Method for generating focal field of optical pipe with independent control of length, orientation and orbital angular momentum

Abstract

This invention belongs to the technical field of optical tube focal field generation. It discloses a method for generating an optical tube focal field with independent control over its length, orientation, and orbital angular momentum. The method includes the following steps: customizing a uniform current line source antenna model and solving for the radiation field distribution of the current line source antenna model; modulating the radiation field with a spiral phase factor; obtaining the entrance pupil field distribution based on the radiation field modulated by the spiral phase factor; and calculating the light field distribution characteristics of the entrance pupil field in the focal region of a 4Pi focusing system based on the Richards-Wolf vector diffraction integral theory. This method enables the optical tube length to be independently determined by the antenna length parameter in the focal region, the optical tube spatial orientation to be determined by the antenna orientation, and the direction and topological charge of the orbital angular momentum carried by the optical tube cross-section to be determined by the introduced spiral phase factor. The magnitude of the topological charge of the spiral phase factor simultaneously determines the radial dimension of the hollow dark region in the optical tube cross-section, thereby achieving independent control over the size of the hollow region.

Description

Technical Field

[0001] This invention relates to the technical field of optical tube focal field generation, and more particularly to a method for generating optical tube focal fields with independent control over length, orientation, and orbital angular momentum. Background Technology

[0002] A hollow optical tube (also known as an optical dark channel) is a special focal field structure in which the focal intensity is distributed in a three-dimensional tubular shape, and the central dark region is surrounded by a high-brightness shell. This type of focal field has significant applications in particle trapping and manipulation, atomic cooling and confinement, optical microscopy, stimulated emission depletion super-resolution imaging, optical wrenches, and high-harmonic generation, and therefore continues to receive widespread attention from researchers both domestically and internationally.

[0003] Regarding methods for generating the focal field of a light tube, existing publicly reported techniques mainly fall into the following categories: First, based on the tight focusing of a vector beam (such as azimuthally polarized or radially polarized beams) through a high numerical aperture objective lens, a uniform dark channel is obtained by means of amplitude apodization filtering or phase modulation (J. Modern Opt., 2013, 60, 378–381; J. Appl. Phys., 2018, 124, 043103); Second, based on focusing a vortex beam (such as a Bessel vortex beam or a circular Airy vortex beam), an ultra-long dark channel is formed in the focal field by adjusting the topological charge (Chin. Opt. Lett., 2023, 21, 072601; Photonics, 2023, 10, 1279); Third, based on the dual-objective opposing focusing structure of a 4Pi focusing system, the virtual antenna radiation field is reverse-focused to generate a light tube focal field with adjustable length and spatial orientation (Opt. Commun., 2022, 506, 127581).

[0004] However, a systematic review of the aforementioned publicly available reports reveals the following common problems in existing technologies for achieving optical tube focal fields: First, the focal fields generated by most solutions are limited to the optical axis (z-axis). Even though the literature (Opt. Commun., 2022, 506, 127581) extends the optical tube orientation to three-dimensional space, the degrees of freedom that can be controlled are still limited to the length and spatial orientation of the optical tube, resulting in a single focal field characteristic of the optical tube cross-section. Second, the cross-section of the optical tube focal field in existing solutions is a pure azimuth polarization field, which does not carry orbital angular momentum (OAM) and cannot impart controllable OAM to trapped or guided particles. Third, the size of the hollow dark area in the cross-section of the optical tube focal field in existing technologies depends on fixed parameters such as the physical aperture of the system. Once the system is designed, it remains fixed and lacks independent and continuous control methods. Summary of the Invention

[0005] The purpose of this invention is to provide a method for generating a focal field of a light tube with independently controllable length, orientation, and orbital angular momentum, so as to solve the problems mentioned in the background art.

[0006] To solve the above-mentioned technical problems, the technical solution adopted by the present invention is: a method for generating a focal field of a light tube with independent control over its length, orientation, and orbital angular momentum, comprising the following steps: Step (1): Customize a uniform current line source antenna model and solve for the radiation field distribution of the current line source antenna model; Step (2): Modulate spiral phase factors with the same direction and specified topological charge number on the radiation field; Step (3): Based on the radiation field modulated by the spiral phase factor, the entrance pupil field distribution is obtained by inverse solution; Step (4): Based on the Richards-Wolf vector diffraction integral theory, calculate the light field distribution characteristics of the entrance pupil field in the focal region of the 4Pi focusing system.

[0007] Preferably, the specific process of step (1) is as follows: First, establish a 4Pi optical focusing system consisting of two objective lenses with confocal regions; Then place the current line source antenna in a three-dimensional spatial coordinate system with the common focal point of the 4Pi optical focusing system as the origin; Based on electromagnetic radiation theory, using vector potential functions and the principle of ray superposition, the radiation field distribution of the current line source antenna in the 4Pi optical focusing system was calculated and solved.

[0008] Preferably, in step (1), the 4Pi optical focusing system consists of two high numerical aperture objectives with identical external dimensions and optical parameters, and the optical axes of the two objectives are on the same straight line and are placed confocally.

[0009] Preferably, a global rectangular coordinate system is established in the 4Pi optical focusing system; wherein the origin O of the global rectangular coordinate system is the common focal point of the two objectives; the direction of the optical axis is taken as the z-axis, and the z-axis is perpendicular to the xy plane; the y-axis is vertically upward, and the x-axis is perpendicular to the yz plane.

[0010] Preferably, the geometric length of the current line source antenna is L0, and its center point is located at the origin O of the global rectangular coordinate system; the spatial orientation of the current line source antenna is... ,in θ 0 represents the angle between the axis of the current-source antenna and the z-axis. The azimuth angle between the projection of the current-source antenna onto the xy-plane and the x-axis.

[0011] Preferably, the analytical expression for the radiation field of the current line source antenna is as follows: (1) in: (2) (3) (4) (5) (6) (7) in Let be the spherical coordinate variables of the radiation field. F u This is a coefficient that is independent of the directionality of the radiation field. The array factor is the array of a current-source antenna as a continuous array of dipoles. and Radiation fields θ Components and Component directionality coefficient factor j The imaginary unit is ω, where ω is the angular frequency of the current as a function of time. k For wave number, Permeability, This represents the amplitude of the antenna carrier current.

[0012] Preferably, in step (2), the calculation process of the spiral phase factor is as follows: First, a local coordinate system is established by rotating the global xyz coordinate system, and then the global coordinate system is... z Unit vector around axis Rotate according to the right-hand rule θ 0 angle x shaft and y Shaft synchronous rotation forms x 'and y 'axis, then ( x '-, y '-, z The '-' axes form the three principal axes of a local rectangular coordinate system; at this time, in x '- y The spiral phase factor of the plane is: (8) in, for x '- y 'Plane Self x The azimuth angle measured counterclockwise from the axis; assuming the local coordinate system... x 'axis and ( x -, y -, z The angle between the -) axis and the () axis is ( α 1, β 1, γ 1), y 'axis and ( x -, y -, z The angle between the -) axis and the () axis is ( α 2, β 2, γ 2), z 'axis and ( x -, y -, z The angle between the -) axis and the () axis is ( α 3, β 3, γ 3) The transformation relationship between the coordinate variables of the global rectangular coordinate system and the rotated local rectangular coordinate system is as follows: (9) in: (10) Thus, the analytical expression for the radiation field modulated with the same spatial orientation spiral phase factor is obtained as follows: (11) in, T n is the topological charge number of the spiral phase factor.

[0013] Preferably, step (3) is the modulated radiation field of equation (11) shown. Combined with the apodization function of the lens, the vector field distribution of the entrance pupil can be directly solved analytically. This reverse solution process does not require complex iterative optimization.

[0014] Preferably, in step (4), the light field obtained from step (3) after being converged by the 4Pi optical focusing system is quantitatively evaluated using the Richards-Wolf vector diffraction integral theory: (12) Where λ is the wavelength. The focal field in cylindrical coordinates. θ max This is the maximum convergence angle of the objective lens, which is related to the numerical aperture of the objective lens; Let be the approximate spherical wavefront of the entrance pupil field after refraction through the lens; the two double integrals on the right side of the equation represent the tightly focused fields of the two lenses, where the factor e in the second integral is... jπ This indicates that the instantaneous polarization directions of the two entrance pupil vector light fields are opposite, that is, the phase difference is π.

[0015] As can be seen from the above description of the structure of the present invention, compared with the prior art, the present invention has the following advantages: This invention proposes to utilize the radiation field of a uniform current line source antenna modulated by a spiral phase factor to inversely solve the entrance pupil vector light field. This vector light field is then used as the incident field and input into a 4Pi optical focusing system for focused focusing. This allows the following objectives to be achieved simultaneously in the focal region: the length of the optical tube is independently determined by the length parameter of the current line source antenna; the spatial orientation of the optical tube is determined by the orientation of the current line source antenna; the direction of the orbital angular momentum and topological charge carried by the cross-section of the optical tube are determined by the introduced spiral phase factor; the magnitude of the topological charge of the spiral phase factor simultaneously determines the radial dimension of the hollow dark area in the cross-section of the optical tube. The larger the topological charge, the larger the diameter of the dark area, thus achieving independent control of the size of the hollow area; and the OAM direction is always consistent with the spatial orientation of the optical tube. Attached Figure Description

[0016] The accompanying drawings, which form part of this application, are used to provide a further understanding of the invention. The illustrative embodiments of the invention and their descriptions are used to explain the invention and do not constitute an undue limitation of the invention. In the drawings: Figure 1 This is a flowchart of the optical tube focal field generation process of the present invention; Figure 2 This is a schematic diagram of the 4Pi optical focusing system of the present invention; Figure 3 This is a three-dimensional intensity profile of the x-axis optical tube according to Embodiment 1 of the present invention; Figure 4 This is a three-dimensional intensity profile of the y-axis optical tube according to Embodiment 1 of the present invention; Figure 5 This is a three-dimensional intensity profile of the z-axis optical tube according to Embodiment 1 of the present invention; Figure 6 This is Embodiment 1 of the present invention (65) ° 75 ° ) Three-dimensional profile of the intensity towards the light tube; Figure 7 (65) is the embodiment of the present invention. ° 75 ° A cross-sectional phase distribution diagram of the x-component towards the focal field of the optical tube passing through the origin; Figure 8 (65) is the embodiment of the present invention. ° 75 ° A cross-sectional phase distribution diagram of the y-component towards the focal field of the optical tube passing through the origin; Figure 9 (65) is the embodiment of the present invention. ° 75 ° A cross-sectional phase distribution diagram of the z-component of the optical tube focal field passing through the origin; Figure 10 This is a three-dimensional intensity profile of the z-axis optical tube in Embodiment 2 of the present invention; Figure 11 This is an xy-plane view of the z-axis optical tube according to Embodiment 2 of the present invention; Figure 12 This is a three-dimensional intensity profile of the y-axis optical tube according to Embodiment 2 of the present invention; Figure 13 This is an xz plan view of the y-axis optical tube according to Embodiment 2 of the present invention. Detailed Implementation

[0017] To better understand the technical solution of the present invention, the technical solution of the present invention will be described in detail below with reference to the accompanying drawings and specific embodiments.

[0018] refer to Figure 1 and Figure 2 As shown, the method for generating the focal field of a light tube with independently controllable length, orientation, and orbital angular momentum includes the following steps: customizing a uniform current line source antenna model and solving for the radiation field distribution of the current line source antenna model; modulating a spiral phase factor with the same direction and a specified topological charge on the radiation field; obtaining the entrance pupil field distribution by inversely solving the radiation field after modulation by the spiral phase factor; and calculating the light field distribution characteristics of the entrance pupil field in the focal region of the 4Pi focusing system based on the Richards-Wolf vector diffraction integral theory.

[0019] This invention proposes to utilize the radiation field of a uniform current line source antenna modulated by a spiral phase factor to inversely solve the entrance pupil vector light field. This vector light field is then used as the incident field and input into a 4Pi optical focusing system for focused focusing. This allows the following objectives to be achieved simultaneously in the focal region: the length of the optical tube is independently determined by the length parameter of the current line source antenna; the spatial orientation of the optical tube is determined by the orientation of the current line source antenna; the direction of the orbital angular momentum and topological charge carried by the cross-section of the optical tube are determined by the introduced spiral phase factor; the magnitude of the topological charge of the spiral phase factor simultaneously determines the radial dimension of the hollow dark area in the cross-section of the optical tube. The larger the topological charge, the larger the diameter of the dark area, thus achieving independent control of the size of the hollow area; and the OAM direction is always consistent with the spatial orientation of the optical tube.

[0020] The specific implementation steps of the method of the present invention will now be described in detail: (1) Customize a uniform current line source antenna model and solve its radiation field: The 4Pi optical focusing system consists of two high numerical aperture objectives with identical dimensions and optical parameters. The optical axes of the two objectives are on the same straight line and are placed confocally. A global rectangular coordinate system is established in the 4Pi optical focusing system; where the origin O of the global rectangular coordinate system is the common focal point of the two objectives; the direction of the optical axis is taken as the z-axis, and the z-axis is perpendicular to the xy plane; the y-axis is vertically upward, and the x-axis is perpendicular to the yz plane; Then place the current line source antenna in a three-dimensional spatial coordinate system with the common focal point of the 4Pi optical focusing system as the origin; The geometric length of the current line source antenna is L0, and its center point is located at the origin O of the global rectangular coordinate system; the spatial orientation of the current line source antenna is... ,in θ 0 represents the angle between the axis of the current-source antenna and the z-axis. The azimuth angle between the projection of the current-line source antenna onto the xy plane and the x-axis; The analytical expression for the radiation field of a current-source antenna is as follows: (1) in: (2) (3) (4) (5) (6) (7) in Let be the spherical coordinate variables of the radiation field. F u This is a coefficient that is independent of the directionality of the radiation field. The array factor is the array of a current-source antenna as a continuous array of dipoles. and Radiation fields θ Components and Component directionality coefficient factor j The imaginary unit is ω, where ω is the angular frequency of the current as a function of time. k For wave number, Permeability, This represents the amplitude of the antenna carrier current.

[0021] (2) Modulating the spiral phase factor in the radiation field: In order to enable the focal field to have adjustable orbital angular momentum (OAM) pointing and topological charge, a spiral phase factor with the same pointing and a specified topological charge is superimposed on the radiation field in step (1); The specific solution process for the spiral phase factor is as follows: First, a local coordinate system is established by rotating the global xyz coordinate system. Then, the global coordinate system... z Unit vector around axis Rotate according to the right-hand rule θ 0 angle x shaft and y Shaft synchronous rotation forms x 'and y 'axis, then (x '-, y '-, z The '-' axes form the three principal axes of a local rectangular coordinate system. At this point, in... x '- y The spiral phase factor of the plane is: (8) in, for x '- y 'Plane Self x The azimuth angle measured counterclockwise from the axis; assuming the local coordinate system... x 'axis and ( x -, y -, z The angle between the -) axis and the () axis is ( α 1, β 1, γ 1), y 'axis and ( x -, y -, z The angle between the -) axis and the () axis is ( α 2, β 2, γ 2), z 'axis and ( x -, y -, z The angle between the -) axis and the () axis is ( α 3, β 3, γ 3) The transformation relationship between the coordinate variables of the global rectangular coordinate system and the rotated local rectangular coordinate system is as follows: (9) in: (10) Thus, the analytical expression for the radiation field modulated with the same spatial orientation spiral phase factor is obtained as follows: (11) in, T n is the topological charge number of the spiral phase factor.

[0022] (3) Based on the radiation field of the modulation spiral phase factor in steps (1) and (2), the entrance pupil field is solved in reverse.

[0023] The modulated radiation field shown in Equation (11), combined with the apodization function of the lens, can be directly and analytically solved for the vector field distribution on the entrance pupil surface. This reverse solution process does not require complex iterative optimization.

[0024] (4) Quantitative evaluation of its 4Pi focused light field characteristics based on Richards-Wolf vector diffraction integral theory: The light field obtained from step (3) after being focused by the 4Pi focusing system is quantitatively evaluated using the Richards-Wolf vector diffraction integral theory: (12) Where λ is the wavelength. The focal field in cylindrical coordinates. θ max This is the maximum convergence angle of the objective lens, which is related to the numerical aperture of the objective lens. This is the approximate spherical wavefront of the entrance pupil field after refraction through the lens. The two double integrals on the right-hand side of the equation represent the tightly focused fields of the two lenses, with the factor e in the second integral formula being... jπ This indicates that the instantaneous polarization directions of the two entrance pupil vector light fields are opposite, that is, the phase difference is π.

[0025] The following specific embodiments demonstrate the effectiveness of the method proposed in this invention.

[0026] To simplify calculations, the examples provided will use parameters independent of the shape and polarization of the optical focal field. F u Normalization; for the overall radiation field of the current line source antenna with convergent modulation spiral phase factor, the convergence angle of the high numerical aperture objective lens is taken. .

[0027] Example 1:

[0028] Set the topology load. T n The length of the line source is 2. L 0 is 2λ, and the spatial orientations of the line source and the spiral phase factor are along the x-axis, y-axis, z-axis, and (65) of the global rectangular coordinate system, respectively. ° 75 ° ) direction. Using the current line source antenna method described above, the focal field value is calculated using equation (12), and the characteristics of the focal field are visualized, such as Figures 3 to 9 As shown.

[0029] Depend on Figures 3 to 6 It can be seen that four hollow optical tubes of the same length, with a length slightly less than 2λ, are positioned along the x-axis, y-axis, z-axis, and (65)... ° 75 ° )direction. Figures 3 to 6 The phase distribution of the cross-section of the optical tubes in the tubes all exhibits a helical phase distribution characteristic with a topological charge of 2. Figure 6 Taking a light tube as an example, the phase distributions of its focal field x, y, and z components across the cross-section at the origin are as follows: Figures 7 to 9 As shown.

[0030] Depend on Figures 7 to 9 It can be clearly seen that the tubular hollow region exhibits a distinct spiral phase distribution with a topological charge of 2. The magnitude of the topological charge in OAM is related to the spiral phase factor. T n Consistent, OAM spatial pointer and parameters The established direction matches, verifying the effectiveness of the proposed method.

[0031] Example 2:

[0032] As can be seen from Example 1, the proposed method can independently control the orientation of the optical tube's focal field. The control of the optical tube's length and topological charge will now be explained using the second example. First, two sets of combined parameters are set as follows: , L 0=12λ、 T n =3 and , L 0 = 10λ T n =5, through the previous current line source antenna steps, the three-dimensional contour of the focal field intensity and its cross-sectional view are obtained as follows: Figure 10 and Figure 13 As shown. It can be measured that the optical tube is aligned with the z-axis and y-axis respectively, matching the set direction parameters; the lengths of the optical tube are slightly less than 12λ and 10λ respectively, as determined by the parameters. L 0 determines the size of the hollow region in the light tube; the size varies with the parameters. T n As the diameter of the hollow part increases, the diameter of the hollow part also increases.

[0033] The above two embodiments demonstrate the effectiveness of the method proposed in this invention. By utilizing the radiation field of a uniform current line source antenna modulated by a spiral phase factor, the entrance pupil vector light field is solved in reverse, and then this vector light field is used as the incident field input into a 4Pi optical focusing system for opposing tight focusing. This allows the following objectives to be achieved simultaneously in the focal region: the length of the optical tube is independently determined by the length parameter of the current line source antenna; the spatial orientation of the optical tube is determined by the orientation of the current line source antenna; and the direction of the orbital angular momentum and topological charge carried by the cross-section of the optical tube are determined by the introduced spiral phase factor. This invention demonstrates significant innovation and creativity in the functional dimensions and controllability of the optical tube focal field.

[0034] The above current line source antenna is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for generating the focal field of a light tube with independently controllable length, orientation, and orbital angular momentum, characterized in that, Includes the following steps: Step (1): Customize a uniform current line source antenna model and solve for the radiation field distribution of the current line source antenna model; Step (2): Modulate spiral phase factors with the same orientation and specified topological charge number on the radiation field; Step (3): Based on the radiation field modulated by the spiral phase factor, the entrance pupil field distribution is obtained by inverse solution; Step (4): Based on the Richards-Wolf vector diffraction integral theory, calculate the light field distribution characteristics of the entrance pupil field in the focal region of the 4Pi focusing system.

2. The method for generating a light tube focal field with independently controllable length, orientation, and orbital angular momentum according to claim 1, characterized in that, The specific process of step (1) is as follows: First, establish a 4Pi optical focusing system consisting of two objective lenses with confocal regions; Then place the current line source antenna in a three-dimensional spatial coordinate system with the common focal point of the 4Pi optical focusing system as the origin; Based on electromagnetic radiation theory, using vector potential functions and the principle of ray superposition, the radiation field distribution of the current line source antenna was calculated and solved.

3. The method for generating a light tube focal field with independently controllable length, orientation, and orbital angular momentum according to claim 2, characterized in that: In step (1), the 4Pi optical focusing system consists of two high numerical aperture objectives with identical external dimensions and optical parameters. The optical axes of the two objectives are on the same straight line and are placed confocally.

4. The method for generating a light tube focal field with independently controllable length, orientation, and orbital angular momentum according to claim 2, characterized in that: A global rectangular coordinate system is established in the 4Pi optical focusing system; wherein, the origin O of the global rectangular coordinate system is the common focal point of the two objectives; the direction of the optical axis is taken as the z-axis, and the z-axis is perpendicular to the xy plane; the y-axis is vertically upward, and the x-axis is perpendicular to the yz plane.

5. The method for generating a light tube focal field with independently controllable length, orientation, and orbital angular momentum according to claim 4, characterized in that: The geometric length of the current line source antenna is L0, and its center point is located at the origin O of the global rectangular coordinate system. The spatial orientation of the current line source antenna is ,in θ 0 represents the angle between the axis of the current-source antenna and the z-axis. The azimuth angle between the projection of the current-source antenna onto the xy-plane and the x-axis.

6. The method for generating a light tube focal field with independently controllable length, orientation, and orbital angular momentum according to claim 5, characterized in that: The analytical expression for the radiation field of a current-source antenna is as follows: (1) in: (2) (3) (4) (5) (6) (7) in Let be the spherical coordinate variables of the radiation field. F u This is a coefficient that is independent of the directionality of the radiation field. The array factor is the array of a current-source antenna as a continuous array of dipoles. and Radiation fields θ Components and Component directionality coefficient factor j The imaginary unit is ω, where ω is the angular frequency of the current as a function of time. k For wave number, Permeability, This represents the amplitude of the antenna carrier current.

7. The method for generating a light tube focal field with independently controllable length, orientation, and orbital angular momentum according to claim 6, characterized in that: In step (2), the calculation process of the spiral phase factor is as follows: First, a local coordinate system is established by rotating the global xyz coordinate system, and then the global coordinate system is... z Unit vector around axis Rotate according to the right-hand rule θ 0 angle x shaft and y Shaft synchronous rotation forms x 'and y 'axis, then ( x '-, y '-, z The '-' axes form the three principal axes of a local rectangular coordinate system; at this time, in x '- y The spiral phase factor of the plane is: (8) in, for x '- y 'Plane Self x The azimuth angle measured counterclockwise from the axis; assuming the local coordinate system... x 'axis and ( x -, y -, z The angle between the -) axis and the () axis is ( α 1, β 1, γ 1), y 'axis and ( x -, y -, z The angle between the -) axis and the () axis is ( α 2, β 2, γ 2), z 'axis and ( x -, y -, z The angle between the -) axis and the () axis is ( α 3, β 3, γ 3) The transformation relationship between the coordinate variables of the global rectangular coordinate system and the rotated local rectangular coordinate system is as follows: (9) in: (10) Thus, the analytical expression for the radiation field modulated with the same spatial orientation spiral phase factor is obtained as follows: (11) in, T n is the topological charge number of the spiral phase factor.

8. The method for generating a light tube focal field with independently controllable length, orientation, and orbital angular momentum according to claim 7, characterized in that: Step (3) is the modulation radiation field shown in Equation (11), which, combined with the apodization function of the lens, can directly and analytically solve the vector field distribution on the entrance pupil surface. This reverse solution process does not involve complex iterative optimization.

9. The method for generating a light tube focal field with independently controllable length, orientation, and orbital angular momentum according to claim 1, characterized in that: In step (4), the light field obtained from step (3) after being converged by the 4Pi optical focusing system is quantitatively evaluated using the Richards-Wolf vector diffraction integral theory: (12) Where λ is the wavelength. The focal field in cylindrical coordinates. θ max This is the maximum convergence angle of the objective lens, which is related to the numerical aperture of the objective lens; Let be the approximate spherical wavefront of the entrance pupil field after refraction through the lens; the two double integrals on the right side of the equation represent the tightly focused fields of the two lenses, where the factor e in the second integral is... jπ This indicates that the instantaneous polarization directions of the two entrance pupil vector light fields are opposite, that is, the phase difference is π.

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