A design method for cycloid optical vortex array mask

Abstract

A method for designing a cycloidal optical vortex array mask comprises the following steps: obtaining an ideal Bessel beam in cylindrical coordinates; generating a cycloidal optical vortex array based on the obtained Bessel beam and applying the Fourier transform displacement theorem to obtain its complex transmittance function; and finally, a mask described by this complex transmittance function is the cycloidal optical vortex array mask. This invention utilizes the principles of computational holography and phase mask technology to generate a functional optical vortex array mask through computer encoding, and the array is obtained in the far field. Due to its advantages of local dynamic control and functionality, it has important application value in the field of micromanipulation.

Description

Technical Field

[0001] The present invention relates to the field of micromanipulation and microstructure, and in particular to a design method for generating a cycloid optical vortex array mask with functionality and local dynamics based on a cycloid equation of a Fourier phase shift method and a coordinate positioning method. Background Art

[0002] In the 30 years since Allen discovered that optical vortices possess orbital angular momentum, they have attracted widespread attention due to their diverse applications, including optical tweezers and optical spanners, spatial mode multiplexing in optical communications, optical sensing and metrology, and high-dimensional quantum information protocols. However, since the information contained in a single optical vortex is limited, optical vortex arrays have been explored due to their greater flexibility and broader application range.

[0003] In terms of methods for generating optical vortex arrays, the main ones include Dammann gratings, multi-beam interferometry, special microstructured materials, and widely used spatial light modulators. Based on Dammann gratings, square optical vortex arrays containing different diffraction orders can be generated simultaneously in the far field [Opt. Lett. 40, 2513 (2015)]. Subsequently, other optical vortex arrays with different structures appeared, such as hexagonal [ACS Photonics 4, 1333 (2017)], triangular [Photon. Res. 6, 641 (2018)], compact [Opt. Express 26, 22965 (2018)], elliptical [Appl. Phys. Lett. 116, (2020)], "bear" [Opt. Express 29, 10811 (2021)], and arbitrary curved path optical vortex arrays [Opt. Express 26, 9798 (2018)]. However, a significant amount of research has focused on structural modulation of optical vortex arrays. Applications of optical vortex arrays for microparticle manipulation are limited. Examples include micro-optomechanical pumps [Opt. Express 12, 1144 (2004)], optical sorting using fractional topological charge optical vortex arrays [Opt. Commun. 283, 1889 (2010)], trapping polystyrene (PS) microspheres [Front. Phys. 9, (2021)], and controlling the growth direction of axons [Nat. Photonics 6, 62 (2012)]. However, these optical vortex arrays were generated out of curiosity and creativity, rather than application requirements. This significantly limits the array's scope of application. Consequently, a functional array structure is lacking. This would further expand the array's application range.

[0004] In summary, the field of micromanipulation still lacks a functional array and a method for generating an array that can be locally dynamically adjusted. Summary of the Invention

[0005] To address the above-mentioned deficiencies, the purpose of the present invention is to provide a method for designing a cycloidal optical vortex array mask. The mask designed by this scheme can produce a functional optical vortex array, whose array structure, vortex radius, array spacing, vortex number, and vortex position are all controllable, and has very important application value in the field of micro-manipulation.

[0006] The technical solution adopted by the present invention is: a design method of a cycloid optical vortex array mask, the steps are as follows:

[0007] S1, in cylindrical coordinates The ideal Bessel beam is expressed as:

[0008]

[0009] Among them, J l is the first-kind Bessel function of order l; k r and k z They are the wave vector components in the radial and optical axis directions, and their relationship with the wave number is: k = (k r 2 +k z 2 ) 1 / 2 =2π / λ, where λ is the wavelength of the electromagnetic radiation forming the Bessel beam;

[0010] S2. Based on the Bessel beam obtained in step S1, a cycloid optical vortex array is generated according to the displacement theorem of Fourier transform. The specific expression of its complex transmittance function is:

[0011]

[0012] Where n′ is the refractive index of the axicon, α is the cone angle, N is the total number of optical vortices, and L n,1 and Ln ,2 are the position matrices of the nth optical vortex respectively;

[0013] S3. The mask described based on the complex transmittance function is the cycloid optical vortex array mask.

[0014] As a preferred embodiment, in the step S2:

[0015] The Bessel beam obtained in step S1 is passed through a convex lens with a focal length of f. The light fields before and after the transformation are and E(r,θ), the Fourier transform formula in the cylindrical coordinate system is:

[0016]

[0017] According to the displacement theorem of Fourier transform, an optical vortex is obtained at a specified position, and then a cycloid optical vortex array is generated.

[0018] As a preferred solution, the parametric equation of the cycloid optical vortex array is expressed as:

[0019]

[0020] Where n is the number of optical vortices; D determines the size of the array; a1 and a2 are the stretching factors of the array in the horizontal and vertical directions; b is the curvature of the cycloid. When the sign before b is "+", the above parametric equation is the hypocycloid equation, and c+1 is the number of peaks in the array; when the sign before b is "-", the above parametric equation is the epicycloid equation, and c-1 is the number of peaks in the array; the real number value of the parameter Q = (D, a1, a2, b, c) determines the cycloid array structure, which can be controlled to obtain a functional optical vortex array.

[0021] This solution also includes the application of the cycloid optical vortex array mask generated by the above-mentioned design method. In the experiment, a parallel light beam is irradiated on a spatial light modulator loaded with a structure-controllable perfect optical vortex array mask. The reflected light beam modulated by the spatial light modulator passes through a convex lens to obtain an optical gear composed of two different structural arrays in the far field.

[0022] Technical effects of the present invention:

[0023] The mask designed in this invention can produce a functional optical vortex array with controllable array structure, vortex radius, array spacing, vortex number, and vortex position. Two arrays with different structures can also be combined to create a structure similar to an optical gear. Compared with traditional generation methods, this method is simpler and more functional. Therefore, it has significant application prospects in the field of micromanipulation. BRIEF DESCRIPTION OF THE DRAWINGS

[0024] Figure 1 This is a cycloid optical vortex array mask with controllable structure and flexible diversity produced by the present invention. From left to right are the mask plates of the heart line, kidney line, tricuspid line, and star line. When constructing the heart line array: a = 2.5, b = 1.2, c = 2, D = 1.2; when constructing the kidney line array: a = 2.5, b = 0.9, c = 3, D = 1.2; when constructing the tricuspid line array: a = 2.5, b = 1.1, c = 2, D = 1.2; when constructing the star line array: a = 2.5, b = 1.0, c = 3, D = 1.2;

[0025] Figure 2 yes Figure 1 The displayed masks generate cycloidal optical vortex arrays with different structures;

[0026] Figure 3 It is an optical gear with a functional structure. DETAILED DESCRIPTION

[0027] To overcome the technical problems described in the background, we need to find a functional structure and arrangement. The cycloid is a common curve in mathematics. Gears are ubiquitous in mechanical engineering. They are one of the most important functional structures in physics. Based on the Fourier phase shift method and coordinate positioning technology, a new design method for cycloid optical vortex array masks has been developed. This mask can produce a functional optical vortex array, which has very important application value in the field of micromanipulation.

[0028] The present invention utilizes the principle of computational holography and phase mask technology to obtain a functional optical vortex array mask through computer coding, and obtains the array in the far field; it has the advantages of local dynamic regulation and functionality.

[0029] Figure 1 The present invention produces a cycloid optical vortex array with flexible and diverse structures. Figure 1-3 The design process of this method is described in detail:

[0030] In cylindrical coordinates Under this condition, the ideal l-th order Bessel beam can be expressed as:

[0031]

[0032] Among them, J l is the first-kind Bessel function of order l; k r and k z They are the wave vector components along the optical axis and radial direction, and their relationship with the wave number is: k=(k r 2 +k z 2 ) 1 / 2 = 2π / λ, where λ is the wavelength of the electromagnetic radiation forming the Bessel beam and j is an imaginary unit;

[0033] The Bessel beam passes through a convex lens with a focal length of f, and the light fields before and after the transformation are and E(r,θ). The Fourier transform formula in the cylindrical coordinate system is:

[0034]

[0035] According to the Fourier phase shift method, the optical vortices can be arranged at equal intervals along the trajectory of the cycloid. The complex amplitude transmittance function of the optical vortex array on the plane of the spatial light modulator is:

[0036]

[0037] Where n′ is the refractive index of the axicon, α is the cone angle, N is the total number of optical vortices, and L n,1 and Ln ,2 are the position matrices of the nth optical vortex respectively; here, the parametric equation of the optical vortex array can be expressed as:

[0038]

[0039] Where n is the number of optical vortices; D determines the size of the array; a1 and a2 are the stretching factors of the array in the horizontal and vertical directions; b is the curvature of the cycloid. When the sign before b is "+", the above parameter equation is the inner cycloid equation, and c+1 is the number of peaks in the array. But when the sign before b is "-", the above parameter equation is the outer cycloid equation, and c-1 is the number of peaks in the array; the real value of the parameter Q = (D, a1, a2, b, c) determines the cycloid structure. In the star-shaped optical vortex array, although the entire structure remains unchanged. The positions of the four optical vortices are determined during the dynamic modulation process. The other four optical vortices rotate on the trajectory of the star-shaped line with the same angular velocity. Dynamic modulation is achieved by adjusting the directional angle matrix of the optical vortices in the array.

[0040] t n =[0,d1,π / 2,d2,π,d3,3π / 2,d4]

[0041] When d1 = π / 4–β, d2 = 3π / 4–β, d3 = 5π / 4–β, d4 = 7π / 4–β, and β∈[0,π], the four optical vortices slide at one-quarter of the trajectory. Subsequent experiments can use different values ​​to design corresponding beam structures and encode them into a mask.

[0042] Example

[0043] The following takes a 512×512 size mask as an example, for a laser with an operating wavelength of 532nm and a power of 100mW. According to the mask transmittance function and parameter selection in the specific implementation method, a cycloid optical vortex array mask with a functional structure is finally obtained. Figure 1 As shown in the figure, this freely structured and diverse cycloid optical vortex array mask can be implemented in the far field of a spatial light modulator. Taking the PLUTO-VIS-016 spatial light modulator from Holoeye, Germany, as an example, it has a pixel size of 8μm × 8μm, a fill factor of 93%, and a resolution of 1920 × 1080 pixels. The proposed controllable cycloid optical vortex array mask was experimentally verified.

[0044] Figure 2As shown in the figure, the experimental light intensity distribution of four controllable cycloid optical vortex array masks on the focal plane of a lens with a focal length of 200mm was obtained. From the figure, it can be seen that from left to right are the cardioid line, kidney line, tricuspid line, and star line optical vortex arrays.

[0045] Figure 3 The invention combines two optical vortex arrays of different structures to create a functional array—an optical gear. This array can transport particles like a traditional gear, acting as a conveyor belt. Experiments have shown that using the proposed cycloidal optical vortex array mask, optical vortex arrays and optical gears with varying structures, radii, curvatures, spacing, displacements, and vortex numbers can be created, providing a richer range of degrees of freedom for micromanipulation.

[0046] In summary, the present invention proposes a specific design scheme and implementation plan for a cycloid optical vortex array mask based on the Fourier phase shift method and coordinate positioning technology. Taking a lens with a focal length of 150 mm as an example, a technical implementation route for a cycloid optical vortex array mask is proposed for a laser with an operating wavelength of 532 nm.

[0047] The cycloid optical vortex array mask generated above based on the Fourier phase shifting method and coordinate positioning method represents only one specific embodiment of the present invention and should not be construed as limiting the scope of protection of the present invention. It should be noted that those skilled in the art may make various variations and improvements to the specific implementation details proposed in this patent without departing from the basic concept of the present invention, and such variations and improvements are within the scope of protection of the present invention.

Claims

1. A method for designing a cycloid optical vortex array mask, characterized by: Here are the steps: S1, in cylindrical coordinates The ideal Bessel beam is expressed as: Among them, J l is the first-kind Bessel function of order l; k r and k z They are the wave vector components in the radial and optical axis directions, and their relationship with the wave number is: k = (k r 2 +k z 2 ) 1 / 2 =2π / λ, where λ is the wavelength of the electromagnetic radiation forming the Bessel beam; S2. Based on the Bessel beam obtained in step S1, a cycloid optical vortex array is generated according to the displacement theorem of Fourier transform. The specific expression of its complex transmittance function is: Where n′ is the refractive index of the axicon, α is the cone angle, N is the total number of optical vortices, and L n,1 and Ln,2 are the position matrices of the nth optical vortex, respectively; S3. The mask described based on the complex transmittance function is the cycloid optical vortex array mask. The parametric equation of the cycloid optical vortex array is expressed as: Where n is the number of optical vortices; D determines the size of the array; a1 and a2 are the stretching factors of the array in the horizontal and vertical directions; b is the curvature of the cycloid. When the sign before b is "+", the above parametric equation is the hypocycloid equation, and c+1 is the number of peaks in the array; when the sign before b is "-", the above parametric equation is the epicycloid equation, and c-1 is the number of peaks in the array; the real number value of the parameter Q = (D, a1, a2, b, c) determines the cycloid array structure.

2. The method for designing a cycloid optical vortex array mask according to claim 1, wherein: In the S2 step: The Bessel beam obtained in step S1 is passed through a convex lens with a focal length of f. The light fields before and after the transformation are and E(r,θ), the Fourier transform formula in the cylindrical coordinate system is: According to the displacement theorem of Fourier transform, an optical vortex is obtained at a specified position, and then a cycloid optical vortex array is generated.

3. Application of a cycloid optical vortex array mask generated by the design method according to any one of claims 1-2, characterized in that: A parallel light beam is irradiated on a spatial light modulator loaded with a mask plate superimposed on two cycloid optical vortex arrays with different structures. The reflected light beam modulated by the spatial light modulator passes through a convex lens to obtain an optical gear composed of the two arrays in the far field.

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