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Inversion of the axial projection of the spin angular momentum in the region of the backward energy flow in sharp focus

Victor V. Kotlyar, Anton G. Nalimov, and Sergey S. Stafeev

Open Access Open Access

Abstract

We show theoretically and numerically that when strongly focusing a circularly polarized optical vortex, the longitudinal component of its spin angular momentum undergoes inversion. A left-handed circularly polarized input beam is found to convert in the focus and near the optical axis to a right-handed circularly polarized beam. Thanks to this effect taking place near the strong focus, where a reverse energy flow is known to occur, the spin angular momentum inversion discovered can be utilized to detect a reverse energy flow.

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

1. Introduction

There have been a number of publications [1–6] reporting on the observation of orbital angular momentum in a strong focus of a conventional circularly polarized Gaussian beam, which is then transferred to a microparticle placed in the focal spot. This phenomenon is associated with the conversion of spin angular momentum (SAM) into the orbital angular momentum (OAM) and referred to as spin-to-orbital conversion. Being devoid of OAM, the original beam only carries the on-axis projection of the SAM vector. In the meantime, in the strong focus, a transverse energy flow is observed leading to a non-zero longitudinal projection of the OAM. Although spin-to-orbital conversion was studied [1–6], no analytical description of SAM and OAM within Richards-Wolf formalism [7] has been proposed so far. The Richards-Wolf theory (RWT) [7], which offers the description of an electromagnetic field near a strong focus, has been extended onto a near-focus planar interface [8]. The RWT has also been extended onto the description of strongly focused cylindrical vector beams [9]. Longitudinal components of SAM and OAM in the strong focus of a circularly polarized light field with topological charge have also been calculated based on the RWT [10]. In Ref. [11], near-focus spin-orbital interaction amplification in a layered medium with two interfaces was studied using the RWT. The Hall spin effect in the strong focus of a Gaussian beam traveling in a stratified medium similar to that in Ref. [11] has also been discussed [12]. However, these publications [8–12] offered no RWT-aided analytical relationships to describe all components of the OAM and SAM, using which it would be possible to identify the presence of a reverse energy flow in the tight focus. The Poynting vector (energy flow) has been represented as a sum of the orbital energy flow and spin flow [13–17]. Although the spin flow carries no energy, it can be measured [4,18]. The most efficient approach to analyzing light fields near the strong focus is through the use of the RWT [7] as it enables all components of such fields to be described in a closed form (i.e., doing without the series) for an arbitrary optical field at the input of an aplanatic optical system. We note that the larger is the focal length when compared with the incident wavelength, the more accurate are the RWT-aided results. The use of the RWT has made it possible to deduce a variety of physical parameters of the field in the strong focus [8–12], including the energy flow density and spin flow density, orbital angular momentum, and spin angular momentum. There are also other, both approximate [19] and rigorous [20–22] methods to describe the electromagnetic field in the vicinity of the strong focus. Although these methods are accurate, they offer solutions for the electric and magnetic field components either in the form of infinite series [20] or a finite number of cumbersome terms containing special functions [21,22]. Theoretical analysis of the exact solutions of Maxwell’s equations in the focus is a challenging problem. In addition, while there is a limited number of known exact solutions of Maxwell’s equations, the RWT makes it possible to derive relatively simple closed expressions to describe the near-focus field for a wide variety of incident light fields, including Gaussian and Bessel-Gaussian beams, light fields with integer topological charge, as well as homogeneously (linearly, circularly, and elliptically) and inhomogeneously (azimuthally and radially) polarized optical fields. The said considerations have prompted us to utilize the RWT in this work. A circularly polarized plane wave, which has only the on-axis projection of SAM, is able to set in motion an absorbing spherical microparticle [23] and generate a magnetic field in non-magnetic dispersive media (magnetization effect) [24]. Optical low-loss levitation of spherical quartz microparticles in high vacuum has been demonstrated [25]. Movement of dielectric microparticles on a circular path in tightly focused laser beams has also been studied [11,26]. An absorbing asymmetric microparticle placed in a side-lobe of the diffraction pattern has been demonstrated to rotate about its center of mass [12]. Thanks to the presence of SAM, the light field is able to rotate arbitrary absorbing and birefrigent microparticles, including spherical ones [23]. Rotation of a weakly absorbing spherical particle around its axis in the focus of a circularly polarized Gaussian beam has been theoretically shown [27]. The decomposition of the Poynting vector into the orbital and spin components makes it possible to offer an interpretation of the effect of a reverse energy flow in the tight focus of a laser beam [28,29]. For example, in Ref. [30], the reverse energy flow (the negative on-axis projection of the Poynting vector) was exclusively associated with a phase (vortex) singularity of the light field. However, this is not quite the case as the reverse energy flow has also been shown to occur in the focus of a non-vortex light beam [29]. The reverse energy flow, which has been known in optics since 1919 [31,7], is a fairly universal optical phenomenon that can occur not only in a tight focus but also in some laser beams, such as vector X-waves [32], nonparaxial Airy beams [33], and fractional vortex Bessel beams [34]. The reverse energy flow occurs in optical field regions where the on-axis projection of the spin flow is negative and larger in magnitude than the always positive on-axis projection of the orbital energy flow. In Ref. [35], the rigorous solution of a non-paraxial Helmholtz equation in spherical coordinates was derived in the waist plane, where the non-paraxial focus was assumed to be located. Topics analyzed included intensity patterns of higher-order modes with vortex phase in the focal region and spin-orbital conversion in multi-mode fibers. However, the behavior of the Poynting vector and SAM in the strong focus was beyond the scope of Ref. [35].

In this work, we discuss another interesting optical phenomenon that has been overlooked so far. This is the inversion of the longitudinal component of the SAM vector in the tight focus that occurs in the presence of the on-axis reverse energy flow. Earlier, the on-axis reverse energy flow was shown to occur in the tight focus of a left-handed circularly polarized optical vortex with topological charge 2 [28,29]. Below, we deduce projections of the SAM vector in the focus with a reverse energy flow, demonstrating that the longitudinal component of SAM is positive, which means that while the incident light has the left-handed circular polarization, the light in the focus is right-handed circularly polarized.

2. Spin angular momentum in the focus of a circularly polarized Gaussian beam

For comparison, let us discuss focusing a conventional circularly polarized Gaussian beam. Using the Richards-Wolf formalism [7], it is possible to derive relationships to describe projections of the electric vector in the tight focus of an aplanatic system. For instance, for an incident circularly polarized light field

$$\textbf{E} = \frac{{A(\theta )}}{{\sqrt 2 }}\left( \begin{array}{l} 1\\ i\sigma \end{array} \right),\quad \textbf{H} = \frac{{A(\theta )}}{{\sqrt 2 }}\left( \begin{array}{l} - i\sigma \\ \;\;1 \end{array} \right), $$
where σ=1 for right-handed circular polarization, σ=−1 for left-handed circular polarization, E and Н are the electric and magnetic field vectors, then, in the focal plane the projections of the E-vector will take the form [6], for right-handed circular polarization:
$$\begin{aligned} {E_{xR}} &= \frac{{ - i}}{{\sqrt 2 }}({{I_{0,0}} + {e^{2i\varphi }}{I_{2,2}}} ),\\ {E_{yR}} &= \frac{1}{{\sqrt 2 }}({{I_{0,0}} - {e^{2i\varphi }}{I_{2,2}}} ),\\ {E_{zR}} &={-} \sqrt 2 {e^{i\varphi }}{I_{1,1}} \end{aligned}$$
and for left-handed circular polarization:
$$\begin{aligned} {E_{xL}} &= \frac{{ - i}}{{\sqrt 2 }}({{I_{0,0}} + {e^{ - 2i\varphi }}{I_{2,2}}} ),\\ {E_{yL}} &= \frac{1}{{\sqrt 2 }}({ - {I_{0,0}} + {e^{ - 2i\varphi }}{I_{2,2}}} ),\\ {E_{zL}} &={-} \sqrt 2 {e^{ - i\varphi }}{I_{1,1}}, \end{aligned}$$
where
$${I_{\nu ,\mu }} = \left( {\frac{{\pi f}}{\lambda }} \right)\int\limits_0^{{\theta _0}} {{{\sin }^{\nu + 1}}(\frac{\theta }{2}){{\cos }^{3 - \nu }}(\frac{\theta }{2})} {\cos ^{1/2}}(\theta )A(\theta ){e^{ikz\cos z}}{J_\mu }(x)d\theta ,$$
where λ is the incident wavelength, f is the focal length of an aplanatic system, x = krsinθ, Jµ(x) is a Bessel function of the first kind, an NA = sinθ0 is the numerical aperture. The (real) initial amplitude function A(θ) may be given by a constant (plane wave) or a Gaussian beam
$$A(\theta ) = \textrm{exp} \left( {\frac{{ - {\gamma^2}{{\sin }^2}\theta }}{{{{\sin }^2}{\theta_0}}}} \right). $$
Let us derive projections of the SAM vector:
$$\textbf{S} = \frac{1}{2}{\mathop{\rm Im}\nolimits} [{\textbf{E}^{\ast}{\times} \textbf{E}} ], $$
where Im is the imaginary part of the number, E* denotes the complex conjugation of the E-vector. Substituting (2) into (6) and considering that the integrals in (4) are real functions, we obtain in the focal plane (z=0), for the right-handed circular polarization:
$$\begin{aligned} {S_{xR}} &={-} ({{I_{0,0}} + {I_{2,2}}} ){I_{1,1}}\sin \varphi ,\\ {S_{yR}} &= ({{I_{0,0}} + {I_{2,2}}} ){I_{1,1}}\cos \varphi ,\\ {S_{zR}} &= \frac{1}{2}({I_{0,0}^2 - I_{2,2}^2} ), \end{aligned}$$
and for the left-handed circular polarization:
$$\begin{aligned} {S_{xL}} &={-} ({{I_{0,0}} + {I_{2,2}}} ){I_{2,2}}\sin \varphi ,\\ {S_{yL}} &= ({{I_{0,0}} + {I_{2,2}}} ){I_{1,1}}\cos \varphi ,\\ {S_{zL}} &={-} \frac{1}{2}({I_{0,0}^2 - I_{2,2}^2} ). \end{aligned}$$
The comparison of (7) and (8) suggests that changing the circular polarization helicity causes no changes in both the helicity and magnitude of the transverse SAM component in the focus. Meanwhile, the longitudinal SAM component only changes sign. What is of interest for us in this situation is that when focusing a left-handed circular polarized light wave, the wave helicity in the focus does not change. For left-handed circular polarization, the polarization vector rotates clockwise, meaning that the longitudinal SAM projection is negative, Eq. (8). Meanwhile, for right-handed circular polarization, the polarization vector rotates anticlockwise, meaning that the longitudinal SAM component is positive, Eq. (7). It should be noted that the input left-handed circular polarization, remains left-handed circular polarization near the focus, while the input right-handed circular polarization likewise remains right-handed in the focal plane. From (7) and (8), it follows that on the optical axis (r=0), the longitudinal SAM projections are maximal in the absolute value. Thus, if a low-absorbing spherical microparticle is placed in the focus on the optical axis, it will be set in motion due to SAM around the optical axis and its center of mass [27]. Left-handed incident polarization of Eq. (8) will rotate it clockwise and right-handed circular polarization of Eq. (7) – anticlockwise.

Below, we shall demonstrate that when focusing a circularly polarized optical vortex, the longitudinal SAM component in the focus changes it sign relative to the sign of the SAM of the incident beam.

3. SAM in the focus of a circularly polarized optical vortex with topological charge 2

Previously, it has been demonstrated [28,29] that the absolute value of the on-axis intensity maximum of the reverse energy flow in the strong focus of a left-handed circularly polarized light can only be attained when focusing an optical vortex with the topological charge m=2. At m>2, the reverse energy flow shapes an optical pipe. At m=0 and m=1, no reverse energy flow occurs in the focus. Because of this, in this section, we analyze focusing an optical vortex with m=2.

If instead of Eq. (1), the incident field at the aplanatic system input is given by

$$\textbf{E} = \frac{{A(\theta ){e^{im\varphi }}}}{{\sqrt 2 }}\left( \begin{array}{l} 1\\ i\sigma \end{array} \right),\quad \textbf{H} = \frac{{A(\theta ){e^{im\varphi }}}}{{\sqrt 2 }}\left( \begin{array}{l} - i\sigma \\ \;\;1 \end{array} \right), $$
where m is the integer topological charge of the vortex, the use of the RWT makes it possible to derive projections of the electric vector in the tight focus plane, for right-handed circular polarization and m=2:
$$\begin{aligned} {E_{xR}} &= \frac{i}{{\sqrt 2 }}({{e^{4i\varphi }}{I_{2,4}} + {e^{2i\varphi }}{I_{0,2}}} ),\\ {E_{yR}} &= \frac{1}{{\sqrt 2 }}({{e^{4i\varphi }}{I_{2,4}} - {e^{2i\varphi }}{I_{0,2}}} ),\\ {E_{zR}} &= \sqrt 2 {e^{3i\varphi }}{I_{1,3}}, \end{aligned}$$
and left-handed circular polarization
$$\begin{aligned} {E_{xL}} &= \frac{i}{{\sqrt 2 }}({{I_{2,0}} + {e^{2i\varphi }}{I_{0,2}}} ),\\ {E_{yL}} &= \frac{1}{{\sqrt 2 }}({ - {I_{2,0}} + {e^{2i\varphi }}{I_{0,2}}} ),\\ {E_{zL}} &={-} \sqrt 2 {e^{i\varphi }}{I_{1,1}}. \end{aligned}$$
The topological charge is taken to be m=2 because only at this value a reverse energy flow will occur near the focus on the optical axis for a left-handed circularly polarized optical vortex [28]. Actually, projections of the Poynting vector (energy flow) $\textbf{P} = \frac{1}{2}{\rm{Re}} [{\textbf{E}^{\ast}{\times} \textbf{H}} ]$ take the following form in the focus of an optical vortex (m=2), for right-handed circular polarization
$$\begin{aligned} {P_{xR}} &={-} {I_{1,3}}({{I_{0,2}} + {I_{2,4}}} )\sin \varphi ,\\ {P_{yR}} &= {I_{1,3}}({{I_{0,2}} + {I_{2,4}}} )\cos \varphi ,\\ {P_{zR}} &= \frac{1}{2}({I_{0,2}^2 - I_{2,4}^2} ), \end{aligned}$$
and for left-handed polarization (m=2):
$$\begin{aligned} {P_{xL}} &={-} {I_{1,1}}({{I_{0,2}} + {I_{2,0}}} )\sin \varphi ,\\ {P_{yL}} &= {I_{1,1}}({{I_{0,2}} + {I_{2,0}}} )\cos \varphi ,\\ {P_{zL}} &= \frac{1}{2}({I_{0,2}^2 - I_{2,0}^2} ). \end{aligned}$$
From (12), it follows that the transverse energy flow in the focus of a right-handed circularly polarized optical vortex rotates anticlockwise, while the longitudinal on-axis projection of the Poynting vector equals zero and is positive near the axis. From (13), it follows that the transverse energy flow from a left-handed circular polarized optical vortex also rotates anticlockwise (as is the case for the right-handed circularly polarized vortex), while the on-axis longitudinal projection of the Poynting vector is maximum in the absolute value and negative (${P_{zL}}(z = r = 0) ={-} I_{2,0}^2/2$). A dielectric (absorbing or non-absorbing) particle placed in the focal spot will rotate on a circle in anticlockwise direction. What is remarkable is that the rotation direction does not change when the helicity of the incident beam is changed [36].

Next, when focusing optical vortices with topological charge m=2, the SAM projections can be written down as, for right-handed circular polarization:

$$\begin{aligned} {S_{xR}} &={-} {I_{1,3}}({{I_{0,2}} + {I_{2,4}}} )\sin \varphi ,\\ {S_{yR}} &= {I_{1,3}}({{I_{0,2}} + {I_{2,4}}} )\cos \varphi ,\\ {S_{zR}} &= \frac{1}{2}({I_{0,2}^2 - I_{2,4}^2} ), \end{aligned}$$
and, for left-handed circular polarization:
$$\begin{aligned} {S_{xL}} &= {I_{1,1}}({{I_{2,0}} + {I_{0,2}}} )\sin \varphi ,\\ {S_{yL}} &={-} {I_{1,1}}({{I_{2,0}} + {I_{0,2}}} )\cos \varphi ,\\ {S_{zL}} &= \frac{1}{2}({I_{2,0}^2 - I_{0,2}^2} ). \end{aligned}$$
From (14), it follows that for the right-handed circular polarization, the transverse component of the SAM vector rotates about optical axis anticlockwise, while the longitudinal on-axis projection of SAM in the focus is zero, being positive near the optical axis. From (15), it follows that for the left-handed circular polarization the transverse component of SAM rotates clockwise, while the longitudinal on-axis projection is positive in the focus: (${S_{zL}}(z = r = 0) = I_{2,0}^2/2$).

Hence, we can infer that there is an optical effect of the longitudinal SAM component inversion, which has gone unnoticed so far. The essence of the effect is that while the incident left-handed circularly polarized light field of Eq. (9) has the negative on-axis SAM projection, in the near-axis focal region it is inverted, becoming positive. In other words, while the polarization vector of the incident left-handed circularly polarized field rotates clockwise, it starts rotating anticlockwise in the focus, which is characteristic of right-handed circular polarization. Occurring in parallel with a reverse energy flow, the SAM inversion effect can be utilized as a means of detecting the former. Actually, from a comparison of the longitudinal projections of the Poynting vector (13) and the SAM vector (15), they are seen to be equal in magnitude but opposite in sign:

$${P_{zL}} ={-} {S_{zL}} ={-} \frac{1}{2}({I_{2,0}^2 - I_{0,2}^2} ).$$
Summing up, it is exactly in the near-axis focal region where a reverse energy flow occurs SAM inversion also takes place. So, under the impact of this SAM, an absorbing spherical microparticle placed in the focus centered on the optical axis may be expected to start rotating about the optical axis anticlockwise, although the incident wave has clockwise circular polarization.

We note that the expressions for the Poynting vector and SAM derived in Sections 2 and 3 are valid for any numerical aperture (any NA = sinθ0 of Eq. (4)) and any real radially symmetric amplitude of the input field (any function A(θ) > 0 of Eq. (4)).

Let us also remind that the Poynting vector is a sum of the orbital energy flow Por and of the spin flow Psp [14]:

$$\textbf{P} = \frac{{\rm{Re} }}{2}({{\textbf{E}^\ast } \times \textbf{H}} )= {\textbf{P}_{or}} + {\textbf{P}_{sp}},$$
$${\textbf{P}_{or}} = \frac{{{\mathop{\rm Im}\nolimits} }}{{2k}}({{\textbf{E}^\ast }\cdot ({\nabla \textbf{E}} )} ),\quad {\textbf{P}_{sp}} = \frac{1}{{4k}}({\nabla \times {\mathop{\rm Im}\nolimits} ({{\textbf{E}^\ast } \times \textbf{E}} )} ).$$
Using the notion of the two flows in Eq. (18), the generation of a reverse energy flow can be explained as follows. Since the orbital (canonical) energy flow is always positive (Por,z >0), the reverse energy flow in Eq. (17) (Pz <0) will be generated if the spin flow is negative (Psp,z <0) and larger in the absolute value than the orbital flow ($|{{P_{sp,z}}} |> {P_{or,z}}$). From Eq. (18), the spin flow is the curl of the SAM vector. Hence, the direction of Psp,z can be identified using a right-hand rule. From Eq. (15), the transverse vector SAM is seen to rotate clockwise in the focal plane (at φ=0, ${S_{yL}}$<0), which means that the on-axis spin flow is negative (Psp,z <0). However, Eqs. (15) and (18) do not answer the question: which on-axis value will be larger, the spin flow $|{{P_{sp,z}}} |$ or the orbital flow ${P_{or,z}}$. In the meantime, what unambiguously follows from Eq. (14) is that the on-axis spin flow in the focus will be positive for right-handed circular polarization of incident light (Psp,z >0). Hence, we infer that when focusing right-handed circularly polarized light, no on-axis reverse energy flow will occur.

4. Numerical simulation

To verify the theoretical predictions, a rigorous numerical modeling was conducted. Maxwell’s equations were numerically solved using a FDTD-aided method in the software RSoft FullWAVE for an incident wavelength of λ=0.633 µm and an incident wave aperture of D=8 µm. The incident field was modeled as a right- and left-handed circularly polarized plane wave with unit-amplitude and phase singularity, which was multiplied by a spherical wave converging at a distance of f=1.31 µm (numerical aperture is NA=0.95). The E-vector projections were, for a left-handed circular wave:

$$\begin{aligned} {E_x} &= \textrm{exp} ({i({kf - m\varphi - \omega t} )} ),\\ {E_y} &= \textrm{exp} (i\left( {kf - m\varphi - \omega t - \frac{\pi }{2}} \right)) \end{aligned}$$
and, for a right-handed circular wave:
$$\begin{aligned} {E_x} &= \textrm{exp} ({i({kf - m\varphi - \omega t} )} ),\\ {E_y} &= \textrm{exp} (i\left( {kf - m\varphi - \omega t + \frac{\pi }{2}} \right)) \end{aligned}$$
Here, m=2 denotes the topological charge of the optical vortex, k=2π/λ, and ω is the cyclic frequency. The unit vectors formed a right-hand triplet.

Figure 1 depicts distributions of the longitudinal SAM projections in the focus and their radial cross-sections for an initial optical vortex with right-handed (a, c) and left-handed (b, d) circular polarizations. Figure 1 suggests that with an incident left-handed circular polarized wave with phase singularity m=2, there occurs a central region of about 0.3-µm in diameter where the SAM vector is positive, which implies that in this micro-region there is a right-handed circularly polarized field. Because of this, a spherical microparticle with a small imaginary component of the refractive index (with absorption) put in this region will rotate anticlockwise about the optical axis and its center, notwithstanding the left-handed circularly polarized incident field. From Figs. 1(b) and 1(d), the longitudinal SAM component is seen to be negative at r>0.3 µm, meaning that the polarization vector rotates clockwise, as is the case with a left-handed circularly polarized wave. If placed in the region where r>0.3 µm, the same microparticle will rotate clockwise.

 figure: Fig. 1.

Fig. 1. On-axis projections of SAM Sz in the focal plane for an incident field with (a) right-handed and (b) left-handed circular polarization and phase singularity of order m=2, (c,d) their respective profiles along the x axis and profiles of the Z-axis projections of the Poynting vector P for right-handed (e) and left-handed (f) circular polarizations. Superimposed on the plot in Fig. 1(d) is the beam intensity pattern ${|E |^2}$ (dashed line).

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Shown in Figs. 1(e) and 1(f) are radial profiles of the on-axis projection Pz of the Poynting vector when focusing a right-handed (Fig. 1(e)) and left-handed (Fig. 1(f)) circularly polarized vortex (m=2). An on-axis reverse energy flow can be seen to occur only for the left-handed circularly polarized incident beam (Fig. 1(f)). In Fig. 1(f), the reverse energy flow is seen to occur in a region of about 300 nm, with the region of SAM inversion (Sz >0) being near-same in size (Fig. 1(d)). We note that earlier [37] the reverse energy flow region was experimentally measured with a microlens with NA=0.95 for an incident wavelength of 532 nm, which was found to be about 300 nm.

Figures 1(a) and 1(b) depicts results of the numerical computation of Sz using the rigorous FDTD-based solution of Maxwell's equations, whereas for comparison purposes, the same patterns derived using Richards-Wolf formulae (7) and (8) are also shown in Figs. 2(a) and 2(b). The patterns in Figs. 1(a), 1(b), 2(a), and 2(b) are seen to be in agreement.

 figure: Fig. 2.

Fig. 2. On-axis projections of SAM Sz in the focal plane for an incident field with (a) right-handed and (b) left-handed circular polarization and phase singularity of order m=2 calculated by Richards-Wolf formulas, and (c,d) their respective profiles along the x axis.

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To validate the above hypothesis, we calculated a torque exerted upon a 250-nm bead whose center was in the focal plane. First, the diffraction of a light wave with initial amplitude (18) by a dielectric bead and all projections of the electromagnetic field near the nano-bead were calculated. The light field was calculated using an FDTD method. The torque was calculated using a Maxwell’s stress tensor:

$$\textbf{M} = \oint\limits_\Omega {[{\textbf{r} \times ({\sigma \cdot \textbf{n}} )} ]d\Omega }, $$
where n is normal to the surface Ω, which embraces the bead, r is the radius-vector drawn to the surface Ω from the bead center, relative to which the torque was calculated, and σ is the Maxwell’s stress tensor, whose components in the SGS system take the form:
$${\sigma _{ik}} = \frac{1}{{4\pi }}\left( {\frac{{{{|\textbf{E} |}^2} + {{|\textbf{H} |}^2}}}{2}{\delta_{ik}} - {E_i}{E_k} - {H_i}{H_k}} \right), $$
where ${E_i},\;{H_i}$ are the E- and H-field components and ${\delta _{ik}}$ is the Kronecker symbol.

The incident beam was assumed to have a 100-mW power, the FDTD-aided computation was conducted on a λ/80 mesh in the particle location region (within the boundaries −1.5 µm < x<1.5 µm; −0.65 µm < y<0.65 µm, 0.3 µm < z<2 µm), being taken as λ/30 elsewhere. The torque exerted on a nano-bead with the refractive index n=1.5 + 0.01i and centered in the focal plane was found to be ${M_z} = 3,23 \cdot {10^{ - 21}}$Nm. For a particle with higher absorption, n=1.5 + 0.3i, the torque was found to be larger: ${M_z} = 7,9 \cdot {10^{ - 20}}$Nm. Shown in Fig. 3 is a plot for the longitudinal torque projection Mz versus the bead center shift in the focal plane along the x axis in the range from 0 to 1 µm with a 50-nm step.

 figure: Fig. 3.

Fig. 3. Torque Mz versus the nano-bead shift in the focal plane along the x axis.

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From Fig. 3, the near-axis torque Mz is seen to be positive, making the nano-bead rotate anticlockwise. This ‘anomalous’ rotation takes place in the region of a reverse energy flow. If the nano-bead center is shifted in the focal plane off the optical axis, the angular momentum changes sign at x=0.19 µm, becoming negative. In the region where the longitudinal projection of the torque is negative, the bead rotates in a normal manner – clockwise, i.e. similar to the rotation of the polarization of a left-handed circularly polarized beam.

5. Conclusion

Using the Richards-Wolf theory, we have derived analytical relationships to describe projections of the spin angular momentum (SAM) and Poynting vector in the tight focus of an incident left (right)-handed circularly polarized Gaussian beam as well as for a left (right)-handed circularly polarized optical vortex with topological charge 2. The on-axis longitudinal projection of SAM in the focus for an incident Gaussian beam has been shown to be negative for left-handed circular polarization and positive for the right-handed circular polarization. And vice versa, when focusing an optical vortex (m=2), the on-axis longitudinal projection of SAM in the focus has been shown to be positive for left-handed circular polarization and positive for right-handed circular polarization. Such an ‘anomalous’ behavior of the longitudinal SAM component is due to the fact that when focusing a left-handed circularly polarized optical vortex (m=2) there occurs a reverse on-axis energy flow in the focus (negative projection of the Poynting vector). In a similar way, it can be demonstrated that when focusing a right-handed circularly polarized optical vortex with m=−2, the longitudinal SAM component will be inverted in the focal plane near the optical axis. However, focusing a left-handed circularly polarized beam does not result in SAM inversion because no reverse energy flow occurs in the focus.

Using a rigorous FDTD-aided numerical computation of a near-focus field, the longitudinal SAM component has been shown to behave in the focus in agreement with the theoretical predictions. In addition, via rigorous computation, i.e. doing without the dipole approximation and using instead a Maxwell’s stress tensor, we have calculated a torque exerted upon a nano-bead with a complex refractive index and its center found in the focal plane center. The nano-bead has been shown to be under the action of a torque with a positive on-axis projection, rotating anticlockwise although the polarization vector of the incident left-handed circular polarized field rotates clockwise. The ‘anomalous’ behavior discovered of the SAM vector in the focus with a reverse energy flow is a physical proof of the reverse energy flow in the focus and can be used as a means for its detection.

Funding

Ministry of Science and Higher Education of the Russian Federation (007-GZ/Ch3363/26); Russian Science Foundation (18-19-00595).

Disclosures

The authors declare no conflicts of interest.

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Figures (3)
Fig. 1.
Fig. 1. On-axis projections of SAM Sz in the focal plane for an incident field with (a) right-handed and (b) left-handed circular polarization and phase singularity of order m=2, (c,d) their respective profiles along the x axis and profiles of the Z-axis projections of the Poynting vector P for right-handed (e) and left-handed (f) circular polarizations. Superimposed on the plot in Fig. 1(d) is the beam intensity pattern ${|E |^2}$ (dashed line).

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Fig. 2.
Fig. 2. On-axis projections of SAM Sz in the focal plane for an incident field with (a) right-handed and (b) left-handed circular polarization and phase singularity of order m=2 calculated by Richards-Wolf formulas, and (c,d) their respective profiles along the x axis.

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Fig. 3.
Fig. 3. Torque Mz versus the nano-bead shift in the focal plane along the x axis.

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Equations (22)

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E = A ( θ ) 2 ( 1 i σ ) , H = A ( θ ) 2 ( − i σ 1 ) ,
E x R = − i 2 ( I 0 , 0 + e 2 i φ I 2 , 2 ) , E y R = 1 2 ( I 0 , 0 − e 2 i φ I 2 , 2 ) , E z R = − 2 e i φ I 1 , 1
E x L = − i 2 ( I 0 , 0 + e − 2 i φ I 2 , 2 ) , E y L = 1 2 ( − I 0 , 0 + e − 2 i φ I 2 , 2 ) , E z L = − 2 e − i φ I 1 , 1 ,
I ν , μ = ( π f λ ) ∫ 0 θ 0 sin ν + 1 ( θ 2 ) cos 3 − ν ( θ 2 ) cos 1 / 2 ( θ ) A ( θ ) e i k z cos ⁡ z J μ ( x ) d θ ,
A ( θ ) = exp ( − γ 2 sin 2 θ sin 2 θ 0 ) .
S = 1 2 Im [ E ∗ × E ] ,
S x R = − ( I 0 , 0 + I 2 , 2 ) I 1 , 1 sin ⁡ φ , S y R = ( I 0 , 0 + I 2 , 2 ) I 1 , 1 cos ⁡ φ , S z R = 1 2 ( I 0 , 0 2 − I 2 , 2 2 ) ,
S x L = − ( I 0 , 0 + I 2 , 2 ) I 2 , 2 sin ⁡ φ , S y L = ( I 0 , 0 + I 2 , 2 ) I 1 , 1 cos ⁡ φ , S z L = − 1 2 ( I 0 , 0 2 − I 2 , 2 2 ) .
E = A ( θ ) e i m φ 2 ( 1 i σ ) , H = A ( θ ) e i m φ 2 ( − i σ 1 ) ,
E x R = i 2 ( e 4 i φ I 2 , 4 + e 2 i φ I 0 , 2 ) , E y R = 1 2 ( e 4 i φ I 2 , 4 − e 2 i φ I 0 , 2 ) , E z R = 2 e 3 i φ I 1 , 3 ,
E x L = i 2 ( I 2 , 0 + e 2 i φ I 0 , 2 ) , E y L = 1 2 ( − I 2 , 0 + e 2 i φ I 0 , 2 ) , E z L = − 2 e i φ I 1 , 1 .
P x R = − I 1 , 3 ( I 0 , 2 + I 2 , 4 ) sin ⁡ φ , P y R = I 1 , 3 ( I 0 , 2 + I 2 , 4 ) cos ⁡ φ , P z R = 1 2 ( I 0 , 2 2 − I 2 , 4 2 ) ,
P x L = − I 1 , 1 ( I 0 , 2 + I 2 , 0 ) sin ⁡ φ , P y L = I 1 , 1 ( I 0 , 2 + I 2 , 0 ) cos ⁡ φ , P z L = 1 2 ( I 0 , 2 2 − I 2 , 0 2 ) .
S x R = − I 1 , 3 ( I 0 , 2 + I 2 , 4 ) sin ⁡ φ , S y R = I 1 , 3 ( I 0 , 2 + I 2 , 4 ) cos ⁡ φ , S z R = 1 2 ( I 0 , 2 2 − I 2 , 4 2 ) ,
S x L = I 1 , 1 ( I 2 , 0 + I 0 , 2 ) sin ⁡ φ , S y L = − I 1 , 1 ( I 2 , 0 + I 0 , 2 ) cos ⁡ φ , S z L = 1 2 ( I 2 , 0 2 − I 0 , 2 2 ) .
P z L = − S z L = − 1 2 ( I 2 , 0 2 − I 0 , 2 2 ) .
P = R e 2 ( E ∗ × H ) = P o r + P s p ,
P o r = Im 2 k ( E ∗ ⋅ ( ∇ E ) ) , P s p = 1 4 k ( ∇ × Im ( E ∗ × E ) ) .
E x = exp ( i ( k f − m φ − ω t ) ) , E y = exp ( i ( k f − m φ − ω t − π 2 ) )
E x = exp ( i ( k f − m φ − ω t ) ) , E y = exp ( i ( k f − m φ − ω t + π 2 ) )
M = ∮ Ω ⁡ [ r × ( σ ⋅ n ) ] d Ω ,
σ i k = 1 4 π ( | E | 2 + | H | 2 2 δ i k − E i E k − H i H k ) ,
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