WO2024016450A1 - Method for calculating tight focusing three-dimensional spin density by using mode decomposition of optical system - Google Patents

Abstract

A method for calculating tight focusing three-dimensional spin density by using mode decomposition of an optical system. The method comprises: enabling partially coherent light emitted by a light source, a tight focusing lens, and a space behind the tight focusing lens to be equivalent to a nonlinear optical system (S1); calculating an output signal of the nonlinear optical system, the output signal being expressed as a 3×3 cross-spectral density function on a focal plane of the tight focusing lens (S2); by using a mode decomposition theory, converting the output signal of the nonlinear optical system into incoherent superposition of output signals of a series of linear optical systems (S3); and finally obtaining three-dimensional spin density represented by the 3×3 cross-spectral density function on the focal plane of the tight focusing lens (S4). The calculation difficulty and the calculation time for tight focusing spin density are reduced, and the research on the interaction of optical spin orbits is promoted.

Description

Method for calculating tightly focused three-dimensional spin density using mode decomposition of optical systems Technical field

The invention relates to the field of optical technology, and in particular to a method for calculating tightly focused three-dimensional spin density using mode decomposition of an optical system.

Background technique

In addition to linear momentum, angular momentum is also an important property of light. Light has two different types of angular momentum, spin angular momentum and orbital angular momentum, which are respectively related to the circular polarization state and spiral phase of light. In the past few decades of research, it has been discovered that when light interacts with matter, it can cause linear momentum to be transferred from light to matter, thereby exerting a force on matter and making technologies such as optical tweezers possible. When the angular momentum of light interacts with a particle, spin angular momentum causes the particle to spin, and orbital angular momentum causes the particle to rotate around the optical axis. Therefore, spin angular momentum and orbital angular momentum can also be distinguished based on the different mechanical effects they exhibit when interacting with particles. The spin angular momentum and orbital angular momentum of light, as two different degrees of freedom, are independent and conserved under free space propagation. In fact, optical spin-orbit interactions occur under various conditions, such as light-matter interactions in anisotropic media, evanescent waves, scattering, and tightly focused systems. The coupling between spin angular momentum and orbital angular momentum is called spin-to-orbit conversion. The coupling between the spin angular momentum and the external orbital angular momentum will cause a spin-related displacement, that is, the spin Hall effect.

The process of focusing a beam of light by a lens with a large numerical aperture is called tight focusing. In the past few decades, the characteristics of a tightly focused light field have been extensively studied and applied to optical microscopy, capture and material processing. Therefore, the spin-orbit interaction in tightly focused systems has received widespread attention. In 2007, Zhao et al. discovered that after tightly focusing circularly polarized light, spin angular momentum can be converted into orbital angular momentum, which is called spin-to-orbit conversion. Recently, Yao et al. discovered that after a vortex beam carrying no spin is tightly focused, the orbital angular momentum can be converted into longitudinal spin angular momentum, which is called orbit-to-spin conversion. The influence of parameters such as incident light topology, beam waist width, and the ratio of pupil radius to beam waist radius on this orbit-to-spin conversion has been extensively studied. The calculation of tightly focused spin density is the basis of these research works. The spin density distribution at the focal plane after tightly focusing completely coherent light can be obtained by calculating only a double integral. However, for partially coherent light after focusing The spin density distribution at the focal plane will involve a quadruple integral, which increases the computational difficulty and time, hindering the study of tightly focused spin-orbit interactions.

Contents of the invention

The technical problem to be solved by the present invention is to provide a method for calculating tightly focused three-dimensional spin density using mode decomposition of an optical system that reduces calculation difficulty and shortens calculation time.

In order to solve the above problems, the present invention provides a method for calculating tightly focused three-dimensional spin density using mode decomposition of an optical system, which includes the following steps:

S1. Equivalent the partially coherent light emitted by the light source, the tight focusing lens and the space behind the tight focusing lens to a nonlinear optical system;

S2. According to the nonlinear system theory and Richard Wolf diffraction theory, use the input signal and pulse function of the nonlinear optical system to calculate the output signal of the nonlinear optical system. The output signal of the nonlinear optical system Expressed as a 3×3 cross-spectral density function on the focal plane of a tightly focused lens;

S3. Use mode decomposition theory to convert the output signal of the nonlinear optical system into an incoherent superposition of the output signals of a series of linear optical systems;

S4. According to the definition of spin angular momentum, calculate the three-dimensional spin density represented by the 3×3 cross-spectral density function on the focal plane of the tight focusing lens.

As a further improvement of the present invention, the 3×3 cross-spectral density function on the focal plane of the tight focusing lens is:

Among them, C(r′1, r′2) is the 3×3 cross spectral density function on the focal plane of the tight focusing lens; C′(r ₁ , r ₂ ) is the cross spectrum of the rear surface of the tight focusing lens. Density function; r ₁ and r ₂ respectively represent the spatial coordinates of two points at the light source, r ₁ = (x ₁ , y ₁ ), r ₁ = (x ₂ , y ₂ ); r′ ₁ and r′ ₂ respectively represent the compact The spatial coordinates of the two points r ₁ and r ₂ after focusing, r′ ₁ = (x ₁ , y′ ₁ ), r′ ₂ = (x′ ₂ , y′ ₂ ); i represents an imaginary number; f and λ represent compact numbers respectively. The focal length of the focusing lens and the wavelength of the partially coherent light emitted by the light source; k ₁ and k ₂ respectively represent the corresponding coordinates of points r ₁ and r ₂ in the wave number space, k ₁ = (k _1x ,k _1y ,k _1z ), k ₂ = (k _2x ,k _2y ,k _2z ); dS ₁ = sinθ ₁ dθ ₁ dφ ₁ and dS ₂ = sinθ ₂ dθ ₂ dφ ₂ represent the integral differential elements at points r ₁ and r ₂ respectively, φ ₁ and φ ₂ are the azimuth angles of points r ₁ and r ₂ respectively, and φ ₁ = arctan (y ₁ /x ₁ ), φ ₂ = arctan (y ₂ /x ₂ ); θ ₁ is the angle between point r ₁ and the focus of the tight focusing lens The angle between the line and the optical axis, θ ₂ is the angle between the line connecting point r ₂ and the focus of the tight focusing lens and the optical axis, and satisfies 0 ≤ θ ₁ ≤ arcsin (NA/n _t ), 0 ≤θ ₂ ≤arcsin(NA/n _t ); NA and n _t respectively represent the numerical aperture of the tight focusing lens and the refractive index of the imaging space.

As a further improvement of the present invention,

sinθ ₁ ＝k _1z /k ₀ , x ₁ ＝-f k _1x /k ₀ , y ₁ ＝-f k _1y /k ₀

sinθ ₂ =k _2z /k ₀ , x ₂ =-f k _1z /k ₀ , y ₁ =-f k _1y /k ₀

in,

Represents the wave number of partially coherent light emitted by the light source; substituted into formula (7), we get:

Let r′ ₁ =r′ ₂ =r′. At this time, C(r′ ₁ , r′ ₂ ) is expressed as:

make:

q ₂ (r′, k ₁ , k ₂ )=C (k ₁ , k ₂ )h ^* (r′, k ₁ )h (r′, k ₂ )

At this time, formula (9) is expressed as:

Among them, q ₂ (r′, k ₁ , k ₂ ) is the pulse function of the nonlinear optical system, and is the response of the output point r′ to the pulses of points k ₁ and k ₂ .

As a further improvement of the present invention, step S3 includes:

S31. Express the cross-spectral density function at the light source in the form of mode decomposition;

S32. Substitute the mode decomposition form of the cross-spectral density function at the light source into the 3×3 cross-spectral density function on the focal plane of the tight focusing lens to convert the output signal of the nonlinear optical system into a series of linear optical systems incoherent superposition of the output signals.

As a further improvement of the present invention, the cross-spectral density function at the light source is expressed in the form of mode decomposition, as follows:

Among them, C (k ₁ , k ₂ ) represents the cross-spectral density function at the light source; k ₁ and k ₂ respectively represent the corresponding coordinates of the two points at the light source in the wave number space;

and

Represents the n-th completely coherent mode decomposed by the cross-spectral density function of points k ₁ and k ₂ respectively, n=1, 2,...,N, N is the total number of completely coherent modes; β _n represents the n-th completely coherent mode Weight, T represents transpose;

S32. Substitute formula (11) into the 3×3 cross-spectral density function on the focal plane of the tight focusing lens to obtain:

Among them, C(r′, r′) is the 3×3 cross-spectral density function on the focal plane of the tight focusing lens;

Represents the response of output point r′ to a single pulse of input point k. The integral term in the absolute value in formula (12) is equivalent to the output signal of the linear system.

As a further improvement of the present invention, the three-dimensional spin density is expressed by the 3×3 cross-spectral density function on the focal plane of the tight focusing lens as:

Among them, S _x (r′), S _y (r′), and S _z (r′) respectively represent the components of the spin density in the three directions of space x, y, and z; double quotes indicate taking the imaginary part; ε ₀ represents the dielectric constant in vacuum; ω is the angular frequency of the light source; C _yz (r′, r′), C _zy (r′, r′), C _zx (r′, r′), C _xz (r ′, r′), C _xy (r′, r′) and C _yx (r′, r′) are elements in the 3×3 cross-spectral density function on the focal plane of the tight focusing lens.

The present invention also provides an electronic device, including 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 any one of the above methods are implemented. .

The present invention also provides a computer-readable storage medium on which a computer program is stored. When the program is executed by a processor, the steps of any one of the above methods are implemented.

The invention also provides a system for calculating tightly focused three-dimensional spin density using mode decomposition of an optical system, which includes the following modules:

The equivalent module is used to equate the partially coherent light emitted by the light source, the tight focusing lens and the space behind the tight focusing lens into a nonlinear optical system;

The output signal calculation module is used to calculate the output signal of the nonlinear optical system using the input signal and pulse function of the nonlinear optical system according to the nonlinear system theory and Richard Wolf diffraction theory. The output signal of the optical system is expressed as a 3×3 cross-spectral density function on the focal plane of the tight focusing lens;

A mode decomposition module for converting the output signal of the nonlinear optical system into an incoherent superposition of the output signals of a series of linear optical systems using mode decomposition theory;

The three-dimensional spin density calculation module is used to calculate the three-dimensional spin density represented by the 3×3 cross-spectral density function on the focal plane of the tight focusing lens according to the definition of spin angular momentum.

Beneficial effects of the present invention:

The present invention equates the partially coherent light emitted by the light source, the tight focusing lens and the space behind the tight focusing lens into a nonlinear optical system, and calculates the output signal of the nonlinear optical system. The output signal is expressed as 3 on the focal plane of the tight focusing lens. ×3 cross spectral density function, and then use the mode decomposition theory to convert the output signal of the nonlinear optical system into an incoherent superposition of the output signals of a series of linear optical systems, and finally obtain a 3 × 3 cross spectral density function on the focal plane of the tight focusing lens Three-dimensional spin density represented by the spectral density function. The invention reduces the calculation difficulty and calculation time of tightly focused spin density and promotes research on optical spin-orbit interaction.

The above description is only an overview of the technical solution of the present invention. In order to have a clearer understanding of the technical means of the present invention, it can be implemented according to the content of the description, and in order to make the above and other objects, features and advantages of the present invention more obvious and understandable. , the following is a detailed description of the preferred embodiments, together with the accompanying drawings.

Description of drawings

Figure 1 is a flow chart of a method for calculating tightly focused three-dimensional spin density using mode decomposition of an optical system in a preferred embodiment 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.

In order to solve the problem of long time and difficulty in calculating tightly focused spin density, the present invention utilizes the mode decomposition theory of optical systems. Generally, an optical system illuminated by completely coherent light is called a linear optical system, and an optical system illuminated by partially coherent light is called a nonlinear optical system. For a linear optical system, the relationship between its input signal and output signal can be obtained through simple linear transformation. For nonlinear optical systems, the calculation of its output signal will involve complex integrals and require a lot of calculation time. By using the mode decomposition theory of nonlinear optical systems, the nonlinear optical system is decomposed into a series of superpositions of linear optical systems. The output signal of the linear optical system is first obtained, and then these output signals are superimposed to obtain the output of the nonlinear optical system. signal, thus reducing the computational difficulty and calculation time of nonlinear optical systems. The partially coherent light tight focusing process is a nonlinear optical system. By using the mode decomposition theory of nonlinear optical systems, the output signal of the tight focusing system is expressed as the superposition of a series of linear signals, and the spin density can be calculated using the output signal. This method reduces the computational difficulty and time of tightly focused spin density and will promote the study of optical spin-orbit interactions. Detailed introduction below:

As shown in Figure 1, it is a method for calculating tightly focused three-dimensional spin density using mode decomposition of an optical system in a preferred embodiment of the present invention, which includes the following steps:

S1. Equivalent the partially coherent light emitted by the light source, the tight focusing lens and the space behind the tight focusing lens to a nonlinear optical system;

S2. According to the nonlinear system theory and Richard Wolf diffraction theory, use the input signal and pulse function of the nonlinear optical system to calculate the output signal of the nonlinear optical system. The output signal of the nonlinear optical system Expressed as a 3×3 cross-spectral density function on the focal plane of a tightly focused lens;

Among them, the input signal of the nonlinear optical system is the transmittance function determined by the tight focusing lens and polarization state, and the pulse function is determined by the cross-spectral density function C(r ₁ , r ₂ ) of the light source and the coordinates of the light source surface (x, y) and the wave number space (k _x , k _y ), the output signal represents the cross-spectral density function C (r′ ₁ , r′ ₂ ) at the focal plane.

The linear optical system only needs to calculate a simple linear transformation, as follows:

Among them, C ^* (x') represents the output signal of the linear system, t ^* (x) represents the input signal, and q (x', x) is the pulse signal of the output point x' at the input point x.

The relationship between the output signal and the input signal of the nonlinear optical system is calculated as:

Among them, C(x′) and t(x) represent the output signal and input signal of the nonlinear optical system respectively, and q ₂ (x′, x ₁ , x ₂ ) is the kernel function of the second-order Volterra series ( Volterra kernel), which represents the double-pulse signal of the output point x′ at the input point x ₁ and x ₂ , that is, the pulse function. q ₂ (x′, x ₁ , x ₂ ) is a 6D complex function, which makes the calculation of nonlinear optical systems difficult. The nonlinear optical system can be expressed as a series of incoherent superpositions of linear systems through mode decomposition theory, thereby simplifying the above calculations. The present invention applies this theory to the calculation of spin density under tight focusing systems.

In the nonlinear optical system, it is known that the partially coherent light emitted by the light source is a transverse electric field propagating along the z-axis. The ensemble average of the cross-spectral density function of this partially coherent light is expressed as:

C(r ₁ , r ₂ )＝<U ^* (r ₁ )U ^T (r ₂ )> (3)

Among them, U(r ₁ ) and U(r ₂ ) represent the optical signals (amplitudes) of the two points r ₁ and r ₂ at the light source respectively, the angle brackets represent the ensemble average, * represents the complex conjugate, and T represents the transformation of the matrix. Set. The relationship between the optical signals on the front and rear surfaces of the tight focusing lens is:

U′(r)＝t ₀ (r)U(r) (4)

Among them, t ₀ (r) represents the transmittance function; U′(r) represents the optical signal on the rear surface of the tightly focused lens, and U(r) represents the optical signal on the front surface of the tightly focused lens. When the polarization state of the incident light is cylindrical vector polarization, its transmittance function can be expressed as:

in,

is the transmittance function of a tightly focused lens; φ ₀ represents the initial phase of cylindrical vector polarization; when φ ₀ =0° and 90° represent radial polarization and angular polarization respectively. The first and second matrices on the right side of formula (5) respectively represent the x, y, and z directions of the partially coherent light of radial polarization (RP) and angular polarization (AP) in the space after passing through the tightly focusing lens. Portion size. Therefore, the cross-spectral density function of the back surface of a tight focusing lens is expressed as:

It is a 3×3 matrix. Using the Richard Wolf diffraction formula, the 3×3 cross-spectral density function at the tightly focused focal plane can be obtained, as follows:

Among them, C (r ₁ ′, r ₂ ′) is the 3×3 cross-spectral density function on the focal plane of the tight focusing lens; C′ (r ₁ , r ₂ ) is the cross spectrum of the rear surface of the tight focusing lens Density function; r ₁ and r ₂ respectively represent the spatial coordinates of two points at the light source, r ₁ = (x ₁ , y ₁ ), r ₁ = (x ₂ , y ₂ ); r′ ₁ and r′ ₂ respectively represent the compact The spatial coordinates of the two points r ₁ and r ₂ after focusing, r′ ₁ = (x′ ₁ , y′ ₁ ), r′ ₂ = (x′ ₂ , y′ ₂ ); i represents an imaginary number; f and λ represent respectively The focal length of the tight focusing lens and the wavelength of the partially coherent light emitted by the light source; k ₁ and k ₂ respectively represent the corresponding coordinates of points r ₁ and r ₂ in the wave number space, k ₁ = (k _1x ,k _1y ,k _1z ), k ₂ = (k _2x ,k _2y ,k _2z ); dS ₁ = sinθ ₁ dθ ₁ dφ ₁ and dS ₂ = sinθ ₂ dθ ₂ dφ ₂ represent the integral differential elements at points r ₁ and r ₂ respectively, φ ₁ and φ ₂ are the azimuth angles of points r ₁ and r ₂ respectively, and φ ₁ =arctan(y ₁ /X ₁ ), φ ₂ =arctan(y ₂ /x ₂ ); θ ₁ is the point r ₁ and the focus of the tight focusing lens The angle between the line connecting point r ₂ and the focus of the tight focusing lens and the optical axis, θ ₂ is the angle between the line connecting point r 2 and the focus of the tight focusing lens and the optical axis, and satisfies 0 ≤ θ ₁ ≤ arcsin (NA/n _t ), 0≤θ ₂ ≤arcsin(NA/n _t ); NA and n _t respectively represent the numerical aperture of the tight focusing lens and the refractive index of the imaging space.

in:

sinθ ₁ ＝k _1z /k ₀ , x ₁ ＝-f k _1x /k ₀ , y ₁ ＝-f k _1y /k ₀

sinθ ₂ =k _2z /k ₀ , x ₂ =-f k _1x /k ₀ , y ₁ =-f k _1y /k ₀

in,

Represents the wave number of partially coherent light emitted by the light source; substituted into formula (7), we get:

Let r′ ₁ =r′ ₂ =r′. At this time, C(r ₁ ′, r ₂ ′) is expressed as:

make:

q ₂ (r′, k ₁ , k ₂ )=C (k ₁ , k ₂ )h ^* (r′, k ₁ )h (r′, k ₂ )

At this time, formula (9) is expressed as:

Among them, q ₂ (r′, k ₁ , k ₂ ) is the pulse function of the nonlinear optical system, and is the response of the output point r′ to the pulses of points k ₁ and k ₂ . Therefore, this nonlinear optical system can also be called a bilinear system.

S3. Use mode decomposition theory to convert the output signal of the nonlinear optical system into an incoherent superposition of the output signals of a series of linear optical systems; specifically including:

S31. Express the cross-spectral density function at the light source in the form of mode decomposition, as follows:

Among them, C (k ₁ , k ₂ ) represents the cross-spectral density function at the light source; k ₁ and k ₂ respectively represent the corresponding coordinates of the two points at the light source in the wave number space;

and

Represents the n-th completely coherent mode decomposed by the cross-spectral density function of points k ₁ and k ₂ respectively, n=1, 2,...,N, N is the total number of completely coherent modes; β _n represents the n-th completely coherent mode Weight, T represents transpose;

S32. Substitute the mode decomposition form of the cross-spectral density function at the light source into the 3×3 cross-spectral density function on the focal plane of the tight focusing lens to convert the output signal of the nonlinear optical system into a series of linear optical systems incoherent superposition of the output signals.

Specifically, formula (11) is substituted into the 3×3 cross-spectral density function on the focal plane of the tight focusing lens to obtain:

Among them, C(r′, r′) is the 3×3 cross-spectral density function on the focal plane of the tight focusing lens;

Represents the response of output point r′ to a single pulse of input point k. The integral term in the absolute value in formula (12):

Equivalent to the output signal of a linear system, that is, a pattern system. Therefore, formula (12) can be considered as a series of impulse responses as

The superposition of modular squares of a system of modes.

S4. According to the definition of spin angular momentum, calculate the three-dimensional spin density represented by the 3×3 cross-spectral density function on the focal plane of the tight focusing lens.

For any time domain harmonic light field with an angular frequency of ω = kc (c is the speed of light in vacuum), its spin density at the ρ point in space can be defined as:

Among them, ε ₀ represents the dielectric constant in vacuum; ω is the angular frequency of the light source. Using mode decomposition theory, the spin density of point r′ at the focal plane can be expressed as:

Among them, ψ _n (r′) represents the expansion mode of the cross-spectral density function at the focal plane. The components of the spin density in the three directions of space x, y, and z are expressed as:

The three-dimensional spin density is expressed by the 3×3 cross-spectral density function on the focal plane of the tight focusing lens as:

Among them, S _x (r′), S _y (r′), and S _z (r′) respectively represent the components of the spin density in the three directions of space x, y, and z; double quotes indicate taking the imaginary part; C _yz (r′, r′), C _zy (r′, r′), C _zx (r′, r′), C _xz (r′, r′), C _xy (r′, r′) and C _yx (r′, r′) is the element in the 3×3 cross-spectral density function on the focal plane of the tight focusing lens.

In one embodiment, the method of calculating tightly focused three-dimensional spin density using mode decomposition of an optical system in the present invention is used to calculate the three-dimensional spin density of a radially polarized Gaussian Scher mode beam after being tightly focused. The specific calculation process is as follows:

The ensemble average of the cross-spectral density function of a radially polarized Gaussian Scher mode beam propagating along the z-axis is expressed as:

in,

and

are the spectral intensities of the incident light in front of the tight focusing lens at points r ₁ and r ₂ respectively,

is the radial polarization unit vector,

is the coherence distribution between the two points r ₁ and r ₂ , * represents the complex conjugate, and T represents the transpose of the matrix. The relationship between the light field amplitudes of the two points r ₁ and r ₂ on the front and rear surfaces of the tight focusing lens is:

Among them, t ₀ (r ₁ ) and t ₀ (r ₂ ) respectively represent the transmittance function of the tight focusing lens to the light source at two points r ₁ and r ₂ . It is known that the polarization state of the incident light is radially polarized, t ₀ (r ₁ ) and t ₀ (r ₂ ) can be expressed as:

and

represent the transmittance functions of the tightly focused lens at θ ₁ and θ ₂ respectively. The matrix on the right side of equation (20) represents the x, y, and z directions of the radially polarized (RP) Gaussian Scher mode beam after passing through the lens. Portion size. Putting the above formula into formula (6), we can get the cross-spectral density function of the back surface of the tight focusing lens:

Abbreviate formula (21) as:

It is a 3×3 matrix, where:

in:

sinθ ₁ ＝k _1z /k ₀ , x ₁ ＝-f k _1x /k ₀ , y ₁ ＝-f k _1y /k ₀

sinθ ₂ =k _2z /k ₀ , x ₂ =-f k _1x /k ₀ , y ₁ =-f k _1y /k ₀

in,

Represents the wave number of the partially coherent light emitted by the light source; then formula (23) can be expressed as:

Substituting formula (23) into formula (8), the cross-spectral density at the tightly focused focal plane can be obtained:

Let r′ ₁ =r′ ₂ =r′, at this time, C _xx (r ₁ ′, r ₂ ′) is expressed as:

make:

Then formula (24) can be written as:

Among them, C _xx (r′, r′) is the output signal after the nonlinear optical system, that is, the first item in the 3×3 cross-spectral density function matrix of the field.

The cross-spectral density function at the light source is expressed in the form of mode decomposition as:

in,

and β _n respectively represent the expansion mode of the incident light and its corresponding weight. Substituting formula (28) into formula (27) we can get:

in,

Among them, C _nxx (r′, r′) is the n-th completely coherent mode decomposed by C _xx (r′, r′), F is the Fourier transform symbol, and C _nxx (r′, r′) can be Fast calculation of Lieye transform. The integral term in the absolute value of formula (29) is the output signal of the linear system with completely coherent light incident, that is, the mode system. Formula (29) can be considered as a series of impulse responses as

Coherent superposition of modular squares of a system of modes. Similarly, the other eight items of the cross-spectral density function matrix are C _xy (r ₁ ′, r ₁ ′), C _xz (r ₁ ′, r ₁ ′), C _yx (r ₁ ′, r ₁ ′), C _yy (r ₁ ′, r ₁ ′), C _yz (r ₁ ′, r ₁ ′), C _zx (r ₁ ′, r ₁ ′), C _zy (r ₁ ′, r ₁ ′), C _zz (r ₁ ′, r ₁ ′) can be obtained in the same way. Using these 9 cross-spectral density formulas, the spin density component can be obtained:

That is formula (17).

The present invention equates the partially coherent light emitted by the light source, the tight focusing lens and the space behind the tight focusing lens into a nonlinear optical system, and calculates the output signal of the nonlinear optical system, which is expressed as a 3×3 signal on the focal plane of the tight focusing lens. Cross spectral density function, and then use mode decomposition theory to convert the output signal of the nonlinear optical system into an incoherent superposition of the output signals of a series of linear optical systems, and finally obtain a 3×3 cross spectral density function on the focal plane of the tight focusing lens represents the three-dimensional spin density. The invention reduces the calculation difficulty and calculation time of tightly focused spin density and promotes research on optical spin-orbit interaction.

A preferred embodiment of the present invention also discloses an electronic 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, it implements what is described in the above embodiment. Method steps.

A preferred embodiment of the present invention also discloses a computer-readable storage medium on which a computer program is stored. When the program is executed by a processor, the steps of the method described in the above embodiment are implemented.

The preferred embodiment of the present invention also discloses a system for calculating tightly focused three-dimensional spin density using mode decomposition of an optical system, which includes the following modules:

The equivalent module is used to equate the partially coherent light emitted by the light source, the tight focusing lens and the space behind the tight focusing lens into a nonlinear optical system;

The output signal calculation module is used to calculate the output signal of the nonlinear optical system using the input signal and pulse function of the nonlinear optical system according to the nonlinear system theory and Richard Wolf diffraction theory. The output signal of the optical system is expressed as a 3×3 cross-spectral density function on the focal plane of the tight focusing lens;

A mode decomposition module for converting the output signal of the nonlinear optical system into an incoherent superposition of the output signals of a series of linear optical systems using mode decomposition theory;

The three-dimensional spin density calculation module is used to calculate the three-dimensional spin density represented by the 3×3 cross-spectral density function on the focal plane of the tight focusing lens according to the definition of spin angular momentum.

The system for calculating tightly focused three-dimensional spin density using mode decomposition of an optical system in the embodiment of the present invention is used to implement the aforementioned method of calculating tightly focused three-dimensional spin density using mode decomposition of an optical system. Therefore, the specific implementation of the system can be seen The foregoing embodiments of a method for calculating tightly focused three-dimensional spin density using mode decomposition of an optical system, therefore, for its specific implementation, refer to the corresponding description of the above method embodiments and will not be introduced here.

In addition, since the system for calculating tightly focused three-dimensional spin density using mode decomposition of an optical system in this embodiment is used to implement the aforementioned method of calculating tightly focused three-dimensional spin density using mode decomposition of an optical system, its function is the same as that of the above method. The functions correspond to each other and will not be described again here.

The above embodiments are only preferred embodiments to fully illustrate the present invention, and the protection scope of the present invention is not limited thereto. Equivalent substitutions or transformations made by those skilled in the art on the basis of the present invention are within the protection scope of the present invention. The protection scope of the present invention shall be determined by the claims.

Claims (9)

1. A method for calculating tightly focused three-dimensional spin density using mode decomposition of an optical system, which is characterized by including the following steps:

   S1. Equivalent the partially coherent light emitted by the light source, the tight focusing lens and the space behind the tight focusing lens to a nonlinear optical system;

   S2. According to the nonlinear system theory and Richard Wolf diffraction theory, use the input signal and pulse function of the nonlinear optical system to calculate the output signal of the nonlinear optical system. The output signal of the nonlinear optical system Expressed as a 3×3 cross-spectral density function on the focal plane of a tightly focused lens;

   S3. Use mode decomposition theory to convert the output signal of the nonlinear optical system into an incoherent superposition of the output signals of a series of linear optical systems;

   S4. According to the definition of spin angular momentum, calculate the three-dimensional spin density represented by the 3×3 cross-spectral density function on the focal plane of the tight focusing lens.
2. The method of calculating tightly focused three-dimensional spin density using mode decomposition of an optical system as claimed in claim 1, characterized in that the 3×3 cross-spectral density function on the focal plane of the tightly focused lens is:

   

   Among them, C (r ₁ ′, r ₂ ′) is the 3×3 cross-spectral density function on the focal plane of the tight focusing lens; C′ (r ₁ , r ₂ ) is the cross spectrum of the rear surface of the tight focusing lens Density function; r ₁ and r ₂ respectively represent the spatial coordinates of two points at the light source, r ₁ = (x ₁ , y ₁ ), r ₁ = (x ₂ , y ₂ ); r′ ₁ and r′ ₂ respectively represent the compact The spatial coordinates of the two points r ₁ and r ₂ after focusing, r′ ₁ = (x′ ₁ , y′ ₁ ), r′ ₂ = (x′ ₂ , y′ ₂ ); i represents an imaginary number; f and λ represent respectively The focal length of the tight focusing lens and the wavelength of the partially coherent light emitted by the light source; k ₁ and k ₂ respectively represent the corresponding coordinates of points r ₁ and r ₂ in the wave number space, k ₁ = (k _1x , k _1y , k _1z ), k ₂ = (k _2x , k _2y , k _2z ); dS ₁ = sinθ ₁ dθ ₁ dφ ₁ and dS ₂ = sinθ ₂ dθ ₂ dφ ₂ respectively represent the integral differential elements at points r ₁ and r ₂ , φ ₁ and φ ₂ are the azimuth angles of points r ₁ and r ₂ respectively, and φ ₁ =arctan(y ₁ /x ₁ ), φ ₂ =arctan(y ₂ /x ₂ ); θ ₁ is the point r ₁ and the focus of the tight focusing lens The angle between the line connecting point r ₂ and the focus of the tight focusing lens and the optical axis, θ ₂ is the angle between the line connecting point r 2 and the focus of the tight focusing lens and the optical axis, and satisfies 0 ≤ θ ₁ ≤ arcsin (NA/n _t ), 0≤θ ₂ ≤arcsin(NA/n _t ); NA and n _t respectively represent the numerical aperture of the tight focusing lens and the refractive index of the imaging space.
3. The method of calculating tightly focused three-dimensional spin density using mode decomposition of an optical system as claimed in claim 2, characterized in that:

   sinθ ₁ ＝k _1z /k ₀ , x ₁ ＝-fk _1x /k ₀ , y ₁ ＝-fk _1y /k ₀

   sinθ ₂ =k _2z /k ₀ , x ₂ =-fk _1x /k ₀ , y ₁ =-fk _1y /k ₀

   in,

   

   Represents the wave number of partially coherent light emitted by the light source; substituted into formula (7), we get:

   

   Let r′ ₁ =r′ ₂ =r′. At this time, C(r ₁ ′, r ₂ ′) is expressed as:

   

   make:

   

   

   q ₂ (r′, k ₁ , k ₂ )=C (k ₁ , k ₂ )h ^* (r′, k ₁ )h (r′, k ₂ )

   At this time, formula (9) is expressed as:

   

   Among them, q ₂ (r′, k ₁ , k ₂ ) is the pulse function of the nonlinear optical system, and is the response of the output point r′ to the pulses of points k ₁ and k ₂ .
4. The method for calculating tightly focused three-dimensional spin density using mode decomposition of an optical system as claimed in claim 1, wherein step S3 includes:

   S31. Express the cross-spectral density function at the light source in the form of mode decomposition;

   S32. Substitute the mode decomposition form of the cross-spectral density function at the light source into the 3×3 cross-spectral density function on the focal plane of the tight focusing lens to convert the output signal of the nonlinear optical system into a series of linear optical systems incoherent superposition of the output signals.
5. The method of calculating tightly focused three-dimensional spin density using mode decomposition of an optical system as claimed in claim 4, characterized in that the cross-spectral density function at the light source is expressed in the form of mode decomposition, as follows:

   

   Among them, C (k ₁ , k ₂ ) represents the cross-spectral density function at the light source; k ₁ and k ₂ respectively represent the corresponding coordinates of the two points at the light source in the wave number space;

   

   and

   

   Represents the n-th completely coherent mode decomposed by the cross-spectral density function of points k ₁ and k ₂ respectively, n=1, 2,...,N, N is the total number of completely coherent modes; β _n represents the n-th completely coherent mode Weight, T represents transpose;

   S32. Substitute formula (11) into the 3×3 cross-spectral density function on the focal plane of the tight focusing lens to obtain:

   

   Among them, C(r′, r′) is the 3×3 cross-spectral density function on the focal plane of the tight focusing lens;

   

   Represents the response of output point r′ to a single pulse of input point k. The integral term in the absolute value in formula (12) is equivalent to the output signal of the linear system.
6. The method for calculating tightly focused three-dimensional spin density using mode decomposition of an optical system as claimed in claim 5, characterized in that the three-dimensional spin density is represented by a 3×3 cross-spectral density function on the focal plane of the tightly focused lens as :

   

   

   

   Among them, S _x (r′), S _y (r′), and S _z (r′) respectively represent the components of the spin density in the three directions of space x, y, and z; double quotes indicate taking the imaginary part; ε ₀ represents the dielectric constant in vacuum; ω is the angular frequency of the light source; C _yz (r′, r′), C _zy (r′, r′), C _zx (r′, r′), C _xz (r ′, r′), C _xy (r′, r′) and C _yx (r′, r′) are elements in the 3×3 cross-spectral density function on the focal plane of the tight focusing lens.
7. An electronic 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 1-6. Describe the steps of the method.
8. 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 1-6 are implemented.
9. A system for calculating tightly focused three-dimensional spin density using mode decomposition of optical systems, which is characterized by including the following modules:

   The equivalent module is used to equate the partially coherent light emitted by the light source, the tight focusing lens and the space behind the tight focusing lens into a nonlinear optical system;

   The output signal calculation module is used to calculate the output signal of the nonlinear optical system using the input signal and pulse function of the nonlinear optical system according to the nonlinear system theory and Richard Wolf diffraction theory. The output signal of the optical system is expressed as a 3×3 cross-spectral density function on the focal plane of the tight focusing lens;

   A mode decomposition module for converting the output signal of the nonlinear optical system into an incoherent superposition of the output signals of a series of linear optical systems using mode decomposition theory;

   The three-dimensional spin density calculation module is used to calculate the three-dimensional spin density represented by the 3×3 cross-spectral density function on the focal plane of the tight focusing lens according to the definition of spin angular momentum.

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