## 2 Results
### Electromagnetic Field  
Let us consider two counter-propagating Gaussian and Laguerre-Gaussian beams (LGB) [13], whose electric fields (\(\vec{e}_{0},\vec{e}_{1}\)) are elliptically polarized, slightly temporally de-phased (\(\tau\,t\)), and with different beam waist:  
\[\vec{e}_{0} =G_{0}(x,y,z)\,exp(i\,(\phi_{0}(x,y,z)+\tau\,t))(\hat{x}+b_{0}\,i \hat{y}) \tag{8}\] \[\vec{e}_{1} =G_{1}(x,y,z)\,exp(i\,(\phi_{1}(x,y,z))(\hat{x}+b_{1}\,i\hat{y}) \tag{9}\]  
The total electric field may be written as:  
\[\vec{E}=\vec{e}_{0}+\vec{e}_{1} \tag{10}\]  
introducing cylindrical coordinates \(r=\sqrt{x^{2}+y^{2}}\) and \(\theta=arctan(\frac{y}{\chi})\), the amplitude and phase of the fields (\(\vec{e}_{0},\vec{e}_{1}\)) given by Equations (8) and (9) can be expressed (assuming a (1-1) Laguerre Gaussian beam) as:  
\[G_{0}(r,\theta,z) =A_{0}\frac{w_{0}}{w_{0}(z)}exp\left(-\frac{r^{2}}{w_{0}^{2}(z)}\right) \tag{11}\] \[G_{1}(r,\theta,z) =A_{0}\frac{w_{1}}{w_{1}(z)}\left(\frac{\sqrt{2}r}{w_{q}(z)} \right)exp\left(-\frac{r^{2}}{w_{1}^{2}(z)}\right)L_{1}^{1}\left(\frac{2r^{2} }{w_{1}^{2}(z)}\right)\] (12) \[\phi_{0}(r,\theta,z) =-k\left(\frac{r^{2}}{2R_{0}(z)}+z\right)+\zeta_{0}(z)\] (13) \[\phi_{1}(r,\theta,z) =k\left(\frac{r^{2}}{2R_{1}(z)}+z\right)+\theta-4\zeta_{1}(z) \tag{14}\]  
where \(L_{1}^{1}\) is the generalized Laguerre (1,1) polynomial, \(k=n\frac{\omega}{c}\) is the wave number inside of a dielectric medium of refractive index \(n\), and \(A_{0}\), \(w_{q}(z)\), \(\zeta_{q}(z)\), and \(R_{q}(z)\) are the electric field amplitude, beam waist, the Gouy phase, and the curvature radius of Gaussian wave fronts for \(q\in\{0,1\}\), respectively [18]. These parameters are given by:  
\[w_{q}(z) =w_{q}\sqrt{1+\frac{z^{2}}{z_{q}^{2}}} \tag{15}\] \[R_{q}(z) =z\left(1+\frac{z_{q}^{2}}{z^{2}}\right)\] (16) \[\zeta_{q}(z) =tan^{-1}\left(\frac{z}{z_{q}}\right)\] (17) \[\vec{e}_{1} =G_{1}(x,y,z)\,exp(i\,(\phi_{1}(x,y,z))(\hat{x}+b_{1}\,i\hat{y}) \tag{9}\]0  
where \(\lambda_{0}\) is the vacuum wavelength. The magnetic field according to the Maxwell equations is given by:  
\[\vec{e}_{1} =G_{1}(x,y,z)\,exp(i\,(\phi_{1}(x,y,z))(\hat{x}+b_{1}\,i\hat{y}) \tag{9}\]1  
\(\omega_{0}\) being the electromagnetic field frequency. Under the condition \(b_{0}=-b_{1}^{-1}\), the polarization of the counter-propagating beams will be right-handed and left-handed, respectively, the degree of circular polarization being different from one if \(b_{1}\
eq 1\). In this case, Equation (6) can be written as:  
\[\vec{e}_{1} =G_{1}(x,y,z)\,exp(i\,(\phi_{1}(x,y,z))(\hat{x}+b_{1}\,i\hat{y}) \tag{9}\]2where:  
\[\vec{e}_{1} =G_{1}(x,y,z)\,exp(i\,(\phi_{1}(x,y,z))(\hat{x}+b_{1}\,i\hat{y}) \tag{9}\]3 \[\vec{e}_{1} =G_{1}(x,y,z)\,exp(i\,(\phi_{1}(x,y,z))(\hat{x}+b_{1}\,i\hat{y}) \tag{9}\]4 (22) \[\vec{e}_{1} =G_{1}(x,y,z)\,exp(i\,(\phi_{1}(x,y,z))(\hat{x}+b_{1}\,i\hat{y}) \tag{9}\]5  
Taking into account that \(\phi_{0}-\phi_{1}\approx-2 kz-\theta\), Equation (20) can therefore be written as:  
\[\vec{e}_{1} =G_{1}(x,y,z)\,exp(i\,(\phi_{1}(x,y,z))(\hat{x}+b_{1}\,i\hat{y}) \tag{9}\]6  
where \(\phi(z,\theta)=\tau\,t-2 kz-\theta\). Equation (24) describes a chiral-dielectric optical trap given by \(W_{0}(r,z)\) and a chiral conveyor given by \(W_{s}(r,z)\,sin(\tau\,t-2 kz-\theta)+W_{c}(r,z)cos(\tau\,t-2 kz-\theta)\).