## 1 Introduction  
CVBs have axisymmetric intensity and polarization distribution, and under strong focusing conditions, a unique and strong local longitudinal electric field is formed in any plane parallel to the optical axis [1, 2]. Relying on this feature, in terms of laser cutting, smaller and stronger light spots can be generated after CVB focusing; thus, the cutting depth-to-width ratio can be significantly improved using CVB cutting, and the cutting speed can be increased to 1.5-2 times that of conventional cutting [3, 4]. CVB cutting has a wide range of applications in thick-plate cutting. In terms of laser drilling, CVB drilling can obtain straight holes with favorable height-width-depth ratios and a high punching rate [5, 6]. Therefore, CVB processing is an effective scheme to improve processing quality and efficiency. In addition, under strong focusing conditions, CVB has a small focusing spot, and its application in high-resolution optical imaging system can greatly improve system resolution [7, 8, 9]. The TM\({}_{01}\) mode generates a large gradient and light field scattering force; therefore, it has a stronger trapping ability than Gaussian light, and can be used for particle trapping [10, 11] and particle acceleration [12, 13]. In the environment constituting strong scattering in water, CVB has a higher transmittance than traditional Gaussian light and can be used for wireless optical communication [14, 15, 16]. Initially, research related to CVBs has mainly focused on solid-state lasers. However, solid-state lasers have serious thermal effects and poor system reliability, coupled with the insertion of specially designed conical elements [17], birefringence elements [18], interference elements [19], sub-wavelength gratings [20], and other spatial devices, which seriously limit the capabilities of CVBs in practical applications. Compared with solid-state lasers, all-fiber lasers are convenient for thermal management and have a flexible structure, high beam quality, and high stability; therefore, all-fiber lasers are likely to become the main development direction for high-power CVBs. After nearly 10 years of development, the realization of mode conversion mainly includes bias-core coupling [21; 22; 23; 24], long-period grating [25], special optical fibers [26; 27; 28], spatial Q-wave chips [29], mode selection couplers [30; 31; 32; 33], etc., among which the use of mode selection couplers is the simplest way to achieve mode conversion and can obtain high mode purity, while offering low production costs, strong stability, and adaptability to complex engineering application environments. To date, a number of institutions have carried out research on asymmetric structural mode selection couplers. In 2014, [PERSON] et al. of the University of Southampton [30] made an asymmetric mode selection coupler, which realized the conversion of an LP\({}_{01}\) (linear polarized) mode to an LP\({}_{11}\) mode when the transmission wavelength was 1550 nm, with a coupling efficiency of 91%, a mode purity of 92%, and an insertion loss of 0.5 dB. In 2017, [PERSON] et al. of Nanjing University of Posts and Telecommunications [31] made an asymmetric mode selection coupler and built a figure-8-shaped mode-locked resonator, obtaining an LP\({}_{11}\) mode output with a central wavelength of 1556.3 nm, and a pulse width of 17 ns, with a mode purity of 94.2% and an insertion loss of 0.65 dB. In 2018, [PERSON] et al. of Shanghai University [32] fabricated an asymmetric mode selection coupler, and built a passive mode-locked fiber laser, obtaining an LP\({}_{11}\) mode output with an average power of 75 mW, a coupling efficiency of 89%, and a central wavelength of 1043 nm. In 2018, [PERSON] et al. of Northwestern Polytechnical University [33] made a 1550 nm band asymmetric mode selection coupler, and built a ring-cavity passive mode-locked laser, which realized an LP\({}_{11}\) mode output with pulse width adjustable from 39.2/31.9 ps to 5.6/5.2 ps. In 2019, [PERSON] et al.