CN115079407A - 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 PDF

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CN115079407A
CN115079407A CN202210864257.6A CN202210864257A CN115079407A CN 115079407 A CN115079407 A CN 115079407A CN 202210864257 A CN202210864257 A CN 202210864257A CN 115079407 A CN115079407 A CN 115079407A
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optical system
tightly
focused
density function
mode decomposition
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CN115079407B (en
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王子璇
陈亚红
王飞
蔡阳健
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Suzhou University
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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

Method for calculating tightly focused three-dimensional spin density using mode decomposition of optical system
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:
Figure 35605DEST_PATH_IMAGE001
(7)
wherein the content of the first and second substances,
Figure 906609DEST_PATH_IMAGE002
is a cross spectral density function of 3 x 3 on the focal plane of the tightly-focused lens;
Figure 247592DEST_PATH_IMAGE003
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 )；
Figure 913059DEST_PATH_IMAGE004
And
Figure 464126DEST_PATH_IMAGE005
respectively representing after tight focusing r 1 And r 2 The spatial coordinates of the two points are,
Figure 873242DEST_PATH_IMAGE006
，
Figure 68731DEST_PATH_IMAGE007
(ii) a i represents an imaginary number; f and
Figure 29734DEST_PATH_IMAGE008
respectively 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,
Figure 943463DEST_PATH_IMAGE009
，
Figure 890691DEST_PATH_IMAGE010
；
Figure 65320DEST_PATH_IMAGE011
and
Figure 338170DEST_PATH_IMAGE012
respectively represent points r 1 And r 2 The integral of (a) is a infinitesimal,
Figure 208037DEST_PATH_IMAGE013
and
Figure 349168DEST_PATH_IMAGE014
are respectively a point r 1 And r 2 Is in an azimuth of
Figure 519249DEST_PATH_IMAGE015
，
Figure 963000DEST_PATH_IMAGE016
；
Figure 320163DEST_PATH_IMAGE017
Is a point r 1 And the angle between the line connecting the focal points of the tight focusing lenses and the optical axis,
Figure 999406DEST_PATH_IMAGE018
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
Figure 23994DEST_PATH_IMAGE019
，
Figure 638646DEST_PATH_IMAGE020
(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,
Figure 607739DEST_PATH_IMAGE021
Figure 966039DEST_PATH_IMAGE022
wherein the content of the first and second substances,
Figure 845134DEST_PATH_IMAGE023
representing the wavenumber of the partially coherent light emitted by the light source; substituting equation (7) yields:
Figure 489742DEST_PATH_IMAGE024
(8)
order to
Figure 87076DEST_PATH_IMAGE025
At this time, the process of the present invention,
Figure 983488DEST_PATH_IMAGE002
expressed as:
Figure 982668DEST_PATH_IMAGE026
(9)
order:
Figure 532598DEST_PATH_IMAGE027
Figure 617229DEST_PATH_IMAGE028
Figure 329050DEST_PATH_IMAGE029
at this time, equation (9) is expressed as:
Figure 307371DEST_PATH_IMAGE030
(10)
wherein the content of the first and second substances,
Figure 169147DEST_PATH_IMAGE031
is a pulse function of the nonlinear optical system and is an output point
Figure 475495DEST_PATH_IMAGE032
To 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:
Figure 838343DEST_PATH_IMAGE033
(11)
wherein the content of the first and second substances,
Figure 812115DEST_PATH_IMAGE034
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;
Figure 110373DEST_PATH_IMAGE035
and
Figure 904016DEST_PATH_IMAGE036
respectively represent point k 1 And k 2 N =1,2, … …, N being the total number of completely coherent modes;
Figure 70555DEST_PATH_IMAGE037
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:
Figure 633255DEST_PATH_IMAGE038
(12)
wherein the content of the first and second substances,
Figure 102414DEST_PATH_IMAGE039
is a cross spectral density function of 3 x 3 on the focal plane of the tightly-focused lens;
Figure 242408DEST_PATH_IMAGE040
indicates an output point
Figure 88004DEST_PATH_IMAGE032
The 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:
Figure 505210DEST_PATH_IMAGE041
wherein the content of the first and second substances,
Figure 269904DEST_PATH_IMAGE042
，
Figure 38140DEST_PATH_IMAGE043
，
Figure 156268DEST_PATH_IMAGE044
components in the three directions x, y and z in space respectively representing spin density; taking an imaginary part by double quote;
Figure 818194DEST_PATH_IMAGE045
represents a dielectric constant in a vacuum;
Figure 363576DEST_PATH_IMAGE046
is the angular frequency of the light source;
Figure 619108DEST_PATH_IMAGE047
，
Figure 931140DEST_PATH_IMAGE048
，
Figure 57359DEST_PATH_IMAGE049
，
Figure 39222DEST_PATH_IMAGE050
，
Figure 641105DEST_PATH_IMAGE051
and
Figure 366615DEST_PATH_IMAGE052
are 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.
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 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 source
Figure 878499DEST_PATH_IMAGE053
Coordinates (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
Figure 890317DEST_PATH_IMAGE054
。
Linear optical systems can be obtained by computing only a simple linear transformation as follows:
Figure 120441DEST_PATH_IMAGE055
(1)
wherein the content of the first and second substances,
Figure 384064DEST_PATH_IMAGE056
representing the output signal of the linear system,
Figure 609509DEST_PATH_IMAGE057
which is representative of the input signal(s),
Figure 198753DEST_PATH_IMAGE058
is an output point
Figure 509649DEST_PATH_IMAGE059
The pulse signal at input point x.
Calculating the relation between the output signal and the input signal of the nonlinear optical system as follows:
Figure 576962DEST_PATH_IMAGE060
(2)
wherein the content of the first and second substances,
Figure 797859DEST_PATH_IMAGE061
and
Figure 417059DEST_PATH_IMAGE062
respectively representing the output signal and the input signal of the non-linear optical system,
Figure 356196DEST_PATH_IMAGE063
is a kernel function of a second order Walteria series (Volterra kernel) which represents the output point
Figure 961621DEST_PATH_IMAGE059
At the input point x 1 And x 2 Two points of double pulse signals, i.e. pulse functions.
Figure 427237DEST_PATH_IMAGE063
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:
Figure 92705DEST_PATH_IMAGE064
(3)
wherein the content of the first and second substances,
Figure 253559DEST_PATH_IMAGE065
and
Figure 787308DEST_PATH_IMAGE066
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:
Figure 982798DEST_PATH_IMAGE067
(4)
wherein, the first and the second end of the pipe are connected with each other,
Figure 819167DEST_PATH_IMAGE068
represents a transmittance function;
Figure 591950DEST_PATH_IMAGE069
an optical signal representing the back surface of the tight focusing lens,
Figure 539178DEST_PATH_IMAGE070
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:
Figure 854753DEST_PATH_IMAGE071
(5)
wherein the content of the first and second substances,
Figure 721078DEST_PATH_IMAGE072
is a transmittance function of the tight focus lens;
Figure 122103DEST_PATH_IMAGE073
an initial phase representing a cylindrical vector polarization; when in use
Figure 861303DEST_PATH_IMAGE074
And
Figure 890438DEST_PATH_IMAGE075
radial 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:
Figure 334189DEST_PATH_IMAGE076
(6)
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:
Figure 815986DEST_PATH_IMAGE001
(7)
wherein the content of the first and second substances,
Figure 370595DEST_PATH_IMAGE002
is a cross spectral density function of 3 x 3 on the focal plane of the tight focus lens;
Figure 395183DEST_PATH_IMAGE003
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 )；
Figure 868890DEST_PATH_IMAGE004
And
Figure 978928DEST_PATH_IMAGE005
respectively representing after tight focusing r 1 And r 2 The spatial coordinates of the two points are,
Figure 806070DEST_PATH_IMAGE006
，
Figure 75377DEST_PATH_IMAGE007
(ii) a i represents an imaginary number; f and
Figure 860931DEST_PATH_IMAGE008
respectively 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,
Figure 192686DEST_PATH_IMAGE009
，
Figure 948153DEST_PATH_IMAGE010
；
Figure 681753DEST_PATH_IMAGE011
and
Figure 638208DEST_PATH_IMAGE012
respectively represent points r 1 And r 2 The integral of (a) is a infinitesimal,
Figure 253997DEST_PATH_IMAGE013
and
Figure 219679DEST_PATH_IMAGE014
are respectively a point r 1 And r 2 Is in an azimuth of
Figure 73366DEST_PATH_IMAGE015
，
Figure 794197DEST_PATH_IMAGE016
；
Figure 631703DEST_PATH_IMAGE017
Is a point r 1 And the angle between the line connecting the focal points of the tight focusing lenses and the optical axis,
Figure 604338DEST_PATH_IMAGE018
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
Figure 437165DEST_PATH_IMAGE019
，
Figure 469843DEST_PATH_IMAGE020
(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:
Figure 388120DEST_PATH_IMAGE021
Figure 430026DEST_PATH_IMAGE022
wherein the content of the first and second substances,
Figure 992725DEST_PATH_IMAGE023
representing the wavenumber of the partially coherent light emitted by the light source; substituting into equation (7) yields:
Figure 320938DEST_PATH_IMAGE077
(8)
order to
Figure 336299DEST_PATH_IMAGE025
At this time, the process of the present invention,
Figure 181895DEST_PATH_IMAGE002
expressed as:
Figure 723735DEST_PATH_IMAGE078
(9)
order:
Figure 363795DEST_PATH_IMAGE027
Figure 991085DEST_PATH_IMAGE028
Figure 374793DEST_PATH_IMAGE029
at this time, equation (9) is expressed as:
Figure 646506DEST_PATH_IMAGE030
(10)
wherein the content of the first and second substances,
Figure 582101DEST_PATH_IMAGE031
is a pulse function of the nonlinear optical system and is an output point
Figure 837633DEST_PATH_IMAGE032
To 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:
Figure 759452DEST_PATH_IMAGE079
(11)
wherein the content of the first and second substances,
Figure 275884DEST_PATH_IMAGE034
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;
Figure 257747DEST_PATH_IMAGE035
and
Figure 734996DEST_PATH_IMAGE036
respectively represent point k 1 And k 2 N =1,2, … …, N being the total number of completely coherent modes;
Figure 585140DEST_PATH_IMAGE037
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:
Figure 97024DEST_PATH_IMAGE038
(12)
wherein the content of the first and second substances,
Figure 984208DEST_PATH_IMAGE039
is a cross spectral density function of 3 x 3 on the focal plane of the tightly-focused lens;
Figure 73387DEST_PATH_IMAGE080
indicates an output point
Figure 602589DEST_PATH_IMAGE032
The response to a single pulse at input point k, the integral term in absolute value in equation (12):
Figure 968979DEST_PATH_IMAGE081
(13)
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 as
Figure 151699DEST_PATH_IMAGE082
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 tightly-focused lens according to the definition of the spin angular momentum.
For angular frequency of
Figure 869119DEST_PATH_IMAGE083
(c is the vacuum speed of light) arbitrary time domain harmonic light field, which is in space
Figure 682572DEST_PATH_IMAGE084
The spin density of a spot can be defined as:
Figure 28102DEST_PATH_IMAGE085
(14)
wherein, the first and the second end of the pipe are connected with each other,
Figure 788248DEST_PATH_IMAGE045
represents a dielectric constant in a vacuum;
Figure 665068DEST_PATH_IMAGE046
is the angular frequency of the light source. Using pattern decomposition theory, focalAt the face
Figure 801651DEST_PATH_IMAGE086
The spin density of a spot can be expressed as:
Figure 877055DEST_PATH_IMAGE087
(15)
wherein the content of the first and second substances,
Figure 73681DEST_PATH_IMAGE088
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:
Figure 359169DEST_PATH_IMAGE089
(16)
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:
Figure 33864DEST_PATH_IMAGE041
(17)
wherein the content of the first and second substances,
Figure 229353DEST_PATH_IMAGE042
，
Figure 331301DEST_PATH_IMAGE043
，
Figure 776189DEST_PATH_IMAGE044
components in the three directions x, y and z in space respectively representing spin density; taking an imaginary part by double quote;
Figure 254575DEST_PATH_IMAGE047
，
Figure 570150DEST_PATH_IMAGE048
，
Figure 577420DEST_PATH_IMAGE049
，
Figure 837500DEST_PATH_IMAGE050
，
Figure 588418DEST_PATH_IMAGE051
and
Figure 758500DEST_PATH_IMAGE052
are 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:
Figure 61305DEST_PATH_IMAGE090
(18)
wherein the content of the first and second substances,
Figure 824993DEST_PATH_IMAGE091
and
Figure 645181DEST_PATH_IMAGE092
incident 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,
Figure 873031DEST_PATH_IMAGE093
in the form of a unit vector of radial polarization,
Figure 753263DEST_PATH_IMAGE094
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:
Figure 128880DEST_PATH_IMAGE095
(19)
wherein the content of the first and second substances,
Figure 221601DEST_PATH_IMAGE096
and
Figure 366275DEST_PATH_IMAGE097
respectively, 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,
Figure 10883DEST_PATH_IMAGE096
and
Figure 608217DEST_PATH_IMAGE097
can be expressed as:
Figure 504629DEST_PATH_IMAGE098
(20)
Figure 503809DEST_PATH_IMAGE099
and
Figure 584898DEST_PATH_IMAGE100
respectively, show tightly-focusing lenses
Figure 669528DEST_PATH_IMAGE101
And
Figure 776156DEST_PATH_IMAGE102
the 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:
Figure 754476DEST_PATH_IMAGE103
(21)
the formula (21) is abbreviated as:
Figure 870113DEST_PATH_IMAGE104
(22)
it is a 3 x 3 matrix in which:
Figure 442040DEST_PATH_IMAGE105
(23)
wherein:
Figure 680254DEST_PATH_IMAGE021
Figure 247502DEST_PATH_IMAGE022
wherein, the first and the second end of the pipe are connected with each other,
Figure 811339DEST_PATH_IMAGE023
representing the wavenumber of the partially coherent light emitted by the light source; equation (23) can be expressed as:
Figure 870561DEST_PATH_IMAGE106
(24)
substituting equation (23) into equation (8) yields the cross-spectral density at the tightly focused focal plane:
Figure 505942DEST_PATH_IMAGE107
Figure 678429DEST_PATH_IMAGE108
Figure 6642DEST_PATH_IMAGE109
(25)
order to
Figure 553161DEST_PATH_IMAGE025
At this time, the process of the present invention,
Figure 398757DEST_PATH_IMAGE110
expressed as:
Figure 973220DEST_PATH_IMAGE111
(26)
order:
Figure 82121DEST_PATH_IMAGE112
Figure 538773DEST_PATH_IMAGE113
Figure 345317DEST_PATH_IMAGE114
Figure 832011DEST_PATH_IMAGE115
then equation (24) can be written as:
Figure 144437DEST_PATH_IMAGE116
(27)
wherein the content of the first and second substances,
Figure 868810DEST_PATH_IMAGE117
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:
Figure 321788DEST_PATH_IMAGE118
(28)
wherein the content of the first and second substances,
Figure 979166DEST_PATH_IMAGE119
and
Figure 836394DEST_PATH_IMAGE120
respectively, representing the unfolded mode of the incident light and its corresponding weight. Substituting equation (28) into equation (27) yields:
Figure 579223DEST_PATH_IMAGE121
(29)
wherein the content of the first and second substances,
Figure 163788DEST_PATH_IMAGE122
Figure 941251DEST_PATH_IMAGE123
wherein the content of the first and second substances,
Figure 94015DEST_PATH_IMAGE124
is that
Figure 58559DEST_PATH_IMAGE125
The nth fully coherent mode of decomposition, F is the fourier transform symbol,
Figure 118919DEST_PATH_IMAGE124
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 of
Figure 16468DEST_PATH_IMAGE126
The coherent superposition of the mode squares of the system. Similarly, the other 8 terms of the cross spectral density function matrix
Figure 340133DEST_PATH_IMAGE127
、
Figure 323133DEST_PATH_IMAGE128
、
Figure 249501DEST_PATH_IMAGE129
、
Figure 470397DEST_PATH_IMAGE130
、
Figure 230543DEST_PATH_IMAGE131
、
Figure 169680DEST_PATH_IMAGE132
、
Figure 306263DEST_PATH_IMAGE133
、
Figure 178405DEST_PATH_IMAGE134
Can be determined by the same method. The spin density component can be obtained by using the 9-term cross spectral density formula:
Figure 109451DEST_PATH_IMAGE135
(30)
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:
Figure 110760DEST_PATH_IMAGE001
(7)
wherein the content of the first and second substances,
Figure 910088DEST_PATH_IMAGE002
is a cross spectral density function of 3 x 3 on the focal plane of the tightly-focused lens;
Figure 105577DEST_PATH_IMAGE003
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 )；
Figure 941946DEST_PATH_IMAGE004
And
Figure 121255DEST_PATH_IMAGE005
respectively representing after tight focusing r 1 And r 2 The spatial coordinates of the two points are,
Figure 334062DEST_PATH_IMAGE006
，
Figure 243112DEST_PATH_IMAGE007
(ii) a i represents an imaginary number; f and
Figure 515961DEST_PATH_IMAGE008
respectively 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,
Figure 651408DEST_PATH_IMAGE009
，
Figure 933484DEST_PATH_IMAGE010
；
Figure 228199DEST_PATH_IMAGE011
and
Figure 140792DEST_PATH_IMAGE012
respectively represent points r 1 And r 2 The integral of (a) is a infinitesimal,
Figure 29113DEST_PATH_IMAGE013
and
Figure 646039DEST_PATH_IMAGE014
are respectively a point r 1 And r 2 Is in an azimuth of
Figure 405048DEST_PATH_IMAGE015
，
Figure 754121DEST_PATH_IMAGE016
；
Figure 988793DEST_PATH_IMAGE017
Is a point r 1 And the angle between the line connecting the focal points of the tight focusing lenses and the optical axis,
Figure 347093DEST_PATH_IMAGE018
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
Figure 226188DEST_PATH_IMAGE019
，
Figure 11741DEST_PATH_IMAGE020
(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,
Figure 733709DEST_PATH_IMAGE021
Figure 36646DEST_PATH_IMAGE022
wherein the content of the first and second substances,
Figure 289686DEST_PATH_IMAGE023
representing the wavenumber of the partially coherent light emitted by the light source; substituting into equation (7) yields:
Figure 370775DEST_PATH_IMAGE024
(8)
order to
Figure 455406DEST_PATH_IMAGE025
At this time, the process of the present invention,
Figure 155508DEST_PATH_IMAGE002
expressed as:
Figure 9195DEST_PATH_IMAGE026
(9)
order:
Figure 136551DEST_PATH_IMAGE027
Figure 833111DEST_PATH_IMAGE028
Figure 71326DEST_PATH_IMAGE029
at this time, equation (9) is expressed as:
Figure 779519DEST_PATH_IMAGE030
(10)
wherein the content of the first and second substances,
Figure 77776DEST_PATH_IMAGE031
is a pulse function of the nonlinear optical system and is an output point
Figure 996054DEST_PATH_IMAGE032
To point k 1 And k 2 Response of two-point pulse.
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:
Figure 37959DEST_PATH_IMAGE033
(11)
wherein the content of the first and second substances,
Figure 866238DEST_PATH_IMAGE034
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;
Figure 194451DEST_PATH_IMAGE035
and
Figure 475390DEST_PATH_IMAGE036
respectively represent point k 1 And k 2 N =1,2, … …, N being the total number of completely coherent modes;
Figure 320987DEST_PATH_IMAGE037
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:
Figure 3772DEST_PATH_IMAGE038
(12)
wherein, the first and the second end of the pipe are connected with each other,
Figure 581515DEST_PATH_IMAGE039
is a cross spectral density function of 3 x 3 on the focal plane of the tightly-focused lens;
Figure 615330DEST_PATH_IMAGE040
indicates an output point
Figure 123672DEST_PATH_IMAGE032
The 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:
Figure 926543DEST_PATH_IMAGE041
wherein the content of the first and second substances,
Figure 737504DEST_PATH_IMAGE042
，
Figure 727457DEST_PATH_IMAGE043
，
Figure 39489DEST_PATH_IMAGE044
components in the three directions x, y and z in space respectively representing spin density; taking an imaginary part by double quote;
Figure 368971DEST_PATH_IMAGE045
represents a dielectric constant in a vacuum;
Figure 350833DEST_PATH_IMAGE046
is the angular frequency of the light source;
Figure 93661DEST_PATH_IMAGE047
，
Figure 943805DEST_PATH_IMAGE048
，
Figure 455689DEST_PATH_IMAGE049
，
Figure 874032DEST_PATH_IMAGE050
，
Figure 369736DEST_PATH_IMAGE051
and
Figure 367779DEST_PATH_IMAGE052
are 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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