## 3 Propagation Constant \(\beta\) and Fundamental Mode HE\({}_{11}\)
From the electric field in the HE\({}_{11}\) mode with rotating polarization, we can easily calculate the total intensity \(\left|E\right|^{2}\) which is given by  
Figure 3:0 Figure 3:1  
Since the total intensity \(\left|E\right|^{2}\) is independent of \(\phi\), the distribution of \(\left|E\right|^{2}\) is cylindrically symmetric as shown in Figure 3:2 Since the polarization of the field breaks the symmetry, in general, the electric field propagating in the fiber is not cylindrically symmetric. However, for the rotating polarization, averaging over a period washes out this symmetry breaking. The regions of \(r<a\) and \(r>a\) can be clearly identified due to the discontinuity at the boundary \(r=a\). This strong discontinuity arises from the large refractive index difference between the bulk and the surrounding medium as well as from the strong radial confinement of the field when \(\lambda>a\). Figure 3:3 Vector field plots of the electric component perpendicular to the fiber axis in the fundamental HE\({}_{11}\) mode for rotating polarization at (**a**) \(t=0\), (**b**) \(\pi/4\omega\), (**c**) \(2\pi/4\omega\), (**d**) \(3\pi/4\omega\) under the conditions of \(a=250\) nm and \(\lambda=780\) nm. Figure 3:4 (**a**) The total intensity \(\left|E\right|^{2}\) of the electric field in the fundamental mode HE\({}_{11}\) with rotating polarization. (**b**) Radial dependence of the total intensity \(\left|E\right|^{2}\) of the electric field. Parameters are same as used in Figure 3:5  
We now present the field equations of the fundamental HE\({}_{11}\) mode with quasi-linear polarization. In the cylindrical coordinates \((r,\phi,z)\), for \(r<a\) (inside the fiber), the solutions of [PERSON]'s equations for the Cartesian components of the electric field can be obtained by [1,45,47]  
Figure 3:6 Figure 3:7 Figure 3:8  
For \(r>a\),  
Figure 3:9 \[E_{r}(r,\phi,z,t) = -iA\frac{\beta}{2h}[(1-s)J_{0}(hr)-(1+s)J_{2}(hr)]e^{i(\omega t \pm\phi-\beta z)},\]0 \[E_{r}(r,\phi,z,t) = -iA\frac{\beta}{2h}[(1-s)J_{0}(hr)-(1+s)J_{2}(hr)]e^{i(\omega t \pm\phi-\beta z)},\]1  
The angle \(\phi_{0}\) accounts for the orientation axis of the polarization of the electric field. Two sets of the solutions corresponding to \(\phi_{0}=0\) and \(\pi/2\) imply \(x\)-polarization and \(y\)-polarization, respectively. On the contrary to the rotational polarization, the polarization of transversed \(E\) field does not depend on time but it is position-dependent as shown in \[E_{r}(r,\phi,z,t) = -iA\frac{\beta}{2h}[(1-s)J_{0}(hr)-(1+s)J_{2}(hr)]e^{i(\omega t \pm\phi-\beta z)},\]2 The total intensity of the electric field can be given by  
\[E_{r}(r,\phi,z,t) = -iA\frac{\beta}{2h}[(1-s)J_{0}(hr)-(1+s)J_{2}(hr)]e^{i(\omega t \pm\phi-\beta z)},\]3 \[E_{r}(r,\phi,z,t) = -iA\frac{\beta}{2h}[(1-s)J_{0}(hr)-(1+s)J_{2}(hr)]e^{i(\omega t \pm\phi-\beta z)},\]4 \[E_{r}(r,\phi,z,t) = -iA\frac{\beta}{2h}[(1-s)J_{0}(hr)-(1+s)J_{2}(hr)]e^{i(\omega t \pm\phi-\beta z)},\]5 \[E_{r}(r,\phi,z,t) = -iA\frac{\beta}{2h}[(1-s)J_{0}(hr)-(1+s)J_{2}(hr)]e^{i(\omega t \pm\phi-\beta z)},\]6  
Figure 9a presents the total intensity of the electric field in a fundamental HE\({}_{11}\) mode with quasi-polarization. We choose the \(x\)-axis as the major orientation of polarization (\(\phi_{0}=0\)). A discontinuity of the field intensity takes place at the fiber surface because of the boundary condition for the radial component of the electric field. It is dramatically induced by the large gap between the  
\[E_{r}(r,\phi,z,t) = -iA\frac{\beta}{2h}[(1-s)J_{0}(hr)-(1+s)J_{2}(hr)]e^{i(\omega t \pm\phi-\beta z)},\]7 Vector field plot of the electric field component perpendicular to the \(z\)-axis in the fundamental HE\({}_{11}\) mode with \(\phi_{0}=0\) (\(x\)-polarization). Parameters are same as used in \[E_{r}(r,\phi,z,t) = -iA\frac{\beta}{2h}[(1-s)J_{0}(hr)-(1+s)J_{2}(hr)]e^{i(\omega t \pm\phi-\beta z)},\]8  
refractive indices \(n_{1}\) and \(n_{2}\). The spatial distribution of the field intensity is not cylindrically symmetric.