Elliptical core main channel multiplexer / demultiplexer
By using the refractive index core of elliptical cores with a small mode optical fiber and pure silica in the optical fiber communication system, the multiplexing and demultiplexing of low crosstalk and high extinction ratios of three modes: LP01, LP11a and LP11b are realized, which solves the problems of crosstalk and loss between modes in the prior art, extends the transmission distance and reduces the system complexity and cost.
Patent Information
- Application Number
- CN202421295205.2
- Authority / Receiving Office
- CN · China
- Patent Type
- Utility models(China)
- Current Assignee / Owner
- Filing Date
- 2024-06-04
- Publication Date
- 2025-03-25
- Estimated Expiration
- 2034-06-04
AI Technical Summary
The crosstalk and loss between modes of the existing small-mode fiber multiplexer/demultiplexer still exist, resulting in limited transmission distance of the communication system, and the complexity and cost of multi-input and multi-output digital signal processing.
The elliptical core small mode optical fiber is used as the main channel, and the refractive index core of pure silica and the step refractive index distribution core are combined to achieve multiplexing and demultiplexing of three modes: LP01, LP11a and LP11b. Through a large effective refractive index difference and reasonable setting of the position and length of the transmission channel, crosstalk between modes and reduce losses.
The mode multiplexing and demultiplexing of three modes: LP01, LP11a and LP11b are realized, and the mode multiplexing and demultiplexing of the three modes of LP01, LP11a and LP11b are supported. The MIMO-FREE operation is supported, which extends the transmission distance of the optical fiber communication system and reduces the complexity and cost of the system.
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Figure CN222674328U_ABST
Abstract
Description
Technical Field
[0001] This patent application relates to an elliptical core main channel multiplexer / demultiplexer, which can be applied to the fields of fiber optics, fiber optic communications, fiber optic wireless access, optical information processing and new generation information technology. Background Art
[0002] In recent years, the traffic volume of various communication services has grown exponentially, and single-mode fiber communications have faced unprecedented challenges. The optical fiber communication industry has achieved a breakthrough in the transmission capacity of communication networks around the physical dimension of space division multiplexing (including core multiplexing and mode division multiplexing and their combination); the mode coupling between mode division multiplexing and few-mode fiber (Few-Mode Fiber, FMF) in space division multiplexing and its related devices and application research have become cutting-edge research hotspots [Sun Hyok Chang, Hwan Seok Chung, Nicolas K. Fontaine, Roland Ryf, Kyung Jun Park, Kwangjoon Kim, Jyung Chan Lee, Jong Hyun Lee, Byoung Yoon Kim, and Young Kie Kim, "Mode division multiplexed optical transmission enabled by all-fiber mode multiplexer," Opt. Express 22, 14229-14236 (2014); Yanlei Li, Xiao Wang, Hongjun Zheng, Xin Li, Chenglin Bai, Weisheng Hu, Yang Liu, Qiuhuan Dong, A novel six-core few-mode fiber with low loss and low crosstalk, Optical Fiber Technology, Volume 57, 2020, 102211, ISSN1068-5200, https: / / doi.org / 10.1016 / j.yoffe.2020.102211; Gao Yan, Li Yanlei, Xing Huadong, Li Xin, Zheng Hongjun*, Bai Chenglin, Hu Weisheng, Xu Hengying, Yin Yingxin, Dong Qiuhuan, Research on Mode Division Multiplexing Optical Transmission Technology, Journal of Liaocheng University (Natural Science Edition), ISSN: 1672-6634, 2022, 35(1): 30-56; Zheng Hongjun, Li Xin, Bai Chenglin, Transmission of chirped pulses in optical fibers, Beijing: Science Press, 2018, 1-184; Dong Qiuhuan, Liu Yang, Zheng Hongjun, Li Xin, Bai Chenglin, Hu Weisheng, Chen Nanguang. Research on few-mode multiplexing (demultiplexing) technology in mode division multiplexing system[J]. Journal of Liaocheng University (Natural Science Edition), 2020, 33(2): 50-67; Wang Xiao, Zheng Hongjun* (corresponding author), Li Xin, Liu Yang, Yu Ruyuan, Bai Chenglin, Hu Weisheng, New progress in the research of few-mode fibers in mode division multiplexing system, Journal of Liaocheng University (Natural Science Edition), 2019.4, 32(2): 69-79]; pure silica core can effectively reduce optical fiber attenuation and fusion loss, and is currently mostly used in single-mode optical fiber [T.Hasegawa et al.2016.Advances in ultra-low loss silica fibers[J].Frontiersin Optics,paper FTu2B.2;S.Ten.2016.Ultra Low-loss Optical Fiber Technology[J].Optical Fiber Communication Conference,paper Th4E.5;Y.Tamura,H.Sakuma,Y.Yamamoto,and T.Hasegawa,″Ultra-low loss silica core fiber for long haul transmission,″in Conference on Lasers and Electro-Optics,OSA Technical Digest(online)(Optica Publishing Group,2018),paper SF2K.3]; for optical components such as few-mode optical fiber transmission and mode division multiplexer / demultiplexer, a mode division multiplexing transmission method for suppressing mode crosstalk is proposed. [ARMay and M.N. Zervas, "Few-mode fibers with improved mode spacing," 2015 European Conference on Optical Communication (ECOC), 2015, pp.1-3, doi: 10.1109 / ECOC.2015.7341706; Tao Hu, Juhao Li, Dawei Ge, Zhongying Wu, Yu Tian, Lei Shen, Yaping Liu, Su Chen, Zhengbin Li, Yongqi He, and Zhangyuan Chen, "Weakly-coupled4-mode step-index FMF and demonstration of IM / DD MDM transmission," Opt.Express 26, 8356-8363 (2018); Jiang, Shoulin and Ma, Lin and Zhang, Zhaopeng and Xu, Xiao and Wang, Shuai and Du, Jiangbing and Yang, Chen and Tong, Weijun and He, Zuyuan, "Design and Characterization of Ring-Assisted Few-Mode Fibers for Weakly Coupled Mode-Division Multiplexing Transmission," in Journal of Lightwave Technology, col. 36, no. 23, pp. 5547-5555, 1 Dec. 1, 2018, doi: 10.1109 / JLT.2018.2874526; Dawei Ge, Yuyang Gao, Yu Yang, Lei Shen, Zhengbin Li, Zhangyuan Chen, Yongqi He, Juhao Li, "A 6-LP-mode ultralow-modal-crosshalk double-ring-core FMF for weakly-coupled MDM transmission," Optics Communications, vol. 451, pp. 97-103, Nov. 2019.]. However, crosstalk and loss between modes of these circular core few-mode fibers still exist. Crosstalk between modes requires the use of multiple-input multiple-output digital signal processing MIMO-DSP. For short-distance communication, the more modes there are, the more serious the complexity, calculation amount, and cost of MIMO-DSP. The loss of optical fiber will affect the transmission distance of the optical communication system. The loss of optical fiber is an important parameter that determines the performance of the optical fiber communication system. In order to solve the crosstalk and loss in the optical fiber, elliptical core few-mode fiber [PARMIGIANI F, JUNGY,. L,etal.Elliptical Corg Few Mode Fibers for Multiple-Input Multiple Output-FreeSpace Division Multiplexing Transmission[J].IEEE Photonics TechnologyLetters,2017,29(21):1764-1767;RIESENN,LOVEJD,ARKWRIGHTJW.Few-ModeElliptical-Core Fiber Data Transmission[J].IEEE Photonics Technology Letters,2012,24(5):344-346;IP E,MILIONE G,LI M,et al.SDM Transmission of Real-Time 10GbETrafficusing Commercial SFP+Transceivers over 0.5km Elliptical-Core Few-ModeFiber[J].Optics Express,2015,23(13):17120-17126;TAN H,ZHANGJ,LIU J,et al.Low-Loss Ring-Core Fiber Supporting 4 Mode Groups.In Conference on Lasers andElectro-Optics,OSA Technical Digest,2019,Paper SM2L.4;SONG W,CHEN H,WANG J,etal.Panda Type Elliptical Ring Core Few-Mode Fiber[J].Optical FiberTechnology,2020,60:1-5;YANG T,ZHANG H,XI L,et al.Design of a Novel Bow-TiePolarization Ring-Core Few-Mode Fiber for MIMO-Free MDM System.In 26thOptoelectronics and Communications Conference,P.Alexander Wai,H.Tam,and C.Yu,eds.,OSA Technical Digest,2021,Paper JS3C.3], elliptical ring core few-mode fiber [WANG L, NEJAD RM, CORSI A, et al. Linearly Polarized Vector Modes: Enabling MIMO-Free Mode-Division multiplexing [J]. Optics Express, 2017, 25(10): 11736-11748; WANG L, LAROCHELLE S. Design of Eight-Mode Polarization-Maintaining Few-Mode Fiber for Multiple-Input Multiple-Output-Free Spatial Division Multiplexing [J]. Optics Letters, 2015, 40(24): 5846-5849] and panda-shaped few-mode fiber [YANH, LI S, XIE Z, et al. Design of PAND Ring-Core Fiber with 10 Polarization-Maintaining Modes [J]. Photonics Research, 2017, 5(1): 1-5; CAO Y, ZHAOY, YU X, et al. al.Design and Characterization of 16-Mode PANDA Polarization-Maintaining Few-Mode Ring-Core Fiber for Spatial Division Multiplexing[J].Optical Engineering, 2017, 56:116102] was proposed and proved to have the ability to separate spatial modes and reduce mode crosstalk. .
[0003] In view of the above problems, the multiplexer combines the advantages of pure silica core and elliptical core few-mode fiber. The present invention proposes an elliptical core main channel multiplexer / demultiplexer; it breaks the mode degeneracy of LP11a / LP11b, realizes mode multiplexing and demultiplexing of three mode outputs of LP01, LP11a and LP11b, and adopts a pure silica refractive index core to achieve low loss performance; adopts a large effective refractive index difference transmission channel and reasonably sets the position and length of the transmission channel to achieve low crosstalk and high extinction ratio performance, can better separate spatial modes and reduce crosstalk between modes, can be used for MIMO-FREE operation, and is expected to solve the current research challenges of few-mode fiber multiplexers / demultiplexers, has important academic value and application value, and has great research significance and broad application prospects. Utility Model Content
[0004] With the support of the National Natural Science Foundation of China (No. 61671227 and 61431009), the Natural Science Foundation of Shandong Province (ZR2011FM015), and the special funds for the "Taishan Scholars" construction project, this patent application proposes an elliptical core main channel multiplexer / demultiplexer; the multiplexer combines the advantages of pure silica core, elliptical core and step-index distribution core, and provides important support for in-depth research in the fields of fiber optics, fiber optic communications, fiber wireless access, optical information processing and new generation information technology. The technical solution adopted by this patent application to solve its technical problems is:
[0005] This patent application proposes an elliptical core main channel multiplexer / demultiplexer, characterized in that: the mode division multiplexer / demultiplexer is composed of a few-mode fiber core and two single-mode fiber cores to form a transmission channel; the elliptical few-mode fiber core transmission channel FMF is the main channel of mode division multiplexing / demultiplexing, supports three non-degenerate modes LP01, LP11a and LP11b, is placed on the z-axis, and its axis coincides with the z-axis; the two single-mode fiber cores are respectively placed on the positive half axis of the x-axis and the positive half axis of the y-axis, and the axis of the circular single-mode fiber core SMF is parallel to the axis of the FMF starting from the y-axis; according to the coupled mode theory, the mode LP01 is transmitted from the left end to the right end along the main channel FMF to realize the multiplexing function; other light fields are respectively transmitted from the left end to the right end of the main channel FMF; The SMF corresponding to LP11a and LP11b is incident from the left end, coupled to the modes LP11a and LP11b in the main channel FMF, and output in the LP11a and LP11b modes at the right end of the main channel FMF; the mode multiplexing of the three modes LP01, LP11a and LP11b in the FMF main channel is realized; if the modes LP01, LP11a and LP11b are incident from the left end of the main channel FMF core, the coupled demultiplexing is transmitted along the z direction; according to the coupled mode theory, the mode LP01 is transmitted from the left end to the right end along the main channel FMF to realize demultiplexing; the modes LP11a and LP11b are incident from the left end of the main channel FMF, respectively, and the coupled demultiplexing is realized. The right end output of the SMF corresponding to the modes LP11a and LP11b is combined to realize the demultiplexing of the three modes LP01, LP11a and LP11b from the FMF main channel mode; according to the relationship between the mode coupling length and the channel spacing, the channel center spacing from the SMF of the positive semi-axis of the x-axis to the main channel FMF is 15.8μm, the channel length is 8.4mm, and the diameter is 10.8μm; the channel center spacing from the SMF of the positive semi-axis of the y-axis to the main channel FMF is 16μm, the channel length is 6.23mm, and the diameter is 10.8μm; the ellipticity of the elliptical core few-mode fiber FMF is 1.5, the major semi-axis radius is 5.4μm, and the minor semi-axis radius is 3.6μm; length 8.4min; the performance of the mode multiplexer can be adjusted by reasonably setting the position and length of the transmission channel; the transmission channel consists of a core and a cladding; the cladding part is made of fluorine-doped silica material with a refractive index of n2=1.4320; the core part is made of pure silica material with a refractive index of n1=1.4440; the refractive index of the SMF core located at the positive half of the x-axis is 1.4385, and the refractive index of the SMF core located at the positive half of the y-axis is 1.4359; the mode field characteristics in the optical fiber can be changed by changing the size and refractive index distribution of the core and cladding; SMF constitutes the coupling channel of the multiplexer / demultiplexer, and the refractive index of the core part is higher than the surrounding refractive index.
[0006] The beneficial effects of this patent application are as follows:
[0007] 1. The multiplexer is composed of directional mode selective couplers. Each mode in the few-mode fiber core has a mode-preserving function. The large effective refractive index difference between the modes ensures low crosstalk between the modes. The refractive index core of pure silica is used to achieve low loss performance.
[0008] 2. The optical fiber used in the multiplexer combines the advantages of pure silica core, step refractive index distribution and elliptical core few-mode core, providing important support for in-depth research in the fields of fiber optics, fiber-optic communications, fiber-optic wireless access and optical information processing, and new generation information technology.
[0009] 3. The mode multiplexing and demultiplexing of three non-degenerate mode outputs, LP01, LP11a and LP11b, are realized, which can be used for MIMO-FREE operation. BRIEF DESCRIPTION OF THE DRAWINGS
[0010] Figure 1 It is a cross-sectional schematic diagram of an elliptical core main channel multiplexer / demultiplexer of the present application, wherein the elliptical core at the coordinate center serves as the main channel for transmission; the SMF fiber core located on the X-coordinate axis corresponds to the coupling mode LP11a; and the SMF fiber core located on the Y-coordinate axis corresponds to the coupling mode LP11b.
[0011] Figure 2 It is a three-dimensional stereogram of an elliptical core main channel multiplexer / demultiplexer of the present application.
[0012] Figure 3 The mode field distribution diagrams of the LP01, LP11a and LP11b modes of the elliptical core fiber at a wavelength of 1.55μm are given.
[0013] Figure 4 The effective refractive index difference of each mode of elliptical core fiber at 1.530μm-1.565μm is given. The circle, asterisk and triangle represent the effective refractive index difference n between LP01 mode and LP11a mode respectively. LP01 -n LP11a、 The effective refractive index difference n between LP11a mode and LP11b mode LP11a -n LP11b and the effective refractive index difference n between the LP11b mode and the cladding LP11b -n Cladding .
[0014] Figure 5 The variation of the coupling efficiency of each spatial mode of the multiplexer / demultiplexer with the incident wavelength in the C band is given. The coupling efficiency of the LP01, LP11a and LP11b modes are represented by the curves of circles, dots and stars respectively.
[0015] Figure 6The variation of the mode extinction ratio of the multiplexer / demultiplexer in the mode channel with the wavelength of the incident light is given. The extinction ratios of the LP01, LP11a and LP11b modes are represented by the curves of circles, dots and stars respectively.
[0016] Figure 7 The intrinsic loss of the main channel of the multiplexer / demultiplexer is given, and the intrinsic losses of the LP01, LP11a and LP11b modes are represented by curves with circles, dots and stars, respectively. DETAILED DESCRIPTION
[0017] The technical solution of the present patent application is described in detail below in conjunction with the embodiments and drawings, but the protection scope is not limited thereto.
[0018] Example 1 Figure 1 This is a cross-sectional schematic diagram of an elliptical core main channel multiplexer / demultiplexer of the present patent application, wherein the elliptical core at the coordinate center serves as the main channel for transmission; the SMF core located on the X-coordinate axis corresponds to the coupling mode LP11a; the SMF core located on the Y-coordinate axis corresponds to the coupling mode LP11b; the three cores are parallel along the propagation Z direction; the major semi-axis diameter of the FMF is d=10.8μm, the minor semi-axis diameter of the FMF is d=7.2μm, and the SMF core corresponding to the positive semi-axis of the X-axis is d=10.8μm. The core distance of the FMF core is 15.8μm, and the core distance of the SMF at the positive half axis of the Y axis is 16μm; except for the core, the rest is the cladding; the refractive indices of the FMF core, cladding, the SMF core at the positive half axis of the X axis, and the SMF core at the positive half axis of the Y axis are n1=1.4440 and n2=1.4320 respectively; all cores are step refractive index distribution; the refractive index of the SMF core at the positive half axis of the x axis is 1.4385, and the refractive index of the SMF core at the positive half axis of the y axis is 1.4359;.
[0019] Figure 2 This is a three-dimensional diagram of an elliptical core main channel multiplexer / demultiplexer of this patent. The core length of the SMF of the positive half axis of the X axis of the coupling channel is L1 = 8.4mm, and the core length of the SMF of the positive half axis of the Y axis of the coupling channel is L2 = 6.23mm. Since the FMF is to be connected to the transmission optical fiber in the transmission link, the length must be at least greater than or equal to the length of the SMF 8.4mm, and the FMF can be lengthened according to actual conditions.
[0020] Figure 3 The mode field distribution diagrams of the LP01, LP11a and LP11b modes of the elliptical core few-mode fiber at a wavelength of 1.55μm are given, and the modes are obtained by the beam propagation method. Each mode has a mode-preserving function.
[0021] Figure 4The effective refractive index difference of each mode of the ring core fiber at 1.530μm-1.565μm is given, where the circle, star and triangle represent the effective refractive index difference n between the LP01 mode and the LP11a mode, respectively. LP01 -n LP11a、 The effective refractive index difference n between LP11a mode and LP11b mode LP11a -n LP11b and the effective refractive index difference n between the LP11b mode and the cladding LP11b -n Cladding .Depend on Figure 4 It can be seen that in the wavelength range of 1.530μm-1.565μm, the effective refractive index difference n between the LP01 mode and the LP11a mode is LP01 -n LP11a Greater than 3.951×10 -3 As the wavelength increases, the refractive index difference gradually increases. The effective refractive index difference between the LP11a mode and the LP11b mode is n LP11a -n LP11b Greater than 2.235×10 -3 , the refractive index difference gradually increases with the increase of wavelength; the effective refractive index difference between the LP11b mode and the cladding n LP11b -n Cladding Greater than 1.901×10 -3 , the refractive index difference decreases gradually with the increase of wavelength; n LP01 -n LP11a At 1.54 μm and 1.55 μm, they are 3.985×10 -3 4.019×10 -3 , n LP11a -n LP11b At 1.54 μm and 1.55 μm, they are 2.241×10 -3 , 2.245×10 -3 , n LP11b -n Cladding At 1.54 μm and 1.55 μm, they are 2.083×10 -3 , 2.01×10 -3 .
[0022] Figure 5The variation of the coupling efficiency of each spatial mode of the multiplexer / demultiplexer with the incident wavelength in the C-band is given. The coupling efficiency of the LP01, LP11a and LP11b modes are represented by the curves of circles, dots and stars, respectively. The coupling efficiency of the LP01 mode is about 0dB in the entire C-band; the coupling efficiency of the modes LP11a and LP11b shows a decreasing trend in the C-band on both sides of 1.55μm, and reaches the maximum value at 1.55μm, which is -0.0041dB and -0.0118dB respectively. In the entire C-band, the coupling efficiency of LP11a is higher than -5.6997dB; the coupling efficiency of LP11b is higher than -4.5308dB.
[0023] Figure 6 The variation of the mode extinction ratio of the multiplexer / demultiplexer in the mode channel with the wavelength of the incident light is given. The extinction ratios of the LP01, LP11a and LP11b modes are represented by the curves of circles, dots and stars, respectively. It can be seen from the figure that in the entire C band, the extinction ratio of the LP01 mode decreases with the increase of the incident wavelength, and the LP11a and LP11b modes both show a decreasing trend on both sides of 1.55μm; the extinction ratio of LP01 can be maintained above 58.188dB, and 1.53 The maximum value of LP11a is 33.65dB at 1.55μm, and the maximum value of LP11b is 30.19dB at 1.55μm.
[0024] Figure 7 The intrinsic loss of the main channel of the multiplexer / demultiplexer is given, and the intrinsic losses of the LP01, LP11a and LP11b modes are represented by circle, dot and star curves respectively. The intrinsic losses of the three modes show a trend of first decreasing and then increasing in the C band. Overall, the intrinsic losses of the three modes are LP11b, LP11a and LP01 from large to small; the intrinsic loss of mode LP01 can be maintained above 0.1474dB / km, and the maximum value is 0.1507dB / km at 1565nm; the intrinsic loss of mode LP11a can be maintained above 0.1524dB / km, and the maximum value is 0.1556dB / km at 1565nm; the intrinsic loss of mode LP11b can be maintained above 0.1592dB / km, and the maximum value is 0.1623dB / km at 1565nm.
[0025] In summary, the proposed multiplexer / demultiplexer realizes mode multiplexing and demultiplexing of three non-degenerate mode outputs, LP01, LP11a and LP11b, and can be used for MIMO-FREE operation.
Claims
1. An elliptical core main channel multiplexer / demultiplexer, characterized in that: The multiplexer / demultiplexer consists of an elliptical few-mode fiber core and two circular single-mode fiber core SMF transmission channels; the few-mode fiber core transmission channel FMF is the main channel of the mode division multiplexer / demultiplexer, supporting three modes LP01, LP11a and LP11b, and is placed on the z-axis with its axis coinciding with the z-axis; the two single-mode fiber cores are placed on the positive half axis of the x-axis and the positive half axis of the y-axis respectively, and the axis of the SMF starts from the y-axis and is parallel to the axis of the FMF; according to the coupled mode theory, the mode LP01 is along the The main channel FMF is transmitted from the left end to the right end to realize the multiplexing function; other light fields are incident from the left end of the SMF corresponding to LP11a and LP11b, respectively, and are coupled to the modes LP11a and LP11b in the main channel FMF, respectively, and output in the LP11a and LP11b modes at the right end of the main channel FMF; the mode multiplexing of the three modes LP01, LP11a and LP11b in the FMF main channel is realized; if the modes LP01, LP11a and LP11b are transmitted from the main channel FMF to the right end, the three modes LP01, LP11a and LP11b are output in the LP11a and LP11b modes. The left end of the channel FMF core is incident, and the transmission coupling demultiplexing is carried out along the z direction; according to the coupled mode theory, the mode LP01 is transmitted from the left end to the right end along the main channel FMF to realize demultiplexing; the modes LP11a and LP11b are incident from the left end of the main channel FMF, and are respectively transmitted and coupled to the right end output of the SMF corresponding to the modes LP11a and LP11b, realizing the demultiplexing of the three modes LP01, LP11a and LP11b from the FMF main channel mode; according to the relationship between the mode coupling length and the channel spacing, the channel center spacing from the SMF of the positive semi-axis of the x-axis to the main channel FMF is 15.8μm, the channel length is 8.4mm, and the diameter is 10.8μm; the channel center spacing from the SMF of the positive semi-axis of the y-axis to the main channel FMF is 16μm, the channel length is 6.23mm, and the diameter is 10.8μm; the ellipticity of the elliptical core few-mode fiber FMF is 1.5, the major semi-axis radius is 5.4μm, and the minor semi-axis radius is 3.6μm; The length is 8.4mm; the performance of the mode multiplexer can be adjusted by reasonably setting the position and length of the transmission channel; the transmission channel is composed of a core and a cladding; the cladding part is made of fluorine-doped silica material with a refractive index of n2=1.432; the core part is made of pure silica material with a refractive index of n1=1.4440; the refractive index of the SMF core located at the positive half axis of the x-axis is 1.4385, and the refractive index of the SMF core located at the positive half axis of the y-axis is 1.4359; the refractive index of the core part is higher than the surrounding refractive index; the mode field characteristics in the optical fiber can be changed by changing the size and refractive index distribution of the core and cladding; the multiplexer combines the advantages of pure silica core and elliptical core few-mode core, breaks the degenerate mode LP11a / LP11b, and realizes the mode multiplexing and demultiplexing of three non-degenerate modes of LP01, LP11a and LP11b.