# Besinc Pseudo-Shell Model Sources with Circular Coherence
## 4 Trapping Dielectric Nanoparticle with Pseudo-Schell Beams
The gradient force pushes the particle towards the region of  
Figure 7: (**a**) evolution of the intensity in the beam axis with propagation distance after the focus of a lens and (**b**) intensity profile at different propagation distances for a source with \(m=1\), \(\lambda=632.8\) nm, \(\delta_{c}=0.1\) mm and \(w_{0}=0.5\) mm. Other parameters are \(f=5\) mm, \(s_{0}=2.50\) m, and \(I_{0}=1\) W/mm\({}^{2}\). maximum (minimum) intensity if the relative refractive index is higher (lower) than 1. This force can be expressed as [38]  
\[\mathbf{F}_{\text{Grad}}(X,Y,z)=\frac{2\,\pi}{c}n_{2}\,a^{3}\,\frac{n_{r}^{2}-1} {n_{r}^{2}+2}\,\boldsymbol{\
abla}I(X,Y,z). \tag{20}\]  
Figure 8a shows the evolution along the beam axis of the \(z\)-component of the gradient force on a dielectric nanoparticle with radius \(a=30\) nm and refractive index \(n_{1}=1.59\) or \(n_{1}=1.00\) immersed in water (\(n_{2}=1.33\)). The on-axis scattering force (that always push the particle away from the focus, \(z=0\)) is shown in Figure 8b. It must be noted that other forces, such as the weight and buoyancy forces, are of the order of \(10^{-7}\) pN for dielectric nanoparticles of that size. On the other hand, the Brownian force has a magnitude \(F_{B}=\sqrt{12\,\pi\,x\,a\,\,k_{B}\,T}\)[39], where \(\kappa\) is the medium viscosity, \(k_{B}\) is the Boltzmann constant and \(T\) the temperature. For water at room temperature (\(T=300\) K), the viscosity is \(\kappa\simeq 7.977\times 10^{-4}\) Pa, so that the Brownian force is of the order of \(2\times 10^{-3}\) pN. It can be noted that the scattering force is about two orders of magnitude lower than the \(z\)-component of the gradient force, so that this is the dominant force, the remaining ones (weight, buoyancy, drag and or Brownian forces) being negligible. Taking these considerations into account, there are two (one) stable equilibrium positions along the \(z\)-axis for particles with higher (lower) refractive index than the surrounding medium. These positions correspond to the propagation distances where the on-axis intensity (see Figure 7a) reaches a relative or absolute maximum (minimum) that is, \(z_{1}\simeq 9.39\)\(\upmu\)m and \(z_{3}\simeq 10.69\)\(\upmu\)m (\(z_{2}\simeq 10.02\)\(\upmu\)m). Figure 9 shows the behavior, as functions of the transverse coordinate, of the \(x\)-component of the gradient force (parts (a) and (c)) and the scattering force (parts (b) and (d)) at the three planes where a maximum or a minimum of the axial intensity is reached. Due to the circular symmetry of the intensity profile across the transverse section, these results are the same along any radius. Again, the scattering force (always directed along the \(z\)-axis) is about two orders of magnitude lower than the gradient force, so that the main effects of the radiation on the particle are due to the gradient force. For the case of dielectric particles with a refractive index lower than that of the surrounding medium (parts (a) and (b) of Figure 9), there is stable equilibrium point in the plane \(z_{2}\simeq 10.02\)\(\upmu\)nm. In the case of a particle with higher refractive index than the medium (parts (c) and (d) of Figure 9), there is a stable equilibrium point in the plane \(z_{1}\simeq 9.39\)\(\upmu\)m and another one in the plane \(z_{3}\simeq 10.69\)\(\upmu\)m.  
Figure 8: (**a**) \(z\)-component of the gradient force and (**b**) scattering force along the beam axis for a dielectric nanoparticle with radius \(a=30\) nm and having a refractive index \(n_{1}\) higher (blue line) or lower (red line) than that of the surrounding medium (\(n_{2}=1.33\)). Other parameters are the same as in Figure 7.