# A FBG Intensity Modulation System Combined with an Optical Whispering Gallery Mode Edge Filter
## 2 The Operating Principles of the Whispering Gallery Mode (WGM) Edge Filter  
### The Bend Loss for Single-Mode Optical Fibers  
In 1976, [PERSON] offered a new kind of method for calculating the bend loss of a bent optical fiber that could also be used to calculate the general bending radiation loss [12]. In 1986, [PERSON] _et al._ also investigated the bend loss of a high-numerical-aperture single-mode fiber as a function of the wavelength and bend radius [13]. In 1992, [PERSON] proposed a simple approach to calculate the bending losses of coated single-mode fibers and indicated that the losses of the fundamental mode in single-mode fibers are influenced by their fiber bending curvatures [14]. In single-mode optical fibers, the light lost from the fundamental mode at a bend comprises two components. One is the transition loss and the other is the pure bend loss. The transition loss arises from the fundamental mode to leaky core modes when there is a variation in curvature of the fiber axis. On the other hand, the pure bend loss results from the continual loss of guidance at the outer portion of the evanescent field of the fundamental mode. Assuming that a bend of radius \(R\) in a step index single-mode fiber with a core of radius \(a\) and refractive index \(n_{\text{co}}\) is surrounded by a cladding of refractive index \(n_{\text{cl}}\), this is shown in Figure 1. The ratio of the output to the input power, neglecting the intrinsic attenuation of the fiber, for the fundamental mode can be presented as [13]:  
\[\frac{P_{o}}{P_{i}}=\frac{P_{o}}{P_{i}}\mid\mathbf{a}\times\frac{P_{o}}{P_{i}}\mid \mathbf{b}\times\frac{P_{o}}{P_{i}}\mid\mathbf{c} \tag{1}\]  
where \(P_{i}\) and \(P_{o}\) are the input and output optical power; the arc \(\mathbf{b}\) expresses the pure bend loss section and the arcs \(\mathbf{a}\) and \(\mathbf{c}\) illustrate the transition loss section.  
Due to the symmetry, the transition components are equal so that  
\[\frac{P_{o}}{P_{i}}\mid_{\mathbf{a}}=\frac{P_{o}}{P_{i}}\mid_{\mathbf{c}} \tag{2}\]  
and the pure bend loss component can be expressed by  
\[\frac{P_{o}}{P_{i}}\mid_{\mathbf{b}}=\exp(-2\alpha L) \tag{3}\]  
where \(L\) is the length of arc b.  
We substitute Equations (2) and (3) into Equation (1) and rewrite it as follows:  
\[\ln\left(\frac{p_{o}}{P_{i}}\right)=-2\alpha L+2\ln\left(\frac{P_{o}}{P_{i}} \right)\mid_{\mathbf{a}} \tag{4}\]  
The pure bend loss coefficient \(2\alpha\) for the fundamental mode is given as follows [12,13]:  
\[2a=\frac{\sqrt{\pi}\kappa^{2}}{2\gamma^{3/2}V^{2}\sqrt{R}K_{+1}^{2}(\gamma a) }\exp(-\frac{2}{3}\frac{\gamma^{3}}{\beta^{2}}R) \tag{5}\]  
where \(K_{+1}(\gamma\;a)\) is a modified Hankel function, \(a\) is the radius of the fiber core and \(R\) is the bend radius. \(\kappa=\left(n_{co}^{2}k^{2}-\beta^{2}\right)^{1/2}\), wherever \(n_{co}\) is the refractive index of the core. \(k=\frac{2\pi}{\lambda}\), wherever \(\lambda\) is the wavelength  
Figure 1: The input power \(P_{i}\) and output power \(P_{0}\) transmission and the optical path formation of the whispering gallery mode generated in a bending single-mode optical fiber.  
in free space. \(\beta=n_{cl}k(1+b\Delta)\), where \(b=\dfrac{(\beta/k)^{2}-n_{cl}^{2}}{m_{co}^{2}-n_{cl}^{2}}\), \(\Delta=(n_{co}^{2}-n_{cl}^{2})/2n_{co}^{2}\), and \(n_{cl}\) is the refractive index of the cladding. \(V=ak(n_{co}^{2}-n_{cl}^{2})^{1/2}\), and \(\gamma=(\beta^{2}-n_{cl}^{2}k^{2})^{1/2}\).  
It can be seen from Equations (1)-(5) that when light propagates through the bend optical fiber, the bend loss will occur and is inversely proportional to the bend radius.