We report on the numerical study to model a lensless optical manipulation trap to investigate some observed features as guiding and modulation effects caused by a micro-sphere. For this we calculate the field distribution and force exerted upon a micrometer-sized spherical dielectric particle in an evanescent field. The method of calculation is based on the integral equation formalism describe by A.A. Maradudin et. al.  and A. Mendoza-Suárez and E.R. Méndez . The numerical experiments were done considering a two-dimensional model.
©2008 Optical Society of America
Single-beam trapping is a well known technique that uses optical radiation pressure to control and manipulate microscopic particles that since its publications by Ashkin et al in 1987 , has became an increasingly useful tool in a variety of applications in biology, micro-fluidics, near-field microscopy, etc. This laser trap consists of a single beam highly focused by a high numerical aperture microscope objective creating a high intensity region, focal region, where a three-dimensional trapping zone is created. Particles falling within the laser beam will experience a force directed towards the focus of the beam. With this technique it is possible to trap one or few particles at the same time.
A different approach was used by Kawata and Sugiura . They observed particle movement in the evanescent field of a laser beam. A laser beam is incident from below through a sapphire semi-cylindrical prism, the evanescent field is created above the plane face of the semi-cylindrical prism using a polarized beam of a given wavelength impinging at an incident angle higher that the critical angle. This technique allows simultaneous optical manipulation of a large number of particles over an extended area.
Recently, guiding and trapping particles using near-field and evanescent optical forces has been a very active field and several configuration have been used, such as guiding dielectric and metallic nano- and micro-particles in an extended homogeneous evanescent field generated at the surface of a prism illuminated under total internal reflection and in the vicinity of a waveguide [4-8], patterned designs of the optical near-field capable of stable trapping [9, 10] and with the use of surface plasmon polariton excitation [11, 12]. Also, there has been a comprehensive theoretical work for calculating the optical forces exerted on particles immersed in an evanescent field [13-21]. Extinction theorem boundary condition formalism has been use by Lester and Nieto-Vesperinas, Arias-González, et al.  and Arias-González and Nieto-Vesperinas [15,16]; the coupled dipole method by Chaumet and Nieto-Vesperinas  and the Green dyadic method by Quidant et al.  for calculating radiation forces in an homogeneous and a patterned optical near field. Particles considered were with radii equal or less than the illuminating wavelength. For larger size particles, Almaas and Brevick  used electromagnetic wave theory and Chang, et al.  used Mie-Debye theory, both of them assumed that multiple reflections between the sphere and the flat surface were unimportant. Later J.Y. Waltz  used a ray optics model for calculating the radiation force.
The purpose of the present paper is to present exact numerical calculations for the field distribution and radiation forces exerted on a micrometer size particle (radius larger than the illumination wavelength) illuminated by an evanescent field created in a flat surface under total internal reflection to aid in understanding the scattering processes in large particles and that do not take place in nano-metric particles (radius smaller than wavelength) and therefore the force exerted on the latter ones show a different behavior that has been recently reported . The method of calculation is based on the integral equation formalism describe by A.A. Maradudin et. al.  and A. Mendoza-Suárez and E.R. Méndez ]; the latter considers some modifications to the former that allows us to deal with multivalued surfaces.
The structure of the paper is as follows. First, in Sect. 2, we present the theoretical model and a brief description of the method of solution. Then, in Sect. 3, we present some representative results, together with a discussion of their implications. Finally, in Sect. 4, we present our main conclusions.
2. The numerical method
In our numerical experiments, we have simplified the physical system to a two-dimensional one. Such simplification reduces both computational cost and complexity behind the mathematical formalism and, excluding depolarizations effects, this calculation permits an understanding of the basic physical processes involved in the scattering of an evanescent field by a particle without loss of generality.
The system we study in this paper is shown in Fig. 2; both, the surface S1 and the particle (surface S2) are invariant along the y-axis, and the plane of incidence of the electromagnetic field is the xz-plane; there is no cross-polarized scattering in it. In our numerical experiments the system is illuminated by a Gaussian beam at an angle of incidence θinc. Also, medium I is characterized by a dielectric constant ε1, in the region z<0, ε 2 in the region z>0 and a dielectric particle (ε3) with a diameter d at a height h above the surface S1.
Illuminating the system with an s-polarized field, with the medium of incidence being the prism in the region z<0 and employing the Green’s integral theorem, the fields in the different regions are:
and inside the particle
where Einc(r⃑) represents the incident field, E(R⃗) is the field at the surface. Here R⃗ stands for the coordinate of a point on the surface. The unit vector n̂(R⃗) is normal to the surface, directed from region z>0 into region z<0, and n̂p(R⃗) is the local outward normal on the particle surface. The Green’s functions for the three regions may be represented in terms of Hankel functions of the first kind
where nl is the complex refractive index.
Once electromagnetic fields have been calculated, the time-averaged force <F> on the surface can be calculated by using the Maxwell’s stress tensor as 
where ∑s2 is the cylinder surface, n is its local outward normal, the asterisks stands for complex conjugates and ε and µ are the electric and magnetic permittivities.
In this section, we present representative results obtained by the numerical technique described in the previous section. In all the numerical experiments the illumination is provided by a Gaussian beam and we used the following parameters: wavelength of the incident beam λ=1.064 µm, incident angle θinc=51.0°, refractive index of the prism nI=1.75, refractive index of water nII=1.33, refractive index of the particle nIII=1.59, and a diameter of the particle d=6.8µm. These parameters are similar to those used in the experimental work by Kawata and Sugiura .
As a first result, Fig. 3 shows the numerical calculation of the near and far-field distribution of light scattered by a spherical particle in an evanescent field at a height from the flat surface S1 of h=0.3µm. For this calculation we have used a narrow Gaussian beam in order to show the behavior of the reflected beam from the flat surface. In the inset we show the field distribution (in arbitrary units) in the far-field region above and beneath the prism surface (S1). It is worth noticing that the scale is different for both sides.
In this case, the incident beam and the total reflected beam generate an interference region where a standing wave is formed, which is known as Wiener’s fringes. Also, it is possible to observe the asymmetry of the evanescent spot with respect to the origin of the system of coordinates and also the lateral translation of the reflected beam with respect to the location predicted by geometrical optics. The above is due to the Goos-Hänschen shift. As we notice in Fig. 3, when the particle is close enough to the flat surface to sense the evanescent field, light is tunneled into the particle where the conditions for propagating are again satisfied, and therefore, this light is propagated within the particle and also scattered by it. Thus, part of the incident light has been coupled to the particle, and the magnitude of the total reflected beam decreases. Since our numerical model takes into account the interaction of the particle and the surface where the evanescent field has been generated, interference effects are observable mainly between the particle and the flat surface on the left. This is also noticeable in the intensity distribution scattered in the far-field above the surface S1, as a ringing feature on the graph.
Figure 4 shows numerical calculations of the near-field distribution and the intensity at the far-field for the spherical particle (d=6µm) in an evanescent field at different heights from the flat surface S1: (a) h=1.0µm, (b) h=0.8µm, (c) h=0.6µm, (d) h=0.4µm, (e) h=0.2µm and (f) h=0.1µm. In this case, the half-width of the Gaussian beam was 50µm. As we can see from the sequence of images, when the particle gets closer to the prism surface, the coupling of the evanescent field to the particle gets larger. A series of interference artifacts are present mainly on left of the particle. The complex pattern is due to the interference of the evanescent field generated by total internal reflection and the field that has been coupled into the particle and then scattered by it. Similar modulated pattern behavior has been found experimentally and numerically by Ishikawa et al (1999)  and Ishikawa et al (2000) , respectively. However, in our model the interaction with the flat surface is included, and not only through the fact that the incident light is the evanescent wave.
It is noticeable that due to the coupling of the evanescent field to the particle, the strength of the reflected intensity (beneath the surface S1) decreases as the particle gets closer to the surface. Compare the insets of Fig. 4(a) and Fig. 4(f). For the same reason, the strength of the scattered light in the far-field above the surface S1 increases also when the particle gets closer to the surface. The angular intensity distribution in the far-field above the flat surface is shown in the insets; the magnitude of the intensity when the particle is the closest (h=0.1µm) to the surface is larger in comparison to the intensity scattered when the particle is at a height of h=1.0µm. The scattering distribution agrees with the fringe pattern observed in front of the particle (right side) by Kawata and Sugiura .
As an illustrative example of the calculations of the force exerted on a particle applying Eq. (5), we show in Fig. 5 the field distribution in the near-field region when the particle is at a height of h=0.2µm from the flat surface S1. The results of computing the force exerted on each segment of the surface of the particle, as well as the direction of the total force acting on the particle (arrow at the center of the particle), are shown. The scale for the magnitude of the force at each segment is not the same than that for the total force on the surface. As can be seen, the direction of the driving force is along the positive x-axis and pulled towards the surface (negative z-axis). This result agrees with the observations made by Kawata and Sugiura  and Kawata and Tani . It is worth noticing the complex pattern within the particle due to the coupled evanescent field and propagating inside the particle.
It is interesting to observe the behavior of the magnitude and direction of the force when the particle is fixed to a horizontal position (in this case x=0) and moved vertically away from the flat surface. To present the results of the latter situation, we used the dimensionless parameters Qx and Qz for the x-direction force and the z-direction force, respectively, and they are defined by (Refs.  and ):
In Fig. 6 we present the numerical results of calculating the dimensionless parameters Qz versus the distance h (see Fig. 2) between the flat surface and a particle of diameter d. In the numerical results the z component of the force is negative, which means that the particle is pulled down toward the surface. Nevertheless, we have plotted the logarithm of |Qz| to show some details of the Qz parameter. An exponential decay behavior was expected due to z dependence of the amplitude of the evanescent field, which decreases exponentially as the distance from the flat surface increases. However, the Qz does not follow the exponential tail expected. In the case of the smaller particle, 1 µm radius, the curve for the ln(|Qz|) parameter gets closer to a straight line but as the radius increases it departs from the straight line. For a Rayleigh particle, the curve for the ln(|Qz|) should follow a straight line. When the size of the particle is much larger than the illuminating wavelength, the force exerted on it results from a more complex process that includes multiple scattering and the coupled evanescent field propagating inside it. This can be seen in the field distribution shown in Fig. 5. Experimentally this behavior has been observed by Volpe, et al , when they used a photonic force microscope to measure forces on micro-particles immersed in an evanescent field. For nano-particles (radius smaller than the wavelength), the negative exponential decay has been obtained [15,17].
To compare the behavior of x and z component of the force, we have plotted the Qx and |Qz| parameters in Fig. 7 for a 4 µm radius particle. The dimensionless parameter Qx is positive, meaning that the particle is pushed in the positive x-direction. For any given height h, the force in the x direction is larger than in the z direction, thus guiding the particle along the surface. Also as Volpe, et al,  pointed out, x component of the force decreases sharper than the z component. To observe this, in the inset of Fig. 7, we have plotted the derivative of the x (continuous line) and z (dotted line) force components. There is a noticeable difference in the way that the components behavior, the changes in the z are slower.
We have presented a numerical study to calculate the near and far field distribution of a dielectric particle interacting with an evanescent field generated above the plane face of a prism by total internal reflection when it has been illuminated with a polarized beam of a given wavelength. This situation models the lensless optical guiding with an evanescent field. As in Refs [1, 2], we have simplified the physical system to a two-dimensional geometry, which is much simpler but none the less useful to study. This model is valid even if the particle size is smaller than the wavelength of the light. Also, by using Maxwell’s stress tensor, we have calculated the force exerted on the particle and presented the results in the form of a non-dimensional parameter.
Numerical results show the way in which a particle interacts with an evanescent field when the particle gets closer to the flat surface where the evanescent field is generated. The closer to the surface, the larger the coupling of the evanescent field to the particle, and thus, the larger the amount of light scattered by the particle into the far-field and also the more complex process results in it. The force exerted on the particle does not increases exponentially when the distance from the flat surface to the particle gets smaller for particles with radius larger than the wavelength. For large particles the total force exerted on the particle results from a more complex process. The z-component of the force pushes the particle towards the surface, while the x-component guides the particle along the surface in the positive direction.
The authors gratefully acknowledge the support of the Consejo Nacional de Ciencia y Tecnología through project F1-61995 and the award of a scholarship to J.P. Vite-Frías.
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