen that ${\mathrm{LP}}_{41}^{\text{even}}$ and ${\mathrm{LP}}_{41}^{\text{odd}}$ have different evolution paths, which can lead to differences in the purity of excited ${\mathrm{LP}}_{41}^{\text{even}}$ and ${\mathrm{LP}}_{41}^{\text{odd}}$.

#### Fig. 3. (a) ERI of eigenmodes along the taper; (b) evolution of ${\mathrm{LP}}_{41}^{\text{even}}$ and ${\mathrm{LP}}_{41}^{\text{odd}}$ .

## 3. Experiments

The PL is fabricated with a Vytran automated glass processor with an integrated cleaver, using graphite filaments. The diameters of the fluorine-doped capillary and the fluorine-doped fibers are larger than that we need. Therefore, the outer diameter of the fluorine-doped fiber is pretapered to 240 µm. Then the diameter of the fiber bundle will be 490 µm. To facilitate the insertion of the fiber bundle into the capillary, the inner diameter of the low refractive index capillary is pretapered to a size slightly larger than the diameter of the fiber bundle. The coating layer of single-mode fibers will be removed and then combined with the fluorine-doped fiber to form a fiber bundle. With the help of anhydrous ethanol, the fiber bundle can be easily inserted into the capillary. Pretapering with the help of a vacuum pump will be made to further fit the inner wall of the capillary with the fiber bundle. Finally, a final taper with a vacuum pump is made to finish the fabrication of the PL. The online cleaving is finished by the integrated cleaver.

The image of the fabricated PL is shown in Fig. 4(a). The taper length of the PL is close to 9 cm. The diameter of the few-mode end of the fabricated PL is 31 µm, which gives a taper ratio of 0.1047. During the taper process, the air holes at the single-mode end of the PL will gradually collapse until they disappear at the few-mode end, as shown in Fig. 4(b). Then, with different single-mode fibers as input, the mode fields of the output at the few-mode end of the PL are shown in Fig. 4(c). It can be seen that only the output with single-mode fiber ${\mathrm{F}}_{3}$ as input has the obvious characteristics of the high-purity ${\mathrm{LP}}_{41}$ mode group, which is consistent with the discussion about Fig. 3.

#### Fig. 4. (a) Image of the fabricated PL; (b) image of the cross section along the taper; (c) output of the PL at 1550 nm with different single-mode fibers as input.

## 4. Results

In order to better analyze the performance of the excitation of the ${\mathrm{LP}}_{41}$ mode group, tunable lasers (Yenista TUNICS-T100S-HP and Agilent Keysight 8164B) and infrared camera (Xi’an Leading Optoelectronic, LD-SW640171550-UC-G) are used to collect the intensity distributions of ${\mathrm{LP}}_{41}^{\text{even}}$ and ${\mathrm{LP}}_{41}^{\text{odd}}$. Due to the lack of a tunable laser at 1360–1460 nm, only the intensity distributions at 1260–1360 nm and 1460–1640 nm are collected. Light from the tunable laser is injected into the single-mode fiber ${\mathrm{F}}_{3}\text{-}1/{\mathrm{F}}_{3}\text{-}2$ at the single-mode end of the PL, respectively. Then, the fundamental mode in the single-mode fiber evolves into ${\mathrm{LP}}_{41}^{\text{even}}$ and ${\mathrm{LP}}_{41}^{\text{odd}}$ at the few-mode end of the PL along the taper. The output of the PL is captured by the infrared camera after being collimated by an objective lens. Intensity distributions of ${\mathrm{LP}}_{41}^{\text{even}}$ and ${\mathrm{LP}}_{41}^{\text{odd}}$ at several typical wavelengths are shown in Fig. 5(a). It can be seen that in a wide wavelength range, the intensity distributions can maintain the characteristics of eight lobes of ${\mathrm{LP}}_{41}^{\text{even}}$ and ${\mathrm{LP}}_{41}^{\text{odd}}$.

#### Fig. 5. (a) Intensity distributions of ${\mathrm{LP}}_{41}^{\text{even}}$ and ${\mathrm{LP}}_{41}^{\text{odd}}$ ; (b) measured image, recovered image, and recovered component of ${\mathrm{LP}}_{41}^{\text{odd}}$ at 1310 and 1565 nm; (c) measured purity of ${\mathrm{LP}}_{41}^{\text{even}}$ and ${\mathrm{LP}}_{41}^{\text{odd}}$ at 1260–1360 nm and 1460–1640 nm.

Through the numerical mode decomposition method of the collected intensity distribution, the purities of ${\mathrm{LP}}_{41}^{\text{even}}$ and ${\mathrm{LP}}_{41}^{\text{odd}}$ were measured. The purity of a mode is written as

The mode purity was measured by detecting the mode fields at the output of the PL and executing mode decomposition^{[16,17]}. First, the intensity distribution was captured by an infrared camera. Second, the real and imaginary parts of the Fourier coefficients from the Fourier expansion were formulated as an equation group. Third, the intensity distribution was reduced to an azimuthal sampling one-dimensional sequence with a certain radius. Finally, we solved the equation group and obtained the amplitudes of each mode component, and then we calculated the mode purity. Based on the recovered mode components, the intensity image can be reconstructed. Figure 5(b) shows measured image, recovered image, and recovered component of ${\mathrm{LP}}_{41}^{\text{odd}}$ at 1310 and 1565 nm. The purity of ${\mathrm{LP}}_{41}^{\text{odd}}$ at 1310 and 1565 nm is measured to be about 95.09% and 93.37%, respectively. The correlations between the measured image and the recovered image of ${\mathrm{LP}}_{41}^{\text{odd}}$ at 1310 and 1565 nm are 0.9500 and 0.9636, which ensures the credibility and accuracy of the recovered results.

With this method, the purity of ${\mathrm{LP}}_{41}^{\text{even}}$ and ${\mathrm{LP}}_{41}^{\text{odd}}$ at 1260–1360 nm and 1460–1640 nm is measured, as shown in Fig. 5(c). The highest mode purity of ${\mathrm{LP}}_{41}^{\text{odd}}$ in the wavelength range of 1260–1360 nm and 1460–1640 nm is 95.09% at 1310 nm and 93.37% at 1565 nm, respectively, and the highest mode purity of ${\mathrm{LP}}_{41}^{\text{even}}$ in the wavelength range of 1260–1360 nm and 1460–1640 nm is 95.26% at 1355 nm and 86.27% at 1470 nm, respectively. The bandwidths of mode purity higher than 85.00% of ${\mathrm{LP}}_{41}^{\text{even}}$ and ${\mathrm{LP}}_{41}^{\text{odd}}$ are 50 and 155 nm, respectively. The bandwidth of mode purity higher than 85.00% of ${\mathrm{LP}}_{41}^{\text{odd}}$ is wider than that of ${\mathrm{LP}}_{41}^{\text{even}}$. This may be because the paths of light injected into single-mode fiber ${\mathrm{F}}_{3}\text{-}1/{\mathrm{F}}_{3}\text{-}2$ are not exactly the same as shown in Fig. 3(b), which causes differences in the purity of excited ${\mathrm{LP}}_{41}^{\text{even}}$ and ${\mathrm{LP}}_{41}^{\text{odd}}$.

The IL of the fabricated PL is also experimentally measured. The IL of an optical device is written as

The measured ILs of ${\mathrm{LP}}_{41}^{\text{even}}$ and ${\mathrm{LP}}_{41}^{\text{odd}}$ are shown in Fig. 6. The IL of ${\mathrm{LP}}_{41}^{\text{odd}}$ is less than 0.2 dB in the wavelength range of 1260–1360 nm and 1460–1640 nm. And the IL of ${\mathrm{LP}}_{41}^{\text{even}}$ is less than 0.4 dB in the wavelength range of 1260–1360 nm and 1460–1640 nm. It can be seen that the ILs of ${\mathrm{LP}}_{41}^{\text{even}}$ and ${\mathrm{LP}}_{41}^{\text{odd}}$ show obvious differences in the wavelength range of 1460–1640 nm. This is due to gravity and the deviation of the position of the fiber bundle in the temperature field created by the graphite filament; the cross section of the tapered region is not symmetrical during the tapering process. In the short wavelength range, light is better confined. However, in the long wavelength range, the mode field area is larger, making ${\mathrm{LP}}_{41}^{\text{even}}$ and ${\mathrm{LP}}_{41}^{\text{odd}}$ more sensitive to defects during the manufacturing process. And the evolution paths of ${\mathrm{LP}}_{41}^{\text{even}}$ and ${\mathrm{LP}}_{41}^{\text{odd}}$ are different, which leads to significant differences in IL within the wavelength range of 1460–1640 nm.

#### Fig. 6. ILs of ${\mathrm{LP}}_{41}^{\text{even}}$ and ${\mathrm{LP}}_{41}^{\text{odd}}$ .

## 5. Conclusion

In summary, a low-loss, high-purity, and ultrabroadband all-fiber ${\mathrm{LP}}_{41}$ mode converter was proposed and fabricated. Selective excitation of ${\mathrm{LP}}_{41}^{\text{even}}$ and ${\mathrm{LP}}_{41}^{\text{odd}}$ was realized by tapering a nine-core single-mode fiber bundle with an outer low refractive index capillary. Using tunable laser sources with a limited spectral range, the operating wavelength of the all-fiber ${\mathrm{LP}}_{41}$ mode converter was measured to cover 1260–1360 nm and 1460–1640 nm. By decomposing the collected mode field intensity, the purity of ${\mathrm{LP}}_{41}^{\text{odd}}$ at 1310 and 1565 nm was measured to be about 95.09% and 93.37%, respectively. The bandwidths of purity higher than 85.00% of ${\mathrm{LP}}_{41}^{\text{even}}$ and ${\mathrm{LP}}_{41}^{\text{odd}}$ were 50 and 155 nm, respectively. The IL of ${\mathrm{LP}}_{41}^{\text{odd}}$ was less than 0.2 dB at 1260–1360 nm and 1460–1640 nm, and the IL of ${\mathrm{LP}}_{41}^{\text{even}}$ was less than 0.4 dB at 1260–1360 nm and 1460–1640 nm. Our work provides a low-loss, high-purity, and ultrabroadband all-fiber ${\mathrm{LP}}_{41}$ mode converter for space-division and wavelength-division multiplexing fiber systems.

Liang Chen, Huiyi Guo, Zekun Shi, Wenzhe Chang, Boyu Chen, Zhi Wang, Yan-ge Liu. Low-loss, high-purity, and ultrabroadband all-fiber LP_{41} mode converter employing a mode-selective photonic lantern [Invited][J]. Chinese Optics Letters, 2023, 21(11): 110008.