## 1 Introduction  
Cylindrical vector beams with cylindrical symmetry in both amplitude and polarization [1], as a special class of beams different from Laguerre-Gaussian and Bessel beams, have drawn considerable attention recently. Due to their unique characteristics, they have been applied in many novel ways, including lithography, particle acceleration, material processing, high-resolution metrology, atom guiding, optical trapping and manipulation [2; 3; 4; 5]. Until now, a variety of methods for generating CVBs have been reported, which are categorized as passive or active depending on whether amplifying media is used. Converting an incident Gaussian beam to a radially polarized beam using a radial polarizer [6] is the simplest passive method. However, this method does not produce very high purity transverse modes. Generally, devices with spatially variant polarization properties, such as axial birefringence and dichroism, can be used to generate CVBs outside the laser cavity. Active methods involve the use of laser intracavity devices, such as an axial birefringent component or an axial dichroic component, to provide mode discrimination and force the laser to oscillate in the CVB modes against the fundamental mode. However, the extracavity or intracavity involvement of devices with axial birefringence, dichroism or spatially variant polarization properties [7; 8] increases the complexity of the system and the laser threshold. The generation of CVBs based on optical fiber is another technique that deserves special attention, due to its excellent performance, such as compactness, excellent flexibility, and high efficiency. [PERSON] et al. [9] has presented two mode converters based on photonic crystal fibers (PCFs) by controlling hole collapse in PCFs or using the ferrule technique to join conventional single-mode fibers to PCFs without splicing, by which mode conversion with high extinction ratio has been realized. However, the used fabrication process was relatively complicated. Recently, the generation of CVBs using few-mode fiber (FMF) or other specially designed fiber attracts increasing interest due to the flexibility of these systems [10]. For example, the two-mode fiber (TMF) can guide the fundamental mode and the first group higher-order modes in the scalar approximation. As the radially and azimuthally polarized beams are the eigenmodes TM01 and TE01, respectively, CVBs can be generated by exciting these modes in TMFs. Fiber lasers with an FMF base have been utilized to generated both continuous wave (CW) and pulsed (Q-switched and mode-locked) CVBs [11; 12; 13]. Regarding mode-locking methods for generating mode-locked CVB, carbon nanotubes (CNTs) and graphene have emerged as promising saturable absorbers [14; 15; 16] with an ultrafast recovery time [17; 18], able to support short pulses [19; 20], and with a number of favorable properties for laser development, such as broadband operation [21], and integration into all-fiber configurations [22; 23]. Moreover, there are few works on carbon nanotube mode-locked CVB fiber lasers.  
In this paper, we propose and demonstrate a carbon nanotube mode-locked CVB all-fiber laser. The lateral offset splicing technique and TM-FBG are utilized to excite and extract CVBs, while the carbon nanotube saturable absorber (CNT-SA) is used to mode-lock the fiber laser. Due to the CNT-SA, the fiber laser can operate with a low threshold. Both radially and azimuthally polarized beams can be obtained from the fiber laser. The polarized states also can be switched through adjusting the polarization controllers. This all-fiber-based CVB pulse laser with a repetition rate of 10.8 MHz can provide a cost-effective CVB source for practical applications.