## 3 Propagation Constant \(\beta\) and Fundamental Mode HE\({}_{11}\)  
In order to investigate the propagation constant \(\beta\) as a function of a diameter \(d\), the Equation (4) can be rewritten in terms of \(\beta\) and \(d\),  
\[ \begin{split}\left(\frac{4l\beta k_{0}(n_{1}^{2}-n_{2}^{2})}{d^{2 }((k_{0}n_{1})^{2}-\beta^{2})(\beta^{2}-(k_{0}n_{2})^{2})}\right)^{2}& =\left(\frac{I_{l-1}(ha)-\frac{l}{ha}J_{l}(ha)}{haJ_{l}(ha)}-\frac{ K_{l-1}(qa)+K_{l+1}(qa)}{2 qaK_{l}(qa)}\right)\\ &\times\left(\frac{n_{1}^{2}(J_{l-1}(ha)-\frac{l}{ha}J_{l}(ha))}{ haJ_{l}(ha)}-\frac{n_{2}^{2}(K_{l-1}(qa)+K_{l+1}(qa))}{2 qaK_{l}(qa)}\right).\end{split}  \tag{6}\]  
Figure 5a presents the propagation constant (or effective refractive index) as a function of \(V\)-number for the first seven modes. The first two excited modes TM\({}_{01}\) and TE\({}_{01}\) lie at \(V\geq 2.405\) so that the single-mode and multi-mode regimes are clearly distinguished. When \(V\)-number decreases, \(\beta/k_{0}\) tends to be close to \(n_{1}\). It means that the field will propagate more into the surrounding medium while decreasing the diameter. Figure 5b shows the dependence of the propagation constant as a function of a diameter at the wavelength of 780 nm. The HE\({}_{11}\) mode only propagates for \(d\leq 550\) nm. Figure 4: Wavelength for single mode condition of a silica-air waveguide as a function of a diameter, \(d=2a\). Figure 3: Solutions for the (**a**) EH modes and (**b**) HE modes of Equation (4) at \(V=11.03\). The right- and left-hand sides of Equation (4) are plotted by the solid-red and black-dashed lines, respectively. We now study a fundamental mode HE\({}_{11}\) with rotating polarization. In the cylindrical coordinates \((r,\phi,z)\), for \(r<a\) (inside the fiber), the solutions of [PERSON]'s equations for the electric field \(E\) can be given by [1; 45; 47]  
\[E_{r}(r,\phi,z,t) = -iA\frac{\beta}{2h}[(1-s)J_{0}(hr)-(1+s)J_{2}(hr)]e^{i(\omega t \pm\phi-\beta z)},\] \[E_{\phi}(r,\phi,z,t) = \pm A\frac{\beta}{2h}[(1-s)J_{0}(hr)+(1+s)J_{2}(hr)]e^{i(\omega t \pm\phi-\beta z)},\] \[E_{z}(r,\phi,z,t) = AJ_{1}(hr)e^{i(\omega t\pm\phi-\beta z)}. \tag{7}\]  
For \(r>a\),  
\[E_{r}(r,\phi,z,t) = -iA\frac{\beta}{2q}\frac{J_{1}(ha)}{K_{1}(qa)}[(1-s)K_{0}(qr)+(1+ s)K_{2}(qr)]e^{i(\omega t\pm\phi-\beta z)},\] \[E_{\phi}(r,\phi,z,t) = \pm A\frac{\beta}{2q}\frac{J_{1}(ha)}{K_{1}(qa)}[(1-s)K_{0}(qr)- (1+s)K_{2}(qr)]e^{i(\omega t\pm\phi-\beta z)},\] \[E_{z}(r,\phi,z,t) = A\frac{J_{1}(ha)}{K_{1}(qa)}K_{1}(qr)e^{i(\omega t\pm\phi-\beta z )}, \tag{8}\]  
where  
\[s=\left[\frac{1}{(ha)^{2}}+\frac{1}{(qa)^{2}}\right]\left[\frac{J_{0}(ha)- \frac{1}{ha}J_{1}(ha)}{haJ_{1}(ha)}-\frac{K_{0}(qa)+K_{2}(qa)}{2 qaK_{1}(qa)} \right]^{-1}. \tag{9}\]  
The \(+(-)\) sign in Equations (7) and (8) accounts for clockwise (counterclockwise) rotation of the polarization around the \(z\)-axis. The coefficient \(A\) is determined by the normalization condition. Figure 6 shows vector plots of the electric field component transversal to the \(z\)-axis at time \(t=0,\pi/4\omega,\pi/2\omega\), and \(3\pi/4\omega\). We perform numerical calculations for a silica-air fiber with a radius \(a=250\) nm at a wavelength of \(\lambda=780\) nm. A red circle describes the fiber surface. With increasing time, the real part of the transversed electric field rotates along the circular or elliptical with respect to the fiber axis. Figure 5: (**a**) Numerical solutions of effective refractive index, \(n_{\rm eff}=\beta/k_{0}\), as a function of \(V\)-number for the first seven modes. (**b**) Numerical solutions of propagation constant, \(\beta\), as a function of diameter at 780 nm. To apply for sensing and trapping of atoms by the optical force of an evanescent wave around an MNF [10; 11; 12], it is necessary to calculate the optical potential which is proportional to the total intensity of the electric field.