CN115079407B - Method for calculating tightly focused three-dimensional spin density using mode decomposition of optical system - Google Patents
Method for calculating tightly focused three-dimensional spin density using mode decomposition of optical system Download PDFInfo
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Abstract
The invention discloses a method for calculating tightly-focused three-dimensional spin density by using mode decomposition of an optical system, which is characterized in that a part of coherent light emitted by a light source, a space behind a tightly-focused lens and the tightly-focused lens are equivalent to a nonlinear optical system, an output signal of the nonlinear optical system is calculated, the output signal is represented as a 3 x 3 cross spectral density function on a focal plane of the tightly-focused lens, then the output signal of the nonlinear optical system is converted into incoherent superposition of output signals of a series of linear optical systems by using a mode decomposition theory, and finally, the three-dimensional spin density represented by the 3 x 3 cross spectral density function on the focal plane of the tightly-focused lens is obtained. The invention reduces the calculation difficulty and the calculation time of the tightly focused spin density and promotes the research on the interaction of the optical spin orbit.
Description
Technical Field
The invention relates to the technical field of optics, in particular to a method for calculating tightly-focused three-dimensional spin density by using mode decomposition of an optical system.
Background
In addition to linear momentum, angular momentum is an important property of light. Light has two different types of angular momentum, spin angular momentum, orbital angular momentum, which are related to the circular polarization state, the helical phase, of the light, respectively. In the past decades, it has been found that the interaction of light with substances results in a linear momentum transfer from the light to the substance, thereby exerting a force on the substance and enabling optical tweezers and other techniques. When the angular momentum of light interacts with a particle, the spin angular momentum causes the particle to spin, and the orbital angular momentum may cause the particle to rotate about the optical axis. The spin angular momentum and the orbital angular momentum can therefore also be distinguished by the different mechanical effects they exhibit when interacting with the particles. The spin angular momentum and orbital angular momentum of light, as two different degrees of freedom, are independent and conserved with respect to each other under free space propagation. In fact, optical spin-orbit interactions occur under a variety of conditions, such as light-substance 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 transfer. The coupling between the spin angular momentum and the external orbital angular momentum causes a spin-dependent displacement, the spin hall effect.
The process of focusing a beam of light by a large numerical aperture lens, which we call tight focusing, has been widely studied for the past few decades and applied to optical microscopy, trapping and material processing, so that spin-orbit interaction under tight focusing systems has gained much attention. 2007. In the years, zhao et al discovered that spin angular momentum can be converted to orbital angular momentum after the circularly polarized light is tightly focused, which is known as spin-to-orbit conversion. More recently, yao et al have discovered that orbital angular momentum can be converted to longitudinal spin angular momentum, referred to as orbital to spin conversion, after the vortex beam, which does not carry the spin, is tightly focused. The effect of parameters such as incident light topology, beam waist width, pupil radius to beam waist radius ratio on this orbit-to-spin conversion has been studied extensively. The calculation of the tightly focused spin density is the basis of the research work, the spin density distribution at the focal plane after the tightly focused completely coherent light can be obtained only by calculating a double integral, but the spin density distribution at the focal plane after the focusing of the partially coherent light relates to a quadruple integral, so that the calculation difficulty and the calculation time are increased, and the research on the interaction of tightly focused spin orbits is hindered.
Disclosure of Invention
The invention aims to provide a method for calculating the tightly-focused three-dimensional spin density by using the mode decomposition of an optical system, which reduces the calculation difficulty and shortens the calculation time.
In order to solve the above problems, the present invention provides a method for calculating a tightly focused three-dimensional spin density using mode decomposition of an optical system, comprising the steps of:
s1, enabling partial 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;
s2, according to a nonlinear system theory and a Richcard Wolff diffraction theory, calculating an output signal of the nonlinear optical system by using an input signal and a pulse function of the nonlinear optical system, wherein the output signal of the nonlinear optical system is expressed as a cross spectral density function of 3 multiplied by 3 on a focal plane of a tight focusing lens;
s3, converting the output signals of the nonlinear optical system into incoherent superposition of the output signals of a series of linear optical systems by using a mode decomposition theory;
and S4, calculating to obtain the three-dimensional spin density represented by a 3 multiplied by 3 cross spectral density function on the focal plane of the tight focusing lens according to the definition of the spin angular momentum.
As a further improvement of the present invention, the 3 × 3 cross spectral density function at the focal plane of the tightly-focused lens is:
wherein, C (r) 1 ′，r 2 ') is a 3 × 3 cross spectral density function at the focal plane of the tightly focused lens; c' (r) 1 ，r 2 ) Is a cross-spectral density function of the back surface of the tight focus lens; r is 1 And r 2 Respectively representing the spatial coordinates of two points at the light source, r 1 ＝(x 1 ,y 1 )，r 2 ＝(x 2 ,y 2 )；r′ 1 And r' 2 Respectively representing after tight focusing r 1 And r 2 Space coordinates of two points, r' 1 ＝(x′ 1 ，y′ 1 )，r′ 2 ＝(x′ 2 ，y′ 2 ) (ii) a i represents an imaginary number; f and λ denote the focal length of the tight focusing lens and the wavelength of the partially coherent light emitted by the light source, respectively; k is a radical of 1 And k 2 Respectively represent points r 1 And r 2 Corresponding coordinate, k, in wavenumber space 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 points r 1 And r 2 Integral infinitesimal of phi 1 And phi 2 Are respectively a point r 1 And r 2 Is in an azimuth of 1 ＝arctan(y 1 /x 1 )，φ 2 ＝arctan(y 2 /x 2 )；θ 1 Is a point r 1 Angle between line of focus of tight focusing lens and optical axis, theta 2 Is a point r 2 And the included angle between the line of the focal point of the tight focusing lens and the optical axis and satisfies the condition that theta is more than or equal to 0 1 ≤arcsin(NA/n t )，0≤θ 2 ≤arcsin(NA/n t ) (ii) a NA and n t Respectively, the numerical aperture of the tight focus lens and the refractive index of the imaging volume.
As a further improvement of the present invention,
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 2x /k 0 ,y 2 ＝-fk 2y /k 0
wherein the content of the first and second substances,
representing the wavenumber of the partially coherent light emitted by the light source; substituting equation (7) yields:
r 'to' 1 ＝r′ 2 = r', then, C (r) 1 ′，r 2 ') is expressed as:
order:
q 2 (r′，k 1 ，k 2 )＝C(k 1 ，k 2 )h * (r′，k 1 )h(r′，k 2 )
at this time, equation (9) is expressed as:
wherein q is 2 (r′，k 1 ，k 2 ) Is a pulse function of the nonlinear optical system, and is that an output point r' is opposite to a point k 1 And k 2 Response of the two-point pulse.
As a further improvement of the present invention, step S3 includes:
s31, representing the cross spectrum density function at the light source into a mode decomposition form;
and S32, substituting the mode decomposition form of the cross spectrum density function at the light source into the 3 x 3 cross spectrum density function on the focal plane of the tight focusing lens so as to convert the output signal of the nonlinear optical system into the incoherent superposition of the output signals of a series of linear optical systems.
As a further refinement of the invention, the cross-spectral density function at the light source is expressed in the form of a mode decomposition as follows:
wherein, C (k) 1 ，k 2 ) Representing a cross-spectral density function at the light source; k is a radical of 1 And k 2 Respectively representing the corresponding coordinates of two points at the light source in the wave number space;
and
respectively represent point k 1 And k 2 N =1,2, … …, N being the total number of complete coherent modes; beta is a n Represents the weight of the nth fully coherent mode, T represents transposition;
and S32, substituting the formula (11) into a 3 multiplied by 3 cross spectral density function on the focal plane of the tight focusing lens to obtain:
wherein C (r ', r') is a cross spectral density function of 3 × 3 at the focal plane of the tightly-focused lens;
representing the response of the output point r' to a single pulse at the input point k, the integral term in the absolute value in equation (12) is equivalent to the output signal of a linear system.
As a further improvement of the present invention, the three-dimensional spin density is represented by a 3 × 3 cross spectral density function at the focal plane of the tightly-focused lens as:
wherein S is x (r′)，S y (r′)，S z (r') represents components of spin density in three directions in space x, y, z, respectively; taking an imaginary part by double quote; epsilon 0 Represents a dielectric constant in a 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') is the element in the 3 x 3 cross spectral density function at the focal plane of the tightly focused lens.
The invention also provides an electronic device comprising a memory, a processor and a computer program stored on the memory and executable on the processor, wherein the processor implements the steps of any one of the above methods when executing the program.
The invention also provides a computer-readable storage medium, on which a computer program is stored which, when being executed by a processor, carries out the steps of any of the methods described above.
The invention also provides a system for calculating tightly focused three-dimensional spin density using mode decomposition of an optical system, comprising the following modules:
the equivalent module is used for enabling the space behind the partially coherent light, the tight focusing lens and the tight focusing lens which are emitted by the light source to be equivalent to a nonlinear optical system;
the output signal calculation module is used for calculating the output signal of the nonlinear optical system by utilizing the input signal and the pulse function of the nonlinear optical system according to the nonlinear system theory and the Richcard-Voff diffraction theory, and the output signal of the nonlinear optical system is expressed as a cross spectral density function of 3 multiplied by 3 on the focal plane of the tight focusing lens;
the mode decomposition module is used for converting the output signals of the nonlinear optical system into incoherent superposition of the output signals of a series of linear optical systems by using a mode decomposition theory;
and the three-dimensional spin density calculation module is used for calculating and obtaining the three-dimensional spin density represented by a 3 x 3 cross spectral density function on the focal plane of the tight focusing lens according to the definition of the spin angular momentum.
The invention has the beneficial effects that:
the method comprises the steps of enabling partial 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, calculating an output signal of the nonlinear optical system, wherein the output signal is represented as a 3 x 3 cross spectral density function on a focal plane of the tight focusing lens, converting the output signal of the nonlinear optical system into incoherent superposition of output signals of a series of linear optical systems by using a mode decomposition theory, and finally obtaining three-dimensional spin density represented by the 3 x 3 cross spectral density function on the focal plane of the tight focusing lens. The invention reduces the calculation difficulty and the calculation time of the tightly focused spin density and promotes the research on the interaction of the optical spin orbit.
The foregoing description is only an overview of the technical solutions of the present invention, and in order to make the technical means of the present invention more clearly understood, the present invention may be implemented in accordance with the content of the description, and in order to make the above and other objects, features, and advantages of the present invention more clearly understood, the following preferred embodiments are described in detail with reference to the accompanying drawings.
Drawings
FIG. 1 is a flow chart of a method for calculating tightly focused three-dimensional spin densities using mode decomposition of an optical system in a preferred embodiment of the invention.
Detailed Description
The present invention is further described below in conjunction with the following figures and specific examples so that those skilled in the art may better understand the present invention and practice it, but the examples are not intended to limit the present invention.
In order to solve the problems of long calculation time and high difficulty of tightly focused spin density, the invention utilizes the mode decomposition theory of an optical system. Generally, the optical system for fully coherent light illumination is collectively called a linear optical system, and the optical system for partially coherent light illumination is collectively called a nonlinear optical system. For a linear optical system, the relationship between its input signal and its output signal can be obtained by a simple linear transformation. For a nonlinear optical system, the calculation of its output signal involves complex integration, which takes a lot of calculation time. By utilizing the mode decomposition theory of the nonlinear optical system, the nonlinear optical system is decomposed into a series of superposition of linear optical systems, output signals of the linear optical system are obtained firstly, and then the output signals are superposed to obtain output signals of the nonlinear optical system, so that the calculation difficulty and the calculation time of the nonlinear optical system are reduced. The partial coherent light tight focusing process is a nonlinear optical system, and the output signal of the tight focusing system is expressed into the superposition of a series of linear signals by utilizing the mode decomposition theory of the nonlinear optical system, and the spin density can be calculated by using the output signal. This approach reduces the computational difficulty and time of tightly focused spin density and facilitates the study of optical spin-orbit interactions. The following is a detailed description:
as shown in FIG. 1, a method for calculating a tightly focused three-dimensional spin density using mode decomposition of an optical system in a preferred embodiment of the present invention comprises the steps of:
s1, enabling partial 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;
s2, according to a nonlinear system theory and a Richcard Wolff diffraction theory, calculating an output signal of the nonlinear optical system by using an input signal and a pulse function of the nonlinear optical system, wherein the output signal of the nonlinear optical system is expressed as a cross spectral density function of 3 multiplied by 3 on a focal plane of a tight focusing lens;
wherein the input signal of the nonlinear optical system is a transmittance function determined by the tight focusing lens and the polarization state, and the pulse function is a cross-spectral density function C (r) of the light source 1 ，r 2 ) Coordinates (x, y) of the light source plane and wavenumber space (k) x ,k y ) The determined output signal represents the cross-spectral density function C (r' 1 ，r′ 2 )。
Linear optical systems can be obtained by computing only a simple linear transformation as follows:
where C (x ') represents the output signal of the linear system, t (x) represents the input signal, and q (x ', x) is the pulse signal at the input point x at the output point x '.
Calculating the relation between the output signal and the input signal of the nonlinear optical system as follows:
wherein C (x') and t (x) represent an output signal and an input signal of the nonlinear optical system, respectively, q 2 (x′，x 1 ，x 2 ) Is a kernel function of a second order Volterra series (Volterra kernel), which is expressed as an output point x' at an input point x 1 And x 2 Two-point double pulse signal, namely pulse function. q. q of 2 (x′，x 1 ，x 2 ) Is a 6D complex function, making the computation of the nonlinear optical system difficult. The nonlinear optical system can be expressed as a series of incoherent superposition of linear systems by the mode decomposition theory, so that the calculation is simplified. The present invention applies this theory to the calculation of spin density in tightly focused systems.
In a nonlinear optical system, the partially coherent light emitted from a known light source is a transverse electric field propagating along the z-axis, and the ensemble average of the cross-spectral density function of the partially coherent light is represented as:
C(r 1 ,r 2 )＝<U * (r 1 )U T (r 2 )> (3)
wherein, U (r) 1 ) And U (r) 2 ) Respectively representing two points r at the light source 1 And r 2 Optical signal (amplitude), angle brackets indicate ensemble averages, a-complex conjugate, and T indicates the transpose of the matrix. The relationship of the optical signals of the front and back surfaces of the tight focus lens is:
U′(r)＝t 0 (r)U(r) (4)
wherein, t 0 (r) represents a transmittance function; u' (r) representsThe optical signal of the back surface of the tight focus lens, and U (r) denotes the optical signal of the front surface of the tight focus lens. When the polarization state of the incident light is cylindrical vector polarization, the transmittance function can be expressed as:
wherein the content of the first and second substances,
is a transmittance function of the tight focus lens; phi is a 0 An initial phase representing a cylindrical vector polarization; when phi is 0 The terms 0 ° and 90 ° denote radial polarization and angular polarization, respectively. The first and second matrices on the right of equation (5) represent the magnitudes of the x, y, z components in space after passing through the tightly focused lens for the partially coherent light with Radial Polarization (RP) and Angular Polarization (AP), respectively. Thus, the cross-spectral density function of the back surface of the tightly focused lens is expressed as:
it is a 3 × 3 matrix, and 3 × 3 cross spectral density function at the tightly focused focal plane can be obtained by using richardov diffraction formula as follows:
wherein, C (r' 1 ，r′ 2 ) Is a cross spectral density function of 3 x 3 on the focal plane of the tightly-focused lens; c' (r) 1 ，r 2 ) Is a cross-spectral density function of the back surface of the tight focusing lens; r is 1 And r 2 Respectively representing the spatial coordinates of two points at the light source, r 1 ＝(x 1 ,y 1 )，r 2 ＝(x 2 ,y 2 )；r′ 1 And r' 2 Respectively after tight focusing 1 And r 2 Space coordinates of two points, r' 1 ＝(x′ 1 ，y′ 1 )，r′ 2 ＝(x′ 2 ，y′ 2 ) (ii) a i represents an imaginary number; f and λ denote the focal length of the tight focusing lens and the wavelength of the partially coherent light emitted by the light source, respectively; k is a radical of formula 1 And k 2 Respectively represent points r 1 And r 2 Corresponding coordinate, k, in wavenumber space 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 points r 1 And r 2 Integral infinitesimal of phi 1 And phi 2 Are respectively a point r 1 And r 2 Is in an azimuth of 1 ＝arctan(y 1 /x 1 )，φ 2 ＝arctan(y 2 /x 2 )；θ 1 Is a point r 1 Angle between line of focus of tight focusing lens and optical axis, theta 2 Is a point r 2 And the included angle between the line of the focal point of the tight focusing lens and the optical axis and satisfies the condition that theta is more than or equal to 0 1 ≤arcsin(NA/n t )，0≤θ 2 ≤arcsin(NA/n t ) (ii) a NA and n t Respectively, the numerical aperture of the tight focus lens and the refractive index of the imaging volume.
Wherein:
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 2x /k 0 ,y 2 ＝-fk 2y /k 0
wherein the content of the first and second substances,
representing the wavenumber of the partially coherent light emitted by the light source; substituting into equation (7) yields:
r 'to' 1 ＝r′ 2 = r, in this case, C (r' 1 ，r′ 2 ) Expressed as:
order:
q 2 (r′，k 1 ，k 2 )＝C(k 1 ，k 2 )h * (r′，k 1 )h(r′，k 2 )
at this time, equation (9) is expressed as:
wherein q is 2 (r′，k 1 ，k 2 ) Is a pulse function of the nonlinear optical system, and is that an output point r' is opposite to a point k 1 And k 2 Response of the two-point pulse. Such non-linear optical systems may therefore also be referred to collectively as bilinear systems.
S3, converting the output signals of the nonlinear optical system into incoherent superposition of the output signals of a series of linear optical systems by using a mode decomposition theory; the method specifically comprises the following steps:
s31, expressing the cross spectral density function at the light source into a mode decomposition form as follows:
wherein, C (k) 1 ，k 2 ) Representing a cross-spectral density function at the light source; k is a radical of 1 And k 2 Respectively representing the corresponding coordinates of two points at the light source in the wave number space;
and
respectively represent points k 1 And k 2 N =1,2, … …, N being the total number of complete coherent modes; beta is a n Represents the weight of the nth fully coherent mode, T represents transposition;
and S32, substituting the mode decomposition form of the cross spectral density function at the light source into the 3 x 3 cross spectral density function on the focal plane of the tightly-focusing lens so as to convert the output signal of the nonlinear optical system into incoherent superposition of the output signals of a series of linear optical systems.
Specifically, substituting equation (11) into the cross spectral density function of 3 × 3 at the focal plane of the tight focus lens yields:
wherein C (r ', r') is a cross spectral density function of 3 × 3 at the focal plane of the tightly-focused lens;
represents the response of the output point r' to a single pulse at the input point k, the integral term in absolute value in equation (12):
output equivalent to linear systemAnd outputting a signal, namely a mode system. Thus, equation (12) can be considered as a series of impulse responses of
The superposition of the squares of the modes of the system.
And S4, calculating to obtain the three-dimensional spin density represented by a 3 multiplied by 3 cross spectral density function on the focal plane of the tight focusing lens according to the definition of the spin angular momentum.
For any time-domain harmonic optical field with an angular frequency ω = kc (c is the vacuum speed of light), its spin density at the point ρ in space can be defined as:
wherein epsilon 0 Represents a dielectric constant in a vacuum; ω is the angular frequency of the light source. Using mode decomposition theory, the spin density at the r' point at the focal plane can be expressed as:
wherein the content of the first and second substances,
representing the unfolded mode of the cross-spectral density function at the focal plane. The components of spin density in three directions in space x, y, z are expressed as:
the three-dimensional spin density is represented by a 3 x 3 cross spectral density function at the focal plane of the tightly-focused lens as:
wherein S is x (r′)，S y (r′)，S z (r') represents components of spin density in three directions in space x, y, z, respectively; taking an imaginary part by double quote; 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 x 3 cross spectral density function at the focal plane of the tightly focused lens.
In one embodiment, the method for calculating the tightly focused three-dimensional spin density by using the mode decomposition of the optical system in the invention is used for calculating the three-dimensional spin density of the radially polarized gaussian schell model light beam after being tightly focused, and the specific calculation process is as follows:
the ensemble average of the cross spectral density function for a radially polarized gaussian schel mode beam propagating along the z-axis is represented as:
wherein the content of the first and second substances,
and
respectively, incident light before the tight focusing lens at r 1 And r 2 The spectral intensities of the two points are,
in the form of a unit vector of radial polarization,
is r 1 And r 2 The distribution of coherence between two points denotes the complex conjugate, and T denotes the transpose of the matrix. r is 1 And r 2 The relationship between the optical field amplitudes of the two points on the front and back surfaces of the tight focusing lens is respectively as follows:
wherein, t 0 (r 1 ) And t 0 (r 2 ) Respectively, the tight focusing lens is on the light source at r 1 And r 2 Two points of the transmittance function, knowing that the polarization state of the incident light is radially polarized, t 0 (r 1 ) And t 0 (r 2 ) Can be expressed as:
the formula (21) is abbreviated as:
it is a 3 x 3 matrix in which:
wherein:
sinθ 1 ＝k 1z /k 0 ，x 1 ＝-fk 1x /k 0 ，y 1 ＝-f k 1y /k 0
sinθ 2 ＝k 2z /k 0 ,x 2 ＝-fk 2x /k 0 ,y 2 ＝-fk 2y /k 0
wherein, the first and the second end of the pipe are connected with each other,
representing the wavenumber of the partially coherent light emitted by the light source; equation (23) can be expressed as:
substituting equation (23) into equation (8) yields the cross-spectral density at the tightly focused focal plane:
r 'to' 1 ＝r′ 2 = r', then, C xx (r 1 ′，r 2 ') is expressed as:
order:
t′ 0 (k 1 )＝cos 3 θ 1 cos 2 φ 1 exp[-i(k 1z z′)]
t′ 0 (k 2 )＝cos 3 θ 2 cos 2 φ 2 exp[-i(k 2z z′)]
then equation (24) can be written as:
wherein, C xx (r ', r') is the first term in the 3 x 3 cross spectral density function matrix of the output signal, i.e. the field, after the nonlinear optical system.
The cross-spectral density function at the light source is expressed in the form of a mode decomposition as:
wherein the content of the first and second substances,
and beta n Respectively, representing the unfolded mode of the incident light and its corresponding weight. Substituting equation (28) into equation (27) yields:
wherein the content of the first and second substances,
wherein, C nxx (r ', r') is C xx (r′，r′) The nth complete coherent mode of decomposition, F is the Fourier transform symbol, C nxx (r ', r') can be computed quickly using Fourier transform. The integral term inside the absolute value of equation (29) is the output signal of a linear system of completely coherent light incidence, i.e., a modal system, and equation (29) can be considered as a series of impulse responses of
The coherent superposition of the mode squares of the system. Similarly, the other 8 terms C of the cross spectral density function matrix xy (r′,r′)、C xz (r′,r′)、C yx (r′,r′)、C yy (r′,r′)、C yz (r′,r′)、 C zx (r′,r′)、C zy (r′,r′)、C zz (r ', r') can be determined in the same manner. The spin density component can be obtained by using the 9-term cross spectral density formula:
namely, equation (17).
The method comprises the steps of enabling partial 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, calculating an output signal of the nonlinear optical system to be represented as a 3 x 3 cross spectral density function on a focal plane of the tight focusing lens, converting the output signal of the nonlinear optical system into incoherent superposition of output signals of a series of linear optical systems by using a mode decomposition theory, and finally obtaining the three-dimensional spin density represented by the 3 x 3 cross spectral density function on the focal plane of the tight focusing lens. The invention reduces the calculation difficulty and the calculation time of the tightly focused spin density and promotes the research on the interaction of the optical spin orbit.
The preferred embodiment of the present invention also discloses an electronic device, which comprises a memory, a processor and a computer program stored in the memory and capable of running on the processor, wherein the processor executes the program to implement the steps of the method in the above embodiments.
The preferred embodiment of the present invention also discloses a computer readable storage medium, on which a computer program is stored, which when executed by a processor implements the steps of the method described in the above embodiments.
The preferred embodiment of the invention also discloses a system for calculating the tightly focused three-dimensional spin density by using the mode decomposition of the optical system, which comprises the following modules:
the equivalent module is used for enabling the space behind the partially coherent light emitted by the light source, the tight focusing lens and the tight focusing lens to be equivalent to a nonlinear optical system;
the output signal calculation module is used for calculating the output signal of the nonlinear optical system by utilizing the input signal and the pulse function of the nonlinear optical system according to the nonlinear system theory and the Richcard-Voff diffraction theory, and the output signal of the nonlinear optical system is expressed as a cross spectral density function of 3 multiplied by 3 on the focal plane of the tight focusing lens;
the mode decomposition module is used for converting the output signals of the nonlinear optical system into incoherent superposition of the output signals of a series of linear optical systems by using a mode decomposition theory;
and the three-dimensional spin density calculation module is used for calculating and obtaining the three-dimensional spin density represented by a 3 x 3 cross spectral density function on the focal plane of the tight focusing lens according to the definition of the spin angular momentum.
The system for calculating the tightly-focused three-dimensional spin density by using the mode decomposition of the optical system in the embodiment of the invention is used for realizing the method for calculating the tightly-focused three-dimensional spin density by using the mode decomposition of the optical system, so the specific implementation of the system can be seen in the previous embodiment part of the method for calculating the tightly-focused three-dimensional spin density by using the mode decomposition of the optical system, and therefore, the specific implementation thereof can refer to the description of the corresponding above embodiment of the method and is not further described herein.
In addition, since the system for calculating the tightly focused three-dimensional spin density by mode decomposition of the optical system of the present embodiment is used to implement the aforementioned method for calculating the tightly focused three-dimensional spin density by mode decomposition of the optical system, the role thereof corresponds to that of the above method, and the description thereof is omitted.
The above embodiments are merely preferred embodiments for fully illustrating the present invention, and the scope of the present invention is not limited thereto. The equivalent substitution or change made by the technical personnel in the technical field on the basis of the invention is all within the protection scope of the invention. The protection scope of the invention is subject to the claims.
Claims (6)
1. A method for calculating a tightly focused three-dimensional spin density using mode decomposition of an optical system, comprising the steps of:
s1, enabling partial 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;
s2, according to a nonlinear system theory and a Richcard Wolff diffraction theory, calculating an output signal of the nonlinear optical system by using an input signal and a pulse function of the nonlinear optical system, wherein the output signal of the nonlinear optical system is expressed as a cross spectral density function of 3 multiplied by 3 on a focal plane of a tightly-focusing lens;
s3, converting the output signals of the nonlinear optical system into incoherent superposition of the output signals of a series of linear optical systems by using a mode decomposition theory;
s4, calculating to obtain a three-dimensional spin density represented by a 3 x 3 cross spectral density function on a focal plane of the tight focusing lens according to the definition of the spin angular momentum;
the step S3 comprises the following steps:
s31, representing the cross spectrum density function at the light source into a mode decomposition form;
s32, substituting the mode decomposition form of the cross spectrum density function at the light source into a 3 x 3 cross spectrum density function on the focal plane of the tight focusing lens so as to convert the output signal of the nonlinear optical system into incoherent superposition of the output signals of a series of linear optical systems;
the cross spectral density function at the light source is expressed in the form of a mode decomposition as follows:
wherein, C (k) 1 ，k 2 ) Representing a cross-spectral density function at the light source; k is a radical of 1 And k 2 Respectively representing the corresponding coordinates of two points at the light source in the wave number space;
and
respectively represent points k 1 And k 2 N =1,2, … …, N being the total number of complete coherent modes; beta is a n Represents the weight of the nth fully coherent mode, T represents transposition;
substituting equation (11) into the 3 × 3 cross spectral density function at the focal plane of the tight-focus lens to obtain:
wherein C (r ', r') is a cross spectral density function of 3 × 3 at the focal plane of the tightly-focused lens;
represents the response of the output point r' to a single pulse of the input point k, and the integral term in the absolute value in equation (12) is equivalent to the output signal of a linear system;
the three-dimensional spin density is represented by a 3 x 3 cross spectral density function at the focal plane of the tightly-focused lens as:
wherein S is x (r′)，S y (r′)，S z (r') represents components of spin density in three directions in space x, y, z, respectively; taking an imaginary part by double quote; epsilon 0 Represents a dielectric constant in a 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') is the element in the 3 x 3 cross spectral density function at the focal plane of the tightly focused lens.
2. The method for calculating the tightly-focused three-dimensional spin density using the mode decomposition of the optical system as claimed in claim 1, wherein the 3 x 3 cross-spectral density function at the focal plane of the tightly-focused lens is:
wherein, C (r) 1 ′，r 2 ') is a 3 x 3 cross spectral density function at the focal plane of the tight focus lens; c' (r) 1 ，r 2 ) Is a cross-spectral density function of the back surface of the tight focusing lens; r is 1 And r 2 Respectively representing the spatial coordinates of two points at the light source, r 1 ＝(x 1 ,y 1 )，r 2 ＝(x 2 ,y 2 )；r′ 1 And r' 2 Respectively representing after tight focusing r 1 And r 2 Spatial coordinates of two points r' 1 ＝(x′ 1 ，y′ 1 )，r′ 2 ＝(x′ 2 ，y′ 2 ) (ii) a i represents an imaginary number; f and λ denote the focal length and light, respectively, of the tightly-focused lensThe wavelength of the partially coherent light emitted by the source; k is a radical of 1 And k 2 Respectively represent points r 1 And r 2 Corresponding coordinate, k, in wavenumber space 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 points r 1 And r 2 Integral infinitesimal of phi 1 And phi 2 Are respectively a point r 1 And r 2 Is in an azimuth of 1 ＝arctan(y 1 /x 1 )，φ 2 ＝arctan(y 2 /x 2 )；θ 1 Is a point r 1 Angle between line of focus of tight focusing lens and optical axis, theta 2 Is a point r 2 And the included angle between the line of the focal point of the tight focusing lens and the optical axis and satisfies the condition that theta is more than or equal to 0 1 ≤arcsin(NA/n t )，0≤θ 2 ≤arcsin(NA/n t ) (ii) a NA and n t Respectively, the numerical aperture of the tight focus lens and the refractive index of the imaging space.
3. The method for calculating the tightly-focused three-dimensional spin density using the mode decomposition of the optical system according to claim 2,
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 2x /k 0 ，y 2 ＝-fk 2y /k 0
wherein the content of the first and second substances,
representing the wavenumber of the partially coherent light emitted by the light source; substituting into equation (7) yields:
r 'to' 1 ＝r′ 2 = r', then, C (r) 1 ′，r 2 ') is expressed as:
order:
q 2 (r′，k 1 ，k 2 )＝C(k 1 ，k 2 )h * (r′，k 1 )h(r′，k 2 )
at this time, equation (9) is expressed as:
wherein, t 0 (k1) And t 0 (k 2 ) Are respectively shown at k 1 And k 2 Transmittance function of electric field at two points, q 2 (r′，k 1 ，k 2 ) Is a pulse function of the nonlinear optical system, and is that an output point r' is opposite to a point k 1 And k 2 Response of the two-point pulse.
4. An electronic device comprising a memory, a processor and a computer program stored on the memory and executable on the processor, characterized in that the steps of the method of any of claims 1-3 are implemented when the program is executed by the processor.
5. A computer-readable storage medium, on which a computer program is stored which, when being executed by a processor, carries out the steps of the method according to any one of claims 1 to 3.
6. A system for calculating a tightly focused three dimensional spin density using mode decomposition of an optical system, comprising the following modules:
the equivalent module is used for enabling the space behind the partially coherent light, the tight focusing lens and the tight focusing lens which are emitted by the light source to be equivalent to a nonlinear optical system;
the output signal calculation module is used for calculating the output signal of the nonlinear optical system by utilizing the input signal and the pulse function of the nonlinear optical system according to the nonlinear system theory and the Richcard-Voff diffraction theory, and the output signal of the nonlinear optical system is expressed as a cross spectral density function of 3 multiplied by 3 on the focal plane of the tight focusing lens;
the mode decomposition module is used for converting the output signals of the nonlinear optical system into incoherent superposition of the output signals of a series of linear optical systems by using a mode decomposition theory;
the three-dimensional spin density calculation module is used for calculating and obtaining the three-dimensional spin density represented by a 3 x 3 cross spectral density function on the focal plane of the tight focusing lens according to the definition of the spin angular momentum;
the pattern decomposition module is used for executing the following steps:
s31, representing the cross spectrum density function at the light source into a mode decomposition form;
s32, substituting the mode decomposition form of the cross spectrum density function at the light source into a 3 x 3 cross spectrum density function on the focal plane of the tight focusing lens so as to convert the output signal of the nonlinear optical system into incoherent superposition of the output signals of a series of linear optical systems;
the cross spectral density function at the light source is expressed in the form of a mode decomposition as follows:
wherein, C (k) 1 ，k 2 ) Representing a cross-spectral density function at the light source; k is a radical of formula 1 And k 2 Respectively representing the corresponding coordinates of two points at the light source in the wave number space;
and
respectively represent point k 1 And k 2 N =1,2, a. Beta is a beta n Represents the weight of the nth fully coherent mode, T represents transposition;
substituting equation (11) into the 3 × 3 cross spectral density function at the focal plane of the tight-focus lens to obtain:
wherein C (r ', r') is a cross spectral density function of 3 × 3 at the focal plane of the tightly-focused lens;
representing the response of the output point r' to a single pulse of the input point k, the integral term in the absolute value in equation (12) is equivalent to the output signal of a linear system;
the three-dimensional spin density is represented by a 3 x 3 cross spectral density function at the focal plane of the tightly-focused lens as:
wherein S is x (r′)，S y (r′)，S z (r') represents components of spin density in three directions in space x, y, z, respectively; taking an imaginary part by double quote; epsilon 0 Represents a dielectric constant in a 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') is an element in the cross spectral density function of 3 x 3 at the focal plane of the tight focus lens.
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| Fast calculation of tightly focused random electromagnetic beams: controlling the focal field by spatial coherence;Ruihuan Tong 等;《Optics Express》;20200330;第1-4部分，第9713-9725页 * |
| 紧聚焦混合阶庞加莱光的自旋密度;孙宏 等;《华东师范大学学报（自然科学版）》;20200331;全文 * |
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