Optical forces on Mie particles in an Airy evanescent field

Yang Yang; Wei-Ping Zang; Zhi-Yu Zhao; Jian-Guo Tian

doi:10.1364/OE.20.025681

1. Introduction

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

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

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

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

2. Theory and description

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

2.1 Expansions of the field components

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

k_{3} = n_{3} k_{0}, k_{2} = n_{2} k_{0} .

where k₀ is the wave number in vacuum. The governing equation for the electromagnetic fields is the vector Helmholtz equation, which can be solved in a standard manner, for instance by making use of the vector potentials approach [9]. Then, we will give the expressions for the coefficients A_lm and B_lm that characterize the incident electric and magnetic fields. These coefficients are defined as follows. We first write the definition of the spherical harmonics:

Y_{l m} (θ, ϕ) = {[\frac{2 l + 1}{4 π} \frac{(l - m)!}{(l + m)!}]}^{1 / 2} P_{l}^{m} (\cos θ) \exp (i m ϕ),

where θ is the polar angle and ϕ the azimuthal angle. The Riccati-Bessel function is defined as

ψ_{l} (x) = x j_{l} (x) = {(\frac{π x}{2})}^{1 / 2} J_{v} (x),

where j_l is the spherical Bessel function and J_v is the ordinary Bessel function of the first kind, with v = l + 1/2. The radial part of the Helmholtz equation permits us to make the expansions

\begin{array}{l} E_{r}^{(i)} = \frac{1}{{\tilde{r}}^{2}} \sum_{l = 1}^{\infty} \sum_{m = - l}^{l} l (l + 1) A_{l m} ψ_{l} (α \tilde{r}) Y_{l m} (θ, ϕ), \\ H_{r}^{(i)} = \frac{1}{{\tilde{r}}^{2}} \sum_{l = 1}^{\infty} \sum_{m = - l}^{l} l (l + 1) B_{l m} ψ_{l} (α \tilde{r}) Y_{l m} (θ, ϕ), \end{array}

where

α = k_{2} a, \tilde{r} = r / a,

a is the radius of spherical particle. Taking into account the orthogonality property of Y_lm, the expressions for the expansion coefficients are derived as follows:

A_{l m} = \frac{1}{l (l + 1) ψ_{l} (α)} \int_{0}^{2 π} \int_{0}^{π} E_{r}^{(i)} (a, θ, ϕ) Y_{l m}^{*} (θ, ϕ) \sin θ d θ d ϕ,

and

B_{l m} = \frac{1}{l (l + 1) ψ_{l} (α)} \int_{0}^{2 π} \int_{0}^{π} H_{r}^{(i)} (a, θ, ϕ) Y_{l m}^{*} (θ, ϕ) \sin θ d θ d ϕ,

The expansions for the incident (i), the scattered (s), and the internal (w) electromagnetic fields of 2D Airy evanescent wave in the spherical coordinate system are given in the Appendix.

2.2. Force components

Assuming a steady-state condition, the net radiation force F on the particle can be determined by integrating the dot product of the outwardly directed normal unit vector $\hat{n}$ and Maxwell’s stress tensor $\overset{\leftrightarrow}{T}$ over a surface enclosing the particle [25]:

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

where

〈 〉

represents a time average. After a great deal of algebra and applying numerous recursion and orthogonality relationships among the spherical harmonic functions, Eq. (8) can be directly integrated, and the net force on the particle can be expressed as a series over the coefficients a_lm, b_lm, A_lm, and B_lm [25]:

\begin{matrix} F_{x} + i F_{y} = \frac{i α^{4}}{16 π k_{2}^{2}} {\sum_{l = 1}^{\infty} \sum_{m = - l}^{l} {[\frac{(l + m + 2) (l + m + 1)}{(2 l + 1) (2 l + 3)}]}^{1 / 2} l (l + 2) (2 n_{2}^{2} a_{l m} a_{l + 1, m + 1}^{*} + n_{2}^{2} a_{l m} A_{l + 1, m + 1}^{*} \\ + n_{2}^{2} A_{l m} a_{l + 1, m + 1}^{*} + 2 b_{l m} b_{l + 1, m + 1}^{*} + b_{l m} B_{l + 1, m + 1}^{*} + B_{l m} b_{l + 1, m + 1}^{*}) + {[\frac{(l - m + 1) (l - m + 2)}{(2 l + 1) (2 l + 3)}]}^{1 / 2} \\ \times l (l + 2) (2 n_{2}^{2} a_{l + 1, m - 1} a_{l m}^{*} + n_{2}^{2} a_{l + 1, m - 1} A_{l m}^{*} + n_{2}^{2} A_{l + 1, m - 1} a_{l m}^{*} + 2 b_{l + 1, m - 1} b_{l m}^{*} + b_{l + 1, m - 1} B_{l m}^{*} \\ + B_{l + 1, m - 1} b_{l m}^{*}) - {[(l + m + 1) (l - m)]}^{1 / 2} n_{2} (- 2 a_{l m} b_{l, m + 1}^{*} + 2 b_{l m} a_{l, m + 1}^{*} - a_{l m} B_{l, m + 1}^{*} \\ + b_{l m} A_{l, m + 1}^{*} + B_{l m} a_{l, m + 1}^{*} - A_{l m} b_{l, m + 1}^{*})}, \end{matrix}

\begin{matrix} F_{z} = - \frac{α^{4}}{8 π k_{2}^{2}} {\sum_{l = 1}^{\infty} \sum_{m = - l}^{l} {[\frac{(l + m + 2) (l + m + 1)}{(2 l + 1) (2 l + 3)}]}^{1 / 2} l (l + 2) Im [2 n_{2}^{2} a_{l + 1, m} a_{l m}^{*} + n_{2}^{2} a_{l + 1, m} A_{l m}^{*} \\ + n_{2}^{2} A_{l + 1, m} a_{l m}^{*} + 2 b_{l + 1, m} b_{l m}^{*} + b_{l + 1, m} B_{l m}^{*} + B_{l + 1, m} b_{l m}^{*} + n_{2} m (2 a_{l m} b_{l m}^{*} + a_{l m} B_{l m}^{*} + A_{l m} b_{l m}^{*})] . \end{matrix}

2.3. Airy evanescent field as an incident field

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

The schematic diagram of the problem is shown in Fig. 1 . The origin of coordinate system coincides with the center of the spherical particle at a distance d from the interface. The media below and above the boundary plane at $z = - d$ have indices of refraction n₁ and n₂ (< n₁), respectively. A focused Airy beam with wave vector k₁ and frequency $ω$ is incident from medium n₁ with an angle of incidence $θ_{1} (> θ_{c r i t}),$ $θ_{c r i t} = \sin^{- 1} (n_{2} / n_{1})$ . The Airy beam center at (x_c, y_c, z_c). We take x-z plane as the plane of incidence and ( $\hat{x}, \hat{y}, \hat{z}$ ) as unit vectors.

Fig. 1 Spherical particle of radius a situated in the evanescent field region $z > - d$ . An Airy beam is incident from below with an angle of incidence $θ_{1} (> θ_{c r i t})$ in the substrate.

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

E = \frac{i}{n_{1}^{2} k_{0}} \nabla \times \nabla \times A, H = \nabla \times A,

where we used the Lorentz gauge condition in the Gaussian system of electromagnetic units. The Fourier spectrum for the two-dimensional finite energy Airy beam in Cartesian coordinate can be expressed as

Φ (s_{x}, s_{y}) \propto \exp [\frac{i}{3} ({\bar{x}}_{0}^{3} s_{x}^{3} + {\bar{y}}_{0}^{3} s_{y}^{3} - 3 a_{0}^{2} {\bar{x}}_{0} s_{x} - 3 a_{0}^{2} {\bar{y}}_{0} s_{y}) - a_{0} ({\bar{x}}_{0}^{2} s_{x}^{2} + {\bar{y}}_{0}^{2} s_{y}^{2})] .

s_{x} = n_{x} \cos θ_{1} - n_{z} \sin θ_{1}, s_{y} = n_{y}, s_{z} = n_{x} \sin θ_{1} + n_{z} \cos θ_{1}, {\bar{x}}_{0} = n_{1} k_{0} x_{0}, {\bar{y}}_{0} = n_{1} k_{0} y_{0} .

where (n_x, n_y, n_z) denote the dimensionless direction cosines of the wave vector k₁ along the Cartesian coordinate axes, x₀ and y₀ are characteristic lengths and a₀ is the aperture coefficient which determines the beam propagation distance. If the vector potential A is perpendicular to the plane of incidence, the vector potential can be expressed as follow:

A = \hat{y} \frac{C}{i k_{0}} \iint d s_{x} d s_{y} Φ (s_{x}, s_{y}) \exp [i f_{i} (x, y, z)],

where

f_{i} (x, y, z) = n_{1} k_{0} [n_{x} (x - x_{c}) + n_{y} (y - y_{c}) + n_{z} (z - z_{c})],

C is a normalization factor and the integration is over the domain of

s_{x}^{2} + s_{y}^{2} < 1

and

n_{z} > 0

.

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

E^{(i)} = C \iint d s_{x} d s_{y} \frac{n_{x} {\hat{s}}_{i} + n_{y} n_{z} {\hat{p}}_{i}}{1 - n_{z}^{2}} Φ (s_{x}, s_{y}) \exp [i f_{i} (x, y, z)],

H^{(i)} = n_{1} C \iint d s_{x} d s_{y} \frac{n_{y} n_{z} {\hat{s}}_{i} - n_{x} {\hat{p}}_{i}}{1 - n_{z}^{2}} Φ (s_{x}, s_{y}) \exp [i f_{i} (x, y, z)],

where we have defined two vectors,

{\hat{s}}_{i}

and

{\hat{p}}_{i}

represent perpendicular and parallel to the plane of incidence, respectively. They can be expressed as follows:

{\hat{s}}_{i} = - n_{y} \hat{x} + n_{x} \hat{y}, {\hat{p}}_{i} = (n_{x} \hat{x} + n_{y} \hat{y}) n_{z} - (1 - n_{z}^{2}) \hat{z} .

The transmitted Fresnel amplitude coefficients at the interface for p polarization and s polarization are

T_{p} = \frac{2 n_{1} n_{z}}{n_{1} ξ + n_{2} n_{z}}, T_{s} = \frac{2 n_{1} n_{z}}{n_{1} n_{z} + n_{2} ξ} .

where

ξ = {[1 - {(n_{1} / n_{2})}^{2} (n_{x}^{2} + n_{y}^{2})]}^{1 / 2} .

On paraxial condition,

n_{x} \approx \sin θ_{1}, n_{y} \approx 0,

if

θ_{1} > \sin^{- 1} (n_{1} / n_{2})

, the parameter

ξ

will be an imaginary number, the incident Airy beam is totally reflected on the interface and an evanescent electromagnetic wave is generated on the side of the medium n₂. As a result, the transmitted electromagnetic field can be written as

E^{(t)} = C \iint d s_{x} d s_{y} \frac{n_{x} T_{s} {\hat{s}}_{t} + n_{y} n_{z} T_{p} {\hat{p}}_{t}}{1 - n_{z}^{2}} Φ (s_{x}, s_{y}) \exp [i f_{t} (x, y, z)],

H^{(t)} = n_{2} C \iint d s_{x} d s_{y} \frac{n_{y} n_{z} T_{p} {\hat{s}}_{t} - n_{x} T_{s} {\hat{p}}_{t}}{1 - n_{z}^{2}} Φ (s_{x}, s_{y}) \exp [i f_{t} (x, y, z)],

Where

{\hat{s}}_{t} = - n_{y} \hat{x} + n_{x} \hat{y}, {\hat{p}}_{t} = (n_{x} \hat{x} + n_{y} \hat{y}) ξ - (n_{1} / n_{2}) (1 - n_{z}^{2}) \hat{z} .

f_{t} (x, y, z) = n_{1} k_{0} [n_{x} (x - x_{c}) + n_{y} (y - y_{c}) - n_{z} (z_{c} + d)] + n_{2} k_{0} (z + d) ξ .

For evanescent field, the integral domain is

{(n_{1} / n_{2})}^{2} (n_{x}^{2} + n_{y}^{2}) > 1

. The Eqs. (19) and (20) can be rewritten as

E^{(t)} = E_{x}^{(t)} \hat{x} + E_{y}^{(t)} \hat{y} + E_{z}^{(t)} \hat{z},

H^{(t)} = H_{x}^{(t)} \hat{x} + H_{y}^{(t)} \hat{y} + H_{z}^{(t)} \hat{z},

The radial field components of the evanescent fields are represented as

E_{r}^{(t)} = E_{x}^{(t)} \sin θ \cos ϕ + E_{y}^{(t)} \sin θ \sin ϕ + E_{z}^{(t)} \cos θ,

H_{r}^{(t)} = H_{x}^{(t)} \sin θ \cos ϕ + H_{y}^{(t)} \sin θ \sin ϕ + H_{z}^{(t)} \cos θ .

Here θ and ϕ denote the usual polar and azimuthal angles. Substitute

E_{r}^{(t)}

and

H_{r}^{(t)}

into Eqs. (6) and (7), after using a great deal of recursions and orthogonal relationships and finishing the integrals about θ and ϕ, we obtain the coefficients

A_{l m}

and

B_{l m}

for the Airy evanescent field as follows:

A_{l m} = A_{c} \iint d s_{x} d s_{y} Φ (s_{x}, s_{y}) \exp [i \cdot g (r_{0})] \frac{{(n_{x} - i n_{y})}^{m}}{{(1 - n_{z}^{2})}^{(m + 1) / 2}} [i n_{x} β_{l m} (ξ) T_{s} + n_{y} n_{z} α_{l m} (ξ) T_{p}],

B_{l m} = B_{c} \iint d s_{x} d s_{y} Φ (s_{x}, s_{y}) \exp [i \cdot g (r_{0})] \frac{{(n_{x} - i n_{y})}^{m}}{{(1 - n_{z}^{2})}^{(m + 1) / 2}} [i n_{x} α_{l m} (ξ) T_{s} + n_{y} n_{z} β_{l m} (ξ) T_{p}],

where

A_{c}

and

B_{c}

are constants:

A_{c} = - C \frac{i^{l + 1}}{l (l + 1) α^{2}} \sqrt{π (2 l + 1) \frac{(l - m)!}{(l + m)!}}, B_{c} = n_{2} C \frac{i^{l}}{l (l + 1) α^{2}} \sqrt{π (2 l + 1) \frac{(l - m)!}{(l + m)!}},

g (r_{0}) = - n_{1} k_{0} [n_{x} x_{c} + n_{y} y_{c} + n_{z} (z_{c} + d)] + n_{2} k_{0} ξ d .

the functions

α_{l m} (ξ)

and

β_{l m} (ξ)

are defined as

α_{l m} (ξ) = ξ [P_{l}^{m + 1} (ξ) + (l + m) (l - m + 1) P_{l}^{m - 1} (ξ)] - 2 m {(1 - ξ^{2})}^{1 / 2} P_{l}^{m} (ξ),

β_{l m} (ξ) = P_{l}^{m + 1} (ξ) - (l + m) (l - m + 1) P_{l}^{m - 1} (ξ) .

3. Results and discussions

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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4. Conclusions

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

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

a_{l m} = \frac{{ψ^{'}}_{l} (\tilde{n} α) ψ_{l} (α) - \tilde{n} ψ_{l} (\tilde{n} α) {ψ^{'}}_{l} (α)}{\tilde{n} ψ_{l} (\tilde{n} α) ξ_{l}^{(1)}^{'} (α) - {ψ^{'}}_{l} (\tilde{n} α) ξ_{l}^{(1)} (α)} A_{l m},

b_{l m} = \frac{\tilde{n} {ψ^{'}}_{l} (\tilde{n} α) ψ_{l} (α) - ψ_{l} (\tilde{n} α) {ψ^{'}}_{l} (α)}{ψ_{l} (\tilde{n} α) ξ_{l}^{(1)}^{'} (α) - \tilde{n} {ψ^{'}}_{l} (\tilde{n} α) ξ_{l}^{(1)} (α)} B_{l m},

c_{l m} = \frac{ξ_{l}^{(1)}^{'} (α) ψ_{l} (α) - ξ_{l}^{(1)} (α) {ψ^{'}}_{l} (α)}{{\tilde{n}}^{2} ψ_{l} (\tilde{n} α) ξ_{l}^{(1)}^{'} (α) - \tilde{n} {ψ^{'}}_{l} (\tilde{n} α) ξ_{l}^{(1)} (α)} A_{l m},

d_{l m} = \frac{ξ_{l}^{(1)}^{'} (α) ψ_{l} (α) - ξ_{l}^{(1)} (α) {ψ^{'}}_{l} (α)}{ψ_{l} (\tilde{n} α) ξ_{l}^{(1)}^{'} (α) - \tilde{n} {ψ^{'}}_{l} (\tilde{n} α) ξ_{l}^{(1)} (α)} B_{l m} .

where $k_{0} = 2 π / λ, α = n_{2} k_{0} a, \tilde{n} = n_{3} / n_{2},$ a, n₃ is radius and refractive index of the particle, respectively. $ξ_{l}^{(1)} = ψ_{l} - i χ_{l}$ , $ψ_{l}$ and $χ_{l}$ are the Riccati-Bessel functions. $Y_{l m} (θ, ϕ)$ is the spherical harmonic function, $\tilde{r} = r / a$ . The expansions for the incident (i), the scattered (s), and the internal (w) electromagnetic fields in the spherical coordinate system are summarized as follows:

Incident field

E_{r}^{(i)} = \frac{1}{{\tilde{r}}^{2}} \sum_{l = 1}^{\infty} \sum_{m = - l}^{l} [l (l + 1) A_{l m} ψ_{l} (α \tilde{r}) Y_{l m} (θ, ϕ)],

E_{θ}^{(i)} = \frac{α}{\tilde{r}} \sum_{l = 1}^{\infty} \sum_{m = - l}^{l} (A_{l m} {ψ^{'}}_{l} (α \tilde{r}) \frac{\partial Y_{l m} (θ, ϕ)}{\partial θ} - \frac{m}{n_{2}} B_{l m} ψ_{l} (α \tilde{r}) \frac{Y_{l m} (θ, ϕ)}{\sin θ}),

E_{ϕ}^{(i)} = \frac{α}{\tilde{r}} \sum_{l = 1}^{\infty} \sum_{m = - l}^{l} (i m A_{l m} {ψ^{'}}_{l} (α \tilde{r}) \frac{Y_{l m} (θ, ϕ)}{\sin θ} - \frac{i}{n_{2}} B_{l m} ψ_{l} (α \tilde{r}) \frac{\partial Y_{l m} (θ, ϕ)}{\partial θ}) .

H_{r}^{(i)} = \frac{1}{{\tilde{r}}^{2}} \sum_{l = 1}^{\infty} \sum_{m = - l}^{l} [l (l + 1) B_{l m} ψ_{l} (α \tilde{r}) Y_{l m} (θ, ϕ)],

H_{θ}^{(i)} = \frac{α}{\tilde{r}} \sum_{l = 1}^{\infty} \sum_{m = - l}^{l} (B_{l m} {ψ^{'}}_{l} (α \tilde{r}) \frac{\partial Y_{l m} (θ, ϕ)}{\partial θ} + m n_{2} A_{l m} ψ_{l} (α \tilde{r}) \frac{Y_{l m} (θ, ϕ)}{\sin θ}),

H_{ϕ}^{(i)} = \frac{α}{\tilde{r}} \sum_{l = 1}^{\infty} \sum_{m = - l}^{l} (i m B_{l m} {ψ^{'}}_{l} (α \tilde{r}) \frac{Y_{l m} (θ, ϕ)}{\sin θ} + i n_{2} A_{l m} ψ_{l} (α \tilde{r}) \frac{\partial Y_{l m} (θ, ϕ)}{\partial θ}) .

Scattered field

E_{r}^{(s)} = \frac{1}{{\tilde{r}}^{2}} \sum_{l = 1}^{\infty} \sum_{m = - l}^{l} [l (l + 1) a_{l m} ξ_{l}^{(1)} (α \tilde{r}) Y_{l m} (θ, ϕ)],

E_{θ}^{(s)} = \frac{α}{\tilde{r}} \sum_{l = 1}^{\infty} \sum_{m = - l}^{l} (a_{l m} ξ_{l}^{(1)}^{'} (α \tilde{r}) \frac{\partial Y_{l m} (θ, ϕ)}{\partial θ} - \frac{m}{n_{2}} b_{l m} ξ_{l}^{(1)} (α \tilde{r}) \frac{Y_{l m} (θ, ϕ)}{\sin θ}),

E_{ϕ}^{(s)} = \frac{α}{\tilde{r}} \sum_{l = 1}^{\infty} \sum_{m = - l}^{l} (i m a_{l m} ξ_{l}^{(1)}^{'} (α \tilde{r}) \frac{Y_{l m} (θ, ϕ)}{\sin θ} - \frac{i}{n_{2}} b_{l m} ξ_{l}^{(1)} (α \tilde{r}) \frac{\partial Y_{l m} (θ, ϕ)}{\partial θ}) .

H_{r}^{(s)} = \frac{1}{{\tilde{r}}^{2}} \sum_{l = 1}^{\infty} \sum_{m = - l}^{l} [l (l + 1) b_{l m} ξ_{l}^{(1)} (α \tilde{r}) Y_{l m} (θ, ϕ)],

H_{θ}^{(s)} = \frac{α}{\tilde{r}} \sum_{l = 1}^{\infty} \sum_{m = - l}^{l} (b_{l m} ξ_{l}^{(1)}^{'} (α \tilde{r}) \frac{\partial Y_{l m} (θ, ϕ)}{\partial θ} + m n_{2} a_{l m} ξ_{l}^{(1)} (α \tilde{r}) \frac{Y_{l m} (θ, ϕ)}{\sin θ}),

H_{ϕ}^{(s)} = \frac{α}{\tilde{r}} \sum_{l = 1}^{\infty} \sum_{m = - l}^{l} (i m b_{l m} ξ_{l}^{(1)}^{'} (α \tilde{r}) \frac{Y_{l m} (θ, ϕ)}{\sin θ} + i n_{2} a_{l m} ξ_{l}^{(1)} (α \tilde{r}) \frac{\partial Y_{l m} (θ, ϕ)}{\partial θ}) .

Internal field

E_{r}^{(w)} = \frac{1}{{\tilde{r}}^{2}} \sum_{l = 1}^{\infty} \sum_{m = - l}^{l} [l (l + 1) c_{l m} ψ_{l} (\tilde{n} α \tilde{r}) Y_{l m} (θ, ϕ)],

E_{θ}^{(w)} = \frac{α}{\tilde{r}} \sum_{l = 1}^{\infty} \sum_{m = - l}^{l} (\tilde{n} c_{l m} {ψ^{'}}_{l} (\tilde{n} α \tilde{r}) \frac{\partial Y_{l m} (θ, ϕ)}{\partial θ} - \frac{m}{n_{2}} d_{l m} ψ_{l} (\tilde{n} α \tilde{r}) \frac{Y_{l m} (θ, ϕ)}{\sin θ}),

E_{ϕ}^{(w)} = \frac{α}{\tilde{r}} \sum_{l = 1}^{\infty} \sum_{m = - l}^{l} (i m \tilde{n} c_{l m} {ψ^{'}}_{l} (\tilde{n} α \tilde{r}) \frac{Y_{l m} (θ, ϕ)}{\sin θ} - \frac{i}{n_{2}} d_{l m} ψ_{l} (\tilde{n} α \tilde{r}) \frac{\partial Y_{l m} (θ, ϕ)}{\partial θ}) .

H_{r}^{(w)} = \frac{1}{{\tilde{r}}^{2}} \sum_{l = 1}^{\infty} \sum_{m = - l}^{l} [l (l + 1) d_{l m} ψ_{l} (\tilde{n} α \tilde{r}) Y_{l m} (θ, ϕ)],

H_{θ}^{(w)} = \frac{α}{\tilde{r}} \sum_{l = 1}^{\infty} \sum_{m = - l}^{l} (\tilde{n} d_{l m} {ψ^{'}}_{l} (\tilde{n} α \tilde{r}) \frac{\partial Y_{l m} (θ, ϕ)}{\partial θ} + m n_{2} {\tilde{n}}^{2} c_{l m} ψ_{l} (\tilde{n} α \tilde{r}) \frac{Y_{l m} (θ, ϕ)}{\sin θ}),

H_{ϕ}^{(w)} = \frac{α}{\tilde{r}} \sum_{l = 1}^{\infty} \sum_{m = - l}^{l} (i m \tilde{n} d_{l m} {ψ^{'}}_{l} (\tilde{n} α \tilde{r}) \frac{Y_{l m} (θ, ϕ)}{\sin θ} + i n_{2} {\tilde{n}}^{2} c_{l m} ψ_{l} (\tilde{n} α \tilde{r}) \frac{\partial Y_{l m} (θ, ϕ)}{\partial θ}) .

Acknowledgments

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

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Optical forces on Mie particles in an Airy evanescent field

Abstract

1. Introduction

2. Theory and description

2.1 Expansions of the field components

2.2. Force components

2.3. Airy evanescent field as an incident field

3. Results and discussions

4. Conclusions

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

Incident field

Scattered field

Internal field

Acknowledgments

References and links

Cited By

Figures (7)

Equations (54)

Optics Express