## 4 Fraction of Power Propagating Inside and Outside a Micro/Nanofiber  
In this section, we investigate the fractional power and effective diameter of the fundamental mode guided in an MNF. In the case of propagation in an optical fiber, the average flux transmitted through the fiber is written by the \(z\)-component of the cycle-averaged Poynting vector [49]  
\[\left\langle\vec{S}\right\rangle=\frac{1}{2}\text{Re}\left[\vec{E}\times\vec{ H}^{*}\right]. \tag{14}\]  
For the HE\({}_{11}\) mode, the \(z\)-components of Poynting vector are obtained by [1, 3]  
\[\left\langle\vec{S}_{z}\right\rangle_{in} = \frac{1}{2}\left(\frac{\epsilon_{0}}{\mu_{0}}\right)^{1/2}\frac{ k_{0}n_{1}^{2}}{\beta I_{1}^{2}(ha)}[a_{1}a_{3}J_{0}^{2}(hr)+a_{2}a_{4}J_{2}^{2}(hr)+ \frac{1-F_{1}F_{2}}{2}J_{0}(hr)J_{2}(hr)\cos\left(2(\phi)\right],\] \[\left\langle\vec{S}_{z}\right\rangle_{out} = \frac{1}{2}\left(\frac{\epsilon_{0}}{\mu_{0}}\right)^{1/2}\frac{ k_{0}n_{1}^{2}}{\beta K_{1}^{2}(qa)}\frac{h^{2}}{q^{2}}[a_{1}a_{5}K_{0}^{2}(qr)+a_{2}a_{6} K_{2}^{2}(qr)+\frac{1-2\Delta-F_{1}F_{2}}{2}K_{0}(qr)K_{2}(qr) \tag{15}\] \[\times\cos\left(2(\phi)\right),\]  
where  
\[F_{1} = \left(\frac{a^{2}hq}{V}\right)^{2}[b_{1}+(1-2\Delta)],F_{2}= \left(\frac{V}{a^{2}hq}\right)^{2}\frac{1}{b_{1}+b_{2}},b_{1}=\frac{1}{2 ah} \left\{\frac{I_{0}(ha)}{I_{1}(ha)}-\frac{I_{2}(ha)}{I_{1}(ha)}\right\},\] \[b_{2} = -\frac{1}{2 aq}\left\{\frac{K_{0}(qa)}{K_{1}(qa)}+\frac{K_{2}(qa)} {K_{1}(qa)}\right\},a_{1}=\frac{F_{2}-1}{2},a_{2}=\frac{F_{2}+1}{2},a_{3}= \frac{F_{1}-1}{2},a_{4}=\frac{F_{1}+1}{2},\] \[a_{5} = \frac{F_{1}-1+2\Delta}{2},a_{6}=\frac{F_{1}+1-2\Delta}{2},\Delta =\frac{n_{1}-n_{2}}{n_{1}}. \tag{16}\]  
Figure 9: (**a**) The total intensity \(|E|^{2}\) of the electric field in a fundamental mode HE\({}_{11}\) with quasi-linear polarization. (**b**) Radial dependences of the total intensity \(|E|^{2}\) of the electric field at \(\phi=0\) (\(x\)-polarization) and \(\phi=\pi/2\) (\(y\)-polarization). Parameters are same as used in Figure 6.  
The radial and azimuthal components transversal to the fiber axis, \(\left\langle\vec{S_{r}}\right\rangle\) and \(\left\langle\vec{S_{\phi}}\right\rangle\), describe the fraction of the energy flow of the electromagnetic field that does not propagate. The fraction of power propagating inside and outside the fiber can be given by \(\eta=\frac{P_{in}}{P_{in}+P_{out}}\), where \(P_{in}=\int_{0}^{2\pi}d\phi\int_{0}^{a}\left\langle\vec{S_{z}}\right\rangle_{ in}rdr\) and \(P_{out}=\int_{0}^{2\pi}d\phi\int_{a}^{\infty}\left\langle\vec{S_{z}}\right\rangle _{out}rdr\)[3]. The fractional power \(\eta\) is related to the diameter-dependent group velocity and waveguide dispersion. Figure 10 shows the fraction of power for the fundamental mode inside the core as a function of the diameter at \(\lambda=780\) nm and \(\lambda=1.5\) um. At \(\lambda=780\) nm (\(1.5\) um), the 90% energy inside the core is confined by \(d=700\) nm (\(1.36\) um) of an MNF whereas the rest of the energy is confined by \(270\) nm (\(515\) nm) of it. Because of the highly confined optical modes in high-index contrast structures, a tight bending radius of less than \(5\) um can be achieved, that results in compact guided optical components while giving more flexibilities in the photonic circuit layout [50; 51; 52; 53]. Now, let us define the effective radius, \(a_{\text{eff}}\), which is a hypothetic radius for possessing \(86.5\%\) (\(=\)\(1-e^{2}\)) of the total power [3]. \(a_{\text{eff}}\) can be obtained from \(\frac{P_{in}}{P_{in}+P_{out}}=86.5\%\) for \(a_{\text{eff}}\leq a\) and \(\frac{P_{in}+P_{\text{eff}}}{P_{in}+P_{out}}=86.5\%\) for \(a_{\text{eff}}>a\), where \(P_{\text{eff}}=\int_{0}^{2\pi}d\phi\int_{a}^{a_{\text{eff}}}\left\langle\vec{ S_{z}}\right\rangle_{in}rdr\). As the diameter decreases, \(\text{d}_{\text{eff}}(=2a_{\text{eff}})\) is getting larger so that it leads most of light energy to propagate in air. In Figure 11, the major power can be confined within \(d=500\) nm at \(\lambda=780\) nm and \(d=1.1\) um at \(\lambda=1.5\) um.