## Abstract

In this paper we discuss the effects of multiple scattering to the optical forces on a particle by an evanescent field. We show that the iterative method to process the effects of the interaction between the particle and a plane surface is invalid when the radius of particle is large or when the structural resonance of the particle occurs. By using the generalized minimum residual method to solve the set of equations directly, the divergence appears in the iterative method can be removed completely. As an illustrative example, we discussed the effects of multiple scattering to optical forces on a particle in an evanescent field from an incident plane wave. The interpretations of numerical results are presented in detail.

© 2013 OSA

## 1. Introduction

Since Ashkin *et al*. proved that a 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. Optical tweezers have been used to manipulate and trap micro-scale objects [1], liquid droplets [2], and even some submicron objects such as viruses [3] and nanoparticles [4]. 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 [5]. Almaas and Brevik theoretically discussed the force upon a micrometer-sized dielectric sphere in a plane evanescent field [6]. Meanwhile, Chang *et al*. theoretically investigated the optical force exerted on a dielectric sphere by a Gaussian evanescent field [7]. Recently, the optical force exerted on a dielectric sphere by the Airy evanescent field has been investigated in detail [8]. Jack Ng *et al*. have proposed a size-selective optical force achieved by utilizing the evanescent wave to excite a microsphere's high-Q whispering gallery mode (WGM) [9, 10].

The optical properties of a particle in the vicinity of a plane surface are known to be rather different from these in free space. The actual calculation of these properties requires a consideration of the boundary conditions both across the surface of the particle and across the plane surface, so that one has to resort to some more complicated approaches than those for isolated particles [1]. Bobbert and Vlieger formulated a theory that succeeds in exactly calculating the scattered field from a sphere near a perfectly reflecting plane surface, and introduced a suitable approximation to get the observed field in the far zone [11]. Videen *et al*. developed an approximate solution by representing the reflected fields as that produced by an 'image' source within the normal incidence approximation (NIA) [12]. Using this approximate method, they discussed the characterization of metallic nano-particle via surface wave scattering [13, 14]. Based on the expansion of the electromagnetic field in terms of vector multipole fields and with the imposition of the boundary conditions, Fucile *et al*. accurately calculated the full scattering pattern from a sphere in the vicinity of a plane surface [15]. Meanwhile, Wriedt and Doicu [16] considered the exact solution for a sphere adjacent to a surface and developed equivalent formulations that are more compact and suitable to numerical evaluation than the formulas presented in [11]. This exact method has been applied to predict the scattering properties of the sphere or sphere clusters on a plane surface [17, 18]. Meanwhile, this problem can be processed by using the boundary element method (BEM), which needs extremely fine meshes for three dimensional calculations [10]. Chang *et al*. theoretically examined the optical force exerted on a dielectric sphere by the evanescent field from a Gaussian beam, and considered the effects of interaction between the sphere and the plane surface [19]. By representing the scattered electromagnetic fields as an integral of plane electromagnetic waves [20], they proposed an iterative procedure to calculate the effects of multiple scattering to the optical force. Inami *et al*. [21] also developed an analytical method, which combined the iterative calculations of the extended Mie theory developed by Barton *et al*. [22, 23] and the plane wave decomposition. This theoretical approach is similar to the iterative procedure proposed by Barton *et al.* [24]. Although the iterative method is easy to grasp, it may diverge when the particle is in structural resonant or when the particle is in touch with the surface. The divergence is attributed to the strong feedback between the particle and the surface. Moreover, Chaumet *et al*. used coupled dipole method to discuss the electromagnetic force on a particle over a flat dielectric substrate [25]. Arias-Gonzalez et.al studied the range of validity of the dipole approximation and interpreted the nature of the electromagnetic force for dielectric and metallic nanocylinders and particles on surface [26, 27]. Coupled dipole method also applied to study optical forces on a magnetodielectric small particle [28]. However, this method can be used only when the radius of the particle less than the wavelength.

In this paper, we followed the method presented by Vidden [12] and Wriedt *et al*. [16], and extended the method to study the radiation force on a particle in an evanescent field excited by an arbitrary shape beam. The interaction between particle and surface is included correctly. As an illustrative example, we discussed the radiation force on a particle in an evanescent field and compare our results with those of Almaas' [6]. In addition, the differences between the iterative method and the directly solving matrix method are pointed out.

## 2. Theory and description

#### 2.1. Derivation of scattering fields

Figure 1 shows the geometry of the coordinate system and the symbols appropriate to our problem. A sphere of radius *a* is located on the *z* axis, with a distance *d* above a plane surface that bounds two media of different refractive indices. The origin of the coordinate system coincides with the center of the sphere. The indices of refraction for the media below and above the plane surface at *z* = *d* are n_{1} and n_{2}, respectively. An arbitrary beam with wave vector **k _{1}** and frequency$\omega $is incident from medium n

_{1}at an angle of incidence $\beta >{\beta}_{crit},$ where${\beta}_{crit}$ is the critical angle.

First, let us find the formal solutions of the problem for interaction between the particle and dielectric interface. We denote the initial input electric field vector and, the transmitted evanescent field by ** E^{inc}** (

**r**) and

**(**

*E*^{eva}**r**), respectively. They can be related by the transmission matrix $\tau $of the plane surface, which determines the surface transmission characteristics, as

**(**

*E*^{eva}**r**) will produce an additional scattered electric field vector

**(**

*E*^{sca}**r**). In addition to the evanescent fields

**(**

*E*^{eva}**r**) and scattering field

**(**

*E*^{sca}**r**), a third field, ${E}^{\mathrm{int}}\left(r\right)$, is incident upon the sphere. This field is a result of the scattered field reflecting off the surface and striking the sphere. This interacting field${E}^{\mathrm{int}}\left(r\right)$and field

**(**

*E*^{sca}**r**) can be related by the reflective matrix of the plane surface

**A**, asThe total field striking the particle can be expressed as follows:

**(**

*E*^{sca}**r**) and the total input field ${E}^{tot}\left(r\right)$can be related by using the transition matrix (

**T**-matrix):where

**T**is the

**T**-matrix of particle that only depends on the shape and physical properties of the particle. Its elements can be determined by the extended boundary condition method [22].

Substituting Eqs. (2) and (3) into Eq. (4), the scattered field can be expressed in terms of the evanescent field ${E}^{eva}\left(r\right)$,

where**I**,

**T**, and

**A**represent the identity matrix, the

**T**-matrix of particle and the reflecting matrix of the plane surface, respectively. Next, we will give the expressions of the

**T**-matrix of particle and the reflection matrix

**A**of the plane surface.

For an arbitrary incidence beam such as a plane wave [6], a focused Gaussian beam [7] and an Airy beam [8], the evanescent field can be obtained by using Eq. (1) and written in terms of vector spherical wave functions (VSWF)

**A**appearing in Eq. (2), and can be expressed as

**A**can be founded in [16].

In the null-field method, the scattered coefficients are related to the expansion coefficients of the fields striking the particle by the $T$-matrix. For an axis-symmetric particle, the equations are uncoupled, and we can pursure a separate solution for each azimuthal mode *m*. For a fixed azimuthal mode *m*, we can truncate the expansions given by Eqs. (7) and (8), and derive the following matrix equations:

1. Starting with${}^{0}e{}_{m{n}_{1}}^{R}={}^{0}f{}_{m{n}_{1}}^{R}=0,$ i.e., neglecting the interaction between the particle and the interface. From Eq. (10) it follows that

#### 2.2. Incident field

For the sake of simplicity, we choose a plane wave as the initial incident field. After a great deal of algebra and by applying numerous recursion, product, and orthogonality relationships among the spherical harmonic functions, the expanding coefficients of the evanescent field in Eq. (6), ${a}_{m{n}_{1}}^{0}$and ${b}_{m{n}_{1}}^{0}$ can be expressed as follows:

#### 2.3. Force calculation

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 $\widehat{n}$ and the Maxwell’s stress tensor$\overleftrightarrow{T}$ over a surface enclosing the particle [23]:

## 3. Results and discussion

The theory that we developed in Section 2 will now be applied to the same system studied by Almaas *et al*. [6]. The parameters of the beam, the particle and the interface used here are same with those in [6].

Due to the differences of the directions of axis between [6] and this paper, the radiation force components between [6] and current paper are related by

where ${Q}_{z}$and${Q}_{x}$represent the horizontal and vertical force components appeared in [6], respectively.First, we consider the effect of the interaction between sphere and surface on the ${Q}_{x}^{c}$components. The particle situates on the surface and the angle of incidence is$\beta ={51}^{0}$ . The refractive indices of the media below and above the plane surface are n_{1} = 1.75 and n_{2} = 1.0, respectively. The refractive index of the sphere is n_{3} = 1.5. Figure 2 shows the variation of the optical force component${Q}_{x}^{c}$ versus the size parameter $\alpha $ from 0.1 to 8.6, when the incident plane wave is polarized (a) orthogonal (s), and (b) parallel (p) to the x-z plane, respectively. The distance *d* from the center of the sphere to the interface equals to the radius of sphere, i.e. $d=a.$ It is the case of contact between the sphere and the interface. The black line represents the zero-order multiple scattering of the evanescent wave by the sphere, i.e. neglecting the interaction between sphere and the plane surface. The result is identical to that shown in Fig. 9 of [6]. The blue lines represent the fourth order correction by using the iteration method and the red dashed lines represent the exact results by solving Eq. (12) by using GMRES method directly.

We can see that the interaction between sphere and interface always reduces ${Q}_{x}^{c}$ for s polarization. For p polarized incidence, the multiple scattering enhances ${Q}_{x}^{c}$ for small radius of sphere ($\alpha <2$), conversely reduces ${Q}_{x}^{c}$ for large radius of sphere ($\alpha >2$).

Figure 3 shows the variation of the optical component${Q}_{z}^{c}$ versus the size parameter$\alpha $ from 0.1 to 8.6, when the incident plane wave is polarized (a) orthogonal (s) and parallel (p) to the x-z plane, respectively.

We can find that the interaction between the sphere and the interface significantly affects the radiation force. When the radius of a sphere is small, such as the radius of sphere$a<\lambda /2$, the sphere is immersed inside the evanescent field, and the multiple scattering affects notably the optical force sensed by the sphere. When the sphere is in structural resonance, the multiple scattering effect is particularly strong, and some of the photons can tunnel into the propagation mode of the interface and the radiation force is reduced distinctly. But for p polarization, the multiple scattering effect always enhances the optical force ${Q}_{z}^{c}$. Meanwhile, Figs. 2 and 3 also show the results from the iteration method, which are identical to the results by solving Eq. (13) directly for spheres with moderate sizes.

Next, we discuss the convergence of the iteration method. The simulation results are shown in Fig. 4. The black lines represent the radiation force without multiple scattering effect. The three weaker peaks in the black line in Fig. 4(b) are the first-order magnetic resonant peaks, at $\alpha =$11.0933, 11.8351 and 12.5744, respectively. The corresponding peaks are stronger for s polarization, as shown in Fig. 4(a). On the other hand, the three strong peaks in Fig. 4(b) are the first-order electric resonant peaks, at $\alpha =$11.4526, 12.2043 and 12.9532, respectively. We can find that the iteration method does not converge and in fact diverge with an increasing number of iteration nearby the positions of TE or TM resonance [note the log scale in Fig. 4]. In this case we must solve Eq. (12) directly. In this paper, we always use GMRES method to solve Eq. (12), and the reliability of results has been validated by other numerical method. In addition, we can see that the iteration method converges for non-resonance cases.

Figure 5 provides the effect of multiple scattering to the horizontal radiation force${Q}_{x}^{c}$as a function of the normalized distance between the sphere and the interface. The normalized distances equal to (*d-a)/a*.

In Fig. 5(a), the size parameter is $\alpha =$12.5744, and a peak of the first-order magnetic resonance for ${\text{TM}}_{15}^{1}.$The incident beam is s polarized. When the sphere touches with the interface, the horizontal radiation force ${Q}_{x}^{c}$ is 0.595, while the result by neglecting the effect of multiple scattering is 2.592. In other words, when the effect of multiple scattering is neglected, the horizontal radiation force ${Q}_{x}^{c}$ is over-estimated by four times. In Fig. 5(b), the size parameter is $\alpha =$12.9532, and a peak of the first-order electric resonance for ${\text{TE}}_{15}^{1}$ and the incident beam is p polarization. When the sphere touches with the interface, the horizontal radiation force ${Q}_{x}^{c}$is 0.3275, while the result by neglecting the effect of multiple scattering is 0.8393. So when the effect of multiple scattering is neglected, the horizontal radiation force ${Q}_{x}^{c}$ is overstated 2.5 times.

## 4. Conclusions

In conclusion, we have discussed the effect of multiple scattering to the optical force on a sphere by an evanescent field. The simulation results show that the effects of multiple scattering are significant when the sphere touches with the interface. When the radius of sphere is not very large or when the structural resonance does not occur, the iteration method can be used to deal with the effect of multiple scattering; if the structural resonance occurs, the iteration method is invalid, while the GMRES method can be used to solve this problem directly. We believe that the theoretical results presented in this paper would be useful for investigations in optical micro-manipulation and near-field optics.

## 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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