224 Optical Fiber Communications a a Common-core area Axial displacement (which is also often called lateral displacement) results when the axes of the two fibers are separated by a distance d. Longitudinal separation occurs when the fibers have the same axis but have a gap s between their end faces. Angular misalignment results when the two axes form an angle so that the fiber end faces are no longer parallel. The most common misalignment occurring in prac- tice, which also causes the greatest power loss, is axial displacement. This axial offset reduces the overlap area of the two fiber-core end faces, as illustrated in Fig. 5.10, and consequently reduces the amount of optical power that can be coupled from one fiber into the other. Fig. 5.10 Axial offset reduces the shaded common core area of the two fiber end faces. To illustrate the effects of axial misalignment, first consider the simple case of two identical step-index fibers of radii a. Suppose that their axes are offset by a separation d as is shown in Fig. 5.10, and assume there is a uniform modal power distribution in the emitting fiber. Since the numerical aperture is constant across the end faces of the two fibers, the optical power coupled from one fiber to another is simply proportional to the common area A comm of the two fiber cores. It is straightforward to show that this is (see Prob. 5.9) d² A = 2a² arccos- comm d 2a - da² - 4212 4 (5.22) For the step-index fiber, the coupling efficiency is simply the ratio of the common-core area to the core end-face area, NF.step = =-arccos. A 2 comm 2 πα 元 d d 1 - 2a πα 2a (5.23)