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Transversely oriented cylindrically polarized optical fields

Xindong Meng, Chenhao Wan, and Qiwen Zhan

Open Access Open Access

Abstract

Cylindrical vector (CV) beams have nonuniform polarization vector distribution with a singularity line directed along the optical axis. In this paper, we propose a method to synthesize transversely oriented cylindrically polarized optical fields in the focal region with a singularity line perpendicular to the optical axis. The scheme is based on the time-reversal method, the vectorial diffraction theory, and the 4Pi optical configuration. Both transversely oriented radially polarized and azimuthally polarized optical fields are demonstrated. The superposition of transverse cylindrically polarized optical fields leads to a peculiar distribution carrying controllable transverse spin angular momentum (SAM) and transverse orbital angular momentum (OAM) that may find applications in optical tweezing, light-matter interaction, and unidirectional beam propagation excitation.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

As a vector solution to the Maxwell equations, cylindrical vector (CV) beams have attracted numerous attentions in the past couple of decades for its peculiar and symmetric spatial distribution of the polarization vector [1]. The emergence of a nonnegligible longitudinal electric field component of CV beams under tight focusing enables intricate focal field manipulation achieving sub-diffraction-limited focal spot [2]. CV beams have also found applications in electron acceleration [3], optical tweezing [4], laser micromachining [5], and high-capacity optical communication [6]. In the field of singular optics and topological photonics, CV beams belong to a category of beams with polarization singularities of index 1 [7]. Compared to phase singularities, polarization singularities are distinct for the vectorial nature that is attractive to the research on the interaction of light with birefringent, nonlinear or magneto-optic materials including spin-orbit coupling [8,9], spin-orbit Hall effect [10], and magneto-lithography [11–13].

The singularity line of CV beams is directed along the optical axis. In the current paper, we build cylindrically polarized optical fields in the focal region with a singularity line perpendicular to the optical axis, i.e. the nonuniform polarization vector is spatially distributed in the x-z plane. Both transversely-oriented radially polarized optical field and azimuthally polarized optical field are generated. The required optical fields in the pupil plane for the generation of such focal fields are calculated as well. With the research and development of structured light in recent years, it is possible to control all degrees of freedom of structured light. Common methods of tailoring complex vector beams include using metasurfaces or spatial light modulators [14–16]. Therefore, the generation method proposed in the current paper can be readily realized in an experimental manner. Furthermore, the superposition of the two different transversely oriented cylindrically polarized optical fields creates an interesting optical field with controllable transverse spin angular momentum (SAM) and transverse orbital angular momentum (OAM).

Different from the longitudinal SAM, the transverse SAM can still exist even in the unpolarized light field [17]. It is not only a component of the spin angular momentum density, but also an independent physical entity. The transverse SAM has been discovered to be tightly connected with unidirectional beam propagation [18]. Transverse OAM are experimentally demonstrated recently and provides a new degree of freedom to applications harnessing photonic OAM [19–25] . The current paper presents a synthesis method of special light field with controllable transverse SAM and transverse OAM, which is different from the above methods. The creation of optical fields with both controllable transverse SAM and transverse OAM is thus appealing and applicable to the research related to optical manipulation, light-matter interaction, and polarization topology.

2. Method

According to the time-reversal theory [26,27] and vectorial diffraction theory [28,29], tightly focusing with one microscope objective the complex conjugate of the radiation fields emitted from an infinitesimal dipole antenna results in a light spot in the shape of an ellipsoid of subwavelength size [30–32]. Taking advantage of two objectives with the 4Pi configuration leads to a nearly spherical spot of linear polarization vibrating along the direction of dipole oscillating direction [33–37]. Focusing the complex conjugate of the collective radiation fields emitted from three individual dipoles oscillating in three orthogonal directions and positioning on one spot, not only can a spherical spot be formed but also the three-dimensional state of polarization of the light spot can be controlled. Through placing a number of infinitesimal dipole antenna sets on a fine grid in the x-z plane of the focal region, collecting all the radiation fields, and focusing the complex conjugate of the radiation fields, it is proven that the amplitude, phase and polarization vector distribution in the x-z plane can be tailored. Figure 1 shows the schematic of the optical setup that consists of two high-NA (numerical aperture) microscope objectives in the 4Pi configuration where the focal point is at the origin. A number of dipole antenna sets are evenly placed on a fine grid with a subwavelength spacing. Each set contains two individual dipoles oscillating in the x- and z-directions respectively. The amplitude and phase distributions in the x-y plane of a traditional cylindrically polarized optical fields are sampled and mapped to the relative amplitude and phase of the dipole antenna sets positioning in the x-z plane.

 figure: Fig. 1.

Fig. 1. Schematic of the proposed method. Dipole antenna sets are uniformly positioned within a circular area of 1.5λ radius in the x-z plane. The radiation fields emitted from dipole antenna sets are collected by two high-NA objective lenses. The complex conjugate of the radiation fields is focused back based on the Richards-Wolf vectorial diffraction theory.

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The dipole antennae are positioned on a grid within a circular region of a radius of 1.5λ in the x-z plane in the focal region. The position of each spot P is expressed as $({{r_0}sin{\varphi_n},0,{r_0}cos{\varphi_n}} )$ where ${\varphi _n} \in [{0,2\pi } )$ is the angle with respect to the z axis as shown in the inset of Fig. 1 and ${r_0} \in [{0,1.5\lambda } ]$ is the radial coordinate in the circular area. The radiation field emitted from each dipole antenna depends on its amplitude, phase, oscillating direction and position. Based on the antenna theory [38], an x-dipole generates both elevation and azimuthal components and a z-dipole generates only an elevation component. The radiation fields emitted from dipoles off the origin are differed by a position-dependent phase term $exp({jk{r_0}\Delta } )$, where Δ is the path length difference between OA and PA given by $\Delta = {r_0}({cos{\varphi_n}cos\theta + sin{\varphi_n}sin\theta cos{\varphi_n}} )$. The combined radiation fields in the curved surface Ω emitted from all dipole antenna sets are thus given by:

$$\begin{array}{l} {{\vec{E}}_\Omega }(\theta ,\phi ) = \sum\limits_{n = 1}^N {[{{\vec{E}}_{xn}} + {{\vec{E}}_{zn}}]} \\ = \sum\limits_{n = 1}^N {j\eta \frac{{k{I_0}\exp ( - jkr)}}{{4\pi r}}} [{A_{xn}}(\cos \theta \cos \phi {{\vec{e}}_\theta } - \sin \phi {{\vec{e}}_\phi }) - {A_{zn}}\sin \theta {{\vec{e}}_\theta }]\exp [jk{r_0}\Delta ] \end{array}$$
where η is the impedance, k is the wave number, ${I_0}$ is the constant electric current, r is radius of curvature of the curved surface Ω, ${\vec{e}_\theta }$ and ${\vec{e}_\phi }$ are unit vectors along the elevation and azimuthal directions in the curved surface. ${\vec{E}_{xn}}$ and ${\vec{E}_{zn}}$ represent the radiation fields of the nth dipole oscillating along the x and z directions, $({\theta ,\phi } )$ are spherical coordinates, and N is the number of sets of dipole antennae. ${A_{xn}}$ and ${A_{zn}}$ represent the relative amplitude and phase of the x-, z-dipole antennae of the nth set. The combined radiation fields in the curved surface in the negative-z sphere can be obtained in a similar way.

The two objective lenses obey the sine condition $r = fsin\theta $, where f is the focal length. The projection function is given by ${\vec{E}_i}({r,\phi } )= {\vec{E}_\Omega }({\theta ,\phi } )/\sqrt {cos\theta } $, where $\theta = si{n^{ - 1}}({r/f} )$ [29]. Finally, the complex conjugate of the radiation fields received from all sets of dipole antennae with two high-NA objectives are focused back through the two objectives. According to Richards-Wolf vectorial diffraction method, the electric fields in the focal region are given by [28,29]:

$$\vec{E}({r_p},\Psi ,{z_p}) = \frac{{ik}}{{2\pi }}\int_0^{{\theta _{\max }}} {\int_0^{2\pi } {{{\vec{E}}_\Omega }(\theta ,\phi )\exp ( - jk{r_p}\sin \theta \cos (\phi - \Psi ) - jk{z_p}\cos \theta )\sin \theta d\theta d\phi } }$$
where ${\vec{E}_\Omega }({\theta ,\phi } )$ are the electric fields in the curved surface, $({{r_p},\Psi ,{z_p}} )$ represents the cylindrical coordinates in the focal region, and ${\theta _{max}} = si{n^{ - 1}}({NA} )$. Based on the 4Pi focusing system [36], the electric fields near the focus can be expressed as:
$${\vec{E}_f}({r_p},\Psi ,{z_p}) = \vec{E}({r_p},\Psi ,{z_p}) + \vec{E}^{\prime}({r_p},\Psi , - {z_p}), $$
where $\vec{E}$ and $\vec{E}^{\prime}$ denote the focal fields of objective lens 1 and lens 2, respectively.

3. Numerical simulations

The relative amplitude and phase of dipole antennae in the x-z plane are sampled from the optical field data of radially and azimuthally polarized optical fields in the x-y plane. These cylindrically polarized optical fields in the x-y plane are expressed as the superposition of orthogonally polarized Hermite-Gauss HG01 and HG10 modes [1]:

$${\vec{E}_{radially}} = H{G_{10}}{\vec{e}_x} + H{G_{01}}{\vec{e}_y}$$
$${\vec{E}_{azimuthally}} ={-} H{G_{01}}{\vec{e}_x} + H{G_{10}}{\vec{e}_y}$$
The left-handed circularly polarized optical field with vortex phase in the x-y plane is expressed as the superposition of the radially and azimuthally polarized optical fields:
$${\vec{E}_{circularly}} = ({\vec{E}_{radially}} + {\vec{E}_{azimuthally}}\exp (i\frac{\pi }{2}))\exp (il\phi )$$
$exp({il\phi } )$ represents the helical phase factor, l is an integer referred to as a topological charge, and ϕ is the azimuthal angle. Figure 2 shows the intensity and polarization distribution of HG10, HG01, radially polarized optical field, azimuthally polarized optical field and the superposition of radially and azimuthally polarized optical fields.

 figure: Fig. 2.

Fig. 2. Intensity of HG modes, cylindrically polarized optical fields and circularly polarized optical field. The figure shows the intensity profiles of HG10, HG01, radially polarized optical field, azimuthally polarized optical field and the superposition of radially and azimuthally polarized optical fields, respectively. The white lines in the intensity profiles represent the polarization vectors.

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In the simulation, the number of dipole antenna sets is 117, the spacing between neighboring antenna sets is 0.3λ, and the NA is set to 1. The first example is the demonstration of a radially polarized optical field in the x-z plane. In a circular area of a radius of 1.5λ in the x-z plane, the dipole antenna sets are placed uniformly with each antenna set containing an x-dipole and a z-dipole.

The intensity and polarization distribution of the front and rear pupil planes are shown in Fig. 3(a) and Fig. 4(a). The intensity is mainly concentrated at the edge of the aperture and there are three polarization singularities of zero intensity on x = 0, which are located at the upper, central and lower positions respectively. The intensity and phase of their x and y components are shown in Fig. 3(b) to 3(e) and Fig. 4(b) to 4(e), respectively. The main intensity distribution of x component is concentrated in the central area, the intensity on the left and right sides is low, and the light intensity at x = 0 is zero. The intensity of y component is mainly distributed at the edge of the aperture, and the light intensity at y = 0 is zero. Logarithmical scale is applied to display the details of the intensity distribution throughout the paper.

 figure: Fig. 3.

Fig. 3. Intensity, phase and polarization vector distribution in the pupil plane of objective lens 1 for creating a transversely oriented radially polarized optical field. (a) The intensity and polarization distributions in the pupil plane 1; The intensity distribution of the pupil plane is logarithmically processed to show the details of the intensity distribution. (b)-(c) The intensity distributions of ${E_x}$ and ${E_y}$ components in the pupil plane 1, respectively; (d)-(e) The phase distributions of ${E_x}$ and ${E_y}$ components in the pupil plane 1, respectively;

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 figure: Fig. 4.

Fig. 4. Intensity, phase and polarization vector distribution in the pupil plane of objective lens 2 for creating a transversely oriented radially polarized optical field. (a) The intensity and polarization distributions in the pupil plane 2; The intensity distribution of the pupil plane is logarithmically processed to show the details of the intensity distribution. (b)-(c) The intensity distributions of ${E_x}$ and ${E_y}$ components in the pupil plane 2, respectively; (d)-(e) The phase distributions of ${E_x}$ and ${E_y}$ components in the pupil plane 2, respectively.

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The focal field distributions in the x-z plane can be calculated by Eq. (2) and Eq. (3). Figure 5(a) shows the intensity and polarization vector distributions in the x-z plane. It can be seen that the central polarization singularity shows a certain polarization state. Theoretically, the states of polarization at the singularity points should be undetermined. Due to the inherent quantization error of numerical calculation, there will be a certain polarization state at the singularity points. However, since the intensities at these points are zero, the states of polarization are trivial. Figure 5(b) and 5(c) show the intensity of the x and z components of the focal fields, respectively. The corresponding phase distributions are shown in Fig. 5(e) and 5(f). Figure 5(d) shows the intensity distribution in the y-z plane. It is clearly shown that an annular intensity distribution and a radial polarization distribution are successfully reconstructed in the x-z plane. Also, the intensity distribution of the x-z plane and the y-z plane can clearly see that the optical field presents a torus-shaped three-dimensional distribution structure in the focal field.

 figure: Fig. 5.

Fig. 5. Intensity, phase and polarization vector distribution of a transversely oriented radially polarized optical field in the focal field. (a) The intensity and polarization distributions in the x-z plane; (b)-(c) The intensity distributions of ${E_x}$ and ${E_z}$ components in the x-z plane, respectively; (d) The intensity distribution in the y-z plane. (e)-(f) The phase distributions of ${E_x}$ and ${E_z}$ components in the x-z plane, respectively.

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An azimuthally polarized optical field in the x-z plane is created in a similar way as the second example. Through adjusting the relative amplitude and phase of the x-dipole and z-dipole of each antenna set, a transversely oriented azimuthally polarized optical field can be built in the focal region. The intensity and polarization distributions of the pupil plane 1 and pupil plane 2 are shown in Fig. 6(a) and Fig. 7(a). The intensity and phase distributions of x and y components in the pupil plane 1 and the pupil plane 2 are shown in Fig. 6(b) to 6(e) and Fig. 7(b) to 7(e), respectively. Different from the transversely oriented radially polarized optical field, in the pupil plane intensity distribution, there are only two polarization singularities and zero intensity points at the upper and lower positions at x = 0, and the intensity is not zero at the central origin. This difference is due to the fact that the phase of the x component of the pupil plane of the azimuthally polarized optical field does not change significantly in the center, while the phase of the x component and the y component of the pupil plane of the radially polarized optical field have sharp changes in the center.

 figure: Fig. 6.

Fig. 6. Intensity, phase and polarization vector distribution in the pupil plane of objective lens 1 for creating a transversely oriented azimuthally polarized optical field. (a) The intensity and polarization distributions in the pupil plane 1; (b)-(c) The intensity distributions of ${E_x}$ and ${E_y}$ components in the pupil plane 1, respectively; (d)-(e) The phase distributions of ${E_x}$ and ${E_y}$ components in the pupil plane 1, respectively.

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 figure: Fig. 7.

Fig. 7. Intensity, phase and polarization vector distribution in the pupil plane of objective lens 2 for creating a transversely oriented azimuthally polarized optical field. (a) The intensity and polarization distributions in the pupil plane 2; (b)-(c) The intensity distributions of ${E_x}$ and ${E_y}$ components in the pupil plane 2, respectively; (d)-(e) The phase distributions of ${E_x}$ and ${E_y}$ components in the pupil plane 2, respectively.

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Figure 8(a) shows the intensity and polarization distributions of the focal field. The quantization error of numerical calculation removes the ambiguity of the state of polarization in the center and displays a certain polarization state. It is shown that an annular light field distribution and an azimuthal polarization distribution are realized in the focal region. Figure 8(b) and 8(c) show the intensity of the x and z components of the focal fields, respectively. Figure 8(e) and 8(f) show the corresponding phase distributions. Figure 8(d) shows the intensity distribution in the y-z plane. The intensity distribution of the x-z plane and the y-z plane can clearly see that the optical field presents a torus-shaped three-dimensional distribution structure in the focal field same as the transversely oriented radially polarized optical field.

 figure: Fig. 8.

Fig. 8. Intensity, phase and polarization vector distribution of a transversely oriented azimuthally polarized optical field in the x-z plane. (a) The intensity and polarization distributions; (b)-(c) The intensity distributions of ${E_x}$ and ${E_z}$ components in the x-z plane, respectively; (d) The intensity distribution in the y-z plane; (e)-(f) The phase distributions of ${E_x}$ and ${E_z}$ components in the x-z plane, respectively.

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The last example is the creation of transversely oriented circularly polarized optical field carrying both transverse SAM and transverse OAM through the superposition of transversely oriented cylindrically polarized optical fields. Figures 9 and 10 show the optical field information in the pupil plane 1 and the pupil plane 2, respectively. It shows that there are polarization vectors with opposite rotation directions in the upper and lower parts of the pupil plane, and there are phase singularities and corresponding zero intensity points in the upper and lower positions. Figure 11 shows the field distribution in the focal region.

 figure: Fig. 9.

Fig. 9. Intensity, phase and polarization vector distribution in the pupil plane of objective lens 1 for creating a transversely oriented circularly polarized optical field with vortex phase. (a) The intensity and polarization distributions in the pupil plane 1; (b)-(c) The intensity distributions of ${E_x}$ and .. components in the pupil plane 1, respectively; (d)-(e) The phase distributions of ${E_x}$ and ${E_y}$ components in the pupil plane 1, respectively; The white lines in the intensity profiles represent the right-handed circular polarization and the red lines represent the left-handed circular polarization;

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 figure: Fig. 10.

Fig. 10. Intensity, phase and polarization vector distribution in the pupil plane of objective lens 2 for creating a transversely oriented circularly polarized optical field with vortex phase. (a) The intensity and polarization distributions in the pupil plane 2; (b)-(c) The intensity distributions of ${E_x}$ and ${E_y}$ components in the pupil plane 2, respectively; (d)-(e) The phase distributions of ${E_x}$ and ${E_y}$ components in the pupil plane 2, respectively; The white lines in the intensity profiles represent the right-handed circular polarization and the red lines represent the left-handed circular polarization;

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 figure: Fig. 11.

Fig. 11. Intensity, phase and polarization vector distribution in the x-z plane of a transversely oriented circularly polarized optical field with vortex phase. (a) The intensity and polarization distributions; (b) The phase distributions.

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As shown in Fig. 11(a), the circularly polarized focal field has an annular intensity distribution in the x-z plane. Figure 11(b) shows the associating spiral phase distribution. The transversely oriented circular polarization and transversely oriented spiral phase indicate that the optical field carries both transverse SAM and transverse OAM. The spacing interval of dipole antennae has an impact on the intensity and polarization distribution. We draw the intensity distribution for two different intervals of 0.3λ and 0.6λ, respectively. The ratio of the semi-minor axis and semi-major axis of the polarization ellipse of points in the direction of 0°, 90°, 180° and 270° are shown in Fig. 12.

 figure: Fig. 12.

Fig. 12. Intensity and polarization vector distributions at different dipole antenna intervals. (a) The intensity distributions at an interval of 0.3λ; (b) The intensity distributions at an interval of 0.6λ; (c) Calculation results of the ratio of the semi-minor axis and semi-major axis of polarization state at 0°, 90°, 180° and 270° position. The white lines in the intensity profiles represent the polarization vectors. The ratio is calculated by $e = b/a$, a is the semi-major axis of the ellipse and b is the semi-minor axis of the ellipse. When e = 1, it is circular polarization; when e = 0, it is linear polarization.

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Figure 12(a) and 12(b) show the intensity distributions for spacing interval of 0.3λ and 0.6λ, respectively. Figure 12(c) shows the ratio of the semi-minor axis and semi-major axis of the polarization ellipse at different positions. It can be found that the ratio and intensity are related to the dipole spacing. The intensity becomes more uniform and the ratio becomes bigger as the interval becomes smaller. The current spacing interval is close to Rayleigh criterion. Continuous decrease in the spacing interval does not improve the uniformity and the ratio significantly but increase the calculation time significantly. Therefore, under comprehensive consideration, we choose the placement interval of 0.3λ in this paper. The spin density $\vec{s}$ and OAM density $\vec{r} \times \vec{p}$ in the x-z plane are calculated based on Eq. (7) and Eq. (8), respectively [39–42]:

$$\vec{s} \propto {\mathop{\rm Im}\nolimits} ({\overrightarrow E ^*} \times \overrightarrow E )$$
$$\vec{r} \times \vec{p} = \vec{r} \times {\mathop{\rm Im}\nolimits} ({\overrightarrow E ^\ast } \cdot (\nabla )\overrightarrow E )$$
where $\textrm{Im}$ represents the imaginary part. The OAM and SAM values of the pupil plane and the x-z plane are shown in Fig. 13. It can be seen from the results that conservation relationship is satisfied.

 figure: Fig. 13.

Fig. 13. The OAM and SAM values of the pupil plane and the x-z plane. The upper row displays the OAM and SAM values of the pupil plane 1 and pupil plane 2. The lower row displays the OAM and SAM values of x-z plane when topological charge number is 1. The unit is ħ where ħ is the Dirac constant. AM: angular momentum. $OA{M_x}$ and $OA{M_z}$ represent OAM values for different polarization components.

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 figure: Fig. 14.

Fig. 14. Intensity, polarization vector, phase, transverse spin density and OAM density in the x-z plane. (a) The intensity and polarization distributions of a left-handed circularly polarized optical field with a topological charge number of 1; (b) The corresponding phase distributions; (c) The spin density distribution; (d) The OAM density distribution; (e) The intensity and polarization distributions of a right-handed circularly polarized optical field with a topological charge number of 2; (f) The corresponding phase distributions; (g) The spin density distribution; (h) The OAM density distribution.

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The upper row of Fig. 14 displays the intensity and polarization distribution, phase distribution, spin density, and OAM density of a transversely-oriented left-handed circularly polarized optical field with a spiral phase of topological charge 1. To illustrate the circling momentum density around the origin, the dominant linear momentum density in the propagation direction is removed in Fig. 14(d) and 14(h). Since the relative amplitude and phase of all the dipole antennae can be accurately controlled, it is straightforward to create optical fields carrying transverse SAM of either sign and transverse OAM of arbitrary integer of topological charge. The lower row of Fig. 14 plots the optical field information for a transversely-oriented right-handed circularly optical field with a spiral phase of topological charge 2. As the topological charge is increased, a larger dark central area appears as expected.

4. Conclusions

In summary, we propose a scheme based on the time-reversal method, the vectorial diffraction theory, and the 4Pi optical configuration to create transversely oriented cylindrically polarized optical fields. Both transversely oriented radially polarized and azimuthally polarized optical fields are demonstrated. The superposition of transverse cylindrically polarized optical fields results in a peculiar optical field carrying controllable transverse SAM and transverse OAM. The creation of such peculiar optical focal fields may find applications in unidirectional beam propagation, optical manipulation, light-matter interaction, and polarization topology.

Funding

National Natural Science Foundation of China (61875245, 92050202); Wuhan Municipal Science and Technology Bureau (2020010601012169).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (14)
Fig. 1.
Fig. 1. Schematic of the proposed method. Dipole antenna sets are uniformly positioned within a circular area of 1.5λ radius in the x-z plane. The radiation fields emitted from dipole antenna sets are collected by two high-NA objective lenses. The complex conjugate of the radiation fields is focused back based on the Richards-Wolf vectorial diffraction theory.

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Fig. 2.
Fig. 2. Intensity of HG modes, cylindrically polarized optical fields and circularly polarized optical field. The figure shows the intensity profiles of HG10, HG01, radially polarized optical field, azimuthally polarized optical field and the superposition of radially and azimuthally polarized optical fields, respectively. The white lines in the intensity profiles represent the polarization vectors.

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Fig. 3.
Fig. 3. Intensity, phase and polarization vector distribution in the pupil plane of objective lens 1 for creating a transversely oriented radially polarized optical field. (a) The intensity and polarization distributions in the pupil plane 1; The intensity distribution of the pupil plane is logarithmically processed to show the details of the intensity distribution. (b)-(c) The intensity distributions of ${E_x}$ and ${E_y}$ components in the pupil plane 1, respectively; (d)-(e) The phase distributions of ${E_x}$ and ${E_y}$ components in the pupil plane 1, respectively;

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Fig. 4.
Fig. 4. Intensity, phase and polarization vector distribution in the pupil plane of objective lens 2 for creating a transversely oriented radially polarized optical field. (a) The intensity and polarization distributions in the pupil plane 2; The intensity distribution of the pupil plane is logarithmically processed to show the details of the intensity distribution. (b)-(c) The intensity distributions of ${E_x}$ and ${E_y}$ components in the pupil plane 2, respectively; (d)-(e) The phase distributions of ${E_x}$ and ${E_y}$ components in the pupil plane 2, respectively.

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Fig. 5.
Fig. 5. Intensity, phase and polarization vector distribution of a transversely oriented radially polarized optical field in the focal field. (a) The intensity and polarization distributions in the x-z plane; (b)-(c) The intensity distributions of ${E_x}$ and ${E_z}$ components in the x-z plane, respectively; (d) The intensity distribution in the y-z plane. (e)-(f) The phase distributions of ${E_x}$ and ${E_z}$ components in the x-z plane, respectively.

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Fig. 6.
Fig. 6. Intensity, phase and polarization vector distribution in the pupil plane of objective lens 1 for creating a transversely oriented azimuthally polarized optical field. (a) The intensity and polarization distributions in the pupil plane 1; (b)-(c) The intensity distributions of ${E_x}$ and ${E_y}$ components in the pupil plane 1, respectively; (d)-(e) The phase distributions of ${E_x}$ and ${E_y}$ components in the pupil plane 1, respectively.

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Fig. 7.
Fig. 7. Intensity, phase and polarization vector distribution in the pupil plane of objective lens 2 for creating a transversely oriented azimuthally polarized optical field. (a) The intensity and polarization distributions in the pupil plane 2; (b)-(c) The intensity distributions of ${E_x}$ and ${E_y}$ components in the pupil plane 2, respectively; (d)-(e) The phase distributions of ${E_x}$ and ${E_y}$ components in the pupil plane 2, respectively.

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Fig. 8.
Fig. 8. Intensity, phase and polarization vector distribution of a transversely oriented azimuthally polarized optical field in the x-z plane. (a) The intensity and polarization distributions; (b)-(c) The intensity distributions of ${E_x}$ and ${E_z}$ components in the x-z plane, respectively; (d) The intensity distribution in the y-z plane; (e)-(f) The phase distributions of ${E_x}$ and ${E_z}$ components in the x-z plane, respectively.

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Fig. 9.
Fig. 9. Intensity, phase and polarization vector distribution in the pupil plane of objective lens 1 for creating a transversely oriented circularly polarized optical field with vortex phase. (a) The intensity and polarization distributions in the pupil plane 1; (b)-(c) The intensity distributions of ${E_x}$ and .. components in the pupil plane 1, respectively; (d)-(e) The phase distributions of ${E_x}$ and ${E_y}$ components in the pupil plane 1, respectively; The white lines in the intensity profiles represent the right-handed circular polarization and the red lines represent the left-handed circular polarization;

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Fig. 10.
Fig. 10. Intensity, phase and polarization vector distribution in the pupil plane of objective lens 2 for creating a transversely oriented circularly polarized optical field with vortex phase. (a) The intensity and polarization distributions in the pupil plane 2; (b)-(c) The intensity distributions of ${E_x}$ and ${E_y}$ components in the pupil plane 2, respectively; (d)-(e) The phase distributions of ${E_x}$ and ${E_y}$ components in the pupil plane 2, respectively; The white lines in the intensity profiles represent the right-handed circular polarization and the red lines represent the left-handed circular polarization;

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Fig. 11.
Fig. 11. Intensity, phase and polarization vector distribution in the x-z plane of a transversely oriented circularly polarized optical field with vortex phase. (a) The intensity and polarization distributions; (b) The phase distributions.

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Fig. 12.
Fig. 12. Intensity and polarization vector distributions at different dipole antenna intervals. (a) The intensity distributions at an interval of 0.3λ; (b) The intensity distributions at an interval of 0.6λ; (c) Calculation results of the ratio of the semi-minor axis and semi-major axis of polarization state at 0°, 90°, 180° and 270° position. The white lines in the intensity profiles represent the polarization vectors. The ratio is calculated by $e = b/a$, a is the semi-major axis of the ellipse and b is the semi-minor axis of the ellipse. When e = 1, it is circular polarization; when e = 0, it is linear polarization.

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Fig. 13.
Fig. 13. The OAM and SAM values of the pupil plane and the x-z plane. The upper row displays the OAM and SAM values of the pupil plane 1 and pupil plane 2. The lower row displays the OAM and SAM values of x-z plane when topological charge number is 1. The unit is ħ where ħ is the Dirac constant. AM: angular momentum. $OA{M_x}$ and $OA{M_z}$ represent OAM values for different polarization components.

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Fig. 14.
Fig. 14. Intensity, polarization vector, phase, transverse spin density and OAM density in the x-z plane. (a) The intensity and polarization distributions of a left-handed circularly polarized optical field with a topological charge number of 1; (b) The corresponding phase distributions; (c) The spin density distribution; (d) The OAM density distribution; (e) The intensity and polarization distributions of a right-handed circularly polarized optical field with a topological charge number of 2; (f) The corresponding phase distributions; (g) The spin density distribution; (h) The OAM density distribution.

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Equations (8)

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E → Ω ( θ , ϕ ) = ∑ n = 1 N [ E → x n + E → z n ] = ∑ n = 1 N j η k I 0 exp ⁡ ( − j k r ) 4 π r [ A x n ( cos ⁡ θ cos ⁡ ϕ e → θ − sin ⁡ ϕ e → ϕ ) − A z n sin ⁡ θ e → θ ] exp ⁡ [ j k r 0 Δ ]
E → ( r p , Ψ , z p ) = i k 2 π ∫ 0 θ max ∫ 0 2 π E → Ω ( θ , ϕ ) exp ⁡ ( − j k r p sin ⁡ θ cos ⁡ ( ϕ − Ψ ) − j k z p cos ⁡ θ ) sin ⁡ θ d θ d ϕ
E → f ( r p , Ψ , z p ) = E → ( r p , Ψ , z p ) + E → ′ ( r p , Ψ , − z p ) ,
E → r a d i a l l y = H G 10 e → x + H G 01 e → y
E → a z i m u t h a l l y = − H G 01 e → x + H G 10 e → y
E → c i r c u l a r l y = ( E → r a d i a l l y + E → a z i m u t h a l l y exp ⁡ ( i π 2 ) ) exp ⁡ ( i l ϕ )
s → ∝ Im ( E → ∗ × E → )
r → × p → = r → × Im ( E → ∗ ⋅ ( ∇ ) E → )
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