Abstract

Using the theory of electromagnetic scattering of a uniaxial anisotropic sphere, we derive the analytical expressions of the radiation forces exerted on a uniaxial anisotropic sphere by an off-axis incident Gaussian beam. The beam’s propagation direction is parallel to the primary optical axis of the anisotropic sphere. The effects of the permittivity tensor elements εt and εz on the axial radiation forces are numerically analyzed in detail. The two transverse components of radiation forces exerted on a uniaxial anisotropic sphere, which is distinct from that exerted on an isotropic sphere due to the two eigen waves in the uniaxial anisotropic sphere, are numerically studied as well. The characteristics of the axial and transverse radiation forces are discussed for different radii of the sphere, beam waist width, and distances from the sphere center to the beam center of an off-axis Gaussian beam. The theoretical predictions of radiation forces exerted on a uniaxial anisotropic sphere are hoped to provide effective ways to achieve the improvement of optical tweezers as well as the capture, suspension, and high-precision delivery of anisotropic particles.

© 2011 OSA

1. Introduction

Since radiation force was first applied for the optical acceleration and trapping of a particle in 1970, as first reported by Ashkin [1], optical tweezers have attracted considerable attention due to advantages such as the capability to hold and manipulate particles, such as biological cell, with micrometer and sub-micrometer dimensions non-intrusively. Optical tweezers have been comprehensively applied in various fields such as physics, biology, engineering dynamics, and so on. Subsequently, Ashkin produced remarkable studies on these [2, 3]. On basis of his works, many other researchers also investigated the radiation forces exerted on spherical particles of various sizes using different methods.

For a particle much smaller than the incident wavelength (d<<λ), the Rayleigh dipole approximation (RDA) is employed to calculate radiation forces. Radiation forces exerted on a dielectric sphere in the Rayleigh scattering regime were studied by Harada et al. [4] in 1996. Conversely, for a particle much larger than the incident wavelength (d/λ>10), many scholars employed the ray optics theory (ROT) as it is most applicable to radiation force calculations. In 1991, Bakker Schut et al. [5] investigated the validity of ROT for calculating the stability of optical traps, making comparisons with experimental results on axial radiation forces. In 1995, Gauthier et al. [6] demonstrated that the axial and radial forces applied to micrometer-sized spheres could be obtained from ROT. In 1996, Wohland et al. [7] researched on the influence of polarization on axial and lateral forces exerted by optical tweezers. Meanwhile, in 1998, Shojiro et al. [8] analyzed the axial force acting on a dielectric sphere in a focused laser beam using ROT. For a particle whose size is of the order of the incident wavelength, both the RDA and the ROT are not applicable because the diffraction phenomena cannot be neglected. The generalized Lorenz-Mie theory has been proposed and developed by Gouesbet et al [9], Ren et al [11,12], and Lock et al [13] to calculate radiation forces. An equivalent alternative is available from Barton et al [10] Using GLMT, Nahmias et al. [14] analyzed the radiation forces in laser trapping and laser-guided direct writing applications. Martinot and Pouligny et al [15] also studied the forces exerted by a Gaussian laser beam on a small dielectric sphere in water as a function of beam-off centre. The GLMT was justified to be a complete beam scattering theory for particles of all sizes. Unlike the RDA and ROT, however, it did not have limitations for particles of intermediate sizes. Nahmias et al. [16] and Mao et al. [17] studied this performance by compared the radiation forces obtained by ROT and GLMT, and described the limitations and error analysis as well. In 2007, Xu et al. [18] theoretically predicted the radiation force exerted on a spheroid by an arbitrarily shaped beam. Some other also studied the radiation force exerted on multiple spheres by two or more beams [19].

With the development of optical tweezer technology, these instruments have become applicable to more fields, such as in the manipulation and capture of cells and large molecules in biology, preparation of photon crystals, facture and process of minitype apparatus, laser refrigeration technology, and so on. In previous literatures, theories and experiments on radiation forces focused on the homogeneous isotropic spherical particles.

Recently, the anisotropic medium has drawn growing attention due to its applicability in the fabrication of microstrip devices, microstrip circuits, and microwave engineering, to name a few. However, published works on radiation forces exerted on anisotropic particles are extremely scarce. The scattering characteristics of a uniaxial anisotropic sphere, by a plane wave were investigated by many scholars [2023] in past decades. Researchers from our group recently studied the scattered, internal, and incident fields of a uniaxial anisotropic sphere by an on-axis, off-axis, and arbitrary orientation incident Gaussian beam [24, 25], and analyzed the electromagnetic scattering by uniaxial anisotropic biospheres as well [26].

In view of the distinct internal field of the anisotropic sphere from that of the isotropic sphere, the axial radiation force exerted on an anisotropic sphere may represent different properties. The two transverse components of radiation forces exerted on an anisotropic sphere may be inconsistent without the symmetrical property in the transverse direction as the isotropic sphere. Therefore, it is critical and interesting to exploit the radiation forces exerted on a uniaxial anisotropic spherical particle by an off-axis incident Gaussian beam.

Based on the GLMT [9] and scattering coefficients of a uniaxial anisotropic sphere scattered by an off-axis Gaussian beam [24, 25], we have studied the axial and transverse radiation forces exerted on a uniaxial anisotropic sphere. Moreover, utilizing the orthogonality of associated Legendre functions and trigonometric functions, the analytical expressions of the two kinds of radiation forces are derived. The effects of radius, beam waist width, and permittivity tensor elements εt and εz on axial radiation forces are numerically analyzed in detail. Some selected results on the transverse radiation are also numerically discussed. Time dependence of the form exp(iωt) is assumed and suppressed, where ω is the circular frequency.

2. Scattering of a uniaxial anisotropic sphere by an off-axis Gaussian beam

Consider a homogeneous, uniaxial anisotropic sphere of radius a centrally located in a spherical coordinate system. The primary optical axis is coincident with the z-axis. As Fig. 1 illustrates, the particle is illuminated by an x-polarized at the waist Gaussian beam propagating in the z-axis direction, while the center of the beam waist O is located at (x0,y0,z0).

 

Fig. 1 Uniaxial anisotropic sphere illuminated by an off-axis Gaussian beam.

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In terms of spherical vector wave functions (SVWFs), the incident Gaussian beam can be expanded in the particle coordinate system Oxyz as [25]

Ei=E0n=1m=nn[amniMmn(1)(r,k0)+bmniNmn(1)(r,k0)],Hi=E0k0iωμ0n=1m=nn[amniNmn(1)(r,k0)+bmniMmn(1)(r,k0)],
where k0=2π/λ and λ is light wavelength in the surrounding medium, μ0 is the permeability of the surrounding medium, E0 is the amplitude of the electric field at beam waist center. Mmn(l) and Nmn(l) are the SVWFs, which are written as
Mmn(l)(kr,θ,ϕ)=zn(l)(kr)[imPnm(cosθ)sinθeimϕθ^dPnm(cosθ)dθeimϕϕ^],Nmn(l)(kr,θ,ϕ)=n(n+1)zn(l)(kr)krPnm(cosθ)eimϕr^+1krd(rzn(l)(kr))dr[dPnm(cosθ)dθθ^+imPnm(cosθ)sinθϕ^]eimϕ,
where zn(l) represents an appropriate kind of spherical Bessel functions: the first kind jn, the second kind yn, or the third kind hn(1) and hn(2), denoted by l=1, 2, 3, or 4, respectively; Pnm(cosθ) is the associated Legendre function of the first kind.amni and bmni are so-called beam shaped coefficients (BSC). Gouesbet et al [27] put forward three approaches to compute the BSC. Wu et al [28] also put forward an improved algorithm of BSC. Comparing with other methods, the localized approximation is a faster and more efficient method to calculate the BSC. The more detailed description on the localized approximation can be found in [29, 30]. Here, we adopt the localized approximations with sum of series given by A. Doicu [31] to obtain expansion coefficients,
[amnibmni]=Cnm(1)m1Knmψ¯00eik0z012[ei(m1)φ0Jm1(2Q¯ρ0ρnw02)ei(m+1)φ0Jm+1(2Q¯ρ0ρnw02)],
where

Cnm={in12n+1n(n+1),m0(1)|m|(n+|m|)!(n|m|)!in12n+1n(n+1),m<0,   Knm={(i)|m|i(n+1/2)|m|1,                     m0n(n+1)n+1/2,                                                                                                                   m=0,
ψ¯00=iQ¯exp(iQ¯ρ02w02)exp(iQ¯(n+1/2)2k02w02),
ρn=(n+1/2)/k0,Q¯=(i2z0/l)1,ρ0=x02+y02,φ0=arctan(x0/y0).

The scattered fields can be expanded with the SVWFs as

Es=E0n=1m=nn[AmnsMmn(3)(r,k0)+BmnsNmn(3)(r,k0)],Hs=E0k0iωμ0n=1m=nn[AmnsNmn(3)(r,k0)+BmnsMmn(3)(r,k0)].

Meanwhile, the permittivity and permeability tensors ε¯¯ and μ¯¯ of uniaxial anisotropic sphere are characterized by

ε¯¯=[εt000εt000εz],            μ¯¯=[μt000μt000μz].

The internal fields can be expanded in terms of SVWFs using the Fourier transform method, as [20, 24, 25]:

EI(r)=E0q=12n=1m=nnn=12πGmnq0π[AmnqeMmn(1)(r,kq)                                                                 +BmnqeNmn(1)(r,kq)+CmnqeLmn(1)(r,kq)]pnm(cosθk)kq2sinθkdθk,
HI(r)=E0q=12n=1m=nnn=12πGmnq0π[AmnqhMmn(1)(r,kq)                                                                       +BmnqhNmn(1)(r,kq)+CmnqhLmn(1)(r,kq)]pnm(cosθk)kq2sinθkdθk,
where the expressions of the coefficientsAmnqe, Bmnqe,Cmnqe,Amnqh,Bmnqh, and Cmnqh can be found in reference [20]. The kq(q=1,2) denotes the two eigen wave numbers in the uniaxial anisotropic sphere and its expressions are,

k12=ω2εtμtμzμtsin2θk+μzcos2θk,     k22=ω2εtεzμtεtsin2θk+εzcos2θk.

In Eqs. (9)-(11), kq, θk and ϕk (included in the coefficients Amnqe, …, Cmnqh) are the spherical coordinate of the eigen wave number in spatial domain which are produced during the Fourier transform. Then, utilizing the continuity of the tangential electric and magnetic field components at r=a, the scattering coefficients can then be derived [20, 24, 25]:

Amns=1hn(1)(k0a)[n=12πGmnqq=120πAmnqejn(kqa)Pnm(cosθk)kq2sinθkdθkamnijn(k0a)],
Bmns=1hn(1)(k0a)[iωμ0k0q=12n=12πGmnq0πAmnqhjn(kqa)Pnm(cosθk)kq2sinθkdθkbmnijn(k0a)],
where the unknown expansion coefficient Gmnq can be solved through incident expansion coefficients [25]
q=12n=12πGmnq0πUmnqPnm(cosθk)kq2sinθkdθk=amnii(k0a)2,
q=12n=12πGmnq0πVmnqPnm(cosθk)kq2sinθkdθk=bmnii(k0a)2,
Where

Umnq={Amnqe1k0rddr[rhn(1)(k0r)]jn(kqr)iωμ0k0[Bmnqh1kqrddr[rjn(kqr)]+Cmnqhjn(kqr)r]hn(1)(k0r)}r=a,
Vmnq={iωμ0k0Amnqh1k0rddr[rhn(1)(k0r)]jn(kqr)[Bmnqe1kqrddr[rjn(kqr)]+Cmnqejn(kqr)r]hn(1)(k0r)}r=a.

3. Radiation forces exerted on a uniaxial anisotropic sphere

According to the classical electromagnetic field theory, laser beams carry energy and momentum. When a strongly convergent laser beam illuminates a particle, part of the momentum will be transferred from the laser beam to the particle due to the process of scattering. It will be represented through the radiation force received by the particle [10],

F=12Re02π0π[ε0ErE+μ0HrH12(ε0E2+μ0H2)r^]dS,
where ε0 and μ0 pertain to the permittivity and permeability of the surrounding medium, respectively. dS denotes the surface element at a close spherical surface of the surrounding particle. The electromagnetic fields indicate the superposition of the incident and scattered electromagnetic fields: E=Ei+Es,       H=Hi+Hs.

Using the relations between coordinates of Cartesian coordinate system and sphere coordinate system, we arrive at

x=rsinθcosϕ,y=rsinθsinϕ,z=rcosθ.

The three components of the radiation forces can be written as

Fx=12Re02π0π{12[ε0(ErEr*EθEθ*EϕEϕ*)+μ0(HrHr*HθHθ*HϕHϕ*)]r^       +(ε0ErEθ*+μ0HrHθ*)θ^+(ε0ErEϕ*+μ0HrHϕ*)ϕ^}r2sin2θcosϕdθdϕ|r>a,
Fy=12Re02π0π{12[ε0(ErEr*EθEθ*EϕEϕ*)+μ0(HrHr*HθHθ*HϕHϕ*)]r^       +(ε0ErEθ*+μ0HrHθ*)θ^+(ε0ErEϕ*+μ0HrHϕ*)ϕ^}r2sin2θsinϕdθdϕ|r>a,
Fz=12Re02π0π{12[ε0(ErEr*EθEθ*EϕEϕ*)+μ0(HrHr*HθHθ*HϕHϕ*)]r^       +(ε0ErEθ*+μ0HrHθ*)θ^+(ε0ErEϕ*+μ0HrHϕ*)ϕ^}r2sinθcosθdθdϕ|r>a,
where the superscript sign ‘*’ denotes the conjugate of the variable. Substituting the expressions of the incident and scattered field into Eqs. (20)(22) and utilizing the orthogonality of associated Legendre functions and trigonometric functions found in the appendix and the approximate expressions of Ricatti-Bessel functions at rλ, we can derive the analytical expressions of the radiation forces:
Fx+iFy=n0P0πck02w02n=1m=nn[(nm)(n+m+1)Nmn1Nm+1n1(amniBm+1nS*                             +bmniAm+1nS*+BmnSam+1ni*+AmnSbm+1ni*+2AmnSBm+1nS*+2BmnSAm+1nS*)                             i(nm1)(nm)(2n1)(2n+1)(n1)(n+1)Nmn1Nm+1n11(amniAm+1n1S*                           +bmniBm+1n1S*+AmnSam+1n1i*+BmnSbm+1n1i*+2AmnSAm+1n1S*+2BmnSBm+1n1S*)                           i(n+m+1)(n+m+2)(2n+1)(2n+3)n(n+2)Nmn1Nm+1n+11(amniAm+1n+1S*+                           bmniBm+1n+1S*+AmnSam+1n+1i*+BmnSbm+1n+1i*+2AmnSAm+1n+1S*+2BmnSBm+1n+1S*)],
Fz=2n0P0πck02w02Ren=1m=nn[in(n+2)(nm+1)(n+m+1)(2n+1)(2n+3)Nmn1Nmn+11(amn+1iAmnS*+                                                   Amn+1Samni*+bmn+1iBmnS*+Bmn+1Sbmni*+2Amn+1SAmnS*+2Bmn+1SBmnS*)                                                     mNmn2(amniBmnS*+bmniAmnS*+2AmnSBmnS*)].
where c is the speed of light in vacuum, n0 is the refractive index of surrounding medium, P0=kπw0E02/4μ0 denotes the power of the incident beam, and

Nmn=(2n+1)(nm)!4π(n+m)!       (m=0,±1,,±n).

4. Numerical results and discussion

3.1 Axial radiation force

To verify the correctness of our method, the axial radiation force exerted on a uniaxial anisotropic sphere reduced to an isotropic sphere is calculated and compared with the experimental results (the curves are denoted by Static measurement and Dynamic measurement) obtained by Schut et al. [5] and ROT results from [5, 8, 16] shown in Fig. 2 . It can be concluded that both the rigorous analytical approach in this paper and the ROT can predict radiation force. However, the results from our method are more agreeable with the experimental results, especially the peak value of the axial radiation force. This may be attributed to the fact that the ROT absolutely neglects the effect of the diffraction while the parameter size of the particle is not very large compared with the incident wavelength. Note that P0=100mW, d denotes the distance of the sphere center from the beam center. The remaining figures possess similar conditions as well.

 

Fig. 2 Comparison of the axial radiation force from the theory when the anisotropic sphere is reduced to an isotropic sphere (solid curve) with the experimental results (the curves are denoted by Static measurement and Dynamic measurement) and Ray Optics results in reference [5].( w0=1.8μm, εt=εz=2.3716εz, μt=μz=μ0, x0=y0=0, λ=0.488μm, a=3.75μm, n0=1.33)

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In Fig. 3 , the axial radiation force Fz exerted on a uniaxial anisotropic sphere with both electric and magnetic anisotropy in vacuum is plotted as a function of d for several values of beam waist widths w0. It can be found that for a strongly focused beam (here w 0 < 0.4μm) the axial radiation force Fz is negative around d = 1μm and positive elsewhere, i.e. in a small region after the beam waist, the uniaxial anisotropic sphere is attracted by the beam to the opposite direction of the propagation. This situation may indicate that the uniaxial anisotropic spherical particle can also be captured by a focus Gaussian beam, which is similar to the capture trait for isotropic sphere [5][8][12]. As w0 increases, the positions where Fz take maximal value shift slightly to the beam waist center and the minimum of Fz decreases, however, the form of the curves exhibits little change.

 

Fig. 3 Variation of the axial radiation force Fz with d for different beam waist widths. (a=2.0μm,εt=2.0ε0, εz=2.4ε0, μt=2.0μ0, μz=2.8μ0, x0=y0=0, λ=0.6328μm,n0=1.0.)

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Figure 4 shows variation of the axial radiation force Fz with d for several values of sphere radius a. The negative axial radiation forces appear as the sphere radius increases when the beam waist width is stated. More calculations and simulations indicate that the negative axial radiation forces appear at several positions and the values of Fz become very large when a5μm. This property of a uniaxial anisotropic sphere is very different from that of an isotropic sphere described in [8] due to their distinct internal electromagnetic fields.

 

Fig. 4 Variation of the axial radiation force Fz with d for different sphere radii.( w0=0.5μm, εt=2.0ε0, εz=2.4ε0, μt=2.0μ0, μz=2.8μ0, x0=y0=0, λ=0.6328μm,n0=1.0.)

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Fig. 7 Variation of the axial radiation force Fz with dfor different εz.(εt=2.0ε0, μt=μz=μ0, x0=y0=0, λ=0.6328μm, w0=0.4μm, a=2.0μm, n0=1.33.)

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Figures 5 -8 shows the variation of axial radiation force Fz with d for several values of εt and εz when n0=1.33 and μt=μz=μ0. The uniaxial anisotropic sphere can be captured under the beam waist center when εt<n02, and above the beam waist center when εt>n02. Such property of a uniaxial anisotropic sphere is very different from that of an isotropic sphere, which can be captured only above the beam waist center. This phenomenon has been studied by many researchers [46,8,12]. By introducing an improved standard scheme to compute the standard BSC, Polaert et al [32] also illustrated a reverse radiation forces exerted on an isotropic sphere with perfect Gaussian beams. However, Gaussian beams do not possess the best structure for producing such forces. As the value of εt increases, the bound position and area denoted by the values of d increases; as the value of εz increases, the extremum of Fz, especially the peak values, obviously increases. Figure 8 shows that the only positive axial radiation forces will be exerted on a uniaxial anisotropic sphere when εt is large, regardless of the size of εz. In fact, careful examination and abundant simulations indicate that the values of axial radiation forces will always be positive when eitherεt or εz is very large. For example, for the conditions shown in Fig. 5, the negative value of Fz will disappear when εz>6.3ε0. In Fig. 6 , the negative value of Fz will disappear when εz>7.0ε0. The values of Fz will become very large because of the massive increase in scattering force. According to dualization between the permittivity and permeability tensors in a uniaxial anisotropic sphere, the effects of permeability tensor components on the axial radiation forces may be similar to that of permittivity tensor components. Thus, these will not be analyzed in detail due to length restrictions.

 

Fig. 5 Variation of the axial radiation force Fz with dfor different εz.(εt=1.4ε0, μt=μz=μ0, x0=y0=0, λ=0.6328μm, w0=0.4μm, a=2.0μm, n0=1.33.)

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Fig. 8 Variation of the axial radiation force Fz with dfor different εz.(εt=4.0ε0, μt=μz=μ0, x0=y0=0, λ=0.6328μm, w0=0.4μm, a=2.0μm, n0=1.33.).

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Fig. 6 Variation of the axial radiation force Fz with dfor different εz.(εt=1.6ε0, μt=μz=μ0, x0=y0=0, λ=0.6328μm, w0=0.4μm, a=2.0μm, n0=1.33.)

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4.2 Transverse radiation forces

In this section, transverse radiation forces are calculated when an off-axis Gaussian beam incident at a uniaxial anisotropic sphere. In Fig. 9 , the transverse radiation forces Fx (Fy) exerted on a uniaxial anisotropic sphere located in water (n0=1.33) by an off-axis Gaussian beam are plotted as a function of x0 (y0). Note that (x0,y0,z0) denotes the coordinate of beam center in particle coordinate system Oxyz. It can be seen that the uniaxial anisotropic sphere with different radii is bound around the axis of the beam center despite the appearance of the secondary extremum of Fx, which is smaller than the extremum. As the radius increases, the bound area increases in both x-axis and y-axis. The variations of Fx with x0 and Fy with y0 are similar, but the secondary extremum of Fx evidently increases compared to that of Fy, resulting from the x-polarization incident field and the two eigen waves in the uniaxial anisotropic sphere. It is different from the properties of isotropic sphere; the variations of Fx with -x 0 and Fy with y0 are almost the same and there is no secondary extremum.

 

Fig. 9 Variation of the transverse radiation force Fx (Fy) with x0 (y0) for different sphere radii. (εt=2.0ε0, εz=2.4ε0, μt=μz=μ0, z0=1μm, y0=0(x0=0), λ=0.6328μm, w0=0.4μm, n0=1.33)

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Figures 10 and 11 show that the transverse radiation force Fx varies with x0 for several coordinates z0 and beam waist widths w0. It can be observed that as the sphere center moves away from the beam waist center along the beam axis, the peak values of Fx decrease when the secondary extremum exhibits little change. Thus, the uniaxial anisotropic sphere will not be bound in the transverse direction when the distance between the sphere center and the beam waist center along the beam axis is very large. As beam waist width w 0 increases, the secondary extremum of Fx gradually disappears, allowing the uniaxial anisotropic to be bound more easily in the transverse direction. A comparison of Fig. 10 with Fig. 3 indicates that the uniaxial anisotropic sphere will be captured more easily when w0 is very small; however, the sphere may be very difficult to be bound when w0 is very small (w0=0.2μm shown in Fig. 10). We need to consider both the axial trapping and the transverse trapping to select a beam waist width for a uniaxial anisotropic sphere with different radii, permittivity, and permeability if we intend to capture a uniaxial anisotropic spherical particle in a practical experiment.

 

Fig. 10 Variation of the transverse radiation force Fx with x0 for different z0.(w0=0.4μm, εt=2.0ε0, εz=2.4ε0, μt=μz=μ0, y0=0, λ=0.6328μm, a=1.0μm, n0=1.33.)

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Fig. 11 Variation of the transverse radiation force Fx with x0 for different w 0. (z0=1μm, εt=2.0ε0, εz=2.4ε0, μt=μz=μ0, y0=0, λ=0.6328μm, a=1.0μm, n0=1.33.)

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5. Conclusion

The analytical expressions of radiation forces exerted on a uniaxial anisotropic sphere by an off-axis Gaussian beam are derived. The primary optical axis is supposed parallel to the propagation direction of the incident beam. The effects of the position, the radius, the permittivity and the permeability tensor elements of the particle as well as beam waist width on the variations of axial and transverse components of radiation forces exerted on a uniaxial anisotropic sphere as function of the distance between the beam center and the sphere center are numerically studied in detail. It reveals that the properties of the radiation forces of a uniaxial anisotropic sphere are different from that of an isotropic one. The developed theory and the numerical code presented in this paper aim to provide a useful tool for the theoretical and experimental study of trapping or manipulating uniaxial anisotropic spherical particles. Radiation forces exerted on a single or multiple uniaxial anisotropic spheres by an arbitrary direction incident Gaussian beam will be discussed in our future research.

Appendix

Associated Legendre functions and trigonometric functions have following orthogonal relations:

02πei(mm')ϕdϕ0π[m'dPnm(cosθ)dθPn'm'(cosθ)sinθ+mPnm(cosθ)sinθdPn'm'(cosθ)dθ]sin2θeiϕdθ=(nm)(n+m+1)Nmn1Nm'n'1δm+1,m'δn,n'
02πei(m'm)ϕdϕ0π[mPnm(cosθ)sinθdPn'm'(cosθ)dθ+m'dPnm(cosθ)dθPn'm'(cosθ)sinθ]sin2θeiϕdθ=(n'm')(n'+m'+1)Nmn1Nm'n'1δm'+1,mδn,n'
02πei(mm')ϕdϕ0π[dPnm(cosθ)dθdPn'm'(cosθ)dθ+mm'Pnm(cosθ)sinθPn'm'(cosθ)sinθ]sin2θeiϕdθ=Nmn1Nm'n'1[(n+m+1)(n+m+2)(2n+1)(2n+3)n(n+2)δn+1,n'(n'm)(n'm+1)(2n'+1)(2n'+3)n'(n'+2)δn,n'+1]δm+1,m'
02πei(m'm)ϕdϕ0π[dPnm(cosθ)dθdPn'm'(cosθ)dθ+mm'Pnm(cosθ)sinθPn'm'(cosθ)sinθ]sin2θeiϕdθ=Nmn1Nm'n'1[(n'+m'+1)(n'+m'+2)(2n'+1)(2n'+3)n'(n'+2)δn'+1,n(nm')(nm'+1)(2n+1)(2n+3)n(n+2)δn',n+1]δm'+1,m
02πei(mm')ϕdϕ0π[m'dPnm(cosθ)dθPn'm'(cosθ)sinθ+mPnm(cosθ)sinθdPn'm'(cosθ)dθ]sinθcosθdθ=mNmn1Nm'n'1δm,m'δn,n'
02πei(mm')ϕdϕ0π[dPnm(cosθ)dθdPn'm'(cosθ)dθ+mm'Pnm(cosθ)sinθPn'm'(cosθ)sinθ]sinθcosθdθ=Nmn1Nm'n'1(n'm+1)(n'+m+1)(2n'+1)(2n'+3)n'(n'+2)δm,m'δn,n'+1               +Nmn1Nm'n'1(nm+1)(n+m+1)(2n+1)(2n+3)n(n+2)δm,m'δn+1,n'
where δm,n is the Kronecker delta.

Acknowledgments

The authors gratefully acknowledge support from the National Natural Science Foundation of China under Grant No. 60771038 and Fundamental Research Funds for the Central Universities.

References and links

1. A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24(4), 156–159 (1970). [CrossRef]  

2. A. Ashkin, “Applications of laser radiation pressure,” Science 210(4474), 1081–1088 (1980). [CrossRef]   [PubMed]  

3. 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]  

4. Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124(5-6), 529–541 (1996). [CrossRef]  

5. T. C. B. Schut, G. Hesselink, B. G. de Grooth, and J. Greve, “Experimental and theoretical investigations on the validity of the geometrical optics model for calculating the stability of optical traps,” Cytometry 12(6), 479–485 (1991). [CrossRef]   [PubMed]  

6. R. C. Gauthier and S. Wallace, “Optical levitation of spheres: analytical development and numerical computations of force equations,” J. Opt. Soc. Am. B 12(9), 1680–1687 (1995). [CrossRef]  

7. T. Wohland, A. Rosin, and E. H. K. Stelzer, “Theoretical determination of the influence of the polarization on forces exerted by optical tweezers,” Optik (Stuttg.) 102, 181–190 (1996).

8. S. Nemoto and H. Togo, “Axial force acting on a dielectric sphere in a focused laser beam,” Appl. Opt. 37(27), 6386–6394 (1998). [CrossRef]   [PubMed]  

9. G. Gouesbet, B. Maheu, and G. Grehan, “Light Scattering from a sphere arbitrarily located in a Gaussian beam,using a Bromwich formulation,” J. Opt. Soc. Am. A 5(9), 1427–1443 (1988). [CrossRef]  

10. J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66(10), 4954–4962 (1989). [CrossRef]  

11. K. F. Ren, G. Gréhan, and G. Gouesbet, “Radiation pressure forces exerted on a particle arbitrarily located in a Gaussian beam by using the generalized Lorenz-Mie theory and associated resonance effects,” Opt. Commun. 108(4-6), 343–354 (1994). [CrossRef]  

12. K. F. Ren, G. Gréhan, and G. Gouesbet, “Prediction of reverse radiation pressure by generalized Lorenz-Mie theory,” Appl. Opt. 35(15), 2702–2710 (1996). [CrossRef]   [PubMed]  

13. J. A. Lock, “Calculation of the radiation trapping force for laser tweezers by use of generalized Lorenz-Mie theory. I. Localized model description of an on-axis tightly focused laser beam with spherical aberration,” Appl. Opt. 43(12), 2532–2544 (2004). [CrossRef]   [PubMed]  

14. Y. K. Nahmias and D. J. Odde, “Analysis of Radiation Forces in Laser Trapping and Laser-Guided Direct Writing Applications,” IEEE J. Quantum Electron. 38(2), 131–141 (2002). [CrossRef]  

15. G. Martinot-Lagarde, B. Pouligny, M. I. Angelova, G. Grehan, and G. Gouesbet, ““Trapping and levitation of a dielectric sphere with off-centred Gaussian beams. II. GLMT analysis,” Pure Appl. Opt. J. Euro. Opt. Soc. Part A 4(5), 571–585 (1995). [CrossRef]  

16. Y. K. Nahmias, B. Z. Gao, and D. J. Odde, “Dimensionless parameters for the design of optical traps and laser guidance systems,” Appl. Opt. 43(20), 3999–4006 (2004). [CrossRef]   [PubMed]  

17. F. L. Mao, Q. R. Xing, K. Wang, L. Y. Lang, L. Chai, and Q. Y. Wang, “Calculation of axial optical forces exerted on medium-sized particles by optical trap,” Opt. Laser Technol. 39(1), 34–39 (2007). [CrossRef]  

18. F. Xu, K. F. Ren, G. Gouesbet, X. S. Cai, and G. Gréhan, “Theoretical prediction of radiation pressure force exerted on a spheroid by an arbitrarily shaped beam,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(2), 026613 (2007). [CrossRef]   [PubMed]  

19. M. Kawano, J. T. Blakely, R. Gordon, and D. Sinton, “Theory of dielectric micro-sphere dynamics in a dual-beam optical trap,” Opt. Express 16(13), 9306–9317 (2008). [CrossRef]   [PubMed]  

20. Y. L. Geng, X. B. Wu, and L. W. Li, “Mie scattering by a uniaxial anisotropic sphere,” physical review E 70, 056609 (2004). [CrossRef]  

21. B. Stout, M. Nevière, and E. Popov, “Mie scattering by an anisotropic object. Part I. Homogeneous sphere,” J. Opt. Soc. Am. A 23(5), 1111–1123 (2006). [CrossRef]   [PubMed]  

22. C. W. Qiu, S. Zouhdi, and A. Razek, “Modified Spherical Wave Functions With Anisotropy Ratio: Application to the Analysis of Scattering by Multilayered Anisotropic Shells,” IEEE Trans. Antenn. Propag. 55(12), 3515–3523 (2007). [CrossRef]  

23. M. Sluijter, D. K. de Boer, and J. J. M. Braat, “General polarized ray-tracing method for inhomogeneous uniaxially anisotropic media,” J. Opt. Soc. Am. A 25(6), 1260–1273 (2008). [CrossRef]   [PubMed]  

24. Z. S. Wu, Q. K. Yuan, Y. Peng, and Z. J. Li, “Internal and external electromagnetic fields for on-axis Gaussian beam scattering from a uniaxial anisotropic sphere,” J. Opt. Soc. Am. A 26(8), 1778–1788 (2009). [CrossRef]  

25. Q. K. Yuan, Z. S. Wu, and Z. J. Li, “Electromagnetic scattering for a uniaxial anisotropic sphere in an off-axis obliquely incident Gaussian beam,” J. Opt. Soc. Am. A 27(6), 1457–1465 (2010). [CrossRef]   [PubMed]  

26. Z. J. Li, Z. S. Wu, and H. Y. Li, “Analysis of electromagnetic scattering by uniaxial anisotropic bispheres,” J. Opt. Soc. Am. A 28(2), 118–125 (2011). [CrossRef]   [PubMed]  

27. G. Gouesbet, G. Grehan, and B. Maheu, “Computations of the gn coefficients in the geneeralized Lorenz-Mie theory using three different methods,” Appl. Opt. 27(23), 4874–4883 (1988). [CrossRef]   [PubMed]  

28. Z. S. Wu, L. X. Guo, K. F. Ren, G. Gouesbet, and G. Gréhan, “Improved algorithm for electromagnetic scattering of plane waves and shaped beams by multilayered spheres,” Appl. Opt. 36(21), 5188–5198 (1997). [CrossRef]   [PubMed]  

29. G. Gousbet, “Localized interpretation to compute all the coefficients gmn in the generalized Lorentz-Mie theory,” J. Opt. Soc. Am. A 7(6), 998–1007 (1990). [CrossRef]  

30. J. A. Lock and G. Gouesbet, “Rigorous justification fo the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory.I. On-axis beams,” J. Opt. Soc. Am. A 11(9), 2503–2515 (1994). [CrossRef]  

31. A. Doicu and T. Wriedt, “Computation of the beam-shape coefficients in the generalized Lorenz-Mie theory by using the translational addition theorem for spherical vector wave functions,” Appl. Opt. 36(13), 2971–2978 (1997). [CrossRef]   [PubMed]  

32. H. Polaert, G. Gréhan, and G. Gouesbet, “Improved standard beams with application to reverse radiation pressure,” Appl. Opt. 37(12), 2435–2440 (1998). [CrossRef]   [PubMed]  

References

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  • |

  1. A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24(4), 156–159 (1970).
    [CrossRef]
  2. A. Ashkin, “Applications of laser radiation pressure,” Science 210(4474), 1081–1088 (1980).
    [CrossRef] [PubMed]
  3. 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]
  4. Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124(5-6), 529–541 (1996).
    [CrossRef]
  5. T. C. B. Schut, G. Hesselink, B. G. de Grooth, and J. Greve, “Experimental and theoretical investigations on the validity of the geometrical optics model for calculating the stability of optical traps,” Cytometry 12(6), 479–485 (1991).
    [CrossRef] [PubMed]
  6. R. C. Gauthier and S. Wallace, “Optical levitation of spheres: analytical development and numerical computations of force equations,” J. Opt. Soc. Am. B 12(9), 1680–1687 (1995).
    [CrossRef]
  7. T. Wohland, A. Rosin, and E. H. K. Stelzer, “Theoretical determination of the influence of the polarization on forces exerted by optical tweezers,” Optik (Stuttg.) 102, 181–190 (1996).
  8. S. Nemoto and H. Togo, “Axial force acting on a dielectric sphere in a focused laser beam,” Appl. Opt. 37(27), 6386–6394 (1998).
    [CrossRef] [PubMed]
  9. G. Gouesbet, B. Maheu, and G. Grehan, “Light Scattering from a sphere arbitrarily located in a Gaussian beam,using a Bromwich formulation,” J. Opt. Soc. Am. A 5(9), 1427–1443 (1988).
    [CrossRef]
  10. J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66(10), 4954–4962 (1989).
    [CrossRef]
  11. K. F. Ren, G. Gréhan, and G. Gouesbet, “Radiation pressure forces exerted on a particle arbitrarily located in a Gaussian beam by using the generalized Lorenz-Mie theory and associated resonance effects,” Opt. Commun. 108(4-6), 343–354 (1994).
    [CrossRef]
  12. K. F. Ren, G. Gréhan, and G. Gouesbet, “Prediction of reverse radiation pressure by generalized Lorenz-Mie theory,” Appl. Opt. 35(15), 2702–2710 (1996).
    [CrossRef] [PubMed]
  13. J. A. Lock, “Calculation of the radiation trapping force for laser tweezers by use of generalized Lorenz-Mie theory. I. Localized model description of an on-axis tightly focused laser beam with spherical aberration,” Appl. Opt. 43(12), 2532–2544 (2004).
    [CrossRef] [PubMed]
  14. Y. K. Nahmias and D. J. Odde, “Analysis of Radiation Forces in Laser Trapping and Laser-Guided Direct Writing Applications,” IEEE J. Quantum Electron. 38(2), 131–141 (2002).
    [CrossRef]
  15. G. Martinot-Lagarde, B. Pouligny, M. I. Angelova, G. Grehan, and G. Gouesbet, ““Trapping and levitation of a dielectric sphere with off-centred Gaussian beams. II. GLMT analysis,” Pure Appl. Opt. J. Euro. Opt. Soc. Part A 4(5), 571–585 (1995).
    [CrossRef]
  16. Y. K. Nahmias, B. Z. Gao, and D. J. Odde, “Dimensionless parameters for the design of optical traps and laser guidance systems,” Appl. Opt. 43(20), 3999–4006 (2004).
    [CrossRef] [PubMed]
  17. F. L. Mao, Q. R. Xing, K. Wang, L. Y. Lang, L. Chai, and Q. Y. Wang, “Calculation of axial optical forces exerted on medium-sized particles by optical trap,” Opt. Laser Technol. 39(1), 34–39 (2007).
    [CrossRef]
  18. F. Xu, K. F. Ren, G. Gouesbet, X. S. Cai, and G. Gréhan, “Theoretical prediction of radiation pressure force exerted on a spheroid by an arbitrarily shaped beam,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(2), 026613 (2007).
    [CrossRef] [PubMed]
  19. M. Kawano, J. T. Blakely, R. Gordon, and D. Sinton, “Theory of dielectric micro-sphere dynamics in a dual-beam optical trap,” Opt. Express 16(13), 9306–9317 (2008).
    [CrossRef] [PubMed]
  20. Y. L. Geng, X. B. Wu, and L. W. Li, “Mie scattering by a uniaxial anisotropic sphere,” physical review E 70, 056609 (2004).
    [CrossRef]
  21. B. Stout, M. Nevière, and E. Popov, “Mie scattering by an anisotropic object. Part I. Homogeneous sphere,” J. Opt. Soc. Am. A 23(5), 1111–1123 (2006).
    [CrossRef] [PubMed]
  22. C. W. Qiu, S. Zouhdi, and A. Razek, “Modified Spherical Wave Functions With Anisotropy Ratio: Application to the Analysis of Scattering by Multilayered Anisotropic Shells,” IEEE Trans. Antenn. Propag. 55(12), 3515–3523 (2007).
    [CrossRef]
  23. M. Sluijter, D. K. de Boer, and J. J. M. Braat, “General polarized ray-tracing method for inhomogeneous uniaxially anisotropic media,” J. Opt. Soc. Am. A 25(6), 1260–1273 (2008).
    [CrossRef] [PubMed]
  24. Z. S. Wu, Q. K. Yuan, Y. Peng, and Z. J. Li, “Internal and external electromagnetic fields for on-axis Gaussian beam scattering from a uniaxial anisotropic sphere,” J. Opt. Soc. Am. A 26(8), 1778–1788 (2009).
    [CrossRef]
  25. Q. K. Yuan, Z. S. Wu, and Z. J. Li, “Electromagnetic scattering for a uniaxial anisotropic sphere in an off-axis obliquely incident Gaussian beam,” J. Opt. Soc. Am. A 27(6), 1457–1465 (2010).
    [CrossRef] [PubMed]
  26. Z. J. Li, Z. S. Wu, and H. Y. Li, “Analysis of electromagnetic scattering by uniaxial anisotropic bispheres,” J. Opt. Soc. Am. A 28(2), 118–125 (2011).
    [CrossRef] [PubMed]
  27. G. Gouesbet, G. Grehan, and B. Maheu, “Computations of the gn coefficients in the geneeralized Lorenz-Mie theory using three different methods,” Appl. Opt. 27(23), 4874–4883 (1988).
    [CrossRef] [PubMed]
  28. Z. S. Wu, L. X. Guo, K. F. Ren, G. Gouesbet, and G. Gréhan, “Improved algorithm for electromagnetic scattering of plane waves and shaped beams by multilayered spheres,” Appl. Opt. 36(21), 5188–5198 (1997).
    [CrossRef] [PubMed]
  29. G. Gousbet, “Localized interpretation to compute all the coefficients gmn in the generalized Lorentz-Mie theory,” J. Opt. Soc. Am. A 7(6), 998–1007 (1990).
    [CrossRef]
  30. J. A. Lock and G. Gouesbet, “Rigorous justification fo the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory.I. On-axis beams,” J. Opt. Soc. Am. A 11(9), 2503–2515 (1994).
    [CrossRef]
  31. A. Doicu and T. Wriedt, “Computation of the beam-shape coefficients in the generalized Lorenz-Mie theory by using the translational addition theorem for spherical vector wave functions,” Appl. Opt. 36(13), 2971–2978 (1997).
    [CrossRef] [PubMed]
  32. H. Polaert, G. Gréhan, and G. Gouesbet, “Improved standard beams with application to reverse radiation pressure,” Appl. Opt. 37(12), 2435–2440 (1998).
    [CrossRef] [PubMed]

2011

2010

2009

2008

2007

F. L. Mao, Q. R. Xing, K. Wang, L. Y. Lang, L. Chai, and Q. Y. Wang, “Calculation of axial optical forces exerted on medium-sized particles by optical trap,” Opt. Laser Technol. 39(1), 34–39 (2007).
[CrossRef]

F. Xu, K. F. Ren, G. Gouesbet, X. S. Cai, and G. Gréhan, “Theoretical prediction of radiation pressure force exerted on a spheroid by an arbitrarily shaped beam,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(2), 026613 (2007).
[CrossRef] [PubMed]

C. W. Qiu, S. Zouhdi, and A. Razek, “Modified Spherical Wave Functions With Anisotropy Ratio: Application to the Analysis of Scattering by Multilayered Anisotropic Shells,” IEEE Trans. Antenn. Propag. 55(12), 3515–3523 (2007).
[CrossRef]

2006

2004

2002

Y. K. Nahmias and D. J. Odde, “Analysis of Radiation Forces in Laser Trapping and Laser-Guided Direct Writing Applications,” IEEE J. Quantum Electron. 38(2), 131–141 (2002).
[CrossRef]

1998

1997

1996

K. F. Ren, G. Gréhan, and G. Gouesbet, “Prediction of reverse radiation pressure by generalized Lorenz-Mie theory,” Appl. Opt. 35(15), 2702–2710 (1996).
[CrossRef] [PubMed]

T. Wohland, A. Rosin, and E. H. K. Stelzer, “Theoretical determination of the influence of the polarization on forces exerted by optical tweezers,” Optik (Stuttg.) 102, 181–190 (1996).

Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124(5-6), 529–541 (1996).
[CrossRef]

1995

G. Martinot-Lagarde, B. Pouligny, M. I. Angelova, G. Grehan, and G. Gouesbet, ““Trapping and levitation of a dielectric sphere with off-centred Gaussian beams. II. GLMT analysis,” Pure Appl. Opt. J. Euro. Opt. Soc. Part A 4(5), 571–585 (1995).
[CrossRef]

R. C. Gauthier and S. Wallace, “Optical levitation of spheres: analytical development and numerical computations of force equations,” J. Opt. Soc. Am. B 12(9), 1680–1687 (1995).
[CrossRef]

1994

J. A. Lock and G. Gouesbet, “Rigorous justification fo the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory.I. On-axis beams,” J. Opt. Soc. Am. A 11(9), 2503–2515 (1994).
[CrossRef]

K. F. Ren, G. Gréhan, and G. Gouesbet, “Radiation pressure forces exerted on a particle arbitrarily located in a Gaussian beam by using the generalized Lorenz-Mie theory and associated resonance effects,” Opt. Commun. 108(4-6), 343–354 (1994).
[CrossRef]

1991

T. C. B. Schut, G. Hesselink, B. G. de Grooth, and J. Greve, “Experimental and theoretical investigations on the validity of the geometrical optics model for calculating the stability of optical traps,” Cytometry 12(6), 479–485 (1991).
[CrossRef] [PubMed]

1990

1989

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66(10), 4954–4962 (1989).
[CrossRef]

1988

1986

1980

A. Ashkin, “Applications of laser radiation pressure,” Science 210(4474), 1081–1088 (1980).
[CrossRef] [PubMed]

1970

A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24(4), 156–159 (1970).
[CrossRef]

Alexander, D. R.

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66(10), 4954–4962 (1989).
[CrossRef]

Angelova, M. I.

G. Martinot-Lagarde, B. Pouligny, M. I. Angelova, G. Grehan, and G. Gouesbet, ““Trapping and levitation of a dielectric sphere with off-centred Gaussian beams. II. GLMT analysis,” Pure Appl. Opt. J. Euro. Opt. Soc. Part A 4(5), 571–585 (1995).
[CrossRef]

Asakura, T.

Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124(5-6), 529–541 (1996).
[CrossRef]

Ashkin, A.

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]

A. Ashkin, “Applications of laser radiation pressure,” Science 210(4474), 1081–1088 (1980).
[CrossRef] [PubMed]

A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24(4), 156–159 (1970).
[CrossRef]

Barton, J. P.

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66(10), 4954–4962 (1989).
[CrossRef]

Bjorkholm, J. E.

Blakely, J. T.

Braat, J. J. M.

Cai, X. S.

F. Xu, K. F. Ren, G. Gouesbet, X. S. Cai, and G. Gréhan, “Theoretical prediction of radiation pressure force exerted on a spheroid by an arbitrarily shaped beam,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(2), 026613 (2007).
[CrossRef] [PubMed]

Chai, L.

F. L. Mao, Q. R. Xing, K. Wang, L. Y. Lang, L. Chai, and Q. Y. Wang, “Calculation of axial optical forces exerted on medium-sized particles by optical trap,” Opt. Laser Technol. 39(1), 34–39 (2007).
[CrossRef]

Chu, S.

de Boer, D. K.

de Grooth, B. G.

T. C. B. Schut, G. Hesselink, B. G. de Grooth, and J. Greve, “Experimental and theoretical investigations on the validity of the geometrical optics model for calculating the stability of optical traps,” Cytometry 12(6), 479–485 (1991).
[CrossRef] [PubMed]

Doicu, A.

Dziedzic, J. M.

Gao, B. Z.

Gauthier, R. C.

Geng, Y. L.

Y. L. Geng, X. B. Wu, and L. W. Li, “Mie scattering by a uniaxial anisotropic sphere,” physical review E 70, 056609 (2004).
[CrossRef]

Gordon, R.

Gouesbet, G.

F. Xu, K. F. Ren, G. Gouesbet, X. S. Cai, and G. Gréhan, “Theoretical prediction of radiation pressure force exerted on a spheroid by an arbitrarily shaped beam,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(2), 026613 (2007).
[CrossRef] [PubMed]

H. Polaert, G. Gréhan, and G. Gouesbet, “Improved standard beams with application to reverse radiation pressure,” Appl. Opt. 37(12), 2435–2440 (1998).
[CrossRef] [PubMed]

Z. S. Wu, L. X. Guo, K. F. Ren, G. Gouesbet, and G. Gréhan, “Improved algorithm for electromagnetic scattering of plane waves and shaped beams by multilayered spheres,” Appl. Opt. 36(21), 5188–5198 (1997).
[CrossRef] [PubMed]

K. F. Ren, G. Gréhan, and G. Gouesbet, “Prediction of reverse radiation pressure by generalized Lorenz-Mie theory,” Appl. Opt. 35(15), 2702–2710 (1996).
[CrossRef] [PubMed]

G. Martinot-Lagarde, B. Pouligny, M. I. Angelova, G. Grehan, and G. Gouesbet, ““Trapping and levitation of a dielectric sphere with off-centred Gaussian beams. II. GLMT analysis,” Pure Appl. Opt. J. Euro. Opt. Soc. Part A 4(5), 571–585 (1995).
[CrossRef]

J. A. Lock and G. Gouesbet, “Rigorous justification fo the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory.I. On-axis beams,” J. Opt. Soc. Am. A 11(9), 2503–2515 (1994).
[CrossRef]

K. F. Ren, G. Gréhan, and G. Gouesbet, “Radiation pressure forces exerted on a particle arbitrarily located in a Gaussian beam by using the generalized Lorenz-Mie theory and associated resonance effects,” Opt. Commun. 108(4-6), 343–354 (1994).
[CrossRef]

G. Gouesbet, G. Grehan, and B. Maheu, “Computations of the gn coefficients in the geneeralized Lorenz-Mie theory using three different methods,” Appl. Opt. 27(23), 4874–4883 (1988).
[CrossRef] [PubMed]

G. Gouesbet, B. Maheu, and G. Grehan, “Light Scattering from a sphere arbitrarily located in a Gaussian beam,using a Bromwich formulation,” J. Opt. Soc. Am. A 5(9), 1427–1443 (1988).
[CrossRef]

Gousbet, G.

Grehan, G.

Gréhan, G.

F. Xu, K. F. Ren, G. Gouesbet, X. S. Cai, and G. Gréhan, “Theoretical prediction of radiation pressure force exerted on a spheroid by an arbitrarily shaped beam,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(2), 026613 (2007).
[CrossRef] [PubMed]

H. Polaert, G. Gréhan, and G. Gouesbet, “Improved standard beams with application to reverse radiation pressure,” Appl. Opt. 37(12), 2435–2440 (1998).
[CrossRef] [PubMed]

Z. S. Wu, L. X. Guo, K. F. Ren, G. Gouesbet, and G. Gréhan, “Improved algorithm for electromagnetic scattering of plane waves and shaped beams by multilayered spheres,” Appl. Opt. 36(21), 5188–5198 (1997).
[CrossRef] [PubMed]

K. F. Ren, G. Gréhan, and G. Gouesbet, “Prediction of reverse radiation pressure by generalized Lorenz-Mie theory,” Appl. Opt. 35(15), 2702–2710 (1996).
[CrossRef] [PubMed]

K. F. Ren, G. Gréhan, and G. Gouesbet, “Radiation pressure forces exerted on a particle arbitrarily located in a Gaussian beam by using the generalized Lorenz-Mie theory and associated resonance effects,” Opt. Commun. 108(4-6), 343–354 (1994).
[CrossRef]

Greve, J.

T. C. B. Schut, G. Hesselink, B. G. de Grooth, and J. Greve, “Experimental and theoretical investigations on the validity of the geometrical optics model for calculating the stability of optical traps,” Cytometry 12(6), 479–485 (1991).
[CrossRef] [PubMed]

Guo, L. X.

Harada, Y.

Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124(5-6), 529–541 (1996).
[CrossRef]

Hesselink, G.

T. C. B. Schut, G. Hesselink, B. G. de Grooth, and J. Greve, “Experimental and theoretical investigations on the validity of the geometrical optics model for calculating the stability of optical traps,” Cytometry 12(6), 479–485 (1991).
[CrossRef] [PubMed]

Kawano, M.

Lang, L. Y.

F. L. Mao, Q. R. Xing, K. Wang, L. Y. Lang, L. Chai, and Q. Y. Wang, “Calculation of axial optical forces exerted on medium-sized particles by optical trap,” Opt. Laser Technol. 39(1), 34–39 (2007).
[CrossRef]

Li, H. Y.

Li, L. W.

Y. L. Geng, X. B. Wu, and L. W. Li, “Mie scattering by a uniaxial anisotropic sphere,” physical review E 70, 056609 (2004).
[CrossRef]

Li, Z. J.

Lock, J. A.

Maheu, B.

Mao, F. L.

F. L. Mao, Q. R. Xing, K. Wang, L. Y. Lang, L. Chai, and Q. Y. Wang, “Calculation of axial optical forces exerted on medium-sized particles by optical trap,” Opt. Laser Technol. 39(1), 34–39 (2007).
[CrossRef]

Martinot-Lagarde, G.

G. Martinot-Lagarde, B. Pouligny, M. I. Angelova, G. Grehan, and G. Gouesbet, ““Trapping and levitation of a dielectric sphere with off-centred Gaussian beams. II. GLMT analysis,” Pure Appl. Opt. J. Euro. Opt. Soc. Part A 4(5), 571–585 (1995).
[CrossRef]

Nahmias, Y. K.

Y. K. Nahmias, B. Z. Gao, and D. J. Odde, “Dimensionless parameters for the design of optical traps and laser guidance systems,” Appl. Opt. 43(20), 3999–4006 (2004).
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Y. K. Nahmias and D. J. Odde, “Analysis of Radiation Forces in Laser Trapping and Laser-Guided Direct Writing Applications,” IEEE J. Quantum Electron. 38(2), 131–141 (2002).
[CrossRef]

Nemoto, S.

Nevière, M.

Odde, D. J.

Y. K. Nahmias, B. Z. Gao, and D. J. Odde, “Dimensionless parameters for the design of optical traps and laser guidance systems,” Appl. Opt. 43(20), 3999–4006 (2004).
[CrossRef] [PubMed]

Y. K. Nahmias and D. J. Odde, “Analysis of Radiation Forces in Laser Trapping and Laser-Guided Direct Writing Applications,” IEEE J. Quantum Electron. 38(2), 131–141 (2002).
[CrossRef]

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Popov, E.

Pouligny, B.

G. Martinot-Lagarde, B. Pouligny, M. I. Angelova, G. Grehan, and G. Gouesbet, ““Trapping and levitation of a dielectric sphere with off-centred Gaussian beams. II. GLMT analysis,” Pure Appl. Opt. J. Euro. Opt. Soc. Part A 4(5), 571–585 (1995).
[CrossRef]

Qiu, C. W.

C. W. Qiu, S. Zouhdi, and A. Razek, “Modified Spherical Wave Functions With Anisotropy Ratio: Application to the Analysis of Scattering by Multilayered Anisotropic Shells,” IEEE Trans. Antenn. Propag. 55(12), 3515–3523 (2007).
[CrossRef]

Razek, A.

C. W. Qiu, S. Zouhdi, and A. Razek, “Modified Spherical Wave Functions With Anisotropy Ratio: Application to the Analysis of Scattering by Multilayered Anisotropic Shells,” IEEE Trans. Antenn. Propag. 55(12), 3515–3523 (2007).
[CrossRef]

Ren, K. F.

F. Xu, K. F. Ren, G. Gouesbet, X. S. Cai, and G. Gréhan, “Theoretical prediction of radiation pressure force exerted on a spheroid by an arbitrarily shaped beam,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(2), 026613 (2007).
[CrossRef] [PubMed]

Z. S. Wu, L. X. Guo, K. F. Ren, G. Gouesbet, and G. Gréhan, “Improved algorithm for electromagnetic scattering of plane waves and shaped beams by multilayered spheres,” Appl. Opt. 36(21), 5188–5198 (1997).
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K. F. Ren, G. Gréhan, and G. Gouesbet, “Prediction of reverse radiation pressure by generalized Lorenz-Mie theory,” Appl. Opt. 35(15), 2702–2710 (1996).
[CrossRef] [PubMed]

K. F. Ren, G. Gréhan, and G. Gouesbet, “Radiation pressure forces exerted on a particle arbitrarily located in a Gaussian beam by using the generalized Lorenz-Mie theory and associated resonance effects,” Opt. Commun. 108(4-6), 343–354 (1994).
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T. Wohland, A. Rosin, and E. H. K. Stelzer, “Theoretical determination of the influence of the polarization on forces exerted by optical tweezers,” Optik (Stuttg.) 102, 181–190 (1996).

Schaub, S. A.

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66(10), 4954–4962 (1989).
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T. C. B. Schut, G. Hesselink, B. G. de Grooth, and J. Greve, “Experimental and theoretical investigations on the validity of the geometrical optics model for calculating the stability of optical traps,” Cytometry 12(6), 479–485 (1991).
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Sluijter, M.

Stelzer, E. H. K.

T. Wohland, A. Rosin, and E. H. K. Stelzer, “Theoretical determination of the influence of the polarization on forces exerted by optical tweezers,” Optik (Stuttg.) 102, 181–190 (1996).

Stout, B.

Togo, H.

Wallace, S.

Wang, K.

F. L. Mao, Q. R. Xing, K. Wang, L. Y. Lang, L. Chai, and Q. Y. Wang, “Calculation of axial optical forces exerted on medium-sized particles by optical trap,” Opt. Laser Technol. 39(1), 34–39 (2007).
[CrossRef]

Wang, Q. Y.

F. L. Mao, Q. R. Xing, K. Wang, L. Y. Lang, L. Chai, and Q. Y. Wang, “Calculation of axial optical forces exerted on medium-sized particles by optical trap,” Opt. Laser Technol. 39(1), 34–39 (2007).
[CrossRef]

Wohland, T.

T. Wohland, A. Rosin, and E. H. K. Stelzer, “Theoretical determination of the influence of the polarization on forces exerted by optical tweezers,” Optik (Stuttg.) 102, 181–190 (1996).

Wriedt, T.

Wu, X. B.

Y. L. Geng, X. B. Wu, and L. W. Li, “Mie scattering by a uniaxial anisotropic sphere,” physical review E 70, 056609 (2004).
[CrossRef]

Wu, Z. S.

Xing, Q. R.

F. L. Mao, Q. R. Xing, K. Wang, L. Y. Lang, L. Chai, and Q. Y. Wang, “Calculation of axial optical forces exerted on medium-sized particles by optical trap,” Opt. Laser Technol. 39(1), 34–39 (2007).
[CrossRef]

Xu, F.

F. Xu, K. F. Ren, G. Gouesbet, X. S. Cai, and G. Gréhan, “Theoretical prediction of radiation pressure force exerted on a spheroid by an arbitrarily shaped beam,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(2), 026613 (2007).
[CrossRef] [PubMed]

Yuan, Q. K.

Zouhdi, S.

C. W. Qiu, S. Zouhdi, and A. Razek, “Modified Spherical Wave Functions With Anisotropy Ratio: Application to the Analysis of Scattering by Multilayered Anisotropic Shells,” IEEE Trans. Antenn. Propag. 55(12), 3515–3523 (2007).
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A. Doicu and T. Wriedt, “Computation of the beam-shape coefficients in the generalized Lorenz-Mie theory by using the translational addition theorem for spherical vector wave functions,” Appl. Opt. 36(13), 2971–2978 (1997).
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K. F. Ren, G. Gréhan, and G. Gouesbet, “Prediction of reverse radiation pressure by generalized Lorenz-Mie theory,” Appl. Opt. 35(15), 2702–2710 (1996).
[CrossRef] [PubMed]

Z. S. Wu, L. X. Guo, K. F. Ren, G. Gouesbet, and G. Gréhan, “Improved algorithm for electromagnetic scattering of plane waves and shaped beams by multilayered spheres,” Appl. Opt. 36(21), 5188–5198 (1997).
[CrossRef] [PubMed]

H. Polaert, G. Gréhan, and G. Gouesbet, “Improved standard beams with application to reverse radiation pressure,” Appl. Opt. 37(12), 2435–2440 (1998).
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G. Gouesbet, G. Grehan, and B. Maheu, “Computations of the gn coefficients in the geneeralized Lorenz-Mie theory using three different methods,” Appl. Opt. 27(23), 4874–4883 (1988).
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[CrossRef] [PubMed]

Y. K. Nahmias, B. Z. Gao, and D. J. Odde, “Dimensionless parameters for the design of optical traps and laser guidance systems,” Appl. Opt. 43(20), 3999–4006 (2004).
[CrossRef] [PubMed]

Cytometry

T. C. B. Schut, G. Hesselink, B. G. de Grooth, and J. Greve, “Experimental and theoretical investigations on the validity of the geometrical optics model for calculating the stability of optical traps,” Cytometry 12(6), 479–485 (1991).
[CrossRef] [PubMed]

IEEE J. Quantum Electron.

Y. K. Nahmias and D. J. Odde, “Analysis of Radiation Forces in Laser Trapping and Laser-Guided Direct Writing Applications,” IEEE J. Quantum Electron. 38(2), 131–141 (2002).
[CrossRef]

IEEE Trans. Antenn. Propag.

C. W. Qiu, S. Zouhdi, and A. Razek, “Modified Spherical Wave Functions With Anisotropy Ratio: Application to the Analysis of Scattering by Multilayered Anisotropic Shells,” IEEE Trans. Antenn. Propag. 55(12), 3515–3523 (2007).
[CrossRef]

J. Appl. Phys.

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66(10), 4954–4962 (1989).
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J. Opt. Soc. Am. A

J. A. Lock and G. Gouesbet, “Rigorous justification fo the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory.I. On-axis beams,” J. Opt. Soc. Am. A 11(9), 2503–2515 (1994).
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G. Gouesbet, B. Maheu, and G. Grehan, “Light Scattering from a sphere arbitrarily located in a Gaussian beam,using a Bromwich formulation,” J. Opt. Soc. Am. A 5(9), 1427–1443 (1988).
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G. Gousbet, “Localized interpretation to compute all the coefficients gmn in the generalized Lorentz-Mie theory,” J. Opt. Soc. Am. A 7(6), 998–1007 (1990).
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B. Stout, M. Nevière, and E. Popov, “Mie scattering by an anisotropic object. Part I. Homogeneous sphere,” J. Opt. Soc. Am. A 23(5), 1111–1123 (2006).
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M. Sluijter, D. K. de Boer, and J. J. M. Braat, “General polarized ray-tracing method for inhomogeneous uniaxially anisotropic media,” J. Opt. Soc. Am. A 25(6), 1260–1273 (2008).
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Z. S. Wu, Q. K. Yuan, Y. Peng, and Z. J. Li, “Internal and external electromagnetic fields for on-axis Gaussian beam scattering from a uniaxial anisotropic sphere,” J. Opt. Soc. Am. A 26(8), 1778–1788 (2009).
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Q. K. Yuan, Z. S. Wu, and Z. J. Li, “Electromagnetic scattering for a uniaxial anisotropic sphere in an off-axis obliquely incident Gaussian beam,” J. Opt. Soc. Am. A 27(6), 1457–1465 (2010).
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J. Opt. Soc. Am. B

Opt. Commun.

Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124(5-6), 529–541 (1996).
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K. F. Ren, G. Gréhan, and G. Gouesbet, “Radiation pressure forces exerted on a particle arbitrarily located in a Gaussian beam by using the generalized Lorenz-Mie theory and associated resonance effects,” Opt. Commun. 108(4-6), 343–354 (1994).
[CrossRef]

Opt. Express

Opt. Laser Technol.

F. L. Mao, Q. R. Xing, K. Wang, L. Y. Lang, L. Chai, and Q. Y. Wang, “Calculation of axial optical forces exerted on medium-sized particles by optical trap,” Opt. Laser Technol. 39(1), 34–39 (2007).
[CrossRef]

Opt. Lett.

Optik (Stuttg.)

T. Wohland, A. Rosin, and E. H. K. Stelzer, “Theoretical determination of the influence of the polarization on forces exerted by optical tweezers,” Optik (Stuttg.) 102, 181–190 (1996).

Phys. Rev. E Stat. Nonlin. Soft Matter Phys.

F. Xu, K. F. Ren, G. Gouesbet, X. S. Cai, and G. Gréhan, “Theoretical prediction of radiation pressure force exerted on a spheroid by an arbitrarily shaped beam,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(2), 026613 (2007).
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G. Martinot-Lagarde, B. Pouligny, M. I. Angelova, G. Grehan, and G. Gouesbet, ““Trapping and levitation of a dielectric sphere with off-centred Gaussian beams. II. GLMT analysis,” Pure Appl. Opt. J. Euro. Opt. Soc. Part A 4(5), 571–585 (1995).
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Figures (11)

Fig. 1
Fig. 1

Uniaxial anisotropic sphere illuminated by an off-axis Gaussian beam.

Fig. 2
Fig. 2

Comparison of the axial radiation force from the theory when the anisotropic sphere is reduced to an isotropic sphere (solid curve) with the experimental results (the curves are denoted by Static measurement and Dynamic measurement) and Ray Optics results in reference [5].( w 0 = 1.8 μ m , ε t = ε z = 2.3716 ε z , μ t = μ z = μ 0 , x 0 = y 0 = 0 , λ = 0.488 μ m , a = 3.75 μ m , n 0 = 1.33 )

Fig. 3
Fig. 3

Variation of the axial radiation force F z with d for different beam waist widths. ( a = 2.0 μ m , ε t = 2.0 ε 0 , ε z = 2.4 ε 0 , μ t = 2.0 μ 0 , μ z = 2.8 μ 0 , x 0 = y 0 = 0 , λ = 0.6328 μ m , n 0 = 1.0 .)

Fig. 4
Fig. 4

Variation of the axial radiation force F z with d for different sphere radii.( w 0 = 0.5 μ m , ε t = 2.0 ε 0 , ε z = 2.4 ε 0 , μ t = 2.0 μ 0 , μ z = 2.8 μ 0 , x 0 = y 0 = 0 , λ = 0.6328 μ m , n 0 = 1.0 .)

Fig. 7
Fig. 7

Variation of the axial radiation force F z with dfor different ε z .( ε t = 2.0 ε 0 , μ t = μ z = μ 0 , x 0 = y 0 = 0 , λ = 0.6328 μ m , w 0 = 0.4 μ m , a = 2.0 μ m , n 0 = 1.33 .)

Fig. 5
Fig. 5

Variation of the axial radiation force F z with dfor different ε z .( ε t = 1.4 ε 0 , μ t = μ z = μ 0 , x 0 = y 0 = 0 , λ = 0.6328 μ m , w 0 = 0.4 μ m , a = 2.0 μ m , n 0 = 1.33 .)

Fig. 8
Fig. 8

Variation of the axial radiation force F z with dfor different ε z .( ε t = 4.0 ε 0 , μ t = μ z = μ 0 , x 0 = y 0 = 0 , λ = 0.6328 μ m , w 0 = 0.4 μ m , a = 2.0 μ m , n 0 = 1.33 .).

Fig. 6
Fig. 6

Variation of the axial radiation force F z with dfor different ε z .( ε t = 1.6 ε 0 , μ t = μ z = μ 0 , x 0 = y 0 = 0 , λ = 0.6328 μ m , w 0 = 0.4 μ m , a = 2.0 μ m , n 0 = 1.33 .)

Fig. 9
Fig. 9

Variation of the transverse radiation force F x ( F y ) with x 0 ( y 0 ) for different sphere radii. ( ε t = 2.0 ε 0 , ε z = 2.4 ε 0 , μ t = μ z = μ 0 , z 0 = 1 μ m , y 0 = 0 ( x 0 = 0 ) , λ = 0.6328 μ m , w 0 = 0.4 μ m , n 0 = 1.33 )

Fig. 10
Fig. 10

Variation of the transverse radiation force F x with x 0 for different z 0 .( w 0 = 0.4 μ m , ε t = 2.0 ε 0 , ε z = 2.4 ε 0 , μ t = μ z = μ 0 , y 0 = 0 , λ = 0.6328 μ m , a = 1.0 μ m , n 0 = 1.33 .)

Fig. 11
Fig. 11

Variation of the transverse radiation force F x with x 0 for different w 0. ( z 0 = 1 μ m , ε t = 2.0 ε 0 , ε z = 2.4 ε 0 , μ t = μ z = μ 0 , y 0 = 0 , λ = 0.6328 μ m , a = 1.0 μ m , n 0 = 1.33 .)

Equations (31)

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E i = E 0 n = 1 m = n n [ a m n i M m n ( 1 ) ( r , k 0 ) + b m n i N m n ( 1 ) ( r , k 0 ) ] , H i = E 0 k 0 i ω μ 0 n = 1 m = n n [ a m n i N m n ( 1 ) ( r , k 0 ) + b m n i M m n ( 1 ) ( r , k 0 ) ] ,
M m n ( l ) ( k r , θ , ϕ ) = z n ( l ) ( k r ) [ i m P n m ( cos θ ) sin θ e i m ϕ θ ^ d P n m ( cos θ ) d θ e i m ϕ ϕ ^ ] , N m n ( l ) ( k r , θ , ϕ ) = n ( n + 1 ) z n ( l ) ( k r ) k r P n m ( cos θ ) e i m ϕ r ^ + 1 k r d ( r z n ( l ) ( k r ) ) d r [ d P n m ( cos θ ) d θ θ ^ + i m P n m ( cos θ ) sin θ ϕ ^ ] e i m ϕ ,
[ a m n i b m n i ] = C n m ( 1 ) m 1 K n m ψ ¯ 0 0 e i k 0 z 0 1 2 [ e i ( m 1 ) φ 0 J m 1 ( 2 Q ¯ ρ 0 ρ n w 0 2 ) e i ( m + 1 ) φ 0 J m + 1 ( 2 Q ¯ ρ 0 ρ n w 0 2 ) ] ,
C n m = { i n 1 2 n + 1 n ( n + 1 ) , m 0 ( 1 ) | m | ( n + | m | ) ! ( n | m | ) ! i n 1 2 n + 1 n ( n + 1 ) , m < 0 ,     K n m = { ( i ) | m | i ( n + 1 / 2 ) | m | 1 ,                       m 0 n ( n + 1 ) n + 1 / 2 ,                                                                                                                     m = 0 ,
ψ ¯ 0 0 = i Q ¯ exp ( i Q ¯ ρ 0 2 w 0 2 ) exp ( i Q ¯ ( n + 1 / 2 ) 2 k 0 2 w 0 2 ) ,
ρ n = ( n + 1 / 2 ) / k 0 , Q ¯ = ( i 2 z 0 / l ) 1 , ρ 0 = x 0 2 + y 0 2 , φ 0 = arctan ( x 0 / y 0 ) .
E s = E 0 n = 1 m = n n [ A m n s M m n ( 3 ) ( r , k 0 ) + B m n s N m n ( 3 ) ( r , k 0 ) ] , H s = E 0 k 0 i ω μ 0 n = 1 m = n n [ A m n s N m n ( 3 ) ( r , k 0 ) + B m n s M m n ( 3 ) ( r , k 0 ) ] .
ε ¯ ¯ = [ ε t 0 0 0 ε t 0 0 0 ε z ] ,              μ ¯ ¯ = [ μ t 0 0 0 μ t 0 0 0 μ z ] .
E I ( r ) = E 0 q = 1 2 n = 1 m = n n n = 1 2 π G m n q 0 π [ A m n q e M m n ( 1 ) ( r , k q )                                                                   + B m n q e N m n ( 1 ) ( r , k q ) + C m n q e L m n ( 1 ) ( r , k q ) ] p n m ( cos θ k ) k q 2 sin θ k d θ k ,
H I ( r ) = E 0 q = 1 2 n = 1 m = n n n = 1 2 π G m n q 0 π [ A m n q h M m n ( 1 ) ( r , k q )                                                                         + B m n q h N m n ( 1 ) ( r , k q ) + C m n q h L m n ( 1 ) ( r , k q ) ] p n m ( cos θ k ) k q 2 sin θ k d θ k ,
k 1 2 = ω 2 ε t μ t μ z μ t sin 2 θ k + μ z cos 2 θ k ,       k 2 2 = ω 2 ε t ε z μ t ε t sin 2 θ k + ε z cos 2 θ k .
A m n s = 1 h n ( 1 ) ( k 0 a ) [ n = 1 2 π G m n q q = 1 2 0 π A m n q e j n ( k q a ) P n m ( cos θ k ) k q 2 sin θ k d θ k a m n i j n ( k 0 a ) ] ,
B m n s = 1 h n ( 1 ) ( k 0 a ) [ i ω μ 0 k 0 q = 1 2 n = 1 2 π G m n q 0 π A m n q h j n ( k q a ) P n m ( cos θ k ) k q 2 sin θ k d θ k b m n i j n ( k 0 a ) ] ,
q = 1 2 n = 1 2 π G m n q 0 π U m n q P n m ( cos θ k ) k q 2 sin θ k d θ k = a m n i i ( k 0 a ) 2 ,
q = 1 2 n = 1 2 π G m n q 0 π V m n q P n m ( cos θ k ) k q 2 sin θ k d θ k = b m n i i ( k 0 a ) 2 ,
U m n q = { A m n q e 1 k 0 r d d r [ r h n ( 1 ) ( k 0 r ) ] j n ( k q r ) i ω μ 0 k 0 [ B m n q h 1 k q r d d r [ r j n ( k q r ) ] + C m n q h j n ( k q r ) r ] h n ( 1 ) ( k 0 r ) } r = a ,
V m n q = { i ω μ 0 k 0 A m n q h 1 k 0 r d d r [ r h n ( 1 ) ( k 0 r ) ] j n ( k q r ) [ B m n q e 1 k q r d d r [ r j n ( k q r ) ] + C m n q e j n ( k q r ) r ] h n ( 1 ) ( k 0 r ) } r = a .
F = 1 2 Re 0 2 π 0 π [ ε 0 E r E + μ 0 H r H 1 2 ( ε 0 E 2 + μ 0 H 2 ) r ^ ] d S ,
x = r sin θ cos ϕ , y = r sin θ sin ϕ , z = r cos θ .
F x = 1 2 Re 0 2 π 0 π { 1 2 [ ε 0 ( E r E r * E θ E θ * E ϕ E ϕ * ) + μ 0 ( H r H r * H θ H θ * H ϕ H ϕ * ) ] r ^         + ( ε 0 E r E θ * + μ 0 H r H θ * ) θ ^ + ( ε 0 E r E ϕ * + μ 0 H r H ϕ * ) ϕ ^ } r 2 sin 2 θ cos ϕ d θ d ϕ | r > a ,
F y = 1 2 Re 0 2 π 0 π { 1 2 [ ε 0 ( E r E r * E θ E θ * E ϕ E ϕ * ) + μ 0 ( H r H r * H θ H θ * H ϕ H ϕ * ) ] r ^         + ( ε 0 E r E θ * + μ 0 H r H θ * ) θ ^ + ( ε 0 E r E ϕ * + μ 0 H r H ϕ * ) ϕ ^ } r 2 sin 2 θ sin ϕ d θ d ϕ | r > a ,
F z = 1 2 Re 0 2 π 0 π { 1 2 [ ε 0 ( E r E r * E θ E θ * E ϕ E ϕ * ) + μ 0 ( H r H r * H θ H θ * H ϕ H ϕ * ) ] r ^         + ( ε 0 E r E θ * + μ 0 H r H θ * ) θ ^ + ( ε 0 E r E ϕ * + μ 0 H r H ϕ * ) ϕ ^ } r 2 sin θ cos θ d θ d ϕ | r > a ,
F x + i F y = n 0 P 0 π c k 0 2 w 0 2 n = 1 m = n n [ ( n m ) ( n + m + 1 ) N m n 1 N m + 1 n 1 ( a m n i B m + 1 n S *                               + b m n i A m + 1 n S * + B m n S a m + 1 n i * + A m n S b m + 1 n i * + 2 A m n S B m + 1 n S * + 2 B m n S A m + 1 n S * )                               i ( n m 1 ) ( n m ) ( 2 n 1 ) ( 2 n + 1 ) ( n 1 ) ( n + 1 ) N m n 1 N m + 1 n 1 1 ( a m n i A m + 1 n 1 S *                             + b m n i B m + 1 n 1 S * + A m n S a m + 1 n 1 i * + B m n S b m + 1 n 1 i * + 2 A m n S A m + 1 n 1 S * + 2 B m n S B m + 1 n 1 S * )                             i ( n + m + 1 ) ( n + m + 2 ) ( 2 n + 1 ) ( 2 n + 3 ) n ( n + 2 ) N m n 1 N m + 1 n + 1 1 ( a m n i A m + 1 n + 1 S * +                             b m n i B m + 1 n + 1 S * + A m n S a m + 1 n + 1 i * + B m n S b m + 1 n + 1 i * + 2 A m n S A m + 1 n + 1 S * + 2 B m n S B m + 1 n + 1 S * ) ] ,
F z = 2 n 0 P 0 π c k 0 2 w 0 2 Re n = 1 m = n n [ i n ( n + 2 ) ( n m + 1 ) ( n + m + 1 ) ( 2 n + 1 ) ( 2 n + 3 ) N m n 1 N m n + 1 1 ( a m n + 1 i A m n S * +                                                     A m n + 1 S a m n i * + b m n + 1 i B m n S * + B m n + 1 S b m n i * + 2 A m n + 1 S A m n S * + 2 B m n + 1 S B m n S * )                                                       m N m n 2 ( a m n i B m n S * + b m n i A m n S * + 2 A m n S B m n S * ) ] .
N m n = ( 2 n + 1 ) ( n m ) ! 4 π ( n + m ) !         ( m = 0 , ± 1 , , ± n ) .
0 2 π e i ( m m ' ) ϕ d ϕ 0 π [ m ' d P n m ( cos θ ) d θ P n ' m ' ( cos θ ) sin θ + m P n m ( cos θ ) sin θ d P n ' m ' ( cos θ ) d θ ] sin 2 θ e i ϕ d θ = ( n m ) ( n + m + 1 ) N m n 1 N m ' n ' 1 δ m + 1 , m ' δ n , n '
0 2 π e i ( m ' m ) ϕ d ϕ 0 π [ m P n m ( cos θ ) sin θ d P n ' m ' ( cos θ ) d θ + m ' d P n m ( cos θ ) d θ P n ' m ' ( cos θ ) sin θ ] sin 2 θ e i ϕ d θ = ( n ' m ' ) ( n ' + m ' + 1 ) N m n 1 N m ' n ' 1 δ m ' + 1 , m δ n , n '
0 2 π e i ( m m ' ) ϕ d ϕ 0 π [ d P n m ( cos θ ) d θ d P n ' m ' ( cos θ ) d θ + m m ' P n m ( cos θ ) sin θ P n ' m ' ( cos θ ) sin θ ] sin 2 θ e i ϕ d θ = N m n 1 N m ' n ' 1 [ ( n + m + 1 ) ( n + m + 2 ) ( 2 n + 1 ) ( 2 n + 3 ) n ( n + 2 ) δ n + 1 , n ' ( n ' m ) ( n ' m + 1 ) ( 2 n ' + 1 ) ( 2 n ' + 3 ) n ' ( n ' + 2 ) δ n , n ' + 1 ] δ m + 1 , m '
0 2 π e i ( m ' m ) ϕ d ϕ 0 π [ d P n m ( cos θ ) d θ d P n ' m ' ( cos θ ) d θ + m m ' P n m ( cos θ ) sin θ P n ' m ' ( cos θ ) sin θ ] sin 2 θ e i ϕ d θ = N m n 1 N m ' n ' 1 [ ( n ' + m ' + 1 ) ( n ' + m ' + 2 ) ( 2 n ' + 1 ) ( 2 n ' + 3 ) n ' ( n ' + 2 ) δ n ' + 1 , n ( n m ' ) ( n m ' + 1 ) ( 2 n + 1 ) ( 2 n + 3 ) n ( n + 2 ) δ n ' , n + 1 ] δ m ' + 1 , m
0 2 π e i ( m m ' ) ϕ d ϕ 0 π [ m ' d P n m ( cos θ ) d θ P n ' m ' ( cos θ ) sin θ + m P n m ( cos θ ) sin θ d P n ' m ' ( cos θ ) d θ ] sin θ cos θ d θ = m N m n 1 N m ' n ' 1 δ m , m ' δ n , n '
0 2 π e i ( m m ' ) ϕ d ϕ 0 π [ d P n m ( cos θ ) d θ d P n ' m ' ( cos θ ) d θ + m m ' P n m ( cos θ ) sin θ P n ' m ' ( cos θ ) sin θ ] sin θ cos θ d θ = N m n 1 N m ' n ' 1 ( n ' m + 1 ) ( n ' + m + 1 ) ( 2 n ' + 1 ) ( 2 n ' + 3 ) n ' ( n ' + 2 ) δ m , m ' δ n , n ' + 1                 + N m n 1 N m ' n ' 1 ( n m + 1 ) ( n + m + 1 ) ( 2 n + 1 ) ( 2 n + 3 ) n ( n + 2 ) δ m , m ' δ n + 1 , n '

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