CN115079407A - Method for calculating tightly focused three-dimensional spin density using mode decomposition of optical system - Google Patents
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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 of research, it has been found that the interaction of light with substances results in a linear momentum transfer from the light to the substances, thereby exerting forces on the substances 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 the orbital angular momentum of light are independent and conservative of each other as two different degrees of freedom under free space propagation. In fact, optical spin-orbit interactions occur under a variety of conditions, such as light-to-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 is a spin-dependent displacement, i.e., the spin hall effect.
The process of focusing a beam of light by a large numerical aperture lens is called tight focusing, and in the past decades, the characteristics of the tightly focused light field have been extensively studied and applied to optical microscopy, trapping and material processing, so that the spin-orbit interaction under the tight focusing system has gained much attention. In 2007, Zhao et al discovered that after circularly polarized light was tightly focused, the spin angular momentum could be converted to orbital angular momentum, referred to 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 the space of the partially coherent light emitted by the light source, the tightly-focusing lens and the tightly-focusing lens to be equivalent to a nonlinear optical system;
s2, according to the nonlinear system theory and the Richcard Wolff diffraction theory, calculating the output signal of the nonlinear optical system by using the input signal and the 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 the focal plane of the tight focusing lens;
s3, converting the output signal 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 tightly-focused 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 the content of the first and second substances,is a cross spectral density function of 3 x 3 on the focal plane of the tightly-focused lens;is the intersection of the back surface of the tightly-focused lensA cross spectrum density function; 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 1 =(x 2 ,y 2 );Andrespectively representing after tight focusing r 1 And r 2 The spatial coordinates of the two points are,,(ii) a i represents an imaginary number; f andrespectively representing the focal length of the tightly-focusing lens and the wavelength of the partially coherent light emitted by the light source; k is a radical of 1 And k 2 Respectively represent points r 1 And r 2 The corresponding coordinates in the wave number space,,;andrespectively represent points r 1 And r 2 The integral of (a) is a infinitesimal,andare respectively a point r 1 And r 2 Is in an azimuth of ,;Is a point r 1 And the angle between the line connecting the focal points of the tight focusing lenses and the optical axis,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,(ii) a NA and n t Representing the numerical aperture of the tight focus lens and the refractive index of the imaging volume, respectively.
As a further improvement of the present invention,
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:
order:
at this time, equation (9) is expressed as:
wherein the content of the first and second substances,is a pulse function of the nonlinear optical system and is an output pointTo 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 the content of the first and second substances,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;andrespectively represent point k 1 And k 2 N =1,2, … …, N being the total number of completely coherent modes;represents the weight of the nth fully coherent mode, T represents transposition;
s32, substituting equation (11) into the cross spectral density function of 3 × 3 at the focal plane of the tightly-focused lens to obtain:
wherein the content of the first and second substances,is a cross spectral density function of 3 x 3 on the focal plane of the tightly-focused lens;indicates an output pointThe integral term in the absolute value in equation (12) is equivalent to the output signal of a linear system for the response of a single pulse at the input point k.
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 the content of the first and second substances,,,components in the three directions x, y and z in space respectively representing spin density; taking an imaginary part by double quote;represents a dielectric constant in a vacuum;is the angular frequency of the light source;,,,,andare elements in a 3 x 3 cross spectral density function at the focal plane of the tight focus 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 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 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.
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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 the space of the partially coherent light emitted by the light source, the tightly-focusing lens and the tightly-focusing lens to be equivalent to a nonlinear optical system;
s2, according to the nonlinear system theory and the Richcard Wolff diffraction theory, calculating the output signal of the nonlinear optical system by using the input signal and the 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 the focal plane of the 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 of the light sourceCoordinates (x, y) of the light source plane and wavenumber space (k) x ,k y ) Determining that the output signal represents a cross-spectral density function at the focal plane。
Linear optical systems can be obtained by computing only a simple linear transformation as follows:
wherein the content of the first and second substances,representing the output signal of the linear system,which is representative of the input signal(s),is an output pointThe pulse signal at input point x.
Calculating the relation between the output signal and the input signal of the nonlinear optical system as follows:
wherein the content of the first and second substances,andrespectively representing the output signal and the input signal of the non-linear optical system,is a kernel function of a second order Walteria series (Volterra kernel) which represents the output pointAt the input point x 1 And x 2 Two points of double pulse signals, i.e. pulse functions.Is a 6D complex function, making the computation of the nonlinear optical system difficult. The nonlinear optical system can be represented as a series of incoherent superposition of linear systems by a 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:
wherein the content of the first and second substances,andrespectively 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:
wherein, the first and the second end of the pipe are connected with each other,represents a transmittance function;an optical signal representing the back surface of the tight focusing lens,representing the optical signal of the front surface of the tight focusing lens. When the polarization state of the incident light is cylindrical vector polarizationThe transmittance function can be expressed as:
wherein the content of the first and second substances,is a transmittance function of the tight focus lens;an initial phase representing a cylindrical vector polarization; when in useAndradial and azimuthal polarizations are indicated, 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 the content of the first and second substances,is a cross spectral density function of 3 x 3 on the focal plane of the tight focus lens;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 1 =(x 2 ,y 2 );Andrespectively representing after tight focusing r 1 And r 2 The spatial coordinates of the two points are,,(ii) a i represents an imaginary number; f andrespectively representing the focal length of the tightly-focusing lens and the wavelength of the partially coherent light emitted by the light source; k is a radical of 1 And k 2 Respectively represent points r 1 And r 2 The corresponding coordinates in the wave number space are,,;andrespectively represent points r 1 And r 2 The integral of (a) is a infinitesimal,andare respectively a point r 1 And r 2 Is in an azimuth of ,;Is a point r 1 And the angle between the line connecting the focal points of the tight focusing lenses and the optical axis,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,(ii) a NA and n t Representing the numerical aperture of the tight focus lens and the refractive index of the imaging volume, respectively.
Wherein:
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:
order:
at this time, equation (9) is expressed as:
wherein the content of the first and second substances,is a pulse function of the nonlinear optical system and is an output pointTo 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 signal 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, representing the cross-spectral density function at the light source in the form of mode decomposition, as follows:
wherein the content of the first and second substances,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;andrespectively represent point k 1 And k 2 N =1,2, … …, N being the total number of completely coherent modes;represents the weight of the nth fully coherent mode, T represents transposition;
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.
Specifically, substituting equation (11) into the cross spectral density function of 3 × 3 at the focal plane of the tight focus lens yields:
wherein the content of the first and second substances,is a cross spectral density function of 3 x 3 on the focal plane of the tightly-focused lens;indicates an output pointThe response to a single pulse at input point k, the integral term in absolute value in equation (12):
equivalent to the output signal of a linear system, i.e. a modal system. Thus, equation (12) can be considered as a series of impulse responses asThe 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 tightly-focused lens according to the definition of the spin angular momentum.
For angular frequency of(c is the vacuum speed of light) arbitrary time domain harmonic light field, which is in spaceThe spin density of a spot can be defined as:
wherein, the first and the second end of the pipe are connected with each other,represents a dielectric constant in a vacuum;is the angular frequency of the light source. Using pattern decomposition theory, focalAt the faceThe spin density of a spot 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 the three directions x, y, z in space are expressed as:
the three-dimensional spin density is expressed by a 3 x 3 cross spectral density function at the focal plane of the tight focus lens as:
wherein the content of the first and second substances,,,components in the three directions x, y and z in space respectively representing spin density; taking an imaginary part by double quote;,,,,andare elements in a 3 x 3 cross spectral density function at the focal plane of the tight focus 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,andincident light in front of the respective tight focusing lenses is at r 1 And r 2 The spectral intensities of the two points are such that,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 Relationship of light field amplitudes of two points on front and back surfaces of tight focusing lensRespectively as follows:
wherein the content of the first and second substances,andrespectively, the tight focusing lens is on the light source at r 1 And r 2 The transmission function at two points, knowing that the polarization state of the incident light is radially polarized,andcan be expressed as:
andrespectively, show tightly-focusing lensesAndthe matrix on the right side of equation (20) represents the x, y, z component size of a Radially Polarized (RP) gaussian schell mode beam after passing through the lens. Substituting the above equation into equation (6) yields the cross-spectral density function of the back surface of the tightly focused lens:
the formula (21) is abbreviated as:
it is a 3 x 3 matrix in which:
wherein:
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:
order:
then equation (24) can be written as:
wherein the content of the first and second substances,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,andrespectively, 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,is thatThe nth fully coherent mode of decomposition, F is the fourier transform symbol,fast calculation using fourier transform is possible. 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 ofThe coherent superposition of the mode squares of the system. Similarly, the other 8 terms of the cross spectral density function matrix、、、、、、、Can be determined by the same method. The spin density component can be obtained by using the 9-term cross spectral density formula:
namely, formula (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 an optical system, which comprises 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 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 (9)
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 the space of the partially coherent light emitted by the light source, the tightly-focusing lens and the tightly-focusing lens to be equivalent to a nonlinear optical system;
s2, according to the nonlinear system theory and the Richcard Wolff diffraction theory, calculating the output signal of the nonlinear optical system by using the input signal and the 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 the focal plane of the tight focusing lens;
s3, converting the output signal 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 the 3 multiplied by 3 cross spectrum density function on the focal plane of the tight focusing lens according to the definition of the spin angular momentum.
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 the content of the first and second substances,is a cross spectral density function of 3 x 3 on the focal plane of the tightly-focused lens;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 1 =(x 2 ,y 2 );Andrespectively representing after tight focusing r 1 And r 2 The spatial coordinates of the two points are,,(ii) a i represents an imaginary number; f andrespectively representing the focal length of the tightly-focusing lens and the wavelength of the partially coherent light emitted by the light source; k is a radical of formula 1 And k 2 Respectively represent points r 1 And r 2 The corresponding coordinates in the wave number space,,;andrespectively represent points r 1 And r 2 The integral of (a) is a infinitesimal,andare respectively a point r 1 And r 2 Is in an azimuth of ,;Is a point r 1 And the angle between the line connecting the focal points of the tight focusing lenses and the optical axis,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,(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 of calculating tightly-focused three-dimensional spin density using mode decomposition of an optical system according to claim 2,
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:
order:
at this time, equation (9) is expressed as:
4. 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 step S3 comprises:
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.
5. The method for calculating the tightly-focused three-dimensional spin density using the mode decomposition of the optical system as set forth in claim 4, wherein the cross-spectral density function at the light source is expressed in the form of the mode decomposition as follows:
wherein the content of the first and second substances,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;andrespectively represent point k 1 And k 2 N =1,2, … …, N being the total number of completely coherent modes;represents the weight of the nth fully coherent mode, T represents transposition;
s32, substituting equation (11) into the cross spectral density function of 3 × 3 at the focal plane of the tightly-focused lens to obtain:
wherein, the first and the second end of the pipe are connected with each other,is a cross spectral density function of 3 x 3 on the focal plane of the tightly-focused lens;indicates an output pointThe integral term in the absolute value in equation (12) is equivalent to the output signal of a linear system for the response of a single pulse at the input point k.
6. The method of calculating tightly-focused three-dimensional spin density using mode decomposition of an optical system according to claim 5, wherein 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 the content of the first and second substances,,,components in the three directions x, y and z in space respectively representing spin density; taking an imaginary part by double quote;represents a dielectric constant in a vacuum;is the angular frequency of the light source;,,,,andare elements in a 3 x 3 cross spectral density function at the focal plane of the tight focus lens.
7. 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 1-6 are implemented when the program is executed by the processor.
8. 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 6.
9. 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 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.
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RUIHUAN TONG 等: "Fast calculation of tightly focused random electromagnetic beams: controlling the focal field by spatial coherence", 《OPTICS EXPRESS》 * |
YIMING DONG 等: "Effect of spatial coherence on propagation,tight focusing,and radiation force of an azimuthally polarized beam", 《PHYSICAL REVIEW A》 * |
孙宏 等: "紧聚焦混合阶庞加莱光的自旋密度", 《华东师范大学学报(自然科学版)》 * |
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