Abstract

Based on the generalized Lorenz-Mie theory and the Maxwell stress tensor approach we present the first rigorous full-wave solution of the optical forces acting on spherical microparticles immersed in a two-dimensional vector Airy beam beyond the paraxial approximation. The critical aspect lies in evaluating efficiently and accurately the partial wave expansion coefficients of the incident Airy beam, which are achieved by using the vector angular spectrum representation for a variety of polarizations. The optical field distributions are then simulated to show the self-accelerating and self-healing effects of the Airy beam. The dielectric and gold microparticles are shown to be trapped within the main lobe or the nearby side-lobes mostly by the transverse gradient optical force while driven forward along the parabolic trajectory of the Airy beam by the longitudinal scattering force. It is thus demonstrated theoretically that the vector Airy beam has the capability of precisely transporting both dielectric and metallic microparticles along the prespecified curved paths.

© 2017 Optical Society of America

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 (xb, yb, zb). 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 > zb reads,

Einc(x,y,z)=A(kx,ky;zb)ei[kxx+kyy+kz(zzb)]dkxdky,
where (kx, ky, kz) is the wave vector of each expanded plane waves propagating along the direction (sin α cos β, sin α sin β, cos α) with kx = k sin α cos β, ky = k sin α sin β, kz = 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 + Azez can be obtained based on the incident electric field in the initial plane z = zb.

Considering the fact that in the initial plane (z = zb) 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,zb)=E0Ai(xxbwx)Ai(yybwy)exp(αxxxbwx+αyyybwy)p(x,y),
with the parameters αx > 0 and αy > 0 measuring the power conveyed by beams, wx and wy determining the size of the main lobe and p(x, y) = px (x, y)ex + py (x, y)ey 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, zb) in the initial plane, the transverse component A reads [17],

A(kx,ky;zb)=E04π2wxwye13(αxikxwx)3e13(αyikywy)3ei(kxxb+kyyb)s(α,β),
and the z-component of vector angular spectrum Az=[kxkzAx(kx,ky)+kykzAy(kx,ky)] is obtained by considering the divergenceless feature of the electric field in background medium, i. e., · Einc = 0. As a result, the vector angular spectrum is given by
A=E04π2wxwye13(αxikxwx)3e13(αyikywy)3ei(kxxb+kyyb)s(α,β),
where the vector s(α, β) = s(α, β) + szez = sxex + syey + szez, 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 sz is determined by sz=(kxkzsx+kykzsy). The factors ex, ey and ez 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 pz ≠ 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, (sx, sy) = (− sin β, cos β) is set and then sz = 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 (sx, sy) = (cos β, sin β) and the z-component reads sz = − 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 Einc can be expanded in terms of the VSWFs including Nnm(1)(k,r) and Mnm(1)(k,r), with the suppressed time harmonic factor eiωt throughout this paper, namely [20, 26–29],

Einc(r,θ,ϕ)=n,miEmn[pmnNnm(1)(k,r)+qmnMnm(1)(k,r)],
where (r, θ, ϕ) is the spherical coordinates corresponding to (x, y, z) with the origin locating on the particle center, Emn=E0inγmn1/2 with γmn=[(2n+1)(nm)!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 Nnm(3)(k,r) and Mnm(3)(k,r) [20, 26–29],
Esca(r,θ,ϕ)=n,miEmn[amnNnm(3)(k,r)+bmnMnm(3)(k,r)],
where the partial wave expansion coefficients of the scattered field amn and bmn can be obtained via the Mie scattering coefficients an and bn, namely [19],
amn=anpmn,bmn=bnqmn.

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=Sr^TdS,
where r^ is the unit normal vector on the closed surface and the time-averaged Maxwell’s stress tensor reads,
T=12Re[εEE+μHH12(εEE+μHH)I],
with I being the unit tensor. Considering the lossless background medium, three components of the optical force Fx, Fy and Fz can be simplified to the form expressed by the partial wave expansion coefficients of the incident and scattered fields [30, 31],
Fx=Re[F1],Fy=Im[F1],Fz=Re[F2],
where the factors F1 and F2 are defined as,
F1=2πεk2|E0|2n,m[c11F1(1)c12F1(2)+c13F1(3)],F2=4πεk2|E0|2n,m[c21F2(1)+c22F2(2)],
where the coefficients are,
c11=[(nm)(n+m+1)n2(n+1)2]1/2,c12[n(n+2)(n+m+1)(n+m+2)(n+1)2(2n+1)(2n+3)]1/2,c13=[n(n+2)(nm)(nm+1)(n+1)2(2n+1)(2n+3)]1/2,c21=[n(n+2)(nm+1)(n+m+1)(n+1)2(2n+1)(2n+3)]1/2,c22=mn(n+1),
and
F1(1)=a˜mnb˜m1n+b˜mna˜m1np˜mnq˜m1nq˜mnp˜m1n,F1(2)=a˜mna˜m1n1+b˜mnb˜m1n1p˜mnp˜m1n1q˜mnq˜m1n1,F1(3)=a˜mn1a˜m1n+b˜mn1b˜m1np˜mn1p˜m1nq˜mn1q˜m1n,F2(1)=a˜mna˜mn1+b˜mnb˜mn1p˜mnp˜mn1q˜mnq˜mn1,F2(2)=a˜mnb˜mn*p˜mnq˜mn*,
where the index m1 = m + 1 and n1 = n + 1, and
a˜mn=amn12pmn,p˜mn=12pmn,b˜mn=bmn12qmn,q˜mn=12qmn.

2.3. Partial wave expansion coefficients of the incident Airy beam

Partial wave expansion coefficients of the incident Airy beam, pmn and qmn, 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

pmn=i1n4πE0γmn1/2krjn(kr)θ=0πϕ=02π[erE(r,θ,ϕ)]Pnm(cosθ)eimϕsinθdθdϕ,qmn=inZ4πE0γmn1/2krjn(kr)θ=0πϕ=02π[erH(r,θ,ϕ)]Pnm(cosθ)eimϕsinθdθdϕ,
where jn (kr) is the spherical Bessel function with an order n, Pnm(cosθ) is the first kind associated Legendre function, er is the unit vector in the spherical coordinates system, and Z=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,
pmn=k24π2wxwyγmn1/2eαx3+αy33α=0π/2dαsinαeikzbcosαek2wx2αx+wy2αy2sin2α×{12(px+ipy)(τmnπmncosα)Ic(m+1)+ 12(pxipy)(τmn+πmncosα)Ic(m1),linear and circular polarizationτmnIc(m),radial polarizationiπmncosαIc(m),azimuthal polarization
qmn=k24π2wxwyγmn1/2eαx3+αy33α=0π/2dαsinαeikzbcosαek2wx2αx+wy2αy2sin2α×{12(px+ipy)(πmnτmncosα)Ic(m+1)+ 12(pxipy)(πmn+τmncosα)Ic(m1),linear and circular polarizationπmnIc(m),radial polarizationiτmncosαIc(m),azimuthal polarization
where the two auxiliary functions are defined by πmn(cosα)=msinαPnm(cosα) and τmn(cosα)=ddαPnm(cosα). The factor Ic (m) is an integration and can be evaluated by,
Ic(m)=β=02πdβeimβei(t1sinβ+s1cosβ+t3sin3β+s3cos3β)+s2cos2β,
with parameters t1=(k3wy3sin3α/4kwyαy2sinαkybsinα), s1=(k3wx3sin3α/4kwxαx2sinαkxbsinα), s2=k2sin2α(wx2αxwy2αy)/2, t3=k3wy3sin3α/12, and s3=k3wx3sin3α/12. We can see that the expansion coefficients qmn can be obtained from the coefficients pmn by exchanging the two auxiliary functions πmn(cos α) and τmn(cos α). Accordingly to the simplified Eqs. (13) and (14), we have developed the Fortran programs, based on which the expansion coefficients pmn and qmn 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, wx = wy = 2λ, αx = αy = 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=z2/4k2wx20.937173wx and y=z2/4k2wy20.937173wy [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 wx = wy = 2λ. In contrast, for the strongly non-paraxial Airy beam with wx = wy = 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 εr = 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.

 figure: Fig. 1

Fig. 1 Self-accelerating and self-healing effects of a linear polarized Airy beam with oscillation along x-axis and wx = wy = 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 εr = 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 wx = wy = 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.

 figure: Fig. 2

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 εr = 2.53. For the longitudinal optical forces Fz, the maximum value of Fz 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 Fz ≃ 3.7 pN within the nearby side-lobes. As a result, these longitudinal forces Fz 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.

 figure: Fig. 3

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 εr = 2.53 is located within the transverse plane z = 10λ. The map of the longitudinal optical force Fz (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 (Fx2+Fy2)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 (Fx2+Fy2)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 εr = −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 Fz 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 Fz ≃ 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.

 figure: Fig. 4

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 εr = −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 Fz (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.

 figure: Fig. 5

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 RT=eUmax/kBT1 [36] with the Boltzmann constant kB and the temperature T = 300 K. And the potential depth Umax is obtained by Umax=ε0Re[α]Emax2/2 [36], where the maximum electric field within in the main lobe is Emax = 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 RT ≃ 8.31 × 10−14 satisfies the condition RT ≪ 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 wx and wy. As an example we consider an Airy beam with wx = wy = 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 εr = 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 Fz ≃ 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 Fz=4πε0Ep02(εr1)2(kR)6/[3(εr+2)2k2]=5.62fN [41], where the amplitude of plane wave is Ep0 = 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 > zb reads,

E(x,y,z)=dkxdkyE04π2wxwye13(αxikxwx)3×e13(αyikywy)3ei(kxkb+kyyb+kzzb)ei(kxx+kyy+kzz)s(α,β),
and the magnetic field can be obtained via,
H(x,y,z)=1iωμ0μb×E(x,y,z).
where dkxdky = k2 sin α cos αdαdβ. Using the relation er = sin θ cos ϕex + sin θ sin ϕey + cos θez, one has er · E = sin θ cos ϕ[ex · E(x, y, z)] + sin θ sin ϕ[ey · E(x, y, z)] + cos θ[ez · E(x, y, z)]. After substituting the factor (er · E) into the Eq. (12), then the partial wave expansion coefficient pmn reads,
pmn=C0ϕ=02πθ=0πα=0π/2β=02πdβdαdθdϕC1C2eimϕeikrcosαcosθeikrsinαsinθcos(βϕ)×sinαsinθPnm(cosθ)×[cosθcosα(sxcosβ+sysinβ)+sxcosϕcosαsinθ+sysinϕcosαsinθ],
where,
C0=i1n4π|E0|γmn1/2krjn(kr),C1=E04π2wxwyk2e13(αxikwxcosβsinα)3+13(αyikwysinβsinα)3,C2=eik[xbsinαcosβ+ybsinαsinβ+zbcosα].

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

qmn=C0ϕ=02πθ=0πα=0π/2β=02πdβdαdθdϕC1C2eimϕeikrcosacosθeikrsinαsinθcos(βϕ)×sinαsinθPnm(cosθ){sinαcosαcosθ(sycosβsxsinβ)cosϕsinθ[sxsin2αsinβcosβ+sy(cos2α+sin2αsin2β)]+sinϕsinθ[sx(cos2α+sin2αcos2β+sysin2αsinβcosβ)]},
where,
C0=inZ4π|E0|γmn1/2krjn(kr)=iZC0,C1=C1kωμ0μb=C11Z,
we can see that C0C1=iC0C1. The integrals pmn and qmn 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,

02πeixcos(βϕ)eimϕ[cosϕsinϕ1]dϕ=πimeimβ[iJm1(x)eiβiJm+1(x)eiβJm1(x)eiβ+Jm+1(x)eiβ2Jm(x)],
which are derived from the integral represent of the Bessel function [42],
Jm(x)=12π02πexp(ixsinϕimϕ)dϕ.

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,

pmn=θ=0πα=0π/2β=02πdβdαdθdC0C1C2eikrcosαcosθsinαsinθPnm(cosθ)(π)imeimβ×{cosθcosα(sxcosβ+sysinβ)(2)Jm(krsinαsinθ)+sxcosαsinθ[iJm1(krsinαsinθ)eiβiJm+1(krsinαsinθ)eiβ]+sycosαsinθ[Jm1(krsinαsinθ)eiβ+Jm+1(krsinαsinθ)eiβ],
and
qmn=θ=0πα=0π/2β=02πdβdαdθdC0C1C2eikrcosαcosθsinαsinθPnm(cosθ)(π)imeimβ×{sinαcosαcosθ(sycosβsxsinβ)(2)Jm(krsinαsinθ)sinθ[sxsin2αsinβcosβ+sy(cos2α+sin2αsin2β)]×[iJm1(krsinαsinθ)eiβiJm+1(krsinαsinθ)eiβ]+sinθ[sx(cos2α+sin2αcos2β+sysin2αsinβcosβ)]×[Jm1(krsinαsinθ)eiβ+Jm+1(krsinαsinθ)eiβ]}.

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

By using the mathematical identities [42],

Jm1(x)=mxJm(x)+Jm(x),Jm+1(x)=mxJm(x)Jm(x),
with x = kr sin α sin θ here, the Bessel functions Jm−1(x) and Jm+1(x) are replaced by the factors Jm(x) and Jm(x). For the factor Jm(x), we have,
Jm(krsinαsinθ)=dJm(krsinαsinθ)d(krsinαsinθ)=1krsinθcosαdJm(krsinαsinθ)dα,
then all the factors Jm(krsinαsinθ) in the expansion coefficients pmn and qmn are replaced via the above mathematical relation. In next step, the following mathematical relation is applied to simplify the expansion coefficients pmn and qmn,
ddα[eikrcosαcosθPnm(cosθ)Jm(krsinαsinθ)]=ikrsinαcosθeikrcosαcosθPnm(cosθ)Jm(krsinαsinθ)+eikrcosαcosθPnm(cosθ)dmJm(krsinαsinθ)dα.

Let Cθ=Jm(krsinαsinθ)Pnm(cosθ)eikrcosαcosθ, then we can obtain the following relation via simplification of the above formula,

dJm(x)dα=ikrsinαcosθJm(x)+dCθdαeikrcosαcosθ/Pnm(cosθ).

After substituting the above formula into the expansion coefficients pmn and qmn, the partial wave expansion coefficients can be simplified into,

pmn=α=0π/2β=02πθ=0πdβdαdθ(π)imC0C1C2eimβ1kr×2sinθ[dCθdαisinα(sxcosβ+sysinβ)+Cθmcosα(sycosβsxsinβ)],
and
qmn=α=0π/2β=02πθ=0πdβdαdθ(π)imC0C1C2eimβ1kr×2sinθ[dCθdαisinαcosα(sxsinβsycosβ)+Cθm(sxcosβ+sysinβ)].

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

0πCθsinθdθ=2inmPnm(cosα)jn(kr),
and then we can obtain the following relation,
ddα0πCθsinθdθ=0πdCθdαsinθdθ=2inmjn(kr)ddαPnm(cosα).

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

pmn=α=0π/2β=02πdβαC0C1C2(π)4ineimβjn(kr)kr×[idPnm(cosα)dαsinα(sxcosβ+sysinβ)+mPnm(cosα)cosα(sxsinβ+sycosβ)],
and
qmn=α=0π/2β=02πdβdαC0C1C2(π)4ineimβjn(kr)kr×[idPnm(cosα)dαsinα(sxsinβsycosβ)+mPnm(cosα)cosα(sxcosβ+sysinβ)].

Two auxiliary functions are introduced and defined as follows,

πmnπmn(cosα)=msinαPnm(cosα),τmnτmn(cosα)=ddαPnm(cosα),
then the partial wave expansion coefficients can be simplified into,
pmn=α=0π/2β=02πdβdαC0C1C2(π)4ineimβjn(kr)kr×sinα[i(sxcosβ+sysinβ)τmn+(sxsinβ+sycosβ)cosαπmn],
and
qmn=α=0π/2β=02πdβdαC0C1C2(π)4ineimβjn(kr)kr×sinα[i(sxsinβsycosβ)cosατmn+(sxcosβ+sysinβ)πmn].

At last, after substituting the values of factor C0 and C1 and applying the relation C0C1=iC0C1, the partial wave expansion coefficients are written as,

pmn=k24π2wxwyγmn1/2α=0π/2β=02πdβdα×sinα[(sxcosβ+sysinβ)τmn+i(sxsinβsycosβ)cosαπmn]×eimβe13(αxikwxcosβsinα)3+13(αyikwysinβsinα)3eik[xbsinαcosβ+ybsinαsinβ+zbcosα],
and
qmn=k24π2wxwyγmn1/2α=0π/2β=02πdβdα×sinα[(sxcosβ+sysinβ)πmn+i(sxsinβsycosβ)cosατmn]×eimβe13(αxikwxcosβsinα)3+13(αyikwysinβsinα)3eik[xbsinαcosβ+ybsinαsinβ+zbcosα].

We can see that the expansion coefficients qmn can be obtained from the coefficients pmn by exchanging the two auxiliary functions πmn(cos α) and τmn(cos α).

Simplifications for different polarizations

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

After expanding the factor eimβe13(αxikwxcosβsinα)3+13(αyikwysinβsinα)3 and using the following relations,

cos2β=1+cos2β2,sin2β=1sin2β2,cos3β=3cosβ+cos3β4,sin3β=3sinβsin3β4,
then the partial wave expansion coefficients read,
pmn=k24π2wxwyγmn1/2α=0π/2β=02πdβdα×sinα[(sxcosβ+sysinβ)τmn+i(sxsinβsycosβ)cosαπmn]×eikzbcosαet0eimβeit1sinβeis1cosβes2cos2βeit3sin3βeis3cos3β,
and
qmn=k24π2wxwyγmn1/2α=0π/2β=02πdβdα×sinα[(sxcosβ+sysinβ)πmn+i(sxsinβsycosβ)cosατmn]×eikzbcosαet0eimβeit1sinβeis1cosβes2cos2βeit3sin3βeis3cos3β,
where the factors t0, t1, s1, s2, t3 and s3 are defined in the main text. By using the factor Ic (m) defined in the main text (see Eq. (15)) and the polarization parameters, the partial wave expansion coefficients pmn and qmn 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).

References and links

1. A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24, 156–159 (1970). [CrossRef]  

2. D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003). [CrossRef]   [PubMed]  

3. A. Ashkin, “History of optical trapping and manipulation of small-neutral particle, atoms, and molecules,” IEEE J. Sel. Top. Quantum Electron. 6, 841–856 (2000). [CrossRef]  

4. A. Ashkin, Optical Trapping and Manipulation of Neutral Particles Using Lasers (World Scientific, 2006). [CrossRef]  

5. K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nat. Photonics 5, 335–342 (2011). [CrossRef]  

6. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987). [CrossRef]  

7. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987). [CrossRef]   [PubMed]  

8. V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002). [CrossRef]   [PubMed]  

9. J. Chen, J. Ng, Z. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photonics 5, 531–534 (2011). [CrossRef]  

10. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007). [CrossRef]  

11. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32, 979–981 (2007). [CrossRef]   [PubMed]  

12. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979). [CrossRef]  

13. K. Unnikrishnan and A. R. P. Rau, “Uniqueness of the Airy packet in quantum mechanics,” Am. J. Phys. 64, 1034–1035 (1996). [CrossRef]  

14. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2, 675–678 (2008). [CrossRef]  

15. Z. Zhao, W. Zang, and J. Tian, “Optical trapping and manipulation of Mie particles with Airy beam,” J. Opt. 18, 025607 (2016). [CrossRef]  

16. H. Cheng, W. Zang, W. Zhou, and J. Tian, “Analysis of optical trapping and propulsion of Rayleigh particles using Airy beam,” Opt. Express 18, 20384–20394 (2010). [CrossRef]   [PubMed]  

17. W. Lu, J. Chen, Z. Lin, and S. Liu, “Driving a dielectric cylindrical particle with a one dimensional Airy beam: a rigorous full wave solution,” Prog. Electromagn. Res. 115, 409–422 (2011). [CrossRef]  

18. Z. Zheng, B. Zhang, H. Chen, J. Ding, and H. Wang, “Optical trapping with focused Airy beams,” Appl. Opt. 50, 43–49 (2011). [CrossRef]   [PubMed]  

19. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley and Sons: New York, 1983).

20. G. Gouesbet and G. Gréhan, Generalized Lorenz-Mie theories (Springer: Berlin, 2011). And references therein. [CrossRef]  

21. J. D. Jackson, Classical Electrodynamics, 3rd ed (John Wiley and Sons, 1999).

22. A. Zangwill, Modern Electrodynamics (Cambridge University Press, 2012).

23. Q. Ye and H. Lin, “On deriving the maxwell stress tensor method for calculating the optical force and torque on an object in harmonic electromagnetic fields,” Eur. J. Phys. 38, 045202 (2017). [CrossRef]  

24. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995). [CrossRef]  

25. L. Novotny and B. Hecht, Principles of Nano-optics (Cambridge University Press, 2006). [CrossRef]  

26. J. Ng, Z. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: Numerical simulations,” Phys. Rev. B 72, 085130 (2005). [CrossRef]  

27. X. Yu, Q. Ye, H. Chen, S. Liu, and Z. Lin, “Simple algorithm for partial wave expansion of plasmonic and evanescent fields,” Opt. Express 25, 4201–4215 (2017). [CrossRef]   [PubMed]  

28. Y. Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl. Opt. 34, 4573–4588 (1995). [CrossRef]   [PubMed]  

29. H. Chen, S. Liu, J. Zi, and Z. Lin, “Fano resonance-induced negative optical scattering force on plasmonic nanoparticles,” ACS Nano 9, 1926–1935 (2015). [CrossRef]   [PubMed]  

30. H. Chen, N. Wang, W. Lu, S. Liu, and Z. Lin, “Tailoring azimuthal optical force on lossy chiral particles in Bessel beams,” Phys. Rev. A 90, 043850 (2014). [CrossRef]  

31. N. Wang, J. Chen, S. Liu, and Z. Lin, “Dynamical and phase-diagram study on stable optical pulling force in Bessel beams,” Phys. Rev. A 87, 063812 (2013). [CrossRef]  

32. A. E. Minovich, A. E. Klein, D. N. Neshev, T. Pertsch, Y. S. Kivshar, and D. N. Christodoulides, “Airy plasmons: non-diffracting optical surface waves,” Laser Photon. Rev. 8, 221–232 (2013). [CrossRef]  

33. A. V. Novitsky and D. V. Novitsky, “Nonparaxial airy beams: role of evanescent waves,” Opt. Lett. 34, 3430–3432 (2009). [CrossRef]   [PubMed]  

34. P. M. Hansen, V. K. Bhatia, N. Harrit, and L. Oddershede, “Expanding the optical trapping range of gold nanoparticles,” Nano Lett. 5, 1937–1942 (2005). [CrossRef]   [PubMed]  

35. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972). [CrossRef]  

36. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12, 3377–3382 (2004). [CrossRef]   [PubMed]  

37. S. B. Wang and C. T. Chan, “Lateral optical force on chiral particles near a surface,” Nat. Commun. 5, 3307 (2014). [PubMed]  

38. F. J. Rodríguez-Fortuño, N. Engheta, A. Martínez, and A. V. Zayats, “Lateral forces on circularly polarizable particles near a surface,” Nat. Commun. 6, 8799 (2015). [CrossRef]   [PubMed]  

39. D. B. Ruffner and D. G. Grier, “Optical conveyors: a class of active tractor beams,” Phys. Rev. Lett. 109, 163903 (2012). [CrossRef]   [PubMed]  

40. H. J. Chen, Q. Ye, Y. W. Zhang, L. Shi, S. Y. Liu, J. Zi, and Z. F. Lin, “Reconfigurable lateral optical force achieved by selectively exciting plasmonic dark modes near Fano resonance,” Phys. Rev. A 96, 023809 (2017). [CrossRef]  

41. J. Chen, J. Ng, S. Liu, and Z. Lin, “Analytical calculation of axial optical force on a Rayleigh particle illuminated by Gaussian beams beyond the paraxial approximation,” Phys. Rev. E 80, 026607 (2009). [CrossRef]  

42. N. M. Temme, Special Functions: An Introduction to the Classical Functions of Mathematical Physics (John Wiley and Sons, 1996). [CrossRef]  

43. A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Analytical results for a Bessel function times legendre polynomials class integrals,” J. Phys. A: Math. Gen. 39, L293–L296 (2006). [CrossRef]  

44. A. A. R. Neves, A. Fontes, L. A. Padilha, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Exact partial wave expansion of optical beams with respect to an arbitrary origin,” Opt. Lett. 31, 2477–2479 (2006). [CrossRef]   [PubMed]  

References

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  1. A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24, 156–159 (1970).
    [Crossref]
  2. D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
    [Crossref] [PubMed]
  3. A. Ashkin, “History of optical trapping and manipulation of small-neutral particle, atoms, and molecules,” IEEE J. Sel. Top. Quantum Electron. 6, 841–856 (2000).
    [Crossref]
  4. A. Ashkin, Optical Trapping and Manipulation of Neutral Particles Using Lasers (World Scientific, 2006).
    [Crossref]
  5. K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nat. Photonics 5, 335–342 (2011).
    [Crossref]
  6. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
    [Crossref]
  7. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [Crossref] [PubMed]
  8. V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
    [Crossref] [PubMed]
  9. J. Chen, J. Ng, Z. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photonics 5, 531–534 (2011).
    [Crossref]
  10. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
    [Crossref]
  11. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32, 979–981 (2007).
    [Crossref] [PubMed]
  12. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
    [Crossref]
  13. K. Unnikrishnan and A. R. P. Rau, “Uniqueness of the Airy packet in quantum mechanics,” Am. J. Phys. 64, 1034–1035 (1996).
    [Crossref]
  14. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2, 675–678 (2008).
    [Crossref]
  15. Z. Zhao, W. Zang, and J. Tian, “Optical trapping and manipulation of Mie particles with Airy beam,” J. Opt. 18, 025607 (2016).
    [Crossref]
  16. H. Cheng, W. Zang, W. Zhou, and J. Tian, “Analysis of optical trapping and propulsion of Rayleigh particles using Airy beam,” Opt. Express 18, 20384–20394 (2010).
    [Crossref] [PubMed]
  17. W. Lu, J. Chen, Z. Lin, and S. Liu, “Driving a dielectric cylindrical particle with a one dimensional Airy beam: a rigorous full wave solution,” Prog. Electromagn. Res. 115, 409–422 (2011).
    [Crossref]
  18. Z. Zheng, B. Zhang, H. Chen, J. Ding, and H. Wang, “Optical trapping with focused Airy beams,” Appl. Opt. 50, 43–49 (2011).
    [Crossref] [PubMed]
  19. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley and Sons: New York, 1983).
  20. G. Gouesbet and G. Gréhan, Generalized Lorenz-Mie theories (Springer: Berlin, 2011). And references therein.
    [Crossref]
  21. J. D. Jackson, Classical Electrodynamics, 3rd ed (John Wiley and Sons, 1999).
  22. A. Zangwill, Modern Electrodynamics (Cambridge University Press, 2012).
  23. Q. Ye and H. Lin, “On deriving the maxwell stress tensor method for calculating the optical force and torque on an object in harmonic electromagnetic fields,” Eur. J. Phys. 38, 045202 (2017).
    [Crossref]
  24. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).
    [Crossref]
  25. L. Novotny and B. Hecht, Principles of Nano-optics (Cambridge University Press, 2006).
    [Crossref]
  26. J. Ng, Z. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: Numerical simulations,” Phys. Rev. B 72, 085130 (2005).
    [Crossref]
  27. X. Yu, Q. Ye, H. Chen, S. Liu, and Z. Lin, “Simple algorithm for partial wave expansion of plasmonic and evanescent fields,” Opt. Express 25, 4201–4215 (2017).
    [Crossref] [PubMed]
  28. Y. Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl. Opt. 34, 4573–4588 (1995).
    [Crossref] [PubMed]
  29. H. Chen, S. Liu, J. Zi, and Z. Lin, “Fano resonance-induced negative optical scattering force on plasmonic nanoparticles,” ACS Nano 9, 1926–1935 (2015).
    [Crossref] [PubMed]
  30. H. Chen, N. Wang, W. Lu, S. Liu, and Z. Lin, “Tailoring azimuthal optical force on lossy chiral particles in Bessel beams,” Phys. Rev. A 90, 043850 (2014).
    [Crossref]
  31. N. Wang, J. Chen, S. Liu, and Z. Lin, “Dynamical and phase-diagram study on stable optical pulling force in Bessel beams,” Phys. Rev. A 87, 063812 (2013).
    [Crossref]
  32. A. E. Minovich, A. E. Klein, D. N. Neshev, T. Pertsch, Y. S. Kivshar, and D. N. Christodoulides, “Airy plasmons: non-diffracting optical surface waves,” Laser Photon. Rev. 8, 221–232 (2013).
    [Crossref]
  33. A. V. Novitsky and D. V. Novitsky, “Nonparaxial airy beams: role of evanescent waves,” Opt. Lett. 34, 3430–3432 (2009).
    [Crossref] [PubMed]
  34. P. M. Hansen, V. K. Bhatia, N. Harrit, and L. Oddershede, “Expanding the optical trapping range of gold nanoparticles,” Nano Lett. 5, 1937–1942 (2005).
    [Crossref] [PubMed]
  35. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
    [Crossref]
  36. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12, 3377–3382 (2004).
    [Crossref] [PubMed]
  37. S. B. Wang and C. T. Chan, “Lateral optical force on chiral particles near a surface,” Nat. Commun. 5, 3307 (2014).
    [PubMed]
  38. F. J. Rodríguez-Fortuño, N. Engheta, A. Martínez, and A. V. Zayats, “Lateral forces on circularly polarizable particles near a surface,” Nat. Commun. 6, 8799 (2015).
    [Crossref] [PubMed]
  39. D. B. Ruffner and D. G. Grier, “Optical conveyors: a class of active tractor beams,” Phys. Rev. Lett. 109, 163903 (2012).
    [Crossref] [PubMed]
  40. H. J. Chen, Q. Ye, Y. W. Zhang, L. Shi, S. Y. Liu, J. Zi, and Z. F. Lin, “Reconfigurable lateral optical force achieved by selectively exciting plasmonic dark modes near Fano resonance,” Phys. Rev. A 96, 023809 (2017).
    [Crossref]
  41. J. Chen, J. Ng, S. Liu, and Z. Lin, “Analytical calculation of axial optical force on a Rayleigh particle illuminated by Gaussian beams beyond the paraxial approximation,” Phys. Rev. E 80, 026607 (2009).
    [Crossref]
  42. N. M. Temme, Special Functions: An Introduction to the Classical Functions of Mathematical Physics (John Wiley and Sons, 1996).
    [Crossref]
  43. A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Analytical results for a Bessel function times legendre polynomials class integrals,” J. Phys. A: Math. Gen. 39, L293–L296 (2006).
    [Crossref]
  44. A. A. R. Neves, A. Fontes, L. A. Padilha, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Exact partial wave expansion of optical beams with respect to an arbitrary origin,” Opt. Lett. 31, 2477–2479 (2006).
    [Crossref] [PubMed]

2017 (3)

Q. Ye and H. Lin, “On deriving the maxwell stress tensor method for calculating the optical force and torque on an object in harmonic electromagnetic fields,” Eur. J. Phys. 38, 045202 (2017).
[Crossref]

X. Yu, Q. Ye, H. Chen, S. Liu, and Z. Lin, “Simple algorithm for partial wave expansion of plasmonic and evanescent fields,” Opt. Express 25, 4201–4215 (2017).
[Crossref] [PubMed]

H. J. Chen, Q. Ye, Y. W. Zhang, L. Shi, S. Y. Liu, J. Zi, and Z. F. Lin, “Reconfigurable lateral optical force achieved by selectively exciting plasmonic dark modes near Fano resonance,” Phys. Rev. A 96, 023809 (2017).
[Crossref]

2016 (1)

Z. Zhao, W. Zang, and J. Tian, “Optical trapping and manipulation of Mie particles with Airy beam,” J. Opt. 18, 025607 (2016).
[Crossref]

2015 (2)

F. J. Rodríguez-Fortuño, N. Engheta, A. Martínez, and A. V. Zayats, “Lateral forces on circularly polarizable particles near a surface,” Nat. Commun. 6, 8799 (2015).
[Crossref] [PubMed]

H. Chen, S. Liu, J. Zi, and Z. Lin, “Fano resonance-induced negative optical scattering force on plasmonic nanoparticles,” ACS Nano 9, 1926–1935 (2015).
[Crossref] [PubMed]

2014 (2)

H. Chen, N. Wang, W. Lu, S. Liu, and Z. Lin, “Tailoring azimuthal optical force on lossy chiral particles in Bessel beams,” Phys. Rev. A 90, 043850 (2014).
[Crossref]

S. B. Wang and C. T. Chan, “Lateral optical force on chiral particles near a surface,” Nat. Commun. 5, 3307 (2014).
[PubMed]

2013 (2)

N. Wang, J. Chen, S. Liu, and Z. Lin, “Dynamical and phase-diagram study on stable optical pulling force in Bessel beams,” Phys. Rev. A 87, 063812 (2013).
[Crossref]

A. E. Minovich, A. E. Klein, D. N. Neshev, T. Pertsch, Y. S. Kivshar, and D. N. Christodoulides, “Airy plasmons: non-diffracting optical surface waves,” Laser Photon. Rev. 8, 221–232 (2013).
[Crossref]

2012 (1)

D. B. Ruffner and D. G. Grier, “Optical conveyors: a class of active tractor beams,” Phys. Rev. Lett. 109, 163903 (2012).
[Crossref] [PubMed]

2011 (4)

W. Lu, J. Chen, Z. Lin, and S. Liu, “Driving a dielectric cylindrical particle with a one dimensional Airy beam: a rigorous full wave solution,” Prog. Electromagn. Res. 115, 409–422 (2011).
[Crossref]

Z. Zheng, B. Zhang, H. Chen, J. Ding, and H. Wang, “Optical trapping with focused Airy beams,” Appl. Opt. 50, 43–49 (2011).
[Crossref] [PubMed]

K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nat. Photonics 5, 335–342 (2011).
[Crossref]

J. Chen, J. Ng, Z. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photonics 5, 531–534 (2011).
[Crossref]

2010 (1)

2009 (2)

J. Chen, J. Ng, S. Liu, and Z. Lin, “Analytical calculation of axial optical force on a Rayleigh particle illuminated by Gaussian beams beyond the paraxial approximation,” Phys. Rev. E 80, 026607 (2009).
[Crossref]

A. V. Novitsky and D. V. Novitsky, “Nonparaxial airy beams: role of evanescent waves,” Opt. Lett. 34, 3430–3432 (2009).
[Crossref] [PubMed]

2008 (1)

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2, 675–678 (2008).
[Crossref]

2007 (2)

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
[Crossref]

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32, 979–981 (2007).
[Crossref] [PubMed]

2006 (2)

A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Analytical results for a Bessel function times legendre polynomials class integrals,” J. Phys. A: Math. Gen. 39, L293–L296 (2006).
[Crossref]

A. A. R. Neves, A. Fontes, L. A. Padilha, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Exact partial wave expansion of optical beams with respect to an arbitrary origin,” Opt. Lett. 31, 2477–2479 (2006).
[Crossref] [PubMed]

2005 (2)

P. M. Hansen, V. K. Bhatia, N. Harrit, and L. Oddershede, “Expanding the optical trapping range of gold nanoparticles,” Nano Lett. 5, 1937–1942 (2005).
[Crossref] [PubMed]

J. Ng, Z. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: Numerical simulations,” Phys. Rev. B 72, 085130 (2005).
[Crossref]

2004 (1)

2003 (1)

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[Crossref] [PubMed]

2002 (1)

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[Crossref] [PubMed]

2000 (1)

A. Ashkin, “History of optical trapping and manipulation of small-neutral particle, atoms, and molecules,” IEEE J. Sel. Top. Quantum Electron. 6, 841–856 (2000).
[Crossref]

1996 (1)

K. Unnikrishnan and A. R. P. Rau, “Uniqueness of the Airy packet in quantum mechanics,” Am. J. Phys. 64, 1034–1035 (1996).
[Crossref]

1995 (1)

1987 (2)

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[Crossref]

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

1979 (1)

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
[Crossref]

1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[Crossref]

1970 (1)

A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24, 156–159 (1970).
[Crossref]

Ashkin, A.

A. Ashkin, “History of optical trapping and manipulation of small-neutral particle, atoms, and molecules,” IEEE J. Sel. Top. Quantum Electron. 6, 841–856 (2000).
[Crossref]

A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24, 156–159 (1970).
[Crossref]

A. Ashkin, Optical Trapping and Manipulation of Neutral Particles Using Lasers (World Scientific, 2006).
[Crossref]

Balazs, N. L.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
[Crossref]

Barbosa, L. C.

A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Analytical results for a Bessel function times legendre polynomials class integrals,” J. Phys. A: Math. Gen. 39, L293–L296 (2006).
[Crossref]

A. A. R. Neves, A. Fontes, L. A. Padilha, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Exact partial wave expansion of optical beams with respect to an arbitrary origin,” Opt. Lett. 31, 2477–2479 (2006).
[Crossref] [PubMed]

Baumgartl, J.

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2, 675–678 (2008).
[Crossref]

Berry, M. V.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
[Crossref]

Bhatia, V. K.

P. M. Hansen, V. K. Bhatia, N. Harrit, and L. Oddershede, “Expanding the optical trapping range of gold nanoparticles,” Nano Lett. 5, 1937–1942 (2005).
[Crossref] [PubMed]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley and Sons: New York, 1983).

Broky, J.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
[Crossref]

Cesar, C. L.

A. A. R. Neves, A. Fontes, L. A. Padilha, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Exact partial wave expansion of optical beams with respect to an arbitrary origin,” Opt. Lett. 31, 2477–2479 (2006).
[Crossref] [PubMed]

A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Analytical results for a Bessel function times legendre polynomials class integrals,” J. Phys. A: Math. Gen. 39, L293–L296 (2006).
[Crossref]

Chan, C. T.

S. B. Wang and C. T. Chan, “Lateral optical force on chiral particles near a surface,” Nat. Commun. 5, 3307 (2014).
[PubMed]

J. Chen, J. Ng, Z. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photonics 5, 531–534 (2011).
[Crossref]

J. Ng, Z. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: Numerical simulations,” Phys. Rev. B 72, 085130 (2005).
[Crossref]

Chen, H.

X. Yu, Q. Ye, H. Chen, S. Liu, and Z. Lin, “Simple algorithm for partial wave expansion of plasmonic and evanescent fields,” Opt. Express 25, 4201–4215 (2017).
[Crossref] [PubMed]

H. Chen, S. Liu, J. Zi, and Z. Lin, “Fano resonance-induced negative optical scattering force on plasmonic nanoparticles,” ACS Nano 9, 1926–1935 (2015).
[Crossref] [PubMed]

H. Chen, N. Wang, W. Lu, S. Liu, and Z. Lin, “Tailoring azimuthal optical force on lossy chiral particles in Bessel beams,” Phys. Rev. A 90, 043850 (2014).
[Crossref]

Z. Zheng, B. Zhang, H. Chen, J. Ding, and H. Wang, “Optical trapping with focused Airy beams,” Appl. Opt. 50, 43–49 (2011).
[Crossref] [PubMed]

Chen, H. J.

H. J. Chen, Q. Ye, Y. W. Zhang, L. Shi, S. Y. Liu, J. Zi, and Z. F. Lin, “Reconfigurable lateral optical force achieved by selectively exciting plasmonic dark modes near Fano resonance,” Phys. Rev. A 96, 023809 (2017).
[Crossref]

Chen, J.

N. Wang, J. Chen, S. Liu, and Z. Lin, “Dynamical and phase-diagram study on stable optical pulling force in Bessel beams,” Phys. Rev. A 87, 063812 (2013).
[Crossref]

W. Lu, J. Chen, Z. Lin, and S. Liu, “Driving a dielectric cylindrical particle with a one dimensional Airy beam: a rigorous full wave solution,” Prog. Electromagn. Res. 115, 409–422 (2011).
[Crossref]

J. Chen, J. Ng, Z. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photonics 5, 531–534 (2011).
[Crossref]

J. Chen, J. Ng, S. Liu, and Z. Lin, “Analytical calculation of axial optical force on a Rayleigh particle illuminated by Gaussian beams beyond the paraxial approximation,” Phys. Rev. E 80, 026607 (2009).
[Crossref]

Cheng, H.

Christodoulides, D. N.

A. E. Minovich, A. E. Klein, D. N. Neshev, T. Pertsch, Y. S. Kivshar, and D. N. Christodoulides, “Airy plasmons: non-diffracting optical surface waves,” Laser Photon. Rev. 8, 221–232 (2013).
[Crossref]

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32, 979–981 (2007).
[Crossref] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
[Crossref]

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[Crossref]

Cižmár, T.

K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nat. Photonics 5, 335–342 (2011).
[Crossref]

Cruz, C. H. B.

A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Analytical results for a Bessel function times legendre polynomials class integrals,” J. Phys. A: Math. Gen. 39, L293–L296 (2006).
[Crossref]

A. A. R. Neves, A. Fontes, L. A. Padilha, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Exact partial wave expansion of optical beams with respect to an arbitrary origin,” Opt. Lett. 31, 2477–2479 (2006).
[Crossref] [PubMed]

Dholakia, K.

K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nat. Photonics 5, 335–342 (2011).
[Crossref]

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2, 675–678 (2008).
[Crossref]

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[Crossref] [PubMed]

Ding, J.

Dogariu, A.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
[Crossref]

Durnin, J.

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[Crossref]

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

Engheta, N.

F. J. Rodríguez-Fortuño, N. Engheta, A. Martínez, and A. V. Zayats, “Lateral forces on circularly polarizable particles near a surface,” Nat. Commun. 6, 8799 (2015).
[Crossref] [PubMed]

Fontes, A.

A. A. R. Neves, A. Fontes, L. A. Padilha, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Exact partial wave expansion of optical beams with respect to an arbitrary origin,” Opt. Lett. 31, 2477–2479 (2006).
[Crossref] [PubMed]

A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Analytical results for a Bessel function times legendre polynomials class integrals,” J. Phys. A: Math. Gen. 39, L293–L296 (2006).
[Crossref]

Garcés-Chávez, V.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[Crossref] [PubMed]

Gouesbet, G.

G. Gouesbet and G. Gréhan, Generalized Lorenz-Mie theories (Springer: Berlin, 2011). And references therein.
[Crossref]

Gréhan, G.

G. Gouesbet and G. Gréhan, Generalized Lorenz-Mie theories (Springer: Berlin, 2011). And references therein.
[Crossref]

Grier, D. G.

D. B. Ruffner and D. G. Grier, “Optical conveyors: a class of active tractor beams,” Phys. Rev. Lett. 109, 163903 (2012).
[Crossref] [PubMed]

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[Crossref] [PubMed]

Hansen, P. M.

P. M. Hansen, V. K. Bhatia, N. Harrit, and L. Oddershede, “Expanding the optical trapping range of gold nanoparticles,” Nano Lett. 5, 1937–1942 (2005).
[Crossref] [PubMed]

Harrit, N.

P. M. Hansen, V. K. Bhatia, N. Harrit, and L. Oddershede, “Expanding the optical trapping range of gold nanoparticles,” Nano Lett. 5, 1937–1942 (2005).
[Crossref] [PubMed]

Hecht, B.

L. Novotny and B. Hecht, Principles of Nano-optics (Cambridge University Press, 2006).
[Crossref]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley and Sons: New York, 1983).

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed (John Wiley and Sons, 1999).

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[Crossref]

Kivshar, Y. S.

A. E. Minovich, A. E. Klein, D. N. Neshev, T. Pertsch, Y. S. Kivshar, and D. N. Christodoulides, “Airy plasmons: non-diffracting optical surface waves,” Laser Photon. Rev. 8, 221–232 (2013).
[Crossref]

Klein, A. E.

A. E. Minovich, A. E. Klein, D. N. Neshev, T. Pertsch, Y. S. Kivshar, and D. N. Christodoulides, “Airy plasmons: non-diffracting optical surface waves,” Laser Photon. Rev. 8, 221–232 (2013).
[Crossref]

Lin, H.

Q. Ye and H. Lin, “On deriving the maxwell stress tensor method for calculating the optical force and torque on an object in harmonic electromagnetic fields,” Eur. J. Phys. 38, 045202 (2017).
[Crossref]

Lin, Z.

X. Yu, Q. Ye, H. Chen, S. Liu, and Z. Lin, “Simple algorithm for partial wave expansion of plasmonic and evanescent fields,” Opt. Express 25, 4201–4215 (2017).
[Crossref] [PubMed]

H. Chen, S. Liu, J. Zi, and Z. Lin, “Fano resonance-induced negative optical scattering force on plasmonic nanoparticles,” ACS Nano 9, 1926–1935 (2015).
[Crossref] [PubMed]

H. Chen, N. Wang, W. Lu, S. Liu, and Z. Lin, “Tailoring azimuthal optical force on lossy chiral particles in Bessel beams,” Phys. Rev. A 90, 043850 (2014).
[Crossref]

N. Wang, J. Chen, S. Liu, and Z. Lin, “Dynamical and phase-diagram study on stable optical pulling force in Bessel beams,” Phys. Rev. A 87, 063812 (2013).
[Crossref]

W. Lu, J. Chen, Z. Lin, and S. Liu, “Driving a dielectric cylindrical particle with a one dimensional Airy beam: a rigorous full wave solution,” Prog. Electromagn. Res. 115, 409–422 (2011).
[Crossref]

J. Chen, J. Ng, Z. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photonics 5, 531–534 (2011).
[Crossref]

J. Chen, J. Ng, S. Liu, and Z. Lin, “Analytical calculation of axial optical force on a Rayleigh particle illuminated by Gaussian beams beyond the paraxial approximation,” Phys. Rev. E 80, 026607 (2009).
[Crossref]

J. Ng, Z. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: Numerical simulations,” Phys. Rev. B 72, 085130 (2005).
[Crossref]

Lin, Z. F.

H. J. Chen, Q. Ye, Y. W. Zhang, L. Shi, S. Y. Liu, J. Zi, and Z. F. Lin, “Reconfigurable lateral optical force achieved by selectively exciting plasmonic dark modes near Fano resonance,” Phys. Rev. A 96, 023809 (2017).
[Crossref]

Liu, S.

X. Yu, Q. Ye, H. Chen, S. Liu, and Z. Lin, “Simple algorithm for partial wave expansion of plasmonic and evanescent fields,” Opt. Express 25, 4201–4215 (2017).
[Crossref] [PubMed]

H. Chen, S. Liu, J. Zi, and Z. Lin, “Fano resonance-induced negative optical scattering force on plasmonic nanoparticles,” ACS Nano 9, 1926–1935 (2015).
[Crossref] [PubMed]

H. Chen, N. Wang, W. Lu, S. Liu, and Z. Lin, “Tailoring azimuthal optical force on lossy chiral particles in Bessel beams,” Phys. Rev. A 90, 043850 (2014).
[Crossref]

N. Wang, J. Chen, S. Liu, and Z. Lin, “Dynamical and phase-diagram study on stable optical pulling force in Bessel beams,” Phys. Rev. A 87, 063812 (2013).
[Crossref]

W. Lu, J. Chen, Z. Lin, and S. Liu, “Driving a dielectric cylindrical particle with a one dimensional Airy beam: a rigorous full wave solution,” Prog. Electromagn. Res. 115, 409–422 (2011).
[Crossref]

J. Chen, J. Ng, S. Liu, and Z. Lin, “Analytical calculation of axial optical force on a Rayleigh particle illuminated by Gaussian beams beyond the paraxial approximation,” Phys. Rev. E 80, 026607 (2009).
[Crossref]

Liu, S. Y.

H. J. Chen, Q. Ye, Y. W. Zhang, L. Shi, S. Y. Liu, J. Zi, and Z. F. Lin, “Reconfigurable lateral optical force achieved by selectively exciting plasmonic dark modes near Fano resonance,” Phys. Rev. A 96, 023809 (2017).
[Crossref]

Lu, W.

H. Chen, N. Wang, W. Lu, S. Liu, and Z. Lin, “Tailoring azimuthal optical force on lossy chiral particles in Bessel beams,” Phys. Rev. A 90, 043850 (2014).
[Crossref]

W. Lu, J. Chen, Z. Lin, and S. Liu, “Driving a dielectric cylindrical particle with a one dimensional Airy beam: a rigorous full wave solution,” Prog. Electromagn. Res. 115, 409–422 (2011).
[Crossref]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).
[Crossref]

Martínez, A.

F. J. Rodríguez-Fortuño, N. Engheta, A. Martínez, and A. V. Zayats, “Lateral forces on circularly polarizable particles near a surface,” Nat. Commun. 6, 8799 (2015).
[Crossref] [PubMed]

Mazilu, M.

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2, 675–678 (2008).
[Crossref]

McGloin, D.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[Crossref] [PubMed]

Melville, H.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[Crossref] [PubMed]

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

Minovich, A. E.

A. E. Minovich, A. E. Klein, D. N. Neshev, T. Pertsch, Y. S. Kivshar, and D. N. Christodoulides, “Airy plasmons: non-diffracting optical surface waves,” Laser Photon. Rev. 8, 221–232 (2013).
[Crossref]

Neshev, D. N.

A. E. Minovich, A. E. Klein, D. N. Neshev, T. Pertsch, Y. S. Kivshar, and D. N. Christodoulides, “Airy plasmons: non-diffracting optical surface waves,” Laser Photon. Rev. 8, 221–232 (2013).
[Crossref]

Neves, A. A. R.

A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Analytical results for a Bessel function times legendre polynomials class integrals,” J. Phys. A: Math. Gen. 39, L293–L296 (2006).
[Crossref]

A. A. R. Neves, A. Fontes, L. A. Padilha, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Exact partial wave expansion of optical beams with respect to an arbitrary origin,” Opt. Lett. 31, 2477–2479 (2006).
[Crossref] [PubMed]

Ng, J.

J. Chen, J. Ng, Z. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photonics 5, 531–534 (2011).
[Crossref]

J. Chen, J. Ng, S. Liu, and Z. Lin, “Analytical calculation of axial optical force on a Rayleigh particle illuminated by Gaussian beams beyond the paraxial approximation,” Phys. Rev. E 80, 026607 (2009).
[Crossref]

J. Ng, Z. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: Numerical simulations,” Phys. Rev. B 72, 085130 (2005).
[Crossref]

Novitsky, A. V.

Novitsky, D. V.

Novotny, L.

L. Novotny and B. Hecht, Principles of Nano-optics (Cambridge University Press, 2006).
[Crossref]

Oddershede, L.

P. M. Hansen, V. K. Bhatia, N. Harrit, and L. Oddershede, “Expanding the optical trapping range of gold nanoparticles,” Nano Lett. 5, 1937–1942 (2005).
[Crossref] [PubMed]

Padilha, L. A.

A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Analytical results for a Bessel function times legendre polynomials class integrals,” J. Phys. A: Math. Gen. 39, L293–L296 (2006).
[Crossref]

A. A. R. Neves, A. Fontes, L. A. Padilha, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Exact partial wave expansion of optical beams with respect to an arbitrary origin,” Opt. Lett. 31, 2477–2479 (2006).
[Crossref] [PubMed]

Pertsch, T.

A. E. Minovich, A. E. Klein, D. N. Neshev, T. Pertsch, Y. S. Kivshar, and D. N. Christodoulides, “Airy plasmons: non-diffracting optical surface waves,” Laser Photon. Rev. 8, 221–232 (2013).
[Crossref]

Rau, A. R. P.

K. Unnikrishnan and A. R. P. Rau, “Uniqueness of the Airy packet in quantum mechanics,” Am. J. Phys. 64, 1034–1035 (1996).
[Crossref]

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H. J. Chen, Q. Ye, Y. W. Zhang, L. Shi, S. Y. Liu, J. Zi, and Z. F. Lin, “Reconfigurable lateral optical force achieved by selectively exciting plasmonic dark modes near Fano resonance,” Phys. Rev. A 96, 023809 (2017).
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Figures (5)

Fig. 1
Fig. 1 Self-accelerating and self-healing effects of a linear polarized Airy beam with oscillation along x-axis and wx = wy = 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 εr = 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 wx = wy = 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).
Fig. 2
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).
Fig. 3
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 εr = 2.53 is located within the transverse plane z = 10λ. The map of the longitudinal optical force Fz (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.
Fig. 4
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 εr = −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 Fz (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.
Fig. 5
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.

Equations (46)

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E inc ( x , y , z ) = 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 ,
E ( x , y , z b ) = E 0 Ai ( x x b w x ) Ai ( y y b w y ) exp ( α x x x b w x + α y y y b w y ) p ( x , y ) ,
A ( k x , k y ; z b ) = E 0 4 π 2 w x w y e 1 3 ( α x i k x w x ) 3 e 1 3 ( α y i k y w y ) 3 e i ( k x x b + k y y b ) s ( α , β ) ,
A = E 0 4 π 2 w x w y e 1 3 ( α x i k x w x ) 3 e 1 3 ( α y i k y w y ) 3 e i ( k x x b + k y y b ) s ( α , β ) ,
E inc ( r , θ , ϕ ) = n , m i E m n [ p m n N n m ( 1 ) ( k , r ) + q m n M n m ( 1 ) ( k , r ) ] ,
E sca ( r , θ , ϕ ) = n , m i E m n [ a m n N n m ( 3 ) ( k , r ) + b m n M n m ( 3 ) ( k , r ) ] ,
a m n = a n p m n , b m n = b n q m n .
F = S r ^ T d S ,
T = 1 2 Re [ ε E E + μ H H 1 2 ( ε E E + μ H H ) I ] ,
F x = Re [ F 1 ] , F y = Im [ F 1 ] , F z = Re [ F 2 ] ,
F 1 = 2 π ε k 2 | E 0 | 2 n , m [ c 11 F 1 ( 1 ) c 12 F 1 ( 2 ) + c 13 F 1 ( 3 ) ] , F 2 = 4 π ε k 2 | E 0 | 2 n , m [ c 21 F 2 ( 1 ) + c 22 F 2 ( 2 ) ] ,
c 11 = [ ( n m ) ( n + m + 1 ) n 2 ( n + 1 ) 2 ] 1 / 2 , c 12 [ n ( n + 2 ) ( n + m + 1 ) ( n + m + 2 ) ( n + 1 ) 2 ( 2 n + 1 ) ( 2 n + 3 ) ] 1 / 2 , c 13 = [ n ( n + 2 ) ( n m ) ( n m + 1 ) ( n + 1 ) 2 ( 2 n + 1 ) ( 2 n + 3 ) ] 1 / 2 , c 21 = [ n ( n + 2 ) ( n m + 1 ) ( n + m + 1 ) ( n + 1 ) 2 ( 2 n + 1 ) ( 2 n + 3 ) ] 1 / 2 , c 22 = m n ( n + 1 ) ,
F 1 ( 1 ) = a ˜ m n b ˜ m 1 n + b ˜ m n a ˜ m 1 n p ˜ m n q ˜ m 1 n q ˜ m n p ˜ m 1 n , F 1 ( 2 ) = a ˜ m n a ˜ m 1 n 1 + b ˜ m n b ˜ m 1 n 1 p ˜ m n p ˜ m 1 n 1 q ˜ m n q ˜ m 1 n 1 , F 1 ( 3 ) = a ˜ m n 1 a ˜ m 1 n + b ˜ m n 1 b ˜ m 1 n p ˜ m n 1 p ˜ m 1 n q ˜ m n 1 q ˜ m 1 n , F 2 ( 1 ) = a ˜ m n a ˜ m n 1 + b ˜ m n b ˜ m n 1 p ˜ m n p ˜ m n 1 q ˜ m n q ˜ m n 1 , F 2 ( 2 ) = a ˜ m n b ˜ m n * p ˜ m n q ˜ m n * ,
a ˜ m n = a m n 1 2 p m n , p ˜ m n = 1 2 p m n , b ˜ m n = b m n 1 2 q m n , q ˜ m n = 1 2 q m n .
p m n = i 1 n 4 π E 0 γ m n 1 / 2 k r j n ( k r ) θ = 0 π ϕ = 0 2 π [ e r E ( r , θ , ϕ ) ] P n m ( cos θ ) e i m ϕ sin θ d θ d ϕ , q m n = i n Z 4 π E 0 γ m n 1 / 2 k r j n ( k r ) θ = 0 π ϕ = 0 2 π [ e r H ( r , θ , ϕ ) ] P n m ( cos θ ) e i m ϕ sin θ d θ d ϕ ,
p m n = k 2 4 π 2 w x w y γ m n 1 / 2 e α x 3 + α y 3 3 α = 0 π / 2 d α sin α e i k z b cos α e k 2 w x 2 α x + w y 2 α y 2 sin 2 α × { 1 2 ( p x + i p y ) ( τ m n π m n cos α ) I c ( m + 1 ) +   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
q m n = k 2 4 π 2 w x w y γ m n 1 / 2 e α x 3 + α y 3 3 α = 0 π / 2 d α sin α e i k z b cos α e k 2 w x 2 α x + w y 2 α y 2 sin 2 α × { 1 2 ( p x + i p y ) ( π m n τ m n cos α ) I c ( m + 1 ) +   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
I c ( m ) = 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 ,
E ( x , y , z ) = d k x d k y E 0 4 π 2 w x w y e 1 3 ( α x i k x w x ) 3 × e 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 ) ,
H ( x , y , z ) = 1 i ω μ 0 μ b × E ( x , y , z ) .
p m n = C 0 ϕ = 0 2 π θ = 0 π α = 0 π / 2 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 ) × sin α sin θ P n m ( cos θ ) × [ 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 ] ,
C 0 = i 1 n 4 π | E 0 | γ m n 1 / 2 k r j n ( k r ) , C 1 = E 0 4 π 2 w x w y k 2 e 1 3 ( α x i k w x cos β sin α ) 3 + 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 ] .
q m n = C 0 ϕ = 0 2 π θ = 0 π α = 0 π / 2 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 ) × 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 ) ] } ,
C 0 = i n Z 4 π | E 0 | γ m n 1 / 2 k r j n ( k r ) = i Z C 0 , C 1 = C 1 k ω μ 0 μ b = C 1 1 Z ,
0 2 π e i x cos ( β ϕ ) e i m ϕ [ cos ϕ sin ϕ 1 ] d ϕ = π i m e i m β [ 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 ) ] ,
J m ( x ) = 1 2 π 0 2 π exp ( i x sin ϕ i m ϕ ) d ϕ .
p m n = θ = 0 π α = 0 π / 2 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 × { 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 ] ,
q m n = θ = 0 π α = 0 π / 2 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 × { 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 ) ] × [ 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 ) ] × [ J m 1 ( k r sin α sin θ ) e i β + J m + 1 ( k r sin α sin θ ) e i β ] } .
J m 1 ( x ) = m x J m ( x ) + J m ( x ) , J m + 1 ( x ) = m x J m ( x ) J m ( x ) ,
J m ( k r sin α sin θ ) = d J m ( k r sin α sin θ ) d ( k r sin α sin θ ) = 1 k r sin θ cos α d J m ( k r sin α sin θ ) d α ,
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 ) d m J m ( k r sin α sin θ ) d α .
d J m ( x ) d α = i k r sin α cos θ J m ( x ) + d C θ d α e i k r cos α cos θ / P n m ( cos θ ) .
p m n = α = 0 π / 2 β = 0 2 π θ = 0 π d β d α d θ ( π ) i m C 0 C 1 C 2 e i m β 1 k r × 2 sin θ [ 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 ) ] ,
q m n = α = 0 π / 2 β = 0 2 π θ = 0 π d β d α d θ ( π ) i m C 0 C 1 C 2 e i m β 1 k r × 2 sin θ [ 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 ) ] .
0 π C θ sin θ d θ = 2 i n m P n m ( cos α ) j n ( k r ) ,
d d α 0 π C θ sin θ d θ = 0 π d C θ d α sin θ d θ = 2 i n m j n ( k r ) d d α P n m ( cos α ) .
p m n = α = 0 π / 2 β = 0 2 π d β α C 0 C 1 C 2 ( π ) 4 i n e i m β j n ( k r ) k r × [ i 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 ) ] ,
q m n = α = 0 π / 2 β = 0 2 π d β d α C 0 C 1 C 2 ( π ) 4 i n e i m β j n ( k r ) k r × [ i 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 ) ] .
π m n π m n ( cos α ) = m sin α P n m ( cos α ) , τ m n τ m n ( cos α ) = d d α P n m ( cos α ) ,
p m n = α = 0 π / 2 β = 0 2 π d β d α C 0 C 1 C 2 ( π ) 4 i n e i m β j n ( k r ) k r × 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 ] ,
q m n = α = 0 π / 2 β = 0 2 π d β d α C 0 C 1 C 2 ( π ) 4 i n e i m β j n ( k r ) k r × 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 ] .
p m n = k 2 4 π 2 w x w y γ m n 1 / 2 α = 0 π / 2 β = 0 2 π d β d α × 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 ] × e i m β e 1 3 ( α x i k w x cos β sin α ) 3 + 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 ] ,
q m n = k 2 4 π 2 w x w y γ m n 1 / 2 α = 0 π / 2 β = 0 2 π d β d α × 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 ] × e i m β e 1 3 ( α x i k w x cos β sin α ) 3 + 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 ] .
cos 2 β = 1 + cos 2 β 2 , sin 2 β = 1 sin 2 β 2 , cos 3 β = 3 cos β + cos 3 β 4 , sin 3 β = 3 sin β sin 3 β 4 ,
p m n = k 2 4 π 2 w x w y γ m n 1 / 2 α = 0 π / 2 β = 0 2 π d β d α × 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 ] × 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 ,
q m n = k 2 4 π 2 w x w y γ m n 1 / 2 α = 0 π / 2 β = 0 2 π d β d α × 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 ] × 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 ,

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