In order to understand the behavior of particle affected by LSPR around the bowtie, we calculate the optical forces acting on the particle. Approximations and numerical analysis are used according to the particle size. Mie’s scattering theory, which can be understood by sense of ray optics and Newton’s third law, has been widely used in many researches of conventional optical tweezers. However, it is only for large particles. Tiny biological particles are always out of its scope. Rayleigh scattering theory is right for calculating optical forces on very tiny particles with size much smaller than the incident wavelength. But it is still not suitable for analyzing forces on the particle in our case. This is because the field is highly concentrated with distribution comparable to the particle size. As a result the particle cannot be regarded as a point dipole in vicinity of the LSPR without affection on the field as required in Rayleigh approximation. Finally only Maxwell stress tensor is suitable for rigorously analyzing our case.
Here we expended the time averaged force density in Cartesian coordinate (in x, y, and z
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For doing integration of the tensor on particle surface, we established a simulation method by exploiting Comsol multiphysics software. It is based on finite-element method which can solve problems of physics without time evolution. So the first advantage is time-saving.
Second, its mesh rule is nun-uniform which fits especially for curved surfaces (Fig. 2-1);
many critical structures could be defined well.
Fig. 2-1 The simulated model(a) before (b) after mesh generation of Comsol multiphysics software.
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Besides, the mesh of each part can be defined individually depending on its importance. So we can largely save computer memories for calculation, and get the simulation results quickly.
Finally its capability in post processing is almost impeccable. By which we can calculate the optical forces exerting on a particle easily. In our simulation we first build the bowtie we design, and put the target object at the position we interest in. Then we launch waves and simulate how optical field would distribute in steady state by using Comsol. Finally by integrating the force density over the surface of object we interest in, we can get the optical force caused by the electric and magnetic fields directly. (Fig. 2-2)
Fig. 2-2 Force integration over object surface of Comsol software.
From the force analysis, we were able to extract a parameter commonly used to describe stability of the trapping site: trapping potential U which is defined as [30][31]
U
rF(r')dr' (2.4) It is the integration of the gradient force with respect to the position of particle perturbation14
around the equilibrium point. The rule of thumb, established by Ashkin in his pioneering work, states that stable trapping requires a potential depth over 10 kBT to compensate for stochastic kicks in the particle’s Brownian motion. However, potential in optical traps are rarely measured directly, the probability for the displacement of a trapped object in a potential well is first determined, then used in conjunction with the thermal energy kBT and given by a Boltzmann distribution trap stiffness α. Trap stiffness is also defined as
e q u i l i b r i um it is the derivative of the restoring force with respect to the position of particle perturbed around the equilibrium point. The stiffness is also a parameter commonly used to evaluate optical traps.
It can be regarded as elastic constants in mechanical system. Intuitively, the higher the stiffness means the tighter the trapped system.
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2-2 Finite Element Method
In our work, wave propagation along the waveguide and finally interacting with the particle is simulated by COMSOL Multiphysics software based on finite element method (FEM). The method is a numerical technique for finding solutions of partial differential equations (PDE).
For complicated optical systems, it can solve the boundary value problem, eigenvalue problem and find the steady state solution of a system by employing variational method. To apply this method, it requires discretizing a continuous domain into a set of discrete sub-domains, usually called elements, and the solution of each element would be approximated by certain characteristic form to solve the problems.
Here, the wave equations in the frequency domain for the magnetic and the electric fields are
where c is the vacuum speed of light. To solve the wave equation for either magnetic or the electric field in frequency domain together with boundary conditions, standard FEM method proceeds in three steps. First, the wave equations are identified as solutions of certain variational problems where boundary condition at the surface has been incorporated as additional terms of lagrangian L. The most general variational formulation for the electric field is given by
where is the magnetic permeability and is dielectric function both may varied in space.
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In addition, denotes the outward normal at the surface and the electric field has to satisfy the Dirchlet boundary condition (nE)0
on S . and U
are known quantities which are used to represent various other types of boundary conditions such as impedance boundary conditions and Sommerfeld radiation conditions. Finally, radiation sources within the computational domain V are described through the spatially varying current density J
.
The second step is the most demanding step which consists of the discretization of the Lagrangian. The computational domain V is divided into a number of small-volume elements, the so-called finite elementary functions with unknown coefficients. It becomes possible to approximately enforce the div-conditions of the electric field within a field element as long as the dielectric function does not vary within this element.
In the final step, these expansions facilitate the transformation of the Lagrangian into a set of linear equations by Galerkin method [32]. This matrix system can subsequently be solved via advanced linear algebra methods, either for obtaining eigenfrequencies and eigennodes of the system of interest or to determine scattering cross sections of complex structure as well as transmittance and reflectance through functional elements.
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