Rigorous full-wave calculation of optical forces on dielectric and metallic microparticles immersed in a vector Airy beam

Wanli Lu; Huajin Chen; Shiyang Liu; Zhifang Lin

doi:10.1364/OE.25.023238

1. Introduction

A dielectric particle or a metal nanoparticle illuminated by an incident shaped light beam could be either trapped or moved along a designed trajectory, called optical manipulation [1, 2], by using the optical force stemmed from light-matter interaction. The optical force can behave as a gradient force due to the intensity gradient or manifests itself as a scattering force based on the momentum transfer from the shaped light beam to the particle via scattering and absorption. Optical manipulation has impacted on a variety of fields, such as physics, biology and soft condensed matter, ranging from the microscopic to atomic scales [3–5]. One of the most famous applications is the optical tweezer firstly introduced by Ashkin by using a single focused Gaussian beam, which generates a larger gradient force to cancel the scattering force and then traps a particle [1]. In addition to the conventional focused Gaussian beams, shaping the light beams also plays an increasingly important role in the optical manipulation [5]. Successful examples are, among many others, the non-diffraction Bessel beams [6–9] and Airy beams [10–14].

The non-diffraction Airy beams [10, 11] are introduced by using phase-modulated Gaussian beam. They were firstly proposed within the context of quantum mechanics [12, 13]. In addition to maintaining the shape of the field profile as a non-diffraction light beams, Airy beam, also known as Airy wavepacket, could accelerate itself, leaving us with a parabolic trajectory as it propagates, thus enabling particle transport along curved paths by utilizing the scattering optical force. This application was first demonstrated experimentally [14] by a two-dimensional (2D) Airy beam nearly ten years ago. The theoretical studies, however, are mostly limited to the simpler case of the one-dimensional (1D) Airy beam including paraxial [15, 16] and non-paraxial [17] light beams or the paraxial 2D Airy beam [18], due largely to the complexity in expanding the 2D Airy beams in terms of the vector spherical wave functions (VSWFs) [19, 20]. Here we proposed a method to study the most general case, where the spherical particle can be of any size and the vector 2D Airy beam is treated, beyond the paraxial approximation and, also, for a variety of polarizations.

In this paper, we present the first rigorous full wave solution for the optical force on spherical particles immersed in a 2D vector Airy beam, based on the generalized Lorenz-Mie theory (GLMT) [19, 20] and the Maxwell stress tensor (MST) approach [21–23]. The incident field profile at an arbitrary point is obtained from the optical field in the initial plane based on the vector angular spectrum representation [24, 25]. The explicit expressions of the partial wave expansion coefficients for 2D vector Airy beam are then derived for different polarizations, including linear, radial, and azimuthal polarization. With these expansion coefficients evaluated, the field profiles are simulated based on the GLMT. Numerical results exhibit the parabolic trajectory of the main lobe of the 2D vector Airy beam as well as the self-healing properties after scattered by an obstacle, corroborating our expansion formulations of the 2D vector Airy beams in terms of the VSWFs [19, 20]. The optical forces on dielectric and metallic particles are then evaluated using the MST approach. The results indicate that both the spherical dielectric particle in air and the metallic particle, like gold, suspended in water can be trapped within the main lobe or nearby side-lobes, largely by the transverse gradient optical force, and accelerated along a curved trajectory by the longitudinal scattering force, facilitating optical micromanipulation, especially, particle transport following predesigned curved paths, for plasmonic as well as dielectric particles.

2. Theory and formulations

2.1. Description of a 2D Airy beam

For simplicity, it is assumed that an incident 2D Airy beam propagates along z-axis with a focused point (x_b, y_b, z_b). Based on the vector angular spectrum of the plane waves [24, 25], the electric field at an arbitrary point (x, y, z) in the region z > z_b reads,

E_{inc} (x, y, z) = \int \int_{- \infty}^{\infty} A (k_{x}, k_{y}; z_{b}) e^{i [k_{x} x + k_{y} y + k_{z} (z - z_{b})]} d k_{x} d k_{y},

where (k_x, k_y, k_z) is the wave vector of each expanded plane waves propagating along the direction (sin α cos β, sin α sin β, cos α) with k_x = k sin α cos β, k_y = k sin α sin β, k_z = k cos α and the wave number k in the background. It should be noted that the evanescent wave components are not taken into account in numerical calculation since only the far field propagation is concerned. And the vector angular spectrum A = A_⊥ + A_ze_z can be obtained based on the incident electric field in the initial plane z = z_b.

Considering the fact that in the initial plane (z = z_b) the fields depend only on the transverse coordinates (x, y) and are independent of the longitudinal coordinate z, the electric field profile in the initial plane is described in the following equation, which can be obtained via solving the paraxial wave equation [11],

E_{⊥} (x, y, z_{b}) = E_{0} Ai (\frac{x - x_{b}}{w_{x}}) Ai (\frac{y - y_{b}}{w_{y}}) \exp (α_{x} \frac{x - x_{b}}{w_{x}} + α_{y} \frac{y - y_{b}}{w_{y}}) p_{⊥} (x, y),

with the parameters α_x > 0 and α_y > 0 measuring the power conveyed by beams, w_x and w_y determining the size of the main lobe and p_⊥(x, y) = p_x (x, y)e_x + p_y (x, y)e_y denoting the polarization states of the Airy beam in the initial plane. We should note that the electric fields out of the initial plane are solved based on the Maxwell’s equations, which are rigorous beyond the paraxial approximation.

By using the inverse Fourier transformation for the optical field E_⊥(x, y, z_b) in the initial plane, the transverse component A_⊥ reads [17],

A_{⊥} (k_{x}, k_{y}; z_{b}) = \frac{E_{0}}{4 π^{2}} w_{x} w_{y} e^{\frac{1}{3} {(α_{x} - i k_{x} w_{x})}^{3}} e^{\frac{1}{3} {(α_{y} - i k_{y} w_{y})}^{3}} e^{- i (k_{x} x_{b} + k_{y} y_{b})} s_{⊥} (α, β),

and the z-component of vector angular spectrum

A_{z} = - [\frac{k_{x}}{k_{z}} A_{x} (k_{x}, k_{y}) + \frac{k_{y}}{k_{z}} A_{y} (k_{x}, k_{y})]

is obtained by considering the divergenceless feature of the electric field in background medium, i. e., ∇ · E_inc = 0. As a result, the vector angular spectrum is given by

A = \frac{E_{0}}{4 π^{2}} w_{x} w_{y} e^{\frac{1}{3} {(α_{x} - i k_{x} w_{x})}^{3}} e^{\frac{1}{3} {(α_{y} - i k_{y} w_{y})}^{3}} e^{- i (k_{x} x_{b} + k_{y} y_{b})} s (α, β),

where the vector s(α, β) = s_⊥(α, β) + s_ze_z = s_xe_x + s_ye_y + s_ze_z, existing only in the wave vector space, denotes the polarization states of each expanded plane wave with the wave vector propagating along the direction of (sin α cos β, sin α sin β, cos α), where the z-component s_z is determined by

s_{z} = - (\frac{k_{x}}{k_{z}} s_{x} + \frac{k_{y}}{k_{z}} s_{y})

. The factors e_x, e_y and e_z are the unit vectors in the Cartesian coordinate system.

For an Airy beams with linear or circular polarization, p(x, y), existing only in the real space, is a constant vector at an arbitrary point, then the factor s(α, β) of each expanded plane wave is also a constant vector. We should note that the z-component p_z ≠ 0 for satisfaction of the Maxwell’s equations. For an azimuthally polarized Airy beam, also termed TE (transverse electric) mode without z-component of the electric field, (s_x, s_y) = (− sin β, cos β) is set and then s_z = 0 for each expanded plane wave. For the Airy beams with radially polarization (also called TM - transverse magnetic - mode), the polarization factor is set to (s_x, s_y) = (cos β, sin β) and the z-component reads s_z = − tan α, then the magnetic field profile can be obtained without z-component based on the Maxwell’s equations.

2.2. Optical force based on the GLMT

Based on the GLMT [19, 20], the incident Airy beam E_inc can be expanded in terms of the VSWFs including $N_{n m}^{(1)} (k, r)$ and $M_{n m}^{(1)} (k, r)$ , with the suppressed time harmonic factor e⁻^iωt throughout this paper, namely [20, 26–29],

E_{inc} (r, θ, ϕ) = - \sum_{n, m} i E_{m n} [p_{m n} N_{n m}^{(1)} (k, r) + q_{m n} M_{n m}^{(1)} (k, r)],

where (r, θ, ϕ) is the spherical coordinates corresponding to (x, y, z) with the origin locating on the particle center,

E_{m n} = E_{0} i^{n} γ_{m n}^{1 / 2}

with

γ_{m n} = {[\frac{(2 n + 1) (n - m)!}{n (n + 1) (n + m)!}]}^{1 / 2}

, and the index n varies from 1 to ∞ and m from −n to n. The partial wave expansion coefficients of the incident Airy beam can be obtained by using the orthogonality of the VSWFs. The scattered field is also expanded in terms of the VSWFs

N_{n m}^{(3)} (k, r)

and

M_{n m}^{(3)} (k, r)

[20, 26–29],

E_{sca} (r, θ, ϕ) = \sum_{n, m} i E_{m n} [a_{m n} N_{n m}^{(3)} (k, r) + b_{m n} M_{n m}^{(3)} (k, r)],

where the partial wave expansion coefficients of the scattered field a_mn and b_mn can be obtained via the Mie scattering coefficients a_n and b_n, namely [19],

a_{m n} = a_{n} p_{m n}, b_{m n} = b_{n} q_{m n} .

Based on the Maxwell’s stress tensor, the time-averaged optical force acting on a spherical particle can be calculated by using the integral over the surface S of the particle and reads [21–23],

F = \oint_{S} \hat{r} \cdot 〈 \overset{\leftrightarrow}{T} 〉 d S,

where

\hat{r}

is the unit normal vector on the closed surface and the time-averaged Maxwell’s stress tensor reads,

〈 \overset{\leftrightarrow}{T} 〉 = \frac{1}{2} Re [ε E E^{*} + μ H H^{*} - \frac{1}{2} (ε E \cdot E^{*} + μ H \cdot H^{*}) \overset{\leftrightarrow}{I}],

with

\overset{\leftrightarrow}{I}

being the unit tensor. Considering the lossless background medium, three components of the optical force F_x, F_y and F_z can be simplified to the form expressed by the partial wave expansion coefficients of the incident and scattered fields [30, 31],

F_{x} = Re [F_{1}], F_{y} = Im [F_{1}], F_{z} = Re [F_{2}],

where the factors F₁ and F₂ are defined as,

F_{1} = \frac{2 π ε}{k^{2}} {| E_{0} |}^{2} \sum_{n, m} [c_{11} F_{1}^{(1)} - c_{12} F_{1}^{(2)} + c_{13} F_{1}^{(3)}], F_{2} = - \frac{4 π ε}{k^{2}} {| E_{0} |}^{2} \sum_{n, m} [c_{21} F_{2}^{(1)} + c_{22} F_{2}^{(2)}],

where the coefficients are,

\begin{array}{l} c_{11} = {[\frac{(n - m) (n + m + 1)}{n^{2} {(n + 1)}^{2}}]}^{1 / 2}, c_{12} {[\frac{n (n + 2) (n + m + 1) (n + m + 2)}{{(n + 1)}^{2} (2 n + 1) (2 n + 3)}]}^{1 / 2}, \\ c_{13} = {[\frac{n (n + 2) (n - m) (n - m + 1)}{{(n + 1)}^{2} (2 n + 1) (2 n + 3)}]}^{1 / 2}, c_{21} = {[\frac{n (n + 2) (n - m + 1) (n + m + 1)}{{(n + 1)}^{2} (2 n + 1) (2 n + 3)}]}^{1 / 2}, \\ c_{22} = \frac{m}{n (n + 1)}, \end{array}

and

\begin{array}{l} F_{1}^{(1)} = {\tilde{a}}_{m n} {\tilde{b}}_{m_{1} n}^{*} + {\tilde{b}}_{m n} {\tilde{a}}_{m_{1} n}^{*} - {\tilde{p}}_{m n} {\tilde{q}}_{m_{1} n}^{*} - {\tilde{q}}_{m n} {\tilde{p}}_{m_{1} n}^{*}, \\ F_{1}^{(2)} = {\tilde{a}}_{m n} {\tilde{a}}_{m_{1} n_{1}}^{*} + {\tilde{b}}_{m n} {\tilde{b}}_{m_{1} n_{1}}^{*} - {\tilde{p}}_{m n} {\tilde{p}}_{m_{1} n_{1}}^{*} - {\tilde{q}}_{m n} {\tilde{q}}_{m_{1} n_{1}}^{*}, \\ F_{1}^{(3)} = {\tilde{a}}_{m n_{1}} {\tilde{a}}_{m_{1} n}^{*} + {\tilde{b}}_{m n_{1}} {\tilde{b}}_{m_{1} n}^{*} - {\tilde{p}}_{m n_{1}} {\tilde{p}}_{m_{1} n}^{*} - {\tilde{q}}_{m n_{1}} {\tilde{q}}_{m_{1} n}^{*}, \\ F_{2}^{(1)} = {\tilde{a}}_{m n} {\tilde{a}}_{m n_{1}}^{*} + {\tilde{b}}_{m n} {\tilde{b}}_{m n_{1}}^{*} - {\tilde{p}}_{m n} {\tilde{p}}_{m n_{1}}^{*} - {\tilde{q}}_{m n} {\tilde{q}}_{m n_{1}}^{*}, \\ F_{2}^{(2)} = {\tilde{a}}_{m n} {\tilde{b}}_{m n}^{*} - {\tilde{p}}_{m n} {\tilde{q}}_{m n}^{*}, \end{array}

where the index m₁ = m + 1 and n₁ = n + 1, and

{\tilde{a}}_{m n} = a_{m n} - \frac{1}{2} p_{m n}, {\tilde{p}}_{m n} = \frac{1}{2} p_{m n}, {\tilde{b}}_{m n} = b_{m n} - \frac{1}{2} q_{m n}, {\tilde{q}}_{m n} = \frac{1}{2} q_{m n} .

2.3. Partial wave expansion coefficients of the incident Airy beam

Partial wave expansion coefficients of the incident Airy beam, p_mn and q_mn, play an important role in evaluating the optical forces for a particle illuminated by the 2D Airy beam. Via the orthogonality of the VSWFs, one has

\begin{array}{l} p_{m n} = \frac{i^{1 - n}}{4 π E_{0}} γ_{m n}^{1 / 2} \frac{k r}{j_{n} (k r)} \int_{θ = 0}^{π} \int_{ϕ = 0}^{2 π} [e_{r} \cdot E (r, θ, ϕ)] P_{n}^{m} (\cos θ) e^{- i m ϕ} \sin θ d θ d ϕ, \\ q_{m n} = \frac{i^{- n} Z}{4 π E_{0}} γ_{m n}^{1 / 2} \frac{k r}{j_{n} (k r)} \int_{θ = 0}^{π} \int_{ϕ = 0}^{2 π} [e_{r} \cdot H (r, θ, ϕ)] P_{n}^{m} (\cos θ) e^{- i m ϕ} \sin θ d θ d ϕ, \end{array}

where j_n (kr) is the spherical Bessel function with an order n,

P_{n}^{m} (\cos θ)

is the first kind associated Legendre function, e_r is the unit vector in the spherical coordinates system, and

Z = \sqrt{μ / ε}

is the impedance with permittivity ε and permeability µ in the background. After some algebra (see appendix for details), the coefficients are simplified into,

\begin{array}{l} p_{m n} = \frac{k^{2}}{4 π^{2}} w_{x} w_{y} γ_{m n}^{1 / 2} e^{\frac{α_{x}^{3} + α_{y}^{3}}{3}} \int_{α = 0}^{π / 2} d α \sin α e^{- i k_{z b} \cos α} e^{- k^{2} \frac{w_{x}^{2} α x + w_{y}^{2} α y}{2}} \sin^{2} α \\ \times {\begin{array}{l} \frac{1}{2} (p_{x} + i p_{y}) (τ_{m n} - π_{m n} \cos α) I_{c} (m + 1) + \\ \frac{1}{2} (p_{x} - i p_{y}) (τ_{m n} + π_{m n} \cos α) I_{c} (m - 1), & linear and circular polarization \\ τ_{m n} I_{c} (m), & radial polarization \\ - i π_{m n} \cos α I_{c} (m), & azimuthal polarization \end{array} \end{array}

\begin{array}{l} q_{m n} = \frac{k^{2}}{4 π^{2}} w_{x} w_{y} γ_{m n}^{1 / 2} e^{\frac{α_{x}^{3} + α_{y}^{3}}{3}} \int_{α = 0}^{π / 2} d α \sin α e^{- i k_{z b} \cos α} e^{- k^{2} \frac{w_{x}^{2} α x + w_{y}^{2} α y}{2}} \sin^{2} α \\ \times {\begin{array}{l} \frac{1}{2} (p_{x} + i p_{y}) (π_{m n} - τ_{m n} \cos α) I_{c} (m + 1) + \\ \frac{1}{2} (p_{x} - i p_{y}) (π_{m n} + τ_{m n} \cos α) I_{c} (m - 1), & linear and circular polarization \\ π_{m n} I_{c} (m), & radial polarization \\ - i τ_{m n} \cos α I_{c} (m), & azimuthal polarization \end{array} \end{array}

where the two auxiliary functions are defined by

π_{m n} (\cos α) = \frac{m}{\sin α} P_{n}^{m} (\cos α)

and

τ_{m n} (\cos α) = \frac{d}{d α} P_{n}^{m} (\cos α)

. The factor I_c (m) is an integration and can be evaluated by,

I_{c} (m) = \int_{β = 0}^{2 π} d β e^{- i m β} e^{i (t_{1} \sin β + s_{1} \cos β + t_{3} \sin 3 β + s_{3} \cos 3 β) + s_{2} \cos 2 β},

with parameters

t_{1} = (k^{3} w_{y}^{3} \sin^{3} α / 4 - k w_{y} α_{y}^{2} \sin α - k y_{b} \sin α)

,

s_{1} = (k^{3} w_{x}^{3} \sin^{3} α / 4 - k w_{x} α_{x}^{2} \sin α - k x_{b} \sin α)

,

s_{2} = - k^{2} \sin^{2} α (w_{x}^{2} α_{x} - w_{y}^{2} α_{y}) / 2

,

t_{3} = - k^{3} w_{y}^{3} \sin^{3} α / 12

, and

s_{3} = k^{3} w_{x}^{3} \sin^{3} α / 12

. We can see that the expansion coefficients q_mn can be obtained from the coefficients p_mn by exchanging the two auxiliary functions π_mn(cos α) and τ_mn(cos α). Accordingly to the simplified Eqs. (13) and (14), we have developed the Fortran programs, based on which the expansion coefficients p_mn and q_mn can be evaluated efficiently and accurately compared to the other methods by directly calculating the Maxwell’s equations such as finite-difference time-domain method and finite element method. Then, the field profiles of 2D Airy beam and the optical forces acting on the particles can be acquired based on the Eqs. (5)–(6) and Eq. (10), respectively.

3. Results and discussion

With the aforementioned full wave method, we can examine the characteristics of the 2D Airy beam and the optical force acting on a particle, for simplicity, only the linear polarization is considered. Firstly, we discuss the main properties of the Airy beam, including the self-accelerating and self-healing effects, see Fig. 1. The Airy beam propagates along z-axis with the focus at (0, 0, 0) and it has wavelength λ = 1.064 µm, w_x = w_y = 2λ, α_x = α_y = 0.08 and E₀ = 1. We can see that the main lobe of the Airy beam propagates along a curved line [see Fig. 1(a)]. Based on this incident field profile, the trajectory of intensity center within the main lobe is obtained beyond the paraxial approximation, as shown in Fig. 1(c), and we can see that it is nearly a parabola. The trajectory of paraxial Airy beam is also shown in Fig. 1(c) based on the formulations $x = z^{2} / 4 k^{2} w_{x}^{2} - 0.937173 w_{x}$ and $y = z^{2} / 4 k^{2} w_{y}^{2} - 0.937173 w_{y}$ [17] and it is an ideal parabola. We can see that the trajectories of paraxial and non-paraxial Airy beams coincide well with each other within the propagating distance z ≤ 60λ and appear difference as propagating. Here, for the paraxial and non-paraxial Airy beams, the accelerating trajectories are slightly different, because the Airy beam used here is slightly non-paraxial [32] due to the parameters w_x = w_y = 2λ. In contrast, for the strongly non-paraxial Airy beam with w_x = w_y = λ, the accelerating trajectories of paraxial and non-paraxial Airy beams exhibit remarkable difference in the region z > 20λ, as shown in Fig. 1(c) denoted with the dash lines. We can see that the non-paraxial Airy beam dissipates as propagating in the region z > 30λ (see the red dash line), where the Airy beam is no longer the non-diffraction light beam [32, 33]. In addition, the non-paraxial Airy beam is the same as the paraxial Airy beam only within a short propagating distance. From Fig. 1(a), we can observe that the non-paraxial Airy beam is non-diffractive within a long propagation distance. Moreover, it exhibits the self-healing effect along the propagation trajectory as demonstrated in Fig. 1(b), where a dielectric spherical particle with the radius R = λ and the permittivity ε_r = 2.53 is located at the position (−1.8λ, −1.8λ, 10λ) close to the intensity center of the main lobe. We can see that the incident Airy beam is disturbed by the obstacle, but the field pattern can be self-healed after a certain propagation distance.

Fig. 1 Self-accelerating and self-healing effects of a linear polarized Airy beam with oscillation along x-axis and w_x = w_y = 2λ. The field profiles |E|² corresponding to (a) the incident 2D Airy beam and (b) the 2D Airy beam perturbed by a spherical particle with the radius R = λ, the permittivity ε_r = 2.53, and the location at (−1.8λ, −1.8λ, 10λ), (c) trajectories of the maximum intensity within the main lobe for both the non-paraxial and paraxial Airy beams. For comparison, trajectories for strongly non-paraxial Airy beam with w_x = w_y = λ are plotted in (c) with dashed lines. Panels (a) and (b) are composed of six transverse sections (constant z planes) and a longitudinal section (x = y plane).

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In order to demonstrate the self-healing effect of the Airy beam quantitatively, the field profile |E|² of the 2D Airy beam perturbed by a dielectric particle should be compared to that of the incident 2D Airy beam at different positions along the propagation trajectory. The simulation results are shown in Fig. 2, where the field profile along the transverse line y = x in three different planes z = 15λ, 40λ, and 80λ are plotted for both the perturbed and unperturbed 2D Airy beam, as indicated by the red and black solid lines, respectively. At the propagating distance z = 15λ close to the particle, the scattering field is quite strong so that the field profile, especially that in the main lobe, is severely distorted. After 25λ propagating distance at z = 40λ, the perturbed 2D Airy beam recovers its profile close to the incident one. At the even far propagating distance z = 80λ, the field profile eventually is self-healed, although it is not the same as the incident field due to the energy dissipation by the scatterer.

Fig. 2 Self-healing effect manifested by observing the field profiles |E|² of the 2D Airy beam perturbed by a scatterer, along the transverse line y = x at three different propagation distances z = 15λ, 40λ, and 80λ, as indicated by the red solid line. The field profiles of the incident 2D Airy beam are shown as well for the convenience of the comparison, as indicated by the black solid line. All the parameters are the same as those in Fig. 1(b).

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When a dielectric particle is illuminated by an Airy beam, it will be trapped within the main lobe or the side-lobes by the transverse optical forces, simultaneously, it will be accelerated along the curved trajectory due to the longitudinal optical forces. As shown in Fig. 3, we present the map of the optical forces versus the positions of the spherical particle on a transverse plane with z = 10λ. The dielectric particle has the radius R = λ and the permittivity ε_r = 2.53. For the longitudinal optical forces F_z, the maximum value of F_z reaches about 11.28 pN within the main lobe, as shown in Fig. 3(a), where the intensity center of the main lobe is normalized to 1 mWµm⁻² in the initial plane z = 0 (corresponding beam power is about 30 mW), i.e., E₀ = 3.54 × 10⁶ V/m here. And there are two notable peaks of optical forces F_z ≃ 3.7 pN within the nearby side-lobes. As a result, these longitudinal forces F_z will accelerate the particle along z-axis, while the transverse trapping positions are changed gradually as the particle moving along the longitudinal direction, so the particle will be accelerated along curved propagation trajectory of the main lobe or the nearby side-lobes.

Fig. 3 Optical forces versus the positions of a dielectric particle under the illumination of a linearly polarized Airy beam with polarization along x-axis, where the maximum intensity is normalized to 1 mWµm⁻² in the initial plane z = 0 within the main lobe. The dielectric particle with the radius R = λ and the permittivity ε_r = 2.53 is located within the transverse plane z = 10λ. The map of the longitudinal optical force F_z (a) and the transverse optical force F_⊥ (b) in this transverse plane are potted, where the thick arrows with red color denote directions and magnitudes ${(F_{x}^{2} + F_{y}^{2})}^{1 / 2}$ of the transverse optical forces via the directions and lengths of the arrows. The transverse optical forces with |F_⊥| < 0.1 pN are not shown due to the negligible small magnitude. And the light gray arrows are the stream lines of the transverse optical forces to only indicate the directions.

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The transverse optical forces F_⊥ are also calculated and denoted by the red thick arrows in the map, as shown in Fig. 3(b), where both the magnitudes ${(F_{x}^{2} + F_{y}^{2})}^{1 / 2}$ and the directions of F_⊥ can be discerned. It should be noted that the arrows with the denoted magnitude less than 0.1 pN are not plotted. The magnitude of transverse optical force reaches its maximum value |F_⊥| ≃ 1.87 pN, and the directions of F_⊥ point to intensity centers of the main lobe or the nearby side-lobes. The stream lines of the transverse optical forces are also demonstrated with the light gray arrows to imply the tendency of the accelerated particles toward the intensity centers of the main lobe or the nearby side-lobes, as shown in Fig. 3(b). Therefore, we can arrive at the conclusion that the particle can be trapped within the main lobe or the nearby side-lobes by the transverse optical forces and the particle is most likely to be trapped within the main lobe.

Airy beam can also trap a metal nanoparticle within the main lobe or the nearby side-lobes, and accelerate it along a curved trajectory. As an example, a gold nanoparticle with the radius R = λ/20 suspended in water [34] is considered. The optical forces are calculated and demonstrated in Fig. 4, where the gold nanoparticle is located on the transverse plane z = 10λ, and the intensity center of the main lobe is normalized to 10 mWµm⁻² in the initial plane z = 0 (corresponding beam power is about 300 mW), i.e., E₀ = 11.188 × 10⁶ V/m. The dielectric function of the gold particle is set to ε_r = −48.45 + 3.6i for the wavelength λ = 1.064 µm [35] and the refractive index of water is 1.33. The map of the transverse optical forces F_⊥ versus the positions of the gold nanoparticle are depicted as shown in Fig. 4(b), where the red thick arrows denote the transverse optical forces F_⊥ with the maximum value of |F_⊥| about 32.1 fN. We should point out that only the transverse optical forces with the magnitude |F_⊥| ≥ 10 fN are demonstrated. And the stream lines of the transverse optical forces are shown with light gray arrows, indicating the similar scenario as the dielectric particle case shown in Fig. 3. With the transverse optical forces F_⊥, as shown in Fig. 4(b), we can see that the gold nanoparticle can be trapped within the main lobe or the near side-lobes of the 2D Airy beam. Fig. 4(a) gives the map of the longitudinal optical forces F_z versus the positions of the nanoparticle, where the maximum longitudinal optical force reaches about 86.2 fN within the main lobe of the Airy beam, and F_z ≃ 32.5 fN for the two nearby side-lobes. So the 2D Airy beam will accelerate the gold nanoparticle along a curved line within the main lobe or nearby side-lobes.

Fig. 4 Optical forces versus the positions of a gold nanoparticle suspended in water with the radius R = λ/20 and positioned on the transverse plane z = 10λ under the illumination of a linearly polarized Airy beam polarization along x-axis, where the maximum intensity is normalized to 10 mWµm⁻² in the initial plane z = 0 within the main lobe. The dielectric function of the gold particle is ε_r = −48.45 + 3.6i at wavelength λ = 1.064 µm and the refractive index of water is 1.33. The map of the longitudinal optical forces F_z (a) and the transverse optical forces F_⊥ (b) are plotted within this transverse plane, without showing |F_⊥| < 10 fN due to negligible small magnitude. And the light gray arrows denote the stream lines of transverse optical forces to only indicate the directions.

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The concept of optical trapping potential provides a clearer picture for the transverse tapping by the 2D Airy beam. The optical trapping potential at any specified position on any z equal to constant plane is defined as the integral of work done by the transverse optical force F_⊥ while the particle starts to move from the specified position to a position with vanishing transverse force (F_⊥ = 0) through a path lying on the constant z plane. It is noted that the value of potential so evaluated will change if the particle moves along different paths, implying the presence of non-conservative component in the transverse optical force. Our numerical results show, however, that the discrepancy is less than 10% for the gold nanoparticle (less than 30% for the dielectric particle) when the integral of work is calculated through any different paths, suggesting the conservative gradient force constitutes the main part in the transverse optical force. One therefore concludes that the particle is transversely confined mostly by the conservative gradient force. In Fig. 5, we show the transverse optical trapping potential landscape on the plane z = 10λ for the dielectric and gold particle, corresponding, respectively, to the cases in Fig. 3 and Fig. 4, where the optical trapping potential is normalized by its maximum value. The integration path selected in Fig. 5 is along x-axis direction firstly, then along y-axis direction, and end at the point (16λ, 16λ), where the transverse force F_⊥ = 0. The potential wells are visualized near the centers of the main lobe and nearby side-lobes, showing clear physical illustrations for the transverse confinement by the 2D vector Airy beam for the dielectric and metallic particles.

Fig. 5 The maps of normalized optical trapping potential for the dielectric particle (a) and the gold nanoparticle (b) within the transverse plane z = 10λ, which are calculated based on the path integral of the transverse optical forces F_⊥. The parameters of the dielectric particle and the gold particle are the same as those in Fig. 3 and Fig. 4.

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In order to establish a stable transverse trapping of a gold nanoparticle, Brownian motion and gravity of the gold nanoparticle are considered here and buoyancy force is ignored. For a stable trapping, the potential well should be deep enough to overcome the kinetic energy of the gold nanoparticle in Brownian motion, satisfying the generally accepted criterion $R_{T} = e^{- U_{\max} / k_{B} T} ≪ 1$ [36] with the Boltzmann constant k_B and the temperature T = 300 K. And the potential depth U_max is obtained by $U_{\max} = ε_{0} Re [α] E_{\max}^{2} / 2$ [36], where the maximum electric field within in the main lobe is E_max = 0.245E₀ for the 2D Airy beam and the parameter α is the polarizability for the gold nanoparticle with radius R = λ/20 = 53.2 nm. Based on the above parameters, the parameter R_T ≃ 8.31 × 10⁻¹⁴ satisfies the condition R_T ≪ 1, and the gravity of the gold nanoparticle is about 0.12 fN (the density of gold is about 19.3 g/cm³ [36]), which is small enough to be ignored in comparison with the transverse and longitudinal optical forces. So the Brownian motion and gravity of the gold nanoparticle can be ignored, and the particle could be stably trapped by the transverse optical force and accelerated via the longitudinal force.

In practice, an Airy beam is usually served to trap, or, confine, a particle transversely so as to transport it along a curved trajectory in the longitudinal direction, so it could be used in optically mediated particle clearing [14]. In some occasions, an Airy beam could be used to trap a particle in three dimensions, where a focused Airy beam [18] is applied to generate a larger gradient force to cancel the scattering force in the longitudinal direction. In this case, the Airy beam is similar to the conventional focused Gaussian beam, which is usually used as an optical tweezer, and will have potential applications in particle trapping, guiding, alignment and so on [18]. More importantly, the high efficiency and accuracy of our method make it possible to investigate more fascinating effects related to the newly emergent lateral optical force [37, 38], negative pulling force [9, 39], and Fano resonance induced light-matter interaction [29, 40]. In addition, the 2D Airy beam possesses unique characteristic, both the main lobe and the nearby side-lobes can be used to transport the microparticles. However, the size dependence of the transport in different lobes has not been examined so far, which might be exploited within the framework of our theory. Besides, the dynamical simulation and the stability analysis on the micromanipulation associated with the 2D Airy beam can possibly be achieved as well.

At last, it should be noted that the 2D Airy beam behaves like a plane wave if we expand the width of its main lobe, which can be achieved simply by increasing the parameters w_x and w_y. As an example we consider an Airy beam with w_x = w_y = 4λ, the corresponding intensity center in the initial plane z = 0 is located roughly at (−3.75λ, −3.75λ, 0), which is obtained based on the trajectory of a paraxial Airy beam. Now, locating a Rayleigh particle with the radius R = λ/20 and the permittivity ε_r = 2.53 at the intensity center, we can calculate the optical force acting on this particle by using our rigorous method, where the transverse optical force is negligible small and the longitudinal force F_z ≃ 5.35 fN. The maximum intensity within the main lobe is also normalized to 1 mWµm⁻² here. Meanwhile, the optical force acting on the same particle by a plane wave can be easily obtained based on the formula $F_{z} = 4 π ε_{0} E_{p 0}^{2} {(ε_{r} - 1)}^{2} {(k R)}^{6} / [3 {(ε_{r} + 2)}^{2} k^{2}] = 5.62 fN$ [41], where the amplitude of plane wave is E_p₀ = 0.868 × 10⁶ V/m, corresponding to the optical intensity of the plane wave equal to 1 mWµm⁻². The analytical result and our numerical calculation coincide with each other, indicating the effectiveness and accuracy of our rigorous full-wave solution.

4. Conclusion

In conclusion, based on the GLMT and Maxwell stress tensor approach, a rigorous solution is presented to evaluate the optical forces and field profiles beyond paraxial approximation for a 2D vector Airy beam with different polarizations. We obtain the partial wave expansion coefficients of the incident Airy beam for linear, radial and azimuthal polarizations by using the angular spectrum representation. Based on these coefficients the field distributions are exactly calculated in terms of the summation over VSWFs, from which the self-accelerating and self-healing effects of the Airy beam are demonstrated. In addition, the optical forces are examined for both a dielectric spherical particle with an arbitrary size and a gold nanoparticle illuminated by a 2D Airy beam with linear polarization. The simulation results indicate that the dielectric particle and the gold nanoparticle are trapped within the main lobe or nearby side-lobes via the transverse optical forces and accelerated along a curved trajectory based on the longitudinal optical forces.

Appendix

In this appendix, the formulations of the partial wave expansion coefficients of an incident 2D Airy beam are derived in detail for different polarizations by using Eq. (12) in the main text based on the vector angular spectrum representation. After substituting the vector angular spectrum (see Eq. (4)) into the Eq. (1), the electrical field in the region z > z_b reads,

\begin{array}{l} E (x, y, z) = \int \int_{- \infty}^{\infty} d k_{x} d k_{y} \frac{E_{0}}{4 π^{2}} w_{x} w_{y} e^{\frac{1}{3} {(α_{x} - i k_{x} w_{x})}^{3}} \\ \times e^{\frac{1}{3} {(α_{y} - i k_{y} w_{y})}^{3}} e^{- i (k_{x} k_{b} + k_{y} y_{b} + k_{z} z_{b})} e^{i (k_{x} x + k_{y} y + k_{z} z)} s (α, β), \end{array}

and the magnetic field can be obtained via,

H (x, y, z) = \frac{1}{i ω μ_{0} μ_{b}} \nabla \times E (x, y, z) .

where dk_xdk_y = k² sin α cos αdαdβ. Using the relation e_r = sin θ cos ϕe_x + sin θ sin ϕe_y + cos θe_z, one has e_r · E = sin θ cos ϕ[e_x · E(x, y, z)] + sin θ sin ϕ[e_y · E(x, y, z)] + cos θ[e_z · E(x, y, z)]. After substituting the factor (e_r · E) into the Eq. (12), then the partial wave expansion coefficient p_mn reads,

\begin{array}{l} p_{m n} = C_{0} \int_{ϕ = 0}^{2 π} \int_{θ = 0}^{π} \int_{α = 0}^{π / 2} \int_{β = 0}^{2 π} d β d α d θ d ϕ C_{1} C_{2} e^{- i m ϕ} e^{i k r \cos α \cos θ} e^{i k r \sin α \sin θ \cos (β - ϕ)} \\ \times \sin α \sin θ P_{n}^{m} (\cos θ) \\ \times [- \cos θ \cos α (s_{x} \cos β + s_{y} \sin β) + s_{x} \cos ϕ \cos α \sin θ + s_{y} \sin ϕ \cos α \sin θ], \end{array}

where,

\begin{array}{l} C_{0} = \frac{i^{1 - n}}{4 π | E_{0} |} γ_{m n}^{1 / 2} \frac{k r}{j_{n} (k r)}, \\ C_{1} = \frac{E_{0}}{4 π^{2}} w_{x} w_{y} k^{2} e^{\frac{1}{3} {(α_{x} - i k w_{x} \cos β \sin α)}^{3} + \frac{1}{3} {(α_{y} - i k w_{y} \sin β \sin α)}^{3}}, \\ C_{2} = e^{- i k [x_{b} \sin α \cos β + y_{b} \sin α \sin β + z_{b} \cos α]} . \end{array}

Similarly, after substituting the factor (e_r · H) into the Eq. (12), the partial wave expansion coefficient q_mn reads,

\begin{array}{l} q_{m n} = C_{0}^{'} \int_{ϕ = 0}^{2 π} \int_{θ = 0}^{π} \int_{α = 0}^{π / 2} \int_{β = 0}^{2 π} d β d α d θ d ϕ C_{1}^{'} C_{2} e^{- i m ϕ} e^{i k r \cos a \cos θ} e^{i k r \sin α \sin θ \cos (β - ϕ)} \\ \times \sin α \sin θ P_{n}^{m} (\cos θ) {\sin α \cos α \cos θ (s_{y} \cos β - s_{x} \sin β) \\ - \cos ϕ \sin θ [s_{x} \sin^{2} α \sin β \cos β + s_{y} (\cos^{2} α + \sin^{2} α \sin^{2} β)] \\ + \sin ϕ \sin θ [s_{x} (\cos^{2} α + \sin^{2} α \cos^{2} β + s_{y} \sin^{2} α \sin β \cos β)]}, \end{array}

where,

\begin{array}{l} C_{0}^{'} = - \frac{i^{- n} Z}{4 π | E_{0} |} γ_{m n}^{1 / 2} \frac{k r}{j_{n} (k r)} = i Z C_{0}, \\ C_{1}^{'} = C_{1} \frac{k}{ω μ_{0} μ_{b}} = C_{1} \frac{1}{Z}, \end{array}

we can see that

C_{0}^{'} C_{1}^{'} = i C_{0} C_{1}

. The integrals p_mn and q_mn are difficult to calculate directly, so we will reduce the multiple integrals into double integrals in the next step, in order to calculate the partial wave expansion coefficients accurately.

Integrals with respect to the variable ϕ over the interval from 0 to 2π

We integrate with respect to the variable ϕ firstly, based on the mathematical relations,

\int_{0}^{2 π} e^{i x \cos (β - ϕ)} e^{- i m ϕ} [\begin{matrix} \cos ϕ \\ \sin ϕ \\ 1 \end{matrix}] d ϕ = - π i^{m} e^{- i m β} [\begin{matrix} i J_{m - 1} (x) e^{i β} - i J_{m + 1} (x) e^{- i β} \\ J_{m - 1} (x) e^{i β} + J_{m + 1} (x) e^{- i β} \\ - 2 J_{m} (x) \end{matrix}],

which are derived from the integral represent of the Bessel function [42],

J_{m} (x) = \frac{1}{2 π} \int_{0}^{2 π} \exp (i x \sin ϕ - i m ϕ) d ϕ .

Substitute the factor x = kr sin α sin θ into the Eq. (22), and compare it with the Eqs. (18) and (20), then the partial wave expansion coefficients read,

\begin{array}{l} p_{m n} = \int_{θ = 0}^{π} \int_{α = 0}^{π / 2} \int_{β = 0}^{2 π} d β d α d θ d C_{0} C_{1} C_{2} e^{i k r \cos α \cos θ} \sin α \sin θ P_{n}^{m} (\cos θ) (- π) i^{m} e^{- i m β} \\ \times {- \cos θ \cos α (s_{x} \cos β + s_{y} \sin β) (- 2) J_{m} (k r \sin α \sin θ) \\ + s_{x} \cos α \sin θ [i J_{m - 1} (k r \sin α \sin θ) e^{i β} - i J_{m + 1} (k r \sin α \sin θ) e^{- i β}] \\ + s_{y} \cos α \sin θ [J_{m - 1} (k r \sin α \sin θ) e^{i β} + J_{m + 1} (k r \sin α \sin θ) e^{- i β}], \end{array}

and

\begin{array}{l} q_{m n} = \int_{θ = 0}^{π} \int_{α = 0}^{π / 2} \int_{β = 0}^{2 π} d β d α d θ d C_{0}^{'} C_{1}^{'} C_{2} e^{i k r \cos α \cos θ} \sin α \sin θ P_{n}^{m} (\cos θ) (- π) i^{m} e^{- i m β} \\ \times {- \sin α \cos α \cos θ (s_{y} \cos β - s_{x} \sin β) (- 2) J_{m} (k r \sin α \sin θ) \\ - \sin θ [s_{x} \sin^{2} α \sin β \cos β + s_{y} (\cos^{2} α + \sin^{2} α \sin^{2} β)] \\ \times [i J_{m - 1} (k r \sin α \sin θ) e^{i β} - i J_{m + 1} (k r \sin α \sin θ) e^{- i β}] \\ + \sin θ [s_{x} (\cos^{2} α + \sin^{2} α \cos^{2} β + s_{y} \sin^{2} α \sin β \cos β)] \\ \times [J_{m - 1} (k r \sin α \sin θ) e^{i β} + J_{m + 1} (k r \sin α \sin θ) e^{- i β}]} . \end{array}

Integrals with respect to the variable θ over the interval from 0 to π

By using the mathematical identities [42],

J_{m - 1} (x) = \frac{m}{x} J_{m} (x) + J_{m}^{'} (x), J_{m + 1} (x) = \frac{m}{x} J_{m} (x) - J_{m}^{'} (x),

with x = kr sin α sin θ here, the Bessel functions J_m₋₁(x) and J_m₊₁(x) are replaced by the factors J_m(x) and

J_{m}^{'} (x)

. For the factor

J_{m}^{'} (x)

, we have,

J_{m}^{'} (k r \sin α \sin θ) = \frac{d J_{m} (k r \sin α \sin θ)}{d (k r \sin α \sin θ)} = \frac{1}{k r \sin θ \cos α} \frac{d J_{m} (k r \sin α \sin θ)}{d α},

then all the factors

J_{m}^{'} (k r \sin α \sin θ)

in the expansion coefficients p_mn and q_mn are replaced via the above mathematical relation. In next step, the following mathematical relation is applied to simplify the expansion coefficients p_mn and q_mn,

\begin{array}{l} \frac{d}{d α} [e^{i k r \cos α \cos θ} P_{n}^{m} (\cos θ) J_{m} (k r \sin α \sin θ)] \\ = - i k r \sin α \cos θ e^{i k r \cos α \cos θ} P_{n}^{m} (\cos θ) J_{m} (k r \sin α \sin θ) \\ + e^{i k r \cos α \cos θ} P_{n}^{m} (\cos θ) \frac{d_{m} J_{m} (k r \sin α \sin θ)}{d α} . \end{array}

Let $C_{θ} = J_{m} (k r \sin α \sin θ) P_{n}^{m} (\cos θ) e^{i k r \cos α \cos θ}$ , then we can obtain the following relation via simplification of the above formula,

\frac{d J_{m} (x)}{d α} = i k r \sin α \cos θ J_{m} (x) + \frac{d C_{θ}}{d α} e^{- i k r \cos α \cos θ} / P_{n}^{m} (\cos θ) .

After substituting the above formula into the expansion coefficients p_mn and q_mn, the partial wave expansion coefficients can be simplified into,

\begin{array}{l} p_{m n} = \int_{α = 0}^{π / 2} \int_{β = 0}^{2 π} \int_{θ = 0}^{π} d β d α d θ (- π) i^{m} C_{0} C_{1} C_{2} e^{- i m β} \frac{1}{k r} \\ \times 2 \sin θ [\frac{d C_{θ}}{d α} i \sin α (s_{x} \cos β + s_{y} \sin β) + C_{θ} m \cos α (s_{y} \cos β - s_{x} \sin β)], \end{array}

and

\begin{array}{l} q_{m n} = \int_{α = 0}^{π / 2} \int_{β = 0}^{2 π} \int_{θ = 0}^{π} d β d α d θ (- π) i^{m} C_{0}^{'} C_{1}^{'} C_{2} e^{- i m β} \frac{1}{k r} \\ \times 2 \sin θ [\frac{d C_{θ}}{d α} i \sin α \cos α (s_{x} \sin β - s_{y} \cos β) + C_{θ} m (s_{x} \cos β + s_{y} \sin β)] . \end{array}

By using the mathematical identity in the following [27, 43, 44],

\int_{0}^{π} C_{θ} \sin θ d θ = 2 i^{n - m} P_{n}^{m} (\cos α) j_{n} (k r),

and then we can obtain the following relation,

\frac{d}{d α} \int_{0}^{π} C_{θ} \sin θ d θ = \int_{0}^{π} \frac{d C_{θ}}{d α} \sin θ d θ = 2 i^{n - m} j_{n} (k r) \frac{d}{d α} P_{n}^{m} (\cos α) .

Based on the above two formulas, we can integrate with respect to the variable θ, and the partial wave expansion coefficients are written as,

\begin{array}{l} p_{m n} = \int_{α = 0}^{π / 2} \int_{β = 0}^{2 π} d β α C_{0} C_{1} C_{2} (- π) 4 i^{n} e^{- i m β} \frac{j_{n} (k r)}{k r} \\ \times [i \frac{d P_{n}^{m} (\cos α)}{d α} \sin α (s_{x} \cos β + s_{y} \sin β) + m P_{n}^{m} (\cos α) \cos α (- s_{x} \sin β + s_{y} \cos β)], \end{array}

and

\begin{array}{l} q_{m n} = \int_{α = 0}^{π / 2} \int_{β = 0}^{2 π} d β d α C_{0}^{'} C_{1}^{'} C_{2} (- π) 4 i^{n} e^{- i m β} \frac{j_{n} (k r)}{k r} \\ \times [i \frac{d P_{n}^{m} (\cos α)}{d α} \sin α (s_{x} \sin β - s_{y} \cos β) + m P_{n}^{m} (\cos α) \cos α (s_{x} \cos β + s_{y} \sin β)] . \end{array}

Two auxiliary functions are introduced and defined as follows,

π_{m n} \equiv π_{m n} (\cos α) = \frac{m}{\sin α} P_{n}^{m} (\cos α), τ_{m n} \equiv τ_{m n} (\cos α) = \frac{d}{d α} P_{n}^{m} (\cos α),

then the partial wave expansion coefficients can be simplified into,

\begin{array}{l} p_{m n} = \int_{α = 0}^{π / 2} \int_{β = 0}^{2 π} d β d α C_{0} C_{1} C_{2} (- π) 4 i^{n} e^{- i m β} \frac{j_{n} (k r)}{k r} \\ \times \sin α [i (s_{x} \cos β + s_{y} \sin β) τ_{m n} + (- s_{x} \sin β + s_{y} \cos β) \cos α π_{m n}], \end{array}

and

\begin{array}{l} q_{m n} = \int_{α = 0}^{π / 2} \int_{β = 0}^{2 π} d β d α C_{0}^{'} C_{1}^{'} C_{2} (- π) 4 i^{n} e^{- i m β} \frac{j_{n} (k r)}{k r} \\ \times \sin α [i (s_{x} \sin β - s_{y} \cos β) \cos α τ_{m n} + (s_{x} \cos β + s_{y} \sin β) π_{m n}] . \end{array}

At last, after substituting the values of factor C₀ and C₁ and applying the relation $C_{0}^{'} C_{1}^{'} = i C_{0} C_{1}$ , the partial wave expansion coefficients are written as,

\begin{array}{l} p_{m n} = \frac{k^{2}}{4 π^{2}} w_{x} w_{y} γ_{m n}^{1 / 2} \int_{α = 0}^{π / 2} \int_{β = 0}^{2 π} d β d α \\ \times \sin α [(s_{x} \cos β + s_{y} \sin β) τ_{m n} + i (s_{x} \sin β - s_{y} \cos β) \cos α π_{m n}] \\ \times e^{- i m β} e^{\frac{1}{3} {(α_{x} - i k w_{x} \cos β \sin α)}^{3} + \frac{1}{3} {(α_{y} - i k w_{y} \sin β \sin α)}^{3}} e^{- i k [x_{b} \sin α \cos β + y_{b} \sin α \sin β + z_{b} \cos α]}, \end{array}

and

\begin{array}{l} q_{m n} = \frac{k^{2}}{4 π^{2}} w_{x} w_{y} γ_{m n}^{1 / 2} \int_{α = 0}^{π / 2} \int_{β = 0}^{2 π} d β d α \\ \times \sin α [(s_{x} \cos β + s_{y} \sin β) π_{m n} + i (s_{x} \sin β - s_{y} \cos β) \cos α τ_{m n}] \\ \times e^{- i m β} e^{\frac{1}{3} {(α_{x} - i k w_{x} \cos β \sin α)}^{3} + \frac{1}{3} {(α_{y} - i k w_{y} \sin β \sin α)}^{3}} e^{- i k [x_{b} \sin α \cos β + y_{b} \sin α \sin β + z_{b} \cos α]} . \end{array}

We can see that the expansion coefficients q_mn can be obtained from the coefficients p_mn by exchanging the two auxiliary functions π_mn(cos α) and τ_mn(cos α).

Simplifications for different polarizations

For the linear or circular polarizations, s_x and s_y are constants, i.e., (s_x, s_y) = (1, 0) for x-polarization, (s_x, s_y) = (0, 1) for y-polarization and (s_x, s_y) = (1, ±i) for circular polarizations. Also we can set (s_x, s_y) = (− sin β, cos β) to obtain the TE mode and (s_x, s_y) = (cos β, sin β) for the TM mode.

After expanding the factor $e^{- i m β} e^{\frac{1}{3} {(α_{x} - i k w_{x} \cos β \sin α)}^{3} + \frac{1}{3} {(α_{y} - i k w_{y} \sin β \sin α)}^{3}}$ and using the following relations,

\cos^{2} β = \frac{1 + \cos 2 β}{2}, \sin^{2} β = \frac{1 - \sin 2 β}{2}, \cos^{3} β = \frac{3 \cos β + \cos 3 β}{4}, \sin^{3} β = \frac{3 \sin β - \sin 3 β}{4},

then the partial wave expansion coefficients read,

\begin{array}{l} p_{m n} = \frac{k^{2}}{4 π^{2}} w_{x} w_{y} γ_{m n}^{1 / 2} \int_{α = 0}^{π / 2} \int_{β = 0}^{2 π} d β d α \\ \times \sin α [(s_{x} \cos β + s_{y} \sin β) τ_{m n} + i (s_{x} \sin β - s_{y} \cos β) \cos α π_{m n}] \\ \times e^{- i k z_{b} \cos α} e^{t_{0}} e^{- i m β} e^{i t_{1} \sin β} e^{i s_{1} \cos β} e^{s_{2} \cos 2 β} e^{i t_{3} \sin 3 β} e^{i s_{3} \cos 3 β}, \end{array}

and

\begin{array}{l} q_{m n} = \frac{k^{2}}{4 π^{2}} w_{x} w_{y} γ_{m n}^{1 / 2} \int_{α = 0}^{π / 2} \int_{β = 0}^{2 π} d β d α \\ \times \sin α [(s_{x} \cos β + s_{y} \sin β) π_{m n} + i (s_{x} \sin β - s_{y} \cos β) \cos α τ_{m n}] \\ \times e^{- i k z_{b} \cos α} e^{t_{0}} e^{- i m β} e^{i t_{1} \sin β} e^{i s_{1} \cos β} e^{s_{2} \cos 2 β} e^{i t_{3} \sin 3 β} e^{i s_{3} \cos 3 β}, \end{array}

where the factors t₀, t₁, s₁, s₂, t₃ and s₃ are defined in the main text. By using the factor I_c (m) defined in the main text (see Eq. (15)) and the polarization parameters, the partial wave expansion coefficients p_mn and q_mn can be simplified into the Eqs. (13) and (14).

Funding

National Natural Science Foundation of China (NSFC) (11404394, 11574055, 11574275); Fundamental Research Funds for the Central Universities (2014QNA60).

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Rigorous full-wave calculation of optical forces on dielectric and metallic microparticles immersed in a vector Airy beam

Abstract

1. Introduction

2. Theory and formulations

2.1. Description of a 2D Airy beam

2.2. Optical force based on the GLMT

2.3. Partial wave expansion coefficients of the incident Airy beam

3. Results and discussion

4. Conclusion

Appendix

Integrals with respect to the variable ϕ over the interval from 0 to 2π

Integrals with respect to the variable θ over the interval from 0 to π

Simplifications for different polarizations

Funding

References and links

Cited By

Figures (5)

Equations (46)

Optics Express