## 2 Light Propagation in Step-Index Cylindrical Wires  
We first consider the geometry of a step-index optical fiber with a circular cross-section as shown in Figure 1a. An MNF has a silica core of radius \(a\) with refractive index \(n_{1}\) and an infinite air-clad of refractive index of \(n_{2}=1\). The corresponding refractive index profile can be written as  
\[n(r)= \begin{cases}n_{1},&\text{if }r<\text{a}\\ n_{2},&\text{if }r>\text{a}\end{cases} , \tag{1}\]  
where \(n_{1}>n_{2}\). According to the Sellmeier equation, the refractive index for fused silica at room temperature can be given by [44]  
\[n^{2}-1=\frac{0.6961663\lambda^{2}}{\lambda^{2}-(0.0684043)^{2}}+\frac{0.4079 426\lambda^{2}}{\lambda^{2}-(0.1162414)^{2}}+\frac{0.8974794\lambda^{2}}{ \lambda^{2}-(9.896161)^{2}}. \tag{2}\]  
Since a short length of the MNF is sufficient for microphotonic scales, we assume that both media are non-absorbing and have a magnetic permeability equal to the vacuum permeability \(\mu_{0}\). The corresponding [PERSON]'s equations can be reduced to Helmholtz equations [1; 3]  
\[ \begin{array}{rcl}(\
abla^{2}+n^{2}k^{2}-\beta^{2})\cdot\mathbf{e}&=&0,\\ (\
abla^{2}+n^{2}k^{2}-\beta^{2})\cdot\mathbf{h}&=&0,\end{array}  \tag{3}\]where \(k=2\pi/\lambda\), \(\lambda\) is the wavelength of the light, \(\beta\) is the propagation constant, \(\mathbf{e}\) and \(\mathbf{h}\) are the electric field vector and the magnetic field vector, respectively. The longitudinal propagation constant \(\beta\) of various modes can be determined by the eigenvalue equations of Equation (3) [1,3,45]:  
\[\frac{J_{l-1}(ha)}{haJ_{l}(ha)}=\left(\frac{n_{1}^{2}+n_{2}^{2}}{2n_{1}^{2}} \right)\frac{K_{l-1}(qa)+K_{l+1}(qa)}{2 qaK_{l}(qa)}+\frac{l}{ha}\pm R \tag{4}\]  
with  
\[R=\left[\left(\frac{n_{1}^{2}-n_{2}^{2}}{2n_{1}^{2}}\right)^{2}\left(\frac{K_{ l-1}(qa)+K_{l+1}(qa)}{2 qaK_{l}(qa)}\right)^{2}+\left(\frac{l\beta}{n_{1}k_{0}} \right)^{2}\left(\frac{1}{(qa)^{2}}+\frac{1}{(ha)^{2}}\right)^{2}\right]^{1/2}, \tag{5}\]  
where \(h=(n_{1}^{2}k_{0}^{2}-\beta^{2})^{1/2}\) and \(q=(\beta^{2}-n_{2}^{2}k_{0}^{2})^{1/2}\) characterize the fields inside and outside the fiber. These two parameters are directly related to the \(V\)-number, \(V=\sqrt{(ha)^{2}+(qa)^{2}}=2\pi a/\lambda\sqrt{n_{1}^{2}-n_{2}^{2}}\) which determines the fraction of the optical power in a certain mode confined to the fiber core. \(J_{l}\) and \(K_{l}\) are the Bessel functions of the first kind and the modified Bessel functions of the second kind, respectively. There are different solutions depending on the value \(l\) in Equation (4). For \(l=0\), the \(\pm\) signs stand for the TM \((+)\) and TE \((-)\) modes that are transversal magnetic and electric modes, respectively. For \(l=1\), the \(\pm\) signs lead to two different states of modes, the HE \((+)\) and EH \((-)\) which imply \(E_{z}\) being larger and smaller than \(H_{z}\), respectively [45,46]. Figure 2 shows the solutions of the TM and TE modes of Equation (4) that the magnetic field (electric field) is perpendicular to the fiber axis. Both modes are placed at \(V>2.405\). That means that any non-fundamental mode is not allowed to propagate inside the fiber below the lowest cut-off value of 2.405. Therefore, the single mode condition for the silica-air waveguide is \(V<2.405\). Figure 3 shows the solutions of the EH and HE modes at \(V=11.03\). There is a lower limit in \(V\)-number for the EH modes so that the EH modes are not allowed below the single mode condition. The HE\({}_{11}\) mode is only the fundamental mode which can always propagate inside the fiber since it has no cut-off value. Figure 1: Step-index mathematical model: (**a**) Geometry of an air-clad cylindrical wire. (**b**) Schematic refractive index profile of an air-clad cylindrical wire. Figure 2: Solutions for the (**a**) TM modes and (**b**) TE 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.