CN117031767A - Method for generating a spatial double-spiral focal field with defined characteristics - Google Patents
Method for generating a spatial double-spiral focal field with defined characteristics Download PDFInfo
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- CN117031767A CN117031767A CN202311049656.8A CN202311049656A CN117031767A CN 117031767 A CN117031767 A CN 117031767A CN 202311049656 A CN202311049656 A CN 202311049656A CN 117031767 A CN117031767 A CN 117031767A
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- 230000005855 radiation Effects 0.000 claims abstract description 40
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- NAWXUBYGYWOOIX-SFHVURJKSA-N (2s)-2-[[4-[2-(2,4-diaminoquinazolin-6-yl)ethyl]benzoyl]amino]-4-methylidenepentanedioic acid Chemical compound C1=CC2=NC(N)=NC(N)=C2C=C1CCC1=CC=C(C(=O)N[C@@H](CC(=C)C(O)=O)C(O)=O)C=C1 NAWXUBYGYWOOIX-SFHVURJKSA-N 0.000 claims description 6
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- 230000003287 optical effect Effects 0.000 description 5
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- 235000012489 doughnuts Nutrition 0.000 description 2
- 230000009977 dual effect Effects 0.000 description 2
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- 238000012938 design process Methods 0.000 description 1
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- 238000012576 optical tweezer Methods 0.000 description 1
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- G—PHYSICS
- G02—OPTICS
- G02B—OPTICAL ELEMENTS, SYSTEMS OR APPARATUS
- G02B27/00—Optical systems or apparatus not provided for by any of the groups G02B1/00 - G02B26/00, G02B30/00
- G02B27/09—Beam shaping, e.g. changing the cross-sectional area, not otherwise provided for
- G02B27/0927—Systems for changing the beam intensity distribution, e.g. Gaussian to top-hat
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- G—PHYSICS
- G02—OPTICS
- G02B—OPTICAL ELEMENTS, SYSTEMS OR APPARATUS
- G02B27/00—Optical systems or apparatus not provided for by any of the groups G02B1/00 - G02B26/00, G02B30/00
- G02B27/09—Beam shaping, e.g. changing the cross-sectional area, not otherwise provided for
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- G02B27/095—Refractive optical elements
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Abstract
The application provides a method for generating a spatial double-spiral focal field with specified characteristics, which comprises a 4 pi focusing system, wherein the 4 pi focusing system consists of two high-numerical aperture objective lenses which are identical, bilaterally symmetrical and coaxially arranged; placing a virtual spiral antenna formed by magnetic units in a focal zone of a 4 pi focusing system, calculating a radiation field generated by the formed virtual spiral antenna, completely collecting and collimating the radiation field on a pupil plane through two high numerical aperture objective lenses which are bilaterally symmetrical, and solving the radiation field generated by the virtual spiral antenna through inversion to obtain incident field distribution on the pupil plane; the incident field is realized and is used as an incident light beam of the whole 4 pi focusing system, and the incident light beam is reversely transmitted through a lens and is converged in a focal zone of the 4 pi focusing system to form a required double-spiral focal field; by the technical scheme, iterative optimization of data is not needed, and the linear double-spiral focal field with controllable direction and length and the annular double-spiral focal field with controllable radius and period can be flexibly customized.
Description
Technical Field
The application relates to the technical field of optical focal field customization, in particular to a method for generating a space double-spiral focal field with specified characteristics.
Background
The doughnut focus surrounded by high intensity light has been widely studied for its use as an erase beam for super-resolution microscopes and for important applications in the fields of dark optical trapping, particle trapping, optical tweezers, etc. In 2011, wang J M et al established an ultralong diffraction-limited hollow light pipe by optimizing the radiation field of three sets of symmetrically placed co-located magneto-electric dipole combinations after amplitude. In 2019 Yu Y Z et al have proposed a method for generating a two-dimensional doughnut focal array and hollow light pipe array with predetermined characteristics using a virtual magnetic current source antenna.
For the study of the double helix structure, a learner has disclosed a related report. For example, barbieri N et al use a + -1 vortex phase plate to create an undiffracted double helix beam in a co-path interferometry system. Samanta K et al formed a double helix structure of a hexagonal array using phase control interference techniques.
In the above and other optical phenomena related to the generation of double helix structures, the methods adopted require parameter optimization and lack of flexibility in the design process, limiting their application in practical demands.
Disclosure of Invention
In view of the above, the present application aims to provide a method for generating a spatial double-spiral focal field with specified characteristics, which can flexibly customize a linear double-spiral focal field with controllable orientation and length and a circular double-spiral focal field with controllable radius and period without iterative optimization of data.
In order to achieve the above purpose, the application adopts the following technical scheme: a method for generating a spatial double helical focal field with defined characteristics, comprising a 4 pi focusing system consisting of two identical, side-to-side symmetrically and coaxially placed high numerical aperture objective lenses; placing a virtual spiral antenna formed by magnetic units in a focal zone of a 4 pi focusing system, calculating a radiation field generated by the formed virtual spiral antenna, completely collecting and collimating the radiation field on a pupil plane through two high numerical aperture objective lenses which are bilaterally symmetrical, and solving the radiation field generated by the virtual spiral antenna through inversion to obtain incident field distribution on the pupil plane; the incidence field is realized and is used as the incidence light beam of the whole 4 pi focusing system, and the incidence light beam is reversely transmitted through the lens and is converged in the focal zone of the 4 pi focusing system, so that the required double-spiral focal field is formed.
In a preferred embodiment, the far field radiation field expression of a single magnetic unit placed along the X-axis, Y-axis and Z-axis is first calculated by using the electromagnetic dual-dipole principle and the antenna electromagnetic radiation theory:
wherein the method comprises the steps ofIs the unit vector in the direction θ in the radiation field,/->Is along +.>A unit vector of directions; then the corresponding spatial direction +.>Radiation field ∈of magnetic unit>Represented by the following formula:
in θ 0 Is the angle between the magnetic element and the XOY plane,is the angle between the projection of the magnetic unit on the XOY plane and the X axis.
In a preferred embodiment, if the N magnetic units are rotated progressively and arranged linearly according to a fixed angle difference, the N magnetic unit is rotated by an angle ω= (N-1) ×15°, and a line is obtainedHelical antenna, the spatial orientation of the helical antenna beingLength is L, and its central point is positioned at the origin of coordinates, in which θ l Is the angle between the helical antenna and the XOY plane, < >>An included angle between the projection of the linear spiral antenna on the XOY plane and the X axis;
the total radiation field of the whole line helical antenna is:
when the linear helical antenna is along the Z axis, namely, theta is satisfied l =0, formula (3) is rewritten as:
when the helical antenna is located on the XOY plane, it is satisfied thatFormula (3) is rewritten as:
in the middle ofThe radiation field representing the nth magnetic element in the helical antenna, combined with equation (1), yields:
wherein the method comprises the steps ofAnd->Represents the spatial orientation of the nth magnetic element and satisfies the following relationship:
if the linear spiral antennas are connected end to end according to a circular track, a circular spiral antenna is obtained; the radiation field of the loop helical antenna is calculated as:
in the method, in the process of the application,for wave number, R is the radius of the loop helical antenna, then +.>Parameters included in-> The following new conditions are satisfied:
where T represents the period of the loop helical antenna.
In a preferred embodiment, a virtual helical antenna is placed near the confocal point of the 4pi focusing system, with its total radiation field completely collected and collimated to the pupil plane by two identical and symmetrical objective lenses; inverse solution is performed on normalizationIncident field distribution on pupil plane for generating target focal fieldAs an incident beam of light for the entire system;
if the objective lens satisfies the Helmholtz condition, the apodization function is thatDistribution of incident fieldCan be obtained from the formula (10):
the distribution condition of the double-spiral focal field is obtained by reversely and tightly focusing the incident field calculated in the formula (10) through a 4 pi focusing system by utilizing the Debye vector theory;
wherein C is 0 Represents the amplitude constant of the wave,
compared with the prior art, the application has the following beneficial effects: the application can be implemented by reverse focusing the radiation field of the helical antenna and easily calculating the incident field distribution required to produce a prescribed double helical focal field. Numerical results show that the direction and the length of the linear double-spiral focal field are determined by parameters of the linear spiral antennaAnd L control. The loop helical antenna parameters R and T determine the radius and period of the loop double helical focal field. The novel double-spiral focal field is used for optical lithography, spiral soliton and optical material treatmentAnd has potential application value in the aspects of optical imaging and the like.
Drawings
FIG. 1 is a schematic diagram of virtual helical antenna focusing in accordance with a preferred embodiment of the present application;
FIG. 2 is a schematic diagram of a virtual helical antenna according to a preferred embodiment of the present application;
FIG. 3 is a 3D view of a linear double helix focal field with a length of 5λ along the Z-axis of a preferred embodiment of the present application;
FIG. 4 is a YZ-side view of FIG. 3 in accordance with a preferred embodiment of the present application;
FIG. 5 is a 3D view of a linear double helix focal field with a length of 5.5λ, with an XOY plane (90, 45) in accordance with a preferred embodiment of the present application;
FIG. 6 is a top view of the XY plane of FIG. 5 in accordance with a preferred embodiment of the present application;
FIG. 7 is a 3D view of a linear double helix focal field of length 6λ with the XOY plane (90, 90) of a preferred embodiment of the present application;
FIG. 8 is a top view of the XY plane of FIG. 7 in accordance with a preferred embodiment of the present application;
FIG. 9 is a 3D view of a circular double helical focal field in accordance with a preferred embodiment of the present application;
fig. 10 is a graph showing the intensity distribution of the annular double spiral focal field on the XOY plane for a radius r=4λ and a period t=4 according to the preferred embodiment of the present application;
fig. 11 is a graph showing the intensity distribution of the annular double spiral focal field with a radius r=6λ and a period t=6 in the XOY plane according to the preferred embodiment of the present application;
FIG. 12 is a plot of the required incident field for a 4 lambda radius annular duplex focal zone in accordance with a preferred embodiment of the application;
FIG. 13 is a plot of the required incident field for a circular double helical focal zone of radius 6λ according to a preferred embodiment of the present application.
Detailed Description
The application will be further described with reference to the accompanying drawings and examples.
It should be noted that the following detailed description is illustrative and is intended to provide further explanation of the application. Unless defined otherwise, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this application belongs.
It is noted that the terminology used herein is for the purpose of describing particular embodiments only and is not intended to be limiting of exemplary embodiments according to the present application; as used herein, the singular is also intended to include the plural unless the context clearly indicates otherwise, and furthermore, it is to be understood that the terms "comprises" and/or "comprising" when used in this specification are taken to specify the presence of stated features, steps, operations, devices, components, and/or combinations thereof.
The application combines electromagnetic dual principle, antenna electromagnetic radiation theory, pattern comprehensive technology and time inversion technology, and introduces a method for flexibly customizing a double-helix focal field with controllable characteristics based on a radiation field of a helical antenna formed by magnetic units in a 4 pi focusing system.
The 4 pi focusing system adopted by the application, as shown in figure 1, consists of two identical high numerical aperture objective lenses which are bilaterally symmetrical and coaxially arranged; the virtual spiral antenna formed by the magnetic units is placed in a focal zone of a 4 pi focusing system, a radiation field generated by the formed virtual spiral antenna is calculated, the radiation field is completely collected and collimated on a pupil plane through two high numerical aperture objective lenses which are bilaterally symmetrical, and the radiation field generated by the virtual spiral antenna is solved through inversion, so that the incident field distribution on the pupil plane can be obtained; the incidence field is realized and is used as the incidence light beam of the whole 4 pi focusing system, and the incidence light beam is reversely transmitted through the lens and is converged in the focal zone of the 4 pi focusing system, so that the required double-spiral focal field is formed.
Firstly, the far field radiation field expression of a single magnetic unit placed along the X axis, the Y axis and the Z axis is deduced by utilizing the electromagnetic dual principle and the antenna electromagnetic radiation theory:
wherein the method comprises the steps ofIs the unit vector in the direction θ in the radiation field,/->Is along +.>Unit vector of direction. Then the corresponding spatial direction +.>Radiation field ∈of magnetic unit>Can be represented by the following formula:
in θ 0 Is the angle between the magnetic element and the XOY plane,is the angle between the projection of the magnetic unit on the XOY plane and the X axis.
If the N magnetic units are rotated in a stepwise manner and arranged linearly according to a fixed angle difference, the N-th magnetic unit is rotated by an angle ω= (N-1) ×15°, and a helical antenna can be obtained as shown in fig. 2. The spatial orientation of the helical antenna isLength is L, and its central point is positioned at the origin of coordinates, in which θ l Is the angle between the helical antenna and the XOY plane, < >>Is the angle between the projection of the line spiral antenna on the XOY plane and the X axis.
The total radiation field of the whole line helical antenna is:
when the helical antenna is along the Z-axis (optical axis), i.e. satisfies θ l =0, equation (3) can be rewritten as:
when the helical antenna is located on the XOY plane, it is satisfied thatEquation (3) is rewritten as:
in the middle ofThe radiation field representing the nth magnetic element in the helical antenna, combined with equation (1), yields:
wherein the method comprises the steps ofAnd->Represents the spatial orientation of the nth magnetic element and satisfies the following relationship:
and if the linear spiral antennas are connected end to end according to a circular track, obtaining the circular spiral antenna. The radiation field of the loop helical antenna can be calculated as:
in the method, in the process of the application,for wave number, R is the radius of the loop helical antenna, then +.>Parameters included in-> The following new conditions are satisfied:
where T represents the period of the loop helical antenna (i.e., the complete rotation of the magnetic element through T times is required for the wire helical antenna to complete the entire circular track arrangement).
Further, a virtual helical antenna is placed near the confocal point of the 4pi focusing system, as shown in the red region of fig. 1. The total radiation field is completely collected and collimated to the pupil plane by two identical and symmetrical objective lenses. Inverse solving for the incident field distribution required to generate the target focal field at the normalized pupil planeServing as an incident beam for the entire system.
If the objective lens satisfies the Helmholtz condition, the apodization function is thatDistribution of incident fieldCan be obtained from the formula (10):
and (3) reversely and tightly focusing the incidence field calculated in the formula (10) through a 4 pi focusing system by using the Debye vector theory, so as to obtain the distribution condition of the double-spiral focal field.
Wherein C is 0 Represents the amplitude constant of the wave,
as shown in fig. 1, a 4 pi focusing system consisting of two high numerical aperture objective lenses that are completely symmetric and confocal from side to side. The virtual helical antenna is placed near the confocal point of the 4pi focusing system (as shown by the red region of the system center). The system first applies to the total radiation field of the virtual helical antennaComplete collection (shown by the dashed arrow in orange) is performed and passed through two objective lenses quasi-to pupil planes on both sides; the calculated incident fields on the two pupil planes are 180 degrees out of phase, and then the incident fields areBack-propagating (as shown by the red dashed arrow) and focusing to the confocal point of the system to obtain the desired double helical focal field at the center of the system.
To simplify the calculation we will have a coefficient C independent of the shape of the radiation field 0 Normalized to 1. Maximum focusing angle theta max Is the key parameter of the whole radiation field of the spiral antenna can be completely collected, so that theta is set max =90°, i.e. na=1,for collecting the entire radiation field. It can be achieved by using a metasurface planar lens or a reflective objective lens.
Example 1 Generation of Linear double spiral focal field
(1.1) Generation of a Z axial Linear double spiral focal field
Let the parameters of the virtual line spiral antenna be theta l ＝0°、Take any value, l=5λ, and substitute it into equation (4). By combining the formulas (6), (7) and (11), a Z-axis linear double-spiral focal field is obtained, the 3D graph of which is shown in FIG. 3, and FIG. 4 is a YZ-plane side view of FIG. 3. It can be seen from fig. 3 and 4 that the optical focus field is formed by two independent single-helix focus fields oriented along the Z-axis and having a length close to 5λ, respectively defined by the parameters +_ of the virtual-line helical antenna>And L.
(1.2) Generation of an XOY planar Linear double spiral focal field
Let two groups of parameters of the virtual line spiral antenna be theta l ＝90°、L=5.5λ and θ l ＝90°、/>L=6λ is substituted into formula (5), respectively. By combining the formulas (6), (7) and (11), linear double-spiral focal fields with different orientations on the XOY plane are obtained, wherein 3D graphs are respectively shown in fig. 5 and 7, fig. 6 is an XY plane top view of fig. 5, and fig. 8 is an XY plane top view of fig. 7. It can be seen from fig. 5, 6, 7 and 8 that the linear double helical focal fields are all located in the XOY plane, oriented at 45 ° and 90 ° to the X axis (i.e. along the Y axis), respectively, and are approximately 5.5λ and 6λ in length, respectively.
Thus, as can be seen from example 1, by varying the parameters of the wire helical antennaAnd L can flexibly adjust the space direction and the length of the linear double-spiral focal field.
Example 2 Generation of circular double helical focal field
Let the parameters of the virtual loop helical antenna be r=4λ, t=4, and substituting them into formula (8). By combining the formulas (6), (9) and (11), a circular double-spiral focal field is obtained, the 3D graph of which is shown in FIG. 9, and FIG. 10 is a light intensity distribution diagram of the focal field on the XOY plane.
When the parameters R and T are set to r=6λ and t=6, they are substituted into formula (8). By combining the formulas (6), (9) and (11), a circular double spiral focal field is obtained, and the light intensity distribution diagram of the focal field on the XOY plane is shown in fig. 11.
Thus, as can be seen from example 2, the radius and period of the annular double helical focal field can be flexibly adjusted by changing the parameters R and T of the annular helical antenna.
Example 3 customization of the incident field distribution required for a double spiral focal zone
Take as an example two circular double helical focal fields (i.e., as shown in fig. 10 and 11) with radii of 4λ and 6λ, respectively, in example 2. When the focusing objective satisfies the helmholtz condition, equations (6), (8), (9) are substituted into equation (10), and the required incident field distribution on the pupil plane is obtained, as shown in fig. 12 and 13, respectively.
Claims (4)
1. A method for generating a spatial double-spiral focal field with defined characteristics, characterized by comprising a 4 pi focusing system consisting of two identical, left-right symmetrically and coaxially placed high numerical aperture objective lenses; placing a virtual spiral antenna formed by magnetic units in a focal zone of a 4 pi focusing system, calculating a radiation field generated by the formed virtual spiral antenna, completely collecting and collimating the radiation field on a pupil plane through two high numerical aperture objective lenses which are bilaterally symmetrical, and solving the radiation field generated by the virtual spiral antenna through inversion to obtain incident field distribution on the pupil plane; the incidence field is realized and is used as the incidence light beam of the whole 4 pi focusing system, and the incidence light beam is reversely transmitted through the lens and is converged in the focal zone of the 4 pi focusing system, so that the required double-spiral focal field is formed.
2. A method for generating a spatial double helical focal field with defined characteristics according to claim 1, characterized by first using electromagnetic duality principle and antenna electromagnetic radiation theory to derive the far field radiation field expression of a single magnetic element placed along X-axis, Y-axis and Z-axis as:
wherein the method comprises the steps ofIs the unit vector in the direction θ in the radiation field,/->Is along +.>A unit vector of directions; then the corresponding spatial direction +.>Radiation field ∈of magnetic unit>Represented by the following formula:
in θ 0 Is the angle between the magnetic element and the XOY plane,is the angle between the projection of the magnetic unit on the XOY plane and the X axis.
3. A method for generating a spatial double helical focal field having defined characteristics according to claim 2, wherein if N of said magnetic elements are rotated progressively and arranged linearly according to a fixed angular difference, wherein the nth magnetic element is rotated by an angle ω= (N-1) ×15 °, a linear helical antenna is obtained, the spatial orientation of which isLength is L, and its central point is positioned at the origin of coordinates, in which θ l Is the angle between the helical antenna and the XOY plane, < >>An included angle between the projection of the linear spiral antenna on the XOY plane and the X axis;
the total radiation field of the whole line helical antenna is:
when the linear helical antenna is along the Z axis, namely, theta is satisfied l =0, formula (3) is rewritten as:
when the helical antenna is located on the XOY plane, it is satisfied thatFormula (3) is rewritten as:
in the middle ofThe radiation field representing the nth magnetic element in the helical antenna, combined with equation (1), yields:
wherein the method comprises the steps ofAnd->Represents the spatial orientation of the nth magnetic element and satisfies the following relationship:
if the linear spiral antennas are connected end to end according to a circular track, a circular spiral antenna is obtained; the radiation field of the loop helical antenna is calculated as:
in the method, in the process of the application,for wave number, R is the radius of the loop helical antenna, then +.>Parameters included in->The following new conditions are satisfied:
where T represents the period of the loop helical antenna.
4. A method for generating a spatial double helical focal field with defined characteristics according to claim 3, characterized in that a virtual helical antenna is placed near the confocal point of a 4pi focusing system, whose total radiation field is completely collected and collimated to the pupil plane by two identical and symmetrical objective lenses; inverse solving for the incident field distribution required to generate the target focal field at the normalized pupil planeAs an incident beam of light for the entire system;
if the objective lens satisfies the Helmholtz condition, the apodization function is thatDistribution of incident fieldCan be obtained from the formula (10):
the distribution condition of the double-spiral focal field is obtained by reversely and tightly focusing the incident field calculated in the formula (10) through a 4 pi focusing system by utilizing the Debye vector theory;
wherein C is 0 Represents the amplitude constant of the wave,
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| CN117741964A (en) * | 2024-02-07 | 2024-03-22 | 泉州师范学院 | Method for generating super-resolution diffraction-free transverse polarized light needle |
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| CN117741964A (en) * | 2024-02-07 | 2024-03-22 | 泉州师范学院 | Method for generating super-resolution diffraction-free transverse polarized light needle |
| CN117741964B (en) * | 2024-02-07 | 2024-05-03 | 泉州师范学院 | Method for generating super-resolution diffraction-free transverse polarized light needle |
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