## Abstract

We introduce the use of hollow micron-sized spheres with a finite-thickness glass shell as individual micromirrors operating by total internal reflection (TIR) when illuminated off-axis. We also demonstrated that this kind of spheres can be optically trapped and manipulated in two dimensions using a Gaussian beam in a conventional optical tweezers setup, which allows the precise positioning of the micromirrors at specific locations within a sample cell. This mirrors constitutes a new micro-tool in the context of the so called lab-on-a-chip

©2005 Optical Society of America

## 1. Introduction

A continuous trend towards miniaturization has arisen in the last few decades in different areas, such as optics and electronics. In optics, this trend is closely connected with the development of optical tweezers and laser micromanipulation techniques. The first optical traps introduced by Ashkin in the 70’s, operated mainly by means of an adequate balance of the scattering forward forces exerted by two laser beams propagating in opposite directions [1]. A fundamental improvement was achieved in 1986 when the single-beam gradient trap was introduced [2], nowadays known as optical tweezers. Furthermore, a complete set of optical micro-tools has been implemented in recent years like, for instance, optical spanners and rotators [3, 4, 5], scissors [6, 7], stretchers [8], sorters [9, 10], among others, which have found a very wide spectrum of applications. The incorporation of micro-mirrors in this optical toolkit, which are essential elements in many optical systems, may open new possibilities and facilitate some tasks at microscopic level. For example, the ability of redirecting the light within a sample cell could help to easily remove residues in optical microsurgery, or it could be used for the independent operation of optical micro-machinery elements with different orientations in space [11, 12].

In fact, two-dimensional fixed arrays of metal-coated micromirrors have been already fabricated using different techniques (photolithography, for example), in those cases, with purposes oriented toward the development of Micro-Opto-Electro-Mechanical Systems (MOEMS) and microresonators. An outstanding example is the Digital Micromirror Device designed by Texas Instruments [13], consisting of an array of individually addressed plane micro-mirrors. Other examples are arrays of concave spherical and tetragonal pyramidal mirrors fixed in a template [14, 15].

In this paper we introduce the use of hollow glass micro-spheres with finite shell thickness as individual and movable dielectric micro-mirrors. The operation principle is based on TIR that arises when the spheres are illuminated slightly off center by a focused Gaussian beam. The incoming beam can be deflected into a wide range of directions by only controlling the point of incidence on the sphere’s surface and the generated light streak is highly directional and intense enough to be used to push other particles or objects around the microsphere or to remove particles stuck to the sphere. We even achieve a double reflection by two hollow spheres placed close to each other. Moreover, we also demonstrated, for the first time to our knowledge, that hollow glass spheres, which are usually repelled from high intensity regions [16, 17, 18, 19, 20], can be optically trapped and manipulated in two-dimensions with a nominal Gaussian beam. The trapping takes place when the particle is located above the beam waist and centered with respect to the propagation axis.

## 2. Theory of hollow spheres as micromirrors

We will start this section with a geometrical ray optics analysis of a single light ray falling upon a hollow sphere with finite shell thickness, as shown in Fig. 1, which will allow us to identify some of the main parameters and limiting conditions. Let *R*
_{1} be the internal radius of the hollow sphere and *R*
_{2} its external radius, so that the shell thickness is given by *t*=*R*
_{2}-*R*
_{1}. The refractive indices of the surrounding medium (region I), the glass shell (region II) and the medium inside the sphere (region III) are *n*
_{1}, *n*
_{2} and *n*
_{3}, respectively, with *n*
_{2}>*n*
_{1}>*n*
_{3}. In our case, the glass microspheres (*n*
_{2}=1.5) are suspended in water (*n*
_{1}=1.33), and we assume that the hollow sphere is filled with air (*n*
_{3}=1.0). The light ray strikes the sphere at point A, forming an incidence angle α_{0} to the normal of the sphere’s surface, and the corresponding refracted angle is denoted by α_{1}. Provided that the light ray reaches the interface glass-air, we denote the incidence point as B and the incidence angle as α_{2}. In that case, the position of point B will be identified by the angle *θ*=π/2-[α_{0}-α_{1}+α_{2}].

TIR will occur at the interface glass-air for angles α_{2}≥α_{2c}=sin^{-1}(*n*
_{3}/*n*
_{2}). In addition, we can see from Fig. 1 that *R*
_{1} sin α_{2}=*R*
_{2} sin α_{1}, and finally, by applying the Snell’s law at the interface water-glass, we can find a condition for the minimum angle α_{0} leading to TIR at the interface glass-air:

This can be expressed also in terms of the minimum displacement of the incoming light ray respect to the sphere’s center (*d*=*R*
_{2} sin α_{0}) in order to obtain TIR for that ray, which is *d*
_{min}=*n*
_{3}
*R*
_{1}/*n*
_{1}.

An equally simple analysis shows that a second total internal reflection at the interface glass-water is not possible, giving rise to a strong deflected ray emerging from the sphere. The total deviation angle of the light ray respect to its incidence direction is found to be δ=2(α_{0}-α_{1}+α_{2}). As an example, for a 20 µm-diameter particle with an internal diameter of 14 µm, the emerging of a light ray perpendicular to the incoming ray direction (δ=90°) requires that α_{0}=31.8o (*d*=5.3µm).

Note that the condition for an incident light ray to reach the interface glass-air is sin α_{0}<*n*
_{2}
*R*
_{1}/*n*
_{1}
*R*
_{2}, which implies that for a strong reflection to take place is necessary that *n*
_{3}
*R*
_{1}/*n*
_{1}≤*d*<*n*
_{2}
*R*
_{1}/*n*
_{1}. Rays who do not satisfy such condition will see the sphere rather as a solid sphere of refractive index *n*
_{2}.

Although the parameters and limiting conditions identified above are useful as a starting point, the description of a strongly focused incident light beam is more complicated and requires a more detailed analysis. For that purpose, we used the integral equation formalism described in Refs. [21, 22] to explicitly calculate and model the incident and scattered electromagnetic field in the vicinity of the hollow particle with finite shell thickness.

As in Refs [21, 22], we have simplified the physical system to a two-dimensional one. Such simplification reduces both computational cost and complexity behind the mathematical formalism. In practice, modelling our system in a two-dimensional geometry is much simpler and it is useful to study it qualitatively. This model is valid even if the particle size is smaller than the light wavelength.

Consider an s-polarized apertured focused incident plane wave of amplitude *U _{inc}*, whose profile at the focal plane is given by

where *a* is the numerical aperture of the system and λ is the wavelength of the light in the medium. By employing the Green’s integral theorem, the fields in the different regions of refractive index (see Fig. 1) can be expressed as

$$+\frac{1}{4\pi}{\int}_{{S}_{2}}\left\{{U}^{2}\left(\mathit{R}\right)\left({\hat{\mathit{n}}}_{2}\left(\mathit{R}\right)\xb7\nabla {G}_{\mathit{II}}(\mathit{r}\mid \mathit{R})\right)-{G}_{\mathit{II}}(\mathit{r}\mid \mathit{R}){V}^{2}\left(\mathit{R}\right)\right\}d{s}_{2},$$

where the surfaces *S*
_{1} and *S*
_{2} are the interfaces between regions I and II and between regions II and III, respectively. The position vector to an observation point is denoted by **r** and **R** stands for the coordinate of a point on the surface. *U _{inc}*(

**r**) represents the incident field and

*U*(

^{m}**r**) (

*m*=I, II, III) is the field at any point at the corresponding region of the space. The unit vector

**n̂**

_{1}(

**R**) is normal to the surface

*S*

_{1}, directed from region II into region I, and

**n̂**

_{2}(

**R**) is normal to surface

*S*

_{2}, from region III into region II. The unknown source functions are the field

*U*(

^{n}*)and the normal derivative of the field at the corresponding surface,*

**R***V*(

^{n}*)=*

**R**

**n̂***(*

_{n}*)·∇*

**R***U*(

^{n}*) (*

**R***n*=1, 2).

The Green’s functions for the three regions are represented in terms of Hankel functions of the first kind [21], namely,

with *k _{o}*=2

*π*/

*λ*, where

_{o}*λ*is the light wavelength at the vacuum and

_{o}*ε*(

_{m}*ω*) is the dielectric function at the different regions (

*m*=I, II, III). The source functions

*U*(

^{n}*)and*

**R***V*(

^{n}*) may be found from a system of coupled integral equations obtained by letting the point of observation, in Eqs. (3)-(5), approach the surfaces. By taking the limit as the point of observation approaches the surface*

**R***S*

_{1}(

*S*

_{2}) from region I (II), we obtain a system of coupled integral equations which are discretized and converted into matrix equations in a similar way to the analysis given in Refs. [21, 22]. By solving the matrix system we found the source functions, that is, the field and the normal derivative of the field on the surface. Knowledge of the source functions allows the calculation of the field in regions I, II and III from Eqs. (2)-(5).

Figure 2 shows typical results of the numerical simulations for a plane wave focused by a microscope objective of numerical aperture N.A.=*a*=1.2, incident on a hollow glass sphere. We found that when the incident beam is focused at the interface glass-air the strongest beam (higher reflectivity) is produced; focusing the beam on any other position produces a weaker beam. On the left column of Fig. 2, frames (a)-(d) show the incident and scattered electric field as a function of the incidence position, defined by the angle θ, for a hollow glass sphere with internal and external radii of *R*
_{1}=7 µm and *R*
_{2}=10 µm, respectively. In (a) the incidence point is located at θ=75°, and in (b) at θ=60°; these cases are under the minimal condition discussed previously to obtain TIR. We can see that in (a), most of the light reaching the glass-air interface is transmitted and forward scattered. In case (b), the amount of light reflected at the second interface increases considerably, but it is still not enough to observe a well defined deflected beam. In the last two cases, (c) θ=45° and (d) θ=15°, there is a strong deflection of the incident beam producing what we will call a light streak effect. Note that, although in (c) there is no TIR for all of the incoming light, the reflectivity at the interface is large enough to produces a strong deflected beam. We can also see from these figures that the light streak can arise in different directions respect to the vertical depending on the incidence point, defining a wide range of polar angles for the reflected beam. It is also evident by symmetry, that the whole range of the azimuthal angle in the horizontal plane [0, 2*π*), can be scanned by the reflected light streak.

Numerical simulations show that the reflectivity of the micromirrors for the beam emerging at 90° respect to the incoming beam is approximately 20%, but the reflectivity can be as high as 35% for grazing incidence (small θ). For a given incidence angle, the reflectivity of the micromirrors is almost independent of the hollow sphere internal diameter. This is true as long as the beam spot size remains larger than the hollow sphere internal diameter, as shown below.

On the other hand, frames (e) to (h) on the right column show the scattered electric field as a function of the shell thickness for a fixed external radius and incidence position, given by *R*
_{2}=10 µm and θ=45°, respectively. In (e) *t*=0, which corresponds to an air bubble, in (f) *t*=1 µm and in (g) *t*=5 µm. We can also look at frame (c) for *t*=3 µm and the same conditions of external radius and incidence angle. In all of those cases the light streak is well defined. The deflection angle changes just slightly with shell thickness, being very close to 90° for (c) and (g) and a little bit larger for a smaller shell thickness. In contrast, we can see that in (h), where *t*=7 µm, the deflected beam has become very weak and wide to be noticed, despite of the light reflection at the second interface. For higher values of *t* the reflection is even weaker, until it obviously disappear in the limit case of a solid sphere *t*=*R*
_{2}. Hence, we can conclude that there is a range for the optimum shell thickness in order to obtain a well defined light streak effect.

We can also note from Fig. 2 that when the light streak effect is generated, the deflected beam is highly directional. This is in agreement with our experimental observations, discussed in the next section.

## 3. Experiments

In our experiments, we used a standard optical tweezers setup operating with a nominal Gaussian beam from an argon ion laser emitting at 514 nm and with output power up to 30 mW. The light beam was focused onto a sample cell by means of an oil-immersion 100× microscope objective with N.A.=1.25. Our sample consisted of hollow glass spheres suspended in water, with external diameters ranging from 5 to 20 microns and different unknown internal diameters (Polysciences Inc.).

We found that the hollow microspheres at the bottom of the sample cell can be optically trapped when the illuminating beam is centered respect to the sphere and provided that the beam waist is below the sphere. The trapping is two-dimensional, in the sense that the particles are pushed against the bottom of the sample cell by the strongly convergent light beam. The case of the hollow spheres could be compared with the case of reflective metallic particles, since both have been demonstrated to exhibit similar properties in interaction with light [23] due, precisely, to the TIR arising in the former case. Therefore, the optical trapping is attributed here to the scattering force, which has a radial inwards component besides the component along the propagation axis when the sphere is located above the focal plane [24, 25]. It is worth to mention that the trapping of air bubbles in a fluid with a Gaussian beam have been reported previously, but in that case it was achieved by means of a hybrid trap involving an adequate balance of optical and convection induced forces, [26] which is clearly not our case.

Figure 3 shows the trapping and manipulation of a glass hollow microsphere with an external diameter of approximately 20 µm, the power at the sample plane was in the range of 10 to 15 mW. The shell thickness of the hollow sphere can be qualitatively appreciated from this figure. The two-dimensional trapping is verified by the horizontal displacement of the microsphere respect to its surroundings in the different consecutive frames. Notice that the minimum shell thickness for a particle to rest at the bottom of the sample cell is given by *t*
_{min}=*R*
_{2} [1-(1-ρ_{w}/ρ_{g})^{1/3}], where ρ_{w} (1.0 g/cm^{3}) and ρ_{g} (2.5 g/cm^{3}) are the water and glass densities, respectively. For instance, a 20 µm diameter sphere at the bottom of the sample cell would have a shell thickness of at least *t*=3.13 µm.

On the other hand, the light streak effect appears when the sphere is illuminated off-center, which is in agreement with our theoretical analysis. This has been observed in microspheres with different external and internal diameter, some of them were resting at the bottom of the sample cell, while others were floating within the water. We also observed that there is an optimum position of the beam focus along the propagation axis in order to observe the sharpest and strongest deflected beam. Based on our theoretical modeling, we assume that the optimum position corresponds to the case when the incident beam is focused at the glass-air interface of the hollow sphere. It was also verified that, for a certain displacement of the incidence position of the beam respect to the sphere’s center, the emerging light streak is approximately perpendicular to the incoming beam, and of course, the full range of the azimuthal angle can be covered as well by controlling the incidence position. The generated normal beam is quite intense and it has been used to push other particles or objects around the microsphere along several tens of microns due to radiation pressure, as it can be observed in the sequence of frames (a) to (d) of Fig. 4.

Finally, we were able to obtain two consecutive reflections of an incident light beam by means of two hollow spheres located close to each other. The reflected beam from the second micromirror was intense enough to push particles located in the vicinity of the spheres. This is illustrated in the frames (a) to (d) of Fig. 5. It is worth to mention that in this case we used a small amount of Rhodamine diluted in the water with the purpose of enhancing the view of the reflected beam, due to fluorescence of the Rhodamine at the laser wavelength. This experiment can be generalized to obtain reflections from an array of micromirrors and microresonators.

## 4. Conclusions

We have demonstrated two new effects here. On the one hand, we showed for the first time to our knowledge, that a hollow glass sphere can be optically trapped in two-dimensions with a single Gaussian beam, provided the beam is centered with respect to the sphere and its focal plane is located below the sphere. The trapping mechanism in this case was identified with the scattering force directed inwards and downwards for the converging region of the beam.

On the other hand, we presented a theoretical analysis and experimental results for the deflection of the incident light beam in different directions due to a strong internal reflection on the inner interface of the hollow particle. The illuminating light beam can be deviated into a wide interval of values of the polar angle (respect to the vertical axis) and the whole range of azimuthal angles in the horizontal plane, by only controlling the point of incidence on the sphere’s surface. The reflected beam was found to be strong enough to push other particles around the hollow sphere along distances of several tens of microns.

Our results allow us to introduce the use of hollow glass spheres as a very simple variety of individually movable micromirrors, which constitute a new component of the so-called optical toolkit. As examples of potential applications we can mention the removal of residues in optical microsurgery and the independent operation of optical micro-machinery elements with different orientations in space. We have also shown that fabrication of microresonators may be practical consequence of our results.

## Acknowledgments

R. R-G acknowledges support from CONACYT through the project 36133-F. K.V-S acknowledges support of DGAPA-UNAM through the project IN103103.

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