CN105589203B

CN105589203B - Produce the method and device of radial polarisation array beams

Info

Publication number: CN105589203B
Application number: CN201610142131.2A
Authority: CN
Inventors: 王静; 朱时军; 李振华
Original assignee: Nanjing University of Science and Technology
Current assignee: Nanjing University of Science and Technology
Priority date: 2016-03-13
Filing date: 2016-03-13
Publication date: 2018-04-03
Anticipated expiration: 2036-03-13
Also published as: CN105589203A

Abstract

The present invention relates to a kind of method and device for producing radial polarisation array beams, method includes：Spatial polarization regulation and control are carried out to linearly polarized laser, obtain radial polarized light beam；The space correlation characteristic of radial polarized light beam is reduced, obtains the radial polarisation Xie Ermo light beams of space correlation characteristic Gaussian distributed；Using the relevance of two-dimensional comb function regulation and control radial polarisation Xie Ermo light beams, Gauss array association radial polarized light beam is obtained；Gauss array associates radial polarized light beam after thin lens focuses on, and radial polarisation array beams are obtained in focal position.Radial polarisation array beams caused by the present invention can greatly improve operating efficiency compared with single-mode beams, in application fields such as particle-capture, high density storages, have very high researching value.

Description

Produce the method and device of radial polarisation array beams

Technical field

The present invention relates to radial polarized light beam, particularly a kind of method and device for producing radial polarisation array beams.

Background technology

In recent years, a kind of spatial non-uniform light beam --- radial polarized light beam receives extensive concern.Radial polarisation Light beam has special polarization properties, relative to even polarization light field, such as linear polarization, elliptical polarization, circular polarization, radial polarisation light What polarization state was unevenly distributed, its electric field oscillation has perfect rotational symmetry characteristic, and light field center intensity is zero.In recent years The theory come and experimental studies have found that, radial polarized light beam, which compares general uniform light beam, has some new features, such as " tightly gathers The minimum light spot area of Jiao " radial polarized light beam is up to 0.16 λ², it is smaller by nearly 50% than the linearly polarized light with the conditions of.In addition, also Very strong long-range salt free ligands axial electric field component and easily excitating surface plasmon etc. can be obtained.Radial polarisation light The production method of beam can be divided mainly into intracavitary and produce and two kinds be produced outside chamber：Intracavitary production method is to learn member by being placed in intracavitary Part, such as conscope, birefringence element and liquid crystal polarized selector, select by these elements or change the inclined of light beam Polarization state produces radial polarized light beam.Rule is produced outside chamber to be mainly superimposed by mode-interference or using some particular polarization patterns turn Element is changed, such as radial polarizer, LCD space light modulator, phase delay element, polarizationselective light fibre and sub-wave length grating Deng acquisition radial polarized light beam.Current research show radial polarized light beam particle-capture, super-resolution measurement, super-resolution into Picture, high density storage and material processing and other fields have important application prospect.

Based on partially coherent partial polarization general theory, light beam can use the rank cross spectrum density matrix of spatial frequency domain 2 × 2 Represent, consider the spatial coherence of light field, the radial polarized light beam of associate feature Gaussian distributed is suggested and tests generation (F.Wang,Y.Cai,Y.Dong,and O.Korotkova,Experimental generation of a radially polarized beam with controllable spatial coherence.Appl.Phys.Lett.100,051108 (2012)), research shows that the focus characteristics of radial polarisation light can be regulated and controled by the spatial coherence of light beam, produces empty The intensity distribution of the heart, flat-top and Gaussian, there is important application prospect in material heat treatment and particle-capture field.Research It has also been found that partially coherent radial polarized light beam compares linear polarization vector beam, it can effectively reduce atmospheric turbulance and optical signal is disturbed It is dynamic, there is important application in terms of free optic communication.

The associate feature of light beam can regulate and control the association of light beam under the conditions of the authenticity of beam configuration is met by design Characteristic carrys out Effective Regulation optical parameter characteristic.Radial polarisation array beams are compared with single-mode beams, at some such as high density storage, super The application fields such as resolution imaging, material heat treatment, particle-capture can greatly improve operating efficiency, have very high Practical Research Value.The generation of array beams can be directly obtained by laser array by beam shaping, or light beam is adjusted by beam splitter Intensity distribution is controlled to realize, but by regulating and controlling to the associate feature of light beam so that light beam realizes battle array after being transmitted by optical system The method of row light beam has no report always.

The content of the invention

It is an object of the invention to provide a kind of method and device for producing radial polarisation array beams.

The technical scheme for realizing the object of the invention is：A kind of method for producing radial polarisation array beams, including following step Suddenly：

Step 1, spatial polarization regulation and control are carried out to linearly polarized laser, obtain radial polarized light beam；

Step 2, the space correlation characteristic for reducing radial polarized light beam, obtain the footpath of space correlation characteristic Gaussian distributed To polarization Xie Ermo light beams；

Step 3, the relevance using two-dimensional comb function regulation and control radial polarisation Xie Ermo light beams, obtain the association of Gauss array Radial polarized light beam；

Step 4, Gauss array associate radial polarized light beam after thin lens focuses on, and radial polarisation is obtained in focal position Array beams.

A kind of device for producing radial polarisation array beams, including linear polarization He-Ne laser, radial polarisation converter, rotation Frosted glass plate, spatial light modulator, collimation lens, G amplitude filter plate, condenser lens and the laser beam analyzer turned；

Linearly polarized laser beam caused by linear polarization He-Ne laser makes linearly polarized laser beam by radial polarisation photoconverter Spatial radial polarization is converted to by the linear polarization of space uniform, the space phase of laser beam is reduced by the frosted glass plate of rotation afterwards Dryness, obtain the radial polarized light beam of coherence's Gaussian distributed；The association of spatial light modulator modulated radial light beam Characteristic, the association of Gauss array is made it have, collimated optical beam is directed at by collimation lens, and light beam is regulated and controled in source by gaussian filtering piece The waist radius of field, the radial polarized light beam with Gauss array associate feature finally is produced in source field, it is saturating by focusing on afterwards Mirror carries out light beam focusing, and is distributed in focal point with laser beam analyzer measurement distribution of light intensity.

Compared with prior art, its remarkable advantage is the present invention：(1) the radial polarisation array that the present invention obtains in focal point Light beam can be regulated and controled by initial parameter.Array size can be regulated and controled by initial parameter N, M；Hot spot spacing can pass through α regulates and controls；Spot intensity distribution can pass through the coherence length σ of source field_gTo carry out being regulated to Gauss array distribution and hollow array Distribution, both light beams can manipulate two kinds of different particles in terms of optical beam manipulation particle：Refractive index is more than surrounding environment Particle and refractive index are less than the particle of surrounding environment, and the size of coherence's also controllable focal point stigma；(2) radial polarisation Array beams can greatly improve work compared with single-mode beams, in some such as particle-capture, high density storage application fields Efficiency, there is very high Practical Research to be worth；(3) present invention to the associate feature of radial polarized light beam by regulating and controlling so that light Beam realizes radial polarisation array beams, simple structure after being transmitted by optical system, and regulation and control parameter is enriched, and cost is compared to sharp Light device array, which closes beam, obvious reduction.

Brief description of the drawings

Fig. 1 is a kind of schematic device for producing radial polarisation array beams provided by the invention.

Light intensity and the polarization state distribution of radial polarized light beam are associated in Fig. 2 (a) for source field Gauss array, is source in Fig. 2 (b) Field intensity is distributed in the profile at y=0, and Fig. 2 (c) and Fig. 2 (d) are on light field any point and central point on y=0 sections One-dimensional degree of coherence distribution map.

Fig. 3 (a) and Fig. 3 (b) is respectively the intensity distribution that the coherence length of laser takes focal point array beams under 10mm and 1mm Two-dimensional silhouette figure.

Fig. 4 (a) is light beam intensity and polarization state distribution map, Fig. 4 at transmission range z=0.8f after lens focus (b) it is intensity and polarization state distribution map at transmission range z=0.95f, Fig. 4 (c) is intensity and polarization state at transmission range z=f Distribution map.

Embodiment

With reference to Fig. 1, a kind of method of generation radial polarisation array beams of the invention, comprise the following steps：

Step 1, spatial polarization regulation and control are carried out to linearly polarized laser, obtain radial polarized light beam；The electricity of radial polarized light beam Field vector mathematical modeling is expressed as x-polarisation TEM₁₀With y-polarisation TEM₀₁Coherent superposition：

Wherein r²=x²+y², x, y are respectively abscissa and ordinate under cartesian coordinate, e_x、e_yRepresent orthogonal Electric field oscillation unit vector；

Step 2, the space correlation characteristic for reducing radial polarized light beam, obtain the footpath of space correlation characteristic Gaussian distributed To polarization Xie Ermo light beams；The space correlation characteristic of radial polarisation Xie Ermo light beams is

Step 3, the relevance using two-dimensional comb function regulation and control radial polarisation Xie Ermo light beams, obtain the association of Gauss array Radial polarized light beam；Specially：

Two-dimensional comb function is write using spatial light modulator, radial polarisation Xie Ermo light beams are modulated, obtain sky Between associate feature inverse Fourier transform, be expressed as：

(υ_x,υ_y) representation space frequency domain coordinate, τ_x、τ_yHorizontal stroke, the ordinate of Gauss array are represented respectively；δ represents dirac Function；

Gauss array association radial polarized light beam 2 × 2 rank cross-spectral density matrixes be：

The association of Gauss array is expressed as：

r₁=(x₁,y₁)、r₂=(x₂,y₂) it is 2 position coordinateses of light field；σ₀For beam waist width；σ_gFor Gauss array Associate the coherence length of radial polarized light beam；A is dimensionless real number, for regulating and controlling lattice distance；N, M represent array size, n_x、 n_yThe respectively position of Gauss array；

Step 4, Gauss array associate radial polarized light beam after thin lens focuses on, and radial polarisation is obtained in focal position Array beams；

The abcd matrix that thin lens focuses on is expressed as：

F is focal distance of thin convex lens, z be inspection surface to the distance of thin lens, focal point z value is f；

2 × 2 rank cross spectrum density matrix of radial polarisation array beams, matrix element are expressed as：

Wherein, α represents x or y；Represent W_xy(ρ₁,ρ₂, z) conjugation；ρ₁=(ρ_1x,ρ_1y) and ρ₂=(ρ_2x, ρ_2y) for any two points in receiving surface；

Wherein, k represents wave number.

As shown in figure 1, a kind of device for producing radial polarisation array beams, it is characterised in that swash including linear polarization He-Ne Light device 1, radial polarisation converter 2, the frosted glass plate 3 of rotation, spatial light modulator 4, collimation lens 5, G amplitude filter plate 6th, condenser lens 7 and laser beam analyzer 8；

Linearly polarized laser beam makes linearly polarized laser by radial polarisation photoconverter 2 caused by linear polarization He-Ne laser 1 Beam is converted to spatial radial polarization by the linear polarization of space uniform, reduces the sky of laser beam by the frosted glass plate 3 of rotation afterwards Between coherence, obtain the radial polarized light beam of coherence's Gaussian distributed；Computer 9 is write spatial light by comb function and adjusted Device 4 processed, the associate feature of modulated radial light beam make it have the association of Gauss array, direct light are directed at by collimation lens 5 Beam, and waist radius of the light beam in source field is regulated and controled by gaussian filtering piece 6, finally being produced in source field has the association of Gauss array special Property radial polarized light beam, afterwards by condenser lens 7 carry out light beam focusing, and focal point with laser beam analyzer 8 measure light Field intensity is distributed.

The distance of the condenser lens 7 and laser beam analyzer 8 is the focal length of condenser lens 7.

Below in conjunction with the accompanying drawings and embodiment the present invention is described further.

Embodiment

Referring to Fig. 1, a linearly polarized laser beam will use linearly polarized laser by radial polarisation photoconverter in the present embodiment Laser beam caused by device obtains radial polarized light beam by radial polarisation converter first, and the spatial coherence of laser beam is very high can It is considered what is be concerned with completely, then the electric field intensity mathematical modeling of radial polarized light beam is represented by x-polarisation TEM₁₀With y-polarisation TEM₀₁Coherent superposition：

Wherein r²=x²+y², σ₀For Beam waist radius, can be regulated and controled by G amplitude filter plate.

According to cross polarization general theory, the space correlation characteristic to radial polarisation light regulates and controls, and partially coherent is radially Light beam can be expressed as with 2 × 2 rank cross spectrum density matrix：

μ(r₁,r₂) describe the associate feature of light field.

Laser beam is after the frosted glass plate of rotation, and the coherence of light field can reduce, and Gaussian distributed, its relevant letter Several inverse Fourier transforms are represented by：

σ_gCoherence length is represented, is determined by rotating ground glass piece degree of roughness, υ representation space frequency domain coordinates.

Two-dimensional comb function is write using spatial light modulator, to being modulated by the light beam of frosted glass plate, obtains height The inverse Fourier transform of this array correlation function, is expressed as：

After frosted glass plate and spatial light modulator regulation and control light beam of the light beam by rotation, coherent function can finally represent For：

Meet in the present embodiment：Superposition for Gaussian function is nonnegativity, andWith the polarization side of light beam To unrelated, i.e.,Meet the authenticity of light beam, to arbitrary function f_α(r) can meet：

Q represents the definition of light beam nonnegativity,And f_β(r₂) represent respectively on the point of light field two r₁、r₂Arbitrary function,^* Complex conjugate is represented, α, β are corresponding with the element of cross spectrum density matrix；

Need to ensure that caused array Gauss associates radial polarized light beam and meets beam condition in far field in the present embodiment.From By far field in space a bit(ρ is distance of the light source to light field,For unit direction vector at far field) spectral concentration Function S^∞(ρ) is：

WhereinFor the four-dimensional welfare leaf transformation of source field, α=x, y,Represent that two dimension is empty Between frequency vector,Represent unit trivectorThe two-dimentional unit vector projected in the plane of source,Represent beDirection Angle, i.e., far field point of observation ρ direction and the angle of source plane normal are pointed to by the origin of source plane.It is remote in order to meet Beam condition, far-field intensity distribution are negligible, i.e. light in the scope in addition in the narrow angle on z directions The initial parameter of beam needs to meet condition：

Max (M, N) is x, the big value of y direction mode numbers.

Referring to Fig. 2, light intensity and the polarization state distribution of radial polarized light beam, Fig. 2 are associated in Fig. 2 (a) for source field Gauss array (b) profile at y=0 is distributed in for source field intensity in, Fig. 2 (c) and Fig. 2 (d) are any one on light field on y=0 sections The one-dimensional degree of coherence distribution map of point and central point, coherence length is respectively σ_g=10mm and σ_g=1mm.Other specification in embodiment It is chosen for：Source field waist width σ₀=1mm；Wavelength X=632.8nm；The λ of α=1000 can be used to lattice distance；N, M are used for Regulate and control the size of array beams, N=M=1 obtains 3 × 3 arrays.

Gauss array association radial polarized light beam caused by the plane of source is transmitted by ABCD paraxial optics system, can be with Characterized with broad sense Collins integral equations：

Wherein ρ₁=(ρ_1x,ρ_1y)；ρ₂=(ρ_2x,ρ_2y) for any two points in receiving surface.

When the ABCD systems of process are the lens focus shown in Fig. 1, abcd matrix is represented by：

F is focal distance of thin convex lens, z be inspection surface to the distance of thin lens, focal point z value is f.

By integral and calculating, 2 × 2 rank cross spectrum density matrix in the far field after ABCD optical systems are transmitted are obtained, Its each matrix element is represented by：

Wherein, α represents x or y；Represent W_xy(ρ₁,ρ₂, z) conjugation；

Wherein, k represents wave number.

Radial polarisation array beams finally are obtained in focal point in the present embodiment, i.e. the intensity distribution of light beam shows as array Form, and the polarization properties of each stigma are radial polarisations, and the light of focal point can be obtained by cross spectrum density matrix It is distributed as by force：

Referring to Fig. 3 (a) and Fig. 3 (b), focal point array beams field strength two-dimensional silhouette figure under light source difference coherence length, just The optical field distribution of beginning is hollow, and the coherence length by regulating and controlling light beam will influence the intensity distribution of array beams, source field phase When dry length is larger, σ is taken as_g=10mm, each point hot spot of array beams is hollow, and this array beams can be used to grasp The particle that refractive index is less than surrounding environment is controlled, source field coherence length is σ_gDuring=1mm, each point hot spot of array beams is high This distribution, this array beams can be used to manipulate the particle that refractive index is more than surrounding environment.The focal length f=of condenser lens 150mm, other specification are chosen consistent with Fig. 2.

It will consider to obtain the polarization properties of array beams in the present embodiment, the polarization state at any point can use polarization in light field Ellipse is described, and based on partially coherent partial polarization general theory, the light of focal point can be obtained by cross spectrum density matrix Three important parameters of field any point polarization ellipse, oval deflection are represented by：

Wherein, Re reals；

The major axis A of polarization ellipse₊And short axle A_-It can be expressed as by cross spectrum density matrix：

Referring to Fig. 4, by taking the hollow radial direction polarization arrays light beam described in Fig. 3 (a) as an example, the black line of light class describes light The distribution property of field polarization state.To illustrate intensity and polarization state distribution of the light beam after lens focus, Fig. 4 value is focusing The focal length f=150mm of lens, coherence length σ_g=10mm, other parameters are chosen consistent with Fig. 2.Pass through lens in Fig. 4 (a) Transmission range z=0.8f after focusing, Fig. 4 (b) transmission range are z=0.95f, and Fig. 4 (c) transmission ranges are z=f.

It can be seen that Gauss array associates radial polarized light beam during such as Fig. 1 optical systems are transmitted, the distribution of its polarization state It is not to obey all the time radially-arranged, light beam is during array beams are split into, the space polarization state and strong of a bit Degree distribution is influenceed by multiple pattern hot spots, is a stack result, but after light beam fully nonlinear water wave is array beams, each The polarization state of stigma is to obey radial polarisation, and the intensity distribution property of each stigma be it is consistent, it is final of the invention To radial polarisation array beams.

Claims

A kind of 1. method for producing radial polarisation array beams, it is characterised in that comprise the following steps：

Step 1, spatial polarization regulation and control are carried out to linearly polarized laser, obtain radial polarized light beam；

Step 2, the space correlation characteristic for reducing radial polarized light beam, the radial direction for obtaining space correlation characteristic Gaussian distributed are inclined Shake Xie Ermo light beams；

Step 3, the relevance using two-dimensional comb function regulation and control radial polarisation Xie Ermo light beams, obtain the association of Gauss array radially Light beam, wherein Gauss array association radial polarized light beam 2 × 2 rank cross-spectral density matrixes be：

<mrow> <mover> <mi>W</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>r</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>r</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <msubsup> <mi>&sigma;</mi> <mn>0</mn> <mn>2</mn> </msubsup> </mrow> </mfrac> <mfenced open = "(" close = ")"> <mtable> <mtr> <mtd> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <msub> <mi>x</mi> <mn>2</mn> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <msub> <mi>y</mi> <mn>2</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>y</mi> <mn>1</mn> </msub> <msub> <mi>x</mi> <mn>2</mn> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>y</mi> <mn>1</mn> </msub> <msub> <mi>y</mi> <mn>2</mn> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mi>exp</mi> <mo>&lsqb;</mo> <mo>-</mo> <mfrac> <mrow> <msubsup> <mi>r</mi> <mn>1</mn> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>r</mi> <mn>2</mn> <mn>2</mn> </msubsup> </mrow> <mrow> <mn>4</mn> <msubsup> <mi>&sigma;</mi> <mn>0</mn> <mn>2</mn> </msubsup> </mrow> </mfrac> <mo>&rsqb;</mo> <mi>&mu;</mi> <mrow> <mo>(</mo> <msub> <mi>r</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>r</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </mrow>

The association of Gauss array is expressed as：

<mrow> <mi>&mu;</mi> <mrow> <mo>(</mo> <msub> <mi>r</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>r</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mi>N</mi> <mi>M</mi> </mrow> </mfrac> <mi>exp</mi> <mo>&lsqb;</mo> <mo>-</mo> <mfrac> <msup> <mrow> <mo>(</mo> <msub> <mi>r</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>r</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mrow> <mn>2</mn> <msubsup> <mi>&sigma;</mi> <mi>g</mi> <mn>2</mn> </msubsup> </mrow> </mfrac> <mo>&rsqb;</mo> <munderover> <mo>&Sigma;</mo> <mrow> <msub> <mi>n</mi> <mi>x</mi> </msub> <mo>=</mo> <mo>-</mo> <mi>N</mi> </mrow> <mi>N</mi> </munderover> <munderover> <mo>&Sigma;</mo> <mrow> <msub> <mi>n</mi> <mi>y</mi> </msub> <mo>=</mo> <mo>-</mo> <mi>M</mi> </mrow> <mi>M</mi> </munderover> <mi>exp</mi> <mo>&lsqb;</mo> <mi>i</mi> <mfrac> <mrow> <msub> <mi>&pi;n</mi> <mi>x</mi> </msub> </mrow> <mi>a</mi> </mfrac> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>i</mi> <mfrac> <mrow> <msub> <mi>&pi;n</mi> <mi>y</mi> </msub> </mrow> <mi>a</mi> </mfrac> <mrow> <mo>(</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mo>&rsqb;</mo> <mo>.</mo> </mrow>

In above formula, r₁=(x₁,y₁)、r₂=(x₂,y₂) it is 2 position coordinateses of light field；σ₀For beam waist width；σ_gFor Gauss battle array The coherence length of row association radial polarized light beam；A is dimensionless real number, for regulating and controlling lattice distance；M represents the line number of array, N For the columns of display, n_x、n_yThe respectively position of Gauss array；

Step 4, Gauss array associate radial polarized light beam after thin lens focuses on, and radial polarisation array is obtained in focal position Light beam.
2. the method according to claim 1 for producing radial polarisation array beams, it is characterised in that in step 1 radially partially The electric field intensity mathematical modeling of light beam of shaking is expressed as x-polarisation TEM₁₀With y-polarisation TEM₀₁Coherent superposition：

<mrow> <mi>E</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>E</mi> <mi>x</mi> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <msub> <mi>e</mi> <mi>x</mi> </msub> <mo>+</mo> <msub> <mi>E</mi> <mi>y</mi> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <msub> <mi>e</mi> <mi>y</mi> </msub> <mo>=</mo> <mo>&lsqb;</mo> <mfrac> <mi>x</mi> <mrow> <mn>2</mn> <msub> <mi>&sigma;</mi> <mn>0</mn> </msub> </mrow> </mfrac> <mi>exp</mi> <mrow> <mo>(</mo> <mo>-</mo> <mfrac> <msup> <mi>r</mi> <mn>2</mn> </msup> <mrow> <mn>4</mn> <msubsup> <mi>&sigma;</mi> <mn>0</mn> <mn>2</mn> </msubsup> </mrow> </mfrac> <mo>)</mo> </mrow> <msub> <mi>e</mi> <mi>x</mi> </msub> <mo>+</mo> <mfrac> <mi>y</mi> <mrow> <mn>2</mn> <msub> <mi>&sigma;</mi> <mn>0</mn> </msub> </mrow> </mfrac> <mi>exp</mi> <mrow> <mo>(</mo> <mo>-</mo> <mfrac> <msup> <mi>r</mi> <mn>2</mn> </msup> <mrow> <mn>4</mn> <msubsup> <mi>&sigma;</mi> <mn>0</mn> <mn>2</mn> </msubsup> </mrow> </mfrac> <mo>)</mo> </mrow> <msub> <mi>e</mi> <mi>y</mi> </msub> <mo>&rsqb;</mo> </mrow>

Wherein r²=x²+y², x, y are respectively abscissa and ordinate under cartesian coordinate, e_x、e_yRepresent orthogonal electric field Unit of vibration vector, E_x(x, y) and E_y(x, y) is respectively any point (x, y) perpendicular to two of direction of beam propagation z-axis Electric field point vector in mutually orthogonal direction.
3. the method according to claim 1 for producing radial polarisation array beams, it is characterised in that to radially in step 2 The space correlation characteristic of polarised light is regulated and controled, and the space correlation characteristic of radial polarisation Xie Ermo light beams is

<mrow> <mi>exp</mi> <mo>&lsqb;</mo> <mo>-</mo> <mfrac> <msup> <mrow> <mo>(</mo> <msub> <mi>r</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>r</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mrow> <mn>2</mn> <msubsup> <mi>&sigma;</mi> <mi>g</mi> <mn>2</mn> </msubsup> </mrow> </mfrac> <mo>&rsqb;</mo> <mo>.</mo> </mrow>
4. the method according to claim 1 for producing radial polarisation array beams, it is characterised in that using sky in step 3 Between optical modulator write two-dimensional comb function, radial polarisation Xie Ermo light beams are modulated, obtain the inverse of space correlation characteristic Fourier transformation, it is expressed as：

<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <mover> <mi>&mu;</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>&upsi;</mi> <mi>x</mi> </msub> <mo>,</mo> <msub> <mi>&upsi;</mi> <mi>y</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <msubsup> <mi>&pi;&sigma;</mi> <mi>g</mi> <mn>2</mn> </msubsup> </mrow> <mrow> <mi>N</mi> <mo>&times;</mo> <mi>M</mi> </mrow> </mfrac> <munderover> <mo>&Sigma;</mo> <mrow> <msub> <mi>n</mi> <mi>x</mi> </msub> <mo>=</mo> <mo>-</mo> <mi>N</mi> </mrow> <mi>N</mi> </munderover> <mi>exp</mi> <mo>&lsqb;</mo> <mo>-</mo> <mn>2</mn> <msup> <mi>&pi;</mi> <mn>2</mn> </msup> <msubsup> <mi>&sigma;</mi> <mi>g</mi> <mn>2</mn> </msubsup> <msup> <mrow> <mo>(</mo> <msub> <mi>&upsi;</mi> <mi>x</mi> </msub> <mo>+</mo> <msub> <mi>&tau;</mi> <mi>x</mi> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>&rsqb;</mo> <mi>&delta;</mi> <mrow> <mo>(</mo> <msub> <mi>&tau;</mi> <mi>x</mi> </msub> <mo>-</mo> <mfrac> <msub> <mi>n</mi> <mi>x</mi> </msub> <mrow> <mn>2</mn> <mi>&alpha;</mi> </mrow> </mfrac> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&times;</mo> <munderover> <mo>&Sigma;</mo> <mrow> <msub> <mi>n</mi> <mi>y</mi> </msub> <mo>=</mo> <mo>-</mo> <mi>M</mi> </mrow> <mi>M</mi> </munderover> <mi>exp</mi> <mo>&lsqb;</mo> <mo>-</mo> <mn>2</mn> <msup> <mi>&pi;</mi> <mn>2</mn> </msup> <msubsup> <mi>&sigma;</mi> <mi>g</mi> <mn>2</mn> </msubsup> <msup> <mrow> <mo>(</mo> <msub> <mi>&upsi;</mi> <mi>y</mi> </msub> <mo>+</mo> <msub> <mi>&tau;</mi> <mi>y</mi> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>&rsqb;</mo> <mi>&delta;</mi> <mrow> <mo>(</mo> <msub> <mi>&tau;</mi> <mi>y</mi> </msub> <mo>-</mo> <mfrac> <msub> <mi>n</mi> <mi>y</mi> </msub> <mrow> <mn>2</mn> <mi>&alpha;</mi> </mrow> </mfrac> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced>

(υ_x,υ_y) representation space frequency coordinate, τ_x、τ_yHorizontal stroke, the ordinate of Gauss array are represented respectively；δ represents Dirac function.
5. the method according to claim 1 for producing radial polarisation array beams, it is characterised in that thin lens in step 4 The abcd matrix of focusing is expressed as：

<mrow> <mfenced open = "(" close = ")"> <mtable> <mtr> <mtd> <mi>A</mi> </mtd> <mtd> <mi>B</mi> </mtd> </mtr> <mtr> <mtd> <mi>C</mi> </mtd> <mtd> <mi>D</mi> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mfenced open = "(" close = ")"> <mtable> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mi>z</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "(" close = ")"> <mtable> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mn>1</mn> <mo>/</mo> <mi>f</mi> </mrow> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "(" close = ")"> <mtable> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mi>f</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mfenced open = "(" close = ")"> <mtable> <mtr> <mtd> <mrow> <mn>1</mn> <mo>-</mo> <mi>z</mi> <mo>/</mo> <mi>f</mi> </mrow> </mtd> <mtd> <mi>f</mi> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mn>1</mn> <mo>/</mo> <mi>f</mi> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> </mrow>

Wherein, f is focal distance of thin convex lens, z be inspection surface to the distance of thin lens, focal point z value is f；

2 × 2 rank cross spectrum density matrix of radial polarisation array beams, matrix element are expressed as：

<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>W</mi> <mrow> <mi>&alpha;</mi> <mi>&alpha;</mi> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>&rho;</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>&rho;</mi> <mn>2</mn> </msub> <mo>,</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mo>&Sigma;</mo> <mrow> <mi>n</mi> <mi>x</mi> <mo>=</mo> <mo>-</mo> <mi>N</mi> </mrow> <mi>N</mi> </munderover> <munderover> <mo>&Sigma;</mo> <mrow> <mi>n</mi> <mi>y</mi> <mo>=</mo> <mo>-</mo> <mi>M</mi> </mrow> <mi>M</mi> </munderover> <mfrac> <mrow> <msubsup> <mi>&sigma;</mi> <mi>g</mi> <mn>2</mn> </msubsup> <msub> <mi>C</mi> <mn>1</mn> </msub> </mrow> <mrow> <msub> <mi>S</mi> <mn>0</mn> </msub> <msubsup> <mi>S</mi> <mn>1</mn> <mn>2</mn> </msubsup> </mrow> </mfrac> <mo>&lsqb;</mo> <msub> <mi>S</mi> <mn>1</mn> </msub> <mo>+</mo> <mfrac> <mrow> <mo>(</mo> <mn>2</mn> <mo>+</mo> <msub> <mi>S</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <msubsup> <mi>&sigma;</mi> <mi>g</mi> <mn>2</mn> </msubsup> </mfrac> <msubsup> <mi>&rho;</mi> <mrow> <mi>&alpha;</mi> <mn>1</mn> </mrow> <mo>&prime;</mo> </msubsup> <msubsup> <mi>&rho;</mi> <mrow> <mi>&alpha;</mi> <mn>2</mn> </mrow> <mo>&prime;</mo> </msubsup> <mo>-</mo> <mn>2</mn> <mrow> <mo>(</mo> <msub> <mi>M</mi> <mn>2</mn> </msub> <msubsup> <mi>&rho;</mi> <mrow> <mi>&alpha;</mi> <mn>1</mn> </mrow> <mrow> <mo>&prime;</mo> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msub> <mi>M</mi> <mn>1</mn> </msub> <msubsup> <mi>&rho;</mi> <mrow> <mi>&alpha;</mi> <mn>2</mn> </mrow> <mrow> <mo>&prime;</mo> <mn>2</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <mo>&rsqb;</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&times;</mo> <mi>exp</mi> <mo>&lsqb;</mo> <mo>-</mo> <mfrac> <mn>1</mn> <msub> <mi>S</mi> <mn>1</mn> </msub> </mfrac> <mo>&lsqb;</mo> <msub> <mi>M</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <msubsup> <mi>&rho;</mi> <mrow> <mi>x</mi> <mn>1</mn> </mrow> <mrow> <mo>&prime;</mo> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>&rho;</mi> <mrow> <mi>y</mi> <mn>1</mn> </mrow> <mrow> <mo>&prime;</mo> <mn>2</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>M</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <msubsup> <mi>&rho;</mi> <mrow> <mi>x</mi> <mn>2</mn> </mrow> <mrow> <mo>&prime;</mo> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>&rho;</mi> <mrow> <mi>y</mi> <mn>2</mn> </mrow> <mrow> <mo>&prime;</mo> <mn>2</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <mo>&rsqb;</mo> <mo>+</mo> <mfrac> <mn>1</mn> <mrow> <msub> <mi>S</mi> <mn>1</mn> </msub> <msubsup> <mi>&sigma;</mi> <mi>g</mi> <mn>2</mn> </msubsup> </mrow> </mfrac> <mrow> <mo>(</mo> <msubsup> <mi>&rho;</mi> <mrow> <mi>x</mi> <mn>1</mn> </mrow> <mo>&prime;</mo> </msubsup> <msubsup> <mi>&rho;</mi> <mrow> <mi>x</mi> <mn>2</mn> </mrow> <mo>&prime;</mo> </msubsup> <mo>+</mo> <msubsup> <mi>&rho;</mi> <mrow> <mi>y</mi> <mn>1</mn> </mrow> <mo>&prime;</mo> </msubsup> <msubsup> <mi>&rho;</mi> <mrow> <mi>y</mi> <mn>2</mn> </mrow> <mo>&prime;</mo> </msubsup> <mo>)</mo> </mrow> <mo>&rsqb;</mo> </mrow> </mtd> </mtr> </mtable> </mfenced>

<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>W</mi> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>&rho;</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>&rho;</mi> <mn>2</mn> </msub> <mo>,</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mo>&Sigma;</mo> <mrow> <mi>n</mi> <mi>x</mi> <mo>=</mo> <mo>-</mo> <mi>N</mi> </mrow> <mi>N</mi> </munderover> <munderover> <mo>&Sigma;</mo> <mrow> <mi>n</mi> <mi>y</mi> <mo>=</mo> <mo>-</mo> <mi>M</mi> </mrow> <mi>M</mi> </munderover> <mfrac> <msub> <mi>C</mi> <mn>1</mn> </msub> <mrow> <msub> <mi>S</mi> <mn>0</mn> </msub> <msubsup> <mi>S</mi> <mn>1</mn> <mn>2</mn> </msubsup> </mrow> </mfrac> <mrow> <mo>(</mo> <mn>2</mn> <msub> <mi>M</mi> <mn>2</mn> </msub> <msubsup> <mi>&sigma;</mi> <mi>g</mi> <mn>2</mn> </msubsup> <msubsup> <mi>&rho;</mi> <mrow> <mi>x</mi> <mn>1</mn> </mrow> <mo>&prime;</mo> </msubsup> <mo>-</mo> <msubsup> <mi>&rho;</mi> <mrow> <mi>x</mi> <mn>2</mn> </mrow> <mo>&prime;</mo> </msubsup> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mn>2</mn> <msub> <mi>M</mi> <mn>1</mn> </msub> <msubsup> <mi>&sigma;</mi> <mi>g</mi> <mn>2</mn> </msubsup> <msubsup> <mi>&rho;</mi> <mrow> <mi>y</mi> <mn>2</mn> </mrow> <mo>&prime;</mo> </msubsup> <mo>-</mo> <msubsup> <mi>&rho;</mi> <mrow> <mi>y</mi> <mn>1</mn> </mrow> <mo>&prime;</mo> </msubsup> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&times;</mo> <mi>exp</mi> <mo>&lsqb;</mo> <mo>-</mo> <mfrac> <mn>1</mn> <msub> <mi>S</mi> <mn>1</mn> </msub> </mfrac> <mo>&lsqb;</mo> <msub> <mi>M</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <msubsup> <mi>&rho;</mi> <mrow> <mi>x</mi> <mn>1</mn> </mrow> <mrow> <mo>&prime;</mo> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>&rho;</mi> <mrow> <mi>y</mi> <mn>1</mn> </mrow> <mrow> <mo>&prime;</mo> <mn>2</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>M</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <msubsup> <mi>&rho;</mi> <mrow> <mi>x</mi> <mn>2</mn> </mrow> <mrow> <mo>&prime;</mo> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>&rho;</mi> <mrow> <mi>y</mi> <mn>2</mn> </mrow> <mrow> <mo>&prime;</mo> <mn>2</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <mo>&rsqb;</mo> <mo>+</mo> <mfrac> <mn>1</mn> <mrow> <msub> <mi>S</mi> <mn>1</mn> </msub> <msubsup> <mi>&sigma;</mi> <mi>g</mi> <mn>2</mn> </msubsup> </mrow> </mfrac> <mrow> <mo>(</mo> <msubsup> <mi>&rho;</mi> <mrow> <mi>x</mi> <mn>1</mn> </mrow> <mo>&prime;</mo> </msubsup> <msubsup> <mi>&rho;</mi> <mrow> <mi>x</mi> <mn>2</mn> </mrow> <mo>&prime;</mo> </msubsup> <mo>+</mo> <msubsup> <mi>&rho;</mi> <mrow> <mi>y</mi> <mn>1</mn> </mrow> <mo>&prime;</mo> </msubsup> <msubsup> <mi>&rho;</mi> <mrow> <mi>y</mi> <mn>2</mn> </mrow> <mo>&prime;</mo> </msubsup> <mo>)</mo> </mrow> <mo>&rsqb;</mo> </mrow> </mtd> </mtr> </mtable> </mfenced>

<mrow> <msub> <mi>W</mi> <mrow> <mi>y</mi> <mi>x</mi> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>&rho;</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>&rho;</mi> <mn>2</mn> </msub> <mo>,</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>=</mo> <msubsup> <mi>W</mi> <mrow> <mi>x</mi> <mi>y</mi> </mrow> <mo>*</mo> </msubsup> <mrow> <mo>(</mo> <msub> <mi>&rho;</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>&rho;</mi> <mn>2</mn> </msub> <mo>,</mo> <mi>z</mi> <mo>)</mo> </mrow> </mrow>

Wherein, α represents x or y；Represent W_xy(ρ₁,ρ₂, z) conjugation；ρ₁=(ρ_1x,ρ_1y) and ρ₂=(ρ_2x,ρ_2y) For any two points in receiving surface；

<mrow> <msub> <mi>C</mi> <mn>1</mn> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <msubsup> <mi>&sigma;</mi> <mn>0</mn> <mn>2</mn> </msubsup> <mi>N</mi> <mi>M</mi> </mrow> </mfrac> <mo>;</mo> <msub> <mi>M</mi> <mn>1</mn> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <msubsup> <mi>&sigma;</mi> <mn>0</mn> <mn>2</mn> </msubsup> </mrow> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <msubsup> <mi>&sigma;</mi> <mi>g</mi> <mn>2</mn> </msubsup> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mi>i</mi> <mi>A</mi> <mi>k</mi> </mrow> <mrow> <mn>2</mn> <mi>B</mi> </mrow> </mfrac> <mo>;</mo> <msub> <mi>M</mi> <mn>2</mn> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <msubsup> <mi>&sigma;</mi> <mn>0</mn> <mn>2</mn> </msubsup> </mrow> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <msubsup> <mi>&sigma;</mi> <mi>g</mi> <mn>2</mn> </msubsup> </mrow> </mfrac> <mo>-</mo> <mfrac> <mrow> <mi>i</mi> <mi>A</mi> <mi>k</mi> </mrow> <mrow> <mn>2</mn> <mi>B</mi> </mrow> </mfrac> <mo>;</mo> </mrow>

<mrow> <msubsup> <mi>&rho;</mi> <mi>&alpha;</mi> <mo>&prime;</mo> </msubsup> <mo>=</mo> <msub> <mi>&rho;</mi> <mi>&alpha;</mi> </msub> <mo>-</mo> <mfrac> <mrow> <msub> <mi>B&pi;n</mi> <mi>&alpha;</mi> </msub> </mrow> <mrow> <mi>k</mi> <mi>a</mi> </mrow> </mfrac> <mo>;</mo> <msub> <mi>S</mi> <mn>0</mn> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mn>4</mn> <msub> <mi>M</mi> <mn>1</mn> </msub> <msub> <mi>M</mi> <mn>2</mn> </msub> <msubsup> <mi>&sigma;</mi> <mi>g</mi> <mn>4</mn> </msubsup> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>;</mo> <msub> <mi>S</mi> <mn>1</mn> </msub> <mo>=</mo> <msup> <mi>A</mi> <mn>2</mn> </msup> <mo>+</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>+</mo> <mfrac> <mrow> <mn>4</mn> <msubsup> <mi>&sigma;</mi> <mn>0</mn> <mn>2</mn> </msubsup> </mrow> <msubsup> <mi>&sigma;</mi> <mi>g</mi> <mn>2</mn> </msubsup> </mfrac> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <mfrac> <mi>B</mi> <mrow> <mn>2</mn> <msubsup> <mi>k&sigma;</mi> <mn>0</mn> <mn>2</mn> </msubsup> </mrow> </mfrac> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow>

Wherein, k represents wave number.