# Besinc Pseudo-Shell Model Sources with Circular Coherence
## 4 Trapping Dielectric Nanoparticle with Pseudo-Schell Beams  
Previous results suggest that a focused pseudo-Schell model beam could be used for trapping dielectric nanoparticles in a similar way to other kinds of partially coherent beams [5; 6; 34; 35; 36; 37]. An advantage of the pseudo-Schell model beams is the sharpening of the intensity profile and the high peak that is reached after the source plane (see Figures 5 and 6), which could increase the gradient force exerted on a dielectric particle. By appropriately choosing the source parameters, traps can be obtained with the same beam for particles with a refractive index \(n_{1}\) either higher or lower than the refractive index \(n_{2}\) of the surrounding medium. Figure 6: Evolution of the intensity profile with propagation distance for a beam with \(m=1\) and different values of the \(\delta_{c}/w_{0}\) ratio [numerical evaluation of Equation (17) for free space propagation]. The insets show the intensity profile across the source plane and across the plane where the maximum intensity is reached. If the source plane is at a distance \(s_{0}\) before a thin converging lens with focal length \(f\), the beam propagated at a distance \(z\) after the lens focus can be calculated applying Equation (17), where the ABCD transfer matrix has to be chosen as  
\[\left( \begin{array}{cc}A&B\\ C&D\end{array} \right)=\left( \begin{array}{cc}1&f+z\\ 0&1\end{array} \right)\left( \begin{array}{cc}1&0\\ -1/f&1\end{array} \right)\left( \begin{array}{cc}1&s_{0}\\ 0&1\end{array} \right)=\left( \begin{array}{cc}-z/f&f+z-zs_{0}/f\\ -1/f&1-s_{0}/f\end{array} \right). \tag{18}\]  
In the following, we consider a pseudo-Schell model source described by Equation (10) with coherence parameter \(\delta_{c}=0.1\) mm, intensity width at the source plane \(w_{0}=0.5\) mm, and \(m=1\) (which correspond to the case depicted in the bottom right-hand part of Figure 6). The usual operation line of a visible He-Ne laser is taken for the wavelength \(\lambda=632.8\) nm. The source plane is located at \(s_{0}=2.50\) m before a converging lens of focal length \(f=5\) mm. These parameters have been chosen to obtain high values of the intensity gradient in a region close to the lens focus, within the paraxial approximation. A factor \(I_{0}=1\) W/mm\({}^{2}\) is taken for calculating the intensity of the generated beam for a propagation distance \(z\) after the focus of the lens. Figure 7a shows the evolution of the intensity at the beam center (\(R=0\)) along the \(z\)-axis in a region after the lens focus. It can be observed that the intensity shows two relative maxima and a relative minimum at given propagation distances in this region. This fast variation of the intensity with propagation distance could be exploited for particle trapping along the \(z\)-direction in different positions, depending on the optical properties of the particle. Figure 7b shows the transverse intensity profile of the beam at three different planes, namely, those where the maximum and the minimum axial intensity is reached. Let us consider a spherical dielectric nanoparticle having a radius smaller than the wavelength of the light (\(a<<\lambda\)). Therefore, the Rayleigh scattering approximation can be used to obtain the radiation forces exerted on the nanoparticle. Two different contributions can be distinguished: the scattering force and the gradient force. The first one is directed along the propagation direction and is proportional to the beam intensity. It can be written as [38]  
\[\mathbf{F}_{\text{Scat}}(X,Y,z)=\frac{8\,\pi}{3\,c}n_{2}\,(k\,a)^{4}\,a^{2}\, \left(\frac{n_{r}^{2}-1}{n_{r}^{2}+2}\right)^{2}\,I(X,Y,z)\mathbf{u}_{z}, \tag{19}\]  
where \(c\) is the speed of light in vacuum, \(n_{r}=n_{1}/n_{2}\) is the relative refractive index of the particle, and \(\mathbf{u}_{z}\) is a unitary vector along \(z\).