Generation of Non-diffraction Vortex Beam and Its Application in Digital Communication

—An axicon-frustum basin-type Bessel resonator is designed for generating non-diffraction vortex beams with opposite propagation directions under different conditions. Based on the theory of geometric optics, the principle of creating vortex beams is analyzed in this axicon-frustum basin-shaped resonant cavity. It is shown that a non-diffraction vortex beam can be produced in the cavity under high-order beam stimulation. In the terahertz wavebands, the dyadic Green's function algorithm is used to numerically evaluate the electromagnetic characteristics in the cavity, and the numerical results demonstrate that the conclusion of the principle analysis is valid. Finally, the application of non-diffraction vortex beams in digital communication is explored.

the fields of optical computing [10], atomic cooling [11], biomedicine [12], quantum communication [13], and biological science. However, the study of vortex beam in microwave and millimeter wave bands lags far behind that of optical vortices [14], [15]. Until 2007, no one other than Thidé [16] had employed the vortex beam characteristics of orbital angular momentum (OAM) to improve the capacity of communication systems. After that, people began to get involved in this field. The vortex beam in microwave and millimeter wave bands is mainly generated by antenna arrays [17], [18], [19], helical surfaces [20], [21], and transmission grating structures [22]. This electromagnetic vortex wave with OAM provides a new degree of freedom that can be used for multiplexing technology. The value of its topological charge is unlimited in theory and its OAM is orthogonal each other, which can be widely used in wireless communication [23], [24] to expand the communication capacity. Therefore, in the present paper we reconstruct an axicon-frustum basin-shaped resonant cavity based on the original resonator. Numerical results demonstrate that the designed cavity can produce a nondiffraction vortex beam at THz band under high-order beam stimulation.

II. DESIGN OF RESONATOR
Three types of Bessel resonators [25] have been built and another type of resonator has been rebuilt in our previous work [26]. Now, an axicon-frustum basin mirror (face mirror) is added to the axiconfrustum resonator [26] to form an axicon-frustum basin-shaped resonator. As shown in Fig. 1, the cavity length of the built-in Bessel resonator satisfies the following condition: where L is the cavity length, q is a positive integer, and q  is the resonant wavelength. Fig. 1 shows an axicon-frustum basin-type Bessel resonator constructed by mirror M1 and mirror M2, where M1 is an axicon-frustum basin-shaped mirror and M2 is a face mirror with radius 2 A . Fig. 1 Bessel beam is also formed, but in the opposite direction of Fig. 1(a). If (1) and (2) are simultaneously satisfied, the cavity will oscillate to forming standing waves. If antinodes are selected on the mirror surface, a Bessel beam will appear. When a partial reflection film is plated on one mirror and a total reflection film is plated on another mirror, a coupled output beam can be obtained. Fig. 1(c) shows the simulation model of the axicon-frustum basin-shaped resonator.

III. THEORY ANALYSIS
Suppose there is a vortex wave beam propagating along the z-axis through the spiral phase plate, satisfying: Using the Fresnel-Kirchhoff diffraction integral formula under cylindrical coordinates, the field after the axicon is [27], [28]: is the wave number in free space,  is the radial coordinate, R is the aperture radius of the axicon, n J is the Bessel function of the nth order, and its order is generally selected the same as the topological charge number, namely n = l . The penetrating rate () t  is written as where  represents refraction vertex angle (see Fig. 2), and arcsin( sin ) (3) and (5) into (4) yields: The vortex beam is confined in the non-diffractive region, as defined by the collimation max z :  Step 1: assuming that there exist an initial vortex excitation on mirror M1; Step 2: when this excitation transits forth from mirror M1 to mirror M2, the intensity distribution on the mirror M2 can be evaluated by using Eqs. (10) and (11); Step 3: then the beam on the M2 transits back from M2 to M1, we can also obtain the intensity distribution on the M1 in the same way; Step 4: repeat steps 2 and 3, until the electromagnetic field in the resonator reaches to the steady state.
Step 5: The stable fields on the mirror are normalized and depicted.
As example, we present the simulation example at terahertz frequency band. The initial excitation is    From this example, we can draw the following conclusions: Firstly, the oscillation can be excited to generate Bessel vortex beam in the axicon-frustum basin-type resonator at terahertz waveband.
Second, the vertex angle  of an axicon is in proportion to the number of topological charge l , that is, the larger l generally leads to the larger  . Third, two output beams on mirrors M1 and M2 are in the contrary directions due to the opposite directions of two mirrors. When the topological charge is 1 = l , two beams are generated and their phase difference is 180  ; when the topological charge is  In fact, the vortex beam has another feature of a fixed phase difference. This can be explored to apply in digital phase shift keying for digital communication. For instance, four different carrier phases are used to represent digital information in quaternary phase shift communication. Each carrier phase represents two bits information, as given in Table1.   TABLE I

 =
), corresponding to two-vortex beams and four-vortex beams, respectively. The Bessel vortex beam generated in this paper may find application in the digital communication of PSK and its spatial multiplexing.