62 Fundamentals of Optical Devices Second, for the light beam propagating at the critical angle of the fiber as shown in Figure 1.3.10(b), the beam has an angle y 1 with respect to the surface normal of the fiber end surface, which is related to f by y 1 ¼ ð p=2 À y c Þ þ f. Then the y angle of the reflected beam from the end surface with respect to the fiber sidewall can be found as y ¼ p À y c À 2y 1 ¼ y c À 2f. This angle also has to be smaller than the critical angle, that is, y c À 2f < y c , or f > 0 In the third case as shown in Figure 1.3.10(c), the light beam propa- gates at the critical angle of the fiber but at the opposite side as compared to the ray trace shown in Figure 1.3.10(b). This produces the smallest y 1 angle, which is, y 1 ¼ f À ð p=2 À y c Þ. This corresponds to the biggest y angle, y ¼ p À y c À 2y 1 ¼ 2p À 3y c À 2f. Again, this y angle has to be smaller than the critical angle, y < y c , that is, f > p À 2y c Since in this example y c ¼ sin À1 n 2 1:467 ¼ sin À1 ¼ 86:34 n 1 1:47 The three constraints given above become f > 1:83 , f > 0, and f > 7:3 , respectively. Obviously, in order to satisfy all these three specific conditions, the required surface angle is f > 7:3 . In fact, as an industry standard, commercial APC connectors usually have the surface tilt angle of approximately f ¼ 8 . 1.3.3 Optical Fiber Attenuation Optical fiber is an ideal medium that can be used to carry optical signals over long distances. Attenuation is one of the most important parameters of an opti- cal fiber; it, to a large extent, determines how far an optical signal can be deliv- ered at a detectable power level. There are several sources that contribute to fiber attenuation, such as absorption, scattering, and radiation. Material absorption is mainly caused by photo-induced molecular vibration, which absorbs signal optical power and turns it into heat. Pure silica molecules absorb optical signals in ultraviolet (UV) and infrared (IR) wavelength bands. At the same time, there are often impurities inside silica material such as OH -