WO2023236323A1

WO2023236323A1 - Method for calculating normalized orbital angular momentum flux density, and device and storage medium

Info

Publication number: WO2023236323A1
Application number: PCT/CN2022/107024
Authority: WO
Inventors: 陈亚红; 严晨成; 王飞; 蔡阳健
Original assignee: 苏州大学
Priority date: 2022-06-10
Filing date: 2022-07-21
Publication date: 2023-12-14
Also published as: CN114722354A; CN114722354B

Abstract

The present invention relates to a method for calculating a normalized orbital angular momentum flux density. The method comprises: determining an orbital angular momentum flux expression of a focal field in a tight-focusing system; obtaining an orbital angular momentum flux density expression according to an integral form of the orbital angular momentum flux expression; decomposing the orbital angular momentum flux density expression into 3×4 decomposition formulas having similar forms; simplifying each decomposition formula into the sum of four convolution operations, and writing the orbital angular momentum flux density expression as the sum of 3×4×4 convolution operations; and normalizing an orbital angular momentum flux density to obtain an orbital angular momentum flux density expression in a fractional structure. In the present invention, the normalized orbital angular momentum flux density is processed to be in a fractional structure, the numerator is represented as the sum of 3×4×4 convolution operations, the denominator is represented as the sum of 3×4 convolution operations, and rapid calculation can be realized by means of software Matlab.

Description

Method, equipment and storage medium for calculating normalized orbital angular momentum flux density

Technical field

The invention relates to the field of optical technology, and in particular, to a method, equipment and storage medium for calculating normalized orbital angular momentum flux density.

Background technique

In 1992, Allen first proposed the vortex beam, which is a special beam with a spiral wavefront and carrying orbital angular momentum (OAM). It has been widely used in optical manipulation, remote sensing, imaging, quantum optics, optical communications and other fields. Develop novel and interesting physical properties. Orbital angular momentum has also become the focus of research work by domestic and foreign scholars. It has many applications in optical tweezers, super-resolution microscopy, etc. The orbital angular momentum (OAM) of vortex beams provides a new dimension for regulating light-matter interactions and also brings practical value to related fields.

At present, most research on orbital angular momentum focuses on the case of completely coherent fields. It has been proven that partially coherent fields have the advantages of reducing scintillation caused by turbulence, reducing beam spread, reducing image noise, etc., and are widely used in free space communications, particle capture, and atomic absorption. It has advantages over coherent beams in other aspects. In particular, adding vortices to partially coherent fields can improve these capabilities. Therefore, it is very meaningful to extend the vortex beam from a completely coherent field to a partially coherent field, and coherence has become a new degree of freedom for regulating the orbital angular momentum flux density of partially coherent beams. In 2022, Wang Haiyun proposed an effective numerical method (Wang H, Yang Z, Liu L, et al. Fast calculation of orbital angular momentum flux density of partially coherent Schell-model beams on propagation[J]. Optics Express, 2022, 30(10):16856-16872.), which only uses two-dimensional Fourier transform to calculate the orbital angular momentum flux density of any partially coherent paraxial ABCD beam, and establishes a fast numerical calculation method for orbital angular momentum flux density. General form. However, this method is only applicable to paraxial beam models and has certain limitations. It cannot calculate non-paraxial situations, such as orbital angular momentum flux density in tightly focused focal fields.

Although the research on the orbital angular momentum of the tightly focused focal field of fully coherent light has developed rapidly, the research on the orbital angular momentum of the tightly focused focal field of partially coherent light has been rarely involved. This is largely due to the inevitable limitations in the calculation process. Partial differentials and quadruple integrals. If the traditional fully coherent expansion method is used to solve the problem, the partially coherent light needs to be decomposed into an incoherent superposition of multiple fully coherent lights, and the orbital angular momentum flux density of each fully coherent light mode in the tightly focused focal field needs to be solved separately, and then The orbital angular momentum flux density of each mode is superimposed to obtain the orbital angular momentum flux density of partially coherent light. However, this approximate calculation method requires a large number of modes, which brings great obstacles to calculation efficiency and result accuracy.

Therefore, it is particularly important to propose an efficient and accurate method for calculating the orbital angular momentum flux density in a tightly focused focal field.

Contents of the invention

To this end, the technical problem to be solved by the present invention is to overcome the problems existing in the existing technology and propose a method, equipment and storage medium for calculating the normalized orbital angular momentum flux density, which will normalize the orbital angular momentum flux Density processing is a fractional structure, the numerator is expressed as the sum of convolution operations, and the denominator is expressed as the sum of convolution operations. Rapid calculation can be achieved with the help of the software Matlab.

In order to solve the above technical problems, the present invention provides a method for calculating normalized orbital angular momentum flux density, which includes the following steps:

Determine the orbital angular momentum flux expression of the focal field in a tight focusing system based on the general expression of the orbital angular momentum flux along the transmission direction of a partially coherent beam and the cross-spectral density matrix in the non-paraxial case;

According to the integral form of the orbital angular momentum flux expression, the orbital angular momentum flux density expression is obtained;

Decompose the orbital angular momentum flux density expression into 3×4 decomposition formulas with similar forms;

Simplify each decomposition expression into the sum of 4 convolution operations, and write the orbital angular momentum flux density expression as the sum of 3×4×4 convolution operations;

The orbital angular momentum flux density is normalized to obtain the fractional structure orbital angular momentum flux density expression.

In one embodiment of the present invention, the orbital angular momentum flux form of the focal field under a tight focusing system is determined based on the general expression of the orbital angular momentum flux of a partially coherent beam along the transmission direction and the cross-spectral density matrix in the non-paraxial case. Methods include:

The general expression for determining the orbital angular momentum flux along the propagation direction of a partially coherent beam is as follows:

where c represents the speed of light in vacuum, S _t represents the average energy through the beam cross-section, which is described by the integral of the Poynting vector S(r, z), and <xθ _y > and <yθ _x > represent the crossed second-order moment ;

In a tight focusing system, a 3×3 cross-spectral density matrix Φ is used to describe the second-order correlation characteristics of a partially coherent vector beam near the focal plane:

where r ₁ = (x ₁ , y ₁ ) and r ₂ = (x ₂ , y ₂ ) represent the cross-sectional coordinates of two observation points on the same section near the focal field, z represents the longitudinal distance from the observation point to the focus, τ (r ₁ , z) and τ (r ₂ , z) represent the electric fields at r ₁ and r ₂ respectively, <> represents the ensemble average,

Represents the transposed complex conjugate, Φ _ij (i, j=1, 2, 3) represents the 9 cross-spectral density matrix elements in the Φ matrix;

Introducing a new coordinate expression form

and r _d =r ₁ -r ₂ = (r _dx , r _dy ), write formula (2) as

The Poynting vector S(r,z) is represented by the polarization matrix element in formula (3):

Then the sum of the two crossed second-order moments in formula (1) are expressed as:

where i is the imaginary unit and k is the incident light wave number; by substituting formula (5) and formula (6) into formula (1), the orbital angular momentum flux expression of the focal field under the tight focusing system is obtained as follows:

In one embodiment of the present invention, a method for obtaining the orbital angular momentum flux density expression based on the integral form of the orbital angular momentum flux expression includes:

The orbital angular momentum flux density M _dz (r, z) along the transmission direction is obtained from the integral form of equation (7) as follows:

Write the matrix elements Φ ₁₁ ( _rs , r _d , z), Φ ₂₂ ( _rs , r _d , z) and Φ ₃₃ ( _rs , r _d , z) of the cross-spectral density matrix in formula (8) as expressions The formula is as follows:

where f is the focal length of the lens, λ is the wavelength of the incident light,

represents the Fourier transform of A _ηj ,

represents the Fourier transform of B _iζ , and * represents the conjugate.

In one embodiment of the present invention, the method of decomposing the orbital angular momentum flux density expression into 3×4 decomposition formulas with similar forms includes:

Write formula (9) as the sum of four integrals of similar form

in

So formula (8) is expressed as

in

In one embodiment of the present invention, each decomposition expression is simplified to the sum of 4 convolution operations, and the orbital angular momentum flux density expression is written as the sum of 3×4×4 convolution operations. include:

Perform partial derivatives, Dirac functions and Fourier transform operations on the decomposition formula respectively, and simplify each decomposition formula into the sum of 4 convolution operations to get:

in

Substituting formulas (23)-(26) into formula (13), the orbital angular momentum flux density M _dz (r, z) along the transmission direction is obtained.

In one embodiment of the present invention, methods for performing partial derivatives, Dirac functions and Fourier transform operations on the decomposition formula include:

against

The first partial derivative inside square brackets in an expression:

Among them, δ() represents the Dirac function, which satisfies δ(s)=∫exp(-2πisv)dv; δ′() represents the first derivative of the Dirac function, which satisfies

Use ∫(v)δ′(vv ₀ )dv=-f′(v ₀ ) to integrate u ₁ of formula (15), and use Fourier transform to integrate ρ _d to get:

in

is the Fourier transform of u _xx ,

for

The Fourier transform of

A known

represents the Fourier transform of A _1j , that is

So:

in

Substituting formula (17) into formula (16), we get:

in

is the convolution operation symbol.

In one embodiment of the present invention, the method of normalizing the orbital angular momentum flux density and obtaining the fractional structure orbital angular momentum flux density expression includes:

Normalizing the orbital angular momentum flux density, the normalized orbital angular momentum flux density at r is expressed as:

in

and ω are the reduced Planck constant and the angular frequency of light respectively, and the expression of the Poynting vector S(r, z) is obtained:

Substituting formulas (23)-(26) and formula (29) into formula (28), the normalized orbital angular momentum flux density m _dx (r) along the transmission direction of the tightly focused focal field is finally obtained.

In one embodiment of the present invention, the numerator in the fractional structure orbital angular momentum flux density expression is expressed as the sum of 3×4×4 convolution operations, and the denominator is expressed as the sum of 3×4 convolution operations.

In addition, the present invention also provides a computer device, which includes a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the program, the steps of the above method are implemented.

Furthermore, the present invention also provides a computer-readable storage medium on which a computer program is stored, and when the program is executed by a processor, the steps of the above method are implemented.

The above technical solution of the present invention has the following advantages compared with the existing technology:

1. The present invention processes the normalized orbital angular momentum flux density into a fractional structure. The numerator is expressed as the sum of convolution operations, and the denominator is expressed as the sum of convolution operations. Rapid calculation can be realized with the help of the software Matlab;

2. The present invention adopts numerical calculation without fuzzy processing such as approximation and error, and the calculation results are accurate and clear;

3. The present invention can change the wavelength, coherence, polarization and lens parameters of the incident light according to the actual situation. The calculation time and result accuracy are not affected, and it has wide applicability.

Description of the drawings

In order to make the content of the present invention easier to understand clearly, the present invention will be described in further detail below based on specific embodiments of the present invention and in conjunction with the accompanying drawings.

Figure 1 is a schematic flow chart of the method for calculating the normalized orbital angular momentum flux density of the present invention.

Detailed ways

The present invention will be further described below in conjunction with the accompanying drawings and specific examples, so that those skilled in the art can better understand and implement the present invention, but the examples are not intended to limit the present invention.

Please refer to Figure 1. An embodiment of the present invention provides a method for calculating normalized orbital angular momentum flux density, which includes the following steps:

S1: Determine the orbital angular momentum flux expression of the focal field in a tight focusing system based on the general expression of the orbital angular momentum flux along the transmission direction of a partially coherent beam and the cross-spectral density matrix in the non-paraxial case;

S2: Obtain the orbital angular momentum flux density expression based on the integral form of the orbital angular momentum flux expression;

S3: Decompose the orbital angular momentum flux density expression into 3×4 decomposition formulas with similar forms;

S4: Simplify each decomposition expression into the sum of 4 convolution operations, and write the orbital angular momentum flux density expression as the sum of 3×4×4 convolution operations;

S5: Normalize the orbital angular momentum flux density to obtain the fractional structure orbital angular momentum flux density expression.

In a method for calculating the normalized orbital angular momentum flux density disclosed in the present invention, the present invention processes the normalized orbital angular momentum flux density into a fractional structure. The numerator is expressed as the sum of convolution operations, and the denominator is expressed as It is the sum of convolution operations and can be quickly calculated with the help of the software Matlab.

Specifically, the orbital angular momentum flux M _z of a partially coherent beam along the transmission direction can be expressed by two crossed second-order moments <xθ _y > and <yθ _x >:

where c is the speed of light in vacuum, which can also be expressed as

ε ₀ and μ ₀ are the dielectric constant and magnetic permeability in vacuum respectively. S _t represents the average energy through the beam cross section, which can be described by the integral of the Poynting vector S (r, z).

In a tightly focused system, a 3×3 cross-spectral density matrix Φ can be used to describe the second-order correlation characteristics of a partially coherent vector beam near the focal plane:

Where r ₁ = (x ₁ , y ₁ ) and r ₂ = (x ₂ , y ₂ ) are the cross-sectional coordinates of two observation points on the same section near the focal field, Z is the longitudinal distance from the observation point to the focus, τ (r ₁ , z) and τ (r ₂ , z) represent the electric fields at r ₁ and r ₂ respectively. <> represents the ensemble average,

Represents transposed complex conjugate. Φ _ij (i, j=1, 2, 3) represents the 9 cross-spectral density matrix elements in the matrix Φ. A new coordinate expression form is introduced here

And r _d =r ₁ -r ₂ = (r _dx , r _dy ), then formula (2) is written as:

Therefore, the Poynting vector S(r, z) in the tightly focused focal field can be expressed by the polarization matrix element in formula (3):

The lower right subscript indicates coordinates r ₁ =r ₂ =r = (x, y). Then the two cross second-order moments <xθ _y ) and <yθ _x > in formula (1) are respectively expressed:

In the above formula, i is the imaginary unit, and k is the incident light wave number.

Substitute formula (5) and formula (6) into formula (1) to obtain the orbital angular momentum flux:

The orbital angular momentum flux density M _dc (r, z) along the transmission direction can be obtained from the integral form of equation (7):

In the tightly focused system of partially coherent Scher mode beam, the matrix elements Φ ₁₁ ( _rs , r _d , z), Φ ₂₂ ( _rs , r _d , z) and Φ of the cross-spectral density matrix in equation (8) can be ₃₃ (rs _s , r _d , z) is written as a unified expression:

where f is the focal length of the lens and λ is the wavelength of the incident light.

represents the Fourier transform of A _ηj ,

Represents the Fourier transform of B _iζ , * indicates conjugation; A _ηj (η=1, 2; j=1, 2, 3) and B _iζ (i=1, 2, 3ζ=1, 2) are recorded as:

ρ = ρ (cosφ, sinφ) is the coordinate of the incident point, where ρ is the distance of the incident point relative to the optical axis, φ∈(0, 2π] is the azimuth angle of the incident point relative to the optical axis. i is the imaginary unit, k is the incident light wave number, z is the longitudinal distance from the observation point to the focus; D(ρ) is the aperture function, determined by the lens parameters. When ρ is less than the maximum radius R of the lens, D(ρ) = 1, otherwise it is 0. θ is the incident point and focus The angle between the connecting line and the optical axis. τ _ix (ρ) and τ _iy (ρ) are the elements in the incident light electric field. ρ ₁ = (ρ _1x , ρ _1y ) and ρ ₂ = (ρ _2x , ρ _2y ) are two coordinates of the incident point, introducing a new coordinate expression form

and ρ _d =ρ ₁ -ρ ₂ =(ρ _dx , ρ _dy ).

Represents the Fourier transform of u _αβ , u _αβ (α, β = x, y) is the incident light coherent structure matrix

Medium element. u ₁ and u ₂ are integral variables.

For simplicity of calculation, formula (9) is written as the sum of four integrals with similar forms:

in

So formula (8) is expressed as:

in

It is not difficult to find that the four expressions shown in formula (14) have the same structure, so we use the first one

Take an example to develop the derivation, and make analogies to the other three formulas.

First target

The first partial derivative within square brackets in an expression

in

is the Fourier transform of u _xx ,

for

The Fourier transform of , defined here

A known

represents the Fourier transform of A _1j , that is

So

in

express

The Fourier transform of , defined here

Substituting formula (17) into formula (16), we get:

is the convolution operation symbol.

Next calculate

The second partial derivative within square brackets in the expression:

Integrate u ₁ and ρ _d in equation (19):

in

is the Fourier transform of u _xx ,

for

The Fourier transform of , defined here

at the same time

in

express

The Fourier transform of , defined here

Substituting formula (21) into formula (20), we get:

Substitute formula (18) and formula (22) into

In the expression, we get

After similar derivation, we also get

and

in

Substituting formulas (23)-(26) into formula (13), the orbital angular momentum flux density M _dz (r, z) along the transmission direction is obtained. Since the orbital angular momentum flux density depends on the field strength and circulation, in order to eliminate the influence of the intensity, the orbital angular momentum flux density is normalized. The normalized orbital angular momentum flux density at r is expressed as:

in

Above, formulas (23)-(26) and formula (29) are substituted into formula (28), and finally the normalized orbital angular momentum flux density m _dz (r) along the transmission direction of the tightly focused focal field is obtained. Observing its fractional structure, the numerator is expressed as the sum of 3×4×4 convolution operations, and the denominator is expressed as the sum of 3×4 convolution operations. Quick calculation can be achieved with the help of software Matlab.

The present invention adopts numerical calculation without fuzzy processing such as approximation and error, and the calculation results are accurate and clear.

The invention can change the wavelength, coherence, polarization and lens parameters of the incident light according to the actual situation, without affecting the calculation time and result accuracy, and has wide applicability.

In this embodiment, the incident beam is a partially coherent x-polarized Laguerre Gaussian vortex beam, with angular quantum number l=1 and radial quantum number p=0. The incident electric field is expressed as:

The beam waist width w ₀ =1mm, and the incident wavelength is λ =632.nm. ! Represents factorial. Coherence is a Gaussian function:

Coherence length δ ₀ =0.2mm. In the tight focusing system, the numerical aperture of the lens N _A =0.95, the focal length f=3mm, the refractive index of the surrounding medium n _t =1, and the focal plane z=0 is selected. Then the elements in the A and B matrices are:

θ∈[0, α] is the angle between the line connecting the incident point and the focus and the optical axis, φ∈(0, 2π] is the azimuth angle of the incident point relative to the optical axis. After calculation, we can get:

in

Poynting vector S(r,z)

in,

Represents the Fourier transform of A _ηj (η=1, 2; j=1, 2, 3),

Represents the Fourier transform of B _iζ (i=1, 2,; ζ=1, 2), * indicates conjugate;

Represents the Fourier transform of u _αβ (α, β = x, y).

Substitute formulas (33)-(36) and formula (38) into the m _dz (r) expression

Above we obtain the normalized orbital angular momentum flux density of a partially coherent x-polarized Laguerre Gaussian vortex beam in a tightly focused focal field. In this fraction structure, the numerator is expressed as the sum of 3×4×4 convolution operations, and the denominator is expressed as the sum of 3×4 convolution operations. Quick calculation can be achieved with the help of software Matlab.

Corresponding to the above method embodiment, an embodiment of the present invention also provides a computer device, including:

memory for storing computer programs;

A processor configured to implement the steps of the method for calculating the normalized orbital angular momentum flux density when executing a computer program.

In the embodiment of the present invention, the processor may be a central processing unit (CPU), an application-specific integrated circuit, a digital signal processor, a field programmable gate array, or other programmable logic devices.

The processor may call a program stored in the memory. Specifically, the processor may perform operations in the embodiment of the method for calculating normalized orbital angular momentum flux density.

The memory is used to store one or more programs. The program may include program code, and the program code may include computer operation instructions.

Additionally, the memory may include high-speed random access memory and may also include non-volatile memory, such as at least one magnetic disk storage device or other volatile solid-state storage device.

Corresponding to the above method embodiments, embodiments of the present invention also provide a computer-readable storage medium. A computer program is stored on the computer-readable storage medium. When the computer program is executed by a processor, the above-mentioned calculation of the normalized orbital angular momentum is implemented. Steps of the Flux Density Method.

Those skilled in the art will understand that embodiments of the present application may be provided as methods, systems, or computer program products. Accordingly, the present application may take the form of an entirely hardware embodiment, an entirely software embodiment, or an embodiment that combines software and hardware aspects. Furthermore, the present application may take the form of a computer program product implemented on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, etc.) having computer-usable program code embodied therein.

The present application is described with reference to flowchart illustrations and/or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the application. It will be understood that each process and/or block in the flowchart illustrations and/or block diagrams, and combinations of processes and/or blocks in the flowchart illustrations and/or block diagrams, can be implemented by computer program instructions. These computer program instructions may be provided to a processor of a general purpose computer, special purpose computer, embedded processor, or other programmable data processing device to produce a machine, such that the instructions executed by the processor of the computer or other programmable data processing device produce a use A device for realizing the functions specified in one process or multiple processes of the flowchart and/or one block or multiple blocks of the block diagram.

These computer program instructions may also be stored in a computer-readable memory that causes a computer or other programmable data processing apparatus to operate in a particular manner, such that the instructions stored in the computer-readable memory produce an article of manufacture including the instruction means, the instructions The device implements the functions specified in a process or processes of the flowchart and/or a block or blocks of the block diagram.

These computer program instructions may also be loaded onto a computer or other programmable data processing device, causing a series of operating steps to be performed on the computer or other programmable device to produce computer-implemented processing, thereby executing on the computer or other programmable device. Instructions provide steps for implementing the functions specified in a process or processes of a flowchart diagram and/or a block or blocks of a block diagram.

Obviously, the above-mentioned embodiments are only examples for clear explanation and are not intended to limit the implementation. For those of ordinary skill in the art, other changes or modifications may be made based on the above description. An exhaustive list of all implementations is neither necessary nor possible. The obvious changes or modifications derived therefrom are still within the protection scope of the present invention.

Claims

A method for calculating normalized orbital angular momentum flux density, characterized by including the following steps:

Determine the orbital angular momentum flux expression of the focal field in a tight focusing system based on the general expression of the orbital angular momentum flux along the transmission direction of a partially coherent beam and the cross-spectral density matrix in the non-paraxial case;

According to the integral form of the orbital angular momentum flux expression, the orbital angular momentum flux density expression is obtained;

Decompose the orbital angular momentum flux density expression into 3×4 decomposition formulas with similar forms;

Simplify each decomposition expression into the sum of 4 convolution operations, and write the orbital angular momentum flux density expression as the sum of 3×4×4 convolution operations;

The orbital angular momentum flux density is normalized to obtain the fractional structure orbital angular momentum flux density expression.
The method for calculating normalized orbital angular momentum flux density according to claim 1, characterized in that, according to the general expression of the orbital angular momentum flux of a partially coherent beam along the transmission direction and the cross spectrum in the non-paraxial case Density matrix methods for determining the orbital angular momentum flux form of the focal field in a tightly focused system include:

The general expression for determining the orbital angular momentum flux along the propagation direction of a partially coherent beam is as follows:

where c represents the speed of light in vacuum, S t represents the average energy through the beam cross-section, which is described by the integral of the Poynting vector S(r, z), and <xθ y > and <yθ x > represent the crossed second-order moment ;

In a tight focusing system, a 3×3 cross-spectral density matrix Φ is used to describe the second-order correlation characteristics of a partially coherent vector beam near the focal plane:

where r 1 = (x 1 , y 1 ) and r 2 = (x 2 , y 2 ) represent the cross-sectional coordinates of two observation points on the same section near the focal field, z represents the longitudinal distance from the observation point to the focus, τ (r 1 , z) and τ (r 2 , z) represent the electric fields at r 1 and r 2 respectively, <> represents the ensemble average,
Represents the transposed complex conjugate, Φ ij (i, j=1, 2, 3) represents the 9 cross-spectral density matrix elements in the Φ matrix;

Introducing a new coordinate expression form
and r d =r 1 -r 2 = (r dx , r dy ), write formula (2) as

The Poynting vector S(r,z) is represented by the polarization matrix element in formula (3):

Then the sum of the two crossed second-order moments in formula (1) are expressed as:

where i is the imaginary unit and k is the incident light wave number; by substituting formula (5) and formula (6) into formula (1), the orbital angular momentum flux expression of the focal field under the tight focusing system is obtained as follows:
The method for calculating normalized orbital angular momentum flux density according to claim 2, characterized in that the method of obtaining the orbital angular momentum flux density expression according to the integral form of the orbital angular momentum flux expression includes:

The orbital angular momentum flux density M dz (r, z) along the transmission direction is obtained from the integral form of equation (7) as follows:

Write the matrix elements Φ 11 ( rs , r d , z), Φ 22 ( rs , r d , z) and Φ 33 ( rs , r d , z) of the cross-spectral density matrix in formula (8) as expressions The formula is as follows:

where f is the focal length of the lens, λ is the wavelength of the incident light,
represents the Fourier transform of A ηj ,
represents the Fourier transform of B iζ , and * represents the conjugate.
The method for calculating normalized orbital angular momentum flux density according to claim 3, characterized in that the method of decomposing the orbital angular momentum flux density expression into 3×4 decomposition formulas with similar forms includes:

Write formula (9) as the sum of four integrals of similar form

in

So formula (8) is expressed as

in
The method for calculating normalized orbital angular momentum flux density according to claim 4, characterized in that each decomposition formula is simplified into the sum of four convolution operations, and the orbital angular momentum flux density expression is Methods written as the sum of 3×4×4 convolution operations include:

Perform partial derivatives, Dirac functions and Fourier transform operations on the decomposition formula respectively, and simplify each decomposition formula into the sum of 4 convolution operations to get:

in

Substituting formulas (23)-(26) into formula (13), the orbital angular momentum flux density M dz (r, z) along the transmission direction is obtained.
The method for calculating normalized orbital angular momentum flux density according to claim 5, characterized in that the method of performing partial derivatives, Dirac functions and Fourier transform operations on the decomposition formula includes:

against
The first partial derivative inside square brackets in an expression:

Among them, δ() represents the Dirac function, which satisfies δ(s)=∫exp(-2πisv)dv; δ′() represents the first derivative of the Dirac function, which satisfies

Use ∫f(v)δ′(vv 0 )dv=-f′(v 0 ) to integrate u 1 of formula (15), and use Fourier transform to integrate ρ d to get:

in
is the Fourier transform of u xx ,
for
The Fourier transform of

A known
represents the Fourier transform of A 1j , that is
So:

in
Substituting formula (17) into formula (16), we get:

in
is the convolution operation symbol.
The method for calculating normalized orbital angular momentum flux density according to claim 6, characterized in that the method of normalizing the orbital angular momentum flux density to obtain a fractional structure orbital angular momentum flux density expression includes: :

Normalizing the orbital angular momentum flux density, the normalized orbital angular momentum flux density at r is expressed as:

in
and ω are the reduced Planck constant and the angular frequency of light respectively, and the expression of the Poynting vector S(r, z) is obtained:

Substituting formulas (23)-(26) and formula (29) into formula (28), the normalized orbital angular momentum flux density m dz (r) along the transmission direction of the tightly focused focal field is finally obtained.
The method for calculating normalized orbital angular momentum flux density according to claim 1 or 7, characterized in that the molecule in the fractional structure orbital angular momentum flux density expression is represented by 3×4×4 convolution operations The denominator is expressed as the sum of 3×4 convolution operations.
A computer device, including a memory, a processor and a computer program stored in the memory and executable on the processor, characterized in that when the processor executes the program, it implements any one of claims 1 to 8 Method steps.
A computer-readable storage medium on which a computer program is stored, characterized in that when the program is executed by a processor, the steps of the method described in any one of claims 1 to 8 are implemented.