## 2 Bending Loss  
To implement an Optical Leaky Feeder (OLF), we devise a method of bending the fiber optics to allow part of the transmitted light to penetrate the cladding area to act as a transmission aperture (slot). These bends will act as an array of light-transmitting points. This will help in reducing the complexity of having multiple transmitters along the fiber. The effect of bending the optical fiber can be modeled as a power loss of the transmitted light signal. At bends, some of the rays escaped the optical fiber due to the incident angle being less than the critical angle. In [4,8], the loss due to bending in the cladding was derived. In [9,10], an experimental formula was investigated. Here, we make use of the results obtained by [11, 12, 13] and study the light power escaped from the fiber bend to the outer surrounding media. Since we need to calculate the emitted light from an OLF, the overall effect can be modeled as a skewness in the light intensity distribution as shown in Figure 1.  
In a straight optical fiber, the intensity distribution can be written as [14]:  
\[I_{1}(r)=\frac{1}{w_{0}\sqrt{2\pi}}e^{-\frac{r^{2}}{2w_{0}^{2}}}, \tag{1}\]  
where \(\mathbf{w_{0}}\) is the Mode Field Diameter (MFD) of the fiber with a core radius of \(\mathbf{a}\) and it can be written as [10]:  
\[w_{0}=2a\bigg{(}0.65+\frac{1.619}{V^{\frac{1}{2}}}+\frac{2.879}{V^{6}}\bigg{)}, \tag{2}\]  
where the Normalized Frequency V for step-indexed fiber is given by:  
\[V=\bigg{(}\frac{2\pi a}{\lambda}\bigg{)}\sqrt{\frac{n_{2}^{2}-n_{1}^{2}}{n_{0 }^{2}}}, \tag{3}\]  
where \(\mathbf{n_{0}}\) is the Refractive Index (RI) of the surrounding medium, \(\mathbf{n_{2}}\) and \(\mathbf{n_{1}}\) are the RI of the core and cladding, respectively. Similarly for the graded index fiber:  
\[V=\bigg{(}\frac{2\pi a}{\lambda}\bigg{)}\sqrt{\frac{n(r)^{2}-n_{1}^{2}}{n_{0} ^{2}}}. \tag{4}\]  
Here, the graded index might have several shapes and is usually a function of \(\tau,\lambda\) is the wavelength of light propagating through the fiber, \(\mathbf{n_{0}}\) is the Refractive Index (RI) of the surrounding medium, \(\mathbf{n_{1}}\) are the RI of the cladding and \(n(r)\) is the RI profile in the core for the graded index fiber. In a curved optical fiber, the intensity distribution can be modeled as a skewed function of the Gaussian distribution in (1). It can be written as [15, 16, 17]:  
\[I_{2}(r,\alpha)=\frac{2}{w_{0}\sqrt{2\pi}}e^{-\frac{r^{2}}{2w_{0}^{2}}}\bigg{(} 1+\text{erf}\bigg{(}\frac{\alpha r}{w_{0}}\bigg{)}\bigg{)}, \tag{5}\]  
Figure 1: Light intensity profile before and inside fiber bending. where \(\alpha\) is the skewness parameter and can be determined following the steps in [18]. There will be micro bending loss as well as macro bending loss, such that, when the bending radius R decreases, the losses will increase. These losses practically vanish when the bending radius is larger than the critical radius Rc [19]. The relation between \(\alpha\) and R can be written as:  
\[\alpha=\frac{\alpha_{0}}{\left(R_{c}-R\right)}\ R<R_{c}, \tag{6}\]  
where \(\alpha_{0}\) is a constant (Unitless) and R\({}_{c}\) is given by [19]:  
\[R_{c}\cong 20\frac{\lambda}{\left(n_{2}-n_{1}\right)^{\frac{3}{2}}}\left(2.748-0. 996\frac{\lambda}{\lambda_{c}}\right)^{-3}, \tag{7}\]  
where \(\lambda\) is the wavelength of the propagated light and \(\lambda_{c}\) is the cutoff wavelength, which is given for step index by:  
\[\lambda_{c}=\left(\frac{2\pi a}{2.405}\right)\sqrt{\frac{n_{2}^{2}-n_{1}^{2}} {n_{0}^{2}}}, \tag{8}\]  
and graded index fiber is given by:  
\[I_{1}(r)=\frac{1}{w_{0}\sqrt{2\pi}}e^{-\frac{r^{2}}{2w_{0}^{2}}}, \tag{1}\]0  
where \(\beta\) is a manufacturing constant which depends on the index profile \(\mathbf{n}(\mathbf{r})\). Usually, \(\beta\) = 2 is a typical number used in industry and \(\mathbf{n_{c}}\) is approximated to be the average of \(\mathbf{n}(\mathbf{r})\).