Abstract

We observed in the optical tweezers experiment that some anisotropic nanorod was stably trapped in an orientation tiled to the beam axis. We explain this trapping with the T-matrix calculation. As the vector spherical wave functions do not individually satisfy the anisotropic vector wave equation, we expand the incident and scattered fields in the isotropic buffer in terms of E, and the internal field in the anisotropic nanoparticle in terms of D, and use the boundary condition for the normal components of D to compute the T-matrix. We found that when the optical axes of an anisotropic nanorod are not aligned to the nanorod axis, the nanorod may be trapped stably at a tilted angle, under which the lateral torque equals to zero and the derivative of the torque is negative.

© 2015 Optical Society of America

1. Introduction

There is increasing interest in trapping and manipulating the nanometer-sized particles by optical tweezers. In the topic of trapping the anisotropic crystal particles, the birefringent particle was trapped in the optical tweezers and rotated at a high rate by rotating the plane of the linear polarization of the trapping beam [1]. Stable trapping of a KNbO3 crystal nanorod of subwavelength-size cross section and micrometer length was demonstrated in [2]. The trapped KNbO3 nanorod showed the nonlinear optics second harmonic generation at 530 nm, which would have a potential application as a nano-scale light source for high resolution imaging. As the trap stiffness in position and orientation would be critical for the imaging resolution in the application, this trapping was analyzed using the T-matrix calculation, but for the isotropic nanorod [3]. The Mie theory gives analytical solutions for the optical scattering of spherical particles [4]. The genelarized Lorentz-Mie theory and the T-matrix method are useful for numerically computing the scattering of the particles with other geometrical shapes. The discrete dipole analysis (DDA) was also used to study the trapping of the subwavelength-size, anisotropic particle, but the study did not carry out for nanorods [5,6].

In most experiments the trapped nanorod is aligned with the trapping beam axis, as shown in [2]. In fact, the numerical analysis showed that the nanorod of large aspect ratio tends to align with the trapping beam axis, and the nanorod of small aspect ratio tends to align with the normal of the beam axis [1,3]. In this paper, we report experimental observations that some anisotropic crystal nanorods of large aspect ratio were trapped stably at a tilt angle with respect to the beam axis. To explain this observation we analyzed the trap of an anisotropic nanorod using the T-Matrix method. The T-matrix method and the vector spherical wave functions (VSWF) expansions represent some difficulties in the anisotropic media, because the VSWFs do not individually satisfy the anisotropic vector wave equation. Alternative approaches proposed using the modified VSWF bases [710]. The conventional VSWF expansions were also used for the T-matrix computation in the anisotropic media [11, 12]. In this paper, we propose a new approach to the trapping problem. As in the optical tweezers, the anisotropic particles are immersed in an isotropic aquatic buffer, we can expand the electrical intensity fields of the incident and scattered fields in the buffer medium, where E=0, to the conventional VSWF basis, and the electrical displacement field in the anisotropic medium, where D=0, to the VSWF basis, such that both VSWF expansions are legitimate. Then, we use the boundary condition at the interfaces for the normal components of D instead of the tangent components of Eas that in [13] to compute the T-matrix. Moreover, we present the internal field D in the anisotropic medium by a summation of several components D, each with the wavenumber corresponding to one non-zero and non-equal element in the tensor of indices of refraction in the anisotropic medium. This provides a flexible presentation of the internal field and more VSWF coefficents to help the convergence of the T-matrix calculation.

After the analysis of the optical scattering, we compute the stress distribution on the anisotropic nanorod surface as in [3] for isotropic nanorod, but using the Abraham stress tensor. The total trapping force and torque are computed by integrating the stress distributions over the total nanorod surfaces. Finally, to explain our experimental obeservation, we compute the lateral torque on the anisotropic nanocylinder and its derivative to demonstrate that the anisotropic crystal nanocylinder with the optical axes not aligned to the nanocylinder axis can be trapped at an equilibrium orientation, which is tilted with respect to the beam axis.

2. Experiment

The experimental set up consisted of a conventional optical tweezers in the inverted microscope configuration. The potassium niobate KNbO3 nanorods were made by crashing the KNbO3 crystals. The nanorods were typically of the length of 0.3 - 2 μm and the radius of 50 - 75 nm, as shown in Fig. 1. As the growth of the KNbO3 crystals and the crushing of the crystals to the nanorods were not well controlled, such that the optical axes of the nanorods were unknown and could be randomly tilted with respect to the nanorod axis.

 figure: Fig. 1

Fig. 1 Scanning electron microscope image of a stake of KNbO3 nanorodes.

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The trapping beam was a linearly polarized ND:YVO4 fiber laser beam at 1070 nm focused by an oil immersion lens of NA = 1.25. The sample was illuminated by a white light fiber source with an oil immersion condensor of NA = 1.20, and was imaged by a CCD camera. The nanorods and their aggregates were suspending in the water between two microscope coverslips. A nanorod rested on the bottom coverslip is shown in Fig. 2(a). When the laser beam was turned on, a nanorod was trapped. In most cases the trapped nanorod was aligned along the trapping beam axis, so that the scattered light formed a circular pattern around the beam axis [2]. However, in some cases we observed that a nanorod was trapped stably at a tilted angle with respect to the beam axis. In these cases the scattered light lost the circular symmetry, such that its image projected to the CCD camera showed a wedged shape, as shown in Fig. 2(b). Although we were not able to measure the exact tilted angle of the trapped nanorod in the experiments and we did not know the orientation of the optical axis in the trapped nanorod, the pictures and video clip in Fig. 2b report a new experimentally observed phenomenon. The following numerical analysis will show that the trapping at a tilt angle would possibly occur for the anisotropic crystal nanorod whose optical axes are not aligned to the trapping beam axis.

 figure: Fig. 2

Fig. 2 (a) A non-trapped nanorod resting on the bottom coverslip; (b) A nanorod trapped at a tilt angle. Its shape be seen to be similar to that shown in (a) in few frames later from the instant when the nanorod escapes in videoclip (Media 1).

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3. Stress in anisotropic media

The stress tensor Γis defined as the rate of the flow of momentum through the interface between two media, such that the stress applied to the interface may be computed by

σ=12Re{(Γ2Γ1)n^}n^
where n^ is the unit normal to the interface. The widely useful Maxwell stress tensor has been deducted from conservation of the linear electromagnetic momentum and is expressed as
Γij=ϵ[EiEj*+1ϵμ0BiBj12(E2+1ϵμ0B2)δij]
where Γij denotes the i,j th component in the stress tensor, ε is the permittivity of the medium. Note that the relation ρ=D=εE has been used in the deduction of Eq. (2), implying that the dielectric function εis a scalar constant, so that the Maxwell stress tensor is valid for the vacunm and isotropic medium only. From Maxwell stress tensor the stress on the interface is expressed as [3, 14, 15]
σ=12Re{n^(ε1ε21)(ε1E1n2+ε2E1t2)}n^
where a factor ½ is necessary to take the average value of the stress over one cyle of the electromagnetic field.

To analyze the stress tensor in anisotropic media, we consider the general total energy-momentum tensor. The correct form of this tensor has been debated for a long time as the Abraham-Minkowski controversy. The momentum density of the electromagnetic field in a material medium takes in the form P=D×B, due to Minkowski or P=(E×H)/c2, due to Abraham [1618]. However, as it is explained in [18], both forms may be used if the material part of the total energy-momemtum tensor is always taken into account. We assume that the material does not present a magnetic discontinuity and does not present magnetization, so that the terms related to the magnetic intensity can be removed. We negleted the material energy-momentum tensor and the electrostrictive effect related to the time derivatives of the dielectric tensor εij/t. We assume that the medium is linear, εijdoes not depend on E, but still depends on the motion, rotation and deformation of the medium. In fact, in the optical trap of nanoparticle at equilibrium state the material tensor related to the speed of the microscopic deformation, rotation or movement within the nanoparticle itself are very weak compared to the forces generated by the rest of the electromagnetic field and are neglegible, as the nanoparticles are neither flexible nor soft deformable objects [16,18]. In the rest of this paper we consider the nanorod as materially inert. Their specific properties, such as the anisotropy and the shape do not actively respond to the electromagnetic field.

In the anisotropic medium From the Minkowski stress tensor and neglecting the material counterpart, we find

Γij=[DiEj*12(D·E)δij]
We compute first
2Γn^=(nx(2ExDxED)+2ny(ExDy)+2nz(ExDz)2nx(EyDx)+ny(2EyDyED)+2nz(EyDz)2nx(EzDx)+2ny(EzDy)+nz(2EzDzED))=((ED)nx+2nxExDx+2DynyEx+2Ex(nzDz)(ED)ny+2nyEyDy+2DxnxEy+2Ey(nzDz)(ED)nz+2nzEzDz+2DynyEz+2Ez(nxDx))=(ED)n^+2(Dn^)E
Substituting Eq. (5) into Eq. (1) the stress applied to the interface is expressed as
σ=12Re(2D2nE2n^(E2D2))12Re(2D1nE1n^(E1D1))
As D2n=D1n andE2t=E1t, so that
σ=12Re{(E2D2E1D1)n^2Dn(E2nE1n)n^}
The Minkowski stress tensor is not diagonally symmetric and therefore would admit violations of angular momentum conservation. The Abraham stress tensor is diagonally symmetric as
Γij=12(EiDj+EjDiE·Dδij)
From Eq. (8) we first compute:
2Γn^=(nx(2ExDxED)+ny(ExDy+EyDx)+nz(ExDz+EzDx)nx(EyDx+ExDy)+ny(2EyDyED)+nz(EyDz+EzDy)nx(EzDx+ExDz)+ny(EzDy+EyDz)+nz(2EzDzED))=(((ED)n)x+2nxExDx+Dx(nyEy+nzEz)+Ex(nyDy+nzDz)((ED)n)y+2nyEyDy+Dy(nxEx+nzEz)+Ey(nxDx+nzDz)((ED)n)z+2nzEzDz+Dz(nxEx+nyEy)+Ez(nxDx+nyDy))=(ED)n^+(En^)D+(Dn^)E
Substituting Eq. (9) into Eq. (1) we express the stress applied to the interface as follows
σ=12Re(E2nD2+D2nE2n(E2D2))12Re(E1nD1+D1nE1n(E1D1))
which was used in the following T-matrix calculation.

4. Vector spherical wave functions in anisotropic medium

Most analysis of the optical trap examines the gradient force, scattering force and absorption force on the trapped particle. Those forces are total forces. An alternative approach is to compute first the optical scattering by the nanoparticle, and then the distributions of the radiation stress on the surfaces of the nanoparticle via the stress tensor. The latter approach provides us with information of the position and orientation of the trap, which is particularly useful for trapping the nanorod [3].

Optical scattering of a particle is solved as a boundary value problem in the generalized Lorentz-Mie theory. The incident, internal and scattered fields can be expanded on the vector spherical wave function (VSWF) basis, respectively. The expansion coefficients can be computed by the transfer matrix (T-matrix) based on the boundary conditions. The VSWFs are constructed from the vector basis functions defined as

L=ψ,M=×ψ,N=(1/k)×M
where the scalar potential function ψ is a solution of the scalar Helmholtz equation and a is an arbitrary constant vector of unit length. Thus, the vectors L, M and N individually satisfy the vector wave equation, and they are not co-planar. When the potential ψnm are chosen as the scalar spherical harmonics
ψnm=iNnhn(1,2)(kr)Ynm(θ,ϕ)
where the normalization constant iNn = (1/n(n+1)), hn(1,2)(kr) are the spherical Hankel functions of first and second kind, respectively, and Ynm(θ,ϕ) are the scalar spherical harmonic functions. The discrete set of ψnm leads to discrete sets of the VSWFs basis {Lnm, Mnm, Nnm}. The scalar spherical harmonics ψnm constitute an orthogonal and complete set, in the sense that they are the eigenfunctions of the Sturm–Liouville problem. Furthermore, it is proved [19] that in the solution space there does not exist any other vectors, which are independent of the triplet of vectors {Lnm, Mnm, Nnm}, and cannot be represented by their linear combinations.

In a source free, dielectric, anisotropic and non-magnetic medium with scalar constant permeability μ = μ0, the Maxwell’s equations give

××Ek02εrE=0
where k02 = ω2/c02, or equivalently
×(×ε1D)k02D=0
with ∇⋅D = 0 and ∇⋅E ≠ 0, as D = ε0εrE where εis the dielectric tensor. At first sight, the VSWF expansion of the solutions of Eqs. (13) and (14) represent some difficulties in the anisotropic media, because the basis vector wave functions L, M, and N does not individually satisfy the anisotropic vector wave equation. Firstly, ∇ × L = 0, so that L does not satisfy Eq. (13). Secondly, M and N do not satisfy Eqs. (13) and (14) because of the presence of the dielectric tensor εr andεr1. The alternative methods have been proposed using the new modified VSWFs. However, some authors still use the conventional VSWF expansions in the anisotropic medium. Geng et al [11] solved the anisotropic wave equation in the Fourier space. The solution is first represented as an integral of the plane waves. Then, the plane waves are expanded into the VSWF series. Any arbitrary field, including the solutions to the anisotropic wave Eq. (13), can be expanded to a superposition of plane waves, provided that its Fourier transform converges, and an arbitrary plane wave can be represented by a convergent series of VSWFs because the VSWFs constitute a complete set [19,20]. In other words, any arbitrary field can be represented by multiple series of VSWFs. Advantage of the approach of Geng is that the characteristic equation Det [K(k)] = 0 was obtained, where K(k) is the tensor built from the permittivity and permeability tensors of the anisotropic medium. This characteristic equation determines the eigenvectors kq with q = 1-4 in the frequency space. In a homogeneous uniaxial anisotropic medium there are only two permissible eigenvectors kq with q = 1, 3 and 2, 4, in the Fourier space. Proposed by Liu et al [12] is the direct expansion of the solutions to the anisotropic wave equation into the {Lnm, Mnm, Nnm} basis. The legitimacy of the direct expansion into the VSWF basis can be further sustained from the inverse Fourier transform, so that the solution of the vector wave equation in a homogeneous and anisotropic medium E(r,θ,ϕ) is represented
E(r,θ,ϕ)=0dk02πdϕk0πsinθkdθkE(k,ϕk,θk)exp(ik·r)
where E(k,ϕk,θk)is the eigenwave angular spectrum of the electric field in the k-space. The plane wave is the integral in Eq. (15) can be expanded to the VSWF basis with [19]

exp(ikr)=n,m[AnmMnm(r,θ,ϕ)+BnmNnm(r,θ,ϕ)+CnmLnm(r,θ,ϕ)]

Substituting Eq. (16) into Eq. (15) and exchanging the order of integration and summation we obtain for the non-zero-divergence field E with three types of VSWFs, Lnm, Mnm, Nnm

E=nm(anmMnm+bnmNnm+cnmLnm)asE0
and
D=nm(anmMnm+bnmNnm)asD=0
for the divergence-less fields D with two types of VSWFs.

5. T-Matrix in anisotropic medium

In the optical tweezers the anisotropic nanorods are immersed in an isotropic buffer medium, where E=0, so that the incident field and the scattered field may be expanded into the regular VSWF basis

Einc(r)=n=1Nmaxm=nnanmRgMnm(kr)+bnmRgNnm(kr)Escat(r)=n=1Nmaxm=nnpnmMnm(1)(kr)+qnmNnm(1)(kr)
Where the incident field Einc is expanded into the regular VSWFs, RgMnm and RgNnm, instead of Mnm and Nnm because at the nano-particle scale the divergence of Mnm and Nnm close to the origin can slow down the convergence of the T-Matrix solver and require a higher number of modes, Nmax. The coefficients anm, bnm were calculated from the incident field. As that in our previous work, the electric field in the tighly focused incident laser beam is modeled as the vector Gaussian beam with 7th order corrections [21], and is expanded into the VSWF series using the point martcing method to compute anm and bnm. On the other hand, pnm, qnm are computed from anm and bnm using the T-Matrix. For the internal field in the anisotropic nanorod where D=0, we used the VSWF expansion for the displacement field as
Dint(r)=l=1Ln=1Nmaxm=nncnmlRgMnm(klr)+dnmlRgNnm(klr)
where L is the number of non-equal elements in the permittivity tensor of the anisotropic medium, and kl is the wavenumber associated to the l-th index of refraction. In Eq. (20) the internal field is represented as a finite series of the VSWF expansions. When the optical axes of the anisotropic crystal nanorod are aligned with the laboratory coordinates (x, y, z) and the trapping beam is along the z-axis, the refractive index tensor is diagonal and has two non-equal indices of refraction, so that L = 2 for unaxial crystals, and three non-equal indices of refraction, so that L = 3 for biaxial crystals. When the optical axes of the anisotropic crystal nanorod are not aligned with the laboratory coordinate axes (x, y, z) the refractive index tensor will have more non-equal elements and L>3. Thus, Dint is the sum of L expansions of the VSWF to ensure an enough number of the VSWF coefficients for a good convergence in the following numerical calculation of the T-matrix.

The coefficients clnm, dlnm and pnm, qnm are computed from anm and bnm using the T-Matrix. The T-matrix is computed with the point matching method for solving an over-determined system of linear equations representing the boundary conditions at a number of points on the surfaces of the nanorod, with the number of matching points higher than the number of unknown coefficients. We used the boundary condition for the normal components of D, instead of the tangential components of Eused in Ref [13]. With the surface charge density ρs = 0 in the dielectric nanoparticle the boundary condition is

n^ϵ0(n12Einc(kr))=n^ϵ0[n12Escat(kr)+lDl(klr)]
Detailed information of the implementation of the T-matrix calculation is presented by an example in Appendix.

6. Trapping of anisotropic nanocylinder

In the T-matrix method the nano-cylinder is always centered at the origin of the coordinate system and aligned with the z-axis. For computing the force and torque on the nano-cylinder, which is shifted and tilted by the random Brownian motion in the aquatic buffer, we shift and tilt the trapping beam instead, so that the T-matrix needs to be computed only once [3,13]. Instead of calculating the total force and total torque using the extended boundary condition method (EBCM), we computed first the optical fields scattered by the nano-cylinder using the T-matrix method, and then the stress distribution on each interface of the particle using Eq. (10). The total force and total torque were obtained by integrating the stresss distributions on the total surface S of the nano-cylinder as

F=SσdSandτ=Sr×σdS+Vr×(E×B)dV

The second term of the torques in Eq. (22) represents the contribution from the angular momentum of the light, referred to as the spin torque. According to the angular momentum conservation law, the spin torque is due to the loss of the angular momentum of the photons by interaction with the medium. For the anisotropic particle, the polarization state of the trapping beam changes when passing through the particle. To calculate the spin torque, one needs to use the symmetric stress tensor, so that the Abraham tensor must be used. It has been shown that for the isotropic nanocylinders the lateral torque due to the stress distribution on the nanocylinder surfaces is 2-3 orders of magnitude higher than the spin torque related to the polarization state change of the trapped beam [3]. In the case of the anisotropic nanorod, the torque due to the angular momentum of the light is still weak compared with the momentum due to the stress distribution on the surface. Moreover, the spin torque contributed more to the torque component τz, but not to the lateral torque related to the orientation of the trapped nanorod, so that the spin torque was not included in the following calculation.

We computed the lateral torque τy, which is related to the orientation of the trapped nanorod with respect to the incident beam. The nanocylinder was centered at the origin and aligned to the z-axis. The trapping beam was propagating in the + z direction, but titlted in the x-z plane at a tilt angle θ with respect to the + z axis. The nanocylinder had the optical axes aligned with the axes (x,y,z) with the refractive indices nx = 1.87, ny = nz = 1.57. Figure 3 shows the stress distribution on the nano-cylinder side surface. As the stress was outward from the nanorod, it generated a torque τy<0, which tended to align the nanocylinder with the trapping beam. Increasing nx can increase the x-component of the stress and therefore increase τy.

 figure: Fig. 3

Fig. 3 (a) Normal stress distribution (N/m2) on an anisotropic nanocylinder with R = 70 nm and H = 0.7μm, refractive indices nx = 1.87, ny = = nz = 1.57 and NA = 1.25, right circular polarization. The nanocylinder is tilted to 37° with respect to the trapping beam in the x-z plane. (b) Torque on a nanocylinder with R = 50 nm, H = 700 nm as a function of the tilt angle in the x-z plane. Red line, nx = nz = 1.57 and ny = 1.87, Green line nx = 1.87 and ny = nz = 1.57. Beam NA = 1.15.

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Now, we consider a single-crystal KNbO3 nanocylinder having the indices of refraction as na = nc = 2.1295 and nb = 2.2576 with the optical axes a, b and c not aligned with the laboratory axes (x, y, z). The nanorod was centered at the origin and aligned to the z-axis, but with the optical axis b inclined by an angle of 25° in the x-z plane with respect to the z-axis. In this case, the refractive index tensor in the (x,y,z) coordinate system is obtained by rotating the refractive index tensor as

n2=[2.011600.895702.257600.938001.9208]
which contains 5 non-equal elements. Thus, there were 5 corresponding wavenumbers kl = nlk0, and the summation Dl(klr) in Eq. (20) consisted of L = 5 series of VSWF expansions. In Eq. (19) we have 2Nmax (Nmax + 2) known VSWF expansion coefficients anm and bnm for the incident field, and 2Nmax (Nmax + 2) unknown VSWF expansion coefficients pnm and qnm for the scattered field. In Eq. (20) the VSWF expansions of the internal field Dinthad 2LNmax (Nmax + 2) unknown expansion coefficients, clnm and dlnm, with l = 1,2,… L. Therefore, we have total 2(L + 1)Nmax (Nmax + 2) unknowns on the right-hand side of Eq. (21), and 2Nmax (Nmax + 2) known coefficients on the left-hand side of Eq. (21). In the point matching method, we need to solve a group of at least 2(L + 1)Nmax (Nmax + 2) equations built from Eq. (21).

We computed the lateral torque τy and τy/θ as a function of the beam tilt angle θ, as shown in Fig. 4. The equilibrium orientation of the nanocylinders is determined by the conditions that the lateral torque τy=0 and the derivative τy/dθ<0. Thus, the equilibrium orientation is at a titl angle θ = 20° for a nanocylinder of height H = 700 nm, 15° for H = 800 nm and only 6° for H = 1000 nm. In all the three cases the nanocylinder with the optical axis at an angle of 25° to the + z axis can be trapped stably with the nanocylinder tilted at an angle θ with respect to the beam axis.

 figure: Fig. 4

Fig. 4 Lateral torque τy and τy/θ as a function of tilt angle. The nanocylinder of radius R = 50 nm and of height H = 700 nm (Blue line); 800 nm (Green line), and 1 μm (Red line)

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7. Conclusion

We have reported experimental observations that some anisotropic single crystal nanorod was trapped stably at a tilt angle with respect to the beam axis. This is a preliminary result, as we cannot measure the tilt anlge of the trapped nanorod, and did not know the optical axis orientation in the anisotropic crystal nanorod. To explain this observation we computed using the T-matrix method the lateral torque due to the stress distribution on the lateral surface of the nanocylinder, which showed that when the optical axes of the anisotropic nanocylinder were at some tilted angles with respect to the nanocyliner axis, the nanocylinder can be trapped at a stable position at a tilted angle with respect to the beam axis.

We have shown that the T-matrix method and the VSWF expansions can be applied to the optical trap for anisotropic particles, as we can expand electrical fields of the incident beam and the scattered field in the VSWF series in the buffer medium surounding the nanoparticle where E=0, and expand the displacement field of the internal field in the anisotropic nanoparticle, where D=0. We then use the boundary condition at the interfaces for the normal component of the displacement vector fields instead of that for the tangent comonents of the electrical fields. Moreover, for computing the T-matrix, the internal field Dint is represented as a sum of a set of compoment fields Dlint in order to give better description with more VSWF expansion coefficients to help the convergence of the T-matrix solver.

Appendix

We show an exemple of calculation for the T-matrix with the VSWF expansions in Eqs. (19) and (20) from the boundary condition Eq. (21). For shortening the mathematical expressions we take L=1 in Eq. (20) and the boundary condition Eq. (21) is written as

Einc(θ,ϕ)=n=1Nmaxm=nnpnmMnm(1)(kr)+qnmNnm(1)(kr)+n=1Nmaxm=nncnmRgMnm(kr)+dnmRgNnm(kr)
Consider a single mode in the VSWF expansion of the incident field in Eq. (21) we have
anmRgMnm(kr)=n=1Nmaxm=nnpnmMnm(1)(kr)+qnmNnm(1)(kr)++n=1Nmaxm=nncnmRgMnm(kr)+dnmRgNnm(kr)bnmRgNnm(kr)=n=1Nmaxm=nnpnmMnm(1)(kr)+qnmNnm(1)(kr)++n=1Nmaxm=nncnmRgMnm(kr)+dnmRgNnm(kr)
Dividing each terms in the equations by anm and bnm respectively, and write them in the matrix form, we have
[RgMnmRgNnm]=[[Mnm(1)][Nnm(1)][RgMnm][RgNnm]][t11t12t21t22t11t12t21t22]
where 4Nmax (Nmax + 2) unknown elements of the T-matrix, t11, …t’11,…t12, …t’12... need to be solved in the point matching method from a group of algebraic equations of a number larger than 4Nmax (Nmax + 2). The scattered and internal fields are computed from Eq. (19) with the solved VSWF expansion coefficients in the T-matrix as

pnm=t11anm+t12bnmqnm=t21anm+t22bnmandcnm=t11anm+t12bnmdnm=t21anm+t22bnm

Acknowledgment

Research is supported by the Discovery Grant of the Natural Science and Engineering Council (NSERC) of Canada. The first author was supported by the NSERC and FQRNT scholarship The KNbO3 nanorods were provided by Dr. Bin Yu at ShenZhen University in China.

References and links

1. M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394(6691), 348–350 (1998). [CrossRef]  

2. Y. Nakayama, P. J. Pauzauskie, A. Radenovic, R. M. Onorato, R. J. Saykally, J. Liphardt, and P. Yang, “Tunable nanowire nonlinear optical probe,” Nature 447(7148), 1098–1101 (2007). [CrossRef]   [PubMed]  

3. P. B. Bareil and Y. Sheng, “Angular and position stability of a nanorod trapped in an optical tweezers,” Opt. Express 18(25), 26388–26398 (2010). [CrossRef]   [PubMed]  

4. J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64(4), 1632–1639 (1988). [CrossRef]  

5. S. H. Simpson and S. Hanna, “Application of the discrete dipole approximation to optical trapping calculations of inhomogeneous and anisotropic particles,” Opt. Express 19(17), 16526–16541 (2011). [CrossRef]   [PubMed]  

6. S. H. Simpson and S. Hanna, “Stability analysis and thermal motion of optically trapped nanowires,” Nanotechnology 23(20), 205502 (2012). [CrossRef]   [PubMed]  

7. A. D. Kiselev, V. Y. Reshetnyak, and T. J. Sluckin, “Light scattering by optically anisotropic scatterers: T-matrix theory for radial and uniform anisotropies,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(5), 056609 (2002). [CrossRef]   [PubMed]  

8. A. Doicu, “Null-field method to electromagnetic scattering from uniaxial anisotropic particles,” Opt. Commun. 218(1-3), 11–17 (2003). [CrossRef]  

9. V. Schmidt and T. Wriedt, “T-matrix method for biaxial anisotropic particles,” J. Quant. Spectrosc. Radiat. Transf. 110(14-16), 1392–1397 (2009). [CrossRef]  

10. V. Loke, T. A. Nieminen, S. J. Parkin, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “FDFD/T-matrix hybrid method,” J. Quant. Spectrosc. Radiat. Transf. 106(1-3), 274–284 (2007). [CrossRef]  

11. Y. L. Geng, X. B. Wu, L. W. Li, and B. R. Guan, “Mie scattering by a uniaxial anisotropic sphere,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(5), 056609 (2004). [CrossRef]   [PubMed]  

12. M. Liu, N. Ji, Z. Lin, and S. T. Chui, “Radiation torque on a birefringent sphere caused by an electromagnetic wave,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(5), 056610 (2005). [CrossRef]   [PubMed]  

13. T. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Calculation of the T-matrix: general considerations and application of the point-matching method,” J. Quant. Spectrosc. Radiat. Transf. 79–80, 1019–1029 (2003). [CrossRef]  

14. L. Yu, Y. Sheng, and A. Chiou, “Three-dimensional light-scattering and deformation of individual biconcave human blood cells in optical tweezers,” Opt. Express 21(10), 12174–12184 (2013). [CrossRef]   [PubMed]  

15. K. Okamoto and S. Kawata, “Radiation Force Exerted on Subwavelength Particles near a Nanoaperture,” Phys. Rev. Lett. 83(22), 4534–4537 (1999). [CrossRef]  

16. F. Robinson, “Electromagnetic stress and momentum in matter,” Phys. Rep. 16(6), 313–354 (1975). [CrossRef]  

17. I. Brevik, “Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor,” Phys. Rep. 52(3), 133–201 (1979). [CrossRef]  

18. R. N. C. Pfeifer, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Colloquium: Momentum of an electromagnetic wave in dielectric media,” Rev. Mod. Phys. 79(4), 1197–1216 (2007). [CrossRef]  

19. D. Sarkar and N. J. Halas, “General vector basis function solution of Maxwell’s equations,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 56(1), 1102–1112 (1997). [CrossRef]  

20. P. B. Bareil and Y. Sheng, “Optical trapping of anisotropic nanocylinder.” Proc. SPIE 8810, paper 88102v (2013). [CrossRef]  

21. P. B. Bareil and Y. Sheng, “Modeling highly focused laser beam in optical tweezers with the vector Gaussian beam in the T-matrix method,” J. Opt. Soc. Am. A 30(1), 1–6 (2013). [CrossRef]   [PubMed]  

References

  • View by:

  1. M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394(6691), 348–350 (1998).
    [Crossref]
  2. Y. Nakayama, P. J. Pauzauskie, A. Radenovic, R. M. Onorato, R. J. Saykally, J. Liphardt, and P. Yang, “Tunable nanowire nonlinear optical probe,” Nature 447(7148), 1098–1101 (2007).
    [Crossref] [PubMed]
  3. P. B. Bareil and Y. Sheng, “Angular and position stability of a nanorod trapped in an optical tweezers,” Opt. Express 18(25), 26388–26398 (2010).
    [Crossref] [PubMed]
  4. J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64(4), 1632–1639 (1988).
    [Crossref]
  5. S. H. Simpson and S. Hanna, “Application of the discrete dipole approximation to optical trapping calculations of inhomogeneous and anisotropic particles,” Opt. Express 19(17), 16526–16541 (2011).
    [Crossref] [PubMed]
  6. S. H. Simpson and S. Hanna, “Stability analysis and thermal motion of optically trapped nanowires,” Nanotechnology 23(20), 205502 (2012).
    [Crossref] [PubMed]
  7. A. D. Kiselev, V. Y. Reshetnyak, and T. J. Sluckin, “Light scattering by optically anisotropic scatterers: T-matrix theory for radial and uniform anisotropies,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(5), 056609 (2002).
    [Crossref] [PubMed]
  8. A. Doicu, “Null-field method to electromagnetic scattering from uniaxial anisotropic particles,” Opt. Commun. 218(1-3), 11–17 (2003).
    [Crossref]
  9. V. Schmidt and T. Wriedt, “T-matrix method for biaxial anisotropic particles,” J. Quant. Spectrosc. Radiat. Transf. 110(14-16), 1392–1397 (2009).
    [Crossref]
  10. V. Loke, T. A. Nieminen, S. J. Parkin, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “FDFD/T-matrix hybrid method,” J. Quant. Spectrosc. Radiat. Transf. 106(1-3), 274–284 (2007).
    [Crossref]
  11. Y. L. Geng, X. B. Wu, L. W. Li, and B. R. Guan, “Mie scattering by a uniaxial anisotropic sphere,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(5), 056609 (2004).
    [Crossref] [PubMed]
  12. M. Liu, N. Ji, Z. Lin, and S. T. Chui, “Radiation torque on a birefringent sphere caused by an electromagnetic wave,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(5), 056610 (2005).
    [Crossref] [PubMed]
  13. T. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Calculation of the T-matrix: general considerations and application of the point-matching method,” J. Quant. Spectrosc. Radiat. Transf. 79–80, 1019–1029 (2003).
    [Crossref]
  14. L. Yu, Y. Sheng, and A. Chiou, “Three-dimensional light-scattering and deformation of individual biconcave human blood cells in optical tweezers,” Opt. Express 21(10), 12174–12184 (2013).
    [Crossref] [PubMed]
  15. K. Okamoto and S. Kawata, “Radiation Force Exerted on Subwavelength Particles near a Nanoaperture,” Phys. Rev. Lett. 83(22), 4534–4537 (1999).
    [Crossref]
  16. F. Robinson, “Electromagnetic stress and momentum in matter,” Phys. Rep. 16(6), 313–354 (1975).
    [Crossref]
  17. I. Brevik, “Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor,” Phys. Rep. 52(3), 133–201 (1979).
    [Crossref]
  18. R. N. C. Pfeifer, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Colloquium: Momentum of an electromagnetic wave in dielectric media,” Rev. Mod. Phys. 79(4), 1197–1216 (2007).
    [Crossref]
  19. D. Sarkar and N. J. Halas, “General vector basis function solution of Maxwell’s equations,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 56(1), 1102–1112 (1997).
    [Crossref]
  20. P. B. Bareil and Y. Sheng, “Optical trapping of anisotropic nanocylinder.” Proc. SPIE 8810, paper 88102v (2013).
    [Crossref]
  21. P. B. Bareil and Y. Sheng, “Modeling highly focused laser beam in optical tweezers with the vector Gaussian beam in the T-matrix method,” J. Opt. Soc. Am. A 30(1), 1–6 (2013).
    [Crossref] [PubMed]

2013 (2)

2012 (1)

S. H. Simpson and S. Hanna, “Stability analysis and thermal motion of optically trapped nanowires,” Nanotechnology 23(20), 205502 (2012).
[Crossref] [PubMed]

2011 (1)

2010 (1)

2009 (1)

V. Schmidt and T. Wriedt, “T-matrix method for biaxial anisotropic particles,” J. Quant. Spectrosc. Radiat. Transf. 110(14-16), 1392–1397 (2009).
[Crossref]

2007 (3)

V. Loke, T. A. Nieminen, S. J. Parkin, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “FDFD/T-matrix hybrid method,” J. Quant. Spectrosc. Radiat. Transf. 106(1-3), 274–284 (2007).
[Crossref]

R. N. C. Pfeifer, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Colloquium: Momentum of an electromagnetic wave in dielectric media,” Rev. Mod. Phys. 79(4), 1197–1216 (2007).
[Crossref]

Y. Nakayama, P. J. Pauzauskie, A. Radenovic, R. M. Onorato, R. J. Saykally, J. Liphardt, and P. Yang, “Tunable nanowire nonlinear optical probe,” Nature 447(7148), 1098–1101 (2007).
[Crossref] [PubMed]

2005 (1)

M. Liu, N. Ji, Z. Lin, and S. T. Chui, “Radiation torque on a birefringent sphere caused by an electromagnetic wave,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(5), 056610 (2005).
[Crossref] [PubMed]

2004 (1)

Y. L. Geng, X. B. Wu, L. W. Li, and B. R. Guan, “Mie scattering by a uniaxial anisotropic sphere,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(5), 056609 (2004).
[Crossref] [PubMed]

2003 (2)

T. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Calculation of the T-matrix: general considerations and application of the point-matching method,” J. Quant. Spectrosc. Radiat. Transf. 79–80, 1019–1029 (2003).
[Crossref]

A. Doicu, “Null-field method to electromagnetic scattering from uniaxial anisotropic particles,” Opt. Commun. 218(1-3), 11–17 (2003).
[Crossref]

2002 (1)

A. D. Kiselev, V. Y. Reshetnyak, and T. J. Sluckin, “Light scattering by optically anisotropic scatterers: T-matrix theory for radial and uniform anisotropies,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(5), 056609 (2002).
[Crossref] [PubMed]

1999 (1)

K. Okamoto and S. Kawata, “Radiation Force Exerted on Subwavelength Particles near a Nanoaperture,” Phys. Rev. Lett. 83(22), 4534–4537 (1999).
[Crossref]

1998 (1)

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394(6691), 348–350 (1998).
[Crossref]

1997 (1)

D. Sarkar and N. J. Halas, “General vector basis function solution of Maxwell’s equations,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 56(1), 1102–1112 (1997).
[Crossref]

1988 (1)

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64(4), 1632–1639 (1988).
[Crossref]

1979 (1)

I. Brevik, “Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor,” Phys. Rep. 52(3), 133–201 (1979).
[Crossref]

1975 (1)

F. Robinson, “Electromagnetic stress and momentum in matter,” Phys. Rep. 16(6), 313–354 (1975).
[Crossref]

Alexander, D. R.

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64(4), 1632–1639 (1988).
[Crossref]

Bareil, P. B.

Barton, J. P.

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64(4), 1632–1639 (1988).
[Crossref]

Brevik, I.

I. Brevik, “Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor,” Phys. Rep. 52(3), 133–201 (1979).
[Crossref]

Chiou, A.

Chui, S. T.

M. Liu, N. Ji, Z. Lin, and S. T. Chui, “Radiation torque on a birefringent sphere caused by an electromagnetic wave,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(5), 056610 (2005).
[Crossref] [PubMed]

Doicu, A.

A. Doicu, “Null-field method to electromagnetic scattering from uniaxial anisotropic particles,” Opt. Commun. 218(1-3), 11–17 (2003).
[Crossref]

Friese, M. E. J.

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394(6691), 348–350 (1998).
[Crossref]

Geng, Y. L.

Y. L. Geng, X. B. Wu, L. W. Li, and B. R. Guan, “Mie scattering by a uniaxial anisotropic sphere,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(5), 056609 (2004).
[Crossref] [PubMed]

Guan, B. R.

Y. L. Geng, X. B. Wu, L. W. Li, and B. R. Guan, “Mie scattering by a uniaxial anisotropic sphere,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(5), 056609 (2004).
[Crossref] [PubMed]

Halas, N. J.

D. Sarkar and N. J. Halas, “General vector basis function solution of Maxwell’s equations,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 56(1), 1102–1112 (1997).
[Crossref]

Hanna, S.

Heckenberg, N. R.

V. Loke, T. A. Nieminen, S. J. Parkin, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “FDFD/T-matrix hybrid method,” J. Quant. Spectrosc. Radiat. Transf. 106(1-3), 274–284 (2007).
[Crossref]

R. N. C. Pfeifer, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Colloquium: Momentum of an electromagnetic wave in dielectric media,” Rev. Mod. Phys. 79(4), 1197–1216 (2007).
[Crossref]

T. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Calculation of the T-matrix: general considerations and application of the point-matching method,” J. Quant. Spectrosc. Radiat. Transf. 79–80, 1019–1029 (2003).
[Crossref]

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394(6691), 348–350 (1998).
[Crossref]

Ji, N.

M. Liu, N. Ji, Z. Lin, and S. T. Chui, “Radiation torque on a birefringent sphere caused by an electromagnetic wave,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(5), 056610 (2005).
[Crossref] [PubMed]

Kawata, S.

K. Okamoto and S. Kawata, “Radiation Force Exerted on Subwavelength Particles near a Nanoaperture,” Phys. Rev. Lett. 83(22), 4534–4537 (1999).
[Crossref]

Kiselev, A. D.

A. D. Kiselev, V. Y. Reshetnyak, and T. J. Sluckin, “Light scattering by optically anisotropic scatterers: T-matrix theory for radial and uniform anisotropies,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(5), 056609 (2002).
[Crossref] [PubMed]

Li, L. W.

Y. L. Geng, X. B. Wu, L. W. Li, and B. R. Guan, “Mie scattering by a uniaxial anisotropic sphere,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(5), 056609 (2004).
[Crossref] [PubMed]

Lin, Z.

M. Liu, N. Ji, Z. Lin, and S. T. Chui, “Radiation torque on a birefringent sphere caused by an electromagnetic wave,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(5), 056610 (2005).
[Crossref] [PubMed]

Liphardt, J.

Y. Nakayama, P. J. Pauzauskie, A. Radenovic, R. M. Onorato, R. J. Saykally, J. Liphardt, and P. Yang, “Tunable nanowire nonlinear optical probe,” Nature 447(7148), 1098–1101 (2007).
[Crossref] [PubMed]

Liu, M.

M. Liu, N. Ji, Z. Lin, and S. T. Chui, “Radiation torque on a birefringent sphere caused by an electromagnetic wave,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(5), 056610 (2005).
[Crossref] [PubMed]

Loke, V.

V. Loke, T. A. Nieminen, S. J. Parkin, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “FDFD/T-matrix hybrid method,” J. Quant. Spectrosc. Radiat. Transf. 106(1-3), 274–284 (2007).
[Crossref]

Nakayama, Y.

Y. Nakayama, P. J. Pauzauskie, A. Radenovic, R. M. Onorato, R. J. Saykally, J. Liphardt, and P. Yang, “Tunable nanowire nonlinear optical probe,” Nature 447(7148), 1098–1101 (2007).
[Crossref] [PubMed]

Nieminen, T.

T. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Calculation of the T-matrix: general considerations and application of the point-matching method,” J. Quant. Spectrosc. Radiat. Transf. 79–80, 1019–1029 (2003).
[Crossref]

Nieminen, T. A.

R. N. C. Pfeifer, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Colloquium: Momentum of an electromagnetic wave in dielectric media,” Rev. Mod. Phys. 79(4), 1197–1216 (2007).
[Crossref]

V. Loke, T. A. Nieminen, S. J. Parkin, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “FDFD/T-matrix hybrid method,” J. Quant. Spectrosc. Radiat. Transf. 106(1-3), 274–284 (2007).
[Crossref]

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394(6691), 348–350 (1998).
[Crossref]

Okamoto, K.

K. Okamoto and S. Kawata, “Radiation Force Exerted on Subwavelength Particles near a Nanoaperture,” Phys. Rev. Lett. 83(22), 4534–4537 (1999).
[Crossref]

Onorato, R. M.

Y. Nakayama, P. J. Pauzauskie, A. Radenovic, R. M. Onorato, R. J. Saykally, J. Liphardt, and P. Yang, “Tunable nanowire nonlinear optical probe,” Nature 447(7148), 1098–1101 (2007).
[Crossref] [PubMed]

Parkin, S. J.

V. Loke, T. A. Nieminen, S. J. Parkin, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “FDFD/T-matrix hybrid method,” J. Quant. Spectrosc. Radiat. Transf. 106(1-3), 274–284 (2007).
[Crossref]

Pauzauskie, P. J.

Y. Nakayama, P. J. Pauzauskie, A. Radenovic, R. M. Onorato, R. J. Saykally, J. Liphardt, and P. Yang, “Tunable nanowire nonlinear optical probe,” Nature 447(7148), 1098–1101 (2007).
[Crossref] [PubMed]

Pfeifer, R. N. C.

R. N. C. Pfeifer, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Colloquium: Momentum of an electromagnetic wave in dielectric media,” Rev. Mod. Phys. 79(4), 1197–1216 (2007).
[Crossref]

Radenovic, A.

Y. Nakayama, P. J. Pauzauskie, A. Radenovic, R. M. Onorato, R. J. Saykally, J. Liphardt, and P. Yang, “Tunable nanowire nonlinear optical probe,” Nature 447(7148), 1098–1101 (2007).
[Crossref] [PubMed]

Reshetnyak, V. Y.

A. D. Kiselev, V. Y. Reshetnyak, and T. J. Sluckin, “Light scattering by optically anisotropic scatterers: T-matrix theory for radial and uniform anisotropies,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(5), 056609 (2002).
[Crossref] [PubMed]

Robinson, F.

F. Robinson, “Electromagnetic stress and momentum in matter,” Phys. Rep. 16(6), 313–354 (1975).
[Crossref]

Rubinsztein-Dunlop, H.

R. N. C. Pfeifer, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Colloquium: Momentum of an electromagnetic wave in dielectric media,” Rev. Mod. Phys. 79(4), 1197–1216 (2007).
[Crossref]

V. Loke, T. A. Nieminen, S. J. Parkin, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “FDFD/T-matrix hybrid method,” J. Quant. Spectrosc. Radiat. Transf. 106(1-3), 274–284 (2007).
[Crossref]

T. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Calculation of the T-matrix: general considerations and application of the point-matching method,” J. Quant. Spectrosc. Radiat. Transf. 79–80, 1019–1029 (2003).
[Crossref]

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394(6691), 348–350 (1998).
[Crossref]

Sarkar, D.

D. Sarkar and N. J. Halas, “General vector basis function solution of Maxwell’s equations,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 56(1), 1102–1112 (1997).
[Crossref]

Saykally, R. J.

Y. Nakayama, P. J. Pauzauskie, A. Radenovic, R. M. Onorato, R. J. Saykally, J. Liphardt, and P. Yang, “Tunable nanowire nonlinear optical probe,” Nature 447(7148), 1098–1101 (2007).
[Crossref] [PubMed]

Schaub, S. A.

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64(4), 1632–1639 (1988).
[Crossref]

Schmidt, V.

V. Schmidt and T. Wriedt, “T-matrix method for biaxial anisotropic particles,” J. Quant. Spectrosc. Radiat. Transf. 110(14-16), 1392–1397 (2009).
[Crossref]

Sheng, Y.

Simpson, S. H.

Sluckin, T. J.

A. D. Kiselev, V. Y. Reshetnyak, and T. J. Sluckin, “Light scattering by optically anisotropic scatterers: T-matrix theory for radial and uniform anisotropies,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(5), 056609 (2002).
[Crossref] [PubMed]

Wriedt, T.

V. Schmidt and T. Wriedt, “T-matrix method for biaxial anisotropic particles,” J. Quant. Spectrosc. Radiat. Transf. 110(14-16), 1392–1397 (2009).
[Crossref]

Wu, X. B.

Y. L. Geng, X. B. Wu, L. W. Li, and B. R. Guan, “Mie scattering by a uniaxial anisotropic sphere,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(5), 056609 (2004).
[Crossref] [PubMed]

Yang, P.

Y. Nakayama, P. J. Pauzauskie, A. Radenovic, R. M. Onorato, R. J. Saykally, J. Liphardt, and P. Yang, “Tunable nanowire nonlinear optical probe,” Nature 447(7148), 1098–1101 (2007).
[Crossref] [PubMed]

Yu, L.

J. Appl. Phys. (1)

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64(4), 1632–1639 (1988).
[Crossref]

J. Opt. Soc. Am. A (1)

J. Quant. Spectrosc. Radiat. Transf. (3)

V. Schmidt and T. Wriedt, “T-matrix method for biaxial anisotropic particles,” J. Quant. Spectrosc. Radiat. Transf. 110(14-16), 1392–1397 (2009).
[Crossref]

V. Loke, T. A. Nieminen, S. J. Parkin, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “FDFD/T-matrix hybrid method,” J. Quant. Spectrosc. Radiat. Transf. 106(1-3), 274–284 (2007).
[Crossref]

T. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Calculation of the T-matrix: general considerations and application of the point-matching method,” J. Quant. Spectrosc. Radiat. Transf. 79–80, 1019–1029 (2003).
[Crossref]

Nanotechnology (1)

S. H. Simpson and S. Hanna, “Stability analysis and thermal motion of optically trapped nanowires,” Nanotechnology 23(20), 205502 (2012).
[Crossref] [PubMed]

Nature (2)

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394(6691), 348–350 (1998).
[Crossref]

Y. Nakayama, P. J. Pauzauskie, A. Radenovic, R. M. Onorato, R. J. Saykally, J. Liphardt, and P. Yang, “Tunable nanowire nonlinear optical probe,” Nature 447(7148), 1098–1101 (2007).
[Crossref] [PubMed]

Opt. Commun. (1)

A. Doicu, “Null-field method to electromagnetic scattering from uniaxial anisotropic particles,” Opt. Commun. 218(1-3), 11–17 (2003).
[Crossref]

Opt. Express (3)

Phys. Rep. (2)

F. Robinson, “Electromagnetic stress and momentum in matter,” Phys. Rep. 16(6), 313–354 (1975).
[Crossref]

I. Brevik, “Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor,” Phys. Rep. 52(3), 133–201 (1979).
[Crossref]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (3)

A. D. Kiselev, V. Y. Reshetnyak, and T. J. Sluckin, “Light scattering by optically anisotropic scatterers: T-matrix theory for radial and uniform anisotropies,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(5), 056609 (2002).
[Crossref] [PubMed]

Y. L. Geng, X. B. Wu, L. W. Li, and B. R. Guan, “Mie scattering by a uniaxial anisotropic sphere,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(5), 056609 (2004).
[Crossref] [PubMed]

M. Liu, N. Ji, Z. Lin, and S. T. Chui, “Radiation torque on a birefringent sphere caused by an electromagnetic wave,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(5), 056610 (2005).
[Crossref] [PubMed]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (1)

D. Sarkar and N. J. Halas, “General vector basis function solution of Maxwell’s equations,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 56(1), 1102–1112 (1997).
[Crossref]

Phys. Rev. Lett. (1)

K. Okamoto and S. Kawata, “Radiation Force Exerted on Subwavelength Particles near a Nanoaperture,” Phys. Rev. Lett. 83(22), 4534–4537 (1999).
[Crossref]

Rev. Mod. Phys. (1)

R. N. C. Pfeifer, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Colloquium: Momentum of an electromagnetic wave in dielectric media,” Rev. Mod. Phys. 79(4), 1197–1216 (2007).
[Crossref]

Other (1)

P. B. Bareil and Y. Sheng, “Optical trapping of anisotropic nanocylinder.” Proc. SPIE 8810, paper 88102v (2013).
[Crossref]

Supplementary Material (1)

Media 1: AVI (464 KB)     

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Figures (4)

Fig. 1
Fig. 1 Scanning electron microscope image of a stake of KNbO3 nanorodes.
Fig. 2
Fig. 2 (a) A non-trapped nanorod resting on the bottom coverslip; (b) A nanorod trapped at a tilt angle. Its shape be seen to be similar to that shown in (a) in few frames later from the instant when the nanorod escapes in videoclip (Media 1).
Fig. 3
Fig. 3 (a) Normal stress distribution (N/m2) on an anisotropic nanocylinder with R = 70 nm and H = 0.7μm, refractive indices nx = 1.87, ny = = nz = 1.57 and NA = 1.25, right circular polarization. The nanocylinder is tilted to 37° with respect to the trapping beam in the x-z plane. (b) Torque on a nanocylinder with R = 50 nm, H = 700 nm as a function of the tilt angle in the x-z plane. Red line, nx = nz = 1.57 and ny = 1.87, Green line nx = 1.87 and ny = nz = 1.57. Beam NA = 1.15.
Fig. 4
Fig. 4 Lateral torque τy and τ y /θ as a function of tilt angle. The nanocylinder of radius R = 50 nm and of height H = 700 nm (Blue line); 800 nm (Green line), and 1 μm (Red line)

Equations (27)

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σ = 1 2 Re{ ( Γ 2 Γ 1 ) n ^ } n ^
Γ ij =ϵ[ E i E j * + 1 ϵ μ 0 B i B j 1 2 ( E 2 + 1 ϵ μ 0 B 2 ) δ ij ]
σ = 1 2 Re{ n ^ ( ε 1 ε 2 1 )( ε 1 E 1n 2 + ε 2 E 1t 2 ) } n ^
Γ ij =[ D i E j * 1 2 ( D·E ) δ ij ]
2Γ n ^ =( n x (2 E x D x E D )+2 n y ( E x D y )+2 n z ( E x D z ) 2 n x ( E y D x )+ n y (2 E y D y E D )+2 n z ( E y D z ) 2 n x ( E z D x )+2 n y ( E z D y )+ n z (2 E z D z E D ) ) =( ( E D ) n x +2 n x E x D x +2 D y n y E x +2 E x ( n z D z ) ( E D ) n y +2 n y E y D y +2 D x n x E y +2 E y ( n z D z ) ( E D ) n z +2 n z E z D z +2 D y n y E z +2 E z ( n x D x ) ) =( E D ) n ^ +2( D n ^ ) E
σ = 1 2 Re( 2 D 2n E 2 n ^ ( E 2 D 2 ) ) 1 2 Re( 2 D 1n E 1 n ^ ( E 1 D 1 ) )
σ = 1 2 Re{ ( E 2 D 2 E 1 D 1 ) n ^ 2 D n ( E 2n E 1n ) n ^ }
Γ ij = 1 2 ( E i D j + E j D i E · D δ ij )
2Γ n ^ =( n x (2 E x D x E D )+ n y ( E x D y + E y D x )+ n z ( E x D z + E z D x ) n x ( E y D x + E x D y )+ n y (2 E y D y E D )+ n z ( E y D z + E z D y ) n x ( E z D x + E x D z )+ n y ( E z D y + E y D z )+ n z (2 E z D z E D ) ) =( (( E D )n) x +2 n x E x D x + D x ( n y E y + n z E z )+ E x ( n y D y + n z D z ) (( E D )n) y +2 n y E y D y + D y ( n x E x + n z E z )+ E y ( n x D x + n z D z ) (( E D )n) z +2 n z E z D z + D z ( n x E x + n y E y )+ E z ( n x D x + n y D y ) ) =( E D ) n ^ +( E n ^ ) D +( D n ^ ) E
σ = 1 2 Re( E 2n D 2 + D 2n E 2 n ( E 2 D 2 ) ) 1 2 Re( E 1n D 1 + D 1n E 1 n ( E 1 D 1 ) )
L=ψ,M=×ψ,N=(1/k)×M
ψ nm =i N n h n (1,2) ( kr ) Y nm (θ,ϕ)
××E k 0 2 ε r E=0
×(× ε 1 D) k 0 2 D=0
E(r,θ,ϕ)= 0 dk 0 2π d ϕ k 0 π sin θ k d θ k E(k, ϕ k , θ k )exp(ik·r)
exp(i k r )= n,m [ A nm M nm (r,θ,ϕ)+ B nm N nm (r,θ,ϕ)+ C nm L nm (r,θ,ϕ)]
E = nm ( a nm M nm + b nm N nm + c nm L nm )as E 0
D = nm ( a nm M nm + b nm N nm )as D =0
E inc ( r )= n=1 Nmax m=n n a nm Rg M nm (k r )+ b nm Rg N nm (k r ) E scat (r)= n=1 Nmax m=n n p nm M nm (1) (k r )+ q nm N nm (1) (k r )
D int (r) = l=1 L n=1 Nmax m=n n c nm l Rg M nm ( k l r )+ d nm l Rg N nm ( k l r )
n ^ ϵ 0 ( n 1 2 E inc (k r ) )= n ^ ϵ 0 [ n 1 2 E scat (k r )+ l D l ( k l r ) ]
F = S σ dSand τ = S r × σ dS + V r ×( E × B )dV
n 2 =[ 2.0116 0 0.8957 0 2.2576 0 0.9380 0 1.9208 ]
E inc (θ,ϕ)= n=1 Nmax m=n n p nm M nm (1) (k r )+ q nm N nm (1) (k r )+ n=1 Nmax m=n n c nm Rg M nm (k r )+ d nm Rg N nm (k r )
a n m Rg M n m (k r )= n=1 Nmax m=n n p nm M nm (1) (k r )+ q nm N nm (1) (k r ) + + n=1 Nmax m=n n c nm Rg M nm (k r )+ d nm Rg N nm (k r ) b n m Rg N n m (k r )= n=1 Nmax m=n n p nm M nm (1) (k r )+ q nm N nm (1) (k r ) + + n=1 Nmax m=n n c nm Rg M nm (k r )+ d nm Rg N nm (k r )
[ Rg M n m Rg N n m ] = [ [ M nm (1) ] [ N nm (1) ] [Rg M nm ] [Rg N nm ] ] [ t 11 t 12 t 21 t 22 t 11 t 12 t 21 t 22 ]
p nm = t 11 a n m + t 12 b n m q nm = t 21 a n m + t 22 b n m and c nm = t 11 a n m + t 12 b n m d nm = t 21 a n m + t 22 b n m

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