## 2 Bending Loss
An empirical formula for \(\mathbf{n}(\mathbf{r})\) is  
\[I_{1}(r)=\frac{1}{w_{0}\sqrt{2\pi}}e^{-\frac{r^{2}}{2w_{0}^{2}}}, \tag{1}\]1  
where \(\xi\) is a constant and a is the core radius. Figure 2 shows the light intensity profile for multiple bending radiuses from R = 100 \(\upmu\)m to R = 1800 \(\upmu\)m. The core radius a = 4.5 \(\upmu\)m, the cladding radius b = 62.5 \(\upmu\)m, \(\mathbf{n_{0}}\) = 1.0 (air), \(\mathbf{n_{1}}\) = 1.48, \(\mathbf{n_{2}}\) =1.5, \(\alpha_{0}=32,780\), and \(\lambda\) = 1550 nm. The figure shows that as the bending radius decreases, the skewness shifts toward the outer region, resulting in increased leakage. \[I_{1}(r)=\frac{1}{w_{0}\sqrt{2\pi}}e^{-\frac{r^{2}}{2w_{0}^{2}}}, \tag{1}\]2 Light intensity profile inside bent fiber for R = 100 \(\upmu\)m to R = 1800 \(\upmu\)m in 10 steps. The total light power that entered the bend can be calculated as [14;20]:  
\[I_{1}(r)=\frac{1}{w_{0}\sqrt{2\pi}}e^{-\frac{r^{2}}{2w_{0}^{2}}}, \tag{1}\]3  
The amount of optical power that escaped from the fiber to the external world can be calculated as:  
\[I_{1}(r)=\frac{1}{w_{0}\sqrt{2\pi}}e^{-\frac{r^{2}}{2w_{0}^{2}}}, \tag{1}\]4  
And hence, the transmitted power efficiency for the OLF at any slot is given by:  
\[I_{1}(r)=\frac{1}{w_{0}\sqrt{2\pi}}e^{-\frac{r^{2}}{2w_{0}^{2}}}, \tag{1}\]5  
To calculate the integral in (12) we may use the following approximation:  
\[I_{1}(r)=\frac{1}{w_{0}\sqrt{2\pi}}e^{-\frac{r^{2}}{2w_{0}^{2}}}, \tag{1}\]6  
Then, Equation (12) becomes:  
\[I_{1}(r)=\frac{1}{w_{0}\sqrt{2\pi}}e^{-\frac{r^{2}}{2w_{0}^{2}}}, \tag{1}\]7  
And this will simplify to:  
\[I_{1}(r)=\frac{1}{w_{0}\sqrt{2\pi}}e^{-\frac{r^{2}}{2w_{0}^{2}}}, \tag{1}\]8  
In seeking simplicity, let  
\[I_{1}(r)=\frac{1}{w_{0}\sqrt{2\pi}}e^{-\frac{r^{2}}{2w_{0}^{2}}}, \tag{1}\]9  
\[w_{0}=2a\bigg{(}0.65+\frac{1.619}{V^{\frac{1}{2}}}+\frac{2.879}{V^{6}}\bigg{)}, \tag{2}\]0  
Then, we can rewrite (16) as:  
\[w_{0}=2a\bigg{(}0.65+\frac{1.619}{V^{\frac{1}{2}}}+\frac{2.879}{V^{6}}\bigg{)}, \tag{2}\]1  
The first integral term can be approximated by:  
\[w_{0}=2a\bigg{(}0.65+\frac{1.619}{V^{\frac{1}{2}}}+\frac{2.879}{V^{6}}\bigg{)}, \tag{2}\]2  
The second integral term is simplified as:  
\[w_{0}=2a\bigg{(}0.65+\frac{1.619}{V^{\frac{1}{2}}}+\frac{2.879}{V^{6}}\bigg{)}, \tag{2}\]3The last integral term can be approximated as:  
\[w_{0}=2a\bigg{(}0.65+\frac{1.619}{V^{\frac{1}{2}}}+\frac{2.879}{V^{6}}\bigg{)}, \tag{2}\]4  
Then, the emitted power is given by:  
\[w_{0}=2a\bigg{(}0.65+\frac{1.619}{V^{\frac{1}{2}}}+\frac{2.879}{V^{6}}\bigg{)}, \tag{2}\]5  
This equation shows the behavior of the escaped light power as:  
\[w_{0}=2a\bigg{(}0.65+\frac{1.619}{V^{\frac{1}{2}}}+\frac{2.879}{V^{6}}\bigg{)}, \tag{2}\]6  
And the transmitted power efficiency can be approximated as:  
\[w_{0}=2a\bigg{(}0.65+\frac{1.619}{V^{\frac{1}{2}}}+\frac{2.879}{V^{6}}\bigg{)}, \tag{2}\]7  
This result shows the loss of transmitted power through emitting light outside of the cladding. Using numerical calculations, Figure 3 shows the radiation efficiency for R = 1 mm to R = 37 mm (R\({}_{\rm c}\) = 20 mm). The used optical fiber has core radius a = 4.5 \(\upmu\)m, the cladding radius b = 62.5 \(\upmu\)m, \(n_{0}\) = 1.0 (air), \(n_{1}\) = 1.48, \(n_{2}\) = 1.5, \(\alpha_{0}\) = 32,780 and \(\lambda\) = 1550 nm. The transmitted power efficiency (radiation efficiency) for R=1 mm was found to be 25% and 0% for R > 20 mm. Considering the power losses in the cladding region due to fiber bending in a single-mode optical fiber (curvature loss), [PERSON] in [10], [PERSON] in [11; 12], and [PERSON] in [13] derived a formula for the loss in the cladding region that can be written as in [18] as:  
\[w_{0}=2a\bigg{(}0.65+\frac{1.619}{V^{\frac{1}{2}}}+\frac{2.879}{V^{6}}\bigg{)}, \tag{2}\]8  
where \(R_{eff}\) is the effective bend radius, \(K_{b}\) and \(\beta\) are constants. This loss was calculated for the loss in the cladding region only at large bending radius, such that, negligible amount of light escaped the cladding area.