Optical backflow for the manipulations of dipolar nanoparticles

Xiangyang Xie; Peng Shi; Changjun Min; Xiaocong Yuan

doi:10.1364/PRJ.561198

1. INTRODUCTION

Backflow, which is a counterintuitive phenomenon that a particle with positive momentum has a non-zero probability of propagating backwards during specific time intervals, was first discovered in quantum physics [1]. To date, significant attention has been attracted to the study of quantum backflow, and the phenomenon has been predicted in various research fields [2 –4]. Despite extensive theoretical investigations, the experimental observation of quantum backflow has faced a significant challenge. Recently, it has been recognized that backflow is a prevalent effect occurring in the interference of coherent waves [5], and several studies have successfully observed the optical backflow [6 –8].

The recent experimental observation of optical azimuthal backflow was achieved by the interference of two Laguerre-Gaussian (LG) beams carrying orbital angular momenta (OAMs) [8], which has sparked considerable research interest due to its broad range of applications in optical manipulations [9 –11], communications [12,13], quantum information [14], spin-orbit interactions [15], and light-matter interactions [16,17]. Optical backflow can be classified into two categories: one involving the inverse wavevector (canonical momentum), such as optical azimuthal backflow [8], and the other involving energy backflow (Poynting vector) [18 –20]. The inverse wavevector is much rarer than the inverse Poynting vector. For instance, in axial energy backflow, the Belinfante’s spin momentum is opposite to the canonical momentum in the backflow region [21 –23]. Moreover, conventional inverse optical forces can only be achieved under specific conditions [24,25], whereas backflow provides a robust mechanism to overcome this limitation, and offers applications in the particle’s path manipulations, the non-conservative forces [26], and the sub-diffraction limit imaging [19].

Previously, optical azimuthal backflow was described by scalar wave theory [8]. In this study, we reconsider optical azimuthal backflow from the perspective of vectorial electromagnetic (EM) theory. We first perform an in-depth analysis of the spin-momentum properties of superimposed OAM beams and confirm the existence of optical azimuthal backflow, even if the magnitudes of the two individual OAMs are equal. The spin-momentum properties are crucial for understanding the underlying physics of the optical backflow, such as the relationship between backflow, off-axis vortex flow, and super-oscillations. In addition to optical azimuthal backflow, we also study the optical axial backflow, which occurs in cylindrical vector vortex beams (CVVBs) under a tightly focusing configuration [27].

In our work, we apply optical backflow to optical manipulations, which have been widely applied to fields such as biomedicine and materials science [28]. Although recent studies on optical magnetism have shown that reverse energy flow can be achieved without OAM [29,30], optical azimuthal backflow generated by OAM beams offers an additional degree of freedom in optical manipulations. Remarkably, by optimizing the material parameters of nanoparticles, we ensure the optical forces parallel to the backflow and mitigate the effects of Brownian motion. Moreover, we analyze the conditions in which optical azimuthal backflow occurs and reveal that the backflow depends on the difference between the topological charges (TCs) of the OAM beams, rather than their absolute values. This implies that the trajectory of trapped particles can be flexibly controlled by adjusting the TCs of the superimposed OAM states, enabling advanced optical manipulation in complex environments. Additionally, by tuning the polarization of CVVBs, we can engineer axial backflow, enabling the control of axial energy flux. Such control is critical for high-precision optical trapping, nanoscale particle manipulation, and advanced imaging techniques.

2. THEORY

A. Optical Azimuthal Backflow

Optical azimuthal backflow occurs in the superposition state of two co-propagating OAM beams. LG beams contain the phase singularity (vortex phase) and OAM characterized by a TC $l$ . In the free space (dielectric constant $ε_{0}$ and magnetic permeability $μ_{0}$ ), the complex amplitude of an LG beam in cylindrical coordinates ( $r$ , $φ$ , $z$ ) can be expressed as [9] $u_{p}^{l} = C_{p l} \frac{w_{0}}{w (z)} {[\frac{\sqrt{2} r}{w (z)}]}^{| l |} L_{p}^{| l |} [\frac{2 r^{2}}{w^{2} (z)}] \exp [- \frac{r^{2}}{w^{2} (z)}] \exp {i [k z - \frac{k r^{2}}{2 R (z)} + φ_{G}]} e^{+ i l φ} e^{- i ω t},$ (1)where $C_{p l}$ is the normalization coefficient with $p$ the radial index and $p = 0$ in the work; $L_{p}^{| l |}$ is the generalized Laguerre polynomial; $z_{0} = π w_{0}^{2} / λ$ is the Rayleigh range with $w_{0}$ the beam waist and $λ$ the wavelength; $k = 2 π / λ$ is the wave number; $R (z) = z [1 + {(z_{0} / z)}^{2}]$ is the curvature radius; $w (z) = w_{0} {[1 + {(z_{0} / z)}^{2}]}^{1 / 2}$ is the beam waist radius, $e^{+ i l φ}$ is the vortex phase, and $φ_{G} = - (2 p + | l | + 1) \arctan (z / z_{0})$ is the Gouy phase. For numerical illustrations, the value $λ = 632.8 nm$ is used throughout the paper, and $w_{0} = 4 λ$ is taken for analysis of the azimuthal backflow case. In Eq. (1), $ω = c k$ is the light frequency, $c$ being the light velocity; in calculations of time-averaged quantities (energy, energy flow, and momentum densities), the time-dependent term $e^{- i ω t}$ can be ignored, and it will be omitted in the following expressions.

Assuming that the two LG beams are $x$ -polarization light, we can thus express the superimposed field as $E_{x} = A_{1} u_{p_{1}}^{l_{1}} + A_{2} u_{p_{2}}^{l_{2}}$ , where $A_{1}$ and $A_{2}$ are the amplitude factors. Here, we primarily consider the properties of superimposed LG modes at the beam waist plane. By employing the electric Gauss law $\nabla \cdot E = 0$ in the paraxial approximation, the electric field can be calculated as $E = [E_{x} \cos φ \hat{r}; - E_{x} \sin φ \hat{φ}; \frac{1}{i k} (\frac{1}{r} \frac{\partial E_{x}}{\partial φ} \sin φ - \frac{\partial E_{x}}{\partial r} \cos φ) \hat{z}] .$ (2)

According to Faraday’s law $H = (\nabla \times E) / i ω μ_{0}$ , the magnetic field can be expressed as $H = Z_{μ ε} [E_{x} \sin φ \hat{r}; E_{x} \cos φ \hat{φ}; \frac{i}{k} (\frac{1}{r} \frac{\partial E_{x}}{\partial φ} \cos φ + \frac{\partial E_{x}}{\partial r} \sin φ) \hat{z}],$ (3)where $Z_{μ ε}$ represents the wave impedance. From Eqs. (2) and (3), the Poynting vector $P = Re {E^{*} \times H} / 2$ can be calculated as $P = \frac{Z_{μ ε}}{2} Re {\frac{E_{x}^{*}}{i k} \frac{\partial E_{x}}{\partial r} \hat{r}; (\frac{\sin 2 φ}{2 i k} \frac{\partial E_{x} E_{x}^{*}}{\partial r} + \frac{E_{x}}{k r} \frac{\partial E_{x}^{*}}{\partial φ}) \hat{φ}; E_{x} E_{x}^{*} \hat{z}},$ (4)and the spin angular momentum (SAM) density $S = Im ε_{0} E^{*} \times E + μ_{0} H^{*} \times H / 4 ω$ is $S = \frac{ε}{4 ω k} Im {(\frac{E_{x}}{r} \frac{\partial E_{x}^{*}}{\partial φ} - \frac{E_{x}^{*}}{r} \frac{\partial E_{x}}{\partial φ}) \hat{r}; - i \frac{\partial E_{x} E_{x}^{*}}{\partial r} \hat{φ}; 0 \hat{z}},$ (5)where the superscript $*$ represents the complex conjugate.

B. Optical Axial Backflow

Then, we consider the optical axial backflow, which occurs in the tightly focused field of CVVBs. CVVBs expressed by $E = [η_{x} \cos (n φ + φ_{0}) \hat{x} + η_{y} \sin (n φ + φ_{0}) \hat{y}] e^{i m φ} e^{- i ω t}$ (6)are a class of structured light with spatially varying polarization structure and helical phase structure. Therefore, they may carry SAM and OAM. Here, $η_{x}$ and $η_{y}$ are the amplitude factor of $x$ - and $y$ -polarization components, respectively, $n$ is the polarization TC, $φ_{0}$ is the initial polarization state, and $m$ is the vortex TC. Like in Section 2.A, the time-dependent multiplier $e^{- i ω t}$ can be omitted during the energy and momentum calculations; for simplicity, $m = 0$ is assumed in our further analysis. However, the axially mechanical characteristics of tightly focused CVVBs are similar to those of tightly focused cylindrical vector beams (CVBs). In this way, we can use the same technique to achieve the reverse axial optical force for the tightly focused CVVBs. In the focal plane, the vectorial diffraction theory is employed to calculate the focused field [31 –35] and the electric field vectors are given by $E (ρ, ϕ, z) = - \frac{i f}{λ} \int_{0}^{θ_{\max}} \int_{0}^{2 π} P (θ) Γ^{e} \cdot e^{i k [z \cos θ + ρ \sin θ \cos (φ - ϕ)]} \sin θ d θ d φ, Γ^{e} = [E_{r} (\begin{matrix} \cos θ \cos φ \hat{x} \\ \cos θ \sin φ \hat{y} \\ - \sin θ \hat{z} \end{matrix}) + E_{ϕ} (\begin{matrix} - \sin φ \hat{x} \\ \cos φ \hat{y} \\ 0 \hat{z} \end{matrix})],$ (7)where ( $ρ$ , $ϕ$ , $z$ ) are the cylindrical coordinates in the focal plane, $f$ is the focal length of the tightly focusing systems, $θ_{\max} = \arcsin (NA)$ is the maximum acceptance angle $θ$ , which is determined by the system’s numerical aperture NA, and $P (θ)$ is the apodization function.

By substituting Eq. (6) into Eq. (7), the electric field distribution near the focus of the focused CVBs can be calculated as $E_{ρ} = \frac{i Φ}{2} {2 [η_{x} \cos (n ϕ + φ_{0}) \cos ϕ + η_{y} \sin (n ϕ + φ_{0}) \sin ϕ] (1 + \cos θ) J_{n} (α) + (η_{x} + η_{y}) (1 - \cos θ) J_{n - 2} (α) \cos [(n - 1) ϕ + φ_{0}] + (η_{x} - η_{y}) (1 - \cos θ) J_{n + 2} (α) \cos [(n + 1) ϕ + φ_{0}]} e^{i k z \cos θ} \hat{ρ},$ (8) $E_{ϕ} = - \frac{i Φ}{2} {2 [η_{x} \cos (n ϕ + φ_{0}) \sin ϕ - η_{y} \sin (n ϕ + φ_{0}) \cos ϕ] (1 + \cos θ) J_{n} (α) + (η_{x} + η_{y}) (1 - \cos θ) J_{n - 2} (α) \sin [(n - 1) ϕ + φ_{0}] - (η_{x} - η_{y}) (1 - \cos θ) J_{n + 2} (α) \sin [(n + 1) ϕ + φ_{0}]} e^{i k z \cos θ} \hat{ϕ},$ (9) $E_{z} = {- (η_{x} + η_{y}) \sin θ J_{n - 1} (α) \cos [(n - 1) ϕ + φ_{0}] + (η_{x} - η_{y}) \sin θ J_{n + 1} (α) \cos [(n + 1) ϕ + φ_{0}]} e^{i k z \cos θ} \hat{z},$ (10)where $Φ = - A i^{n} f P (θ) \sin θ / λ$ , and $J_{n} (α)$ are the first Bessel functions of order $n$ with argument $α = k ρ \sin θ$ . According to Faraday’s law $H = (\nabla \times E) / i ω μ_{0}$ , the magnetic field near the focus of the focused CVBs can be expressed as $H_{ρ} = \frac{k Φ i}{2 ω μ} {2 [η_{x} \cos (n ϕ + φ_{0}) \sin ϕ - η_{y} \sin (n ϕ + φ_{0}) \cos ϕ] (1 + \cos θ) J_{n} (α) - (η_{x} + η_{y}) (1 - \cos θ) J_{n - 2} (α) \sin [(n - 1) ϕ + φ_{0}] + (η_{x} - η_{y}) (1 - \cos θ) J_{n + 2} (α) \sin [(n + 1) ϕ + φ_{0}]} e^{i k z \cos θ} \hat{ρ},$ (11) $H_{ϕ} = \frac{k Φ i}{2 ω μ} {2 [η_{x} \cos (n ϕ + φ_{0}) \cos ϕ + η_{y} \sin (n ϕ + φ_{0}) \sin ϕ] (1 + \cos θ) J_{n} (α) - (η_{x} + η_{y}) (1 - \cos θ) J_{n - 2} (α) \cos [(n - 1) ϕ + φ_{0}] - (η_{x} - η_{y}) (1 - \cos θ) J_{n + 2} (α) \cos [(n + 1) ϕ + φ_{0}]} e^{i k z \cos θ} \hat{ϕ},$ (12) $H_{z} = {(η_{x} + η_{y}) \sin θ J_{n - 1} (α) \sin [(n - 1) ϕ + φ_{0}] + (η_{x} - η_{y}) \sin θ J_{n + 1} α \sin [(n + 1) ϕ + φ_{0}]} e^{i k z \cos θ} \hat{z} .$ (13)

From Eq. (8) to Eq. (13), for optical axial backflow, we will focus on the analysis of only the axial component of the Poynting vector $P_{z} = Re {E_{ρ}^{*} H_{ϕ} - E_{ϕ}^{*} H_{ρ}} / 2$ , which can be calculated as $P_{z} = \frac{k {| Φ |}^{2}}{8 ω μ} Re {2 (η_{x}^{*} η_{x} + η_{y}^{*} η_{y}) {(1 + \cos θ)}^{2} J_{n}^{2} (α) - {(1 - \cos θ)}^{2} (η_{x}^{*} + η_{y}^{*}) (η_{x} + η_{y}) J_{n - 2}^{2} (α) - {(1 - \cos θ)}^{2} (η_{x}^{*} - η_{y}^{*}) (η_{x} - η_{y}) J_{n + 2}^{2} (α) + 2 (η_{x}^{*} η_{x} - η_{y}^{*} η_{y}) \cos (2 n ϕ + 2 φ_{0}) {(1 + \cos θ)}^{2} J_{n}^{2} (α) - 2 (η_{x}^{*} η_{x} - η_{y}^{*} η_{y}) \cos (2 n ϕ + 2 φ_{0}) {(1 - \cos θ)}^{2} J_{n - 2} (α) J_{n + 2} (α)},$ (14)and the SAM density $S = Im {ε_{0} E^{*} \times E + μ_{0} H^{*} \times H} / 4 ω$ is $S_{ρ} = \frac{ε {| Φ |}^{2}}{4 ω} Re {- (η_{x}^{*} η_{x} - η_{y}^{*} η_{y}) \sin θ (1 + \cos θ) J_{n - 1} (α) J_{n} (α) - (η_{x}^{*} η_{x} - η_{y}^{*} η_{y}) \sin θ (1 + \cos θ) J_{n} (α) J_{n + 1} (α) + (η_{x}^{*} η_{x} - η_{y}^{*} η_{y}) \sin θ (1 - \cos θ) J_{n - 2} (α) J_{n + 1} (α) + (η_{x}^{*} η_{x} - η_{y}^{*} η_{y}) \sin θ (1 - \cos θ) J_{n - 1} (α) J_{n + 2} (α)} \sin (2 n ϕ + 2 φ_{0}) \hat{ρ},$ (15) $S_{ϕ} = \frac{ε {| Φ |}^{2}}{4 ω} Re {[- (η_{x}^{*} + η_{y}^{*}) (η_{x} + η_{y}) \sin θ (1 + \cos θ) J_{n - 1} (α) J_{n} (α) - (η_{x}^{*} + η_{y}^{*}) (η_{x} + η_{y}) \sin θ (1 - \cos θ) J_{n - 2} (α) J_{n - 1} (α) + (η_{x}^{*} - η_{y}^{*}) (η_{x} - η_{y}) \sin θ (1 + \cos θ) J_{n} (α) J_{n + 1} (α) + (η_{x}^{*} - η_{y}^{*}) (η_{x} - η_{y}) \sin θ (1 - \cos θ) J_{n + 1} (α) J_{n + 2} (α)] + [- (η_{x}^{*} η_{x} - η_{y}^{*} η_{y}) \sin θ (1 - \cos θ) J_{n - 1} (α) J_{n + 2} (α) - (η_{x}^{*} η_{x} - η_{y}^{*} η_{y}) \sin θ (1 + \cos θ) J_{n - 1} (α) J_{n} (α) + (η_{x}^{*} η_{x} - η_{y}^{*} η_{y}) \sin θ (1 - \cos θ) J_{n - 2} (α) J_{n + 1} (α) + (η_{x}^{*} η_{x} - η_{y}^{*} η_{y}) \sin θ (1 + \cos θ) J_{n} (α) J_{n + 1} (α)] \cos (2 n ϕ + 2 φ_{0})} \hat{ϕ},$ (16) $S_{z} = \frac{ε {| Φ |}^{2}}{4 ω} Im {(η_{y}^{*} η_{x} - η_{x}^{*} η_{y}) [{- (1 + \cos θ)}^{2} J_{n} (α) J_{n} (α) + {(1 - \cos θ)}^{2} J_{n - 2} (α) J_{n + 2} (α)] \sin (2 n ϕ + 2 φ_{0})} \hat{z} .$ (17)

The theoretical results presented above provide the foundation for the subsequent analysis and will be applied in the following sections to interpret the results.

3. RESULTS AND DISCUSSION

A. Optical Azimuthal Backflow

We first consider the case of the optical azimuthal backflow. Since the incident light is linearly polarized along the $x$ -axis (i.e., $E_{y} = 0$ ), it follows from Faraday’s law that $H_{x} = 0$ . Consequently, the $z$ -component of the SAM vanishes, i.e., $S_{z} = 0$ . The Poynting vector $P$ is proportional to the kinetic Poynting momentum $p$ , i.e., $p = P / c^{2}$ [22,36,37], which can be decomposed into spin momentum $p_{s}$ and canonical momentum $p_{o}$ , i.e., $p = p_{s} + p_{o}$ . In the paraxial approximation $(\partial A / \partial z ≪ k A$ ), the horizontal components ( $r$ - and $φ$ -components) of the spin momentum, given by $p_{s} = \nabla \times S / 2$ , are negligible. This implies that the horizontal component of the Poynting momentum $p$ is equal to the canonical momentum $p_{o}$ . Additionally, the Minkowski-type canonical momentum can be re-expressed as $p_{o} = ℏ k$ [37 –40]. Therefore, the optical azimuthal backflow in superposed LG beams can be quantitatively described by the Poynting momentum $p$ . To identify the conditions for optical azimuthal backflow, we assume that the OAM beam is radially uniform (i.e., $\partial u_{p}^{l} / \partial r = 0$ ) and has the form $u_{p}^{l} = A e^{i l φ}$ , where $A$ is a real constant (this assumption is only used in analyzing the occurrence of optical azimuthal backflows). Substituting these conditions into Eq. (4), the azimuthal component of Poynting momentum simplifies to $p_{φ} \propto A_{1}^{2} l_{1} + A_{2}^{2} l_{2} + A_{2} A_{1} (l_{1} + l_{2}) \cos (l_{1} - l_{2}) φ .$ (18)

When the amplitudes of the two superimposed beams are equal (i.e., $A_{1} = A_{2}$ ), the azimuthal component of the Poynting momentum satisfies $p_{φ} \propto [l_{1} + l_{2} + (l_{1} + l_{2}) \cos (l_{1} - l_{2}) φ] < 0$ (where $l_{1} < 0$ and $l_{2} < 0$ ). The azimuthal Poynting momentum of the superimposed state is consistent with that of individual beams, meaning that no backflow phenomenon will occur. When $A_{1} \neq A_{2}$ , for simplicity we set $A_{1} = 1$ , and then Eq. (18) can be transformed into $p_{φ} \propto [1 + A_{2} \cos (l_{1} - l_{2}) φ] l_{1} + [A_{2}^{2} + A_{2} \cos (l_{1} - l_{2}) φ] l_{2} .$ (19)

When $\cos (l_{1} - l_{2}) φ = 0$ , the right-hand side of Eq. (18) simplifies to $l_{1} + A_{2}^{2} l_{2}$ . Since the beams have negative TCs, the value is negative. However, when $\cos (l_{1} - l_{2}) φ = - 1$ , the right-hand side of Eq. (19) becomes $(1 - A_{2}) l_{1} + (A_{2}^{2} - A_{2}) l_{2}$ . The sign of the expression can be changed under the conditions $1 - A_{2} > 0$ and $(A_{2}^{2} - A_{2}) > (1 - A_{2}) l_{1} / l_{2}$ (e.g., $A_{2} = 0.5$ and $l_{1} / l_{2} < 1 / 2$ ); the right-hand side becomes positive, which indicates the presence of optical azimuthal backflow, as discussed in Ref. [8].

To intuitively understand the optical azimuthal backflow phenomenon, we present the distributions of the Poynting momentum, canonical momentum, spin momentum, and wavevector at the waist plane (Fig. 1). As depicted in Figs. 1(b) and 1(e), the azimuthal momentum exhibits anomalously positive values in the regions where optical azimuthal backflow occurs. By comparing Figs. 1(c), 1(f), and 1(g), it is evident that the axial component of the $p_{s}$ is small, while the canonical momentum is approximately parallel to the Poynting momentum. When the two beams are superimposed, the Poynting momentum and OAM are not consistently concentrated at the beam center, leading to the formation of off-axis vortex flow [Fig. 1(h)] [19]. This off-axis vortex flow generates a local phase gradient, which increases the local wavevector. The phase gradient further amplifies the local oscillation frequency, resulting in a wavenumber that exceeds the value of a single beam’s wavenumber. As observed in Fig. 1(i), the local normalized wavenumber exceeds $k = ω / c$ in the regions where optical azimuthal backflow occurs, which indicates a necessary condition for super-oscillations [5,20]. In addition, even when $A_{1} = A_{2}$ , the off-axis vortex flow and the wavenumber still exceed $k = ω / c$ , indicating that the optical azimuthal backflow phenomenon still occurs.

Figure 1.Normalized momentum density and wavevector at the waist plane. The (a) $r$ , (b) $φ$ , (c) $z$ components of Poynting momentum $p$ can be decomposed into the (d) $r$ , (e) $φ$ , (f) $z$ components of $p_{o}$ and (g) the $z$ components of $p_{s}$ . Here, the horizontal components of $p_{s}$ are zero. (h) The normalized $φ$ component of $k / W$ indicates off-axis vortexes and (i) the normalized wavenumber $| k | / W$ is larger than one, which is a necessary condition for super-oscillations. Here, $l_{1} = - 1$ , $l_{2} = - 3$ , $A_{1} = A_{2}$ throughout the paper.

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Then, we apply the concept of optical azimuthal backflow to optical manipulation. For dipolar particles (with chiral parameter $κ$ , dielectric constant $ε_{p}$ , and permeability $μ_{p}$ ), the time-averaged optical force is expressed as [41 –43] $F = \frac{1}{2} Re {d \cdot (\nabla) E^{*} + m \cdot (\nabla) H^{*} - \frac{c k_{0}^{4}}{6 π} (d \times m^{*})} .$ (20)

Here, the electric and magnetic dipole moments can be expressed as $d = α_{e e} E + i α_{e m} H$ , $m = α_{m m} H - i α_{e m} E$ , where $α_{e e}$ , $α_{e m}$ , and $α_{m m}$ are the polarizabilities of the particle. Among these, $α_{e m}$ is associated with the chiral parameter $κ$ , and for a non-chiral particle, $α_{e m} = 0$ .

By substituting the constitutive relations of the particle and the expressions for the EM dipole moments into Eq. (20), the force can be rewritten as [26,44] $F = F_{grad} + F_{rad} + F_{spin} + F_{curl} + F_{flow} .$ (21)

Here, $F_{grad} = \nabla U$ represents the optical gradient force, where $U = Re α_{e e} w^{e} / ε_{0} + Re α_{m m} w^{m} / μ_{0} + Re α_{e m} C / ω ε_{0} μ_{0}$ is associated with the electric energy density $w^{e} = ε_{0} {| E |}^{2} / 4$ , the magnetic energy density $w_{m} = μ_{0} {| H |}^{2} / 4$ , and the chiral density $C = - ω ε_{0} μ_{0} Im E^{*} \cdot H$ . Under the paraxial approximation, the gradient force in the $z$ -component is zero. The radiation force, $F_{rad} = σ P / c$ , is caused by the momentum flux of light and is proportional to the Poynting vector or the momentum density. Here, the speed of light in vacuum is $c = 1 / {(ε_{0} μ_{0})}^{1 / 2}$ , and $σ = σ_{e} + σ_{m} + σ^{'}$ , where $σ_{e} = k Im {α_{e e}} / Re {ε_{p}}$ , $σ_{m} = k Im {α_{m m}} / Re {μ_{p}}$ , and $σ^{'} = - c k^{4} Re {α_{e m} α_{e m}^{*} + α_{e e} α_{m m}^{*}} / 6 π$ . According to the previous analysis, the canonical momentum plays a dominant role in the total momentum. Therefore, the radiation force should resemble the distribution of canonical momentum. The term $F_{spin} = ω γ_{e} S^{e} + ω γ_{m} S^{m}$ is the spin force, which vanishes because both the coefficients $γ_{e} = 2 (ω Im {α_{e m}} - Re {c k^{4} α_{em} α_{e m}^{*}} / 6 π ε_{0})$ and $γ_{m} = 2 (ω Im {α_{e m}} - Re {c k^{4} α_{e m} α_{e m}^{*}} / 6 π μ_{0})$ are related to $α_{em}$ , which is equal to zero here. The term $F_{curl} = - 2 ω^{2} Im {α_{e m}} S_{T} - 2 v (σ_{e} p_{s}^{e} + σ_{m} p_{s}^{m})$ is the curl force, which is related to the optical transverse spin $S_{T} = \nabla \times P / 2 ω^{2}$ [45 –47], and the spin momentum $p_{s} = \nabla \times S / 2$ [37 –40]. Obviously, there is only the axial component in the optical curl force here, since $α_{e m} = 0$ and the horizontal components of $p_{s}$ vanish. The last term, $F_{flow} = σ^{'} Im P / c$ , represents the force associated with the imaginary Poynting vector [48]. Here, the $F_{flow}$ force is very small, less than $10^{- 7}$ of the total force, and can therefore be neglected.

Motivated by the growing interest in levitated optomechanics [49], Fig. 2 shows the distributions of optical force exerted by the azimuthal optical backflow on achiral gold dipolar particles with a radius of 50 nm in vacuum. As is shown in Fig. 2(b), a sign reversal appears in the azimuthal component of the total optical force $F_{φ}$ within the backflow region. This means that if the particle moves along the azimuthal direction [i.e., along a circle depicted in Fig. 2(b)], it experiences the azimuthal force of opposite directions [shown by yellow arrows in Fig. 2(b)] at different segments of its trajectory, and it originates from the combined contribution of the gradient force $F_{grad}$ and the radiation force $F_{rad}$ . However, the presence of the gradient force $F_{grad}$ will cause the misalignment between the Poynting momentum in Fig. 1(b) and optical force in Fig. 2(b), which will make readers confused about the origin of backflow force, particularly in experiment. Therefore, the influence of gradient force and the misalignment should be mitigated by optimizing the material’s properties. On the other hand, the $z$ -component of $F_{curl}$ is much smaller than the total force, and thus the $F_{z}$ aligns parallelly with the axial Poynting momentum.

Figure 2.The distributions of optical force as the optical azimuthal backflow acts on gold dipolar nanoparticles [50]. The (a) $r$ , (b) $φ$ , (c) $z$ components of total optical force $F$ . The total optical force can be divided into the (d) $r$ , (e) $φ$ , (f) $z$ components of radiation $F_{rad}$ related to Poynting momentum, the (g) $r$ , (h) $φ$ components of gradient force $F_{grad}$ related to energy density, and (i) the $z$ component of $F_{curl}$ related to the spin momentum. Here, $ε_{p} = - 11.74 + 1.2611 i$ and $μ_{p} = 1.0$ ; $l_{1} = - 1$ and $l_{2} = - 3$ . All the force is normalized by the maximal value of total optical force throughout the paper.

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Next, we reconsider the $F_{grad}$ , which is related to the optical potential $U = Re {α_{e e}} w^{e} / ε_{0} + Re {α_{m m}} w^{m} / μ_{0}$ . Here, the electric polarizability $α_{e e}$ and the magnetic polarizability $α_{m m}$ coexist, and the electric energy density $w^{e}$ and magnetic energy density $w^{m}$ are of the same order of magnitude. However, since $ε_{0} / μ_{0} ≪ 1 A^{4} s^{6} / ({kg}^{2} m^{4})$ , the normalized electric polarizability $α_{e e} / ε_{0}$ is much larger than the magnetic polarizability $α_{m m} / μ_{0}$ , indicating that the electric polarizability dominates the $F_{grad}$ . In addition, Brownian motion significantly affects optical trapping stability. For dipolar particles, Brownian motion can be evaluated by the random thermal energy $k_{B} T$ , where $k_{B}$ is the Boltzmann constant and $T$ is the ambient temperature. The optical trapping potential well is given by $V = - Re {α_{e e}} {| E |}^{2} / 2$ [51]. Provided that $V > k_{B} T$ , the trapping can be considered stable against the influence of Brownian motion.

Therefore, to align the optical force with the backflow parallelly and mitigate the influence of Brownian motion, we optimize the nanoparticle permittivity by maximizing $ξ_{1} = Im α_{e e} / Re α_{e e}$ , thereby ensuring that $F_{rad}$ becomes dominant over $F_{grad}$ . Meanwhile, by increasing the trapping parameter $ξ_{2} = - Re {α_{e e}} | E |^{2} / 2 k_{B} T I$ to ensure that the optical potential well exceeds the thermal energy, we can achieve the stable trapping. Here, the trapping parameter $ξ_{2}$ represents the potential $V / k_{B} T$ normalized by the intensity ( $| E |^{2}$ ), and it is dimensional. To simplify the symbols in the following calculations and figures, we assume that the quantity $I$ only extracts the numerical part of $| E |^{2}$ to make the $ξ_{2}$ dimensionless (or setting $| E | = 1 V / μm$ uniformly). The corresponding results are shown in Figs. 3(a) and 3(b).

Figure 3.The optimization of material’s permittivity for eliminating the misalignment between force and Poynting momentum and achieving stable optical trapping. Since the $F_{rad} / F_{grad}$ is related to the $ξ_{1} = Im {α_{e e}} / Re {α_{e e}}$ , we scan the permittivity of nanoparticles for maximal (a) $\log_{10} | ξ_{1} |$ to eliminate the misalignment between force and Poynting momentum. The trapping potential well is related to $Re {α_{e e}}$ and we scan the permittivity of nanoparticles for maximal (b) $\log_{10} | ξ_{2} |$ . Here, $μ_{p} = 1.0$ .

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From Fig. 3(a), the maximal value of $| ξ_{1} |$ ( $ξ_{1} = - 1.467 \times 10^{7}$ ) is attained when $ε_{p} = - 1.586 + 1.653 i$ , corresponding to point A. The total optical force is presented in Figs. 4(a)–4(c), which clearly demonstrate that the misalignment is eliminated. However, the trapping potential well ( $V / k_{B} T$ ) is much smaller than one [Fig. 4(d)], indicating that the nanoparticle cannot be trapped stably due to Brownian motion.

Figure 4.The total optical force and the trapping potential well $V / k_{B} T$ . When $ε_{p} = - 1.586 + 1.653 i$ , the (a) $r$ , (b) $φ$ , (c) $z$ components of $F$ match well with the Poynting momentum given by Figs. 1(a)–1(c). However, the (d) potential well is much smaller than one and the particle cannot be trapped stably. When $ε_{p} = - 2.609 - 0.339 i$ , the (d) $r$ , (e) $φ$ , (f) $z$ components of $F$ are shown and the (h) potential well is much larger than one. When $ε_{p} = - 2.7458 + 0.23171 i$ , the (i) $r$ , (j) $φ$ , (k) $z$ components of $F$ are shown and the (l) potential well is much larger than one. Here, $l_{1} = - 1, l_{2} = - 3$ .

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According to Fig. 3(b), the maximal value of $| ξ_{2} | (ξ_{2} = - 1.51 \times 10^{4}$ ) is attained when $ε_{p} = - 2.609 - 0.339 i$ , corresponding to point B. The trapping potential well is much larger than one, indicating that the stable trapping of nanoparticles can be achieved in this instance. However, the distribution of total optical force shown in Fig. 4(e)–4(g) exhibits a misalignment between the Poynting momentum and optical force in the horizontal components. Although the inverse azimuthal force can also be observed, it is not dominant, and one may argue that this backflow force is not originated from the backflow. It can be observed that there is a tradeoff between the larger radiation force $F_{rad}$ (related to $Im {α_{e e}}$ ) and the deeper trapping potential well (related to $Re {α_{e e}}$ ).

To resolve this problem, we re-check Fig. 3 and find a proper material parameter to achieve the relatively large radiation force and trapping potential well. When $ε_{p} = - 2.7458 + 0.23171 i$ [the permittivity of silver at $λ \approx 368 nm$ , and corresponding to point C in Fig. 3(a)], $ξ_{1} = 4.828$ , and $ξ_{2} = - 2.514$ , in this instance [Figs. 4(i)–4(l)], the influence of the gradient force is effectively mitigated, and the effect of Brownian motion is simultaneously suppressed. Here, we ignore the influence of multipoles on the optical force, because we find that the magnitude of quadrupole mode is two orders smaller than that of dipole mode, and the magnitude of hexapole modes is two orders smaller than that of quadrupole mode [52,53].

In addition, the number of optical azimuthal backflow occurrences is related to the difference in the TC of OAM states, rather than their absolute values. According to Eq. (19), the azimuthal Poynting momentum reverses direction when the sign of $\cos (l_{1} - l_{2}) φ$ changes. Given the periodic nature of the cosine function, the sign changes $| l_{1} - l_{2} |$ times in one period, corresponding to the number of azimuthal backflows equal to $| l_{1} - l_{2} |$ . As shown in Fig. 5, we present the variation of $F_{φ}$ with different TCs, confirming that the number of azimuthal backflow occurrences matches the above analysis. This indicates that the topological structure of beams plays a crucial role in the distribution and manipulation of the optical forces.

Figure 5.The azimuthal optical force $F_{φ}$ for different TCs of OAM states. The appearance of optical azimuthal backflow is determined by $| l_{1} - l_{2} |$ , which can be verified in the $F_{φ}$ when (a) $l_{1} = - 1$ and $l_{2} = - 4$ , (b) $l_{1} = - 1$ and $l_{2} = - 5$ , (c) $l_{1} = - 1$ and $l_{2} = - 6$ , (d) $l_{1} = - 2$ and $l_{2} = - 5$ , (e) $l_{1} = - 2$ and $l_{2} = - 6$ , (f) $l_{1} = - 3$ and $l_{2} = - 6$ . Here, $ε_{p} = - 2.7458 + 0.23171 i$ and $μ_{p} = 1.0$ .

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B. Optical Axial Backflow

Then, we consider the case of the optical axial backflow. In the axial case, we are interested in identifying the regions where Eq. (14) takes negative values, corresponding to areas of axial backflow. For simplicity, we set $η_{x} = η_{y} = 1$ , and the azimuthal mode number $n = 2$ and $\cos θ \neq 1$ , which allows $P_{z}$ to be simplified as $P_{z} \propto {(1 + \cos θ)}^{2} J_{2}^{2} (α) - {(1 - \cos θ)}^{2} J_{0}^{2} (α) .$ (22)

Clearly, in the central region where $α = 0$ , we have $J_{2} (0) = 0$ and $J_{0} (0) = 1$ , resulting in a negative value of $P_{z}$ . Since the polarization components are set to $η_{x} = η_{y} = 1$ , we can derive that $S_{z} = 0$ from Eq. (17). On the other hand, the energy flow is predominant along the $z$ -axis, and this longitudinal energy flow varying in the transverse plane will give rise to the transverse SAM [46,47], indicating that the SAM has only transverse components.

As previously concluded, the Poynting vector $P$ is proportional to the kinetic Poynting momentum $p$ , which can be decomposed into spin momentum $p_{s}$ and canonical momentum $p_{o}$ . The spin momentum is given by $p_{s} = \nabla \times S / 2$ . Since the SAM has only transverse components, $p_{s}$ has only axial components. Moreover, on the focal plane, the Poynting vector has only a $z$ -component, implying that the $p_{o}$ also has only an axial component. Consequently, the axial backflow is governed solely by the contributions of the spin and canonical momentum in the $z$ -direction.

To intuitively understand the optical axial backflow phenomenon, we present the distributions of the Poynting momentum, canonical momentum, and spin momentum at the focal plane (Fig. 6). As shown in Figs. 6(a) and 6(c), both the axial kinetic momentum $p$ and spin momentum $p_{s}$ exhibit anomalous negative values in the central region and in a concentric ring. These features are particularly prominent in Fig. 6(c) and correspond to the regions where optical axial backflow occurs. In contrast, as shown in Fig. 6(b), the canonical momentum $p_{o}$ remains unaffected by the backflow.

Figure 6.Normalized momentum density at the focal plane. (a) The $z$ components of Poynting momentum $p$ can be decomposed into the (b) $z$ components of $p_{o}$ and (c) the $z$ components of $p_{s}$ . The horizontal components of $p$ , $p_{o}$ , and $p_{s}$ are zero. Here, $η_{x} = η_{y} = 1$ , numerical aperture $NA = 0.95$ , the beam waist $w_{0} = 0.26 λ$ , and $A = 1$ throughout the paper.

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Similarly, we apply the optical axial backflow to optical manipulation. The optical gradient force $F_{grad} = \nabla U$ in the $z$ -direction is negligible due to the small gradient of the light field along this axis when the beam is focused. The optical radiation force $F_{rad} = σ P / c$ is related to the Poynting vector. According to the previous analysis, the Poynting vector $P$ has only an axial component, which leads to the conclusion that the radiation force also has only an axial component. The optical spin force $F_{spin} = ω γ_{e} S^{e} + ω γ_{m} S^{m}$ vanishes because both the coefficients $γ_{e}$ and $γ_{m}$ are related to $α_{e m}$ , which is equal to zero here. The optical curl force $F_{curl} = - 2 ω^{2} Im {α_{e m}} S_{T} - 2 v \cdot (σ_{e} p_{s}^{e} + σ_{m} p_{s}^{m})$ is determined by the transverse spin $S_{T}$ and spin momentum $p_{s}$ . Since $α_{e m} = 0$ and the spin angular momentum $p_{s}$ has only an axial component, $F_{curl}$ is therefore purely axial. The optical flow force $F_{flow} = σ^{'} Im {P} / c$ is related to the imaginary Poynting vector and provides an additional contribution.

We present the distributions of optical forces acting on achiral gold dipolar nanoparticles with a radius of 70 nm, as shown in Fig. 7. In Figs. 7(a), 7(d), and 7(g), as well as in Figs. 7(b), 7(e), and 7(h), it can be observed that the transverse optical force is primarily contributed by the gradient force $F_{grad}$ , whereas the flow force $F_{flow}$ accounts for only 2% of the total force and can therefore be neglected. According to the previous analysis, the distribution of the optical force $F$ closely resembles the Poynting vector $P$ . However, as shown in Fig. 7(c), the $z$ -component of the total optical force $F_{z}$ does not exhibit any backward force. Further analysis of Figs. 7(f) and 7(i) reveals that the two components contributing to $F_{z}$ , namely, the radiation force $F_{rad, z}$ and the curl force $F_{curl, z}$ , form an oppositely directed force pair in the central region. This counterbalance prevents $F_{z}$ from reversing its direction. To investigate the conditions under which optical force reversal may occur, we perform a theoretical reanalysis of $F_{z}$ , which can be expressed as $F_{z} = F_{rad, z} + F_{curl, z} = c (σ_{e} + σ_{m} + σ^{'}) p_{z} - 2 c (σ_{e} p_{s, z}^{e} + σ_{m} p_{s, z}^{m}) = c σ_{e} (p_{z} - 2 p_{s, z}^{e}) + c σ_{m} (p_{z} - 2 p_{s, z}^{m}) + c σ^{'} p_{z} .$ (23)

Figure 7.The distributions of optical force as the optical axial backflow acts on dipolar nanoparticles. The (a) $r$ , (b) $φ$ , (c) $z$ components of total optical force $F$ . The total optical force can be divided into the (d) $r$ , (e) $φ$ components of gradient force $F_{grad}$ , (f) the $z$ component of radiation force $F_{rad}$ related to Poynting momentum, the (g) $r$ , (h) $φ$ components of flow force $F_{flow}$ related to imaginary Poynting vector, and (i) the $z$ component of $F_{curl}$ related to the spin momentum. Here $, ε_{p} = - 11.74 + 1.2611 i$ and $μ_{p} = 1.0$ .

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In the central region, due to the symmetry and mutual coupling between the electric and magnetic fields, their contributions to the spin are equal (i.e., $p_{s, z}^{e} = p_{s, z}^{m} \approx 0.5 p_{s, z}$ ). This indicates that the first two terms in Eq. (23) are approximately equal to $c (σ_{e} + σ_{m}) p_{o}$ in the central region. As shown in Fig. 6(b), the canonical momentum $p_{o}$ is always positive, indicating that the sign of the first two terms is determined by $σ_{e} + σ_{m}$ . Meanwhile, the last term of Eq. (23) is proportional to the Poynting momentum $p_{z}$ and $σ^{'}$ . Although Brownian motion may influence stable trapping, the conclusion that its influence can be mitigated by optimizing the nanoparticle properties remains valid in this case. Therefore, to stably generate a reverse optical force and mitigate the influence of Brownian motion, we optimize the permittivity and permeability of the nanoparticle as follows: increase the trapping parameter $ξ_{2}$ , ensuring stable optical trapping by maintaining the trapping potential greater than the thermal energy; $ξ_{3} = | (σ_{e} + σ_{m}) / \max (σ_{e}, σ_{m}) |$ ; decrease the impact of the $p_{o}$ on the axial force $F_{z}$ ; and $ξ_{4} = σ^{'} / \max (| σ_{e} |, | σ_{m} |, | σ_{e} + σ_{m} |) > 0$ , maximizing $ξ_{4}$ to ensure the consistency between the axial force $F_{z}$ and the Poynting momentum $p_{z}$ . The results are shown in Fig. 8.

Figure 8.The optimization of material’s nanoparticle properties to ensure stable optical trapping and stably generate a reverse optical force. Since the trapping potential well is related to the $ξ_{2}$ and the $F_{z}$ is related to $ξ_{3} = | (σ_{e} + σ_{m}) / \max (σ_{e}, σ_{m}) |$ and $ξ_{4} = σ^{'} / \max (| σ_{e} |, | σ_{m} |, | σ_{e} + σ_{m} |)$ , we scan the permittivity of nanoparticles for maximal (a), (d) $\log_{10} | ξ_{2} |$ to ensure stable optical trapping and scan the permittivity of nanoparticles for minimal (b), (e) $\log_{10} ξ_{3}$ and maximal (c), (f) $\log_{10} ξ_{4}$ to stably generate a reverse optical force. Here, in (a)–(c) $μ_{p} = 1.0 + 5.0 i$ and in (d)–(f), $μ_{p} = 1.0 + 10.0 i$ .

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Our analysis primarily focuses on the case of $μ_{p} = 1.0 + 5.0 i$ . As shown in Fig. 8(a), most regions satisfy $| ξ_{2} | > 1$ , indicated by the red areas in the figure. To ensure stable axial optical trapping while generating a reverse axial optical force, we exclude parameters that fail to support stable trapping. Consequently, the subsequent discussion is restricted to conditions that guarantee stable trapping.

According to Fig. 8(b), the minimum value of $ξ_{3} (ξ_{3} = 1.297 \times 10^{- 7})$ is obtained at point A, which corresponds to $ε_{p} = - 7.795 - 2.7825 i$ . As shown in Fig. 9(a), a significant reverse axial optical force appears in the central region. In this case, the axial optical force is influenced by the curl force $F_{curl, z}$ , which exhibits a significantly different distribution from the axial Poynting vector $p_{z}$ . According to Fig. 8(c), the value of $ξ_{4} = 0.572$ , which confirms the proportional relationship between the axial radiation force and the curl force, as illustrated in Figs. 9(b) and 9(c). Therefore, a trade-off between $ξ_{3}$ and $ξ_{4}$ is required to effectively suppress the effect of the curl force. Such an optimization ensures that the generated reverse axial optical force is consistent with the axial backflow. Furthermore, as shown in Fig. 9(d), the trapping potential well is larger than one, indicating that Brownian motion is effectively suppressed. Once the particles are trapped in the potential well, they will move in the $- z$ -direction under the influence of $F_{z}$ .

Figure 9.The axial optical force and the trapping potential well $V / k_{B} T$ . When $ε_{p} = - 7.795 - 2.7825 i$ and $μ_{p} = 1.0 + 5.0 i$ , a significant reverse axial optical force emerges in the (a) $z$ components of $F$ . But $F_{z}$ have a markedly different distribution from the (b) $z$ components of $F_{rad}$ , due to the influence of the (c) $z$ components of $F_{curl}$ and (d) potential well is larger than one. When $ε_{p} = 0.535 - 3.715 i$ and $μ_{p} = 1.0 + 5.0 i$ , the distribution of the (e) $z$ components of $F$ aligns with the (f) $z$ components of $F_{rad}$ with tiny (g) $z$ components of $F_{curl}$ and (h) potential well is also larger than one. When $ε_{p} = 0.085 - 3.6525 i$ and $μ_{p} = 1.0 + 10.0 i$ , the $z$ components of (i) $F$ , (j) $F_{rad}$ , (k) $F_{curl}$ and (l) the potential well have similar distributions and results to (e)–(h).

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Based on the previous analysis, we further explore the parameter space to optimize both the radiation force and the trapping potential well. Specifically, by re-examining Fig. 8, we identify material parameters that enable the achievement of a relatively large radiation force and the suppression of Brownian motion. As shown in Figs. 9(e)–9(h), when $ε_{p} = 0.535 - 3.715 i$ [point B in Fig. 8(c)], we obtain $ξ_{2} = - 5.074$ , $ξ_{3} = 1.972$ , and $ξ_{4} = 29.866$ ; the influences of curl force and the Brownian motion are eliminated simultaneously. Consequently, the distribution of the axial optical force $F_{z}$ aligns with the axial radiation force $F_{rad, z}$ (axial Poynting momentum $p_{z}$ ). Similarly, as shown in Figs. 9(i)–9(l), when $μ_{p} = 1.0 + 10.0 i$ and $ε_{p} = 0.085 - 3.6525 i$ [point C in Fig. 8(f)], we obtain $ξ_{2} = - 10.594$ , $ξ_{3} = 3.307$ , and $ξ_{4} = 364.364$ . Under these conditions, a similar distribution of the axial optical force $F_{z}$ is observed.

The aforementioned material parameters may be achieved by an artificial nanoparticle with its permittivity and permeability described by the Drude-Lorentz model [54,55], $ε_{p} = 1 - \frac{ω_{pe}^{2}}{ω^{2} + i γ_{e} ω} .$ (24)

By substituting the desired real and imaginary parts of $ε_{p}$ into this expression, the corresponding plasma frequency $ω_{pe}$ and damping factor $γ_{e}$ can be numerically determined. A similar formulation applies to the magnetic permeability $μ_{p}$ . Besides, although it is difficult to fabricate such nanoparticles at optical frequencies, the scale law of EM theory allows us to realize the design at microwave frequencies [56]. These artificial particles enable the exploration of back-transport phenomena under controlled conditions, potentially offering new capabilities in long-range particle manipulation and directional transport in tailored photonic environments.

4. CONCLUSIONS

In summary, we investigate the spin-momentum properties of both optical azimuthal and axial backflows in the superimposed fields of LG beams and tightly focused CVVBs, respectively. The results demonstrate a close relationship between optical azimuthal backflow, off-axis vortex flow, and super-oscillations. Particularly, the optical azimuthal backflow occurs even when the magnitudes of the two individual OAMs are equal, with the number of azimuthal backflows determined solely by the difference $| l_{1} - l_{2} |$ . This indicates that the topological structure of beams plays a crucial role in the generation, control, and application such as optical manipulation with backflow forces. For optical axial backflow, we show that although the axial Poynting momentum $p_{z}$ is inverted, the spin momentum induced by optical spin-orbit couplings will cause the optical force parallel to the canonical momentum and no backflow force appears for conventional non-magnetic nanoparticles.

Then, we applied both optical azimuthal and axial backflows to optical manipulations. For azimuthal backflow, we optimized the material’s permittivity to eliminate the misalignment between the force and Poynting momentum and mitigate the Brownian motion further. There is a tradeoff between achieving a stronger radiation force and a larger trapping potential well, and thus a proper material parameter must be selected to balance both the radiation force and trapping potential well. In the case of axial backflow, optimization of the material properties was restricted to conditions that guarantee stable trapping. Subsequently, by balancing the electric/magnetic responses and increasing the magnitude of axial optical force, the influence of canonical momentum is suppressed, and the axial optical force is aligned with the Poynting momentum, thereby enabling stable reverse optical force. Our study enriches the in-depth understanding of optical backflow and provides applications in optical trapping, manipulation, and imaging.

Category: Physical Optics

Received: Mar. 4, 2025

Accepted: Apr. 28, 2025

Published Online: Jul. 18, 2025

The Author Email: Peng Shi (pittshiustc@gmail.com)

DOI:10.1364/PRJ.561198

CSTR:32188.14.PRJ.561198