Using vector potential and spectrum representation, we derive the expressions of the Airy evanescent field existed at the interface. Utilizing these expressions and the Arbitrary Beam Theory, the optical forces exerted on a Mie dielectric particle in the Airy evanescent field were theoretically investigated in detail. Numerical results show that the optical forces exhibit strong oscillations which are corresponding to the distributions of the evanescent field. With the increasing the size of particle radius, Morphology Dependent Resonance occurs for the particle with specific refractive index.
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Since Ashkin et al. proved that three-dimensional trapping of a dielectric particle is possible by use of a single, highly focused laser beam , optical tweezers have become an indispensable tool for manipulating small particles without any mechanical contact. The optical tweezers have been used to manipulate and trap micro-scale particles even viruses , liquid droplets , silver nanoparticles , uniaxial anisotropic spheres , and magneto dielectric particles . Kawata and Sugiura demonstrated the movement and trapping of micrometer-sized particles in the evanescent field formed at the sapphire-air interface by total internal reflection of a laser beam . E.Almaas and I.Brevik considered theoretically the force upon a micrometer-sized spherical dielectric particle in the plane evanescent field . Meanwhile, S.Chang et al investigated theoretically the optical force exerted on a dielectric sphere in the Gaussian evanescent field . Conventional tweezers usually utilize Gaussian light beams, which suffer from strong divergence off the focal plane, trapping particles with only a few micrometers apart in the axial direction. The “non-diffracting” beams, especially Bessel beams, do not spread while propagating, even if the beam diameter is reduced to the size of a tightly focused Gaussian laser beam. It has been used to trap atoms and microscopic particles in multiple planes , construct particle conveyor belts , sort microfluidic cells and transfect cells .
A second type of “non-diffracting” beam, the Airy beam, was observed in experiment [13, 14]. Its key difference from the Bessel beam is that it additionally experiences a transverse acceleration and can self bend even in free space. Due to its unique properties, Airy beam has recently attracted a lot of attentions because of its potential applications in plasma guidance , vacuum electron acceleration , and generation of three dimensional optical bullets . Jörg Baumgartl and colleagues [18, 19] experimentally demonstrate the first use of the Airy beam in optical micromanipulation. As opposed to Bessel beams, it can transport microparticles along curved self-healed paths, and remove particles or cells from a section of a sample chamber. Its novelty is that the trapping potential landscape tends to freely self-bend during propagation. Moreover, the diffraction free distance and the bend degree of Airy beam can be controlled, and the acceleration direction can be switched by a nonlinear optical process . These tunable properties make the Airy beam a versatile and powerful tool for optical manipulation.
Up to now, the optical micromanipulation by Airy light beam has been experimentally demonstrated, however, only qualitative theoretical analysis has been presented [18, 19]. Meanwhile, we have quantitatively analyzed the radiation forces and trajectories of Rayleigh particles in Airy beam . Further quantitative theoretical analysis is necessary to guide optical micromanipulation. The micromanipulation by Airy beam reported to date is related to Mie particles [22, 23], whose radii are larger than the wavelength; there has been a considerable amount of interest in this kind of interaction, both from a fundamental and from a practical viewpoint.
In this paper, using vector potential and spectrum representation, we derive the expressions for the evanescent field induced by total internal reflection of an Airy beam at the interface. Utilizing this expression and Arbitrary-Beam theory (ABT) of Barton and associates [24, 25], series-form expressions for the optical force of a spherical particle in the Airy evanescent field are presented. As an application of these equations, the internal and external electromagnetic fields for a dielectric sphere interacted with the Airy evanescent field have been discussed totally. The optical forces exerted on Mie dielectric particles by the Airy evanescent field were theoretically analyzed. The interpretations of numerical results are also presented in detail.
2. Theory and description
In this section, using vector potential approach , we derive the expressions for the evanescent field induced by total internal reflection of an Airy beam at the interface. Then by utilizing ABT, the radiation forces exerted on Mie particles by the Airy evanescent field are derived. Also, the internal and external electromagnetic fields for a dielectric particle illuminated by the Airy evanescent field are presented.
2.1 Expansions of the field components
We first consider the general formalism for the case when there is an arbitrary monochromatic electromagnetic field incident upon a dielectric spherical particle. The origin of coordinates x, y and z is laid at the center of the sphere, with x axis pointing in the horizontal direction. The sphere is taken to be isotropic, homogeneous, nonmagnetic, and nonconducting; its refractive index is n3 and can be complex index of refraction. The refractive index of surrounding medium is n2. The internal and external wave numbers are9]. Then, we will give the expressions for the coefficients Alm and Blm that characterize the incident electric and magnetic fields. These coefficients are defined as follows. We first write the definition of the spherical harmonics:
2.2. Force components
Assuming a steady-state condition, the net radiation force F on the particle can be determined by integrating the dot product of the outwardly directed normal unit vector and Maxwell’s stress tensor over a surface enclosing the particle :Eq. (8) can be directly integrated, and the net force on the particle can be expressed as a series over the coefficients alm, blm, Alm, and Blm :
2.3. Airy evanescent field as an incident field
In this section, using vector potential approach , we get the expressions for the evanescent field formed at the plane interface by total internal reflection of an Airy beam. Then we derive the coefficients Alm and Blm in Eqs. (6) and (7) for the evanescent field of Airy beam.
The schematic diagram of the problem is shown in Fig. 1 . The origin of coordinate system coincides with the center of the spherical particle at a distance d from the interface. The media below and above the boundary plane at have indices of refraction n1 and n2 (< n1), respectively. A focused Airy beam with wave vector k1 and frequency is incident from medium n1 with an angle of incidence . The Airy beam center at (xc, yc, zc). We take x-z plane as the plane of incidence and () as unit vectors.
For Airy beam, if the vector potential A has been determined, the electromagnetic fields E and H can be derived from A as follows:Eqs. (19) and (20) can be rewritten asEqs. (6) and (7), after using a great deal of recursions and orthogonal relationships and finishing the integrals about θ and ϕ, we obtain the coefficients and for the Airy evanescent field as follows:
3. Results and discussions
First, we illustrate the dynamics of the Airy evanescent field at interface. In the following simulations, the refractive indices below and above interface are n1 = 1.5 and n2 = 1.0. The incident beam is considered as a two-dimensional (2D) Airy beam, whose parameters are chosen as: λ = 514.5 nm, a0 = 0.1 and x0 = y0 = 2 μm, respectively. The power of input Airy beam is P = 1 W.
Figure 2 shows the distributions of electric field magnitude of 2D Airy evanescent wave above the interface: (a) in the x-z plane (y = 0), while xc = yc = 0; (b) the distributions at the origin point, which is as a function of the beam center’s displacements (xc, yc) in the x-y plane (z = 0). Incident angle θ1 = 0.85 rad, exceeding the critical angle, d = 0.8λ, beam center zc = −(d + 0.8λ) = −2d.
As shown in Fig. 2(a), the Airy evanescent wave dies away several wavelengths away from the interface. In Fig. 2(b), the distributions of Airy evanescent field illustrate the typical characteristic of 2D Airy beam. Field of maximal intensity is located in the vicinity of the point (xc = yc = 0). With the increasing of the displacements, field intensity decays gradually.
In the following, calculations were performed for the Airy evanescent field incident upon a spherical dielectric particle. The knowledge of the electric field energy density is often referred as source function and their distribution inside the particle is important in the field of Raman or fluorescence scattering from molecules in a microparticle. In Fig. 3(a) we present the source function of a spherical particle in the x-z plane (y = 0) with the radius a = d = 2λ and refractive index is n3 = 1.59, it is refractive index of polystyrene; in Fig. 3(b), the refractive index of the sphere is n3 = 1.5 + 3.1i, it is refractive index of nickel. The Airy beam center at xc = yc = 0, zc = −(d + 0.8λ), other parameters are the same as in Fig. 2(a).
In Fig. 3(a), it is noticeable that most of the energy is coupled into the particle. The wave undergoes total internal reflection inside the particle and a traveling wave along the surface occurs. Therefore most of the energy is confined near the surface. The arrows denote the directions of Poynting vectors. For nickel sphere, the refractive index is complex, most of the energy is absorbed and the wave tunneled into the particle has vanished. It is shown in Fig. 3(b).
Now we discuss the optical forces exerted on a Mie particle in an Airy evanescent field. The distribution of evanescent field is illustrated in Fig. 2(b). A polystyrene sphere with radius a = d = 0.8λ and refractive index n3 = 1.59 is considered. The sphere touches the interface and the refractive index of the surrounding medium is n2 = 1. Figure 4(a) shows the variations of the optical forces as a function of beam center displacements xc, while we let yc = 0, zc = −2d; Fig. 4(b) shows the variations of the optical forces as a function of beam center displacements yc, while xc = 0, other parameters are same as in Fig. 4(a).
We can see from Fig. 4 that the x-component of optical forces Fx is positive, the particle will moves along x direction; the z-component of optical force Fz is negative, the particle will be attracted toward the interface. The characteristics of the optical forces are similar with the distributions of Airy evanescent field, the oscillation peaks of the force curves correspond to the lobes of the evanescent field in Fig. 2(b) exactly. The magnitudes of Fx and Fz are larger than Fy exceeding one order of magnitude, that is because the non-uniform momentum transfer to the particle by the evanescent field.
In Fig. 5 , we evaluate the optical forces exerted on a nickel sphere with refractive index n3 = 1.5 + 3.1i. Other parameters are same as in Fig. 4. Numerical results show that Fx decrease with the absorption coefficient increasing, and Fz is positive for this situation. It is obvious that a bigger repulsive force has caused by absorption and it has suppressed the attractive force toward the interface caused by refractive index gradient.
Then we investigate the behaviors of the optical forces as a function of the radius of particle, while d = 0.55 μm, the center of Airy beam are xc = yc = 0, zc = −(d + 0.8λ), incident angle θ1 = π/3 rad. Particle radius varies in the range (0.15<a<0.55) μm. Numerical results are shown in Fig. 6 : (a) refractive index n3 = 1.59, polystyrene particle, (b) n3 = 1.5 + 3.1i, nickel particle.
As shown in Fig. 6, all the forces increase monotonously with the size of particle radius increasing. In Fig. 6(a), ripple structures of oscillations appear in each optical force curve for polystyrene particle and the oscillation strength increase gradually. But there is no oscillation for the complex refractive index particle in the whole variations of particle radius in Fig. 6(b).
The reason for the occurrence of oscillations for specific refractive index is due to the Morphology Dependent Resonances (MDR) [26, 27]. MDR originates from the interference of an electromagnetic wave propagating inside a dielectric particle confined by total internal reflection, as the wave propagating inside the particle, the constructive interference leads to a series of peaks in the optical force curves for appropriate particle size.
Clearly can be seen from Fig. 3(a), most energy of the evanescent fields is trapped inside the particle due to total internal reflection and the wave circling along the surface. The interference of the internal field would generate MDRs for certain radius particle for n3 = 1.59. But for nickel particle shown in Fig. 3(b), the internal field vanished due to strong absorption for its complex refractive index, so there will be no interference inside the particle, nor the resonances.
Finally, we analyze the impacts of incident angle on the optical forces as a function of particle radius (0.2<a<1) μm. Numerical results are shown in Fig. 7 : (a) θ1 = 0.85 rad, (b) θ1 = π/3 rad, while the distance d is varied with particle radius as d = a in this case, n3 = 1.59, xc = yc = 0, zc = −(d + 0.8λ). As we can see, the oscillations due to the MDR become evident as the particle radius increases, the larger particle size the sharper resonance peaks. And as expected, smaller incident angle would generate larger optical forces, but basically, different incident angles won’t change the shape of the force curves. The magnitude of Fy is much smaller than Fx and Fz for the non-uniform distribution of the evanescent field, we didn’t show them for clarity.
In conclusion, using vector potential and spectrum representation, the expressions for the Airy evanescent field are derived. The internal and near-surface electric fields that Mie particles interacted with the evanescent field are investigated. Particle with complex refractive index would generate strong absorption effect and change the direction of optical force. By using the Arbitrary Beam Theory, the optical forces exerted on the Mie particles in the evanescent field are evaluated. Optical forces for the particles with specific refractive index would exhibit oscillations that come from MDRs. We believe that the theoretical results presented in this paper would be useful for investigations in optical micro-manipulation and near-field optics.
Appendix: expansions of the evanescent fields of 2D Airy beam
where a, n3 is radius and refractive index of the particle, respectively. , and are the Riccati-Bessel functions. is the spherical harmonic function, . The expansions for the incident (i), the scattered (s), and the internal (w) electromagnetic fields in the spherical coordinate system are summarized as follows:
We acknowledge financial supports from the Natural Science Foundation of China (grant 11074130, 61275148), Chinese National Key Basic Research Special Fund (2011CB922003), and 111 Project (B07013).
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