Gloge (ref 6) writes of radiation loss and mode coupling induced by curvature of tlie 

 fiber. A relationsliip tliat determines attenuation loss of a given mode as a function of bend 

 radius for a tv/o-dimensional analysis is given by: 



9 2a 3/2 



ce = 2 nk (d/ - 6^6) exp [-2/3 nk R (6^- - 6^6) -— ] 



where 



a - attenuation for mode defined by 6 in dB/km in graded index fiber 



n - index of refraction at core of fiber 



k = free space propagation constant - 2ir/X 



X = wavelength of light in m 



0J, = critical angle of waveguide = (2A)'''^ 



A = index difference of cladding and core 



a = radius of fiber core in m 



R = radius of bend curvature in m. 



Assuming a fiber with core radius of 25 /zm, an index of refraction of 1 .5 at the core, 

 an index difference of 2% between core and cladding, and propagating light of a 1-pim wave- 

 length, the bending loss for a mode can be calculated as a function of bending radius. Figure 

 9 is a plot for several modes as an illustration of the sensitivity of these modes to curvature. 

 For a given mode, the loss increases logarithmically with bend radius. In general, it is pri- 

 marily the high-order modes that are lost and this is borne out by the calculated numbers. 

 However, the mechanism of mode coupling works to replenish these high-order modes with 

 lower-order mode energy in an effort to reach mode equiUbrium in the waveguide. Thus a 

 spool of fiber composed of a multitude of coils presents a structure that induces an additional, 

 continuous loss. The attenuation numbers in figure 9, however, suggest that by limiting the 

 bend radius to a large value, bending losses can be kept to a very low level. Gloge determined 

 that a graded-index fiber would lose a proportion of its transmission modes equal to 



y - 2a/RA. 



Therefore, the radius at which one-half of all modes is lost is equal to 0.5 cm. Based 

 on this analysis, a minimum spool-winding diameter of 5 inches was selected as a compromise 

 between packing efficiency and low bending loss. 



Given a close-packed, precision-wound spool, the OD and length of the spool can be 

 related by the following equation: 



4 m2 (cos 30°) 39.36 

 L = 



n (0D2 - Id2) 



where 



L = spool length in inches 

 C = cable length in km 

 d = cable diameter in inches 

 OD = outside diameter of spool in inches 

 ID = inside diameter of spool in inches. 



23 



