Analysis of the radiation force and torque exerted on a chiral sphere by a Gaussian beam

Qing-Chao Shang; Zhen-Sen Wu; Tan Qu; Zheng-Jun Li; Lu Bai; Lei Gong

doi:10.1364/OE.21.008677

1. Introduction

Chiral media were first discovered as optical active media in the 19th century for the rotation of the polarization plane of a linearly polarized light that travels through them. Subsequent studies reveal that chiral microstructures are present in these media. The solutions that contain chiral molecules, such as grape sugar and tartaric acid, are optical active media. By imbedding micro chiral objects, such as metallic helices, randomly in an ordinary dielectric, researchers manufactured chiral material in microwave region. Accordingly, many biological particles may be modeled by chiral particles as they contain chiral microstructures inside.

Since Ashkin reported optical levitation [1] and optical trapping [2, 3], the techniques have been extensively applied in many fields. Especially in biology, optical tweezers are widely used for manipulating and classifying biological particles as they handle the particles without mechanical contact. Considerable effort has been devoted to studying the theory of interactions between lasers and particles. Researchers have developed several theoretical models for calculating the radiation force and torque exerted on a particle by a laser beam. The geometrical optics (GO) model, first used by Roosen [4] and later by many other researchers [5–10], conveniently calculates radiation force as it treats a beam as a bundle of rays. However, GO model adapts only to large particles with sizes much larger than the incident wavelength. For particles with sizes much smaller than the wavelength, the Rayleigh regime is appropriate [11]. Although primarily proposed as beam scattering theory [12], the generalized Lorenz Mie theory (GLMT) was applied to calculate radiation force by Gouesbet [13], Ren [14, 15] and Lock [16, 17]. Kim [18, 19] and Barton [20] studied the radiation force by using a similar method in the 1980s. As analytical solutions, GLMT-based models generate more accurate numerical results than GO model and Rayleigh model and can be used for spheres of arbitrary sizes, though calculating a very large sphere may be time consuming. The numerical methods, such as Extended Boundary Condition Method (EBCM, i.e. T-matrix method) [21, 22], finite difference time-domain method [23], and discrete-dipole approximation [24], are also used in calculating radiation forces exerted on particles.

So far, most of the studies on radiation force and torque generally consider particles made of isotropic media, which does not always correspond with reality. Effects of different media should be taken into account. We calculated and analyzed the radiation force and torque exerted on a uniaxial anisotropic sphere by a Gaussian beam in [25, 26]. Ambrosio studied the radiation pressure imposed on spherical particles of negative refractive index [27, 28]. The radiation force acting on a chiral sphere in a circularly polarized plane wave was analyzed by Guzatov [29]. This paper discusses the radiation force and torque exerted on a chiral sphere by a Gaussian beam, particularly the effects of the chirality and polarization of the beam. We hope that the study may lend support to research on the optical trapping of biological particles.

The theoretical scheme of this paper is based on GLMT. The problem of plane wave scattering from a chiral sphere was first solved by Bohren [30], who expanded the internal field of a chiral sphere in terms of spherical vector wave functions (SVWFs). In our previous work, we extended the theory and proposed improved algorithms for beam scattering from chiral spheres [31]. The improved algorithms are applicable to arbitrarily shaped beam scattering if the expansion coefficients of the incident wave are available. The formulations of radiation force and torque are derived from the momentum theorem of electromagnetic field, as is done by Kim [19] and Barton [20]. Section 2 presents the improved algorithms of beam scattering by a chiral sphere and formulations of radiation force and torque. Section 3 presents numerical results of radiation forces and torque exerted on the chiral sphere, including numerical verification of the theory and the numerical analysis. Section 4 is the conclusion. A time dependence exp(-iωt) is assumed throughout this paper.

2. Theoretical formulations

We summarize the theory of beam scattering by a chiral sphere in Section 2.1. The improved expressions of scattering coefficients, developed in our previous work, are presented. Then, the expansion coefficients of a Gaussian beam are introduced on the basis of literature. In Section 2.2, the expressions of radiation force and torque with respect to the scattering coefficients and expansion coefficients of the incident beam are presented in accordance with the momentum theorem of electromagnetic field.

2.1. Scattering of a Gaussian beam by a chiral sphere

Consider that a chiral sphere of radius $a$ is illuminated by a Gaussian beam propagating along the z-axis. The media of the chiral sphere can be described by the following constitutive relations:

D = ε_{c} E + i κ \sqrt{ε_{0} μ_{0}} H, B = - i κ \sqrt{ε_{0} μ_{0}} E + μ_{c} H,

where

ε_{c}

,

μ_{c}

, and

κ

are permittivity, permeability, and chirality parameter of the sphere, respectively;

ε_{0}

and

μ_{0}

denote the permittivity and permeability of free space, respectively. An electromagnetic wave with angular frequency ω in chiral media is always decomposed into two modes: the right-handed circularly polarized (RCP) wave with wave number

k_{1} = ω (\sqrt{μ_{c} ε_{c}} + κ \sqrt{ε_{0} μ_{0}})

and the left-handed circularly polarized (LCP) wave with wave number

k_{2} = ω (\sqrt{μ_{c} ε_{c}} - κ \sqrt{ε_{0} μ_{0}})

.

On the basis of Bohren’s method [30] and our previous work [31], we provide a brief introduction to beam scattering by a chiral sphere. The internal field of a chiral sphere can be expanded in terms of SVWFs [32] in the following forms:

E^{int} = \sum_{n = 1}^{\infty} \sum_{m = - n}^{n} [A_{m n} M_{m n}^{(1)} (r, k_{1}) + A_{m n} N_{m n}^{(1)} (r, k_{1}) + B_{m n} M_{m n}^{(1)} (r, k_{2}) - B_{m n} N_{m n}^{(1)} (r, k_{2})],

H^{int} = Q \sum_{n = 1}^{\infty} \sum_{m = - n}^{n} [A_{m n} N_{m n}^{(1)} (r, k_{1}) + A_{m n} M_{m n}^{(1)} (r, k_{1}) + B_{m n} N_{m n}^{(1)} (r, k_{2}) - B_{m n} M_{m n}^{(1)} (r, k_{2})],

where

Q = - i \sqrt{ε_{c} / μ_{c}}

.

A_{m n}

and

B_{m n}

represent the unknown expansion coefficients of the internal field. The incident beam and scattered field can be expanded as follows:

E_{}^{i p} = E_{0} \sum_{n = 1}^{\infty} \sum_{m = - n}^{n} [a_{m n}^{i p} M_{m n}^{(1)} (r, k) + b_{m n}^{i p} N_{m n}^{(1)} (r, k)],

H_{}^{i p} = \frac{k E_{0}}{i ω μ} \sum_{n = 1}^{\infty} \sum_{m = - n}^{n} [a_{m n}^{i p} N_{m n}^{(1)} (r, k) + b_{m n}^{i p} M_{m n}^{(1)} (r, k)],

E^{s} = E_{0} \sum_{n = 1}^{\infty} \sum_{m = - n}^{n} [A_{m n}^{s} M_{m n}^{(3)} (r, k) + B_{m n}^{s} N_{m n}^{(3)} (r, k)],

H^{s} = \frac{k E_{0}}{i ω μ} \sum_{n = 1}^{\infty} \sum_{m = - n}^{n} [A_{m n}^{s} N_{m n}^{(3)} (r, k) + B_{m n}^{s} M_{m n}^{(3)} (r, k)],

where

k = ω \sqrt{ε μ}

is the wave number of the surrounding medium;

ε

and

μ

are the permittivity and permeability of the surrounding medium, respectively;

a_{m n}^{i p}

and

b_{m n}^{i p}

denote the expansion coefficients of the incident beam;

A_{m n}^{s}

and

B_{m n}^{s}

denote the scattering coefficients.

E_{0}

is the amplitude of the electric field at the beam center. The notation

i p

in the equations above indicates the x-polarized (linearly polarized in the x-direction), y-polarized (linearly polarized in the y-direction), RCP and LCP wave incidences when

i p

is

i x

,

i y

,

i R

, and

i L

, respectively.

Substituting Eqs. (2)-(7) into the boundary conditions at the spherical interface yields the relationship between $a_{m n}^{i p}$ , $b_{m n}^{i p}$ and $A_{m n}^{s}$ , $B_{m n}^{s}$ as follows [31]:

A_{m n}^{s} = A_{n}^{s a} a_{m n}^{i p} + A_{n}^{s b} b_{m n}^{i p}, B_{m n}^{s} = B_{n}^{s a} a_{m n}^{i p} + B_{n}^{s b} b_{m n}^{i p},

where

A_{n}^{s a} = \frac{ψ_{n} (x_{0})}{ξ_{n}^{} (x_{0})} \frac{\frac{D_{n}^{(1)} (x_{1}) - η_{r} D_{n}^{(1)} (x_{0})}{η_{r} D_{n}^{(1)} (x_{1}) - D_{n}^{(3)} (x_{0})} + \frac{D_{n}^{(1)} (x_{2}) - η_{r} D_{n}^{(1)} (x_{0})}{η_{r} D_{n}^{(1)} (x_{2}) - D_{n}^{(3)} (x_{0})}}{\frac{η_{r} D_{n}^{(3)} (x_{0}) - D_{n}^{(1)} (x_{1})}{η_{r} D_{n}^{(1)} (x_{1}) - D_{n}^{(3)} (x_{0})} + \frac{η_{r} D_{n}^{(3)} (x_{0}) - D_{n}^{(1)} (x_{2})}{η_{r} D_{n}^{(1)} (x_{2}) - D_{n}^{(3)} (x_{0})}},

A_{n}^{s b} = \frac{ψ_{n} (x_{0})}{ξ_{n}^{} (x_{0})} \frac{\frac{η_{r} D_{n}^{(1)} (x_{1}) - D_{n}^{(1)} (x_{0})}{η_{r} D_{n}^{(1)} (x_{1}) - D_{n}^{(3)} (x_{0})} - \frac{η_{r} D_{n}^{(1)} (x_{2}) - D_{n}^{(1)} (x_{0})}{η_{r} D_{n}^{(1)} (x_{2}) - D_{n}^{(3)} (x_{0})}}{\frac{η_{r} D_{n}^{(3)} (x_{0}) - D_{n}^{(1)} (x_{1})}{η_{r} D_{n}^{(1)} (x_{1}) - D_{n}^{(3)} (x_{0})} + \frac{η_{r} D_{n}^{(3)} (x_{0}) - D_{n}^{(1)} (x_{2})}{η_{r} D_{n}^{(1)} (x_{2}) - D_{n}^{(3)} (x_{0})}},

B_{n}^{s a} = A_{n}^{s b},

B_{n}^{s b} = \frac{ψ_{n} (x_{0})}{ξ_{n} (x_{0})} \frac{\frac{η_{r} D_{n}^{(1)} (x_{1}) - D_{n}^{(1)} (x_{0})}{D_{n}^{(1)} (x_{1}) - η_{r} D_{n}^{(3)} (x_{0})} + \frac{η_{r} D_{n}^{(1)} (x_{2}) - D_{n}^{(1)} (x_{0})}{D_{n}^{(1)} (x_{2}) - η_{r} D_{n}^{(3)} (x_{0})}}{\frac{D_{n}^{(3)} (x_{0}) - η_{r} D_{n}^{(1)} (x_{1})}{D_{n}^{(1)} (x_{1}) - η_{r} D_{n}^{(3)} (x_{0})} + \frac{D_{n}^{(3)} (x_{0}) - η_{r} D_{n}^{(1)} (x_{2})}{D_{n}^{(1)} (x_{2}) - η_{r} D_{n}^{(3)} (x_{0})}} .

In the expressions above,

D_{n}^{(1)} (z) = {ψ^{'}}_{n} (z) / ψ_{n} (z)

and

D_{n}^{(3)} (z) = {ξ^{'}}_{n} (z) / ξ_{n} (z)

are the logarithmic derivatives of the Riccati-Bessel functions;

ψ_{n} (z) = z j_{n} (z)

and

ξ_{n} (z) = z h_{n}^{(1)} (z)

are the Riccati-Bessel functions of the first kind and third kind, respectively.

D_{n}^{(1)} (z)

,

D_{n}^{(3)} (z)

and

ψ_{n} (z) / ξ_{n}^{} (z)

can be calculated by their recurrence relations [33–35]. The other notations in Eqs. (9)-(12) represent

x_{0} = k a

,

x_{1} = k_{1} a

,

x_{2} = k_{2} a

, and

η_{r} = \sqrt{ε / μ} / \sqrt{ε_{c} / μ_{c}}

.

Now consider the expansion coefficients of the incident Gaussian beam. We use a convenient expression presented in [36] by Doicu and Wriedt for the Davis Gaussian beam [37]. The expression is derived by using a localized approximation for on-axis beams associating with the translational addition theorem for spherical vector wave functions [36, 38]. For an x-polarized Gaussian beam (x-polarized at the waist) propagating in the positive z direction with a beam waist radius $w_{0}$ and a beam center $(- x_{0}, - y_{0}, - z_{0})$ relative to the sphere center, the expansion coefficients are [36, 39]

a_{m n}^{i x} = C_{n m} (i g_{n, T E}^{m}), b_{m n}^{i x} = C_{n m} g_{n, T M}^{m},

where

C_{n m}

is a normalization factor, expressed as follows:

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} .

g_{n, TE}^{m}

and

g_{n, TM}^{m}

are the beam shape coefficients, written as follows:

[\begin{array}{l} g_{n, T M}^{m} \\ i g_{n, T E}^{m} \end{array}] = {(- 1)}^{m - 1} K_{n m} ψ_{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}}) \pm e^{i (m + 1) φ_{0}} J_{m + 1} (2 \frac{\bar{Q} ρ_{0} ρ_{n}}{w_{0}^{2}})],

where

ψ_{0}^{} = i \bar{Q} \exp (\frac{- i \bar{Q} ρ_{0}^{2}}{w_{0}^{2}}) \exp (\frac{- i \bar{Q} {(n + 0.5)}^{2}}{k_{0}^{2} w_{0}^{2}}),

K_{n m} = {\begin{matrix} {(- i)}^{| m |} \frac{i}{{(n + 0.5)}^{| m | - 1}}, m \neq 0 \\ \frac{n (n + 1)}{n + 0.5}, m = 0 \end{matrix},

ρ_{n} = (n + 0.5) / k_{0}, \bar{Q} = {(i - 2 z_{0} / l)}^{- 1}, ρ_{0} = \sqrt{x_{0}^{2} + y_{0}^{2}}, φ_{0} = arc \tan (x_{0} / y_{0}), l = k w_{0}^{2} .

If the incident Gaussian beam is on-axis ( $x_{0} = y_{0} = 0$ ), Eq. (15) will degenerate into a simple form and will be more convenient in numerical calculations. Note that the above formulations may be inapplicable to a strongly focused beam as a paraxial approximation is used in Davis’s beam theory. If necessary, expansion coefficients of an arbitrary beam, such as a strongly focused beam and a Bessel beam, can be used here. For years many researchers [40–42] have been working on the expansion of an arbitrary beam.

The expansion coefficients of a y-polarized Gaussian beam can be readily obtained from those of an x-polarized Gaussian beam: $a_{m n}^{i y} = - i b_{m n}^{i x}$ , $b_{m n}^{i y} = - i a_{m n}^{i x}$ . Thus, the coefficients of an RCP Gaussian beam with the same power can be obtained as follows:

a_{m n}^{i R} = \sqrt{2} (a_{m n}^{i x} + b_{m n}^{i x}) / 2, b_{m n}^{i R} = \sqrt{2} (b_{m n}^{i x} + a_{m n}^{i x}) / 2.

Similarly, for a LCP Gaussian beam, the expansion coefficients are

a_{m n}^{i L} = \sqrt{2} (a_{m n}^{i x} - b_{m n}^{i x}) / 2

and

b_{m n}^{i L} = \sqrt{2} (b_{m n}^{i x} - a_{m n}^{i x}) / 2

.

2.2. Expressions of radiation force and torque

The radiation force and torque on a sphere illuminated by a beam were studied by many scholars several decades ago [13–20]. On the basis of their methods, we derived the expressions of radiation force and torque with respect to expansion coefficients of incident beam and scattering coefficients [25, 26]. According to electrodynamics, the radiation force exerted on the sphere is equal to the average rate of change in momentum obtained from the beam. This force can be expressed as

F = 〈 - \oint_{S} \hat{n} \cdot \overset{\leftrightarrow}{T} d S 〉,

where

\overset{\leftrightarrow}{T} = - ε E (t) E (t) - μ H (t) H (t) + 1 / 2 [ε E^{2} (t) + μ H^{2} (t)] \overset{\leftrightarrow}{I}

is called the Maxwell stress tensor; S is an arbitrary surface that encloses the sphere; and

\hat{n}

denotes the outwardly directed normal unit vector of S; and

\overset{\leftrightarrow}{I}

represents the unit dyad tensor. The operator

〈 〉

represents a time average. Similarly, the radiation torque on the sphere can be expressed as

N = - 〈 \oint_{S} \hat{n} \cdot (\overset{\leftrightarrow}{T} \times r) d S 〉

, where

r

is the position vector of the point on surface S.

Suppose S is a large spherical surface with the center at the origin. Thus, $\hat{n}$ becomes the unit radial vector $\hat{r}$ and the radiation force becomes

F = \frac{1}{2} Re \int_{0}^{2 π} \int_{0}^{π} [ε E_{r} E + μ H_{r} H - \frac{1}{2} (ε E^{2} + μ H^{2}) \hat{r}] r^{2} \sin θ d θ d ϕ,

where

E = E^{i p} + E^{s}

and

H = H^{i p} + H^{s}

are the external fields of the chiral sphere. Substituting the expressions of

E^{i p}

,

E^{s}

,

H^{i p}

, and

H^{s}

in Eqs. (4)-(7) into Eq. (21), as well as using several approximations, yields the radiation force as follows [25]:

\begin{array}{l} F_{x} + i F_{y} = \frac{n_{0} P}{π c k^{2} w_{0}^{2}} \sum_{n = 1}^{\infty} \sum_{m = - n}^{n} N_{m n}^{- 1} \times [\sqrt{(n - m) (n + m + 1)} N_{m + 1 n}^{- 1} \times \\ (a_{m n}^{i x} B_{m + 1 n}^{S *} + b_{m n}^{i x} A_{m + 1 n}^{S *} + a_{m + 1 n}^{i x *} B_{m n}^{S} + b_{m + 1 n}^{i x *} A_{m n}^{S} + 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 + 1 n - 1}^{- 1} \times, \\ (a_{m n}^{i x} A_{m + 1 n - 1}^{S *} + b_{m n}^{i x} B_{m + 1 n - 1}^{S *} + a_{m + 1 n - 1}^{i x *} A_{m n}^{S} + b_{m + 1 n - 1}^{i x *} B_{m n}^{S} + 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 + 1 n + 1}^{- 1} \times \\ (a_{m n}^{i x} A_{m + 1 n + 1}^{S *} + b_{m n}^{i x} B_{m + 1 n + 1}^{S *} + a_{m + 1 n + 1}^{i x *} A_{m n}^{S} + b_{m + 1 n + 1}^{i x *} B_{m n}^{S} + 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}{π c k^{2} w_{0}^{2}} \sum_{n = 1}^{\infty} \sum_{m = - n}^{n} Re [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} \times \\ (A_{m n}^{S *} a_{m n + 1}^{i x} + a_{m n}^{i x *} A_{m n + 1}^{S} + B_{m n}^{S *} b_{m n + 1}^{i x} + b_{m n}^{i x *} B_{m n + 1}^{S} + 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 x} B_{m n}^{S *} + b_{m n}^{i x} A_{m n}^{S *} + 2 A_{m n}^{S} B_{m n}^{S *})] \end{array}

The expressions of radiation torque are obtained in a similar way [26]:

\begin{array}{l} N_{x} + i N_{y} = \frac{n_{0} P}{π c k^{3} w_{0}^{2}} \sum_{n = 1}^{\infty} \sum_{m = - n}^{n} n (n + 1) \sqrt{(n - m) (n + m + 1)} N_{m n}^{- 1} N_{m + 1 n}^{- 1} \times, \\ [(b_{m n}^{i} b_{m + 1 n}^{s *} + a_{m n}^{i} a_{m + 1 n}^{s *}) + (b_{m n}^{s} b_{m + 1 n}^{i *} + a_{m n}^{s} a_{m + 1 n}^{i *}) + 2 (b_{m n}^{s} b_{m + 1 n}^{s *} + a_{m n}^{s} a_{m + 1 n}^{s *})] \end{array}

N_{z} = - \frac{2 n_{0} P}{π c k^{3} w_{0}^{2}} \sum_{n = 1}^{\infty} \sum_{m = - n}^{n} m n (n + 1) N_{m n}^{- 1} N_{m n}^{- 1} [Re (a_{m n}^{i} a_{m n}^{s *} + b_{m n}^{i} b_{m n}^{s *}) + (| a_{m n}^{s} |^{2} + | b_{m n}^{s} |^{2})],

where

n_{0} = \sqrt{ε / ε_{0}}

denotes the refractive index of the surrounding medium; c is the light velocity in vacuum;

N_{m n} = \sqrt{(2 n + 1) (n - m)! / 4 π / (n + m)!}

,

m = 0, \pm 1, \cdot \cdot \cdot, \pm n

; and

P = π w_{0}^{2} \sqrt{ε / μ} E_{0}^{2} / 4

is the power of the incident beam.

3. Numerical results and discussions

According to the expressions above, radiation force and torque are calculated numerically in this section. We focus primarily on the effects of the chirality parameter and beam polarization on the radiation force and torque. Therefore, for simplicity, the permittivity, permeability and chirality parameter are set as real quantities in the succeeding calculations, unless we want to examine the effects of their imaginary parts. In this section, the beam center is regarded as the coordinate origin and the sphere center is located at $(x_{0}, y_{0}, z_{0})$ .

3.1. Verification of radiation force

When the chirality parameter $κ$ vanishes, the chiral sphere degenerates into an isotropic sphere. In this case, the expressions and codes can be verified by comparing the results with those from literature. Figure 1 shows a comparison of the axial radiation force between our numerical results and those in [6]. The triangles and the line with squares denote the experimental data measured by Bakker Schut [6] by using static method and those measured by using dynamic method, respectively. The line with stars denotes Schut’s theoretical results, obtained by using the GO model. The red solid line represents results calculated using our method. The parameters used in Fig. 1 are as follows: the refractive index of the sphere is $n = 1.54$ ; the refractive index of the surrounding medium is $n_{0} = 1.33$ ; the radius of the sphere is a = 3.75 µm; the power of the x-polarized incident Gaussian beam is P = 0.1 W; the wavelength of the beam in vacuum is λ = 0.488 µm; and the beam waist radius is w₀ = 1.8 µm. The experimental results and our numerical results are in good agreement, confirming the validity of the theory and codes in this paper.

Fig. 1 Comparison with results of axial radiation force from literature.

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3.2. Analysis of radiation force

When the sphere is located on the z-axis, it always experiences a zero transverse radiation force. Figure 2 shows axial radiation force Fz versus sphere location z₀ for different chirality parameters, assuming a sphere with radius a = 1.5 μm and refractive index n = 1.59 in a surrounding medium with refractive index n₀ = 1.33. The beam center of the x-polarized incident Gaussian beam with wavelength λ = 0.488 μm in vacuum, beam waist radius w₀ = 0.5 μm, and power P = 0.1 W is located at the origin. The chirality parameter of the sphere is −0.3, 0.0, 0.3, 0.5, and 0.7, respectively. As shown in Fig. 2, the axial radiation force curve of a chiral sphere is similar to that of an isotropic one. As the sphere moves from the negative z-axis to the positive z-axis, the axial radiation force initially decreases and then increases, during which a stable equilibrium point that can trap the sphere stably appears if negative force occurs. However, all the axial radiation forces in Fig. 2 are positive; thus, the sphere cannot be trapped by this single beam. It seems that a large chirality parameter increases the axial radiation force. Therefore, realizing a negative axial radiation force for a chiral sphere may be more difficult. We find that the curves with chirality parameters −0.3 and 0.3 are coincident. The succeeding numerical results will continue to focus on this and we will try to give some explanations then.

Fig. 2 Axial radiation force exerted on a chiral sphere versus z₀.

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The case of transverse radiation force versus x₀ is depicted in Fig. 3 for the same conditions as Fig. 2, except that the sphere now moves along the x-axis. The x-component Fx of transverse radiation force is shown in Fig. 3(a) and the y-component Fy in Fig. 3(b). Figure 3(a) shows that as the chirality parameter increases, Fx varies from positive to negative when the sphere is located on the negative x-axis, and varies from negative to positive when the sphere is on positive x-axis. These findings suggest that transverse radiation force Fx toward the beam axis (z-axis) becomes a repellant force that pushes the sphere away from the z-axis when the chirality parameter is large enough. Similar to the case of axial radiation force in Fig. 2, the curves of transverse radiation force Fx versus x₀ with chirality parameters −0.3 and 0.3 are coincident. As we known, the y-component of the radiation force Fy of an isotropic sphere at x-axis is always zero. However, as shown in Fig. 3(b), the absolute value of Fy for a chiral sphere does not vanish but increases as the chirality parameter increases. This result indicates that a chiral sphere at the x-axis experiences a force that pushes it away from the x-axis. Besides, in contrast to the Fx and Fz curves, the Fy curves with chirality parameters −0.3 and 0.3 are not equal but opposite. Thus, the corresponding spheres will be pushed toward different y-directions. We conclude from Fig. 3 that trapping a chiral sphere transversely may be more difficult than trapping an isotropic one.

Fig. 3 Transverse radiation force exerted on a chiral sphere versus x₀. (a) Fx; (b) Fy.

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Radiation forces, as functions of the absolute value of the chirality parameter |κ|, are shown in Figs. 4 and 5 for differently polarized beam incidences. The x-polarized, LCP, and RCP incident Gaussian beams are denoted by “XP,” “LP,” and “RP” in the figures, respectively. Figures 4(a) and 4(b) present axial radiation force Fz versus the chirality parameter at sphere locations of (0 μm, 0 μm, 0 μm) and (0 μm, 0 μm, 10 μm), respectively. Figure 5(a) shows transverse radiation force Fx for a sphere located at (1.0 μm, 0 μm, 0 μm), and Fig. 5(b) shows transverse radiation force Fy for a sphere located at (0.5 μm, 0 μm, 0 μm). Chirality parameter κ varies from −0.9 to 0.9. The other parameters are the same as those above. It can be observed that the axial radiation forces on a chiral sphere are no longer the same for different circularly polarized incident beams. For the chiral sphere located at the beam center, as shown in Fig. 4(a), Fz basically increases with the chirality parameter in all cases. However, Fig. 4(b) shows that for the RCP beam incidence with a negative chirality parameter and the LCP beam incidence with a positive chirality parameter, Fz decreases to its minimum and then increases as chirality |κ| varies from 0.0 to 0.9. For a chiral sphere with an appropriate chirality parameter, therefore, it may be easier to realize a negative axial radiation force by using a circularly polarized beam. Surprisingly, the other Fz curves in Fig. 4(b) show obvious oscillations, which may be due to the special scattering properties of a chiral sphere for circularly polarized incident waves. Similar curves of scattering cross-sections versus the size parameter of a chiral sphere are found in [29].

Fig. 4 Axial radiation force versus chirality parameter. (a) z₀ = 0 μm; (b) z₀ = 10 μm.

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Fig. 5 Transverse radiation force versus chirality parameter. (a) x₀ = 1.0μm; (b) x₀ = 0.5μm.

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From Fig. 5(a) a phenomenon similar to that in Fig. 3(a) can be observed that for the x-polarized beam incidence, RCP beam incidence with negative chirality, and LCP beam incidence with positive chirality, the transverse force Fx on a chiral sphere at x₀ = 1.0 μm varies from negative to positive as |κ| increases. In these cases, the chiral sphere with a sufficiently large chirality parameter is pushed away from the x-axis, as the situation shown in Fig. 3(a). However, for the RCP beam incidence with positive chirality, and the LCP beam incidence with negative chirality, Fx is always negative, implying that the force always pulls the chiral sphere toward the beam axis. Thus, to transversely trap a chiral sphere effectively, we should use a beam with appropriate polarization in accordance with the chirality. As shown in Fig. 5(b), a chiral sphere at the x-axis experiences a force in the y-direction for all three types of polarized beam incidences (XP, RP, and LP). As analyzed in Fig. 3(b), this force disrupts the transverse trapping of the sphere, especially when the chirality parameter is large.

We now focus on the radiation forces exerted on chiral spheres with opposite chirality parameters and make an attempt to understand them. Figures 4 and 5 show that, for an RCP beam that illuminates a sphere with chirality parameter κ and a LCP incident beam with -κ, the Fxs and Fzs are equal, respectively, but the Fys are opposite. For an x-polarized beam incidence with opposite chirality parameters, the phenomenon observed is consistent with that depicted in Figs. 2 and 3. It seems that symmetry exists between the radiation forces on chiral spheres with opposite chirality parameters. In fact, as introduced in Section 2, the relationship between the wave numbers of the RCP and LCP waves in chiral media is $k_{1} (κ) = k_{2} (- κ)$ , which indeed exhibits symmetry with respect to opposite chirality parameters. Thus, an RCP wave scattering from a chiral sphere with chirality parameter κ is symmetric physically to a LCP wave scattering from a chiral sphere with chirality parameter -κ. Therefore, the scattering fields in the two cases are the same in magnitude but contrary in RCP components and LCP components. This attribute may account for the symmetry of radiation forces discussed above.

3.3. Analysis of radiation torque

Figure 6 presents the axial radiation torques exerted on on-axis ( $x_{0} = y_{0} = 0$ ) chiral and isotropic spheres as functions of axial position z₀ for differently polarized Gaussian beam incidences. The parameters are as follows: the refractive index of the sphere is n = 1.59 + 0.0003i; the refractive index of the surrounding medium is n = 1.33; the radius of the sphere is a = 1.5 µm; the power of the incident Gaussian beam is P = 0.1 W; the wavelength of the beam in vacuum is λ = 0.488 µm; and the beam waist radius is w₀ = 1.0 μm. It can be observed that in contrast to radiation force, the axial radiation torque increases to its maximum at z₀ = 0 and then decreases as the sphere moves from the negative to the positive z-axis. The radiation torques on a chiral sphere with κ = 0.2 and an isotropic one (κ = 0) are almost the same, indicating that the effect of chirality parameter on axial radiation torque is minimal. Instead, polarization of the incident beam has a great influence on the torque. RCP beam incidence and LCP beam incidence rotates the sphere toward different direction, respectively. However, if the chirality parameter is a complex quantity (κ = 0.2 + 0.0002i, for example), as shown in the figure, the curve will be quite different from that for κ = 0.2. Loss of a sphere considerably influences on radiation force and torque. According to the expressions of the wave numbers in chiral media $k_{1} = ω (\sqrt{μ_{c} ε_{c}} + κ \sqrt{ε_{0} μ_{0}})$ and $k_{2} = ω (\sqrt{μ_{c} ε_{c}} - κ \sqrt{ε_{0} μ_{0}})$ , the imaginary part of a chirality parameter also affects the absorption characteristics of the media. Thus, the imaginary part of chirality parameter exerts noticeable effects on the radiation torque.

Fig. 6 Axial radiation torque versus z₀.

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Figure 7 shows three components of radiation torque versus transverse position x₀ for differently polarized beam incidences. All the parameters are the same as previously stated, except for the refractive index of the sphere, which is n = 1.59 + 0.0001i. It can be seen that axial torque Nz increases to its maximum at x₀ = 0 and then decreases as the sphere moves from the negative to the positive x-axis. However, values of Nz are tiny, compared with transverse torque Nx or Ny. Again it is found that a real-valued chirality parameter minimally influences both axial and transverse radiation torques. For both a general isotropic sphere and a chiral sphere located at the x-axis in an RCP or a LCP beam, the sphere experiences a transverse torque in the x-direction. The sign of torque Nx depends on beam polarization and sphere location. The curves of transverse torque Ny in all cases are close to one another and have large values compared with Nx and Nz as Nys are attributed to the asymmetric intensity of the beam with respect to the sphere.

Fig. 7 Radiation torque versus transverse position x₀. (a) Nz; (b) Nx; (c) Ny.

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4. Conclusion

This paper presents the theory and numerical results for radiation force and torque exerted on a chiral sphere by a Gaussian beam. In some cases, the chirality parameter may enhance the axial radiation force and reduce the transverse force toward the beam axis. Thus, trapping a chiral sphere may be more difficult than trapping a general isotropic sphere. As the scattering patterns of a chiral sphere differ for RCP and LCP beam incidences, polarization of the beam strongly influences radiation force. Thus, an appropriately polarized beam should be considered in trapping a chiral sphere. Symmetry may exist between the radiation forces on chiral spheres with opposite chirality parameters. It is found that variation of a real-valued chirality parameter slightly affects the radiation torque. However, if a complex-valued chirality parameter is considered, its imaginary part noticeably influences the torque.

Acknowledgment

The authors gratefully acknowledge supports from the National Natural Science Foundation of China under Grant No. 61172031 and the Fundamental Research Funds for the Central Universities.

References and links

1. A. Ashkin and J. Dziedzic, “Optical levitation by radiation pressure,” Appl. Phys. Lett. 19(8), 283–285 (1971). [CrossRef]

2. 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]

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

4. G. Roosen, “A theoretical and experimental study of the stable equilibrium positions of spheres levitated by two horizontal laser beams,” Opt. Commun. 21(1), 189–194 (1977). [CrossRef]

5. A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61(2), 569–582 (1992). [CrossRef] [PubMed]

6. 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]

7. R. Gussgard, T. Lindmo, and I. Brevik, “Calculation of the trapping force in a strongly focused laser beam,” J. Opt. Soc. Am. B 9(10), 1922–1930 (1992). [CrossRef]

8. W. H. Wright, G. J. Sonek, and M. W. Berns, “Radiation trapping forces on microspheres with optical tweezers,” Appl. Phys. Lett. 63(6), 715–717 (1993). [CrossRef]

9. R. C. Gauthier, “Theoretical investigation of the optical trapping force and torque on cylindrical micro-objects,” J. Opt. Soc. Am. B 14(12), 3323–3333 (1997). [CrossRef]

10. R. C. Gauthier and S. Wallace, “Optical levitation of spheres: analytical development and numerical computations of the force equations,” J. Opt. Soc. Am. B 12(9), 1680–1686 (1995). [CrossRef]

11. 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]

12. G. Gouesbet, B. Maheu, and G. Gréhan, “Scattering of a Gaussian Beam by a Sphere Using a Bromwich Formulation: Case of an arbitrary location,” Particle & Particle Systems Characterization 5(1), 1–8 (1988). [CrossRef]

13. G. Gouesbet, B. Maheu, and G. Gréhan, “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]

14. K. F. Ren, G. Gréha, 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]

15. 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]

16. J. A. Lock, “Calculation of the Radiation Trapping Force for Laser Tweezers by Use of Generalized Lorenz-Mie theory. II. On-Axis Trapping Force,” Appl. Opt. 43(12), 2545–2554 (2004). [CrossRef] [PubMed]

17. 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]

18. J. S. Kim and S. S. Lee, “Radiation pressure on a dielectric sphere in a gaussian laser beam,” Opt. Acta (Lond.) 29(6), 801–806 (1982). [CrossRef]

19. J. S. Kim and S. S. Lee, “Scattering of laser beams and the optical potential well for a homogeneous sphere,” J. Opt. Soc. Am. 73(3), 303–312 (1983). [CrossRef]

20. 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), 4594–4602 (1989). [CrossRef]

21. T. Nieminen, H. Rubinsztein-Dunlop, N. Heckenberg, and A. I. Bishop, “Numerical modelling of optical trapping,” Comput. Phys. Commun. 142(1-3), 468–471 (2001). [CrossRef]

22. T. A. Nieminen, V. L. Y. Loke, G. Knöner, and A. Branczyk, “Toolbox for calculation of optical forces and torques,” Piers Online 3(3), 338–342 (2007). [CrossRef]

23. R. Gauthier, “Computation of the optical trapping force using an FDTD based technique,” Opt. Express 13(10), 3707–3718 (2005). [CrossRef] [PubMed]

24. A. G. Hoekstra, M. Frijlink, L. B. F. M. Waters, and P. M. A. Sloot, “Radiation forces in the discrete-dipole approximation,” J. Opt. Soc. Am. A 18(8), 1944–1953 (2001). [CrossRef] [PubMed]

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

26. Z. J. Li, Z. S. Wu, Q. C. Shang, L. Bai, and C. H. Cao, “Calculation of radiation force and torque exerted on a uniaxial anisotropic sphere by an incident Gaussian beam with arbitrary propagation and polarization directions,” Opt. Express 20(15), 16421–16435 (2012). [CrossRef]

27. L. A. Ambrosio and H. E. Hernández-Figueroa, “Radiation pressure cross sections and optical forces over negative refractive index spherical particles by ordinary Bessel beams,” Appl. Opt. 50(22), 4489–4498 (2011). [CrossRef] [PubMed]

28. L. A. Ambrosio and H. E. Hernández-Figueroa, “Fundamentals of negative refractive index optical trapping: forces and radiation pressures exerted by focused Gaussian beams using the generalized Lorenz-Mie theory,” Biomed. Opt. Express 1(5), 1284–1301 (2010). [CrossRef] [PubMed]

29. D. Guzatov and V. V. Klimov, “Chiral particles in a circularly polarised light field: new effects and applications,” Quantum Electron. 41(6), 526–533 (2011). [CrossRef]

30. F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett. 29(3), 458–462 (1974). [CrossRef]

31. Z.-S. Wu, Q.-C. Shang, and Z.-J. Li, “Calculation of electromagnetic scattering by a large chiral sphere,” Appl. Opt. 51(27), 6661–6668 (2012). [CrossRef] [PubMed]

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

33. A. L. Aden and M. Kerker, “Scattering of electromagnetic wave from concentric sphere,” J. Appl. Phys. 22(10), 1242–1246 (1951). [CrossRef]

34. C. F. Bohren and D. R. Huffman, Absorption and scattering of light by small particles (Wiley, 1998).

35. Z. S. Wu and Y. P. Wang, “Electromagnetic scattering for multilayered sphere: Recursive algorithms,” Radio Sci. 26(6), 1393–1401 (1991). [CrossRef]

36. 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]

37. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19(3), 1177–1179 (1979). [CrossRef]

38. G. Gouesbet, J. A. Lock, and G. Gréhan, “Generalized Lorenz–Mie theories and description of electromagnetic arbitrary shaped beams: Localized approximations and localized beam models, a review,” J. Quant. Spectrosc. Radiat. Transf. 112(1), 1–27 (2011). [CrossRef]

39. G. Gouesbet and J. A. Lock, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory. II. Off-axis beams,” J. Opt. Soc. Am. A 11(9), 2516–2525 (1994). [CrossRef]

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

41. K. F. Ren, G. Gouesbet, and G. Gréhan, “Integral Localized Approximation in Generalized Lorenz-Mie Theory,” Appl. Opt. 37(19), 4218–4225 (1998). [CrossRef] [PubMed]

42. J. A. Lock and G. Gouesbet, “Rigorous justification of 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]

Analysis of the radiation force and torque exerted on a chiral sphere by a Gaussian beam

Abstract

1. Introduction

2. Theoretical formulations

2.1. Scattering of a Gaussian beam by a chiral sphere

2.2. Expressions of radiation force and torque

3. Numerical results and discussions

3.1. Verification of radiation force

3.2. Analysis of radiation force

3.3. Analysis of radiation torque

4. Conclusion

Acknowledgment

References and links

Cited By

Figures (7)

Equations (25)

Optics Express