Effects of Coating on the Optical Trapping Efficiency of Microspheres

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Effects of Coating on the Optical Trapping Efficiency of Microspheres using Geometrical Optics Approximation Bum Jun Park, and Eric M. Furst Langmuir, Just Accepted Manuscript • DOI: 10.1021/la502632h • Publication Date (Web): 25 Aug 2014 Downloaded from http://pubs.acs.org on August 27, 2014

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Effects of Coating on the Optical Trapping Efficiency of Microspheres using Geometrical Optics Approximation Bum Jun Park1* and Eric M. Furst2 1

Department of Chemical Engineering and Industrial Liaison Research Institute, Kyung Hee

University, Yongin-si, Gyeonggi-do, 446-701, South Korea 2

Department of Chemical and Biomolecular Engineering and Center for Molecular & Engineering

Thermodynamics, University of Delaware, Newark, DE 19716, United States

ABSTRACT: We present the optical trapping forces that are generated when a single laser beam strongly focuses on a coated dielectric microsphere. Based on geometrical optics approximation (GOA), in which a particle intercepts all of the rays that make up a single laser beam, we calculate the trapping forces with varying coating thickness and refractive index values. To increase the optical trapping efficiency, the refractive index (nb) of the coating is selected such that na < nb < nc, where na and nc are the refractive indices of the medium and the core material, respectively. The thickness of the coating also increases the trapping efficiency. Importantly, we find that the trapping forces for the coated particles are predominantly determined by two rays: the incident ray and the first refracted ray to the medium.

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INTRODUCTION Optical trapping with a single laser beam, or optical laser tweezers, has been employed to quantitatively measure interaction forces in biological systems1-5 as well as in colloid physics. 611

This technique has also been used in active microrheology to impart local stresses and induce

shear thinning or thickening.12-14 In the ray optics regime where the size of an object is larger than the wavelength of light (2a >> λ0), it is assumed that the object intercepts all of the rays that make up a single laser beam (i.e., geometrical optics approximation, GOA).15-16,17 In this regime, light rays entering an object bend at the medium-object interface where there is a difference in refractive index. The direction change of the light propagation through the object induces a change in momentum, which leads to generate an optical force in the opposite direction. The resulting force can be separated into two components: a scattering force in the direction of the light propagation and a gradient force in the direction of the spatial light gradient. The gradient force tends to move the object toward the center of the beam focus, generating an optical trap. In order to successfully trap the object, the gradient force must be larger than the scattering force. In practice, these trapping conditions can be achieved by using an objective with a high numerical aperture (NA) and by ensuring an appropriate mismatch in the refractive indices between the object and medium. The efficiency of optical trapping can be improved by coating an object with a material that possesses a refractive index lower than that of the core of the object. For instance, the trapping force of a polystyrene particle coated with silica is increased two or three times compared to that of a bare polystyrene particle.18 This enhanced trapping efficiency is likely due to an increase in the convergence angle of the beam (ϕmax), which is analogous to using a higher NA objective. However, a more systematic study is necessary to optimize the coating conditions in terms of the refractive index and thickness of the coating.


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In this paper, we use GOA to analytically investigate the effects of different coatings on the optical trapping efficiency. We describe the detailed methods for calculations of the optical trapping force based on GOA. In order to verify our calculations, we compare the calculated trapping forces to the experimental Stokes drag force for uncoated particles. We also schematically describe the differences in the forces exerted on the coated and uncoated particles in order to provide direct understanding of the role of the coating in enhancement of the trapping efficiency. Finally, we characterize the contribution of ray propagation for each ray that influences the optical trapping force.

THEORETICAL BASIS

Figure 1. Geometry of single ray propagation incident on a coated microsphere.


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Trapping forces for a single ray. We consider the geometry in which a single ray of power P, incident on a coated dielectric sphere, generates a momentum per unit time, naP/c, where na is the refractive index of the medium, and c is the speed of light (Figure 1).15, 19 The momentum change corresponding to the light propagation produces the force 𝐹 = 𝑄(𝑛𝑎 𝑃/𝑐) , where Q is the dimensionless efficiency. The total force acting on the particle is given by the sum of the contributions of the incident ray with power P, the first reflected ray with power 𝑃𝑅1, and the sequential refracted rays with powers 𝑃(𝑇1 𝑇2 )2 , 𝑃(𝑇1 𝑇2 )2 𝑅2 , … , 𝑃(𝑇1 𝑇2 )2 𝑅2𝑛 , where R and T are the Fresnel reflection and transmission coefficients, respectively, depending on the polarization of the incident ray. The subscripts 1 and 2 denote the medium-coating and coating-core interfaces, respectively (Figure 1). The reflection coefficients Rs and Rp for the perpendicular and parallel polarizations with respect to the incident plane are respectively given by −𝜃 ) 2

sin⁡(𝜃

𝑅𝑠,1 = [sin⁡(𝜃𝑎𝑏 +𝜃𝑎)] , 𝑎𝑏

𝑎

−𝜃 ) 2

sin⁡(𝜃

𝑅𝑠,2 = [sin⁡(𝜃𝑏𝑐+𝜃𝑐)] , 𝑏𝑐

𝑐

−𝜃 ) 2

tan⁡(𝜃

𝑅𝑝,1 = [tan⁡(𝜃𝑎𝑏+𝜃𝑎)] 𝑎𝑏

𝑎

−𝜃 ) 2

tan⁡(𝜃

𝑅𝑝,2 = [tan⁡(𝜃𝑏𝑐+𝜃𝑐)] . 𝑏𝑐

𝑐

(1)

Note that the difference between Rs and Rp becomes negligible when the incident angle is sufficiently small (e.g., a ray entering nearly normal to the surface). The transmission coefficients in each case are given by Ts = 1 - Rs and Tp = 1 - Rp. The two refracted angles, θab into the coating region and θc into the core, are obtained from the incident angle θa and the subsequent incident angle θbc (Figure 1) using Snell's law, 𝑛𝑏 sin 𝜃𝑎𝑏 = 𝑛𝑎 sin 𝜃𝑎 , and 𝑛𝑐 sin 𝜃𝑐 = 𝑛𝑏 sin 𝜃𝑏𝑐 . The subscripts a, b, and c denote the medium, coating, and core, respectively. Based on a geometric


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relation, 𝜃𝑎𝑏 is given by sin 𝜃𝑎𝑏 =

𝑎0 −𝛿 𝑎0

sin 𝜃𝑏𝑐 , where a0 and δ indicate the radius of the core and

thickness of the coating, respectively. As shown in Figure 1, based on geometric relations, the first reflected ray with power PR1 and the successive refracted rays (n = 0, 1, 2,…) to the medium have angles 𝜋 + 2𝜃𝑎 , 𝛼, 𝛼 + 𝛽, ⋯ , 𝛼 + 𝑛𝛽 with respect to the incident ray, where 𝛼 = 2(𝜑 − 𝜃𝑐 ), 𝛽 = 𝜋 − 2𝜃𝑐 , and 𝜑 = 𝜃𝑎 − 𝜃𝑎𝑏 + 𝜃𝑏𝑐 . The scattering and gradient components of the dimensionless efficiency, Qs and Qg, which are parallel and perpendicular to the single incident ray, respectively, are given by 𝑛 𝑄𝑠 = 1 − [𝑅1 cos(𝜋 + 2𝜃𝑎 ) + (𝑇1 𝑇2 )2 ∑∞ 𝑛=0 𝑅2 cos⁡(𝛼 + 𝑛𝛽)]

𝑛 𝑄𝑔 = −[𝑅1 sin(𝜋 + 2𝜃𝑎 ) + (𝑇1 𝑇2 )2 ∑∞ 𝑛=0 𝑅2 sin⁡(𝛼 + 𝑛𝛽)].

(2)

Considering the forces in the complex plane, the total efficiency is 1

𝑄𝑡𝑜𝑡 = 𝑄𝑠 + 𝑖𝑄𝑔 = 1 + 𝑅1 cos 2θ𝑎 + 𝑖𝑅1 sin 2𝜃𝑎 − 𝑒 𝑖𝛼 (1−𝑅

2𝑒

𝑖𝛽

).

This is once again separated into two components:

𝑄𝑠 = 1 + 𝑅1 cos 2𝜃𝑎 − 𝑄𝑔 = 𝑅1 sin 2𝜃𝑎 −

(𝑇1 𝑇2 )2 [cos(2𝜑−2𝜃𝑐 )+𝑅2 cos 2𝜑] 1+𝑅22 +2𝑅2 cos 2𝜃𝑐

(𝑇1 𝑇2 )2 [sin(2𝜑−2𝜃𝑐 )+𝑅2 sin 2𝜑] 1+𝑅22 +2𝑅2 cos 2𝜃𝑐

.

(3)

Note that eq 3 shows the full equations of the dimensionless efficiencies for the scattering and gradient forces of a single ray incident on a coated particle. If the incident rays are nearly normal to the surface (θa ≈ 0 and R1 ≈ 0), the term of the first reflection with power PR1 at the mediumcoating interface becomes negligible such that


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𝑄𝑠 = 1 −

𝑄𝑔 = −

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⁡.

(4)

Note that the dimensionless efficiency for an uncoated particle is given by15

𝑄𝑠 = 1 + 𝑅 cos 2𝜃𝑎 −

𝑄𝑔 = 𝑅 sin 2𝜃𝑎 −

𝑇 2 [cos(2𝜃𝑎 −2𝜃𝑏 )+𝑅 cos 2𝜃𝑎 ] 1+𝑅 2 +2𝑅 cos 2𝜃𝑏

𝑇 2 [sin(2𝜃𝑎 −2𝜃𝑏 )+𝑅 sin 2𝜃𝑎 ] 1+𝑅 2 +2𝑅 cos 2𝜃𝑏

,

(5)

where θa and θb are the incident angle and refracted angle at the surface of the uncoated particle, respectively. In the absence of coating, φ and θc in eq 3 correspond to θa and θb in eq 5, respectively, and therefore, eqs 3 and 5 are identical. A portion of the ray refracted into the coating region with power 𝑃𝑇1 is reflected with power 𝑃𝑇1 𝑅2 ⁡at the coating-core interface. This is subsequently refracted through the medium with power 𝑃𝑇12 𝑅2 at an angle of 𝜃′𝑎 = 𝜋 + 2𝜑. Considering this reflection effect in the coating region, eq 3 becomes 𝑄𝑠 = 1 + 𝑅1 cos 2𝜃𝑎 + 𝑇12 𝑅2 cos 2𝜑 −

𝑄𝑔 = 𝑅1 sin 2𝜃𝑎 + 𝑇12 𝑅2 sin 2𝜑 −



.

(6)

However, this effect is expected to be negligible due to the fact that the incident angle is nearly normal to the particle surface.


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Figure 2. Schematics for integrating over all rays incident on a microsphere.

Integration over all incident rays. In order to calculate the total force (i.e., the lateral trapping force and the radiation force) exerted on a microsphere by a highly focused laser beam, the forces generated by individual rays must be integrated over all incident rays. As shown in Figure 2a, we consider a beam focused on point B along the Y-axis, located Δy away from the center of sphere O.15 An arbitrary ray from D hits the sphere at E with an incident angle θa and forms an angle ϕ with the beam axis AB (or Z-axis), which is the half-angle of the light cone. The maximum value of ϕ is determined by 𝑁𝐴 = 𝑛𝑎 sin 𝜙𝑚𝑎𝑥 , where NA is the numerical aperture of an objective lens, and na is the refractive index of the medium surrounding the lens. AD forms an azimuthal angle β' with the Y-axis.


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In the vertical plane ABCD (Figure 2b), the incident ray (the diagonal BD) forms a polar 𝜋

angle of α′ = 2 − 𝜙 with respect to BC. The distances from E to the two perpendicular lines, AB and BC, are 𝑟 ′ = 𝑘cos⁡𝛼′ and 𝑧 = 𝑘sin⁡𝛼′, respectively, where k is the distance between B and E, given by 𝑘 = 𝑎(cos 𝜃𝑎 + cot 𝛾 ′ sin 𝜃𝑎 ). The incident plane OBE in Figure 2c is defined by a vector 𝑛⃗ normal to the sphere with |𝑛⃗| = OE = a and the incident ray. The Y-axis forms an angle γ' with the ray, leading to the two geometric relations cos 𝛾′ = cos 𝛼′ cos 𝛽′ and 𝑎 sin 𝜃𝑎 = Δ𝑦 sin 𝛾′. Finally, the scattering (Qs) and gradient (Qg) forces are defined by the parallel and perpendicular components of the first reflected and successive refracted rays, respectively, with respect to the incident ray (Figure 2c). The trapping force is obtained by integrating with respect to the surface area (S) at which the rays enter the sphere, given by 𝐹 = ∫𝑆 𝐼(𝑟 ′ , 𝑧)𝑄

𝑛1 𝑐

𝑑𝐴. Here, I(r', z) is the intensity of the 2𝑃

Gaussian beam at radial distance (r') from the beam axis, 𝐼(𝑟 ′ , 𝑧) = 𝜋𝜔2 exp

−2𝑟′2 𝜔2

, where P is the

laser power delivered to the sphere.20 The beam waist ω(z) of a Gaussian beam is defined as a 1

𝜆0 𝑧 2 2

2

radius of the 1/e irradiance contour at a distance z of light propagation 𝜔(𝑧) = 𝜔0 [1 + (𝜋𝜔2 ) ] , 0

where λ0 is the vacuum wavelength of the light.21 ω0 is the radius of the 1/e2 irradiance contour at 𝜆

the plane in which the wavefront is flat and is determined by 𝜔0 = 𝑁𝐴𝑛0 𝜋. The new Cartesian 𝑚

coordinate (X', Y', Z') with origin E is specified by the radius k, the azimuthal angle β', and the polar angle α' in the spherical coordinates such that 𝑋 ′ = 𝑘cos⁡α′ sin 𝛽 ′ , 𝑌 ′ = 𝑘cos⁡α′ cos 𝛽 ′ , and 𝑍 ′ = 𝑘sin⁡α′ . Using these relations, the surface integral is parameterized with respect to α' and β': 𝐹 = ∫𝑆 𝐼(𝑟 ′ , 𝑧)𝑄

𝑛1 𝑐

𝑑𝐴 = ∬ 𝐼(𝑟 ′ , 𝑧)𝑄

𝑛1 𝑐

⃗ 𝛼′ × 𝑇 ⃗ 𝛽′ |𝑑𝛼′𝑑𝛽′, |𝑇

(7)


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𝜕𝑌′ ⃗⃗⃗ 𝜕𝑍′ ⃗⃗⃗ ⃗ 𝛼′ and 𝑇 ⃗ 𝛽′ are tangent vectors given by 𝑇 ⃗ 𝛼′ = 𝜕𝑋′ ⃗⃗⃗ ⃗ 𝛽′ = 𝜕𝑋′ ⃗⃗⃗ where 𝑇 𝑋′ + 𝑌′ + 𝑍′ and 𝑇 𝑋′ + 𝜕𝛼′ 𝜕𝛼′ 𝜕𝛼′ 𝜕𝛽′ 𝜕𝑌′ ⃗⃗⃗

𝜕𝛽′

𝜕𝑍′

⃗⃗⃗ , respectively. Their cross product is 𝑌′ + 𝜕𝛽′ 𝑍′

𝜕(𝑋 ′ ,⁡𝑌 ′ )

2

𝜕(𝑌 ′ ,⁡𝑍 ′ )

2

1

𝜕(𝑍 ′ ,⁡𝑋 ′ )

2 2

⃗ 𝛼′ × 𝑇 ⃗ 𝛽′ | = [( ′ ′ ) + ( ′ ′ ) + ( ′ ′ ) ] , |𝑇 𝜕(𝛼 ,⁡𝛽 ) 𝜕(𝛼 ,⁡𝛽 ) 𝜕(𝛼 ,⁡𝛽 ) where each term inside the bracket on the right side of the equation represents the Jacobian given by the formula 𝜕𝑋′ 𝜕(𝑋 ′ ,⁡𝑌 ′ ) 𝜕(𝛼′ ,⁡𝛽 ′ )

=

𝜕𝛼′ |𝜕𝑋′ 𝜕𝛽′

𝜕𝑌′ 𝜕𝛼′ 𝜕𝑌′ |. 𝜕𝛽′

The boundary conditions for integrating eq 7 are 0 < β' < 2π and α'min < α' < π/2, where 𝜋

𝛼′𝑚𝑖𝑛 = 2 − 𝜙𝑚𝑎𝑥 . Using these boundary conditions, the integrals are 𝜋

𝜋

𝐹𝑌 = 2 ∫0 ∫𝛼′2

𝑚𝑖𝑛

𝜋

𝜋

𝐹𝑍 = 2 ∫0 ∫𝛼′2

𝑚𝑖𝑛

𝐼(𝑟 ′ , 𝑧)(𝑄𝑔 sin 𝛾 ′ − 𝑄𝑠 cos 𝛾 ′ )

𝑛1

𝐼(𝑟 ′ , 𝑧)(𝑄𝑠 sin 𝛾 ′ − 𝑄𝑔 cos 𝛾 ′ )

𝑛1

𝑐

𝑐

𝑘 2 cos 𝛼′ 𝑑𝛼′ 𝑑𝛽′

𝑘 2 cos 𝛼′ 𝑑𝛼′ 𝑑𝛽′,

(8)

where FY and FZ indicate the lateral trapping force and the radiation force, perpendicular and parallel to the beam axis (Z-axis), respectively. Based on these methods, we numerically calculate the lateral trapping forces and the radiation forces generated by a laser beam strongly focused on a particle. Before discussing the results, we review the experimental materials and methods used to compare the calculations.

EXPERIMENTAL METHODS


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The laser tweezers apparatus is constructed around an inverted microscope.9 The laser is generated by a 4-W CW Nd:YAG laser (λ0 = 1064 nm, Coherent Compass 1064-4000M). A water immersion objective (63×, NA 1.2, Zeiss C-Apochromat) is used to trap a microsphere, allowing a working distance of approximately 200-300 μm. A second objective (20×, NA 0.45, Zeiss 𝑇

(𝜆 )

Achroplan) is used to calibrate the actual laser power delivered to the sphere, 𝑃63× = 𝑃20× 𝑇63×(𝜆0 ). 20×

0

P20× is directly measured using an optical power meter (Newport, 1815-c) from the 20× objective, and the transmittances T63× and T20× can be measured using a dual objective technique (i.e., T63× ≈ 30% and T20× ≈ 80% for λ0 = 1064 nm).19-20, 22-24 For a detailed description of the laser tweezers setup, we refer the reader to the reference.9 We use two kinds of particles: silica (SiOx, a = 1.35 μm, Polysciences, Inc.) and polystyrene (PS, a = 1.4, 1.7, and 3.7 μm, Interfacial Dynamics Corporation). The size of the particles is measured with a scanning electron microscope (SEM, JSM-7400F). To obtain the trap stiffness, which is the ratio of the Stokes drag force FS to the lateral displacement Δy, a particle in the aqueous phase is held and subjected to drag forces by translating the microscope stage at several constant velocities between u = 3 and 7 μm/s.9, 19-20 The value of Δy from the equilibrium position in the optical trap is measured in response to the Stokes force 𝐹𝑆 = 6𝜋𝑎𝜂𝑢, where η is the viscosity of the aqueous solution.

RESULTS AND DISCUSSION Uncoated particles. We validate our calculations for uncoated particles before proceeding to coated particles. In the ray optics regime, assuming that all rays enter a particle with an incident angle nearly normal to the surface (θa ≈ 0°), the gradient force FY in the radial Y-direction and the


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radiation force FZ in the axial Z-direction have relations with the refractive indices of the particle (np) and medium (nm)18 𝐹𝑌 ~

𝑛𝑝 −𝑛𝑚 𝑛𝑝

𝑛𝑝 −𝑛𝑚

⁡,⁡⁡⁡⁡⁡𝐹𝑍 ~ (𝑛

𝑝 +𝑛𝑚

2

) .

(9)

Based on these relations, we examine the dependence of the trapping forces on the refractive index of the particles. As shown in Figure 3, we calculate the lateral trapping forces (FY) and the vertical radiation forces (FZ) of a particle by varying its refractive index (np) using eqs 5 and 8. These calculation results, normalized by the factors of eq 9, are consistent with one another for the different values of np, demonstrating the minor effect of the assumption of θa ≈ 0° on the trapping force calculations and validity of the calculations. Nevertheless, in our optical trapping force calculations, the incident angle (θa) is varied, depending on the geometric parameters, as described previously.

Figure 3. Effect of the refractive index np of the particles on (a) FY and (b) FZ for an uncoated particle with a = 1.5 μm. The refractive index of the medium is nm = 1.326. The laser power for calculations is P = 6.75 mW.

The Stokes drag forces of uncoated PS and SiOx particles are compared with the lateral trapping forces (FY) calculations. As shown in Figure 4a, both the experimental results for PS and


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SiOx particles show good agreement with the calculations using eqs 5 and 8, and the force exerted on the PS particle is approximately twice that exerted on the SiOx particle. Notably, the calculated lateral trapping forces slightly deviate from the measured trapping forces. This small deviation can be attributed to out-of-plane displacement of the particle during dragging. It has been previously reported that such deviation decreases when the particle is trapped and dragged at the oil-water interface.19 The pinning of the particle at the interface leads to reduction of the out-of-plane displacement of the particle due to the high energetic penalty of deforming the interface. We also find that, for the SiOx particle, the calculated radiation forces FZ in the axial direction are significantly smaller than those of PS (Figure 4b). These results show that our optical trapping force calculations are justified.

Figure 4. Comparison of drag experiments and calculations for (a) FY and (b) FZ. The refractive indices of the aqueous solution, polystyrene (PS, a = 1.4 μm), and silica (SiOx, a = 1.35 μm) particles are nm = 1.326, np = 1.57, and 1.45, respectively. The calibrated laser power is P63× = 6.75 mW.

Coated particles. Optical trapping forces for coated particles are numerically integrated over all incident rays using eqs 3 and 8. Lateral trapping forces and radiation forces in an aqueous solution


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(refractive index na = 1.326) are calculated with different refractive indices in the coated region nb, as shown in Figure 5. Note that the maximum value of Δy is smaller than a0/2 for all coated particle calculations such that the laser only focuses on the core region. It is found that the lateral trapping force for the coated particle increases as nb decreases. However, when nb is larger than nc = 1.57 (core PS), the trapping force becomes weaker than that of uncoated PS (cf. diamond symbols in Figure 5a). Additionally, the radiation force decreases when nb is in the range of na < nb < nc but increases when nb ≈ na or nb ≈ nc.

Figure 5. Effect of the refractive index of the coating (nb) on (a) FY and (b) FZ for coated polystyrene particles. The radius of the PS core is a0 = 1 μm, and the coating thickness is δ = 0.5 μm. The refractive indices of the medium, PS, and SiOx are 1.326, 1.57, and 1.45, respectively. For comparison, the solid and dashed lines indicate uncoated PS and SiOx with a = 1.5 μm, respectively. The laser power is P = 6.75 mW.


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Figure 6. Effect of the coating thickness (δ) on (a) FY and (b) FZ for SiOx-coated polystyrene particles with the radius of the entire particle, a = a0 + δ = 1.5 μm. The refractive indices of the medium, PS, and SiOx are 1.326, 1.57, and 1.45, respectively. For comparison, the solid and dashed lines indicate uncoated PS and SiOx with a = 1.5 μm, respectively. The laser power is P = 6.75 mW.

The trapping forces also depend on the thickness of coating δ. The value of the core radius (a0) is changed, but the total particle radius (a = a0 + δ) is kept constant in order to eliminate any size dependence on the trapping force.25-27 As shown in Figure 6a, the lateral trapping force (FY) for SiOx-coated polystyrene particles increases with δ, compared to those of the uncoated PS and SiOx particles. It is interesting that FY increases with δ as the relative portion of SiOx in the particle increases, whereas FY for the uncoated SiOx particle is the weakest. This suggests that a better trapping efficiency can be achieved when the rays pass the core region with a higher refractive index than the coating. Additionally, it is also found that the coating effectively decreases the radiation force compared to the radiation force of the uncoated PS particle (Figure 6b).

Figure 7. Effect of ray propagations on the trapping forces. (a) Effect of the second reflection of the ray at the coating-core interface (eq 3 versus eq 6). (b) Effect of the first reflected ray and the number of successive refracted rays n (eq 3 versus eq 4 and eq 2). The core radius and the coating thickness are a0 = 1 μm and δ = 0.5 μm, respectively. The laser power is P = 6.75 mW.


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We also examine the effect of ray propagations through the boundaries of a coated particle. A portion of the incident ray is refracted through the coating region and is subsequently reflected at the coating-core boundary (Figure 1). This reflected ray is refracted through the outermost boundary of the particle with power 𝑃𝑇12 𝑅2 . The dimensionless efficiencies of the trapping force in eq 6 account for the effect of this reflected ray, while eq 3 excludes this effect. However, figure 7a shows that these results are consistent with one another, suggesting that the second reflection at the coating-core interface does not significantly affect the trapping force. The trapping forces obtained from eq 4 (which excludes the first reflected ray term) with power PR1 at the medium-coating boundary show good agreement with those from eq 3 (which include all propagations of rays, Figure 7b). As mentioned earlier, this demonstrates that the incident rays hit the particle with an incident angle of θa ≈ 0 such that R1 ≈ 0. Moreover, the effect of successively refractive rays when n > 0 in eq 3 (see Figure 1) is also negligible. As shown in Figure 7b, the trapping forces calculated from eq 2 at n = 0, 1, 2 are in accord with those obtained from eq 3. These results indicate that the trapping forces for coated particles are predominantly determined by two rays: the incident ray with power P and the first refracted ray with power 𝑃𝑇12 𝑇22 (n = 0) to the medium. Notably, similar results for uncoated particles have been reported that the trapping force exclusively depends on the incident ray and the first refracted ray to the medium.19

Schematic description. The optical trapping efficiency for a coated particle can be schematically described by considering the two rays incident on a particle and their corresponding first refracted rays to the medium.15, 20 As shown in Figure 8, the incident rays with momenta 𝑃⃗1,𝑖𝑛


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and 𝑃⃗2,𝑖𝑛 are focused on a point that is laterally displaced as much as Δy from the center of particle. The inner and outer circles indicate the medium-coating and coating-core interfaces, respectively. Based on Snell's law, the propagation of rays in Figure 8 is drawn with an accuracy ± 0.5°. Values of na = 1.326, nb = 1.45, and nc = 1.57 are used for the refractive indices of the medium, coating, and core, respectively. The incident rays are sequentially transmitted to the particle and to the medium, where the refracted rays in the latter case have momenta 𝑃⃗𝐴1,𝑜𝑢𝑡 and 𝑃⃗𝐴2,𝑜𝑢𝑡 . The solid dots indicate the locations at which the rays hit. For comparison, the blue lines show the pathways of rays through an uncoated particle with a refractive index nc, in which the momenta of their refracted rays to the medium are 𝑃⃗𝐵1,𝑜𝑢𝑡 and 𝑃⃗𝐵2,𝑜𝑢𝑡 . For the coated particle, the momenta changes (Δ𝑃⃗𝐴1 , Δ𝑃⃗𝐴2 ) resulting from two sets of incident and refracted rays create counter-forces (𝐹𝐴1 , 𝐹𝐴2 ) that act on the particle (Figure 8). The Y-direction component 𝐹𝐴,𝑌 of their vector sum produces a gradient force (or lateral trapping force) that moves the particle toward the beam focus. The angle between 𝑃⃗1,𝑖𝑛 and 𝑃⃗𝐴1,𝑜𝑢𝑡 is identical to the value of α in Figure 1 and is given by 𝛼 = 2(𝜑 − 𝜃𝑐 ) and 𝜑 = 𝜃𝑎 − 𝜃𝑎𝑏 + 𝜃𝑏𝑐 . Similarly, for the uncoated particle, the momenta changes (Δ𝑃⃗𝐵1 , Δ𝑃⃗𝐵2 ) induce a relatively smaller gradient force 𝐹𝐵,𝑌 (Figure 8), due to the smaller angle difference 2(𝜃𝑎 − 𝜃𝑎𝑏 ) between 𝑃⃗1,𝑖𝑛 and 𝑃⃗𝐵1,𝑜𝑢𝑡 compared to α (i.e., 𝜃𝑏𝑐 > 𝜃𝑐 ). This verifies that the trap stiffness in the lateral direction increases when a coating with a refractive index intermediate to those of the medium and core is used.


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Figure 8. Schematic comparison of lateral trapping forces between coated and uncoated particles.

Similarly, the effect of the coating thickness on the trapping force can be schematically described. As shown in Figure 9, the two incident rays enter the coated particle with δ or δ', where δ' > δ and a0' < a0. When the incident rays refracted to the coating region hit the coating-core ′ interface (solid dots in Figure 9), the corresponding angle (𝜃𝑏𝑐 and 𝜃𝑏𝑐 in Figure 9) with respect

to the vector normal to the coating-core interface increases as the core size decreases (or the coating 𝑎′

𝑎

′ thickness increases) due to the geometric relation, sin 𝜃0 ′ = sin 𝜃0 , such that 𝜃𝑏𝑐 > 𝜃𝑏𝑐 . In this case, 𝑏𝑐

𝑏𝑐

the angle between the incident ray and the first refracted ray increases as the core thickness increases (𝛼 ′ > 𝛼 in Figure 9), resulting in stronger trapping force of the particle with δ'.


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Figure 9. Schematic description of the lateral trapping force depending on the coating thickness.

CONCLUSIONS In this work, we have shown that the optical trapping force for coated microspheres depends on the refractive index and thickness of the coating. In general, optimal trapping conditions can be achieved when the refractive index of the coating is higher than that of the medium but lower than that of the core. The trapping efficiency can also be improved by increasing the coating thickness when the laser beam is focused on the core region. Importantly, the trapping forces are predominantly determined by the incident ray and the first refracted ray to the medium. This result suggests that the vector components of the two rays and the summation of the vector properties over all incident rays enable to numerically calculate the trapping forces. This is potentially useful for estimating the trapping forces of non-spherical objects. The comparison of experimental results with the optical trapping force calculations for coated particles with various geometries and materials warrants future investigation.


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Corresponding Author *E-mail: [email protected]. Notes The authors declare no competing financial interest. ACKNOWLEDGMENT This work is supported by a grant from the Kyung Hee University in 2013 (KHU-20130387) and Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (2014R1A1A1005727).

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14. Sriram, I.; Meyer, A.; Furst, E. M. Active microrheology of a colloidal suspension in the direct collision limit. Phys. Fluids 2010, 22 (6), 062003. 15. Ashkin, A. Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime. Biophys. J. 1992, 61 (2), 569-582. 16. Walz, J. Y.; Prieve, D. C. Prediction and measurement of the optical trapping forces on a microscopic dielectric sphere. Langmuir 1992, 8 (12), 3073-3082. 17. Note that Walz and Prieve studied the optical trapping forces for the case when the spot size of the laser beam is larger than the particle size (see ref. 16). 18. Bormuth, V.; Jannasch, A.; Ander, M.; van Kats, C. M.; van Blaaderen, A.; Howard, J.; Schńffer, E. Optical trapping of coated microspheres. Opt. Express 2008, 16 (18), 13831-13844. 19. Park, B. J.; Furst, E. M. Optical Trapping Forces for Colloids at the Oil− Water Interface. Langmuir 2008, 24 (23), 13383-13392. 20. Neuman, K. C.; Block, S. M. Optical trapping. Rev. Sci. Instrum. 2004, 75 (9), 27872809. 21. Griot, M. Optics Guide 4; Melles Griot: Irvine CA, 1988. 22. Misawa, H.; Koshioka, M.; Sasaki, K.; Kitamura, N.; Masuhara, H. Three-dimensional optical trapping and laser ablation of a single polymer latex particle in water. J. Appl. Phys. 1991, 70 (7), 3829-3836. 23. Viana, N. B.; Rocha, M. S.; Mesquita, O. N.; Mazolli, A.; Maia Neto, P. A. Characterization of objective transmittance for optical tweezers. Appl. Opt. 2006, 45 (18), 42634269. 24. Neuman, K. C.; Chadd, E. H.; Liou, G. F.; Bergman, K.; Block, S. M. Characterization of Photodamage to Escherichia coli in Optical Traps. Biophys. J 1999, 77 (5), 2856-2863. 25. Simmons, R. M.; Finer, J. T.; Chu, S.; Spudich, J. A. Quantitative measurements of force and displacement using an optical trap. Biophys. J. 1996, 70 (4), 1813. 26. Mazolli, A.; Neto, M.; Nussenzveig, H. Theory of trapping forces in optical tweezers. Proc. R. Soc. London. Ser. A 2003, 459 (2040), 3021-3041. 27. Park, B.; Furst, E. Optical trapping forces depending on size of dielectric polystyrene microspheres. Macromol. Res. 2013, 21 (11), 1167-1170.


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Table of Contents


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Effects of Coating on the Optical Trapping Efficiency of Microspheres

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