Effect of multiple scattering to optical forces on a sphere near an optical waveguide

Jian Xu; Wei-Ping Zang; Jian-Guo Tian

doi:10.1364/OE.23.004195

1. Introduction

After the pioneering work of Ashkin et al. in optical trapping and manipulation in 1970s [1], optical tweezers have become excellent tool to manipulate particles ranging from a few hundred nanometers to several micrometers in size [1–4]. In 1992, Kawata and Sugiura [5] demonstrated the manipulation of microparticles in the evanescent field produced by total internal reflection of a laser beam on a prism surface. Then the evanescent field produced by an optical waveguide was used to guide dielectric microparticles [6] and gold Rayleigh particles [7] along a channel waveguide. In recent years, researchers have focused on the use of evanescent fields on planar integrated waveguide to optically manipulate microparticles and biological cells on the waveguide surface [8, 9]. The development of optical manipulation devices based on waveguide opens up the possibility of combining optical trapping with integrated optics to manipulate, detect and sort microparticles [10, 11].

To understand optical trapping, it is essential to calculate the radiation forces acting on a particle. Almaas and Brevik [12] theoretically discussed the radiation forces upon a micrometer-sized dielectric sphere in a plane evanescent field on a prism surface by Arbitrary Beam Theory (ABT). 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) [13, 14] and Li et.al discussed the resonant light pressure effects in optical manipulation with microparticles [15]. Afterwards, Jaising and Hellesø [16, 17] extended ABT to calculate the radiation forces on a dielectric spherical particle in the evanescent field of an optical waveguide. However, their analysis didn’t consider the effect of multiple scattering between the sphere and the plane interface. In practical applications of the optical waveguide trapping, effect of multiple scattering between the sphere and the interface has remarkable influence on optical forces with the sphere close to the surface of the waveguide. To figure out the multiple scattering effect, Videen et al. developed an approximate solution by representing the reflected fields as that produced by an 'image' source within the normal incidence approximation [18]. By the method, they discussed the feature of metallic nano-particle via surface wave scattering [19, 20]. One assumption of his method is that the scattered field from the sphere, reflecting off the surface and interacting with the sphere, is incident on the surface at near-normal incident. The advantage of his method is that the elements of the reflection matrix are the translation coefficients and can be derived by using the addition theorem. As a result, the amount of computer time required to solve the scattering problem is significantly reduced. Wriedt and Doicu [21] considered the exact solutions for a sphere adjacent to a surface and developed equivalent formulations. They are more compact and suitable than the formulas presented in [22] to numerical evaluation. The elements of the reflection matrix can be obtained by numerical integrations. The exact method had been applied to predict the scattering properties of a sphere or sphere clusters on a plane surface [23, 24]. Our group proved that the method presented by Videen [18] and Wriedt et al. [21] can be applied to study the radiation forces on a particle in an evanescent field excited by an arbitrary shape beam [25].

In this paper, we explicitly investigate the radiation forces on a Mie particle in an evanescent field excited by an optical waveguide. We take the interaction between particle and surface of waveguide into consideration accurately. We find that the effect of multiple scattering to the optical forces can’t be ignored, especially when the structure resonance of the particle arises. Moreover, the effect of multiple scattering to optical forces is also studied in detail on the condition that the distance between the sphere and the waveguide is within the effective operating distance.

2. Theory and description

2.1. Derivation of scattering fields

Figure 1 shows the geometry of the coordinate system and the symbols related to our problem. A three-layer slab optical waveguide with refractive indices n₁, n₂ and n₃. Its structure is uniform in the x and y directions. The waves propagate along the x-direction in a cartesian coordinate system. A sphere with radius a is located on the z axis, with a distance d above the upper surface of the waveguide. The center of the sphere coincides with the origin of the coordinate system. The refractive index of sphere is n₀.

Fig. 1 Particle located in the evanescent field produced by a waveguide.

Download Full Size | PDF

First, let us find the formal solution of the problem about the interaction between the particle and dielectric interface. When a particle is exposed to an evanescent field E^eva (r), an additional scattered electric field E^sca (r) appears. With the scattered electric field E^sca (r) reflecting off the plane surface, another field E^int (r) is incident upon the sphere. The interacting field E^int (r) can be expressed by field E^sca (r) as,

E^{i n t} (r) = A \cdot E^{s c a} (r)

Where A is the reflection matrix of the plane surface. According to the analysis, the total field striking the particle consists of two fields E^eva (r) and E^int (r), by,

E^{t o t} (r) = E^{e v a} (r) + E^{i n t} (r) = E^{e v a} (r) + A \cdot E^{s c a} (r)

Meanwhile, the scattered field E^sca (r) and the total input field E^tot (r) can be expressed as,

E^{s c a} (r) = T \cdot E^{t o t} (r)

Where T is the transition matrix (T-matrix) of particle, which only depends on the shape and physical properties of the particle. Its elements can be determined by the extended boundary condition method [26]. Substituting Eq. (2) into Eq. (3), the scattered field and the evanescent field satisfy the following expression,

(I - T \cdot A) E^{s c a} (r) = T \cdot E^{e v a} (r)

Where I, T, and A are respectively the identity matrix, the T-matrix of the particle and the reflection matrix of the plane surface.

Next, the specific expressions of the T-matrix of particle and the reflection matrix A of the plane surface will be given. For the waveguide trapping condition, the evanescent field can be written in terms of vector spherical wave functions (VSWFs),

\begin{array}{l} E^{e v a} (r) = \sum_{l_{1} = 1}^{\infty} \sum_{m = - l_{1}}^{l_{1}} [a_{m l_{1}}^{0} M_{m l_{1}}^{(1)} (k_{1} r) + b_{m l_{1}}^{0} N_{m l_{1}}^{(1)} (k_{1} r)] \\ H^{e v a} (r) = \frac{k}{i ω μ} \sum_{l_{1} = 1}^{\infty} \sum_{m = - l_{1}}^{l_{1}} [b_{m l_{1}}^{0} M_{m l_{1}}^{(1)} (k_{1} r) + a_{m l_{1}}^{0} N_{m l_{1}}^{(1)} (k_{1} r)] \end{array}

The coefficients $a_{m l_{1}}^{0}$ and $b_{m l_{1}}^{0}$ depends on the feature of the special evanescent field. The radial functions used in the expansion of the vector functions for the incident field are the spherical Bessel functions $j_{l} (k_{1} r)$ . The scattered field is expanded as,

\begin{array}{l} E^{s c a} (r) = \sum_{l = 1}^{\infty} \sum_{m = - l}^{l} [e_{m l}^{} M_{m l}^{(3)} (k_{1} r) + f_{m l}^{} N_{m l}^{(3)} (k_{1} r)] \\ H^{s c a} (r) = \frac{k}{i ω μ} \sum_{l = 1}^{\infty} \sum_{m = - l}^{l} [f_{m l}^{} M_{m l}^{(3)} (k_{1} r) + e_{m l}^{} N_{m l}^{(3)} (k_{1} r)] \end{array}

Here the radial functions are the spherical Hankel functions of the first kind $h_{l} (k_{1} r)$ . The interacting field E^int (r) can be a superposition of terms in the expansion [18, 21],

\begin{array}{l} E^{i n t} (r) = \sum_{l_{1} = 1}^{\infty} \sum_{m = - l_{1}}^{l_{1}} [e_{m l_{1}}^{R} M_{m l_{1}}^{(1)} + f_{m l_{1}}^{R} N_{m l_{1}}^{(1)}] \\ H^{i n t} (r) = \frac{k}{i ω μ} \sum_{l_{1} = 1}^{\infty} \sum_{m = - l_{1}}^{l_{1}} [f_{m l_{1}}^{R} M_{m l_{1}}^{(1)} + e_{m l_{1}}^{R} N_{m l_{1}}^{(1)}] \end{array}

Where

[\begin{matrix} e_{m l_{1}}^{R} \\ f_{m l_{1}}^{R} \end{matrix}] = [A_{m l l_{1}}^{}] [\begin{matrix} e_{m l}^{} \\ f_{m l}^{} \end{matrix}]

Here $[A_{m l l_{1}}]$ is the specific expression of reflection matrix A in Eq. (1). The explicit form of the reflection matrix A can be founded in [21].

The T-matrix related to the incident and scattered field coefficients can be calculated in the framework of the null-field method. For an axis-symmetric particle, the equations are uncoupled, and we can pursue a separate solution for each azimuthal mode m. For a fixed azimuthal mode m, we can truncate the expansions according to Eqs. (6) and (7), and derive the following matrix equations,

[\begin{matrix} e_{m l}^{} \\ f_{m l}^{} \end{matrix}] = [T_{m l l_{1}}^{}] ([\begin{matrix} a_{m l_{1}}^{0} \\ b_{m l_{1}}^{0} \end{matrix}] + [\begin{matrix} e_{m l_{1}}^{R} \\ f_{m l_{1}}^{R} \end{matrix}])

Here $m = \bar{- M, M}$ and $l, l_{1} = \bar{1, N}$ , where M is the number of azimuthal modes and N is the truncation index. The explicit expression of the transition matrix $[T_{m l l_{1}}^{}]$ is provided in [27]. Since we know the specific expressions of the T-matrix of particle and the reflection matrix A of the plane surface, the system of equations that determines the scattering field coefficient can be expressed as,

(I - [T_{m l l_{1}}^{}] [A_{m l l_{1}}^{}]) [\begin{matrix} e_{m l}^{} \\ f_{m l}^{} \end{matrix}] = [T_{m l l_{1}}^{}] [\begin{matrix} a_{m l_{1}}^{0} \\ b_{m l_{1}}^{0} \end{matrix}]

This is the explicit expression of Eq. (4). Where I is identity matrix. The total field striking the particle can be expressed as,

\begin{array}{l} E^{t o t} (r) = \sum_{l_{1} = 1}^{\infty} \sum_{m = - l_{1}}^{l_{1}} [a_{m l_{1}}^{} M_{m l_{1}}^{(1)} (k_{1} r) + b_{m l_{1}}^{} N_{m l_{1}}^{(1)} (k_{1} r)] \\ H^{t o t} (r) = \frac{k}{i ω μ} \sum_{l_{1} = 1}^{\infty} \sum_{m = - l_{1}}^{l_{1}} [b_{m l_{1}}^{} M_{m l_{1}}^{(1)} (k_{1} r) + a_{m l_{1}}^{} N_{m l_{1}}^{(1)} (k_{1} r)] \end{array}

Where the expansion coefficients of the total incident field is:

[\begin{matrix} a_{m l}^{} \\ b_{m l}^{} \end{matrix}] = [\begin{matrix} a_{m l_{1}}^{0} \\ b_{m l_{1}}^{0} \end{matrix}] + [A_{m l l_{1}}^{}] [\begin{matrix} e_{m l}^{} \\ f_{m l}^{} \end{matrix}]

We can get the scattering coefficients by solving Eq. (9) directly. In our previous work, the generalized minimum residual method (GMRES) [28] was found to give reliable solutions [25].

2.2. Incident field

Evanescent field produced by the optical waveguide serves as the initial incident field. In the TE mode, the electric field is not in the longitudinal direction but in the transverse direction. The TE evanescent field components in the cover region (z < d) are [29],

\begin{array}{l} E_{y} (x, z) = E_{0}^{s} e^{\sqrt{k_{x}^{2} - k_{1}^{2}} (z - d)} e^{i k_{x} x} \\ H_{z} (x, z) = \frac{- k_{x}}{ω μ} E_{y} \\ H_{x} (x, z) = \frac{i \sqrt{k_{x}^{2} - k_{1}^{2}}}{ω μ} E_{y} \end{array}

In the TM mode, the magnetic field component is not in the longitudinal direction but in the transverse direction. The TM evanescent field components are,

\begin{array}{l} H_{y} (x, z) = E_{0}^{p} e^{\sqrt{k_{x}^{2} - k_{1}^{2}} (z - d)} e^{i k_{x} x} \\ E_{z} (x, z) = \frac{k_{x}}{ω ε} H_{y} \\ H_{x} (x, z) = \frac{- i \sqrt{k_{x}^{2} - k_{1}^{2}}}{ω ε} H_{y} \end{array}

$E_{0}^{s}$ and $E_{0}^{p}$ are amplitude of the evanescent field in the cover region for TE and TM polarization, respectively. $k_{x} = k \cdot n_{e f f}$ and $n_{e f f}$ is the effective refractive index, obtained from the guidance condition of the waveguide.

Evanescent field produced by the optical waveguide serves as the initial incident field. The path to the solution of the expanded coefficients is outlined in [30]. The expanding coefficients of the evanescent field in Eq. (5), $a_{m l_{1}}^{0}$ and $b_{m l_{1}}^{0}$ can be expressed as,

\begin{array}{l} a_{m l_{1}}^{0} = - i^{l_{1}} \frac{2 l_{1} + 1}{l_{1} (l_{1} + 1)} \frac{(l_{1} - m)!}{(l_{1} + m)!} τ_{l_{1}}^{m} (ξ) E_{0}^{s} e^{i d (\sqrt{k_{1}^{2} - k_{x}^{2}} - \sqrt{k_{2}^{2} - k_{x}^{2}})} \\ b_{m l_{1}}^{0} = - n_{1} m i^{l_{1}} \frac{2 l_{1} + 1}{l_{1} (l_{1} + 1)} \frac{(l_{1} - m)!}{(l_{1} + m)!} π_{l_{1}}^{m} (ξ) E_{0}^{s} e^{i d (\sqrt{k_{1}^{2} - k_{x}^{2}} - \sqrt{k_{2}^{2} - k_{x}^{2}})} \end{array}

When the wave is polarized orthogonal to the x-z plane (s-polarized).

\begin{array}{l} a_{m l_{1}}^{0} = - m i^{l_{1}} \frac{2 l_{1} + 1}{l_{1} (l_{1} + 1)} \frac{(l_{1} - m)!}{(l_{1} + m)!} π_{l_{1}}^{m} (ξ) E_{0}^{p} e^{i d (\sqrt{k_{1}^{2} - k_{x}^{2}} - \sqrt{k_{2}^{2} - k_{x}^{2}})} \\ b_{m l_{1}}^{0} = - n_{1} i^{l_{1}} \frac{2 l_{1} + 1}{l_{1} (l_{1} + 1)} \frac{(l_{1} - m)!}{(l_{1} + m)!} τ_{l_{1}}^{m} (ξ) E_{0}^{p} e^{i d (\sqrt{k_{1}^{2} - k_{x}^{2}} - \sqrt{k_{2}^{2} - k_{x}^{2}})} \end{array}

When the wave is polarized parallel to the x-z plane (p-polarized).

Where $τ_{l_{1}}^{m} (ξ) = - \sqrt{1 - ξ^{2}} d P_{l_{1}}^{m} (ξ) / d ξ, π_{l_{1}}^{m} (ξ) = P_{l_{1}}^{m} (ξ) / \sqrt{1 - ξ^{2}}$ , $ξ = \sqrt{1 - n_{e f f}^{2} / n_{1}^{2}}$ .

2.3. Force calculation

Assuming a steady-state condition exists, 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 the Maxwell’s stress tensor $\overset{\leftrightarrow}{T}$ over a surface enclosing the particle [31],

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

After a great deal of algebra and applying numerous recursion, product, and orthogonality relationships among the spherical harmonic functions, Eq. (17) can be integrated directly. The net forces on the particle in non-dimensional form can be expressed as [32],

\begin{array}{l} Q_{x}^{c} + i Q_{y}^{c} = \frac{π}{α^{2}} \sum_{l, m} \frac{1}{2 l + 1} \frac{(l - m)!}{(l + m)!} \times {(l - m) (l + m + 1) Γ_{1} (l, m) + Γ_{2} (l, m) \\ \begin{matrix} \begin{matrix}  \end{matrix} & - i \frac{l (l + 2)}{2 l + 3} \times [(l + m + 1) (l + m + 2) Γ_{2} (l, m) + Γ_{4} (l, m)]} \end{matrix} \\ Q_{z}^{c} = - \frac{2 π}{α^{2}} Re \sum_{l, m} \frac{1}{2 l + 1} \frac{(l - m)!}{(l + m)!} \times {m Γ_{5} (l, m) + i \frac{l (l + 1)}{2 l + 3} (l + m + 1) Γ_{6} (l, m)} \end{array}

Where $α = k_{1} a$ is the size parameter of the sphere, and the non-dimensional radiation forces $Q_{x}^{c}$ , $Q_{y}^{c}$ and $Q_{z}^{c}$ are defined as follows:

Q + i Q_{y}^{c} = \frac{F_{x} + i F_{y}}{ε_{0} E_{0}^{2} a^{2}}, Q_{z}^{c} = \frac{F_{z}}{ε_{0} E_{0}^{2} a^{2}}

Where $E_{0}$ is the field amplitude of the incident evanescent field generated by a waveguide. The parameters $Γ_{j}$ (j = 1-6) in Eqs. (16) are defined as,

\begin{array}{l} Γ_{1} (l, m) = e_{m l} b_{m + 1, l}^{*} + (a_{m l} + 2 e_{m l}) f_{m + 1, l}^{*} \\ Γ_{2} (l, m) = e_{m l}^{*} b_{m - 1, l}^{} + (a_{m l}^{*} + 2 e_{m l}^{*}) f_{m - 1, l}^{} \\ Γ_{3} (l, m) = e_{m l}^{} a_{m + 1, l + 1}^{*} + f_{m l}^{} b_{m + 1, l + 1}^{*} + (a_{m l}^{} + 2 f_{m l}^{}) e_{m + 1, l + 1}^{*} + (b_{m l}^{} + 2 f_{m l}^{}) f_{m + 1, l + 1}^{*} \\ Γ_{4} (l, m) = e_{m l}^{*} a_{m - 1, l + 1}^{} + f_{m l}^{*} b_{m - 1, l + 1}^{} + (a_{m l}^{*} + 2 e_{m l}^{*}) e_{m - 1, l + 1}^{} + (b_{m l}^{*} + 2 f_{m l}^{*}) f_{m - 1, l + 1}^{} \\ Γ_{5} (l, m) = e_{m l} b_{m, l}^{*} + (a_{m l} + 2 e_{m l}) f_{m, l}^{*} \\ Γ_{6} (l, m) = e_{m l}^{} a_{m, l + 1}^{*} + f_{m l}^{} b_{m, l + 1}^{*} + (a_{m l}^{} + 2 e_{m l}^{}) e_{m, l + 1}^{*} + (b_{m l}^{} + 2 f_{m l}^{}) f_{m, l + 1}^{*} \end{array}

3. Numerical calculations and discussions

The schematic structure we simulate is a microsphere immersed in ambient medium trapped by a waveguide, as shown in Fig. 1. The radius and the refractive index of microsphere are denoted by a and n₀, respectively. In addition, n₁ is the refractive index of the ambient medium. The waveguide is made of Ta₂O₅ with refractive index n₂ = 2.1. The waveguide thickness h is optimized to be 0.3 μm, which allows only one propagation mode at λ = 1.064 μm in the waveguide for TE and TM polarization. Thinner waveguide will approach cut-off and thicker waveguide will have less evanescent field. The width of the waveguide is W. The substrate of the waveguide is glass with refractive index n₃ = 1.5. Since the z-axis in our coordinates in Fig. 1 goes downward, by convention, we regard $- Q_{z}^{c}$ as the optical force component along z-direction instead of.

First, we simulate the radiation forces on a polystyrene microspheres (with refractive index n₀ = 1.59) trapped by the waveguide and the ambient medium is water (with refractive index n₁ = 1.33). We define $Δ n = n_{0} - n_{1}$ , which means the refractive index contrast between the sphere and the ambient medium, namely $Δ n = n_{0} - n_{1} = 0.26$ . The distance between the center of the sphere and the surface of the waveguide, denoted by d, is set to be d = a, which means that there is no gap between particle and waveguide. Figures 2(a) and 2(c) show the variation of the optical force components $Q_{x}^{c}$ and $- Q_{z}^{c}$ versus the size parameter $α$ when the incident wave is s-polarized. Figures 2(b) and 2(d) correspond to p-polarized incident wave. The black curves represent the case that multiple scattering effect is ignored, i.e. neglecting the interaction between sphere and plane surface of the waveguide. The red lines show the exact results when the multiple scattering effect is considered by solving Eq. (10) with the GMRES method.

Fig. 2 Plots of the optical force components $Q_{x}^{c}$ and $- Q_{z}^{c}$ versus the size parameter $α$ when the incident waves are s-polarized in (a) and (c), and p-polarized in (b) and (d), respectively. The black curves represent the case that multiple scattering effect is ignored, i.e. neglecting the interaction between sphere and plane surface of the waveguide. The red lines show the exact results when the multiple scattering effect is considered. Parameters used here are n₀ = 1.59, n₁ = 1.33, n₂ = 2.1, n₃ = 1.5, h = 0.3 μm, $λ =1 .064 μm$ and d = a.

Download Full Size | PDF

In Fig. 2, there is no obvious difference between red lines and black lines. It suggests that, with the parameters we used $Δ n = n_{0} - n_{1} = 0.26$ , effect of multiple scattering has little influence on optical forces. So as to accurately measure the effect of multiple scattering to optical forces, we define the difference ratio $ρ = \max | (Q^{1} - Q^{2}) / Q^{1} |$ to describe it, where $Q^{1}$ and $Q^{2}$ are the optical forces with considering multiple scattering effect and without considering multiple scattering effect, respectively. The difference ratio $ρ$ in Figs. 2(a)-2(d) are 0.35, 0.12, 0.05, and 0.1 respectively, which reveals effect of multiple scattering to optical forces is not obvious, i.e. the interaction between sphere and plane surface of the waveguide is weak enough.

As is known, we can increase the optical forces in waveguide trapping by increasing $Δ n$ . In order to investigate the effect of multiple scattering to optical forces when ∆n is increased, it is necessary to increase ∆n. We can increase it in two ways: I. Decrease the refractive index of the ambient medium n₁; II. Increase the refractive index of the ambient medium n₀.

We regard air instead of water to be the ambient medium, namely n₁ = 1. With the decreased n₁, ∆n turns increased, where $Δ n = n_{0} - n_{1} = 0.59$ . Figures 3(a) and 3(c) show the variation of the optical force components $Q_{x}^{c}$ and $- Q_{z}^{c}$ versus the size parameter $α$ when the polarization of incident wave is s-polarized. Figures 3(b) and 3(d) correspond to p-polarized incident wave.

Fig. 3 Plots of the optical force components $Q_{x}^{c}$ and $- Q_{z}^{c}$ versus the size parameter $α$ , when the incident waves are s-polarized in (a) and (c), and p-polarized in (b) and (d), respectively. The black curves represent the case that multiple scattering effect is ignored, i.e. neglecting the interaction between sphere and plane surface of the waveguide. The red lines show the exact results when the multiple scattering effect is considered. Parameter used here is n₁ = 1 and the other parameters are same as that in Fig. 1.

Download Full Size | PDF

The figures suggest that, with ∆n increasing, the optical forces components increase obviously. Compared with figures in Fig. 2, the structure resonance of the sphere excited by evanescent field of the waveguide arises. The difference ratio $ρ$ in Figs. 3(a)-3(d) are 3.1, 2.4, 0.56, and 0.4 respectively. According to them, the difference between red lines and black lines is more obvious than that in Fig. 2. It indicates that, with the parameters we used ∆n = n₀ – n₁ = 0.59, the effect of multiple scattering to optical forces is apparent, i.e. the interaction between sphere and plane surface of the waveguide is strong. So the effect of multiple scattering to optical forces can’t be ignored.

Next we consider waveguide trapping microsphere with high refractive index. Here we choose Ta₂O₅ microsphere (with refractive index n₀ = 2.1) to replace polystyrene microsphere. The ambient medium is water and $Δ n = n_{0} - n_{1} = 0.77$ . Figures 4(a) and 4(c) show the variation of the optical force components $Q_{x}^{c}$ and $- Q_{z}^{c}$ versus the size parameter $α$ when the polarization of incident wave is s-polarized. Figures 4(b) and 4(d) correspond to p-polarized illumination. The difference ratio $ρ$ in Figs. 4(a)-4(d) are 2.6, 1.3, 1.2, and 0.23 respectively. There is strong structure resonance with high quality factor Q. The figures in Fig. 4 are similar to that in Fig. 3.

Fig. 4 Plots of the optical force components $Q_{x}^{c}$ and $- Q_{z}^{c}$ versus the size parameter $α$ when the incident waves are s-polarized in (a) and (c), and p-polarized in (b) and (d), respectively. The black curves represent the case that multiple scattering effect is ignored, i.e. neglecting the interaction between sphere and plane surface of the waveguide. The red lines show the exact results when the multiple scattering effect is considered. Parameter used here is n₀ = 2.1 and the other parameters are the same as that in Fig. 1.

Download Full Size | PDF

So when the refractive index contrast ∆n is large enough, no matter which way to increase ∆n, strong structure resonance appears and the effect of multiple scattering to optical forces becomes greater. We must attach much importance to the effect of multiple scattering to optical forces. Taking this effect into consideration provides us an accurate presentation of optical forces on micro particles trapped by a waveguide.

It is worth to note that, as shown in Fig. 4(c), when size parameter $α > 8$ , the turning point appears where the negative value of $- Q {}_{z}^{c}$ turns to be positive. This transition suggests a turning point from the attractive force between sphere and waveguide to repulsive force between them. It provides us a creative method to separate different particles according to their different sizes.

Figure 5 provides that, when the incident wave is s-polarized, the optical forces components as a function of the normalized distance D, where D is defined as $D = (d - a) / a$ .

Fig. 5 (a) Plots of the optical force components $Q_{x}^{c}$ versus the normalized distance D with s-polarized incident wave. The black curve: ignoring multiple scattering effect and $α = 9.89$ ; the red curve: considering multiple scattering effect and $α = 9.89$ ; the blue curve: ignoring multiple scattering effect and $α = 9.62$ ; the green curve: considering multiple scattering effect and $α = 9.62$ . Inset of Fig. 5(a) shows amplification of the blue and green curves. (b) Plots of the optical force components $- Q_{z}^{c}$ versus the normalized distance D with s-polarized incident wave. The black curve: ignoring multiple scattering effect and $α = 9.86$ ; the red curve: considering multiple scattering effect and $α = 9.86$ ; the blue curve: ignoring multiple scattering effect and $α = 9.24$ ; the green curve: considering multiple scattering effect and $α = 9.24$ . Parameter used here is n₀ = 2.1 and the other parameters are same as that in Fig. 1.

Download Full Size | PDF

According to the different size parameter $α$ , we divide all the curves in Fig. 5(a) into two groups. The first group including the black curve and red curve corresponds to the size parameter $α = 9.89$ , which is the peak of the curve marked in Fig. 4(a). The second group including the blue curve and green curve corresponds to the size parameters $α = 9.62$ , which is the bottom of the curve marked in Fig. 4(a). The optical force $Q_{x}^{c}$ with $α = 9.89$ is much larger than that with $α = 9.62$ . The difference between the curves in the first group is much larger than that in the second group. It means that much larger the optical forces are, much greater influence the multiple scattering has on optical forces. In addition, the difference between curves in each group decreases as the normalized distance increases, which means that, the effect of multiple scattering to optical forces is getting weaker as the microsphere is moving further from the waveguide. The reason is that the evanescent field is concentrated in a small range above the surface of the waveguide, namely the effective operating distance. Within the effective operating distance, the effect of multiple scattering plays an important role in the optical trapping and propulsion of spheres near a waveguide. Figure 5(b) shows the optical forces $- Q_{z}^{c}$ as a function of the normalized distance D. The size parameters are $α = 9.86$ and $α = 9.24$ , which correspond to the peak and bottom of curve noted in Fig. 4(c), respectively. What we have discussed about the Fig. 5(a) can be applied well to Fig. 5(b). Moreover, we can apply the conclusions in Fig. 5 to the case with p-polarized incident wave.

4. Conclusion

In conclusion, we have discussed the effect of multiple scattering to the optical forces on a particle in the evanescent field generated by an optical waveguide. The simulation results show when the refractive index contrast between sphere and ambient medium gets large enough, the effect of multiple scattering is significant. We find that for particles with some size parameters ( $α > 8$ ) and with large refractive contrast (∆n = 0.77) can be repelled from the waveguide, which is key to sort particles of different sizes. Moreover, on the condition that d is within the effective operating distance, the effect of multiple scattering to optical forces is essential to consider.

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).

References and links

1. 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]

2. P. H. Jones, E. Stride, and N. Saffari, “Trapping and manipulation of microscopic bubbles with a scanning optical tweezer,” Appl. Phys. Lett. 89(8), 081113 (2006). [CrossRef]

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

4. L. Bosanac, T. Aabo, P. M. Bendix, and L. B. Oddershede, “Efficient optical trapping and visualization of silver nanoparticles,” Nano Lett. 8(5), 1486–1491 (2008). [CrossRef] [PubMed]

5. S. Kawata and T. Sugiura, “Movement of micrometer-sized particles in the evanescent field of a laser beam,” Opt. Lett. 17(11), 772–774 (1992). [CrossRef] [PubMed]

6. S. Kawata and T. Tani, “Optically driven Mie particles in an evanescent field along a channeled waveguide,” Opt. Lett. 21(21), 1768–1770 (1996). [CrossRef] [PubMed]

7. L. N. Ng, B. J. Luff, M. N. Zervas, and J. S. Wilkinson, “Propulsion of gold nanoparticles on optical waveguides,” Opt. Commun. 208(1-3), 117–124 (2002). [CrossRef]

8. B. S. Ahluwalia, P. McCourt, T. Huser, and O. G. Hellesø, “Optical trapping and propulsion of red blood cells on waveguide surfaces,” Opt. Express 18(20), 21053–21061 (2010). [CrossRef] [PubMed]

9. B. S. Schmidt, A. H. J. Yang, D. Erickson, and M. Lipson, “Optofluidic trapping and transport on solid core waveguides within a microfluidic device,” Opt. Express 15(22), 14322–14334 (2007). [CrossRef] [PubMed]

10. A. H. J. Yang, S. D. Moore, B. S. Schmidt, M. Klug, M. Lipson, and D. Erickson, “Optical manipulation of nanoparticles and biomolecules in sub-wavelength slot waveguides,” Nature 457(7225), 71–75 (2009). [CrossRef] [PubMed]

11. O. G. Hellesø, P. Løvhaugen, A. Z. Subramanian, J. S. Wilkinson, and B. S. Ahluwalia, “Surface transport and stable trapping of particles and cells by an optical waveguide loop,” Lab Chip 12(18), 3436–3440 (2012). [CrossRef] [PubMed]

12. E. Almass and I. Brevik, “Radiation forces on a micrometer-sized sphere in an evanescent field,” J. Opt. Soc. Am. B 12(12), 2429–2438 (1995). [CrossRef]

13. J. Ng and C. T. Chan, “Size-selective optical forces for microspheres using evanescent wave excitation of whispering gallery modes,” Appl. Phys. Lett. 92(25), 251109 (2008). [CrossRef]

14. J. J. Xiao, J. Ng, Z. F. Lin, and C. T. Chan, “Whispering gallery mode enhanced optical force with resonant tunneling excitation in the Kretschmann geometry,” Appl. Phys. Lett. 94(1), 011102 (2009).

15. Y. Li, O. V. Svitelskiy, A. V. Maslov, D. Carnegie, E. Rafailov, and V. N. Astratov, “Giant resonant light forces in microspherical photonics,” Light: Sci. Appl. 2(4), e64 (2013). [CrossRef]

16. H. Y. Jaising and O. G. Hellesø, “Radiation forces on a Mie particle in the evanescent field of an optical waveguide,” Opt. Commun. 246(4-6), 373–383 (2005). [CrossRef]

17. H. Jaising, K. Grujić, and O. G. H. Tomita, “Simulations and Velocity Measurements for a Microparticle in an Evanescent Field,” Opt. Rev. 12(1), 4–6 (2005). [CrossRef]

18. G. Videen, “Light scattering from a sphere on or near a surface,” J. Opt. Soc. Am. A 8, 483–489 (1991). [CrossRef]

19. G. Videen, M. M. Aslan, and M. P. Mengüç, “Characterization of metallic nano-particles via surface wave scattering: A. Theoretical framework and formulation,” J. Quant. Spectrosc. Radiat. Transf. 93(1–3), 195–206 (2005). [CrossRef]

20. M. Aslan, M. P. Mengüç, and G. Videen, “Characterization of metallic nano-particles via surface wave scattering: B. Physical concept and numerical experiments,” J. Quant. Spectrosc. Radiat. Transf. 93(1–3), 207–217 (2005). [CrossRef]

21. T. Wriedt and A. Doicu, “Light scattering from a particle on or near a surface,” Opt. Commun. 152(4–6), 376–384 (1998). [CrossRef]

22. P. A. Bobbert and J. Vlieger, “Light scattering by a sphere on a substrate,” Physica 137A(1-2), 209–242 (1986). [CrossRef]

23. D. W. Mackowski, “A generalization of image theory to predict the interaction of multipole fields with plane surfaces,” J. Quant. Spectrosc. Radiat. Transf. 111(5), 802–809 (2010). [CrossRef]

24. D. W. Mackowski, “Exact solution for the scattering and absorption properties of sphere clusters on a plane surface,” J. Quant. Spectrosc. Radiat. Transf. 109(5), 770–788 (2008). [CrossRef]

25. W. P. Zang, Y. Yang, Z. Y. Zhao, and J. G. Tian, “The effects of multiple scattering to optical forces on a sphere in an evanescent field,” Opt. Express 21(10), 12373–12384 (2013). [CrossRef] [PubMed]

26. J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64(4), 1632–1639 (1988). [CrossRef]

27. A. Doicu, T. Wriedt, and Y. A. Eremin, Light Scattering by Systems of Particles (Springer, 2006).

28. Y. Saad and M. H. Schultz, “GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems,” SIAM J. Sci. Stat. Comput. 7(3), 856–869 (1986). [CrossRef]

29. L. Donald, Lee, Electromagnetic principles of integrated optics (John Wiley, 1986).

30. L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (John Wiley & Sons, 1985).

31. 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]

32. S. Chang, J. T. Kim, J. H. Jo, and S. S. Lee, “Optical force on a sphere caused by the evanescent field of a Gaussian beam; effects of multiple scattering,” Opt. Commun. 139(4–6), 252–261 (1997). [CrossRef]

Effect of multiple scattering to optical forces on a sphere near an optical waveguide

Abstract

1. Introduction

2. Theory and description

2.1. Derivation of scattering fields

2.2. Incident field

2.3. Force calculation

3. Numerical calculations and discussions

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (5)

Equations (20)

Optics Express