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Spin-orbit Hall effect in the tight focusing of a radially polarized vortex beam

Hehe Li, Chenghao Ma, Jingge Wang, Miaomiao Tang, and Xinzhong Li

Open Access Open Access

Abstract

When the first-order radially polarized vortex beam propagates in an uniaxial crystal, the spin and the orbital angular momentum parts can be separated. It is called the optical spin-orbit Hall effect. In this study, we investigate the tight focusing of the radially polarized vortex beam theoretically and find the spatial separation of the spin and the orbital angular momentum parts occurs in the focal plane when the polarization order equals 1 and the vortex charge equals 1 (or -1). Moreover, when the initial phase of the polarization state takes π/2, the spatial separation of intensity in the focal plane corresponds to the spatial separation of the spin and the orbital angular momentum parts. This phenomenon can be considered as a manifestation of the optical spin-orbit Hall effect in the tight focusing of radially polarized vortex beam. Also, we show that, when the polarization order is greater than 1, the initial phase change of polarization state just leads to the rotation of the focal field and the spin and the orbital angular momentum density in the focal plane. Our results provide the potential application in the field of optical micro-manipulation.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Generally, optical angular momentum can be classified into three types [1], spin angular momentum originating from the circular polarization, intrinsic orbital angular momentum originating from the vortex phase [2] or inhomogeneous polarization distribution [3], and extrinsic orbital angular momentum originating from the curve of propagation trajectory of light. Optical angular momentum can interact with each other when the light beam is tightly focused [4–8] or propagates in inhomogeneous or anisotropic medium [9–20]. Optical spin Hall effect and the spin-to-orbital angular momentum conversion are two important manifestations induced by the spin-orbit interaction [20], which have attracted intensive attention because of its potential applications in nano-optics, photonics, and plasmonics.

The optical spin Hall effect describes the spatial separation of different spin states of light, and it displays the propagation trajectory split of the right-hand and left-hand circularly polarized light. It knows that, when a circularly polarized light propagates in inhomogeneous medium (or is reflected by an interface of different optical media), the inhomogeneity of medium will lead to the spin-orbit interaction and induce the optical spin Hall effect [9–19]. Furthermore, when a circularly polarized optical vortex beam is tightly focused, the spin angular momentum of incident beam is partly converted into the orbital angular momentum of the focused field [4–8], which is called the spin-to-orbital angular momentum conversion. Meanwhile, arbitrary spin-to-orbital angular momentum conversion can be realized with help of optical element (J-plate) [20]. The converse process, orbital-to-spin angular momentum conversion also has been investigated in the tight focusing of linearly polarized vortex beam or azimuthally polarized vortex beam [21,22].

The optical spin-orbit Hall effect also is an interesting phenomenon induced by the spin-orbit interaction and displays the optical angular momentum separation in terms of the spin and the orbital parts of light [23]. Unlike the optical spin Hall effect, the optical spin-orbit Hall effect can’t be achieved through a conventional scalar light beam because of its homogeneous polarization state distribution. Fu etc. show the optical spin-orbit Hall effect can be obtained when the first-order radially vortex beam propagates in uniaxial crystal along the optical axis, and the separation of the spin and the orbital angular momentum parts can be modulated by the initial phase of polarization states [23]. The spin-orbit interaction also exists in the tight focusing of the optical vector beams. Then, is there the optical spin-orbit Hall effect in the tight focusing of the radially polarized vortex beam?

Here, we investigate the tight focusing of the radially polarized vortex beam theoretically, and find the spatial separation of the spin and the orbital angular momentum density can be obtained in the focal plane when the polarization order equals to 1 and the vortex charge equals to 1 (or -1). It can be considered as a manifestation of the optical spin-orbit Hall effect in the tight focusing of radially polarized vortex beam. We find that, both the spin and the orbital angular momentum density are modulated by the initial phase of polarization state of incident vector beam, the spin angular momentum density will concentrate in the central region of the focal plane and the orbital angular momentum density will concentrate in the outer ring region of the focal plane when the initial phase of polarization state takes $\pi /2$. The optical angular momentum of radially polarized beam is determined by its vortex charge. We show that the spin-to-orbital angular momentum conversion occurs in the tight focusing of the beam, and the total optical angular momentum is conserved. We know that, because of the rotational symmetry of polarization distribution, when the polarization order of incident radially polarized vortex beam is greater than 1, both the intensity and optical angular momentum distribution in the focal plane just rotate around the optical axis with the change of initial phase of polarization state. Our results provide the potential application in the field of optical micro-manipulation.

2. Theoretical model

The polarization state of radially polarized beam can be described by a vectorial coefficient matrix $[{{c_x}(\phi ),{c_y}(\phi )} ]= [{\cos (m\phi + {\varphi_0}),\sin (m\phi + {\varphi_0})} ]$, m is the polarization order, ${\varphi _0}$ is the initial phase of the polarization state. The change of the polarization state induced by the initial phase can be expressed as the evolution of the polarization state along the equator on the Poincaré sphere, $2{\varphi _0}$ is the rotation angle around the axis of the Poincaré sphere, as shown in Fig. 1. It can be described by the following form [24],

$$\left[ {\begin{array}{c} {{c_x}(\phi )}\\ {{c_y}(\phi )} \end{array}} \right] = R({{\varphi_0}} )\left[ {\begin{array}{c} {\cos (m\phi )}\\ {\sin (m\phi )} \end{array}} \right] = \left[ {\begin{array}{cc} {\cos {\varphi_0}}&{ - \sin {\varphi_0}}\\ {\sin {\varphi_0}}&{\cos {\varphi_0}} \end{array}} \right]\left[ {\begin{array}{c} {\cos (m\phi )}\\ {\sin (m\phi )} \end{array}} \right],$$
where $R({{\varphi_0}} )$ is the rotation matrix describing the rotation operation on the polarization unit vector. This rotation operation corresponds to the evolution of polarization state along the equator on the Poincaré sphere. Figure 1 also shows that, when the polarization order is equal to 1, the polarization state along the equator on the Poincaré sphere is changed from the radially polarized state to azimuthally polarized state; yet when the polarization order is greater than 1, it just is rotated around the optical axis in whole.

 figure: Fig. 1.

Fig. 1. Polarization state evolution of radially polarized beam along the equator on Poincaré sphere.

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The tight focusing of optical vector beam can be analyzed by means of the Richards-Wolf diffraction integral which has the form as [25]

$${\mathbf E}(r,\varphi ,z) ={-} \frac{{if}}{\lambda }\int_0^\alpha {} \int_0^{2\pi } {T(\theta )} F(\theta ,\phi ){{\mathbf P}_E}(\theta ,\phi )Exp\{{ik[{r\sin \theta \cos ({\phi - \varphi } )+ z\cos \theta } ]} \}\sin \theta d\theta d\phi ,$$
where
$${{\mathbf P}_E}(\theta ,\phi ) = \left[ {\begin{array}{cc} {A(\theta ,\phi )}&{C(\theta ,\phi )}\\ {C(\theta ,\phi )}&{B(\theta ,\phi )}\\ { - D(\theta ,\phi )}&{ - E(\theta ,\phi )} \end{array}} \right]R(t)\left[ {\begin{array}{c} {{c_x}(\phi )}\\ {{c_y}(\phi )} \end{array}} \right],$$
and
$$A(\theta ,\phi ) = 1 + {\cos ^2}\phi (\cos \theta - 1), B(\theta ,\phi ) = 1 + {\sin ^2}\phi (\cos \theta - 1),$$
$$C(\theta ,\phi ) = \sin \phi \cos \phi (\cos \theta - 1), D(\theta ,\phi ) = \cos \phi \sin \theta , E(\theta ,\phi ) = \sin \phi \sin \theta ,$$
where $T(\theta ) = \sqrt {\cos (\theta )} $ is the apodization function, $F(\theta ,\phi )$ is the complex amplitude of incident field, f is the focal length, $\lambda $ is the incident light wavelength. For simplicity, we consider the amplitude of incident field is concentrated in a narrow annular region whose central angle is ${\theta _0}$, the complex amplitude is taken as $F({\theta _0},\phi ) = \exp (il\phi )$, l is the vortex charge of incident beam. Obviously, the total optical angular momentum is determined by the vortex charge of incident beam. In order to demonstrate the optical angular momentum conversion in the tight focusing of radially polarized vortex beam clearly, the incident beam can be expressed in a form of superposition of two antipodal circular polarization vortex states [26],
$${{\mathbf E}_{in}} = F({{\theta_0},\phi } )R({{\varphi_0}} )\left[ {\begin{array}{c} {{c_x}(\phi )}\\ {{c_y}(\phi )} \end{array}} \right] = \frac{1}{2}R({{\varphi_0}} )\left\{ {\exp [{i({l - m} )\phi } ]\left[ {\begin{array}{c} 1\\ i \end{array}} \right] + \exp [{i({l + m} )\phi } ]\left[ {\begin{array}{c} 1\\ { - i} \end{array}} \right]} \right\}.$$

The first term in Eq. (3) is the right-hand circularly polarized vortex component, its optical angular momentum can be expressed as $({l - m + {\sigma^ + }} )\hbar $; the second term in Eq. (3) is the left-hand circularly polarized vortex component, its optical angular momentum can be expressed as $({l + m + {\sigma^ - }} )\hbar $, ${\sigma ^ + } = 1$ and ${\sigma ^ - } ={-} 1$ denote the wave helicity corresponding to right and left circular polarization. Then, the total optical angular momentum can be expressed as $l\hbar $ and is determined by the vortex charge of incident beam. Substituting Eq. (1) into the Richards-Wolf diffraction integral, the focal field components are obtained as following,

$${E_x}(r,\varphi ,z) = {E_{l - m}} + {E_{l + m}} + {E_{l - m + 2}} + {E_{l + m - 2}},$$
$${E_y}(r,\varphi ,z) = i({{E_{l - m}} - {E_{l + m}}} )- i({{E_{l - m + 2}} - {E_{l + m - 2}}} ),$$
$${E_z}(r,\varphi ,z) ={-} {E_{l - m + 1}} + {E_{l + m - 1}},$$
and
$${E_{l - m}} = \frac{1}{2}\pi (\cos {\theta _0} + 1)p(z){i^{l - m}}{J_{l - m}}(kr\sin {\theta _0})\exp [i(l - m)\varphi ]\exp ( - i{\varphi _0}),$$
$${E_{l + m}} = \frac{1}{2}\pi (\cos {\theta _0} + 1)p(z){i^{l + m}}{J_{l + m}}(kr\sin {\theta _0})\exp [i(l + m)\varphi ]\exp (i{\varphi _0}),$$
$${E_{l - m + 2}} = \frac{1}{2}\pi (\cos {\theta _0} - 1)p(z){i^{l - m + 2}}{J_{l - m + 2}}(kr\sin {\theta _0})\exp [i(l - m + 2)\varphi ]\exp ( - i{\varphi _0}),$$
$${E_{l + m - 2}} = \frac{1}{2}\pi (\cos {\theta _0} - 1)p(z){i^{l + m - 2}}{J_{l + m - 2}}(kr\sin {\theta _0})\exp [i(l + m - 2)\varphi ]\exp (i{\varphi _0}),$$
$${E_{l - m + 1}} = \pi \sin {\theta _0}p(z){i^{l - m + 1}}{J_{l - m + 1}}(kr\sin {\theta _0})\exp [i(l - m + 1)\varphi ]\exp ( - i{\varphi _0}),$$
$${E_{l + m - 1}} = \pi \sin {\theta _0}p(z){i^{l + m - 1}}{J_{l + m - 1}}(kr\sin {\theta _0})\exp [i(l + m - 1)\varphi ]\exp (i{\varphi _0}),$$
where $p(z) ={-} if\sqrt {\cos {\theta _0}} \sin {\theta _0}\exp (ikz\sin {\theta _0})/\lambda $. By comparing the focal field expression with the incident field (Eq. (3)), the terms ${E_{l - m}}$, ${E_{l - m + 2}}$ and ${E_{l - m + 1}}$ are derived from the right-hand circularly vortex term (the first term in Eq. (3)), the terms ${E_{l\textrm{ + }m}}$, ${E_{l\textrm{ + }m - 2}}$ and ${E_{l\textrm{ + }m - 1}}$ are derived from the left-hand circularly vortex term (the second term in Eq. (3)), the subscript symbol indicates the vortex charge of focal field components. The change of the vortex charge in the focal field components indicates the change of the orbital angular momentum, and it also means that there is the orbital-to-spin angular momentum conversion in the tight focusing of the beam.

Figure 2 shows the intensity evolution in focal plane of tight focusing of radially polarized beams with the different polarization order ($m$) and vortex charge ($l$). The calculation parameters are taken as, ${\theta _0} = {80^ \circ }$, $f = 120\lambda $. According to Eqs. (4)-(6), the focal field distribution is determined by the Bessel function with the different order which is related to the polarization order m and vortex charge l. Especially, the polarization order m and vortex charge l satisfy different specific relations, such as $l - m = 0$, or $l + m = 0$, or $l - m + 2 = 0$, or $l + m - 2 = 0$, the tight focusing of radially polarized vortex beam will display the different focal field properties. When the polarization order equals to 1 ($m = 1$), the intensity changes with the variation of initial phase ${\varphi _0}$, this is because the polarization state of incident vector beam changes from the radially polarized state to azimuthally polarized state; when the polarization order is greater than 1 ($m > 1$), the intensity just rotates with the variation of initial phase ${\varphi _0}$, this is because the polarization state just rotates which has been shown in Fig. 1. The intensity property in the focal plane is determined by the polarization state of incident vector beam. In next, we will study the separation of spin and the orbital angular momentum in the tight focusing of first-order radially polarized vortex beam.

 figure: Fig. 2.

Fig. 2. Intensity properties in the focal plane of tight focusing of radially polarized beams with the different polarization order, vortex charge $l$ and initial phase ${\varphi _0}$, (a1)-(a5) $m = 1$, $l = 1$, (b1)-(b5) $m = 1$, $l ={-} 1$, (c1)-(c5) $m = 2$, $l = 2$, (d1)-(d5) $m = 2$, $l ={-} 2$, (e1)-(e5) $m = 3$, $l = 1$, (f1)-(f5) $m = 3$, $l ={-} 1$, initial phase ${\varphi _0}\textrm{ = }0$, $\pi /4$, $\pi /2$, $3\pi /4$, $\pi $ (columns 1-5, respectively).

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3. Optical spin-orbit Hall effect in the tight focusing of a radially polarized vortex beam

The radially polarized vortex beam just possesses the orbital angular momentum which is determined by the vortex charge and doesn’t possess the spin angular momentum. From Eqs. (4)–(6), the transverse and the longitudinal components of the focal field can be rewritten as the form,

$$\begin{aligned} {{\mathbf E}_ \bot } &= {E_x}(r,\varphi ,z){{\mathbf e}_x} + {E_y}(r,\varphi ,z){{\mathbf e}_y}\\ &= {E_{l - m}}({{{\mathbf e}_x} + i{{\mathbf e}_y}} )+ {E_{l + m}}({{{\mathbf e}_x} - i{{\mathbf e}_y}} )+ {E_{l - m + 2}}({{{\mathbf e}_x} - i{{\mathbf e}_y}} )+ {E_{l + m - 2}}({{{\mathbf e}_x} + i{{\mathbf e}_y}} ), \end{aligned}$$
$${{\mathbf E}_\parallel } = ({ - {E_{l - m + 1}} + {E_{l + m - 1}}} ){{\mathbf e}_z}.$$

The optical angular momentum of four terms in the transverse field component can be expressed as $({l - m + {\sigma^ + }} )\hbar $, $({l\textrm{ + }m + {\sigma^ - }} )\hbar $, $({l - m + 2\textrm{ + }{\sigma^ - }} )\hbar $, $({l\textrm{ + }m - 2 + {\sigma^ + }} )\hbar $, respectively. The optical angular momentum of two terms in the longitudinal field component can be expressed as $({l - m + {\sigma^ + }} )\hbar $ and $({l\textrm{ + }m - {\sigma^ - }} )\hbar $ respectively, which are same as the optical angular momentum of incident vector beam. Though there is the optical angular momentum conversion between the field components, the total optical angular momentum is conserved. Furthermore, we find that, when the polarization order equals to 1 ($m = 1$), the longitudinal field component possesses the same optical angular momentum as the incident beam. Namely, when $m = 1$, the optical angular momentum conversion just occurs between the transverse focal field components.

According to the definition of the spin angular momentum density ${J_s} \propto {\mathop{\rm Im}\nolimits} [{{{\mathbf E}^\ast } \times {\mathbf E}} ]$ and the orbital angular momentum density ${J_L} \propto {\mathop{\rm Im}\nolimits} [{{{\mathbf E}^\ast } \cdot (\nabla ){\mathbf E}} ]$ [27], Fig. 3 shows the longitudinal spin (${J_{SZ}}$) and orbital (${J_{LZ}}$) angular momentum density in the focal plane when the incident radially polarized beams possess the different polarization order and vortex charge. The incident radially polarized vortex beam just possesses the orbital angular momentum. hence the spin angular momentum in the focal plane comes from the orbital-to-spin angular momentum conversion. It is worth noting that, when the polarization order equals to 1, the peak of the spin and the orbital angular momentum density locate in the central area and inner ring area of the focal plane, its magnitude is modulated by the initial phase ${\varphi _0}$; yet, when the polarization order is greater than 1, the spin and the orbital angular momentum density just rotate around the axis with the variation of initial phase ${\varphi _0}$, its magnitude and whole distribution property is unchanged.

 figure: Fig. 3.

Fig. 3. Longitudinal spin angular momentum density (first and third rows) and orbital angular momentum density (second and fourth rows) in the focal plane of tight focusing of radially polarized beams with the different polarization order, vortex charge, (a1)-(a5) and (b1)-(b5), $m = 1$, $l = 1$, (c1)-(c5) and (d1)-(d5) $m = 2$, $l = 2$, initial phase ${\varphi _0}\textrm{ = }0$, $\pi /4$, $\pi /2$, $3\pi /4$, $\pi $ (columns 1-5, respectively).

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The realization of the optical spin-orbit Hall effect requires the spatial separation of the spin and orbital angular momentum density firstly. Figure 3 has shown that, when the polarization order equals to 1, the spatial separation of the spin and the orbital angular momentum density in the focal plane can be obtained. Especially, when the polarization order and the vortex charge satisfy the specific relations simultaneously, $l - m = 0$ and $l + m - 2 = 0$ (or $l + m = 0$ and $l - m + 2 = 0$), the transverse field components are determined by the zero-order and the second-order Bessel functions. Because of the distribution property of the zero-order and the second-order Bessel functions, the longitudinal spin and orbital angular momentum density will be separated in the focal plane. When the polarization order is greater than 1, such as $m = 2$ and $l = 2$ in Fig. 3, we see that the spin and the orbital angular momentum just rotate with the change of the initial phase ${\varphi _0}$. Though the separation of the spin and the orbital angular momentum density also is shown in Fig. 3, it can’t correspond to the separation of the intensity distribution in the focal plane. Then, the spin-orbit Hall effect can’t be obtained when the polarization order of incident polarized vortex beam is greater than 1.

Figure 4 shows the evolution of normalized longitudinal spin and orbital angular momentum density in the focal plane with the change of initial phase ${\varphi _0}$ when $m = 1$ and $l = 1$ (or $- 1$), the sign of the maximum value of the spin and the orbital angular momentum is determined by the vortex charge. According to Eqs. (13) and (14), we know, when the polarization order equals to 1, the spin-to-orbital angular momentum just occurs in the transverse focal field components, and the total optical angular momentum is conserved. We see that, the spin angular momentum density is concentrated in the central area; the orbital angular momentum density is concentrated in the inner ring area. These two areas are not overlap. From the results shown in Fig. 3, the magnitudes of longitudinal spin and orbital angular momentum density in the focal plane are different. Though, both two kinds of optical angular momentum density can be effectively modulated by adjusting the initial phase ${\varphi _0}$. This provides the essential conditions for the occurrence of the optical spin-orbit Hall effect in the tight focusing of the first-order polarized vortex beam.

 figure: Fig. 4.

Fig. 4. Evolution of normalized longitudinal spin angular momentum density (left column) and orbital angular momentum (right column) with the change of initial phase ${\varphi _0}$, red line ${\varphi _0} = 0$, green line ${\varphi _0} = \pi /4$, orange line ${\varphi _0} = \pi /2$, black dotted line ${\varphi _0} = 3\pi /4$, blue dotted line ${\varphi _0} = \pi $.

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In order to observe the spin-orbit Hall effect experimentally, the spatial separation of the spin and the orbital angular momentum density should correspond to the spatial separation of the intensity distribution. Figure 5 shows that, when the initial phase takes $\pi /2$, the separation of the longitudinal spin and orbital angular momentum density corresponds to the intensity separation in the focal plane. Obviously, the central area of intensity distribution corresponds to the longitudinal spin angular momentum density part; the first inner ring of intensity corresponds to the longitudinal orbital angular momentum density. Then, the optical spin-orbit Hall effect is realized in the tight focusing of the first-order polarized vortex beam. It is well-known that, when the spin angular momentum interacts with the small particle, it induces the particle spinning around its own axis; when the orbital angular momentum interacts with the small particle, it induces the particle orbiting around the beam axis. It indicates that both two kinds of particle manipulation effects can be realized in the tight focusing of radially polarized vortex beam.

 figure: Fig. 5.

Fig. 5. Split of longitudinal spin (red line) and orbital (yellow line) angular momentum density distribution in the focal plane when the initial phase ${\varphi _0} = \pi /2$ (a)$m = 1$, $l = 1$, (b) $m = 1$, $l ={-} 1$.

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4. Conclusion

We have reexamined the tight focusing of radially polarized vortex beam, and showed the separation of the longitudinal spin and orbital angular momentum density occurs in the tight focusing of the first-order polarized vortex beam. Just like the optical spin Hall effect which displays the spatial separation of the right-hand and left-hand circular polarization parts, the separation of the spin and the orbital angular momentum parts can be considered as the manifestation of the optical spin-orbit Hall effect in the tight focusing of the optical vector beams. We found both the longitudinal spin and orbital angular momentum density in the focal plane can be modulated by the initial phase of polarization state of incident beam, and the spatial separation of the longitudinal spin and orbital angular momentum density in the focal plane corresponds to the spatial separation of the intensity distribution in the focal plane when the initial phase takes $\pi /2$. Besides, we show that, when the polarization order is greater than 1, the initial phase change of polarization state just leads to the rotation of the focal field, so are the spin and the orbital angular momentum density in the focal plane. It means the optical spin-orbit Hall effect just occurs in the tight focusing of the first-order radially polarized vortex beams. The spatial separation of spin and orbital angular momentum density can be utilized in the field of the particle manipulation.

Funding

National Natural Science Foundation of China (11974101, 11974102, 121274089).

Disclosures

The authors declare no conflicts of interest.

Data availability

No data were generated or analyzed in the presented research.

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Data availability

No data were generated or analyzed in the presented research.

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Figures (5)
Fig. 1.
Fig. 1. Polarization state evolution of radially polarized beam along the equator on Poincaré sphere.

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Fig. 2.
Fig. 2. Intensity properties in the focal plane of tight focusing of radially polarized beams with the different polarization order, vortex charge $l$ and initial phase ${\varphi _0}$, (a1)-(a5) $m = 1$, $l = 1$, (b1)-(b5) $m = 1$, $l ={-} 1$, (c1)-(c5) $m = 2$, $l = 2$, (d1)-(d5) $m = 2$, $l ={-} 2$, (e1)-(e5) $m = 3$, $l = 1$, (f1)-(f5) $m = 3$, $l ={-} 1$, initial phase ${\varphi _0}\textrm{ = }0$, $\pi /4$, $\pi /2$, $3\pi /4$, $\pi $ (columns 1-5, respectively).

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Fig. 3.
Fig. 3. Longitudinal spin angular momentum density (first and third rows) and orbital angular momentum density (second and fourth rows) in the focal plane of tight focusing of radially polarized beams with the different polarization order, vortex charge, (a1)-(a5) and (b1)-(b5), $m = 1$, $l = 1$, (c1)-(c5) and (d1)-(d5) $m = 2$, $l = 2$, initial phase ${\varphi _0}\textrm{ = }0$, $\pi /4$, $\pi /2$, $3\pi /4$, $\pi $ (columns 1-5, respectively).

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Fig. 4.
Fig. 4. Evolution of normalized longitudinal spin angular momentum density (left column) and orbital angular momentum (right column) with the change of initial phase ${\varphi _0}$, red line ${\varphi _0} = 0$, green line ${\varphi _0} = \pi /4$, orange line ${\varphi _0} = \pi /2$, black dotted line ${\varphi _0} = 3\pi /4$, blue dotted line ${\varphi _0} = \pi $.

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Fig. 5.
Fig. 5. Split of longitudinal spin (red line) and orbital (yellow line) angular momentum density distribution in the focal plane when the initial phase ${\varphi _0} = \pi /2$ (a)$m = 1$, $l = 1$, (b) $m = 1$, $l ={-} 1$.

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Equations (17)

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[ c x ( ϕ ) c y ( ϕ ) ] = R ( φ 0 ) [ cos ⁡ ( m ϕ ) sin ⁡ ( m ϕ ) ] = [ cos ⁡ φ 0 − sin ⁡ φ 0 sin ⁡ φ 0 cos ⁡ φ 0 ] [ cos ⁡ ( m ϕ ) sin ⁡ ( m ϕ ) ] ,
E ( r , φ , z ) = − i f λ ∫ 0 α ∫ 0 2 π T ( θ ) F ( θ , ϕ ) P E ( 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 ) ] R ( t ) [ 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 ,
E i n = F ( θ 0 , ϕ ) R ( φ 0 ) [ c x ( ϕ ) c y ( ϕ ) ] = 1 2 R ( φ 0 ) { exp ⁡ [ i ( l − m ) ϕ ] [ 1 i ] + exp ⁡ [ i ( l + m ) ϕ ] [ 1 − i ] } .
E x ( r , φ , z ) = E l − m + E l + m + E l − m + 2 + E l + m − 2 ,
E y ( r , φ , z ) = i ( E l − m − E l + m ) − i ( E l − m + 2 − E l + m − 2 ) ,
E z ( r , φ , z ) = − E l − m + 1 + E l + m − 1 ,
E l − m = 1 2 π ( cos ⁡ θ 0 + 1 ) p ( z ) i l − m J l − m ( k r sin ⁡ θ 0 ) exp ⁡ [ i ( l − m ) φ ] exp ⁡ ( − i φ 0 ) ,
E l + m = 1 2 π ( cos ⁡ θ 0 + 1 ) p ( z ) i l + m J l + m ( k r sin ⁡ θ 0 ) exp ⁡ [ i ( l + m ) φ ] exp ⁡ ( i φ 0 ) ,
E l − m + 2 = 1 2 π ( cos ⁡ θ 0 − 1 ) p ( z ) i l − m + 2 J l − m + 2 ( k r sin ⁡ θ 0 ) exp ⁡ [ i ( l − m + 2 ) φ ] exp ⁡ ( − i φ 0 ) ,
E l + m − 2 = 1 2 π ( cos ⁡ θ 0 − 1 ) p ( z ) i l + m − 2 J l + m − 2 ( k r sin ⁡ θ 0 ) exp ⁡ [ i ( l + m − 2 ) φ ] exp ⁡ ( i φ 0 ) ,
E l − m + 1 = π sin ⁡ θ 0 p ( z ) i l − m + 1 J l − m + 1 ( k r sin ⁡ θ 0 ) exp ⁡ [ i ( l − m + 1 ) φ ] exp ⁡ ( − i φ 0 ) ,
E l + m − 1 = π sin ⁡ θ 0 p ( z ) i l + m − 1 J l + m − 1 ( k r sin ⁡ θ 0 ) exp ⁡ [ i ( l + m − 1 ) φ ] exp ⁡ ( i φ 0 ) ,
E ⊥ = E x ( r , φ , z ) e x + E y ( r , φ , z ) e y = E l − m ( e x + i e y ) + E l + m ( e x − i e y ) + E l − m + 2 ( e x − i e y ) + E l + m − 2 ( e x + i e y ) ,
E ∥ = ( − E l − m + 1 + E l + m − 1 ) e z .
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