## 1 Introduction  
Since the 1990s, when the concept of orbital angular momentum (OAM) was proposed by [PERSON] [1], structured beams carrying OAM have been widely used in optical fields. Instances are optical manipulation [2], imaging [3], and optical communication [4; 5]. Furthermore, in the particular case of a MIMO system, beams with different OAM modes can potentially increase the capacity of radio communication systems under certain particular conditions [6; 7]. Therefore, there have been many studies on the application of vortex beams in antenna design [8; 9], radar imaging [10] and target detection [11]. It was shown that the transmission performance of vortex beams is better than that of plane waves [12] and the information carried by vortex beams is more easily obtained by the receiver [13]. This is mainly due to the scattering and transmission characteristics of typical vortex beams. Techniques based on optics are often used for detecting the physical and chemical properties of substances [14]. The spiral spectrum of the reflected or transmitted field of the vortex beam obtained at the receiving end can be also used for inferring and inversing the information of the transmission channel or target. Most of the current research on the spiral spectrum characteristics of the OAM beams are focused on the macro transmission channels, such as turbulent atmosphere [15; 16; 17; 18; 19; 20] and underwater environment [21; 22; 23; 24; 25], which are related to the modelling of random media. There are, however, few studies on the scattering characteristics and OAM spectrum of vortex beams irradiating a specific target. In 2012, [PERSON] et al. [26] obtained the spiral spectra scattered off a transparent sphere illuminated by a Laguerre-Gaussian beam. They showed that the geometric properties of the sphere, e.g., the position, can be determined based on the scattering spiral spectrum data. Furthermore, the intensity distribution and OAM mode distortion of a Laguerre-Gaussian vortex beam reflected and transmitted by a non-uniform plasma plate were also examined by [PERSON] [27]. Later in 2020, [PERSON] [28] used the physical optical algorithm to investigate the backscattering characteristics of an ideal conductive sphere and a PEC cone illuminated by a vortex beam. They then used phase sampling along the circumference of the scattered field profile and concluded that the scattering field of a symmetric object remains a vortex field, and the topological charge is the same as that of the incident field. Using the spiral spectrum expansion method, some of the above works take samples along the circumference of the field profile, whereas others sample the whole profile. In this paper, we investigate these two sampling approaches and compare their computational differences and their pros and cons. Amongst many existing types of vortex beams, the high-order Bessel beam is often used. This is due to the properties, including non-diffraction [29; 30] and self-healing [31; 32]. The energy of a high-order Bessel beam is, however, infinite in radial distribution [33], hence unattainable in practice. To address this issue, instead of the high-order Bessel beam, an approximated non-diffraction beam, namely, the high-order Bessel-Gaussian beam, is used. Characteristics of the Bessel-Gaussian vortex beams, including transmission in turbulent media [34; 35; 36], self-healing [37; 38], and tight focusing [39] are discussed. Nevertheless, to the best of our knowledge, the interactions between vector high-order Bessel-Gaussian beams and small particles have not been investigated. In our previous works, we derived the beam shape coefficients of polarized Bessel-Gaussian vortex beams [40]. Further, in [41], we studied the scattering of vector Bessel-Gaussian vortex beams caused by spherical particles with non-uniform structure and electric charges. In this paper, we further investigate transmission information of the vector Bessel-Gaussian vortex beam and explore a wider range of potential applications.