Calculation of radiation forces exerted on a uniaxial anisotropic sphere by an off-axis incident Gaussian beam

Zheng-Jun Li; Zhen-Sen Wu; Qing-Chao Shang

doi:10.1364/OE.19.016044

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 [20–23] 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 $ε_{t}$ and $ε_{z}$ 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 $\exp (- i ω t)$ 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 $(x_{0}, y_{0}, z_{0})$ .

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 $O x y z$ as [25]

\begin{array}{l} E^{i} = E_{0} \sum_{n = 1}^{\infty} \sum_{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} \frac{k_{0}}{i ω μ_{0}} \sum_{n = 1}^{\infty} \sum_{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})], \end{array}

where

k_{0} = 2 π / λ

and λ is light wavelength in the surrounding medium,

μ_{0}

is the permeability of the surrounding medium,

E_{0}

is the amplitude of the electric field at beam waist center.

M_{m n}^{(l)}

and

N_{m n}^{(l)}

are the SVWFs, which are written as

\begin{array}{l} M_{m n}^{(l)} (k r, θ, ϕ) = z_{n}^{(l)} (k r) [i m \frac{P_{n}^{m} (\cos θ)}{\sin θ} e^{i m ϕ} \hat{θ} - \frac{d P_{n}^{m} (\cos θ)}{d θ} e^{i m ϕ} \hat{ϕ}], \\ N_{m n}^{(l)} (k r, θ, ϕ) = n (n + 1) \frac{z_{n}^{(l)} (k r)}{k r} P_{n}^{m} (\cos θ) e^{i m ϕ} \hat{r} \\ + \frac{1}{k r} \frac{d (r z_{n}^{(l)} (k r))}{d r} [\frac{d P_{n}^{m} (\cos θ)}{d θ} \hat{θ} + i m \frac{P_{n}^{m} (\cos θ)}{\sin θ} \hat{ϕ}] e^{i m ϕ}, \end{array}

where

z_{n}^{(l)}

represents an appropriate kind of spherical Bessel functions: the first kind jn, the second kind yn, or the third kind

h_{n}^{(1)}

and

h_{n}^{(2)}

, denoted by l=1, 2, 3, or 4, respectively;

P_{n}^{m} (\cos θ)

is the associated Legendre function of the first kind.

a_{m n}^{i}

and

b_{m n}^{i}

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,

[\begin{array}{l} a_{m n}^{i} \\ b_{m n}^{i} \end{array}] = C_{n m} {(- 1)}^{m - 1} K_{n m} {\bar{ψ}}_{0}^{0} e^{- i k_{0} z_{0}} \frac{1}{2} [e^{i (m - 1) φ_{0}} J_{m - 1} (2 \frac{\bar{Q} ρ_{0} ρ_{n}}{w_{0}^{2}}) \mp e^{i (m + 1) φ_{0}} J_{m + 1} (2 \frac{\bar{Q} ρ_{0} ρ_{n}}{w_{0}^{2}})],

where

C_{n m} = {\begin{matrix} i^{n - 1} \frac{2 n + 1}{n (n + 1)} \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} & , m \geq 0 \end{matrix} \\ {(- 1)}^{| m |} \frac{(n + | m |)!}{(n - | m |)!} i^{n - 1} \frac{2 n + 1}{n (n + 1)}, m < 0 \end{matrix}, K_{n m} = {\begin{matrix} {(- i)}^{| m |} \frac{i}{{(n + 1 / 2)}^{| m | - 1}}, m \neq 0 \\ \frac{n (n + 1)}{n + 1 / 2}, m = 0 \end{matrix},

{\bar{ψ}}_{0}^{0} = i \bar{Q} \exp (\frac{- i \bar{Q} ρ_{0}^{2}}{w_{0}^{2}}) \exp (\frac{- i \bar{Q} {(n + 1 / 2)}^{2}}{k_{0}^{2} w_{0}^{2}}),

ρ_{n} = (n + 1 / 2) / k_{0}, \bar{Q} = {(i - 2 z_{0} / l)}^{- 1}, ρ_{0} = \sqrt{x_{0}^{2} + y_{0}^{2}}, φ_{0} = \arctan (x_{0} / y_{0}) .

The scattered fields can be expanded with the SVWFs as

\begin{array}{l} E^{s} = E_{0} \sum_{n = 1}^{\infty} \sum_{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} \frac{k_{0}}{i ω μ_{0}} \sum_{n = 1}^{\infty} \sum_{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})] . \end{array}

Meanwhile, the permittivity and permeability tensors $\bar{\bar{ε}}$ and $\bar{\bar{μ}}$ of uniaxial anisotropic sphere are characterized by

\bar{\bar{ε}} = [\begin{array}{l} ε_{t} 0 0 \\ 0 ε_{t} 0 \\ 0 0 ε_{z} \end{array}], \bar{\bar{μ}} = [\begin{array}{l} μ_{t} 0 0 \\ 0 μ_{t} 0 \\ 0 0 μ_{z} \end{array}] .

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

\begin{array}{l} E^{I} (r) = E_{0} \sum_{q = 1}^{2} \sum_{n = 1}^{\infty} \sum_{m = - n}^{n} \sum_{n^{'} = 1}^{\infty} 2 π G_{m n^{'} q} \int_{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}, \end{array}

\begin{array}{l} H^{I} (r) = E_{0} \sum_{q = 1}^{2} \sum_{n = 1}^{\infty} \sum_{m = - n}^{n} \sum_{n^{'} = 1}^{\infty} 2 π G_{m n^{'} q} \int_{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}, \end{array}

where the expressions of the coefficients

A_{m n q}^{e}

,

B_{m n q}^{e}

,

C_{m n q}^{e}

,

A_{m n q}^{h}

,

B_{m n q}^{h}

, and

C_{m n q}^{h}

can be found in reference [20]. The

k_{q} (q = 1, 2)

denotes the two eigen wave numbers in the uniaxial anisotropic sphere and its expressions are,

k_{1}^{2} = \frac{ω^{2} ε_{t} μ_{t} μ_{z}}{μ_{t} \sin^{2} θ_{k} + μ_{z} \cos^{2} θ_{k}}, k_{2}^{2} = \frac{ω^{2} ε_{t} ε_{z} μ_{t}}{ε_{t} \sin^{2} θ_{k} + ε_{z} \cos^{2} θ_{k}} .

In Eqs. (9)-(11), $k_{q}$ , $θ_{k}$ and $ϕ_{k}$ (included in the coefficients $A_{m n q}^{e}$ , …, $C_{m n q}^{h}$ ) 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]:

A_{m n}^{s} = \frac{1}{h_{n}^{(1)} (k_{0} a)} [\sum_{n^{'} = 1}^{\infty} 2 π G_{m n^{'} q} \sum_{q = 1}^{2} \int_{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} = \frac{1}{h_{n}^{(1)} (k_{0} a)} [\frac{i ω μ_{0}}{k_{0}} \sum_{q = 1}^{2} \sum_{n^{'} = 1}^{\infty} 2 π G_{m n^{'} q} \int_{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)],

where the unknown expansion coefficient

G_{m n^{'} q}

can be solved through incident expansion coefficients [25]

\sum_{q = 1}^{2} \sum_{n^{'} = 1}^{\infty} 2 π G_{m n^{'} q} \int_{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} \frac{i}{{(k_{0} a)}^{2}},

\sum_{q = 1}^{2} \sum_{n^{'} = 1}^{\infty} 2 π G_{m n^{'} q} \int_{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} \frac{i}{{(k_{0} a)}^{2}},

Where

U_{m n q} = {A_{m n q}^{e} \frac{1}{k_{0} r} \frac{d}{d r} [r h_{n}^{(1)} (k_{0} r)] j_{n} (k_{q} r) {- \frac{i ω μ_{0}}{k_{0}} [B_{m n q}^{h} \frac{1}{k_{q} r} \frac{d}{d r} [r j_{n} (k_{q} r)] + C_{m n q}^{h} \frac{j_{n} (k_{q} r)}{r}] h_{n}^{(1)} (k_{0} r)}}_{r = a},

V_{m n q} = {\frac{i ω μ_{0}}{k_{0}} A_{m n q}^{h} \frac{1}{k_{0} r} \frac{d}{d r} [r h_{n}^{(1)} (k_{0} r)] j_{n} (k_{q} r) {- [B_{m n q}^{e} \frac{1}{k_{q} r} \frac{d}{d r} [r j_{n} (k_{q} r)] + C_{m n q}^{e} \frac{j_{n} (k_{q} r)}{r}] h_{n}^{(1)} (k_{0} r)}}_{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 = \frac{1}{2} Re \int_{0}^{2 π} \int_{0}^{π} [ε_{0} E_{r} E + μ_{0} H_{r} H - \frac{1}{2} (ε_{0} E^{2} + μ_{0} H^{2}) \hat{r}] d S,

where

ε_{0}

and

μ_{0}

pertain to the permittivity and permeability of the surrounding medium, respectively.

d S

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 = E^{i} + E^{s}, H = H^{i} + H^{s}

.

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

x = r \sin θ \cos ϕ, y = r \sin θ \sin ϕ, z = r \cos θ .

The three components of the radiation forces can be written as

\begin{array}{l} F_{x} = \frac{1}{2} Re \int_{0}^{2 π} \int_{0}^{π} {\frac{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}^{*})] \hat{r} \\ \begin{matrix} + (ε_{0} E_{r}^{} E_{θ}^{*} + μ_{0} H_{r}^{} H_{θ}^{*}) \hat{θ} + (ε_{0} E_{r}^{} E_{ϕ}^{*} + μ_{0} H_{r}^{} H_{ϕ}^{*}) \hat{ϕ}} r^{2} \sin^{2} θ {\cos ϕ d θ d ϕ |}_{r > a} \end{matrix}, \end{array}

\begin{array}{l} F_{y} = \frac{1}{2} Re \int_{0}^{2 π} \int_{0}^{π} {\frac{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}^{*})] \hat{r} \\ \begin{matrix} + (ε_{0} E_{r}^{} E_{θ}^{*} + μ_{0} H_{r}^{} H_{θ}^{*}) \hat{θ} + (ε_{0} E_{r}^{} E_{ϕ}^{*} + μ_{0} H_{r}^{} H_{ϕ}^{*}) \hat{ϕ}} r^{2} \sin^{2} θ \sin ϕ {d θ d ϕ |}_{r > a} \end{matrix}, \end{array}

\begin{array}{l} F_{z} = \frac{1}{2} Re \int_{0}^{2 π} \int_{0}^{π} {\frac{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}^{*})] \hat{r} \\ \begin{matrix} + (ε_{0} E_{r}^{} E_{θ}^{*} + μ_{0} H_{r}^{} H_{θ}^{*}) \hat{θ} + (ε_{0} E_{r}^{} E_{ϕ}^{*} + μ_{0} H_{r}^{} H_{ϕ}^{*}) \hat{ϕ}} r^{2} \sin θ \cos θ {d θ d ϕ |}_{r > a} \end{matrix}, \end{array}

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

r ≫ λ

, we can derive the analytical expressions of the radiation forces:

\begin{array}{l} F_{x} + i F_{y} = \frac{n_{0} P_{0}}{π c k_{0}^{2} w_{0}^{2}} \sum_{n = 1}^{\infty} \sum_{m = - n}^{n} [\sqrt{(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 \sqrt{\frac{(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 \sqrt{\frac{(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 *})], \end{array}

\begin{array}{l} F_{z} = \frac{2 n_{0} P_{0}}{π c k_{0}^{2} w_{0}^{2}} Re \sum_{n = 1}^{\infty} \sum_{m = - n}^{n} [i n (n + 2) \sqrt{\frac{(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 *})] . \end{array}

where c is the speed of light in vacuum,

n_{0}

is the refractive index of surrounding medium,

P_{0} = k π w_{0} E_{0}^{2} / 4 μ_{0}

denotes the power of the incident beam, and

N_{m n} = \sqrt{\frac{(2 n + 1) (n - m)!}{4 π (n + m)!}} (m = 0, \pm 1, \cdot \cdot \cdot, \pm 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 $P_{0} = 100 mW$ , 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].( $w_{0} = 1.8 μ m$ , $ε_{t} = ε_{z} = 2.3716 ε_{z}$ , $μ_{t} = μ_{z} = μ_{0}$ , $x_{0} = y_{0} = 0$ , $λ = 0.488 μ m$ , $a = 3.75 μ m$ , $n_{0} = 1.33$ )

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In Fig. 3 , the axial radiation force $F_{z}$ 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 $w_{0}$ . It can be found that for a strongly focused beam (here w ₀ < 0.4μm) the axial radiation force F_z 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 $w_{0}$ increases, the positions where F_z take maximal value shift slightly to the beam waist center and the minimum of F_z decreases, however, the form of the curves exhibits little change.

Fig. 3 Variation of the axial radiation force $F_{z}$ with d for different beam waist widths. ( $a = 2.0 μ m$ , $ε_{t} = 2.0 ε_{0}$ , $ε_{z} = 2.4 ε_{0}$ , $μ_{t} = 2.0 μ_{0}$ , $μ_{z} = 2.8 μ_{0}$ , $x_{0} = y_{0} = 0$ , $λ = 0.6328 μ m$ , $n_{0} = 1.0$ .)

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Figure 4 shows variation of the axial radiation force $F_{z}$ 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 $F_{z}$ become very large when $a \geq 5 μm$ . 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 $F_{z}$ with d for different sphere radii.( $w_{0} = 0.5 μ m$ , $ε_{t} = 2.0 ε_{0}$ , $ε_{z} = 2.4 ε_{0}$ , $μ_{t} = 2.0 μ_{0}$ , $μ_{z} = 2.8 μ_{0}$ , $x_{0} = y_{0} = 0$ , $λ = 0.6328 μ m$ , $n_{0} = 1.0$ .)

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Fig. 7 Variation of the axial radiation force $F_{z}$ with dfor different $ε_{z}$ .( $ε_{t} = 2.0 ε_{0}$ , $μ_{t} = μ_{z} = μ_{0}$ , $x_{0} = y_{0} = 0$ , $λ = 0.6328 μ m$ , $w_{0} = 0.4 μ m$ , $a = 2.0 μ m$ , $n_{0} = 1.33$ .)

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Figures 5 -8 shows the variation of axial radiation force $F_{z}$ with d for several values of $ε_{t}$ and $ε_{z}$ when $n_{0} = 1.33$ and $μ_{t} = μ_{z} = μ_{0}$ . The uniaxial anisotropic sphere can be captured under the beam waist center when $ε_{t} < n_{0}^{2}$ , and above the beam waist center when $ε_{t} > n_{0}^{2}$ . 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 [4–6,8,12]. 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 $ε_{t}$ increases, the bound position and area denoted by the values of d increases; as the value of $ε_{z}$ increases, the extremum of $F_{z}$ , 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 $ε_{t}$ is large, regardless of the size of $ε_{z}$ . In fact, careful examination and abundant simulations indicate that the values of axial radiation forces will always be positive when either $ε_{t}$ or $ε_{z}$ is very large. For example, for the conditions shown in Fig. 5, the negative value of $F_{z}$ will disappear when $ε_{z} > 6.3 ε_{0}$ . In Fig. 6 , the negative value of $F_{z}$ will disappear when $ε_{z} > 7.0 ε_{0}$ . The values of $F_{z}$ 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 $F_{z}$ with dfor different $ε_{z}$ .( $ε_{t} = 1.4 ε_{0}$ , $μ_{t} = μ_{z} = μ_{0}$ , $x_{0} = y_{0} = 0$ , $λ = 0.6328 μ m$ , $w_{0} = 0.4 μ m$ , $a = 2.0 μ m$ , $n_{0} = 1.33$ .)

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Fig. 8 Variation of the axial radiation force $F_{z}$ with dfor different $ε_{z}$ .( $ε_{t} = 4.0 ε_{0}$ , $μ_{t} = μ_{z} = μ_{0}$ , $x_{0} = y_{0} = 0$ , $λ = 0.6328 μ m$ , $w_{0} = 0.4 μ m$ , $a = 2.0 μ m$ , $n_{0} = 1.33$ .).

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Fig. 6 Variation of the axial radiation force $F_{z}$ with dfor different $ε_{z}$ .( $ε_{t} = 1.6 ε_{0}$ , $μ_{t} = μ_{z} = μ_{0}$ , $x_{0} = y_{0} = 0$ , $λ = 0.6328 μ m$ , $w_{0} = 0.4 μ m$ , $a = 2.0 μ m$ , $n_{0} = 1.33$ .)

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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 $F_{x}$ ( $F_{y}$ ) exerted on a uniaxial anisotropic sphere located in water ( $n_{0} = 1.33$ ) by an off-axis Gaussian beam are plotted as a function of $- x_{0}$ ( $- y_{0}$ ). Note that $(x_{0}, y_{0}, z_{0})$ denotes the coordinate of beam center in particle coordinate system $O x y z$ . 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 $F_{x}$ , which is smaller than the extremum. As the radius increases, the bound area increases in both x-axis and y-axis. The variations of $F_{x}$ with $- x_{0}$ and $F_{y}$ with $- y_{0}$ are similar, but the secondary extremum of $F_{x}$ evidently increases compared to that of $F_{y}$ , 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 $F_{x}$ with -x ₀ and $F_{y}$ with $- y_{0}$ are almost the same and there is no secondary extremum.

Fig. 9 Variation of the transverse radiation force $F_{x}$ ( $F_{y}$ ) with $- x_{0}$ ( $- y_{0}$ ) for different sphere radii. ( $ε_{t} = 2.0 ε_{0}$ , $ε_{z} = 2.4 ε_{0}$ , $μ_{t} = μ_{z} = μ_{0}$ , $z_{0} = - 1 μ m$ , $y_{0} = 0 (x_{0} = 0)$ , $λ = 0.6328 μ m$ , $w_{0} = 0.4 μ m$ , $n_{0} = 1.33$ )

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Figures 10 and 11 show that the transverse radiation force $F_{x}$ varies with $- x_{0}$ for several coordinates $z_{0}$ and beam waist widths $w_{0}$ . It can be observed that as the sphere center moves away from the beam waist center along the beam axis, the peak values of $F_{x}$ 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 $F_{x}$ 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 $w_{0}$ is very small; however, the sphere may be very difficult to be bound when $w_{0}$ is very small ( $w_{0} = 0.2 μm$ 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 $F_{x}$ with $- x_{0}$ for different $z_{0}$ .( $w_{0} = 0.4 μ m$ , $ε_{t} = 2.0 ε_{0}$ , $ε_{z} = 2.4 ε_{0}$ , $μ_{t} = μ_{z} = μ_{0}$ , $y_{0} = 0$ , $λ = 0.6328 μ m$ , $a = 1.0 μ m$ , $n_{0} = 1.33$ .)

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Fig. 11 Variation of the transverse radiation force $F_{x}$ with $- x_{0}$ for different w ₀. ( $z_{0} = - 1 μ m$ , $ε_{t} = 2.0 ε_{0}$ , $ε_{z} = 2.4 ε_{0}$ , $μ_{t} = μ_{z} = μ_{0}$ , $y_{0} = 0$ , $λ = 0.6328 μ m$ , $a = 1.0 μ m$ , $n_{0} = 1.33$ .)

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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:

\begin{array}{l} \int_{0}^{2 π} e^{i (m}^{- m') ϕ} d ϕ \int_{0}^{π} [m' \frac{d P_{n}^{m} (\cos θ)}{d θ} \frac{P_{n'}^{m'} (\cos θ)}{\sin θ} + m \frac{P_{n}^{m} (\cos θ)}{\sin θ} \frac{d P_{n'}^{m'} (\cos θ)}{d θ}] \sin^{2} θ e^{i ϕ} d θ \\ = - \sqrt{(n - m) (n + m + 1)} N_{m n}^{- 1} N_{m' n'}^{- 1} δ_{m + 1, m'} δ_{n, n'} \end{array}

\begin{array}{l} \int_{0}^{2 π} e^{i (m'}^{- m) ϕ} d ϕ \int_{0}^{π} [m \frac{P_{n}^{m} (\cos θ)}{\sin θ} \frac{d P_{n'}^{m'} (\cos θ)}{d θ} + m' \frac{d P_{n}^{m} (\cos θ)}{d θ} \frac{P_{n'}^{m'} (\cos θ)}{\sin θ}] \sin^{2} θ e^{i ϕ} d θ \\ = - \sqrt{(n' - m') (n' + m' + 1)} N_{m n}^{- 1} N_{m' n'}^{- 1} δ_{m' + 1, m} δ_{n, n'} \end{array}

\begin{array}{l} \int_{0}^{2 π} e^{i (m}^{- m') ϕ} d ϕ \int_{0}^{π} [\frac{d P_{n}^{m} (\cos θ)}{d θ} \frac{d P_{n'}^{m'} (\cos θ)}{d θ} + m m' \frac{P_{n}^{m} (\cos θ)}{\sin θ} \frac{P_{n'}^{m'} (\cos θ)}{\sin θ}] \sin^{2} θ e^{i ϕ} d θ \\ = N_{m n}^{- 1} N_{m' n'}^{- 1} [\sqrt{\frac{(n + m + 1) (n + m + 2)}{(2 n + 1) (2 n + 3)}} n (n + 2) δ_{n + 1, n'} - \sqrt{\frac{(n' - m) (n' - m + 1)}{(2 n' + 1) (2 n' + 3)}} n' (n' + 2) δ_{n, n' + 1}] δ_{m + 1, m'} \end{array}

\begin{array}{l} \int_{0}^{2 π} e^{i (m}^{' - m) ϕ} d ϕ \int_{0}^{π} [\frac{d P_{n}^{m} (\cos θ)}{d θ} \frac{d P_{n'}^{m'} (\cos θ)}{d θ} + m m' \frac{P_{n}^{m} (\cos θ)}{\sin θ} \frac{P_{n'}^{m'} (\cos θ)}{\sin θ}] \sin^{2} θ e^{i ϕ} d θ \\ = N_{m n}^{- 1} N_{m' n'}^{- 1} [\sqrt{\frac{(n' + m' + 1) (n' + m' + 2)}{(2 n' + 1) (2 n' + 3)}} n' (n' + 2) δ_{n' + 1, n} - \sqrt{\frac{(n - m') (n - m' + 1)}{(2 n + 1) (2 n + 3)}} n (n + 2) δ_{n', n + 1}] δ_{m' + 1, m} \end{array}

\begin{array}{l} \int_{0}^{2 π} e^{i (m}^{- m') ϕ} d ϕ \int_{0}^{π} [m' \frac{d P_{n}^{m} (\cos θ)}{d θ} \frac{P_{n'}^{m'} (\cos θ)}{\sin θ} + m \frac{P_{n}^{m} (\cos θ)}{\sin θ} \frac{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'} \end{array}

\begin{array}{l} \int_{0}^{2 π} e^{i (m}^{- m') ϕ} d ϕ \int_{0}^{π} [\frac{d P_{n}^{m} (\cos θ)}{d θ} \frac{d P_{n'}^{m'} (\cos θ)}{d θ} + m m' \frac{P_{n}^{m} (\cos θ)}{\sin θ} \frac{P_{n'}^{m'} (\cos θ)}{\sin θ}] \sin θ \cos θ d θ \\ = N_{m n}^{- 1} N_{m' n'}^{- 1} \sqrt{\frac{(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} \sqrt{\frac{(n - m + 1) (n + m + 1)}{(2 n + 1) (2 n + 3)}} n (n + 2) δ_{m, m'} δ_{n + 1, n'} \end{array}

where

δ_{m, 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.

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Calculation of radiation forces exerted on a uniaxial anisotropic sphere by an off-axis incident Gaussian beam

Abstract

1. Introduction

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

3. Radiation forces exerted on a uniaxial anisotropic sphere

4. Numerical results and discussion

3.1 Axial radiation force

4.2 Transverse radiation forces

5. Conclusion

Appendix

Acknowledgments

References and links

Cited By

Figures (11)

Equations (31)

Optics Express