Abstract

Using vector potential and spectrum representation, we derive the expressions of the Airy evanescent field existed at the interface. Utilizing these expressions and the Arbitrary Beam Theory, the optical forces exerted on a Mie dielectric particle in the Airy evanescent field were theoretically investigated in detail. Numerical results show that the optical forces exhibit strong oscillations which are corresponding to the distributions of the evanescent field. With the increasing the size of particle radius, Morphology Dependent Resonance occurs for the particle with specific refractive index.

© 2012 OSA

1. Introduction

Since Ashkin et al. proved that three-dimensional trapping of a dielectric particle is possible by use of a single, highly focused laser beam [1], optical tweezers have become an indispensable tool for manipulating small particles without any mechanical contact. The optical tweezers have been used to manipulate and trap micro-scale particles even viruses [2], liquid droplets [3], silver nanoparticles [4], uniaxial anisotropic spheres [5], and magneto dielectric particles [6]. Kawata and Sugiura demonstrated the movement and trapping of micrometer-sized particles in the evanescent field formed at the sapphire-air interface by total internal reflection of a laser beam [7]. E.Almaas and I.Brevik considered theoretically the force upon a micrometer-sized spherical dielectric particle in the plane evanescent field [8]. Meanwhile, S.Chang et al investigated theoretically the optical force exerted on a dielectric sphere in the Gaussian evanescent field [9]. Conventional tweezers usually utilize Gaussian light beams, which suffer from strong divergence off the focal plane, trapping particles with only a few micrometers apart in the axial direction. The “non-diffracting” beams, especially Bessel beams, do not spread while propagating, even if the beam diameter is reduced to the size of a tightly focused Gaussian laser beam. It has been used to trap atoms and microscopic particles in multiple planes [10], construct particle conveyor belts [11], sort microfluidic cells and transfect cells [12].

A second type of “non-diffracting” beam, the Airy beam, was observed in experiment [13, 14]. Its key difference from the Bessel beam is that it additionally experiences a transverse acceleration and can self bend even in free space. Due to its unique properties, Airy beam has recently attracted a lot of attentions because of its potential applications in plasma guidance [15], vacuum electron acceleration [16], and generation of three dimensional optical bullets [17]. Jörg Baumgartl and colleagues [18, 19] experimentally demonstrate the first use of the Airy beam in optical micromanipulation. As opposed to Bessel beams, it can transport microparticles along curved self-healed paths, and remove particles or cells from a section of a sample chamber. Its novelty is that the trapping potential landscape tends to freely self-bend during propagation. Moreover, the diffraction free distance and the bend degree of Airy beam can be controlled, and the acceleration direction can be switched by a nonlinear optical process [20]. These tunable properties make the Airy beam a versatile and powerful tool for optical manipulation.

Up to now, the optical micromanipulation by Airy light beam has been experimentally demonstrated, however, only qualitative theoretical analysis has been presented [18, 19]. Meanwhile, we have quantitatively analyzed the radiation forces and trajectories of Rayleigh particles in Airy beam [21]. Further quantitative theoretical analysis is necessary to guide optical micromanipulation. The micromanipulation by Airy beam reported to date is related to Mie particles [22, 23], whose radii are larger than the wavelength; there has been a considerable amount of interest in this kind of interaction, both from a fundamental and from a practical viewpoint.

In this paper, using vector potential and spectrum representation, we derive the expressions for the evanescent field induced by total internal reflection of an Airy beam at the interface. Utilizing this expression and Arbitrary-Beam theory (ABT) of Barton and associates [24, 25], series-form expressions for the optical force of a spherical particle in the Airy evanescent field are presented. As an application of these equations, the internal and external electromagnetic fields for a dielectric sphere interacted with the Airy evanescent field have been discussed totally. The optical forces exerted on Mie dielectric particles by the Airy evanescent field were theoretically analyzed. The interpretations of numerical results are also presented in detail.

2. Theory and description

In this section, using vector potential approach [9], we derive the expressions for the evanescent field induced by total internal reflection of an Airy beam at the interface. Then by utilizing ABT, the radiation forces exerted on Mie particles by the Airy evanescent field are derived. Also, the internal and external electromagnetic fields for a dielectric particle illuminated by the Airy evanescent field are presented.

2.1 Expansions of the field components

We first consider the general formalism for the case when there is an arbitrary monochromatic electromagnetic field E(i)(r)exp(iωt)incident upon a dielectric spherical particle. The origin of coordinates x, y and z is laid at the center of the sphere, with x axis pointing in the horizontal direction. The sphere is taken to be isotropic, homogeneous, nonmagnetic, and nonconducting; its refractive index is n3 and can be complex index of refraction. The refractive index of surrounding medium is n2. The internal and external wave numbers are

k3=n3k0,k2=n2k0.
where k0 is the wave number in vacuum. The governing equation for the electromagnetic fields is the vector Helmholtz equation, which can be solved in a standard manner, for instance by making use of the vector potentials approach [9]. Then, we will give the expressions for the coefficients Alm and Blm that characterize the incident electric and magnetic fields. These coefficients are defined as follows. We first write the definition of the spherical harmonics:
Ylm(θ,ϕ)=[2l+14π(lm)!(l+m)!]1/2Plm(cosθ)exp(imϕ),
where θ is the polar angle and ϕ the azimuthal angle. The Riccati-Bessel function is defined as
ψl(x)=xjl(x)=(πx2)1/2Jv(x),
where jl is the spherical Bessel function and Jv is the ordinary Bessel function of the first kind, with v = l + 1/2. The radial part of the Helmholtz equation permits us to make the expansions
Er(i)=1r˜2l=1m=lll(l+1)Almψl(αr˜)Ylm(θ,ϕ),Hr(i)=1r˜2l=1m=lll(l+1)Blmψl(αr˜)Ylm(θ,ϕ),
where
α=k2a,r˜=r/a,
a is the radius of spherical particle. Taking into account the orthogonality property of Ylm, the expressions for the expansion coefficients are derived as follows:
Alm=1l(l+1)ψl(α)02π0πEr(i)(a,θ,ϕ)Ylm*(θ,ϕ)sinθdθdϕ,
and
Blm=1l(l+1)ψl(α)02π0πHr(i)(a,θ,ϕ)Ylm*(θ,ϕ)sinθdθdϕ,
The expansions for the incident (i), the scattered (s), and the internal (w) electromagnetic fields of 2D Airy evanescent wave in the spherical coordinate system are given in the Appendix.

2.2. Force components

Assuming a steady-state condition, the net radiation force F on the particle can be determined by integrating the dot product of the outwardly directed normal unit vector n^ and Maxwell’s stress tensor Tover a surface enclosing the particle [25]:

F=Sn^TdS,
where represents a time average. After a great deal of algebra and applying numerous recursion and orthogonality relationships among the spherical harmonic functions, Eq. (8) can be directly integrated, and the net force on the particle can be expressed as a series over the coefficients alm, blm, Alm, and Blm [25]:

Fx+iFy=iα416πk22l=1m=ll{[(l+m+2)(l+m+1)(2l+1)(2l+3)]1/2l(l+2)(2n22almal+1,m+1*+n22almAl+1,m+1*+n22Almal+1,m+1*+2blmbl+1,m+1*+blmBl+1,m+1*+Blmbl+1,m+1*)+[(lm+1)(lm+2)(2l+1)(2l+3)]1/2×l(l+2)(2n22al+1,m1alm*+n22al+1,m1Alm*+n22Al+1,m1alm*+2bl+1,m1blm*+bl+1,m1Blm*+Bl+1,m1blm*)[(l+m+1)(lm)]1/2n2(2almbl,m+1*+2blmal,m+1*almBl,m+1*+blmAl,m+1*+Blmal,m+1*Almbl,m+1*)},
Fz=α48πk22l=1m=ll{[(l+m+2)(l+m+1)(2l+1)(2l+3)]1/2l(l+2)Im[2n22al+1,malm*+n22al+1,mAlm*+n22Al+1,malm*+2bl+1,mblm*+bl+1,mBlm*+Bl+1,mblm*+n2m(2almblm*+almBlm*+Almblm*)].

2.3. Airy evanescent field as an incident field

In this section, using vector potential approach [9], we get the expressions for the evanescent field formed at the plane interface by total internal reflection of an Airy beam. Then we derive the coefficients Alm and Blm in Eqs. (6) and (7) for the evanescent field of Airy beam.

The schematic diagram of the problem is shown in Fig. 1 . The origin of coordinate system coincides with the center of the spherical particle at a distance d from the interface. The media below and above the boundary plane at z=dhave indices of refraction n1 and n2 (< n1), respectively. A focused Airy beam with wave vector k1 and frequency ωis incident from medium n1 with an angle of incidenceθ1(>θcrit), θcrit=sin1(n2/n1). The Airy beam center at (xc, yc, zc). We take x-z plane as the plane of incidence and (x^,y^,z^) as unit vectors.

 

Fig. 1 Spherical particle of radius a situated in the evanescent field regionz>d. An Airy beam is incident from below with an angle of incidence θ1(>θcrit)in the substrate.

Download Full Size | PPT Slide | PDF

For Airy beam, if the vector potential A has been determined, the electromagnetic fields E and H can be derived from A as follows:

E=in12k0××A,H=×A,
where we used the Lorentz gauge condition in the Gaussian system of electromagnetic units. The Fourier spectrum for the two-dimensional finite energy Airy beam in Cartesian coordinate can be expressed as
Φ(sx,sy)exp[i3(x¯03sx3+y¯03sy33a02x¯0sx3a02y¯0sy)a0(x¯02sx2+y¯02sy2)].
sx=nxcosθ1nzsinθ1,sy=ny,sz=nxsinθ1+nzcosθ1,x¯0=n1k0x0,y¯0=n1k0y0.
where (nx, ny, nz) denote the dimensionless direction cosines of the wave vector k1 along the Cartesian coordinate axes, x0 and y0 are characteristic lengths and a0 is the aperture coefficient which determines the beam propagation distance. If the vector potential A is perpendicular to the plane of incidence, the vector potential can be expressed as follow:
A=y^Cik0dsxdsyΦ(sx,sy)exp[ifi(x,y,z)],
wherefi(x,y,z)=n1k0[nx(xxc)+ny(yyc)+nz(zzc)],C is a normalization factor and the integration is over the domain of sx2+sy2<1and nz>0.

Substituting Eq. (14) into Eq. (11), the incident electromagnetic fields of Airy beam can be expressed as follows:

E(i)=Cdsxdsynxs^i+nynzp^i1nz2Φ(sx,sy)exp[ifi(x,y,z)],
H(i)=n1Cdsxdsynynzs^inxp^i1nz2Φ(sx,sy)exp[ifi(x,y,z)],
where we have defined two vectors, s^iand p^irepresent perpendicular and parallel to the plane of incidence, respectively. They can be expressed as follows:
s^i=nyx^+nxy^,p^i=(nxx^+nyy^)nz(1nz2)z^.
The transmitted Fresnel amplitude coefficients at the interface for p polarization and s polarization are
Tp=2n1nzn1ξ+n2nz,Ts=2n1nzn1nz+n2ξ.
whereξ=[1(n1/n2)2(nx2+ny2)]1/2. On paraxial condition, nxsinθ1,ny0, if θ1>sin1(n1/n2), the parameter ξ will be an imaginary number, the incident Airy beam is totally reflected on the interface and an evanescent electromagnetic wave is generated on the side of the medium n2. As a result, the transmitted electromagnetic field can be written as
E(t)=CdsxdsynxTss^t+nynzTpp^t1nz2Φ(sx,sy)exp[ift(x,y,z)],
H(t)=n2CdsxdsynynzTps^tnxTsp^t1nz2Φ(sx,sy)exp[ift(x,y,z)],
Where
s^t=nyx^+nxy^,p^t=(nxx^+nyy^)ξ(n1/n2)(1nz2)z^.
ft(x,y,z)=n1k0[nx(xxc)+ny(yyc)nz(zc+d)]+n2k0(z+d)ξ.
For evanescent field, the integral domain is(n1/n2)2(nx2+ny2)>1. The Eqs. (19) and (20) can be rewritten as
E(t)=Ex(t)x^+Ey(t)y^+Ez(t)z^,
H(t)=Hx(t)x^+Hy(t)y^+Hz(t)z^,
The radial field components of the evanescent fields are represented as
Er(t)=Ex(t)sinθcosϕ+Ey(t)sinθsinϕ+Ez(t)cosθ,
Hr(t)=Hx(t)sinθcosϕ+Hy(t)sinθsinϕ+Hz(t)cosθ.
Here θ and ϕ denote the usual polar and azimuthal angles. Substitute Er(t)and Hr(t) into Eqs. (6) and (7), after using a great deal of recursions and orthogonal relationships and finishing the integrals about θ and ϕ, we obtain the coefficients Almand Blm for the Airy evanescent field as follows:
Alm=AcdsxdsyΦ(sx,sy)exp[ig(r0)](nxiny)m(1nz2)(m+1)/2[inxβlm(ξ)Ts+nynzαlm(ξ)Tp],
Blm=BcdsxdsyΦ(sx,sy)exp[ig(r0)](nxiny)m(1nz2)(m+1)/2[inxαlm(ξ)Ts+nynzβlm(ξ)Tp],
where Acand Bcare constants:
Ac=Cil+1l(l+1)α2π(2l+1)(lm)!(l+m)!,Bc=n2Cill(l+1)α2π(2l+1)(lm)!(l+m)!,
g(r0)=n1k0[nxxc+nyyc+nz(zc+d)]+n2k0ξd.
the functions αlm(ξ)and βlm(ξ)are defined as

αlm(ξ)=ξ[Plm+1(ξ)+(l+m)(lm+1)Plm1(ξ)]2m(1ξ2)1/2Plm(ξ),
βlm(ξ)=Plm+1(ξ)(l+m)(lm+1)Plm1(ξ).

3. Results and discussions

First, we illustrate the dynamics of the Airy evanescent field at interface. In the following simulations, the refractive indices below and above interface are n1 = 1.5 and n2 = 1.0. The incident beam is considered as a two-dimensional (2D) Airy beam, whose parameters are chosen as: λ = 514.5 nm, a0 = 0.1 and x0 = y0 = 2 μm, respectively. The power of input Airy beam is P = 1 W.

Figure 2 shows the distributions of electric field magnitude |E| of 2D Airy evanescent wave above the interface: (a) in the x-z plane (y = 0), while xc = yc = 0; (b) the distributions at the origin point, which is as a function of the beam center’s displacements (xc, yc) in the x-y plane (z = 0). Incident angle θ1 = 0.85 rad, exceeding the critical angle, d = 0.8λ, beam center zc = −(d + 0.8λ) = −2d.

 

Fig. 2 Plots of electric field magnitude |(E)| of 2D Airy evanescent wave above the interface: (a) in the x-z plane (y = 0), (b) at the origin point, as a function of the beam center’s displacements (xc, yc) in the x-y plane (z = 0), while d = 0.8λ, zc = −2d, θ1 = 0.85 rad.

Download Full Size | PPT Slide | PDF

As shown in Fig. 2(a), the Airy evanescent wave dies away several wavelengths away from the interface. In Fig. 2(b), the distributions of Airy evanescent field illustrate the typical characteristic of 2D Airy beam. Field of maximal intensity is located in the vicinity of the point (xc = yc = 0). With the increasing of the displacements, field intensity decays gradually.

In the following, calculations were performed for the Airy evanescent field incident upon a spherical dielectric particle. The knowledge of the electric field energy density |E|2is often referred as source function and their distribution inside the particle is important in the field of Raman or fluorescence scattering from molecules in a microparticle. In Fig. 3(a) we present the source function of a spherical particle in the x-z plane (y = 0) with the radius a = d = 2λ and refractive index is n3 = 1.59, it is refractive index of polystyrene; in Fig. 3(b), the refractive index of the sphere is n3 = 1.5 + 3.1i, it is refractive index of nickel. The Airy beam center at xc = yc = 0, zc = −(d + 0.8λ), other parameters are the same as in Fig. 2(a).

 

Fig. 3 Plots of the source function of Airy evanescent field in the x-z plane with a dielectric spherical particle situating on the interface: (a) n3 = 1.59, a polystyrene sphere, (b) n3 = 1.5 + 3.1i, a nickel sphere. The arrows in plots represent power flow of Airy evanescent field. Particle radius a = d = 2λ, xc = yc = 0, zc = −(d + 0.8λ), θ1 = 0.85 rad.

Download Full Size | PPT Slide | PDF

In Fig. 3(a), it is noticeable that most of the energy is coupled into the particle. The wave undergoes total internal reflection inside the particle and a traveling wave along the surface occurs. Therefore most of the energy is confined near the surface. The arrows denote the directions of Poynting vectors. For nickel sphere, the refractive index is complex, most of the energy is absorbed and the wave tunneled into the particle has vanished. It is shown in Fig. 3(b).

Now we discuss the optical forces exerted on a Mie particle in an Airy evanescent field. The distribution of evanescent field is illustrated in Fig. 2(b). A polystyrene sphere with radius a = d = 0.8λ and refractive index n3 = 1.59 is considered. The sphere touches the interface and the refractive index of the surrounding medium is n2 = 1. Figure 4(a) shows the variations of the optical forces as a function of beam center displacements xc, while we let yc = 0, zc = −2d; Fig. 4(b) shows the variations of the optical forces as a function of beam center displacements yc, while xc = 0, other parameters are same as in Fig. 4(a).

 

Fig. 4 Plots of optical forces as a function of beam center’s displacements: (a) xc, while yc = 0, (b) yc, while xc = 0. Particle refractive index n3 = 1.59, a = d = 0.8λ, zc = −2d, θ1 = 0.85 rad.

Download Full Size | PPT Slide | PDF

We can see from Fig. 4 that the x-component of optical forces Fx is positive, the particle will moves along x direction; the z-component of optical force Fz is negative, the particle will be attracted toward the interface. The characteristics of the optical forces are similar with the distributions of Airy evanescent field, the oscillation peaks of the force curves correspond to the lobes of the evanescent field in Fig. 2(b) exactly. The magnitudes of Fx and Fz are larger than Fy exceeding one order of magnitude, that is because the non-uniform momentum transfer to the particle by the evanescent field.

In Fig. 5 , we evaluate the optical forces exerted on a nickel sphere with refractive index n3 = 1.5 + 3.1i. Other parameters are same as in Fig. 4. Numerical results show that Fx decrease with the absorption coefficient increasing, and Fz is positive for this situation. It is obvious that a bigger repulsive force has caused by absorption and it has suppressed the attractive force toward the interface caused by refractive index gradient.

 

Fig. 5 Plots of optical forces as a function of beam center’s displacements: (a) xc, while yc = 0, (b) yc, while xc = 0. Particle refractive index n3 = 1.5 + 3.1i. Other parameters are the same as in Fig. 4.

Download Full Size | PPT Slide | PDF

Then we investigate the behaviors of the optical forces as a function of the radius of particle, while d = 0.55 μm, the center of Airy beam are xc = yc = 0, zc = −(d + 0.8λ), incident angle θ1 = π/3 rad. Particle radius varies in the range (0.15<a<0.55) μm. Numerical results are shown in Fig. 6 : (a) refractive index n3 = 1.59, polystyrene particle, (b) n3 = 1.5 + 3.1i, nickel particle.

 

Fig. 6 Plots of optical forces as a function of particle radius: (a) n3 = 1.59, (b) n3 = 1.5 + 3.1i. While d = 0.55 μm, xc = yc = 0, zc = −(d + 0.8λ), θ1 = π/3.

Download Full Size | PPT Slide | PDF

As shown in Fig. 6, all the forces increase monotonously with the size of particle radius increasing. In Fig. 6(a), ripple structures of oscillations appear in each optical force curve for polystyrene particle and the oscillation strength increase gradually. But there is no oscillation for the complex refractive index particle in the whole variations of particle radius in Fig. 6(b).

The reason for the occurrence of oscillations for specific refractive index is due to the Morphology Dependent Resonances (MDR) [26, 27]. MDR originates from the interference of an electromagnetic wave propagating inside a dielectric particle confined by total internal reflection, as the wave propagating inside the particle, the constructive interference leads to a series of peaks in the optical force curves for appropriate particle size.

Clearly can be seen from Fig. 3(a), most energy of the evanescent fields is trapped inside the particle due to total internal reflection and the wave circling along the surface. The interference of the internal field would generate MDRs for certain radius particle for n3 = 1.59. But for nickel particle shown in Fig. 3(b), the internal field vanished due to strong absorption for its complex refractive index, so there will be no interference inside the particle, nor the resonances.

Finally, we analyze the impacts of incident angle on the optical forces as a function of particle radius (0.2<a<1) μm. Numerical results are shown in Fig. 7 : (a) θ1 = 0.85 rad, (b) θ1 = π/3 rad, while the distance d is varied with particle radius as d = a in this case, n3 = 1.59, xc = yc = 0, zc = −(d + 0.8λ). As we can see, the oscillations due to the MDR become evident as the particle radius increases, the larger particle size the sharper resonance peaks. And as expected, smaller incident angle would generate larger optical forces, but basically, different incident angles won’t change the shape of the force curves. The magnitude of Fy is much smaller than Fx and Fz for the non-uniform distribution of the evanescent field, we didn’t show them for clarity.

 

Fig. 7 Plots of optical forces as a function of particle radius with different incident angles: (a) θ1 = 0.85 rad, (b) θ1 = π/3 rad. While n3 = 1.59, d = a, xc = yc = 0, zc = −(d + 0.8λ).

Download Full Size | PPT Slide | PDF

4. Conclusions

In conclusion, using vector potential and spectrum representation, the expressions for the Airy evanescent field are derived. The internal and near-surface electric fields that Mie particles interacted with the evanescent field are investigated. Particle with complex refractive index would generate strong absorption effect and change the direction of optical force. By using the Arbitrary Beam Theory, the optical forces exerted on the Mie particles in the evanescent field are evaluated. Optical forces for the particles with specific refractive index would exhibit oscillations that come from MDRs. We believe that the theoretical results presented in this paper would be useful for investigations in optical micro-manipulation and near-field optics.

Appendix: expansions of the evanescent fields of 2D Airy beam

alm=ψl(n˜α)ψl(α)n˜ψl(n˜α)ψl(α)n˜ψl(n˜α)ξl(1)(α)ψl(n˜α)ξl(1)(α)Alm,
blm=n˜ψl(n˜α)ψl(α)ψl(n˜α)ψl(α)ψl(n˜α)ξl(1)(α)n˜ψl(n˜α)ξl(1)(α)Blm,
clm=ξl(1)(α)ψl(α)ξl(1)(α)ψl(α)n˜2ψl(n˜α)ξl(1)(α)n˜ψl(n˜α)ξl(1)(α)Alm,
dlm=ξl(1)(α)ψl(α)ξl(1)(α)ψl(α)ψl(n˜α)ξl(1)(α)n˜ψl(n˜α)ξl(1)(α)Blm.

where k0=2π/λ,α=n2k0a,n˜=n3/n2,a, n3 is radius and refractive index of the particle, respectively. ξl(1)=ψliχl, ψland χl are the Riccati-Bessel functions. Ylm(θ,ϕ)is the spherical harmonic function, r˜=r/a. The expansions for the incident (i), the scattered (s), and the internal (w) electromagnetic fields in the spherical coordinate system are summarized as follows:

Incident field

Er(i)=1r˜2l=1m=ll[l(l+1)Almψl(αr˜)Ylm(θ,ϕ)],
Eθ(i)=αr˜l=1m=ll(Almψl(αr˜)Ylm(θ,ϕ)θmn2Blmψl(αr˜)Ylm(θ,ϕ)sinθ),
Eϕ(i)=αr˜l=1m=ll(imAlmψl(αr˜)Ylm(θ,ϕ)sinθin2Blmψl(αr˜)Ylm(θ,ϕ)θ).
Hr(i)=1r˜2l=1m=ll[l(l+1)Blmψl(αr˜)Ylm(θ,ϕ)],
Hθ(i)=αr˜l=1m=ll(Blmψl(αr˜)Ylm(θ,ϕ)θ+mn2Almψl(αr˜)Ylm(θ,ϕ)sinθ),
Hϕ(i)=αr˜l=1m=ll(imBlmψl(αr˜)Ylm(θ,ϕ)sinθ+in2Almψl(αr˜)Ylm(θ,ϕ)θ).

Scattered field

Er(s)=1r˜2l=1m=ll[l(l+1)almξl(1)(αr˜)Ylm(θ,ϕ)],
Eθ(s)=αr˜l=1m=ll(almξl(1)(αr˜)Ylm(θ,ϕ)θmn2blmξl(1)(αr˜)Ylm(θ,ϕ)sinθ),
Eϕ(s)=αr˜l=1m=ll(imalmξl(1)(αr˜)Ylm(θ,ϕ)sinθin2blmξl(1)(αr˜)Ylm(θ,ϕ)θ).
Hr(s)=1r˜2l=1m=ll[l(l+1)blmξl(1)(αr˜)Ylm(θ,ϕ)],
Hθ(s)=αr˜l=1m=ll(blmξl(1)(αr˜)Ylm(θ,ϕ)θ+mn2almξl(1)(αr˜)Ylm(θ,ϕ)sinθ),
Hϕ(s)=αr˜l=1m=ll(imblmξl(1)(αr˜)Ylm(θ,ϕ)sinθ+in2almξl(1)(αr˜)Ylm(θ,ϕ)θ).

Internal field

Er(w)=1r˜2l=1m=ll[l(l+1)clmψl(n˜αr˜)Ylm(θ,ϕ)],
Eθ(w)=αr˜l=1m=ll(n˜clmψl(n˜αr˜)Ylm(θ,ϕ)θmn2dlmψl(n˜αr˜)Ylm(θ,ϕ)sinθ),
Eϕ(w)=αr˜l=1m=ll(imn˜clmψl(n˜αr˜)Ylm(θ,ϕ)sinθin2dlmψl(n˜αr˜)Ylm(θ,ϕ)θ).
Hr(w)=1r˜2l=1m=ll[l(l+1)dlmψl(n˜αr˜)Ylm(θ,ϕ)],
Hθ(w)=αr˜l=1m=ll(n˜dlmψl(n˜αr˜)Ylm(θ,ϕ)θ+mn2n˜2clmψl(n˜αr˜)Ylm(θ,ϕ)sinθ),
Hϕ(w)=αr˜l=1m=ll(imn˜dlmψl(n˜αr˜)Ylm(θ,ϕ)sinθ+in2n˜2clmψl(n˜αr˜)Ylm(θ,ϕ)θ).

Acknowledgments

We acknowledge financial supports from the Natural Science Foundation of China (grant 11074130, 61275148), Chinese National Key Basic Research Special Fund (2011CB922003), and 111 Project (B07013).

References and links

1. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11(5), 288–290 (1986). [CrossRef]   [PubMed]  

2. A. Ashkin and J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science 235(4795), 1517–1520 (1987). [CrossRef]   [PubMed]  

3. P. H. Jones, E. Stride, and N. Saffari, “Trapping and manipulation of microscopic bubbles with a scanning optical tweezer,” Appl. Phys. Lett. 89(8), 081113 (2006). [CrossRef]  

4. L. Bosanac, T. Aabo, P. M. Bendix, and L. B. Oddershede, “Efficient optical trapping and visualization of silver nanoparticles,” Nano Lett. 8(5), 1486–1491 (2008). [CrossRef]   [PubMed]  

5. Z.-J. Li, Z.-S. Wu, and Q.-C. Shang, “Calculation of radiation forces exerted on a uniaxial anisotropic sphere by an off-axis incident Gaussian beam,” Opt. Express 19(17), 16044–16057 (2011). [CrossRef]   [PubMed]  

6. M. Nieto-Vesperinas and J. J. Saenz, “Optical forces from an evanescent wave on a magnetodielectric small particle,” Opt. Lett. 35(23), 4078–4080 (2010). [CrossRef]   [PubMed]  

7. S. Kawata and T. Sugiura, “Movement of micrometer-sized particles in the evanescent field of a laser beam,” Opt. Lett. 17(11), 772–774 (1992). [CrossRef]   [PubMed]  

8. E. Almass and I. Brevik, “Radiation forces on a micrometer-sized sphere in an evanescent field,” J. Opt. Soc. Am. B 12(12), 2429–2438 (1995). [CrossRef]  

9. S. Chang, J. H. Jo, and S. S. Lee, “Theoretical calculations of optical force exerted on a dielectric sphere in the evanescent field generated with a totally-reflected focused gaussian beam,” Opt. Commun. 108(1-3), 133–143 (1994). [CrossRef]  

10. 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(6903), 145–147 (2002). [CrossRef]   [PubMed]  

11. T. Cizmar, V. Garces-Chavez, K. Dholakia, and P. Zemanek, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett. 86(17), 174101 (2005). [CrossRef]  

12. X. Tsampoula, V. Garces-Chavez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, “Femtosecond cellular transfection using a nondiffracting light beam,” Appl. Phys. Lett. 91(5), 053902 (2007). [CrossRef]  

13. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007). [CrossRef]   [PubMed]  

14. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007). [CrossRef]   [PubMed]  

15. P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy Beams,” Science 324(5924), 229–232 (2009). [CrossRef]   [PubMed]  

16. J. X. Li, W. P. Zang, and J. G. Tian, “Vacuum laser-driven acceleration by Airy beams,” Opt. Express 18(7), 7300–7306 (2010). [CrossRef]   [PubMed]  

17. A. Chong, W. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010). [CrossRef]  

18. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008). [CrossRef]  

19. J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip 9(10), 1334–1336 (2009). [CrossRef]   [PubMed]  

20. I. Dolev, T. Ellenbogen, and A. Arie, “Switching the acceleration direction of Airy beams by a nonlinear optical process,” Opt. Lett. 35(10), 1581–1583 (2010). [CrossRef]   [PubMed]  

21. H. Cheng, W. Zang, W. Zhou, and J. Tian, “Analysis of optical trapping and propulsion of Rayleigh particles using Airy beam,” Opt. Express 18(19), 20384–20394 (2010). [CrossRef]   [PubMed]  

22. H. I. Sztul and R. R. Alfano, “The Poynting vector and angular momentum of Airy beams,” Opt. Express 16(13), 9411–9416 (2008). [CrossRef]   [PubMed]  

23. J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008). [CrossRef]   [PubMed]  

24. 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]  

25. J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a sphereical particle illuminated by a focused laser beam,” J. Appl. Phys. 66(10), 4594–4602 (1989). [CrossRef]  

26. B. R. Johnson, “Theory of morphology-dependent resonances: shape resonances and width formulas,” J. Opt. Soc. Am. A 10(2), 343–352 (1993). [CrossRef]  

27. R. Quidant, D. Petrov, and G. Badenes, “Radiation forces on a Rayleigh dielectric sphere in a patterned optical near field,” Opt. Lett. 30(9), 1009–1011 (2005). [CrossRef]   [PubMed]  

References

  • View by:
  • |
  • |
  • |

  1. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett.11(5), 288–290 (1986).
    [CrossRef] [PubMed]
  2. A. Ashkin and J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science235(4795), 1517–1520 (1987).
    [CrossRef] [PubMed]
  3. P. H. Jones, E. Stride, and N. Saffari, “Trapping and manipulation of microscopic bubbles with a scanning optical tweezer,” Appl. Phys. Lett.89(8), 081113 (2006).
    [CrossRef]
  4. L. Bosanac, T. Aabo, P. M. Bendix, and L. B. Oddershede, “Efficient optical trapping and visualization of silver nanoparticles,” Nano Lett.8(5), 1486–1491 (2008).
    [CrossRef] [PubMed]
  5. Z.-J. Li, Z.-S. Wu, and Q.-C. Shang, “Calculation of radiation forces exerted on a uniaxial anisotropic sphere by an off-axis incident Gaussian beam,” Opt. Express19(17), 16044–16057 (2011).
    [CrossRef] [PubMed]
  6. M. Nieto-Vesperinas and J. J. Saenz, “Optical forces from an evanescent wave on a magnetodielectric small particle,” Opt. Lett.35(23), 4078–4080 (2010).
    [CrossRef] [PubMed]
  7. S. Kawata and T. Sugiura, “Movement of micrometer-sized particles in the evanescent field of a laser beam,” Opt. Lett.17(11), 772–774 (1992).
    [CrossRef] [PubMed]
  8. E. Almass and I. Brevik, “Radiation forces on a micrometer-sized sphere in an evanescent field,” J. Opt. Soc. Am. B12(12), 2429–2438 (1995).
    [CrossRef]
  9. S. Chang, J. H. Jo, and S. S. Lee, “Theoretical calculations of optical force exerted on a dielectric sphere in the evanescent field generated with a totally-reflected focused gaussian beam,” Opt. Commun.108(1-3), 133–143 (1994).
    [CrossRef]
  10. V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature419(6903), 145–147 (2002).
    [CrossRef] [PubMed]
  11. T. Cizmar, V. Garces-Chavez, K. Dholakia, and P. Zemanek, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett.86(17), 174101 (2005).
    [CrossRef]
  12. X. Tsampoula, V. Garces-Chavez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, “Femtosecond cellular transfection using a nondiffracting light beam,” Appl. Phys. Lett.91(5), 053902 (2007).
    [CrossRef]
  13. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett.32(8), 979–981 (2007).
    [CrossRef] [PubMed]
  14. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett.99(21), 213901 (2007).
    [CrossRef] [PubMed]
  15. P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy Beams,” Science324(5924), 229–232 (2009).
    [CrossRef] [PubMed]
  16. J. X. Li, W. P. Zang, and J. G. Tian, “Vacuum laser-driven acceleration by Airy beams,” Opt. Express18(7), 7300–7306 (2010).
    [CrossRef] [PubMed]
  17. A. Chong, W. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics4(2), 103–106 (2010).
    [CrossRef]
  18. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics2(11), 675–678 (2008).
    [CrossRef]
  19. J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip9(10), 1334–1336 (2009).
    [CrossRef] [PubMed]
  20. I. Dolev, T. Ellenbogen, and A. Arie, “Switching the acceleration direction of Airy beams by a nonlinear optical process,” Opt. Lett.35(10), 1581–1583 (2010).
    [CrossRef] [PubMed]
  21. H. Cheng, W. Zang, W. Zhou, and J. Tian, “Analysis of optical trapping and propulsion of Rayleigh particles using Airy beam,” Opt. Express18(19), 20384–20394 (2010).
    [CrossRef] [PubMed]
  22. H. I. Sztul and R. R. Alfano, “The Poynting vector and angular momentum of Airy beams,” Opt. Express16(13), 9411–9416 (2008).
    [CrossRef] [PubMed]
  23. J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express16(17), 12880–12891 (2008).
    [CrossRef] [PubMed]
  24. 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]
  25. J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a sphereical particle illuminated by a focused laser beam,” J. Appl. Phys.66(10), 4594–4602 (1989).
    [CrossRef]
  26. B. R. Johnson, “Theory of morphology-dependent resonances: shape resonances and width formulas,” J. Opt. Soc. Am. A10(2), 343–352 (1993).
    [CrossRef]
  27. R. Quidant, D. Petrov, and G. Badenes, “Radiation forces on a Rayleigh dielectric sphere in a patterned optical near field,” Opt. Lett.30(9), 1009–1011 (2005).
    [CrossRef] [PubMed]

2011 (1)

2010 (5)

2009 (2)

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip9(10), 1334–1336 (2009).
[CrossRef] [PubMed]

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy Beams,” Science324(5924), 229–232 (2009).
[CrossRef] [PubMed]

2008 (4)

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics2(11), 675–678 (2008).
[CrossRef]

L. Bosanac, T. Aabo, P. M. Bendix, and L. B. Oddershede, “Efficient optical trapping and visualization of silver nanoparticles,” Nano Lett.8(5), 1486–1491 (2008).
[CrossRef] [PubMed]

H. I. Sztul and R. R. Alfano, “The Poynting vector and angular momentum of Airy beams,” Opt. Express16(13), 9411–9416 (2008).
[CrossRef] [PubMed]

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express16(17), 12880–12891 (2008).
[CrossRef] [PubMed]

2007 (3)

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett.32(8), 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(21), 213901 (2007).
[CrossRef] [PubMed]

X. Tsampoula, V. Garces-Chavez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, “Femtosecond cellular transfection using a nondiffracting light beam,” Appl. Phys. Lett.91(5), 053902 (2007).
[CrossRef]

2006 (1)

P. H. Jones, E. Stride, and N. Saffari, “Trapping and manipulation of microscopic bubbles with a scanning optical tweezer,” Appl. Phys. Lett.89(8), 081113 (2006).
[CrossRef]

2005 (2)

T. Cizmar, V. Garces-Chavez, K. Dholakia, and P. Zemanek, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett.86(17), 174101 (2005).
[CrossRef]

R. Quidant, D. Petrov, and G. Badenes, “Radiation forces on a Rayleigh dielectric sphere in a patterned optical near field,” Opt. Lett.30(9), 1009–1011 (2005).
[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,” Nature419(6903), 145–147 (2002).
[CrossRef] [PubMed]

1995 (1)

1994 (1)

S. Chang, J. H. Jo, and S. S. Lee, “Theoretical calculations of optical force exerted on a dielectric sphere in the evanescent field generated with a totally-reflected focused gaussian beam,” Opt. Commun.108(1-3), 133–143 (1994).
[CrossRef]

1993 (1)

1992 (1)

1989 (1)

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a sphereical particle illuminated by a focused laser beam,” J. Appl. Phys.66(10), 4594–4602 (1989).
[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]

1987 (1)

A. Ashkin and J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science235(4795), 1517–1520 (1987).
[CrossRef] [PubMed]

1986 (1)

Aabo, T.

L. Bosanac, T. Aabo, P. M. Bendix, and L. B. Oddershede, “Efficient optical trapping and visualization of silver nanoparticles,” Nano Lett.8(5), 1486–1491 (2008).
[CrossRef] [PubMed]

Agate, B.

X. Tsampoula, V. Garces-Chavez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, “Femtosecond cellular transfection using a nondiffracting light beam,” Appl. Phys. Lett.91(5), 053902 (2007).
[CrossRef]

Alexander, D. R.

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a sphereical particle illuminated by a focused laser beam,” J. Appl. Phys.66(10), 4594–4602 (1989).
[CrossRef]

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]

Alfano, R. R.

Almass, E.

Arie, A.

Ashkin, A.

Badenes, G.

Barton, J. P.

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a sphereical particle illuminated by a focused laser beam,” J. Appl. Phys.66(10), 4594–4602 (1989).
[CrossRef]

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]

Baumgartl, J.

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip9(10), 1334–1336 (2009).
[CrossRef] [PubMed]

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics2(11), 675–678 (2008).
[CrossRef]

Bendix, P. M.

L. Bosanac, T. Aabo, P. M. Bendix, and L. B. Oddershede, “Efficient optical trapping and visualization of silver nanoparticles,” Nano Lett.8(5), 1486–1491 (2008).
[CrossRef] [PubMed]

Bjorkholm, J. E.

Bosanac, L.

L. Bosanac, T. Aabo, P. M. Bendix, and L. B. Oddershede, “Efficient optical trapping and visualization of silver nanoparticles,” Nano Lett.8(5), 1486–1491 (2008).
[CrossRef] [PubMed]

Brevik, I.

Broky, J.

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express16(17), 12880–12891 (2008).
[CrossRef] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett.99(21), 213901 (2007).
[CrossRef] [PubMed]

Brown, C. T. A.

X. Tsampoula, V. Garces-Chavez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, “Femtosecond cellular transfection using a nondiffracting light beam,” Appl. Phys. Lett.91(5), 053902 (2007).
[CrossRef]

Chang, S.

S. Chang, J. H. Jo, and S. S. Lee, “Theoretical calculations of optical force exerted on a dielectric sphere in the evanescent field generated with a totally-reflected focused gaussian beam,” Opt. Commun.108(1-3), 133–143 (1994).
[CrossRef]

Cheng, H.

Chong, A.

A. Chong, W. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics4(2), 103–106 (2010).
[CrossRef]

Christodoulides, D. N.

A. Chong, W. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics4(2), 103–106 (2010).
[CrossRef]

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy Beams,” Science324(5924), 229–232 (2009).
[CrossRef] [PubMed]

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express16(17), 12880–12891 (2008).
[CrossRef] [PubMed]

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett.32(8), 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(21), 213901 (2007).
[CrossRef] [PubMed]

Chu, S.

Cizmar, T.

T. Cizmar, V. Garces-Chavez, K. Dholakia, and P. Zemanek, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett.86(17), 174101 (2005).
[CrossRef]

Comrie, M.

X. Tsampoula, V. Garces-Chavez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, “Femtosecond cellular transfection using a nondiffracting light beam,” Appl. Phys. Lett.91(5), 053902 (2007).
[CrossRef]

Day, D.

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip9(10), 1334–1336 (2009).
[CrossRef] [PubMed]

Dholakia, K.

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip9(10), 1334–1336 (2009).
[CrossRef] [PubMed]

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics2(11), 675–678 (2008).
[CrossRef]

X. Tsampoula, V. Garces-Chavez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, “Femtosecond cellular transfection using a nondiffracting light beam,” Appl. Phys. Lett.91(5), 053902 (2007).
[CrossRef]

T. Cizmar, V. Garces-Chavez, K. Dholakia, and P. Zemanek, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett.86(17), 174101 (2005).
[CrossRef]

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature419(6903), 145–147 (2002).
[CrossRef] [PubMed]

Dogariu, A.

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express16(17), 12880–12891 (2008).
[CrossRef] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett.99(21), 213901 (2007).
[CrossRef] [PubMed]

Dolev, I.

Dziedzic, J. M.

Ellenbogen, T.

Garces-Chavez, V.

X. Tsampoula, V. Garces-Chavez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, “Femtosecond cellular transfection using a nondiffracting light beam,” Appl. Phys. Lett.91(5), 053902 (2007).
[CrossRef]

T. Cizmar, V. Garces-Chavez, K. Dholakia, and P. Zemanek, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett.86(17), 174101 (2005).
[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,” Nature419(6903), 145–147 (2002).
[CrossRef] [PubMed]

Gu, M.

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip9(10), 1334–1336 (2009).
[CrossRef] [PubMed]

Gunn-Moore, F.

X. Tsampoula, V. Garces-Chavez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, “Femtosecond cellular transfection using a nondiffracting light beam,” Appl. Phys. Lett.91(5), 053902 (2007).
[CrossRef]

Hannappel, G. M.

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip9(10), 1334–1336 (2009).
[CrossRef] [PubMed]

Jo, J. H.

S. Chang, J. H. Jo, and S. S. Lee, “Theoretical calculations of optical force exerted on a dielectric sphere in the evanescent field generated with a totally-reflected focused gaussian beam,” Opt. Commun.108(1-3), 133–143 (1994).
[CrossRef]

Johnson, B. R.

Jones, P. H.

P. H. Jones, E. Stride, and N. Saffari, “Trapping and manipulation of microscopic bubbles with a scanning optical tweezer,” Appl. Phys. Lett.89(8), 081113 (2006).
[CrossRef]

Kawata, S.

Kolesik, M.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy Beams,” Science324(5924), 229–232 (2009).
[CrossRef] [PubMed]

Lee, S. S.

S. Chang, J. H. Jo, and S. S. Lee, “Theoretical calculations of optical force exerted on a dielectric sphere in the evanescent field generated with a totally-reflected focused gaussian beam,” Opt. Commun.108(1-3), 133–143 (1994).
[CrossRef]

Li, J. X.

Li, Z.-J.

Mazilu, M.

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics2(11), 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,” Nature419(6903), 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,” Nature419(6903), 145–147 (2002).
[CrossRef] [PubMed]

Moloney, J. V.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy Beams,” Science324(5924), 229–232 (2009).
[CrossRef] [PubMed]

Nieto-Vesperinas, M.

Oddershede, L. B.

L. Bosanac, T. Aabo, P. M. Bendix, and L. B. Oddershede, “Efficient optical trapping and visualization of silver nanoparticles,” Nano Lett.8(5), 1486–1491 (2008).
[CrossRef] [PubMed]

Petrov, D.

Polynkin, P.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy Beams,” Science324(5924), 229–232 (2009).
[CrossRef] [PubMed]

Quidant, R.

Renninger, W.

A. Chong, W. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics4(2), 103–106 (2010).
[CrossRef]

Saenz, J. J.

Saffari, N.

P. H. Jones, E. Stride, and N. Saffari, “Trapping and manipulation of microscopic bubbles with a scanning optical tweezer,” Appl. Phys. Lett.89(8), 081113 (2006).
[CrossRef]

Schaub, S. A.

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a sphereical particle illuminated by a focused laser beam,” J. Appl. Phys.66(10), 4594–4602 (1989).
[CrossRef]

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]

Shang, Q.-C.

Sibbett, W.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature419(6903), 145–147 (2002).
[CrossRef] [PubMed]

Siviloglou, G. A.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy Beams,” Science324(5924), 229–232 (2009).
[CrossRef] [PubMed]

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express16(17), 12880–12891 (2008).
[CrossRef] [PubMed]

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett.32(8), 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(21), 213901 (2007).
[CrossRef] [PubMed]

Stevenson, D. J.

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip9(10), 1334–1336 (2009).
[CrossRef] [PubMed]

X. Tsampoula, V. Garces-Chavez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, “Femtosecond cellular transfection using a nondiffracting light beam,” Appl. Phys. Lett.91(5), 053902 (2007).
[CrossRef]

Stride, E.

P. H. Jones, E. Stride, and N. Saffari, “Trapping and manipulation of microscopic bubbles with a scanning optical tweezer,” Appl. Phys. Lett.89(8), 081113 (2006).
[CrossRef]

Sugiura, T.

Sztul, H. I.

Tian, J.

Tian, J. G.

Tsampoula, X.

X. Tsampoula, V. Garces-Chavez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, “Femtosecond cellular transfection using a nondiffracting light beam,” Appl. Phys. Lett.91(5), 053902 (2007).
[CrossRef]

Wise, F. W.

A. Chong, W. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics4(2), 103–106 (2010).
[CrossRef]

Wu, Z.-S.

Zang, W.

Zang, W. P.

Zemanek, P.

T. Cizmar, V. Garces-Chavez, K. Dholakia, and P. Zemanek, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett.86(17), 174101 (2005).
[CrossRef]

Zhou, W.

Appl. Phys. Lett. (3)

P. H. Jones, E. Stride, and N. Saffari, “Trapping and manipulation of microscopic bubbles with a scanning optical tweezer,” Appl. Phys. Lett.89(8), 081113 (2006).
[CrossRef]

T. Cizmar, V. Garces-Chavez, K. Dholakia, and P. Zemanek, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett.86(17), 174101 (2005).
[CrossRef]

X. Tsampoula, V. Garces-Chavez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, “Femtosecond cellular transfection using a nondiffracting light beam,” Appl. Phys. Lett.91(5), 053902 (2007).
[CrossRef]

J. Appl. Phys. (2)

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. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a sphereical particle illuminated by a focused laser beam,” J. Appl. Phys.66(10), 4594–4602 (1989).
[CrossRef]

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

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

Lab Chip (1)

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip9(10), 1334–1336 (2009).
[CrossRef] [PubMed]

Nano Lett. (1)

L. Bosanac, T. Aabo, P. M. Bendix, and L. B. Oddershede, “Efficient optical trapping and visualization of silver nanoparticles,” Nano Lett.8(5), 1486–1491 (2008).
[CrossRef] [PubMed]

Nat. Photonics (2)

A. Chong, W. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics4(2), 103–106 (2010).
[CrossRef]

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics2(11), 675–678 (2008).
[CrossRef]

Nature (1)

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature419(6903), 145–147 (2002).
[CrossRef] [PubMed]

Opt. Commun. (1)

S. Chang, J. H. Jo, and S. S. Lee, “Theoretical calculations of optical force exerted on a dielectric sphere in the evanescent field generated with a totally-reflected focused gaussian beam,” Opt. Commun.108(1-3), 133–143 (1994).
[CrossRef]

Opt. Express (5)

Opt. Lett. (6)

Phys. Rev. Lett. (1)

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett.99(21), 213901 (2007).
[CrossRef] [PubMed]

Science (2)

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy Beams,” Science324(5924), 229–232 (2009).
[CrossRef] [PubMed]

A. Ashkin and J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science235(4795), 1517–1520 (1987).
[CrossRef] [PubMed]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

Spherical particle of radius a situated in the evanescent field region z>d . An Airy beam is incident from below with an angle of incidence θ 1 ( > θ crit ) in the substrate.

Fig. 2
Fig. 2

Plots of electric field magnitude |(E)| of 2D Airy evanescent wave above the interface: (a) in the x-z plane (y = 0), (b) at the origin point, as a function of the beam center’s displacements (xc, yc) in the x-y plane (z = 0), while d = 0.8λ, zc = −2d, θ1 = 0.85 rad.

Fig. 3
Fig. 3

Plots of the source function of Airy evanescent field in the x-z plane with a dielectric spherical particle situating on the interface: (a) n3 = 1.59, a polystyrene sphere, (b) n3 = 1.5 + 3.1i, a nickel sphere. The arrows in plots represent power flow of Airy evanescent field. Particle radius a = d = 2λ, xc = yc = 0, zc = −(d + 0.8λ), θ1 = 0.85 rad.

Fig. 4
Fig. 4

Plots of optical forces as a function of beam center’s displacements: (a) xc, while yc = 0, (b) yc, while xc = 0. Particle refractive index n3 = 1.59, a = d = 0.8λ, zc = −2d, θ1 = 0.85 rad.

Fig. 5
Fig. 5

Plots of optical forces as a function of beam center’s displacements: (a) xc, while yc = 0, (b) yc, while xc = 0. Particle refractive index n3 = 1.5 + 3.1i. Other parameters are the same as in Fig. 4.

Fig. 6
Fig. 6

Plots of optical forces as a function of particle radius: (a) n3 = 1.59, (b) n3 = 1.5 + 3.1i. While d = 0.55 μm, xc = yc = 0, zc = −(d + 0.8λ), θ1 = π/3.

Fig. 7
Fig. 7

Plots of optical forces as a function of particle radius with different incident angles: (a) θ1 = 0.85 rad, (b) θ1 = π/3 rad. While n3 = 1.59, d = a, xc = yc = 0, zc = −(d + 0.8λ).

Equations (54)

Equations on this page are rendered with MathJax. Learn more.

k 3 = n 3 k 0 , k 2 = n 2 k 0 .
Y lm ( θ,ϕ )= [ 2l+1 4π ( lm )! ( l+m )! ] 1/2 P l m ( cosθ )exp( imϕ ),
ψ l ( x )=x j l ( x )= ( πx 2 ) 1/2 J v ( x ),
E r ( i ) = 1 r ˜ 2 l=1 m=l l l( l+1 ) A lm ψ l ( α r ˜ ) Y lm ( θ,ϕ ), H r ( i ) = 1 r ˜ 2 l=1 m=l l l( l+1 ) B lm ψ l ( α r ˜ ) Y lm ( θ,ϕ ),
α= k 2 a, r ˜ =r/a,
A lm = 1 l( l+1 ) ψ l ( α ) 0 2π 0 π E r ( i ) ( a,θ,ϕ ) Y lm * ( θ,ϕ )sinθdθdϕ ,
B lm = 1 l( l+1 ) ψ l ( α ) 0 2π 0 π H r ( i ) ( a,θ,ϕ ) Y lm * ( θ,ϕ )sinθdθdϕ ,
F = S n ^ T dS ,
F x +i F y = i α 4 16π k 2 2 l=1 m=l l { [ ( l+m+2 )( l+m+1 ) ( 2l+1 )( 2l+3 ) ] 1/2 l( l+2 )( 2 n 2 2 a lm a l+1,m+1 * + n 2 2 a lm A l+1,m+1 * + n 2 2 A lm a l+1,m+1 * +2 b lm b l+1,m+1 * + b lm B l+1,m+1 * + B lm b l+1,m+1 * )+ [ ( lm+1 )( lm+2 ) ( 2l+1 )( 2l+3 ) ] 1/2 ×l( l+2 )( 2 n 2 2 a l+1,m1 a lm * + n 2 2 a l+1,m1 A lm * + n 2 2 A l+1,m1 a lm * +2 b l+1,m1 b lm * + b l+1,m1 B lm * + B l+1,m1 b lm * ) [ ( l+m+1 )( lm ) ] 1/2 n 2 ( 2 a lm b l,m+1 * +2 b lm a l,m+1 * a lm B l,m+1 * + b lm A l,m+1 * + B lm a l,m+1 * A lm b l,m+1 * ) },
F z = α 4 8π k 2 2 l=1 m=l l { [ ( l+m+2 )( l+m+1 ) ( 2l+1 )( 2l+3 ) ] 1/2 l( l+2 )Im [ 2 n 2 2 a l+1,m a lm * + n 2 2 a l+1,m A lm * + n 2 2 A l+1,m a lm * +2 b l+1,m b lm * + b l+1,m B lm * + B l+1,m b lm * + n 2 m( 2 a lm b lm * + a lm B lm * + A lm b lm * ) ].
E= i n 1 2 k 0 ××A,H=×A,
Φ( s x , s y )exp[ i 3 ( x ¯ 0 3 s x 3 + y ¯ 0 3 s y 3 3 a 0 2 x ¯ 0 s x 3 a 0 2 y ¯ 0 s y ) a 0 ( x ¯ 0 2 s x 2 + y ¯ 0 2 s y 2 ) ].
s x = n x cos θ 1 n z sin θ 1 , s y = n y , s z = n x sin θ 1 + n z cos θ 1 , x ¯ 0 = n 1 k 0 x 0 , y ¯ 0 = n 1 k 0 y 0 .
A= y ^ C i k 0 d s x d s y Φ( s x , s y )exp[ i f i ( x,y,z ) ],
E ( i ) =C d s x d s y n x s ^ i + n y n z p ^ i 1 n z 2 Φ( s x , s y )exp[ i f i ( x,y,z ) ],
H ( i ) = n 1 C d s x d s y n y n z s ^ i n x p ^ i 1 n z 2 Φ( s x , s y )exp[ i f i ( x,y,z ) ],
s ^ i = n y x ^ + n x y ^ , p ^ i =( n x x ^ + n y y ^ ) n z ( 1 n z 2 ) z ^ .
T p = 2 n 1 n z n 1 ξ+ n 2 n z , T s = 2 n 1 n z n 1 n z + n 2 ξ .
E ( t ) =C d s x d s y n x T s s ^ t + n y n z T p p ^ t 1 n z 2 Φ( s x , s y )exp[ i f t ( x,y,z ) ],
H ( t ) = n 2 C d s x d s y n y n z T p s ^ t n x T s p ^ t 1 n z 2 Φ( s x , s y )exp[ i f t ( x,y,z ) ],
s ^ t = n y x ^ + n x y ^ , p ^ t =( n x x ^ + n y y ^ )ξ( n 1 / n 2 )( 1 n z 2 ) z ^ .
f t ( x,y,z )= n 1 k 0 [ n x ( x x c )+ n y ( y y c ) n z ( z c +d ) ]+ n 2 k 0 ( z+d )ξ.
E ( t ) = E x ( t ) x ^ + E y ( t ) y ^ + E z ( t ) z ^ ,
H ( t ) = H x ( t ) x ^ + H y ( t ) y ^ + H z ( t ) z ^ ,
E r ( t ) = E x ( t ) sinθcosϕ+ E y ( t ) sinθsinϕ+ E z ( t ) cosθ,
H r ( t ) = H x ( t ) sinθcosϕ+ H y ( t ) sinθsinϕ+ H z ( t ) cosθ.
A lm = A c d s x d s y Φ( s x , s y )exp[ ig( r 0 ) ] ( n x i n y ) m ( 1 n z 2 ) ( m+1 )/2 [ i n x β lm ( ξ ) T s + n y n z α lm ( ξ ) T p ] ,
B lm = B c d s x d s y Φ( s x , s y )exp[ ig( r 0 ) ] ( n x i n y ) m ( 1 n z 2 ) ( m+1 )/2 [ i n x α lm ( ξ ) T s + n y n z β lm ( ξ ) T p ] ,
A c =C i l+1 l( l+1 ) α 2 π( 2l+1 ) ( lm )! ( l+m )! , B c = n 2 C i l l( l+1 ) α 2 π( 2l+1 ) ( lm )! ( l+m )! ,
g( r 0 )= n 1 k 0 [ n x x c + n y y c + n z ( z c +d ) ]+ n 2 k 0 ξd.
α lm ( ξ )=ξ[ P l m+1 ( ξ )+( l+m )( lm+1 ) P l m1 ( ξ ) ]2m ( 1 ξ 2 ) 1/2 P l m ( ξ ),
β lm ( ξ )= P l m+1 ( ξ )( l+m )( lm+1 ) P l m1 ( ξ ).
a lm = ψ l ( n ˜ α ) ψ l ( α ) n ˜ ψ l ( n ˜ α ) ψ l ( α ) n ˜ ψ l ( n ˜ α ) ξ l ( 1 ) ( α ) ψ l ( n ˜ α ) ξ l ( 1 ) ( α ) A lm ,
b lm = n ˜ ψ l ( n ˜ α ) ψ l ( α ) ψ l ( n ˜ α ) ψ l ( α ) ψ l ( n ˜ α ) ξ l ( 1 ) ( α ) n ˜ ψ l ( n ˜ α ) ξ l ( 1 ) ( α ) B lm ,
c lm = ξ l ( 1 ) ( α ) ψ l ( α ) ξ l ( 1 ) ( α ) ψ l ( α ) n ˜ 2 ψ l ( n ˜ α ) ξ l ( 1 ) ( α ) n ˜ ψ l ( n ˜ α ) ξ l ( 1 ) ( α ) A lm ,
d lm = ξ l ( 1 ) ( α ) ψ l ( α ) ξ l ( 1 ) ( α ) ψ l ( α ) ψ l ( n ˜ α ) ξ l ( 1 ) ( α ) n ˜ ψ l ( n ˜ α ) ξ l ( 1 ) ( α ) B lm .
E r ( i ) = 1 r ˜ 2 l=1 m=l l [ l( l+1 ) A lm ψ l ( α r ˜ ) Y lm ( θ,ϕ ) ] ,
E θ ( i ) = α r ˜ l=1 m=l l ( A lm ψ l ( α r ˜ ) Y lm ( θ,ϕ ) θ m n 2 B lm ψ l ( α r ˜ ) Y lm ( θ,ϕ ) sinθ ) ,
E ϕ ( i ) = α r ˜ l=1 m=l l ( im A lm ψ l ( α r ˜ ) Y lm ( θ,ϕ ) sinθ i n 2 B lm ψ l ( α r ˜ ) Y lm ( θ,ϕ ) θ ) .
H r ( i ) = 1 r ˜ 2 l=1 m=l l [ l( l+1 ) B lm ψ l ( α r ˜ ) Y lm ( θ,ϕ ) ] ,
H θ ( i ) = α r ˜ l=1 m=l l ( B lm ψ l ( α r ˜ ) Y lm ( θ,ϕ ) θ +m n 2 A lm ψ l ( α r ˜ ) Y lm ( θ,ϕ ) sinθ ) ,
H ϕ ( i ) = α r ˜ l=1 m=l l ( im B lm ψ l ( α r ˜ ) Y lm ( θ,ϕ ) sinθ +i n 2 A lm ψ l ( α r ˜ ) Y lm ( θ,ϕ ) θ ) .
E r ( s ) = 1 r ˜ 2 l=1 m=l l [ l( l+1 ) a lm ξ l ( 1 ) ( α r ˜ ) Y lm ( θ,ϕ ) ] ,
E θ ( s ) = α r ˜ l=1 m=l l ( a lm ξ l ( 1 ) ( α r ˜ ) Y lm ( θ,ϕ ) θ m n 2 b lm ξ l ( 1 ) ( α r ˜ ) Y lm ( θ,ϕ ) sinθ ) ,
E ϕ ( s ) = α r ˜ l=1 m=l l ( im a lm ξ l ( 1 ) ( α r ˜ ) Y lm ( θ,ϕ ) sinθ i n 2 b lm ξ l ( 1 ) ( α r ˜ ) Y lm ( θ,ϕ ) θ ) .
H r ( s ) = 1 r ˜ 2 l=1 m=l l [ l( l+1 ) b lm ξ l ( 1 ) ( α r ˜ ) Y lm ( θ,ϕ ) ] ,
H θ ( s ) = α r ˜ l=1 m=l l ( b lm ξ l ( 1 ) ( α r ˜ ) Y lm ( θ,ϕ ) θ +m n 2 a lm ξ l ( 1 ) ( α r ˜ ) Y lm ( θ,ϕ ) sinθ ) ,
H ϕ ( s ) = α r ˜ l=1 m=l l ( im b lm ξ l ( 1 ) ( α r ˜ ) Y lm ( θ,ϕ ) sinθ +i n 2 a lm ξ l ( 1 ) ( α r ˜ ) Y lm ( θ,ϕ ) θ ) .
E r ( w ) = 1 r ˜ 2 l=1 m=l l [ l( l+1 ) c lm ψ l ( n ˜ α r ˜ ) Y lm ( θ,ϕ ) ] ,
E θ ( w ) = α r ˜ l=1 m=l l ( n ˜ c lm ψ l ( n ˜ α r ˜ ) Y lm ( θ,ϕ ) θ m n 2 d lm ψ l ( n ˜ α r ˜ ) Y lm ( θ,ϕ ) sinθ ) ,
E ϕ ( w ) = α r ˜ l=1 m=l l ( im n ˜ c lm ψ l ( n ˜ α r ˜ ) Y lm ( θ,ϕ ) sinθ i n 2 d lm ψ l ( n ˜ α r ˜ ) Y lm ( θ,ϕ ) θ ) .
H r ( w ) = 1 r ˜ 2 l=1 m=l l [ l( l+1 ) d lm ψ l ( n ˜ α r ˜ ) Y lm ( θ,ϕ ) ] ,
H θ ( w ) = α r ˜ l=1 m=l l ( n ˜ d lm ψ l ( n ˜ α r ˜ ) Y lm ( θ,ϕ ) θ +m n 2 n ˜ 2 c lm ψ l ( n ˜ α r ˜ ) Y lm ( θ,ϕ ) sinθ ) ,
H ϕ ( w ) = α r ˜ l=1 m=l l ( im n ˜ d lm ψ l ( n ˜ α r ˜ ) Y lm ( θ,ϕ ) sinθ +i n 2 n ˜ 2 c lm ψ l ( n ˜ α r ˜ ) Y lm ( θ,ϕ ) θ ) .

Metrics