Bright Ring Lattice Generated by Nesting Spiral-Pinhole-Arrays Plate

: A nesting spiral-pinhole-arrays plate (NSP) is designed to produce bright ring lattice (BRL) formed by the superposition of two different orbital angular momentum (OAM) modes. The OAM modes of the BRL can be manipulated by the propagation distance and the spiral arrays of the NSP. We have studied the influences of rotating NSP on the BRL, and found that the rotational angle and direction of the BRL are not consistent with the results of the NSP.


Introduction
Optical vortex with a helical phase front of ( ) carry a well-defined orbital angular momentum (OAM), where is known as topological charge (TC) and φ is azimuthal angle in the cylindrical coordinate system [1]. The application about OAM beams have been studied in many field, such as micromanipulations [2], optical pump [3], communications [4][5], detection [6]. In the past 20 years, researchers have proposed many methods to generate OAM beams, for instance spiral phase plates [7], computer-generated holograms [8], Q-plate [9], metasurface [10][11][12][13], coordinate transformation [14] and spiral pinhole plates [15][16]. The OAM beams made by the above methods are simple to manipulate a single OAM mode or simple superposition of several OAM modes, while, the OAM-based applications are not limited to a single mode in some occasions [17][18][19]. In 2019, Yang et al. [20] proposed a binary mask to generate photonic gears with a superposition of two OAM modes with opposite sign.
In this paper, we presented a NSP constructed by the inner and outer Fermat spiral pinholes (FSP) [21] arrays to generate the superposition of two different OAM modes, namely, bright ring lattice (BRL) [22], whose OAM modes can be controlled by adjusting the structure of NSP and the propagation distance. In addition, the expression for the rotational angle of BRL is derived.
Numerical calculations are performed to analyze the influences of OAM modes of NSP and rotating NSP on the BRL. It is found that the BRL does not keep pace with the NSP, which provides a scheme for increasing the accuracy of micromanipulation [23].

The generation of BRL
FSP is selected as the basic unit of the diffractive plate. As shown in Figure 1(a), a single counterclockwise FSP is given. The azimuthal angle and radial coordinates of the nth pinhole of FSP are expressed as = 2 ⁄ , = ( 0 + / ) 1 2 ⁄ [21], where is the numbers of pinholes, z is the propagation distance and 0 , denote the initial radius, central topological and wavelength, respectively. The optical field generated by a single FSP includes wide OAM spectrum centered at central topological value [20] and the complex amplitude of the field reads as ( ) is amplitude corresponding to index . These OAM modes can be refined to obtain some special modes by adding several identical FSPs equally spaced in the azimuthal angular domain to construct FSP arrays (FSPA). As shown in Figure 1(b), a FSPA containing 8 arms is given.
Considering that the FSPA containing arms is illuminated vertically by monochromatic plane wave (All the beams used in this paper are monochromatic plane waves, and it will not be described later ), λ = 532nm, the complex amplitude of the diffractive field is The OAM mode in E( , ) can be expressed as From the equation (3), we know ( , ) = ( ) for = and ( , ) = 0 for ≠ , where is an integer. It means that some special OAM modes with a multiple of remain and other modes vanish. Significantly, the proportion of central OAM mode is absolutely dominant among these special modes [21].
In order to generate BRL, two different FSPA are nested together. As shown in Figure 2 field-intensity pattern of 9 bright petals with , = −3 and , = 6 is given in Figure 3(c).

The influences of rotating NSP on the BRL
Considering diffractive field of NSP, the complex amplitude generated by the inner and outer It can be seen from equation (4) that the rotational angle of BRL is Equation (5) shows that ∆ is determined simultaneously by , , and , . It means that the rotational angle and direction of BRL are different from that of the inner FSPA. this effect can be used in the field of high-accuracy manipulations [22].

Conclusions
In this paper, we have designed NSP to generate produce BRL with controllable OAM modes by adjusting the structure of the NSP and propagation distance. Furthermore, the expression for the rotational angle of BRL is given, and the effects of rotating NSP on the BRL are studied in detailed.
It is found that the BRL does not keep pace with the NSP, where the rotational direction and angle of the BRL are also related to the construction of NSP. The results of this work may provide a scheme for increasing the accuracy of micromanipulation.