Waveguide trapping of hollow glass spheres

Balpreet Singh Ahluwalia; Pål Løvhaugen; Olav Gaute Hellesø

doi:10.1364/OL.36.003347

Optics Letters
Vol. 36,
Issue 17,
pp. 3347-3349
(2011)
•https://doi.org/10.1364/OL.36.003347

Waveguide trapping of hollow glass spheres

Balpreet Singh Ahluwalia, Pål Løvhaugen, and Olav Gaute Hellesø

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Balpreet Singh Ahluwalia, Pål Løvhaugen, and Olav Gaute Hellesø, "Waveguide trapping of hollow glass spheres," Opt. Lett. 36, 3347-3349 (2011)

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About this Article
History
- Original Manuscript: April 19, 2011
- Revised Manuscript: July 19, 2011
- Manuscript Accepted: July 28, 2011
- Published: August 22, 2011
Virtual Issues
October 18, 2011 Spotlight on Optics

Abstract

Microparticles can be trapped and propelled by the evanescent field of optical waveguides. As the evanescent field only stretches $100 - 200 nm$ 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 $60 nm$ 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.

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 $150 nm$ 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 $3 μm$ and $200 nm$ , respectively. The optical field is TE polarized with wavelength $1070 nm$ . 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 $1 μm$ 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 $h = 150 nm$ 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 $1 W$ are shown in Fig. 2 for a radius $r = 1.5 μm$ as a function of the shell thickness t [13]. For spheres with $t > 60 nm$ , 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 $t > 60 nm$ 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 $500 nm < t < 1500 nm$ , $F_{z}$ increases very slowly, as the cap volume interacting with the evanescent field remains constant. For spheres where $t < 60 nm$ , $F_{x}$ is positive, repelling spheres vertically away from the waveguide surface. Consequently, spheres with $t < 60 nm$ 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 ( $n_{hollow}$ ) depends on the refractive indices of glass ( $n_{glass}$ ) and air ( $n_{air}$ ) and the volumes of the sphere ( $V_{sphere}$ ) and the encapsulated air ( $V_{air}$ ):

n_{hollow} = n_{glass} \frac{V_{sphere} - V_{air}}{V_{sphere}} + n_{air} \frac{V_{air}}{V_{sphere}} .

The average refractive index is seen to be linearly dependent on the void fraction,

V_{air} / V_{sphere}

. Likewise, the average specific density of the hollow sphere (

ρ_{hollow}

) is a linear function of

V_{air} / V_{sphere}

. The interaction between the sphere and the field is approximated using a constant field of height h. For radii

r < h / 2

, the whole sphere interacts with the field. For

r \geq h / 2

, the volume

V_{cap}

of the field–sphere intersection is

V_{cap} = \frac{π h^{2} (3 r - h)}{3} .

We can now define the average refractive index of the cap (

n_{cap}

n_{cap} = n_{glass} \frac{V_{cap} - V_{cap:air}}{V_{cap}} + n_{air} \frac{V_{cap:air}}{V_{cap}},

where

V_{cap:air}

is found by replacing r by

r - t

and h by

h - t

in Eq. (2):

V_{cap:air} = \frac{π (h - t)^{2} (3 (r - t) - (h - t))}{3} .

Our hypothesis is that waveguides will attract hollow spheres for all

n_{cap} > n_{w}

(refractive index of water), independently of

n_{hollow}

. For

n_{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_{hollow} > n_{w}

. For the transition where

n_{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_{cap} = n_{w} = 1.33

in Eq. (3), the waveguide trap transition points, where

F_{x}

is zero, are found. Finally,

ρ_{hollow} = ρ_{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 \geq 0.23 r

(

n_{hollow} \geq n_{w}

and

ρ_{hollow} \geq ρ_{water}

): spheres are trapped by both Gaussian beams and waveguides. II)

0.16 r \leq t \leq 0.23 r

(

n_{hollow} \leq n_{w} \leq n_{cap}

and

ρ_{hollow} \geq ρ_{water}

): spheres are trapped by a waveguide, but not by Gaussian beams. III)

60 nm \leq t \leq 0.16 r

(

n_{w} \leq n_{cap}

and

ρ_{hollow} \leq ρ_{water}

): spheres float but can be trapped by the attractive forces from the waveguide and not by Gaussian beams. IV)

t < 60 nm

(

n_{w} \geq n_{cap}

and

ρ_{hollow} \leq ρ_{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 (

ρ_{hollow} \geq ρ_{water}

) and in regimes III and IV float (

ρ_{hollow} \leq ρ_{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_{cap} \leq n_{w} \leq n_{hollow}

). In regime VI (

n_{hollow} \leq n_{w}

and

n_{cap} \leq 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 nm < r < 350 nm

and

20 nm < t < 60 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 $2 - 15 μm$ and a density of $~ 2.5 g / {cm}^{3}$ . The hollow spheres have specified diameters of $2 - 20 μm$ with a mean of $8 μm$ and a nominal density of $1.1 g / {cm}^{3}$ . We used optical microscopy to measure the shell thickness (t) and diameter ( $2 r$ ) 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 $2 r$ and t. For particles with $r = 1 - 8 μm$ , the thickness was in the range $t = 400 - 1400 nm$ , with an estimated measurement error of $250 nm$ . Figure 4 shows the distribution of the effective refractive index of spheres calculated from the measured values of $2 r$ and t. A scanning electron microscope was also used to confirm the shell thickness variation of the crushed hollow spheres. For spheres with $r < 2 μm$ , t was $300 - 600 nm$ , and for $r > 3 μm$ , we measured $t > 500 nm$ .

The experimental setup and the waveguides employed in this work have been described in previous papers [9, 10]. A $5 W$ Ytterbium fiber laser of $1070 nm$ wavelength is employed with a $3 μm$ wide and $200 nm$ thick tantalum pentoxide waveguide. The input and guided power were $1500 mW$ and $15 mW$ , 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 ( $3 μm$ ), both the hollow and the solid spheres of $4 - 6 μm$ diameter are propelled faster than smaller and larger spheres of corresponding type. The lower velocities can be explained by the smaller spheres ( $2 r < 4 μm$ ) not interacting with the entire width of the waveguide field and by the larger spheres ( $2 r > 4 μm$ ) experiencing larger drag forces. Solid spheres were propelled faster than hollow spheres of similar diameters. The propulsion velocities of solid and hollow spheres of $r = 1.5 μm$ are $15 μm / s$ and $5 μm / s$ , respectively (Fig. 5), giving a velocity ratio of 3. Simulations of solid spheres of $r = 1.5 μm$ give $F_{z} = 13 pN$ and of hollow spheres with $t = 350 nm$ give $F_{z} = 7.5 pN$ (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 ( $ρ > 1$ ). 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.

Fig. 1 Waveguide trapping of a hollow sphere together with the simulated optical field.

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Fig. 2 Gradient ( $F_{x}$ ) and propagation ( $F_{z}$ ) optical forces on a hollow sphere of $1.5 μm$ radius for different shell thicknesses (t). The power guided in the waveguide is kept constant at $1 W$ .

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Fig. 3 Optical trapping regimes of hollow spheres as a function of radius (r) and shell thickness (t).

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Fig. 4 Refractive index and specific density of a batch of hollow spheres used in the experiment.

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Fig. 5 Optical propulsion of hollow and solid spheres of different diameters on a waveguide of $width = 3 μm$ and for an input $power = 1500 mW$ .

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