Anomalous optical forces on the anisotropic Rayleigh particles

Y. X. Ni; J. K. Chen; L. Gao

doi:10.1364/OE.22.027355

1. Introduction

A revolution in our understanding of the light-matter interaction has taken place in the last thirty years. Light carries energy, linear and angular momenta that can be transferred to atoms, molecules and particles. Askin et al demonstrated the levitation and trapping of micron-size particles experimentally by radiation pressure of light [1]. The pioneering of the work led to the development of optical trapping of microscopic particles including biological cells [2, 3], and the wide applications of light force in biology, chemistry, and physics [4, 5].

Usually, light forces on small particles are often small, and how to increase the force and to get steady manipulation on the particles have been a hot topic [6–8]. Recently, a theory of optical forces on small magnetodielectric particles was developed [9, 10]. Later, Silicon spheres were shown to present both dipolar magnetic and electric responses [11, 12], and the optical force on such particles from an incident plane wave was addressed [13]. More recently, optical pulling force-the backward pulling of small solid spheres [14,15] and coated spheres [16] with a forward propagating beam received intense attention. In addition, the momentum exchange between a backward wave and a whispering gallery mode (WGM) resonator was investigated, which provides a new way of creating an optical pulling force [17]. Nevertheless, most of the previous research on optical forces focused on isotropic particles. In general, the majority of solid materials in nature are anisotropic. For instance, radial anisotropy was indeed found in many nanostructures such as biological cells containing mobile charges and real phospholipid vesicles [18–20], and may be easily established from graphitic multishells or spherically stratified medium [21, 22]. It was also suggested that the radial anisotropy can be transformed from the Cartesian anisotropy [23,24]. Along this line, the interactions of electromagnetic wave with anisotropic nanoparticles and nanowires were widely investigated [25–32].

In this paper, we investigate the optical forces on radially anisotropic spheres from an incident plane wave based on Maxwell stress tensor formulation and our previously derived scattering theory for radial anisotropy. Especially, the asymptotic behavior of the optical force for Rayleigh anisotropic sphere under different physical parameters is analytically derived, and non-Rayleigh behavior is exhibited. We shall show both analytically and numerically that the forces on the anisotropic particles under certain conditions can be much larger or much smaller than the conventional case.

This paper is organized as follows. In Section 2, we review the generalized scattering theory of the light scattering by the radially anisotropic spheres, and obtain the conventional expressions for the optical force on the spherically anisotropic particles in the Rayleigh limit. Unconventional dependence of the optical force on the size parameter and wave number is analyzed, and numerical results has also been given in Section 3. In the end, we summarize our main results and make the conclusions.

2. Optical forces on anisotropic spherical particles

2.1. The scattering theory for spheres with radial anisotropy

We consider a radially anisotropic sphere of radius a surrounded by the free space with permittivity ε₀ = 1 and permeability μ₀ = 1. For simplicity, Gaussian unit is used throughout. Radial anisotropy means that the permittivity (or permeability) tensor is diagonal in spherical coordinates, and the element along the radial direction differs from the one along the tangential direction. Here the permittivity and the permeability tensors are written in the spherical coordinates as, ε̿ = (ε_re_re_r +ε_t e_θe_θ + ε_t e_ϕe_ϕ) and μ̿ = (μ_re_re_r + μ_t e_θe_θ + μ_t e_θe_θ). The anisotropic sphere is illuminated by the polarized plane wave with unit amplitude, E_i = e_xe^ik₀z, where k₀ = ω/c is the wave number in vacuum with ω being the frequency.

Based on the Lorenz-Mie scattering theory, light scattering by a spherical particle can be expressed through Debye potentials [33, 34]. Here the electric ψ_TM and magnetic ψ_TE Debye potentials are written as,

\frac{ε_{r}}{ε_{t}} \frac{\partial^{2} ψ_{T M}}{\partial r^{2}} + \frac{1}{r^{2} sin θ} \frac{\partial}{\partial θ} (sin θ \frac{\partial ψ_{T M}}{\partial θ}) + \frac{1}{r^{2} {sin}^{2} θ} \frac{\partial^{2} ψ_{T M}}{\partial ϕ^{2}} + k_{0}^{2} ε_{r} μ_{t} ψ_{T M} = 0,

\frac{μ_{r}}{μ_{t}} \frac{\partial^{2} ψ_{T E}}{\partial r^{2}} + \frac{1}{r^{2} sin θ} \frac{\partial}{\partial θ} (sin θ \frac{\partial ψ_{T E}}{\partial θ}) + \frac{1}{r^{2} {sin}^{2} θ} \frac{\partial^{2} ψ_{T E}}{\partial ϕ^{2}} + k_{0}^{2} ε_{t} μ_{r} ψ_{T E} = 0 .

After some algebraic manipulations, we can obtain the scattering coefficients a_n for the TE mode and b_n for the TM mode [26],

a_{n} = \frac{\sqrt{ε_{t}} ψ_{n}^{'} (q) ψ_{ν_{1}^{n}} (m q) - \sqrt{μ_{t}} ψ_{n} (q) ψ_{ν_{1}^{n}}^{'} (m q)}{\sqrt{ε_{t}} ξ_{n}^{'} (q) ψ_{ν_{1}^{n}} (m q) - \sqrt{μ_{t}} ξ_{n} (q) ψ_{ν_{1}^{n}}^{'} (m q)},

b_{n} = \frac{\sqrt{ε_{t}} ψ_{n} (q) ψ_{ν_{2}^{n}}^{'} (m q) - \sqrt{μ_{t}} ψ_{n}^{'} (q) ψ_{ν_{2}^{n}} (m q)}{\sqrt{ε_{t}} ξ_{n} (q) ψ_{ν_{2}^{n}}^{'} (m q) - \sqrt{μ_{t}} ξ_{n}^{'} (q) ψ_{ν_{2}^{n}} (m q)},

where

m = \sqrt{ε_{t}} \sqrt{μ_{t}}

is the refractive index of the sphere, and q = k₀a is the size parameter. The functions ψ_n(x) and ξ_n(x) are the Ricatti-Bessel functions defined as,

ψ_{n} (x) = \sqrt{\frac{π x}{2}} J_{n + \frac{1}{2}} (x), ξ_{n} (x) = \sqrt{\frac{π x}{2}} H_{n + \frac{1}{2}}^{(1)} (x) .

In above equations,

J_{n + \frac{1}{2}} (x)

and

H_{n + \frac{1}{2}}^{(1)} (x)

represent the (

n + \frac{1}{2}

)th-order Bessel function and the (

n + \frac{1}{2}

)th-order Hankel function of the first kind. And the primes denote the derivative with respect to the argument. From Eqs. (3) and (4), we find that the information on the physical anisotropy is presented by the orders of spherical Bessel functions [26],

ν_{1}^{n} = \sqrt{n (n + 1) Ae + \frac{1}{4}} - \frac{1}{2} and ν_{2}^{n} = \sqrt{n (n + 1) Am + \frac{1}{4}} - \frac{1}{2},

where Ae = ε_t/ε_r and Am = μ_t/μ_r are, respectively, the electric and magnetic anisotropy ratios. For an isotropic case, Ae = Am = 1, Eqs. (3) and (4) are naturally reduced to the exact formulae of Mie theory.

2.2. Optical forces on the anisotropic Rayleigh particles

The total time-averaged optical force on the particle is the integration over any surface S with unit normal n̂ surrounding the particle [35],

< F > = \frac{1}{8 π} ℜ (\oint_{S} d S \cdot T̿) .

The bracket < ··· > corresponds to the time average over an optical cycle, and ℜ means the real part of a complex number. T̿ in Eq. (7) is the Maxwell stress tensor with the element,

T_{i j} = E_{i} E_{j}^{*} + H_{i} H_{j}^{*} - \frac{1}{2} δ_{i j} (E_{k} E_{k}^{*} + H_{k} H_{k}^{*}),

where we sum over the repeating indices and the superscript star corresponds to the complex conjugate. The time-averaged force on a dipolar sphere illuminated by the plane wave can be deduced as the sum of three terms [10],

< F > = < F_{e} > + < F_{m} > + < F_{e - m} > = \frac{k_{0}}{2} [ℑ (α_{e} + α_{m}) - \frac{2 k_{0}^{3}}{3} (ℜ α_{e} ℜ α_{m} + ℑ α_{e} ℑ α_{m})],

where α_e and α_m, respectively, represent the electric and magnetic polarizabilities of the anisotropic spherical particles, and ℑ stands for the imaginary part. The dynamic polarizabilities for the dipolar sphere are given by the first two Mie coefficients a₁ and b₁ [33],

α_{e} = i \frac{3 ε_{0}}{2 k_{0}^{3}} a_{1} and α_{m} = i \frac{3}{2 μ_{0} k_{0}^{3}} b_{1} .

Note that a₁ and b₁ can be obtain from Eqs. (3) and (4) with n = 1.

In the Rayleigh limit, i.e., q ≪ 1 and mq ≪ 1, the Riccati-Bessel functions and their derives for n = 1 with small arguments are approximately to be,

ψ_{1} (x) ~ \frac{x^{2}}{3} - \frac{x^{4}}{30}, ξ_{1} (x) ~ - \frac{i}{x} - \frac{i x}{2} + \frac{x^{2}}{3} .

ψ_{1}^{'} (x) ~ \frac{2 x}{3} - \frac{x^{3}}{6}, ξ_{1}^{'} (x) ~ - \frac{i}{2} + \frac{2 x}{3} + \frac{i}{x^{2}} .

In addition, the Riccati-Bessel functions for the non-integral orders are expressed as the Euler gamma function Γ(· · ·),

ψ_{ν} (x) = \sqrt{\frac{π x}{2}} J_{ν + \frac{1}{2}} (x), with J_{ν} (x) = {(\frac{x}{2})}^{ν} \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k}}{k! Γ (ν + k + 1)} {(\frac{x}{2})}^{2 k},

and for small x, we have,

\begin{array}{l} ψ_{ν} (x) & ~ & \frac{\sqrt{π}}{Γ (ν + \frac{3}{2})} {(\frac{x}{2})}^{ν + 1} - \frac{\sqrt{π}}{Γ (ν + \frac{5}{2})} {(\frac{x}{2})}^{ν + 3}, \\ ψ_{ν}^{'} (x) & ~ & \frac{\sqrt{π} (ν + 1)}{Γ (ν + \frac{1}{2}) (2 ν + 1)} {(\frac{x}{2})}^{ν} - \frac{\sqrt{π} (ν + 3)}{Γ (ν + \frac{3}{2}) (2 ν + 3)} {(\frac{x}{2})}^{ν + 2} . \end{array}

For small arguments q and mq, the scattering coefficients a₁ are approximately to be [30],

a_{1} ~ \frac{\frac{2 C_{1} \sqrt{ε_{t}} - C_{3} \sqrt{μ_{t}}}{3} q^{ν_{1}^{1} + 2} - \frac{4 C_{2} \sqrt{ε_{t}} + C_{1} \sqrt{ε_{t}} - 2 C_{4} \sqrt{μ_{t}}}{6} q^{ν_{1}^{1} + 4}}{i (C_{1} \sqrt{ε_{t}} + C_{4} \sqrt{μ_{t}}) q^{ν_{1}^{1} - 1} - i (\frac{C_{1} \sqrt{ε_{t}}}{2} + C_{2} \sqrt{ε_{t}} + C_{4} \sqrt{μ_{t}} - \frac{C_{3} \sqrt{μ_{t}}}{2}) q^{ν_{1}^{1} + 1} + \frac{2 C_{1} \sqrt{ε_{t}} - C_{3} \sqrt{μ_{t}}}{3} q^{ν_{1}^{1} + 2}},

where the parameters C_k (k = 1, · · ·, 4) are written as,

\begin{array}{r} C_{1} = \frac{\sqrt{π}}{Γ (ν_{1}^{1} + \frac{3}{2})} {(\frac{m}{2})}^{ν_{1}^{1} + 1}, & C_{2} = \frac{\sqrt{π}}{Γ (ν_{1}^{1} + \frac{5}{2})} {(\frac{m}{2})}^{ν_{1}^{1} + 3}, \\ C_{3} = \frac{\sqrt{π} (ν_{1}^{1} + 1)}{Γ (ν_{1}^{1} + \frac{1}{2}) (2 ν_{1}^{1} + 1)} {(\frac{m}{2})}^{ν_{1}^{1}}, & C_{4} = \frac{\sqrt{π} (ν_{1}^{1} + 3)}{Γ (ν_{1}^{1} + \frac{3}{2}) (2 ν_{1}^{1} + 3)} {(\frac{m}{2})}^{ν_{1}^{1} + 2} . \end{array}

b₁ can be similarly obtained by ε_i → μ_i (i = t or r) and

ν_{1}^{1} \to ν_{2}^{1}

.

For the ordinary anisotropic Rayleigh sphere with positive permittivity and permeability, when the radiative correction is taken into account, a₁ and b₁ can be reduced to [36],

a_{1} ≃ \frac{1}{1 + \frac{3 i (ε_{t} + ν_{1}^{1} + 1)}{[2 ε_{t} - (ν_{1}^{1} + 1)] q^{3}}} and b_{1} ≃ \frac{1}{1 + \frac{3 i (μ_{t} + ν_{2}^{1} + 1)}{[2 μ_{t} - (ν_{2}^{1} + 1)] q^{3}}} .

Then the corresponding electric and magnetic complex polarizabilities may be written as,

α_{e} = \frac{α_{e}^{0}}{1 - \frac{2}{3} i k_{0}^{3} α_{e}^{0}} and α_{m} = \frac{α_{m}^{0}}{1 - \frac{2}{3} i k_{0}^{3} α_{m}^{0}},

where

α_{e}^{0} = \frac{a^{3}}{2} \frac{2 ε_{t} - (ν_{1}^{1} + 1)}{ε_{t} + (ν_{1}^{1} + 1)}

and

α_{m}^{0} = \frac{a^{3}}{2} \frac{2 μ_{t} - (ν_{2}^{1} + 1)}{μ_{t} + (ν_{2}^{1} + 1)}

are the static polarizabilities of the anisotropic spheres. After substituting Eq. (18) into Eq. (9), the optical force on the non-absorbing anisotropic sphere is found to be,

< F > \approx e_{z} \frac{k_{0}^{4} a^{6}}{12} {{[\frac{2 ε_{t} - ν_{1}^{1} - 1}{ε_{t} + ν_{1}^{1} + 1}]}^{2} + {[\frac{2 μ_{t} - ν_{2}^{1} - 1}{μ_{t} + ν_{2}^{1} + 1}]}^{2} - [\frac{(2 ε_{t} - ν_{1}^{1} - 1) (2 μ_{t} - ν_{2}^{1} - 1)}{(ε_{t} + ν_{1}^{1} + 1) (μ_{t} + ν_{2}^{1} + 1)}]} .

Generally, Eq. (19) is just the optical force on the anisotropic Rayleigh particles with the incident plane wave. It shows that the behavior for the optical force exhibits

F ~ k_{0}^{4} a^{6}

, which is similar to the case for isotropic particles [10]. However, with the prosperous progress of “metamaterials”, it is accepted by scientific community that the materials with negative permittivity or/and negative permeability can be fabricated. For instance, negative permittivity at microwave frequencies can be accessed by making use of thin metallic wire meshes near the plasma frequency, and arrays of split-ring resonators have negative permeability close to the magnetic resonant frequency [37]. It was also reported that the real part of permeability can be tuned from 0 to −10 by using digitally addressable split-ring resontors [38]. Moreover, a double negative metamaterial constructed from two sets of all-dielectric spheres was developed, and loss reduction and bandwidth were discussed as well [39]. In this connection, if we consider various values of the permeabilites and permittivities, anomalous behavior for the optical forces may arise.

3. Anomalous optical forces on the anisotropic Rayleigh spheres

We now turn our attention to the cases where unusual behaviors for the optical forces are found.

A. If the following condition is satisfied,

ε_{t} = - ν_{1}^{1} - 1,

the first term of the denominator in Eq. (15) will equal 0, which means the electric dipole resonance occurs for anisotropic particles. It is evident that for isotropic particles, one yields ε_t = ε_r = −2. As a consequence, the scattering coefficients can’t be expressed as Eq. (17), and we should look back into the origin expression and take the second and third terms of the denominator in Eq. (15). Then, we obtain the new form for a₁,

\begin{array}{l} a_{1} & ~ & \frac{\frac{2 C_{1} \sqrt{ε_{t}} - C_{3} \sqrt{μ_{t}}}{3} q^{ν_{1}^{1} + 2}}{i (- \frac{C_{1} \sqrt{ε_{t}}}{2} - C_{2} \sqrt{ε_{t}} - C_{4} \sqrt{μ_{t}} + \frac{C_{3} \sqrt{μ_{t}}}{2}) q^{ν_{1}^{1} + 1} + \frac{2 C_{1} \sqrt{ε_{t}} - C_{3} \sqrt{μ_{t}}}{3} q^{ν_{1}^{1} + 2}} \\ ~ & \frac{q^{3}}{- i q^{2} (1 + \frac{μ_{t}}{2 ν_{1}^{1} + 3}) + q^{3}} . \end{array}

Note that b₁ in Eq. (17) keeps unchanged, hence we still have ℜα_m ∼ a³ and ℑα_m ∼ a³q³. On the other hand, substituting the above expression of a₁ into Eq. (10) results in ℜα_e ∼ a³/q², ℑα_e ∼ a³/q. Therefore, for the electric dipole resonance with

ε_{t} = {- ν}_{1}^{1} - 1

, the optical force on the anisotropic sphere reads,

F ~ \frac{3 a^{2}}{4 f^{2}},

with

f = 1 + μ_{t} / (2 ν_{1}^{1} + 3)

. On the other hand, if the magnetic dipole resonance (

μ_{t} = - ν_{2}^{1} - 1

) takes place, F has the same form with

f = 1 + ε_{t} / (2 ν_{2}^{1} + 3)

. Therefore, for the electric (or magnetic) dipole resonance, the asymptotic behavior for the optical force acting on the particle exhibits

F ~ k_{0}^{0} a^{2}

(it is independent of k₀), which is quite different from the normal Rayleigh case

F ~ k_{0}^{4} a^{6}

.

To verify our theoretical predictions, we plot the numerical results of the optical force as a function of k₀ in Fig. 1 based on the generalized Mie theory and Maxwell stress tensor integration method. Under the non-resonant condition (see the pink dash-dotted line), the force on such conventional spheres is proportional to $k_{0}^{4}$ , as predicted from Eq. (19). However, when the electrical dipole resonant condition is satisfied, the optical force is much larger than that under non-resonant condition. Note that under both electric resonant and non-resonant conditions, the distribution of the electric field is quite similar and resembles the electric dipole excitation but with different magnitudes. But what is changing inside the sphere is the the distribution of the magnetic field: the magnetic field inside the sphere is almost zero at $ε_{t} \neq - ν_{1}^{1} - 1$ , while the magnetic field behaves like the magnetic dipole excitation with small magnitude at $ε_{t} = - ν_{1}^{1} - 1$ [40]. To one’s interest, the magnitude of the force can be enlarged one step further through tuning the anisotropic ratio Ae or increasing the value of $ν_{1}^{1}$ (which lead to the decreasing of f) under the electric dipole resonant condition, the value of the force will be increased. In addition, the force for the electric dipole resonance remains unchanged with varying the incident wave number k₀, as expected. In other words, the force on such a Rayleigh sphere will not be sensitive to the incident wave number, it is a broadband enhancing force. Therefore, the optical force is largely enhanced due to the electric (or magnetic) resonance, in comparison with the non-resonant case, in which the force is proportional to (k₀a)⁴a² (note that q = k₀a ≪ 1).

Fig. 1 Optical force versus k₀ for the nonmagnetic anisotropic sphere with μ_r = μ_t = 1 and a = 100 nm based on the generalized Mie theory and Maxwell stress tensor integration method.

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We shall show that if the components of the permeability tensor for the anisotropic sphere satisfies certain condition in addition to the electric dipole resonant condition, the optical force can be further enlarged due to the enhancement of the magnetic field inside the particles [40].

B. If the following conditions

ε_{t} = - ν_{1}^{1} - 1 and μ_{t} = - (2 ν_{1}^{1} + 3) = 2 ε_{t} - 1

are simultaneously satisfied, the first and second terms in the denominator of Eq. (15) equal 0, resulting in a₁ ≈ 1. Actually, the first relation in Eq. (23) is just the electric dipole resonant one, at which the electric field distribution resembles the electric dipole excitation with large enhancement in the magnitude. When the second relation

μ_{t} = - (2 ν_{1}^{1} + 3)

is satisfied, the magnitude of the magnetic field becomes on the order of unity instead of the small magnetic field for case A [40]. Therefore, the enhancement of electric and magnetic field will contribute to the Maxwell stress tensor Eq. (8), and we may predict the huge increase of the force. The second condition can be reduced to μ_t = μ_r = −5 for the isotropic sphere [40, 41]. After some tedious derivations, we have ℑα_e ≫ ℜα_e (ℜα_m, ℑα_m) due to

α_{e} \approx 3 i / 2 k_{0}^{3}

. As a consequence, the force can be simplified as,

F ~ \frac{3}{4 k_{0}^{2}} = \frac{3 a^{2}}{4 k_{0}^{2} a^{2}} .

This result is much larger than 3a²/(4f²) for case A if q ≪ 1. It may also be of interest to note that the optical force is independent of both the optical properties of the sphere and its size. In Fig. 2, the optical forces are, respectively, plotted as a function of radius a and wave number k₀ by using the generalized Mie theory and the Maxwell stress tensor integration method. It is evident that the force on the sphere keeps unchanged with increasing a [see Fig. 2(a)]. And the force acting on the anisotropic sphere exhibits a rapid increase with decreasing k₀, and obeys the unusual relation

F ~ 1 / k_{0}^{2}

, as shown in the inset of Fig. 2(b) (the slope of the line is −2). Hence, those asymptotic behavior of all the curves is consistent with our analytical predictions.

Fig. 2 Optical force versus a (a) and k₀ (b) with the generalized Mie theory and Maxwell stress tensor integration method for ε_r = −1, ε_t = −3, and μ_r = μ_t = −7.

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C. In the end, we study the following conditions:

ε_{t} = (ε_{r} + 1) / (2 ε_{r}) and μ_{t} = (μ_{r} + 1) / (2 μ_{r}) .

In this case, we have predicted the unusual small scattering from the anisotropic particles [30]. Here, we shall show that the optical forces under such conditions might be changed totally. For instance, the first term in the numerator and the third term in the denominator of Eq. (15) will vanish for ε_t = (ε_r + 1)/(2ε_r). As a result, a₁ admits the form

a_{1} ~ \frac{- \frac{4 C_{2} \sqrt{ε_{t}} + C_{1} \sqrt{ε_{t}} - 2 C_{4} \sqrt{μ_{t}}}{6} q^{ν_{1}^{1} + 4}}{i (C_{1} \sqrt{ε_{t}} + C_{4} \sqrt{μ_{t}}) q^{ν_{1}^{1} - 1} + (\frac{- 2 C_{2} \sqrt{ε_{t}} + C_{4} \sqrt{μ_{t}}}{3}) q^{ν_{1}^{1} + 4}},

and b₁ has the similar form with ε_i →μ_i (i = t or r) and

ν_{1}^{1} \to ν_{2}^{1}

. After some algebraic manipulations, we can get the form a₁ = Aq¹⁰ + Bq⁵i (b₁ = A′q¹⁰ + B′q⁵i), where the coefficients A(A′) and B(B′) represent the constants decided by ε_t and μ_t. Substituting Eq. (26) into Eq. (10) and then Eq. (9) leads to,

F ~ k_{0}^{8} a^{10} = {(k_{0} a)}^{8} a^{2} .

To illustrate the asymptotic behavior for this case, F on the nonmagnetic sphere (μ_r = μ_t = 1) as a function of k₀ is given in Fig. 3. Again, for the sphere with ordinary parameters ε_r = ε_t = 3, the optical force obeys the Rayleigh law at k₀a ≪ 1 [see Eq. (19)]. However, for ε_r = 1/5, ε_t = 3, which satisfy Eq. (25), one observes unusual asymptotic behavior of optical force, as predicted from Eq. (27). Therefore, we may predict a tremendous decrease of the force on the sphere in comparison with the conventional case [F ∼ (k₀a)⁴a²] for k₀a ≪ 1. Physically, in the long-wavelength and low-frequency limits, the anisotropic particle can be regarded as an isotropic one with the equivalent permittivity $ε_{e} = ε_{r} ν_{1}^{1}$ and the equivalent permeability $μ_{e} = μ_{r} ν_{2}^{1}$ [26,42]. When Eq. (25) is satisfied, it is easy to obtain ε_e = 1 and μ_e = 1. In other words, such an anisotropic sphere behaves as an isotropic sphere made of vacuum approximately. One may obtain low scattering efficiency by small particles with radial anisotropy [30]. As a consequence, the optical force (or actually radiation pressure mainly due to absorption and reflection) on such an anisotropic sphere will be largely suppressed, and even can be neglected. Incidentally, the approximate zero forces were also found from the spherical cloak [43, 44].

Fig. 3 Optical force versus k₀ for the nonmagnetic anisotropic sphere with a = 100 nm based on the generalized Mie theory and Maxwell stress tensor integration method.

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4. Conclusions

In this paper, we have studied the optical forces on nanospheres with radial anisotropy by using the generalized Mie theory and Maxwell stress tensor integration method for the incident plane wave. Both analytical and numerical results show that the forces on Rayleigh spheres may exhibit anomalous behaviors instead of the well-known dependence on the size parameter and wave number $F ~ k_{0}^{4} a^{6}$ . When the electric dipole resonance takes place, i.e., $ε_{t} = - (ν_{1}^{1} + 1)$ , we find such a dependent form $F ~ k_{0}^{0} a^{2}$ . This indicates that the optical force is independent of the incident wave number k₀ and will be totally enhanced. Moreover, the force exhibits $F ~ 3 / (4 k_{0}^{2})$ when both $ε_{t} = - ν_{1}^{1} - 1$ and μ_t = 2ε_t − 1 are satisfied. It is expected that the force will be tremendously increased further in principle due to the additional enhancement of the magnetic field at μ_t = 2ε_t − 1. On the other hand, the optical force acting on the anisotropic particle can also be totally suppressed with the asymptotic behavior $F ~ k_{0}^{8} a^{10}$ for ε_t = (ε_r + 1)/(2ε_r) and μ_t = (μ_r + 1)/(2μ_r).

Some comments are in order. All results in this paper were derived for the nondissipative case. In general, it is quite difficult to observe the anomalous behavior of the optical force in real dissipative systems. We note that preliminary studies show that it is still possible to observe such unusual behavior if the absorptive terms are smaller than the critical ones, dependent on the size a and the wave number k₀. As a consequence, one should resort to the materials with much small absorption, or to the realistic materials by introducing the optical gain, in order to realize such unusual asymptotic properties experimentally. We believe our results may stimulate further experimental and theoretical works on this fields, since they suggest intriguing possibilities in optical trapping and particle manipulations.

Acknowledgments

This work was supported by the NNSF of China (No. 11074183, 11347105, No. 11104194), the National Basic Research Program (No. 2012CB921501), PAPD of Jiangsu Higher Education Institutions, the Natural Science Foundation for the Youth of Jiangsu Province (No. BK20130284), and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (No. 13KJB140015).

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Anomalous optical forces on the anisotropic Rayleigh particles

Abstract

1. Introduction

2. Optical forces on anisotropic spherical particles

2.1. The scattering theory for spheres with radial anisotropy

2.2. Optical forces on the anisotropic Rayleigh particles

3. Anomalous optical forces on the anisotropic Rayleigh spheres

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (3)

Equations (27)

Optics Express