## Abstract

Microparticles can be trapped and propelled by the evanescent field of optical waveguides. As the evanescent field only stretches $100\u2013200\text{\hspace{0.17em}}\mathrm{nm}$ from the surface of the waveguide, only the lower caps of the microparticles interact directly with the field. This is taken advantage of by trapping hollow glass spheres on waveguides in the same way as solid glass spheres. For the chosen waveguide, numerical simulations show that hollow microspheres with a shell thickness above $60\text{\hspace{0.17em}}\mathrm{nm}$ can be stably trapped, while spheres with thinner shells are repelled. The average refractive index of the sphere–field intersection volume is used to explain the result in a qualitative way.

© 2011 Optical Society of America

Hollow microspheres consist of a thin glass shell encapsulating air. Optical trapping of hollow microspheres using free-space beams has been the subject of several previous studies [1, 2, 3, 4, 5, 6, 7]. As hollow microspheres are repelled away from Gaussian beams [2, 3], they must be trapped with other types of beams, for example, vortex beams [2, 3], higher order Bessel beams [4], self-imaged bottle beams [5], or circularly scanning laser beams [6]. These trapping techniques using such beams are based on repulsive optical forces where the sphere is repelled from the annular bright region and trapped in the central dark region.

Trapping of microparticles is also feasible using the evanescent field of optical waveguides [7, 8, 9, 10]. Waveguide trapping has demonstrated attractive optical forces experimentally, while repulsive optical forces from a waveguide only have been theoretically predicted for whispering gallery mode resonances [11, 12]. In this Letter, the evanescent field of a tantalum pentoxide waveguide [9, 10] is used for optical trapping and propulsion of hollow spheres. In particular, we investigate how the shell thickness influences the attraction and repulsion forces. The evanescent field extends about $150\text{\hspace{0.17em}}\mathrm{nm}$ into the cover medium (water) such that only the lower caps of the spheres interact with the evanescent field. Thus, the optical forces should depend on the average refractive index of the lower cap. This is different from trapping with propagating beams, where the forces depend on the average refractive index of the whole sphere. Numerical simulations and simple analytical expressions for the average refractive index and density of the spheres are used to get a quantitative and qualitative understanding of the different trapping regimes.

Simulations of a hollow sphere on a waveguide surface were performed with a commercial finite element method (FEM) software (Comsol Multiphysics 3.5a). A three-dimensional (3D) model was employed to compute the optical forces using Maxwell’s stress tensor [10, 13]. The simulation parameters correspond to the experimental values described in the next section, with waveguide, water, and glass refractive indices of 2.1, 1.33, and 1.5, respectively, and waveguide width and thickness of $3\text{\hspace{0.17em}}\mathrm{\mu m}$ and $200\text{\hspace{0.17em}}\mathrm{nm}$, respectively. The optical field is TE polarized with wavelength $1070\text{\hspace{0.17em}}\mathrm{nm}$. The spheres’ shell thicknesses and radii were chosen independent of the spheres used in the experiment. Figure 1 shows a cross section of the 3D model, with the simulated waveguide field and a hollow sphere of diameter $1\text{\hspace{0.17em}}\mathrm{\mu m}$ on the waveguide surface. We see how only the lower cap of the sphere interacts with the evanescent field. The simulated evanescent field decayed to 1/e at a distance $h=150\text{\hspace{0.17em}}\mathrm{nm}$ from the surface. The distance *h* depends predominantly on the refractive indices of the waveguide and the immersion medium, and weakly on the width of the waveguide.

The calculated optical forces (gradient and propagation forces) on a hollow sphere for a guided power of $1\text{\hspace{0.17em}}\mathrm{W}$ are shown in Fig. 2 for a radius $r=1.5\text{\hspace{0.17em}}\mathrm{\mu m}$ as a function of the shell thickness *t* [13]. For spheres with $t>60\text{\hspace{0.17em}}\mathrm{nm}$, the gradient force (${F}_{x}$) is negative, pulling the spheres toward the waveguide. The propagation force (${F}_{z}$) propels the hollow sphere along the waveguide. Thus, spheres where $t>60\text{\hspace{0.17em}}\mathrm{nm}$ are trapped and propelled in the same way as solid spheres. The propagation force increases gradually with the shell thickness and is maximum for solid spheres. For spheres where $500\text{\hspace{0.17em}}\mathrm{nm}<t<1500\text{\hspace{0.17em}}\mathrm{nm}$, ${F}_{z}$ increases very slowly, as the cap volume interacting with the evanescent field remains constant. For spheres where $t<60\text{\hspace{0.17em}}\mathrm{nm}$, ${F}_{x}$ is positive, repelling spheres vertically away from the waveguide surface. Consequently, spheres with $t<60\text{\hspace{0.17em}}\mathrm{nm}$ cannot be trapped on the waveguide.

A simple analytical model can be used to explain the simulated gradient force results. The average refractive index of the hollow sphere (${n}_{\text{hollow}}$) depends on the refractive indices of glass (${n}_{\text{glass}}$) and air (${n}_{\text{air}}$) and the volumes of the sphere (${V}_{\text{sphere}}$) and the encapsulated air (${V}_{\text{air}}$):

*h*. For radii $r<h/2$, the whole sphere interacts with the field. For $r\ge h/2$, the volume ${V}_{\text{cap}}$ of the field–sphere intersection is We can now define the average refractive index of the cap (${n}_{\text{cap}}$):

*r*by $r-t$ and

*h*by $h-t$ in Eq. (2): Our hypothesis is that waveguides will attract hollow spheres for all ${n}_{\text{cap}}>{n}_{w}$ (refractive index of water), independently of ${n}_{\text{hollow}}$. For ${n}_{\text{cap}}<{n}_{w}$, a hollow sphere should be repelled away from the waveguide surface. Gaussian beams, which interact with the whole sphere, will trap the sphere for which ${n}_{\text{hollow}}>{n}_{w}$. For the transition where ${n}_{\text{hollow}}={n}_{w}=1.33$, the gradient force ${F}_{x}=0$. This condition is used in Eq. (1) to study how Gaussian beam traps depend on

*r*and

*t*. Similarly, by letting ${n}_{\text{cap}}={n}_{w}=1.33$ in Eq. (3), the waveguide trap transition points, where ${F}_{x}$ is zero, are found. Finally, ${\rho}_{\text{hollow}}={\rho}_{\text{water}}$ determines the transition points between floating and sinking spheres. Figures 3a, 3b show how the transitions depend on

*r*and

*t*and show the simulated waveguide trapping transition points by FEM (${F}_{x}=0$) for three different radii. The simulation results are seen to be consistent with the analytical model. Figure 3a visualizes four hollow sphere trapping regimes. I) $t\ge 0.23r$ (${n}_{\text{hollow}}\ge {n}_{w}$ and ${\rho}_{\text{hollow}}\ge {\rho}_{\text{water}}$): spheres are trapped by both Gaussian beams and waveguides. II) $0.16r\le t\le 0.23r$ (${n}_{\text{hollow}}\le {n}_{w}\le {n}_{\text{cap}}$ and ${\rho}_{\text{hollow}}\ge {\rho}_{\text{water}}$): spheres are trapped by a waveguide, but not by Gaussian beams. III) $60\text{\hspace{0.17em}}\mathrm{nm}\le t\le 0.16r$ (${n}_{w}\le {n}_{\text{cap}}$ and ${\rho}_{\text{hollow}}\le {\rho}_{\text{water}}$): spheres float but can be trapped by the attractive forces from the waveguide and not by Gaussian beams. IV) $t<60\text{\hspace{0.17em}}\mathrm{nm}$ (${n}_{w}\ge {n}_{\text{cap}}$ and ${\rho}_{\text{hollow}}\le {\rho}_{\text{water}}$): spheres float and are repelled away from waveguides. Spheres in regimes I, II, and III are trapped by the waveguide, but a Gaussian beam can only trap spheres in regime I. Spheres in regimes I, II, V, and VI sink (${\rho}_{\text{hollow}}\ge {\rho}_{\text{water}}$) and in regimes III and IV float (${\rho}_{\text{hollow}}\le {\rho}_{\text{water}}$). Figure 3b shows the plot close to the transition region. Interestingly, Fig. 3b shows two small regimes. In regime V spheres are trapped by Gaussian beams and not by waveguides (${n}_{\text{cap}}\le {n}_{w}\le {n}_{\text{hollow}}$). In regime VI (${n}_{\text{hollow}}\le {n}_{w}$ and ${n}_{\text{cap}}\le {n}_{w}$), spheres cannot be trapped by both Gaussian beams and waveguide traps, which is similar to regime IV. However, spheres in regime VI sink, whereas they float in regime IV. Regimes V and VI are small and occur only for $75\text{\hspace{0.17em}}\mathrm{nm}<r<350\text{\hspace{0.17em}}\mathrm{nm}$ and $20\text{\hspace{0.17em}}\mathrm{nm}<t<60\text{\hspace{0.17em}}\mathrm{nm}$ for the chosen waveguide parameters.

Hollow and solid spheres were trapped and propelled experimentally on a waveguide surface. The solid and hollow glass spheres were bought from Polysciences Inc. The solid spheres have specified diameters of $2\u201315\text{\hspace{0.17em}}\mathrm{\mu m}$ and a density of $~2.5\text{\hspace{0.17em}}\mathrm{g}/{\mathrm{cm}}^{3}$. The hollow spheres have specified diameters of $2\u201320\text{\hspace{0.17em}}\mathrm{\mu m}$ with a mean of $8\text{\hspace{0.17em}}\mathrm{\mu m}$ and a nominal density of $1.1\text{\hspace{0.17em}}\mathrm{g}/{\mathrm{cm}}^{3}$. We used optical microscopy to measure the shell thickness (*t*) and diameter ($2r$) and estimate the density for a batch of spheres. Hollow spheres were dried on a substrate and imaged with a $100\times $ objective lens for measuring $2r$ and *t*. For particles with $r=1\u20138\text{\hspace{0.17em}}\mathrm{\mu m}$, the thickness was in the range $t=400\u20131400\text{\hspace{0.17em}}\mathrm{nm}$, with an estimated measurement error of $250\text{\hspace{0.17em}}\mathrm{nm}$. Figure 4 shows the distribution of the effective refractive index of spheres calculated from the measured values of $2r$ and *t*. A scanning electron microscope was also used to confirm the shell thickness variation of the crushed hollow spheres. For spheres with $r<2\text{\hspace{0.17em}}\mathrm{\mu m}$, *t* was $300\u2013600\text{\hspace{0.17em}}\mathrm{nm}$, and for $r>3\text{\hspace{0.17em}}\mathrm{\mu m}$, we measured $t>500\text{\hspace{0.17em}}\mathrm{nm}$.

The experimental setup and the waveguides employed in this work have been described in previous papers [9, 10]. A $5\text{\hspace{0.17em}}\mathrm{W}$ Ytterbium fiber laser of $1070\text{\hspace{0.17em}}\mathrm{nm}$ wavelength is employed with a $3\text{\hspace{0.17em}}\mathrm{\mu m}$ wide and $200\text{\hspace{0.17em}}\mathrm{nm}$ thick tantalum pentoxide waveguide. The input and guided power were $1500\text{\hspace{0.17em}}\mathrm{mW}$ and $15\text{\hspace{0.17em}}\mathrm{mW}$, respectively. To reduce surface adhesion of the hollow spheres, it was necessary to add a small amount of surfactant (Triton X-100) to the solution. This was not necessary for the solid spheres. Figure 5 shows propulsion velocities for hollow and solid glass spheres.

Both types of spheres were trapped and propelled on the waveguide surface. For the chosen waveguide width ($3\text{\hspace{0.17em}}\mathrm{\mu m}$), both the hollow and the solid spheres of $4\u20136\text{\hspace{0.17em}}\mathrm{\mu m}$ diameter are propelled faster than smaller and larger spheres of corresponding type. The lower velocities can be explained by the smaller spheres ($2r<4\text{\hspace{0.17em}}\mathrm{\mu m}$) not interacting with the entire width of the waveguide field and by the larger spheres ($2r>4\text{\hspace{0.17em}}\mathrm{\mu m}$) experiencing larger drag forces. Solid spheres were propelled faster than hollow spheres of similar diameters. The propulsion velocities of solid and hollow spheres of $r=1.5\text{\hspace{0.17em}}\mathrm{\mu m}$ are $15\text{\hspace{0.17em}}\mathrm{\mu m}/\mathrm{s}$ and $5\text{\hspace{0.17em}}\mathrm{\mu m}/\mathrm{s}$, respectively (Fig. 5), giving a velocity ratio of 3. Simulations of solid spheres of $r=1.5\text{\hspace{0.17em}}\mathrm{\mu m}$ give ${F}_{z}=13\text{\hspace{0.17em}}\mathrm{pN}$ and of hollow spheres with $t=350\text{\hspace{0.17em}}\mathrm{nm}$ give ${F}_{z}=7.5\text{\hspace{0.17em}}\mathrm{pN}$ (Fig. 3), giving a force ratio of 1.7. Viscous drag forces are equal for hollow and solid spheres of the same diameter, and the velocity is proportional to ${F}_{z}$ and to the power in the waveguide. The higher ratio of the measured velocities compared to the simulations can be due to the experimental spheres having thinner shells than the simulated spheres. The small amount of surfactant necessary for hollow sphere propulsion also gives reason to believe the hollow spheres have higher surface adhesion. The hollow spheres trapped experimentally (Fig. 5) belong to regimes I and II ($\rho >1$). To trap floating particles (regime III), the particles either have to be pushed onto the waveguide by applying an external force, or the waveguide has to be turned upside down.

Trapping of hollow and solid glass spheres in the evanescent field of an optical waveguide has been studied. The average refractive index of the sphere–field intersection volume is calculated with an analytical expression. A hollow sphere is attracted to the waveguide if the average index of the sphere–field intersection volume is higher than that of the surrounding medium. The attractive or repelling optical forces are dependent on the shell thickness. Stable lift by an optical beam has recently been demonstrated [14]. Future work will show if attractive and repulsive forces of an evanescent field can be balanced to create a stable lift.

The authors wish to thank J. S. Wilkinson, A. Subramanian, S. Gétin, and D. Néel for their help. This work is supported by the Research Council of Norway.

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