Abstract

The controlled and continuous negative energy flow (from negative to positive) on the optical axis in the focal region is obtained by adjusting the polarization distribution of the input second-order radially polarized beam (the polarization topological charge is equal to 2). Moreover, the similar evolution of negative energy flow also can be achieved for the tightly focused vector beams with polarization topological charge −2. It is because both the beams with polarization topological charges 2 and −2 can possess the same polarization and spin flow density distributions with the help of the polarization modulation. The results provide a potential method for modulating the effects induced by the spin-orbit coupling in tight focusing of optical beam.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

In recent years, considerable research interests have been attracted in the study of negative energy flow of light beams because of its potential applications in the field of optical micro-manipulation. As the name implies, the negative energy flow means the direction of energy flow is opposite to the propagation direction of the beam, it can exist in some specific cases, such as the tightly focused vortex beams [1,2] and vector beams [36], the propagation of vector Bessel beam [7,8], non-paraxial Airy beam [9], and quantum optical system [10] etc..

In previous works [13], researchers showed that, the negative energy flow can be observed on the optical axis in the focal plane of the tightly focused cylindrical vector beams with polarization charge 2 or vortex beams with topological charge 2; otherwise, the energy flow on the optical axis in the focal plane is zero. Namely, the on-axis negative energy flow can be modulated by changing the polarization charge or vortex charge. One knows the controlled and continuous change of negative energy flow can provide a more flexible manipulation in real optical application. Yet, it is hard to obtain a continuous change (from negative to positive gradually) of on-axis negative energy flow in the focal region by changing the polarization charge or vortex charge. It is necessary to present a method to control the on-axis energy flow in the focal region.

The polarization is an important parameter of the optical field [11,12]. The different polarization states correspond to different kinds of optical angular momentum, such as the circular polarization is related to spin angular momentum [1315], the hybrid polarization is related to a kind of intrinsic orbital angular momentum [16]. Moreover, the polarization determines the spin angular momentum and spin flow density distributions of vector beam [1719]. And it has known the polarization also plays a crucial role in the field of optical micro-particle manipulation [2024], super-resolution imaging [2527], and optical topological structure [28,29]. This means that, in the tight focusing of vector beam, some effects induced by the spin-orbit coupling may be controlled by adjusting the polarization of input vector beam because the change of polarization can induce the change of intrinsic structure of optical angular momentum.

Here, we introduce a coefficient matrix to modulate the polarization distribution and intrinsic optical angular momentum structure of input vector beam, and obtain the controlled and continuous evolution of negative energy flow on the optical axis in the focal region of the tightly focused radially polarized beams. We find that, by adjusting the polarization distribution of input vector beam, the controllable negative energy flow on the optical axis in the focal region of the tightly focused vector beam can be obtained when the polarization topological charge is equal to 2, which can change from the negative to positive continuously. Moreover, if the input vector beams possess the same intrinsic structure (polarization and spin flow density) as the beam with polarization topological charge 2, such as the vector beam with polarization topological charge −2, there also is a continuous on-axis negative energy flow in the focal region. It means our results can provide a potential way to modulate the effects induced by the spin-orbit coupling in the tight focusing of the optical beam.

2. Theoretical analysis

Let us consider the tight focusing of $m$-order radially polarized beams, $m$ is the polarization topological charge of input vector beam. The Richards-Wolf diffraction integral has the form as following [30],

$$\begin{aligned}\left[ {\begin{array}{c} {\textbf{E}(r,\varphi ,z)}\\ {\textbf{H}(r,\varphi ,z)} \end{array}} \right] &={-} \frac{{if}}{\lambda }\int_0^\alpha {} \int_0^{2\pi } {T(\theta )} A(\theta ,\phi )\left[ {\begin{array}{c} {{\textbf{P}_E}(\theta ,\phi )}\\ {{\textbf{P}_H}(\theta ,\phi )} \end{array}} \right]\\ & \quad \times Exp\{{ik[{r\sin \theta \cos ({\phi - \varphi } )+ z\cos \theta } ]} \}\sin \theta d\theta d\phi ,\end{aligned}$$
where
$${\textbf{P}_E}(\theta ,\phi ) = \left[ {\begin{array}{cc} {A(\theta ,\phi )}&{C(\theta ,\phi )}\\ {C(\theta ,\phi )}&{B(\theta ,\phi )}\\ { - D(\theta ,\phi )}&{ - E(\theta ,\phi )} \end{array}} \right]\left[ {\begin{array}{c} {{{\tilde{c}}_x}(\phi )}\\ {{{\tilde{c}}_y}(\phi )} \end{array}} \right],\,\,\,{\textbf{P}_H}(\theta ,\phi ) = \left[ {\begin{array}{cc} {C(\theta ,\phi )}&{ - A(\theta ,\phi )}\\ {B(\theta ,\phi )}&{ - C(\theta ,\phi )}\\ { - E(\theta ,\phi )}&{D(\theta ,\phi )} \end{array}} \right]\left[ {\begin{array}{c} {{{\tilde{c}}_x}(\phi )}\\ {{{\tilde{c}}_y}(\phi )} \end{array}} \right],$$
and
$$\begin{aligned}A(\theta ,\phi ) &= 1 + {{\cos }^2}\phi (\cos \theta - 1),\,\,\,B(\theta ,\phi ) = 1 + {{\sin }^2}\phi (\cos \theta - 1),\\ C(\theta ,\phi ) &= \sin \phi \cos \phi (\cos \theta - 1),\,\,\,D(\theta ,\phi ) = \cos \phi \sin \theta ,\,\,\,E(\theta ,\phi ) = \sin \phi \sin \theta ,\end{aligned}$$
where $T(\theta ) = \sqrt {\cos \theta } $ is the apodization function, $A(\theta ,\phi )$ is the amplitude of incident field. We consider the amplitude of incident field $A(\theta ,\phi )$ is concentrated in a narrow annular region with the Gaussian envelope whose central angle ${\theta _0}$ and width $\Delta \theta $, where $\Delta \theta $ is a small parameter. For the sake of simplicity, we take $A({\theta _0},\phi ) = 1$ in the calculation and simulation of the energy flow in next.

Generally, the polarization distribution of the radially polarized beam is described by the vectorial coefficient matrix ${\left[ {\begin{array}{cc} {{{\tilde{c}}_x}(\phi )}&{{{\tilde{c}}_y}(\phi )} \end{array}} \right]^T}$ in Eq. (1), and ${\tilde{c}_x}(\phi ) = \cos (m\phi )$, ${\tilde{c}_y}(\phi ) = \sin (m\phi )$. According to the Richards-Wolf diffraction integral, the components of tightly focused vector beam can be derived directly. According to the definition of time averaged energy flow density $\textbf{S} = (c/8\pi ){\mathop{\rm Re}\nolimits} ({\textbf{E}^\ast } \times \textbf{H})$, the longitudinal component of the energy flow density in the focal plane can be expressed as ${S_z} = {\mathop{\rm Re}\nolimits} ({E_x^\ast {H_y} - E_y^\ast {H_x}} )$. Figure 1(a) shows the energy flow density properties in the focal plane ($z = 0$) for different polarization topological charges. The calculation parameters are taken as, ${\theta _0} = {80^ \circ }$, $f = 120\lambda $, $\lambda $ is the wave length of incident light. We know that the energy flow density on the optical axis in the focal plane is negative when the polarization topological charge is equal to 2, and it is zero when the polarization topological charge is unequal to 2, which is consistent with the results obtained in previous works [13]. Figure 1(b) shows the evolution of energy flow density in the center (on-axis) of the focal plane as the change of polarization topological charge. Obviously, the consecutive evolution of energy flow on the optical axis can’t be obtained by changing the polarization topological charge.

 figure: Fig. 1.

Fig. 1. Energy flow density (${S_z}$) in the focal plane when ${\phi _{01}} = {\phi _{02}} = 0$, (a) ${S_z}$ along the radial direction for vector beams with the polarization topological charge m, (b) evolution of on-axis energy flow ${S_{z0}}$ as the change of the polarization topological charge m.

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It has known the polarization distribution determines the intrinsic spin angular momentum and spin flow density of vector beams [18], and it affects some effects induced by the spin-orbit coupling in the tightly focused optical beams [1314,28]. Experimentally, the spatial polarization distribution of vector beams can be well modulated by means of the spatial light modulator and computer-generated hologram [16,3132]. Here, in order to obtain the controlled and continuous change of the negative energy flow on the optical axis in the focal region, a coefficient matrix $\textrm{R(}{\phi _0}\textrm{)}$ is introduced to adjust the polarization distribution and intrinsic structure of optical angular momentum of input vector beam, and the effective polarization matrix ${\left[ {\begin{array}{cc} {{{\tilde{c}}_x}(\phi )}&{{{\tilde{c}}_y}(\phi )} \end{array}} \right]^T}$ in Eq. (1) can be expressed as,

$$\left[ {\begin{array}{c} {{{\tilde{c}}_x}(\phi )}\\ {{{\tilde{c}}_y}(\phi )} \end{array}} \right]\textrm{ = R(}{\phi _0}\textrm{)}\left[ {\begin{array}{c} {\cos (m\phi )}\\ {\sin (m\phi )} \end{array}} \right],$$
where
$$\textrm{R(}{\phi _0}\textrm{) = }\left[ {\begin{array}{cc} {\cos {\phi_{01}}}&{ - \sin {\phi_{01}}}\\ {\sin {\phi_{02}}}&{\cos {\phi_{02}}} \end{array}} \right].$$
$R({\phi _0})$ is the coefficient matrix for adjusting the polarization distribution ${\left[ {\begin{array}{cc} {\cos (m\phi )}&{\sin (m\phi )} \end{array}} \right]^T}$, and it describes the rotation of polarization vector, the first and second row of $R({\phi _0})$ indicate the rotation of two polarization unit vector (${\textbf{e}_x}$ and ${\textbf{e}_y}$) with the angle ${\phi _{01}}$ and ${\phi _{02}}$, respectively. Just as shown in Fig. 2, the intensity, the polarization distribution and intrinsic spin flow density of vector beams can be well modulated by the parameters ${\phi _{01}}$ and ${\phi _{02}}$. In the numerical simulation of Fig. 2, the normalized spin flow density of vector beams is calculated using the expression ${\textbf{P}_s} = {\mathop{\rm Im}\nolimits} [{\nabla \times ({{\textbf{E}^ \ast } \times \textbf{E} + {\textbf{H}^ \ast } \times \textbf{H}} )} ]$, and we consider the input beam has a narrow annular with the amplitude distribution, $\exp [{ - {{(\rho - {\rho_0})}^2}/{w^2}} ]$, where ${\rho _0}$ is the radius of the narrow annular region and ${\rho _0} = 7\lambda $, w is the width of narrow annular region and $w = 0.5\lambda $. For the small w, $\exp [{ - {{(\rho - {\rho_0})}^2}/{w^2}} ]$ can be reduced to $\delta (\rho - {\rho _0})$ [33], and it satisfies the assumption we proposed about the input vector beams approximatively. It shows that, though the polarization topological charge is unchanged, the polarization distribution and the spin flow density are well modulated by the parameters ${\phi _{01}}$ and ${\phi _{02}}$. It implies the consecutive change of on-axis negative energy flow may be obtained by means of the polarization modulation of input vector beam.

 figure: Fig. 2.

Fig. 2. Polarization and normalized spin flow density distributions of input vector beams with polarization topological charge 2, (a) and (e) ${\phi _{01}} = 0$, ${\phi _{02}} = 0$, (b) and (f) ${\phi _{01}} = 0$, ${\phi _{02}} = \pi /2$, (c) and (g) ${\phi _{01}} = \pi /2$, ${\phi _{02}} = 0$, (d) and (h) ${\phi _{01}} = \pi /2$, ${\phi _{02}} = \pi /2$.

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3. Controlled negative energy flow in the focal field

Based on the Richards-Wolf diffraction integral, the field components in the focal region can be derived directly. In previous works, researchers showed the negative energy flow can be obtained in the tight focusing of vector beam with polarization topological charge 2 [13]. Figure 3 shows the evolution of normalized energy flow density in the focal plane ($z = 0$) when the input vector beam with polarization topological charge 2. Obviously, the energy flow in the focal plane is well modulated by changing of the parameter ${\phi _{01}}$. Interestingly, the energy flow density on the optical axis can change from negative value to positive value with the variation of parameter ${\phi _{01}}$.

 figure: Fig. 3.

Fig. 3. Normalized energy flow density (${S_z}$) in the focal plane when input vector beam with polarization topological charge 2, ${\phi _{02}} = 0$, (a) ${\phi _{01}}\textrm{ = }0$, (b) ${\phi _{01}}\textrm{ = }\pi \textrm{/4}$, (c) ${\phi _{01}}\textrm{ = 3}\pi \textrm{/4}$, (d) ${\phi _{01}}\textrm{ = }\pi $, (e) ${\phi _{01}}\textrm{ = 5}\pi \textrm{/4}$, (f) ${\phi _{01}}\textrm{ = 7}\pi \textrm{/4}$, the calculation parameters are same as those in Fig. 1.

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Figures 4(a) and (c) show the evolution of the energy flow density in the focal plane with the change of parameters ${\phi _{01}}$ and ${\phi _{02}}$ when the input vector beam with polarization topological charge 2, the calculation parameters are same as Fig. 1. We find that, when the parameter ${\phi _{01}}$ or ${\phi _{02}}$ is modulated independently, both the on-axis energy flow density evolution are symmetrical about ${\phi _{01}} = \pi $ (or ${\phi _{02}} = \pi $), such as the properties of energy flow density are same when ${\phi _{01}}\textrm{ = }\pi /4$ and ${\phi _{01}}\textrm{ = }7\pi /4$ (or ${\phi _{02}} = \pi /4$ and ${\phi _{02}}\textrm{ = }7\pi /4$), which is shown in Figs. 4(b) and (d). It is worth noting that, the on-axis energy flow can change from the negative to positive when the parameter ${\phi _{01}}$ is modulated independently, and it is negative value when the parameter ${\phi _{02}}$ is modulated independently. It means the negative energy flow on the optical axis can be well controlled by means of the polarization modulation of input vector beam.

 figure: Fig. 4.

Fig. 4. Evolution of energy flow density (${S_z}$) in the focal plane with the change of ${\phi _{01}}$ and ${\phi _{02}}$ when polarization topological charge $m = 2$, (a) and (c) in the focal plane ($z = 0$), (a) ${\phi _{01}} \ne 0$, ${\phi _{02}}\textrm{ = }0$, (c) ${\phi _{01}}\textrm{ = }0$, ${\phi _{02}} \ne 0$, (b) and (d) on the optical axis (${S_{z0}}$), (b) ${\phi _{01}} \ne 0$, ${\phi _{02}}\textrm{ = }0$, (d) ${\phi _{01}}\textrm{ = }0$, ${\phi _{02}} \ne 0$, the calculation parameters are same as those in Fig. 1.

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In previous works, researchers showed the negative energy flow can be obtained when the input vector beam with polarization topological charge 2 [13]. We have shown that, by adjusting the polarization distribution, the continuous energy flow change (from negative to positive) can be obtained when the input vector beam with polarization topological charge 2. One knows the spin flow density of vector beam is determined by the polarization topological charge [18], the change of polarization distribution will induce the change of spin flow density of the vector beam. Then, there is a question: if the input vector beam possesses the same polarization and spin angular momentum distributions as the vector beam with polarization topological charge 2, but its polarization topological charge is unequal to 2, does the on-axis negative energy flow occurs in the focal region of the vector beam?

By modulating the polarization distribution, the vector beam with polarization topological charge −2 can possesses the same polarization and spin angular momentum distributions as the beam with polarization topological charge 2. It means the negative energy flow should be obtained in the tight focusing of the vector beam with polarization topological charge −2. Figure 5 shows the evolution of the energy flow density in the focal plane ($z = 0$) and on the optical axis with the change of ${\phi _{01}}$ and ${\phi _{02}}$ when the input vector beam with polarization topological charge −2. We find that, by means of the polarization modulation, the continuous on-axis negative energy flow in the focal plane also can be obtained in the case that the polarization topological charge of input vector beam is −2, which is a novel phenomenon and different from the results in Refs. [13].

 figure: Fig. 5.

Fig. 5. Evolution of energy flow density (${S_z}$) in the focal plane with the change of ${\phi _{01}}$ and ${\phi _{02}}$ when polarization topological charge $m ={-} 2$, (a) and (c) in the focal plane ($z = 0$), (a) ${\phi _{01}} \ne 0$, ${\phi _{02}}\textrm{ = }0$, (c) ${\phi _{01}}\textrm{ = }0$, ${\phi _{02}} \ne 0$, (b) and (d) on the optical axis (${S_{z0}}$), (b) ${\phi _{01}} \ne 0$, ${\phi _{02}}\textrm{ = }0$, (d) ${\phi _{01}}\textrm{ = }0$, ${\phi _{02}} \ne 0$, the calculation parameters are same as those in Fig. 1.

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The study of on-axis energy flow is important for the optical manipulation. Figure 6 shows the evolution of on-axis energy flow in the focal plane when the polarization topological charges of input vector beams are equal to 2 and −2. We find, by modulating the parameters ${\phi _{01}}$ and ${\phi _{02}}$, the on-axis energy flow possesses similar evolution properties in both two cases. We also know that, for the input vector beam with polarization topological charge 2, the maximal on-axis negative energy flow can be obtained when ${\phi _{01}}\textrm{ = }0$, ${\phi _{02}}\textrm{ = }0$ or ${\phi _{01}}\textrm{ = } \pm \pi $, ${\phi _{02}}\textrm{ = } \pm \pi $; for the input vector beam with polarization topological charge −2, the maximal on-axis negative energy flow can be obtained when ${\phi _{01}}\textrm{ = 0}$, ${\phi _{02}}\textrm{ = } \pm \pi $ or ${\phi _{01}}\textrm{ = } \pm \pi $, ${\phi _{02}}\textrm{ = 0}$.

 figure: Fig. 6.

Fig. 6. Evolution of on-axis energy flow (${S_{z0}}$) in the focal plane as the change of parameters ${\phi _{01}}$ and ${\phi _{02}}$ when the input vector beam with polarization topological charges 2 (a) and −2 (b), the calculation parameters are same as those in Fig. 1.

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In order to verify the influence of polarization distribution in the modulation of negative energy flow, Fig. 7 shows the polarization distribution and the normalized spin flow density of input vector beams when the maximal on-axis negative energy flow in the focal plane is obtained. We find that, when the maximal on-axis negative energy flow is obtained, the polarization and spin flow density distribution are same for the input vector beams with polarization topological charges 2 and −2. Generally, the negative energy flow is obtained by the change of the focal field components, which can be realized by modulating the polarization distribution of incident beam. Intrinsically, because the polarization determines the spin angular momentum and spin flow density distributions of vector beam, the change of polarization distribution induce the change of the intrinsic structure of optical angular momentum. The occurrence of negative energy flow is a manifestation of spin-orbit coupling in the tight focusing of vector beams. Namely, the effects induced by the spin-orbit coupling in the tight focusing of vector beam can be modulated with the help of the adjustment of the polarization distribution.

 figure: Fig. 7.

Fig. 7. Polarization and normalized spin flow density distributions of input vector beam with polarization topological charge 2 and −2 when the maximal negative energy flow is obtained, (a) and (e) ${\phi _{01}} = 0$, ${\phi _{02}} = 0$, (b) and (f) ${\phi _{01}} = \pi $, ${\phi _{02}} = \pi $, (c) and (g) ${\phi _{01}} = 0$, ${\phi _{02}} = \pi $, (d) and (h) ${\phi _{01}} = \pi $, ${\phi _{02}} = 0$.

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4. Conclusions

In previous works, though the on-axis negative energy flow can be obtained in the tight focusing of vector beam with polarization topological charge 2, it can’t realize continuous change from the negative to positive. In our study, by introducing a coefficient matrix to adjust the polarization distribution of input vector beam, we obtained the controlled on-axis negative energy flow in the focal plane for the beams with polarization topological charges 2 and −2 theoretically, which can change from the negative to positive continuously. We found that, by adjusting the polarization distribution, the vector beams with polarization topological charges 2 and −2 can have the same intrinsic structure including the polarization and spin angular momentum distributions. It further verifies the significance of polarization in the tight focusing of vector beams. The occurrence of the negative energy flow is a manifestation of the spin-orbit coupling in the tight focusing of vector beams. It means that the effects induced by the spin-orbit coupling can be modulated with the help of the polarization modulation. Our results provide a potential way to modulate the effects induced by the spin-orbit coupling in the tight focusing of the optical beam.

Funding

National Natural Science Foundation of China (11974101, 11974102, 11704098).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

References

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3. S. N. Khonina, A. V. Ustinov, and S. A. Degtyarev, “Inverse energy flux of focused radially polarized optical beams,” Phys. Rev. A 98(4), 043823 (2018). [CrossRef]  

4. V. V. Kotlyar, S. S. Stafeev, and A. A. Kovalev, “Reverse and toroidal flux of light fields with both phase and polarization higher-order singularities in the sharp focus area,” Opt. Express 27(12), 16689–16702 (2019). [CrossRef]  

5. V. V. Kotlyar, A. G. Nalimov, S. S. Stafeev, and L. O’Faolain, “Single metalens for generating polarization and phase singularities leading to a reverse flow of energy,” J. Opt. 21(5), 055004 (2019). [CrossRef]  

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12. Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, M. Gong, and X. Yuan, “Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities,” Light: Sci. Appl. 8(1), 90 (2019). [CrossRef]  

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References

  • View by:

  1. V. V. Kotlyar, A. A. Kovalev, and A. G. Nalimov, “Energy density and energy flux in the focus of an optical vortex: reverse flux of light energy,” Opt. Lett. 43(12), 2921–2924 (2018).
    [Crossref]
  2. V. V. Kotlyar, S. S. Stafeev, and A. G. Nalimov, “Energy backflow in the focus of a light beam with phase or polarization singularity,” Phys. Rev. A 99(3), 033840 (2019).
    [Crossref]
  3. S. N. Khonina, A. V. Ustinov, and S. A. Degtyarev, “Inverse energy flux of focused radially polarized optical beams,” Phys. Rev. A 98(4), 043823 (2018).
    [Crossref]
  4. V. V. Kotlyar, S. S. Stafeev, and A. A. Kovalev, “Reverse and toroidal flux of light fields with both phase and polarization higher-order singularities in the sharp focus area,” Opt. Express 27(12), 16689–16702 (2019).
    [Crossref]
  5. V. V. Kotlyar, A. G. Nalimov, S. S. Stafeev, and L. O’Faolain, “Single metalens for generating polarization and phase singularities leading to a reverse flow of energy,” J. Opt. 21(5), 055004 (2019).
    [Crossref]
  6. V. V. Kotlyar and A. A. Kovalev, “Near-field backflow of energy,” J. Opt. 21(4), 045603 (2019).
    [Crossref]
  7. A. V. Novitsky and D. V. Novitsky, “Negative propagation of vector Bessel beams,” J. Opt. Soc. Am. A 24(9), 2844–2849 (2007).
    [Crossref]
  8. F. G. Mitri, “Reverse propagation and negative angular momentum density flux of an optical nondiffracting nonparaxial fractional Bessel vortex beam of progressive waves,” J. Opt. Soc. Am. A 33(9), 1661–1667 (2016).
    [Crossref]
  9. P. Vaveliuk and O. Martinez-Matos, “Negative propagation effect in nonparaxial Airy beams,” Opt. Express 20(24), 26913–26921 (2012).
    [Crossref]
  10. Y. Eliezer, T. Zacharias, and A. Bahabad, “Observation of optical backflow,” Optica 7(1), 72–76 (2020).
    [Crossref]
  11. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).
    [Crossref]
  12. Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, M. Gong, and X. Yuan, “Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities,” Light: Sci. Appl. 8(1), 90 (2019).
    [Crossref]
  13. A. Bekshaev, K. Y. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13(5), 053001 (2011).
    [Crossref]
  14. K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin–orbit interactions of light,” Nat. Photonics 9(12), 796–808 (2015).
    [Crossref]
  15. H. Li, J. Wang, M. Tang, X. Li, and J. Tang, “Changes of phase structure of a paraxial beam due to spin-orbit coupling,” Phys. Rev. A 97(5), 053843 (2018).
    [Crossref]
  16. X. L. Wang, J. Chen, Y. Li, J. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105(25), 253602 (2010).
    [Crossref]
  17. V. V. Kotlyar and A. A. Kovalev, “Controlling orbital angular momentum of an optical vortex by varying its ellipticity,” Opt. Commun. 410(1), 202–205 (2018).
    [Crossref]
  18. P. Shi, L. P. Du, and X. C. Yuan, “Structured spin angular momentum in highly focused cylindrical vector vortex beams for optical manipulation,” Opt. Express 26(18), 23449–23459 (2018).
    [Crossref]
  19. P. Meng, Z. Man, A. P. Konijnenberg, and H. P. Urbach, “Angular momentum properties of hybrid cylindrical vector vortex beams in tightly focused optical systems,” Opt. Express 27(24), 35336–35348 (2019).
    [Crossref]
  20. Q. Zhan, “Radiation forces on a dielectric sphere produced by highly focused cylindrical vector beams,” J. Opt. A: Pure Appl. Opt. 5(3), 229–232 (2003).
    [Crossref]
  21. M. Li, S. Yan, Y. Liang, P. Zhang, and B. Yao, “Transverse spinning of particles in highly focused vector vortex beams,” Phys. Rev. A 95(5), 053802 (2017).
    [Crossref]
  22. Y. Zhang, X. Dou, Y. Dai, X. Wang, C. Min, and X. Yuan, “All-optical manipulation of micrometer-sized metallic particles,” Photonics Res. 6(2), 66–71 (2018).
    [Crossref]
  23. M. Li, Y. Cai, S. Yan, Y. Liang, P. Zhang, and B. Yao, “Orbit-induced localized spin angular momentum in strong focusing of optical vectorial vortex beams,” Phys. Rev. A 97(5), 053842 (2018).
    [Crossref]
  24. H. Hu, Q. Gan, and Q. Zhan, “Generation of a Nondiffracting superchiral optical needle for circular dichroism imaging of sparse subdiffraction objects,” Phys. Rev. Lett. 122(22), 223901 (2019).
    [Crossref]
  25. Y. Kozawa and S. Sato, “Numerical analysis of resolution enhancement in laser scanning microscopy using a radially polarized beam,” Opt. Express 23(3), 2076–2084 (2015).
    [Crossref]
  26. F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a subwavelength needle with ultra-long focal length by focusing azimuthally polarized light,” Sci. Rep. 5(1), 9977 (2015).
    [Crossref]
  27. Y. Kozawa, D. Matsunaga, and S. Sato, “Superresolution imaging via superoscillation focusing of a radially polarized beam,” Optica 5(2), 86–92 (2018).
    [Crossref]
  28. L. Du, A. Yang, A. V. Zayats, and X. Yuan, “Deep-subwavelength features of photonic skyrmions in a confined electromagnetic field with orbital angular momentum,” Nat. Phys. 15(7), 650–654 (2019).
    [Crossref]
  29. S. Tsesses, E. Ostrovsky, K. Cohen, B. Gjonaj, N. H. Lindner, and G. Bartal, “Optical skyrmion lattice in evanescent electromagnetic fields,” Science 361(6406), 993–996 (2018).
    [Crossref]
  30. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the aplanatic system,” Proc. R. Soc. London 253(1274), 358–379 (1959).
    [Crossref]
  31. Z. Chen, T. Zeng, B. Qian, and J. Ding, “Complete shaping of optical vector beams,” Opt. Express 23(14), 17701–17710 (2015).
    [Crossref]
  32. C. Chang, Y. Gao, J. Xia, S. Nie, and J. Ding, “Shaping of optical vector beams in three dimensions,” Opt. Lett. 42(19), 3884–3887 (2017).
    [Crossref]
  33. P. Vaity and L. Rusch, “Perfect vortex beam: Fourier transformation of Bessel beam,” Opt. Lett. 40(4), 597–600 (2015).
    [Crossref]

2020 (1)

2019 (8)

Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, M. Gong, and X. Yuan, “Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities,” Light: Sci. Appl. 8(1), 90 (2019).
[Crossref]

V. V. Kotlyar, S. S. Stafeev, and A. A. Kovalev, “Reverse and toroidal flux of light fields with both phase and polarization higher-order singularities in the sharp focus area,” Opt. Express 27(12), 16689–16702 (2019).
[Crossref]

V. V. Kotlyar, A. G. Nalimov, S. S. Stafeev, and L. O’Faolain, “Single metalens for generating polarization and phase singularities leading to a reverse flow of energy,” J. Opt. 21(5), 055004 (2019).
[Crossref]

V. V. Kotlyar and A. A. Kovalev, “Near-field backflow of energy,” J. Opt. 21(4), 045603 (2019).
[Crossref]

V. V. Kotlyar, S. S. Stafeev, and A. G. Nalimov, “Energy backflow in the focus of a light beam with phase or polarization singularity,” Phys. Rev. A 99(3), 033840 (2019).
[Crossref]

P. Meng, Z. Man, A. P. Konijnenberg, and H. P. Urbach, “Angular momentum properties of hybrid cylindrical vector vortex beams in tightly focused optical systems,” Opt. Express 27(24), 35336–35348 (2019).
[Crossref]

H. Hu, Q. Gan, and Q. Zhan, “Generation of a Nondiffracting superchiral optical needle for circular dichroism imaging of sparse subdiffraction objects,” Phys. Rev. Lett. 122(22), 223901 (2019).
[Crossref]

L. Du, A. Yang, A. V. Zayats, and X. Yuan, “Deep-subwavelength features of photonic skyrmions in a confined electromagnetic field with orbital angular momentum,” Nat. Phys. 15(7), 650–654 (2019).
[Crossref]

2018 (9)

S. Tsesses, E. Ostrovsky, K. Cohen, B. Gjonaj, N. H. Lindner, and G. Bartal, “Optical skyrmion lattice in evanescent electromagnetic fields,” Science 361(6406), 993–996 (2018).
[Crossref]

Y. Kozawa, D. Matsunaga, and S. Sato, “Superresolution imaging via superoscillation focusing of a radially polarized beam,” Optica 5(2), 86–92 (2018).
[Crossref]

Y. Zhang, X. Dou, Y. Dai, X. Wang, C. Min, and X. Yuan, “All-optical manipulation of micrometer-sized metallic particles,” Photonics Res. 6(2), 66–71 (2018).
[Crossref]

M. Li, Y. Cai, S. Yan, Y. Liang, P. Zhang, and B. Yao, “Orbit-induced localized spin angular momentum in strong focusing of optical vectorial vortex beams,” Phys. Rev. A 97(5), 053842 (2018).
[Crossref]

V. V. Kotlyar and A. A. Kovalev, “Controlling orbital angular momentum of an optical vortex by varying its ellipticity,” Opt. Commun. 410(1), 202–205 (2018).
[Crossref]

P. Shi, L. P. Du, and X. C. Yuan, “Structured spin angular momentum in highly focused cylindrical vector vortex beams for optical manipulation,” Opt. Express 26(18), 23449–23459 (2018).
[Crossref]

S. N. Khonina, A. V. Ustinov, and S. A. Degtyarev, “Inverse energy flux of focused radially polarized optical beams,” Phys. Rev. A 98(4), 043823 (2018).
[Crossref]

V. V. Kotlyar, A. A. Kovalev, and A. G. Nalimov, “Energy density and energy flux in the focus of an optical vortex: reverse flux of light energy,” Opt. Lett. 43(12), 2921–2924 (2018).
[Crossref]

H. Li, J. Wang, M. Tang, X. Li, and J. Tang, “Changes of phase structure of a paraxial beam due to spin-orbit coupling,” Phys. Rev. A 97(5), 053843 (2018).
[Crossref]

2017 (2)

M. Li, S. Yan, Y. Liang, P. Zhang, and B. Yao, “Transverse spinning of particles in highly focused vector vortex beams,” Phys. Rev. A 95(5), 053802 (2017).
[Crossref]

C. Chang, Y. Gao, J. Xia, S. Nie, and J. Ding, “Shaping of optical vector beams in three dimensions,” Opt. Lett. 42(19), 3884–3887 (2017).
[Crossref]

2016 (1)

2015 (5)

K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin–orbit interactions of light,” Nat. Photonics 9(12), 796–808 (2015).
[Crossref]

P. Vaity and L. Rusch, “Perfect vortex beam: Fourier transformation of Bessel beam,” Opt. Lett. 40(4), 597–600 (2015).
[Crossref]

Z. Chen, T. Zeng, B. Qian, and J. Ding, “Complete shaping of optical vector beams,” Opt. Express 23(14), 17701–17710 (2015).
[Crossref]

Y. Kozawa and S. Sato, “Numerical analysis of resolution enhancement in laser scanning microscopy using a radially polarized beam,” Opt. Express 23(3), 2076–2084 (2015).
[Crossref]

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a subwavelength needle with ultra-long focal length by focusing azimuthally polarized light,” Sci. Rep. 5(1), 9977 (2015).
[Crossref]

2012 (1)

2011 (1)

A. Bekshaev, K. Y. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13(5), 053001 (2011).
[Crossref]

2010 (1)

X. L. Wang, J. Chen, Y. Li, J. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105(25), 253602 (2010).
[Crossref]

2009 (1)

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).
[Crossref]

2007 (1)

2003 (1)

Q. Zhan, “Radiation forces on a dielectric sphere produced by highly focused cylindrical vector beams,” J. Opt. A: Pure Appl. Opt. 5(3), 229–232 (2003).
[Crossref]

1959 (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the aplanatic system,” Proc. R. Soc. London 253(1274), 358–379 (1959).
[Crossref]

Bahabad, A.

Bartal, G.

S. Tsesses, E. Ostrovsky, K. Cohen, B. Gjonaj, N. H. Lindner, and G. Bartal, “Optical skyrmion lattice in evanescent electromagnetic fields,” Science 361(6406), 993–996 (2018).
[Crossref]

Bekshaev, A.

A. Bekshaev, K. Y. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13(5), 053001 (2011).
[Crossref]

Bliokh, K. Y.

K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin–orbit interactions of light,” Nat. Photonics 9(12), 796–808 (2015).
[Crossref]

A. Bekshaev, K. Y. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13(5), 053001 (2011).
[Crossref]

Cai, Y.

M. Li, Y. Cai, S. Yan, Y. Liang, P. Zhang, and B. Yao, “Orbit-induced localized spin angular momentum in strong focusing of optical vectorial vortex beams,” Phys. Rev. A 97(5), 053842 (2018).
[Crossref]

Chang, C.

Chen, J.

X. L. Wang, J. Chen, Y. Li, J. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105(25), 253602 (2010).
[Crossref]

Chen, Z.

Cohen, K.

S. Tsesses, E. Ostrovsky, K. Cohen, B. Gjonaj, N. H. Lindner, and G. Bartal, “Optical skyrmion lattice in evanescent electromagnetic fields,” Science 361(6406), 993–996 (2018).
[Crossref]

Dai, Y.

Y. Zhang, X. Dou, Y. Dai, X. Wang, C. Min, and X. Yuan, “All-optical manipulation of micrometer-sized metallic particles,” Photonics Res. 6(2), 66–71 (2018).
[Crossref]

Degtyarev, S. A.

S. N. Khonina, A. V. Ustinov, and S. A. Degtyarev, “Inverse energy flux of focused radially polarized optical beams,” Phys. Rev. A 98(4), 043823 (2018).
[Crossref]

Ding, J.

Dou, X.

Y. Zhang, X. Dou, Y. Dai, X. Wang, C. Min, and X. Yuan, “All-optical manipulation of micrometer-sized metallic particles,” Photonics Res. 6(2), 66–71 (2018).
[Crossref]

Du, L.

L. Du, A. Yang, A. V. Zayats, and X. Yuan, “Deep-subwavelength features of photonic skyrmions in a confined electromagnetic field with orbital angular momentum,” Nat. Phys. 15(7), 650–654 (2019).
[Crossref]

Du, L. P.

Eliezer, Y.

Fu, X.

Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, M. Gong, and X. Yuan, “Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities,” Light: Sci. Appl. 8(1), 90 (2019).
[Crossref]

Gan, Q.

H. Hu, Q. Gan, and Q. Zhan, “Generation of a Nondiffracting superchiral optical needle for circular dichroism imaging of sparse subdiffraction objects,” Phys. Rev. Lett. 122(22), 223901 (2019).
[Crossref]

Gao, Y.

Gjonaj, B.

S. Tsesses, E. Ostrovsky, K. Cohen, B. Gjonaj, N. H. Lindner, and G. Bartal, “Optical skyrmion lattice in evanescent electromagnetic fields,” Science 361(6406), 993–996 (2018).
[Crossref]

Gong, M.

Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, M. Gong, and X. Yuan, “Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities,” Light: Sci. Appl. 8(1), 90 (2019).
[Crossref]

Guo, C. S.

X. L. Wang, J. Chen, Y. Li, J. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105(25), 253602 (2010).
[Crossref]

Hong, M.

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a subwavelength needle with ultra-long focal length by focusing azimuthally polarized light,” Sci. Rep. 5(1), 9977 (2015).
[Crossref]

Hu, H.

H. Hu, Q. Gan, and Q. Zhan, “Generation of a Nondiffracting superchiral optical needle for circular dichroism imaging of sparse subdiffraction objects,” Phys. Rev. Lett. 122(22), 223901 (2019).
[Crossref]

Huang, K.

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a subwavelength needle with ultra-long focal length by focusing azimuthally polarized light,” Sci. Rep. 5(1), 9977 (2015).
[Crossref]

Jiao, J.

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a subwavelength needle with ultra-long focal length by focusing azimuthally polarized light,” Sci. Rep. 5(1), 9977 (2015).
[Crossref]

Khonina, S. N.

S. N. Khonina, A. V. Ustinov, and S. A. Degtyarev, “Inverse energy flux of focused radially polarized optical beams,” Phys. Rev. A 98(4), 043823 (2018).
[Crossref]

Konijnenberg, A. P.

Kotlyar, V. V.

V. V. Kotlyar, S. S. Stafeev, and A. A. Kovalev, “Reverse and toroidal flux of light fields with both phase and polarization higher-order singularities in the sharp focus area,” Opt. Express 27(12), 16689–16702 (2019).
[Crossref]

V. V. Kotlyar, S. S. Stafeev, and A. G. Nalimov, “Energy backflow in the focus of a light beam with phase or polarization singularity,” Phys. Rev. A 99(3), 033840 (2019).
[Crossref]

V. V. Kotlyar, A. G. Nalimov, S. S. Stafeev, and L. O’Faolain, “Single metalens for generating polarization and phase singularities leading to a reverse flow of energy,” J. Opt. 21(5), 055004 (2019).
[Crossref]

V. V. Kotlyar and A. A. Kovalev, “Near-field backflow of energy,” J. Opt. 21(4), 045603 (2019).
[Crossref]

V. V. Kotlyar, A. A. Kovalev, and A. G. Nalimov, “Energy density and energy flux in the focus of an optical vortex: reverse flux of light energy,” Opt. Lett. 43(12), 2921–2924 (2018).
[Crossref]

V. V. Kotlyar and A. A. Kovalev, “Controlling orbital angular momentum of an optical vortex by varying its ellipticity,” Opt. Commun. 410(1), 202–205 (2018).
[Crossref]

Kovalev, A. A.

Kozawa, Y.

Li, H.

H. Li, J. Wang, M. Tang, X. Li, and J. Tang, “Changes of phase structure of a paraxial beam due to spin-orbit coupling,” Phys. Rev. A 97(5), 053843 (2018).
[Crossref]

Li, M.

M. Li, Y. Cai, S. Yan, Y. Liang, P. Zhang, and B. Yao, “Orbit-induced localized spin angular momentum in strong focusing of optical vectorial vortex beams,” Phys. Rev. A 97(5), 053842 (2018).
[Crossref]

M. Li, S. Yan, Y. Liang, P. Zhang, and B. Yao, “Transverse spinning of particles in highly focused vector vortex beams,” Phys. Rev. A 95(5), 053802 (2017).
[Crossref]

Li, X.

H. Li, J. Wang, M. Tang, X. Li, and J. Tang, “Changes of phase structure of a paraxial beam due to spin-orbit coupling,” Phys. Rev. A 97(5), 053843 (2018).
[Crossref]

Li, Y.

X. L. Wang, J. Chen, Y. Li, J. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105(25), 253602 (2010).
[Crossref]

Liang, Y.

M. Li, Y. Cai, S. Yan, Y. Liang, P. Zhang, and B. Yao, “Orbit-induced localized spin angular momentum in strong focusing of optical vectorial vortex beams,” Phys. Rev. A 97(5), 053842 (2018).
[Crossref]

M. Li, S. Yan, Y. Liang, P. Zhang, and B. Yao, “Transverse spinning of particles in highly focused vector vortex beams,” Phys. Rev. A 95(5), 053802 (2017).
[Crossref]

Lindner, N. H.

S. Tsesses, E. Ostrovsky, K. Cohen, B. Gjonaj, N. H. Lindner, and G. Bartal, “Optical skyrmion lattice in evanescent electromagnetic fields,” Science 361(6406), 993–996 (2018).
[Crossref]

Liu, Q.

Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, M. Gong, and X. Yuan, “Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities,” Light: Sci. Appl. 8(1), 90 (2019).
[Crossref]

Luo, X.

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a subwavelength needle with ultra-long focal length by focusing azimuthally polarized light,” Sci. Rep. 5(1), 9977 (2015).
[Crossref]

Man, Z.

Martinez-Matos, O.

Matsunaga, D.

Meng, P.

Min, C.

Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, M. Gong, and X. Yuan, “Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities,” Light: Sci. Appl. 8(1), 90 (2019).
[Crossref]

Y. Zhang, X. Dou, Y. Dai, X. Wang, C. Min, and X. Yuan, “All-optical manipulation of micrometer-sized metallic particles,” Photonics Res. 6(2), 66–71 (2018).
[Crossref]

Mitri, F. G.

Nalimov, A. G.

V. V. Kotlyar, S. S. Stafeev, and A. G. Nalimov, “Energy backflow in the focus of a light beam with phase or polarization singularity,” Phys. Rev. A 99(3), 033840 (2019).
[Crossref]

V. V. Kotlyar, A. G. Nalimov, S. S. Stafeev, and L. O’Faolain, “Single metalens for generating polarization and phase singularities leading to a reverse flow of energy,” J. Opt. 21(5), 055004 (2019).
[Crossref]

V. V. Kotlyar, A. A. Kovalev, and A. G. Nalimov, “Energy density and energy flux in the focus of an optical vortex: reverse flux of light energy,” Opt. Lett. 43(12), 2921–2924 (2018).
[Crossref]

Nie, S.

Nori, F.

K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin–orbit interactions of light,” Nat. Photonics 9(12), 796–808 (2015).
[Crossref]

Novitsky, A. V.

Novitsky, D. V.

O’Faolain, L.

V. V. Kotlyar, A. G. Nalimov, S. S. Stafeev, and L. O’Faolain, “Single metalens for generating polarization and phase singularities leading to a reverse flow of energy,” J. Opt. 21(5), 055004 (2019).
[Crossref]

Ostrovsky, E.

S. Tsesses, E. Ostrovsky, K. Cohen, B. Gjonaj, N. H. Lindner, and G. Bartal, “Optical skyrmion lattice in evanescent electromagnetic fields,” Science 361(6406), 993–996 (2018).
[Crossref]

Qian, B.

Qin, F.

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a subwavelength needle with ultra-long focal length by focusing azimuthally polarized light,” Sci. Rep. 5(1), 9977 (2015).
[Crossref]

Qiu, C.

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a subwavelength needle with ultra-long focal length by focusing azimuthally polarized light,” Sci. Rep. 5(1), 9977 (2015).
[Crossref]

Richards, B.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the aplanatic system,” Proc. R. Soc. London 253(1274), 358–379 (1959).
[Crossref]

Rodríguez-Fortuño, F. J.

K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin–orbit interactions of light,” Nat. Photonics 9(12), 796–808 (2015).
[Crossref]

Rusch, L.

Sato, S.

Shen, Y.

Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, M. Gong, and X. Yuan, “Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities,” Light: Sci. Appl. 8(1), 90 (2019).
[Crossref]

Shi, P.

Soskin, M.

A. Bekshaev, K. Y. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13(5), 053001 (2011).
[Crossref]

Stafeev, S. S.

V. V. Kotlyar, A. G. Nalimov, S. S. Stafeev, and L. O’Faolain, “Single metalens for generating polarization and phase singularities leading to a reverse flow of energy,” J. Opt. 21(5), 055004 (2019).
[Crossref]

V. V. Kotlyar, S. S. Stafeev, and A. G. Nalimov, “Energy backflow in the focus of a light beam with phase or polarization singularity,” Phys. Rev. A 99(3), 033840 (2019).
[Crossref]

V. V. Kotlyar, S. S. Stafeev, and A. A. Kovalev, “Reverse and toroidal flux of light fields with both phase and polarization higher-order singularities in the sharp focus area,” Opt. Express 27(12), 16689–16702 (2019).
[Crossref]

Tang, J.

H. Li, J. Wang, M. Tang, X. Li, and J. Tang, “Changes of phase structure of a paraxial beam due to spin-orbit coupling,” Phys. Rev. A 97(5), 053843 (2018).
[Crossref]

Tang, M.

H. Li, J. Wang, M. Tang, X. Li, and J. Tang, “Changes of phase structure of a paraxial beam due to spin-orbit coupling,” Phys. Rev. A 97(5), 053843 (2018).
[Crossref]

Tsesses, S.

S. Tsesses, E. Ostrovsky, K. Cohen, B. Gjonaj, N. H. Lindner, and G. Bartal, “Optical skyrmion lattice in evanescent electromagnetic fields,” Science 361(6406), 993–996 (2018).
[Crossref]

Urbach, H. P.

Ustinov, A. V.

S. N. Khonina, A. V. Ustinov, and S. A. Degtyarev, “Inverse energy flux of focused radially polarized optical beams,” Phys. Rev. A 98(4), 043823 (2018).
[Crossref]

Vaity, P.

Vaveliuk, P.

Wang, H. T.

X. L. Wang, J. Chen, Y. Li, J. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105(25), 253602 (2010).
[Crossref]

Wang, J.

H. Li, J. Wang, M. Tang, X. Li, and J. Tang, “Changes of phase structure of a paraxial beam due to spin-orbit coupling,” Phys. Rev. A 97(5), 053843 (2018).
[Crossref]

Wang, X.

Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, M. Gong, and X. Yuan, “Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities,” Light: Sci. Appl. 8(1), 90 (2019).
[Crossref]

Y. Zhang, X. Dou, Y. Dai, X. Wang, C. Min, and X. Yuan, “All-optical manipulation of micrometer-sized metallic particles,” Photonics Res. 6(2), 66–71 (2018).
[Crossref]

Wang, X. L.

X. L. Wang, J. Chen, Y. Li, J. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105(25), 253602 (2010).
[Crossref]

Wolf, E.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the aplanatic system,” Proc. R. Soc. London 253(1274), 358–379 (1959).
[Crossref]

Wu, J.

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a subwavelength needle with ultra-long focal length by focusing azimuthally polarized light,” Sci. Rep. 5(1), 9977 (2015).
[Crossref]

Xia, J.

Xie, Z.

Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, M. Gong, and X. Yuan, “Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities,” Light: Sci. Appl. 8(1), 90 (2019).
[Crossref]

Yan, S.

M. Li, Y. Cai, S. Yan, Y. Liang, P. Zhang, and B. Yao, “Orbit-induced localized spin angular momentum in strong focusing of optical vectorial vortex beams,” Phys. Rev. A 97(5), 053842 (2018).
[Crossref]

M. Li, S. Yan, Y. Liang, P. Zhang, and B. Yao, “Transverse spinning of particles in highly focused vector vortex beams,” Phys. Rev. A 95(5), 053802 (2017).
[Crossref]

Yang, A.

L. Du, A. Yang, A. V. Zayats, and X. Yuan, “Deep-subwavelength features of photonic skyrmions in a confined electromagnetic field with orbital angular momentum,” Nat. Phys. 15(7), 650–654 (2019).
[Crossref]

Yao, B.

M. Li, Y. Cai, S. Yan, Y. Liang, P. Zhang, and B. Yao, “Orbit-induced localized spin angular momentum in strong focusing of optical vectorial vortex beams,” Phys. Rev. A 97(5), 053842 (2018).
[Crossref]

M. Li, S. Yan, Y. Liang, P. Zhang, and B. Yao, “Transverse spinning of particles in highly focused vector vortex beams,” Phys. Rev. A 95(5), 053802 (2017).
[Crossref]

Yuan, X.

Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, M. Gong, and X. Yuan, “Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities,” Light: Sci. Appl. 8(1), 90 (2019).
[Crossref]

L. Du, A. Yang, A. V. Zayats, and X. Yuan, “Deep-subwavelength features of photonic skyrmions in a confined electromagnetic field with orbital angular momentum,” Nat. Phys. 15(7), 650–654 (2019).
[Crossref]

Y. Zhang, X. Dou, Y. Dai, X. Wang, C. Min, and X. Yuan, “All-optical manipulation of micrometer-sized metallic particles,” Photonics Res. 6(2), 66–71 (2018).
[Crossref]

Yuan, X. C.

Zacharias, T.

Zayats, A. V.

L. Du, A. Yang, A. V. Zayats, and X. Yuan, “Deep-subwavelength features of photonic skyrmions in a confined electromagnetic field with orbital angular momentum,” Nat. Phys. 15(7), 650–654 (2019).
[Crossref]

K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin–orbit interactions of light,” Nat. Photonics 9(12), 796–808 (2015).
[Crossref]

Zeng, T.

Zhan, Q.

H. Hu, Q. Gan, and Q. Zhan, “Generation of a Nondiffracting superchiral optical needle for circular dichroism imaging of sparse subdiffraction objects,” Phys. Rev. Lett. 122(22), 223901 (2019).
[Crossref]

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).
[Crossref]

Q. Zhan, “Radiation forces on a dielectric sphere produced by highly focused cylindrical vector beams,” J. Opt. A: Pure Appl. Opt. 5(3), 229–232 (2003).
[Crossref]

Zhang, P.

M. Li, Y. Cai, S. Yan, Y. Liang, P. Zhang, and B. Yao, “Orbit-induced localized spin angular momentum in strong focusing of optical vectorial vortex beams,” Phys. Rev. A 97(5), 053842 (2018).
[Crossref]

M. Li, S. Yan, Y. Liang, P. Zhang, and B. Yao, “Transverse spinning of particles in highly focused vector vortex beams,” Phys. Rev. A 95(5), 053802 (2017).
[Crossref]

Zhang, Y.

Y. Zhang, X. Dou, Y. Dai, X. Wang, C. Min, and X. Yuan, “All-optical manipulation of micrometer-sized metallic particles,” Photonics Res. 6(2), 66–71 (2018).
[Crossref]

Adv. Opt. Photonics (1)

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).
[Crossref]

J. Opt. (3)

V. V. Kotlyar, A. G. Nalimov, S. S. Stafeev, and L. O’Faolain, “Single metalens for generating polarization and phase singularities leading to a reverse flow of energy,” J. Opt. 21(5), 055004 (2019).
[Crossref]

V. V. Kotlyar and A. A. Kovalev, “Near-field backflow of energy,” J. Opt. 21(4), 045603 (2019).
[Crossref]

A. Bekshaev, K. Y. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13(5), 053001 (2011).
[Crossref]

J. Opt. A: Pure Appl. Opt. (1)

Q. Zhan, “Radiation forces on a dielectric sphere produced by highly focused cylindrical vector beams,” J. Opt. A: Pure Appl. Opt. 5(3), 229–232 (2003).
[Crossref]

J. Opt. Soc. Am. A (2)

Light: Sci. Appl. (1)

Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, M. Gong, and X. Yuan, “Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities,” Light: Sci. Appl. 8(1), 90 (2019).
[Crossref]

Nat. Photonics (1)

K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin–orbit interactions of light,” Nat. Photonics 9(12), 796–808 (2015).
[Crossref]

Nat. Phys. (1)

L. Du, A. Yang, A. V. Zayats, and X. Yuan, “Deep-subwavelength features of photonic skyrmions in a confined electromagnetic field with orbital angular momentum,” Nat. Phys. 15(7), 650–654 (2019).
[Crossref]

Opt. Commun. (1)

V. V. Kotlyar and A. A. Kovalev, “Controlling orbital angular momentum of an optical vortex by varying its ellipticity,” Opt. Commun. 410(1), 202–205 (2018).
[Crossref]

Opt. Express (6)

Opt. Lett. (3)

Optica (2)

Photonics Res. (1)

Y. Zhang, X. Dou, Y. Dai, X. Wang, C. Min, and X. Yuan, “All-optical manipulation of micrometer-sized metallic particles,” Photonics Res. 6(2), 66–71 (2018).
[Crossref]

Phys. Rev. A (5)

M. Li, Y. Cai, S. Yan, Y. Liang, P. Zhang, and B. Yao, “Orbit-induced localized spin angular momentum in strong focusing of optical vectorial vortex beams,” Phys. Rev. A 97(5), 053842 (2018).
[Crossref]

V. V. Kotlyar, S. S. Stafeev, and A. G. Nalimov, “Energy backflow in the focus of a light beam with phase or polarization singularity,” Phys. Rev. A 99(3), 033840 (2019).
[Crossref]

S. N. Khonina, A. V. Ustinov, and S. A. Degtyarev, “Inverse energy flux of focused radially polarized optical beams,” Phys. Rev. A 98(4), 043823 (2018).
[Crossref]

M. Li, S. Yan, Y. Liang, P. Zhang, and B. Yao, “Transverse spinning of particles in highly focused vector vortex beams,” Phys. Rev. A 95(5), 053802 (2017).
[Crossref]

H. Li, J. Wang, M. Tang, X. Li, and J. Tang, “Changes of phase structure of a paraxial beam due to spin-orbit coupling,” Phys. Rev. A 97(5), 053843 (2018).
[Crossref]

Phys. Rev. Lett. (2)

X. L. Wang, J. Chen, Y. Li, J. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105(25), 253602 (2010).
[Crossref]

H. Hu, Q. Gan, and Q. Zhan, “Generation of a Nondiffracting superchiral optical needle for circular dichroism imaging of sparse subdiffraction objects,” Phys. Rev. Lett. 122(22), 223901 (2019).
[Crossref]

Proc. R. Soc. London (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the aplanatic system,” Proc. R. Soc. London 253(1274), 358–379 (1959).
[Crossref]

Sci. Rep. (1)

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a subwavelength needle with ultra-long focal length by focusing azimuthally polarized light,” Sci. Rep. 5(1), 9977 (2015).
[Crossref]

Science (1)

S. Tsesses, E. Ostrovsky, K. Cohen, B. Gjonaj, N. H. Lindner, and G. Bartal, “Optical skyrmion lattice in evanescent electromagnetic fields,” Science 361(6406), 993–996 (2018).
[Crossref]

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Figures (7)

Fig. 1.
Fig. 1. Energy flow density (${S_z}$) in the focal plane when ${\phi _{01}} = {\phi _{02}} = 0$, (a) ${S_z}$ along the radial direction for vector beams with the polarization topological charge m, (b) evolution of on-axis energy flow ${S_{z0}}$ as the change of the polarization topological charge m.
Fig. 2.
Fig. 2. Polarization and normalized spin flow density distributions of input vector beams with polarization topological charge 2, (a) and (e) ${\phi _{01}} = 0$, ${\phi _{02}} = 0$, (b) and (f) ${\phi _{01}} = 0$, ${\phi _{02}} = \pi /2$, (c) and (g) ${\phi _{01}} = \pi /2$, ${\phi _{02}} = 0$, (d) and (h) ${\phi _{01}} = \pi /2$, ${\phi _{02}} = \pi /2$.
Fig. 3.
Fig. 3. Normalized energy flow density (${S_z}$) in the focal plane when input vector beam with polarization topological charge 2, ${\phi _{02}} = 0$, (a) ${\phi _{01}}\textrm{ = }0$, (b) ${\phi _{01}}\textrm{ = }\pi \textrm{/4}$, (c) ${\phi _{01}}\textrm{ = 3}\pi \textrm{/4}$, (d) ${\phi _{01}}\textrm{ = }\pi $, (e) ${\phi _{01}}\textrm{ = 5}\pi \textrm{/4}$, (f) ${\phi _{01}}\textrm{ = 7}\pi \textrm{/4}$, the calculation parameters are same as those in Fig. 1.
Fig. 4.
Fig. 4. Evolution of energy flow density (${S_z}$) in the focal plane with the change of ${\phi _{01}}$ and ${\phi _{02}}$ when polarization topological charge $m = 2$, (a) and (c) in the focal plane ($z = 0$), (a) ${\phi _{01}} \ne 0$, ${\phi _{02}}\textrm{ = }0$, (c) ${\phi _{01}}\textrm{ = }0$, ${\phi _{02}} \ne 0$, (b) and (d) on the optical axis (${S_{z0}}$), (b) ${\phi _{01}} \ne 0$, ${\phi _{02}}\textrm{ = }0$, (d) ${\phi _{01}}\textrm{ = }0$, ${\phi _{02}} \ne 0$, the calculation parameters are same as those in Fig. 1.
Fig. 5.
Fig. 5. Evolution of energy flow density (${S_z}$) in the focal plane with the change of ${\phi _{01}}$ and ${\phi _{02}}$ when polarization topological charge $m ={-} 2$, (a) and (c) in the focal plane ($z = 0$), (a) ${\phi _{01}} \ne 0$, ${\phi _{02}}\textrm{ = }0$, (c) ${\phi _{01}}\textrm{ = }0$, ${\phi _{02}} \ne 0$, (b) and (d) on the optical axis (${S_{z0}}$), (b) ${\phi _{01}} \ne 0$, ${\phi _{02}}\textrm{ = }0$, (d) ${\phi _{01}}\textrm{ = }0$, ${\phi _{02}} \ne 0$, the calculation parameters are same as those in Fig. 1.
Fig. 6.
Fig. 6. Evolution of on-axis energy flow (${S_{z0}}$) in the focal plane as the change of parameters ${\phi _{01}}$ and ${\phi _{02}}$ when the input vector beam with polarization topological charges 2 (a) and −2 (b), the calculation parameters are same as those in Fig. 1.
Fig. 7.
Fig. 7. Polarization and normalized spin flow density distributions of input vector beam with polarization topological charge 2 and −2 when the maximal negative energy flow is obtained, (a) and (e) ${\phi _{01}} = 0$, ${\phi _{02}} = 0$, (b) and (f) ${\phi _{01}} = \pi $, ${\phi _{02}} = \pi $, (c) and (g) ${\phi _{01}} = 0$, ${\phi _{02}} = \pi $, (d) and (h) ${\phi _{01}} = \pi $, ${\phi _{02}} = 0$.

Equations (5)

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[ E ( r , φ , z ) H ( r , φ , z ) ] = i f λ 0 α 0 2 π T ( θ ) A ( θ , ϕ ) [ P E ( θ , ϕ ) P H ( θ , ϕ ) ] × E x p { i k [ r sin θ cos ( ϕ φ ) + z cos θ ] } sin θ d θ d ϕ ,
P E ( θ , ϕ ) = [ A ( θ , ϕ ) C ( θ , ϕ ) C ( θ , ϕ ) B ( θ , ϕ ) D ( θ , ϕ ) E ( θ , ϕ ) ] [ c ~ x ( ϕ ) c ~ y ( ϕ ) ] , P H ( θ , ϕ ) = [ C ( θ , ϕ ) A ( θ , ϕ ) B ( θ , ϕ ) C ( θ , ϕ ) E ( θ , ϕ ) D ( θ , ϕ ) ] [ c ~ x ( ϕ ) c ~ y ( ϕ ) ] ,
A ( θ , ϕ ) = 1 + cos 2 ϕ ( cos θ 1 ) , B ( θ , ϕ ) = 1 + sin 2 ϕ ( cos θ 1 ) , C ( θ , ϕ ) = sin ϕ cos ϕ ( cos θ 1 ) , D ( θ , ϕ ) = cos ϕ sin θ , E ( θ , ϕ ) = sin ϕ sin θ ,
[ c ~ x ( ϕ ) c ~ y ( ϕ ) ]  = R( ϕ 0 ) [ cos ( m ϕ ) sin ( m ϕ ) ] ,
R( ϕ 0 ) =  [ cos ϕ 01 sin ϕ 01 sin ϕ 02 cos ϕ 02 ] .

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