d (i, j) dispersion-flattened fibers. Reprinted from Ref. [2], p. 125, courtesy of Irwin. where b is the phase constant of the particular LP mode, k is the propagation constant in vacuum, and N1 and n2 are the core and cladding refractive indexes, respectively. Marcuse et al. [4] have shown that to a very good accu- Equation (6.1.5) is very cumbersome to use because racy the following formulas can be used to calculate b for b has to be calculated from Eq. (6.1.3). For LP modes different LPlm modes: LP01:b01 = (6.1.6) lm=01 Fig 6.1-1 Typical fiber geometry. Reprinted from Ref. [1], p. 12, courtesy of Academic Press. Optical fiber, cable and connectors CHAPTER 6.1 Exact solution Gaussian approx. cladding Norm. radius Fig. 6.1-4 Cutoff frequencies for the lowest order LP modes. Fig. 6.1-3 Comparison between exact and Gaussian approxima- Reprinted from Ref. [1], p. 15, courtesy of Academic Press. tion of the radial mode field distribution for the lowest order mode. Reprinted from Ref. [1], p. 14, courtesy of Academic Press. mode is larger than the geometrical diameter in a single- mode (SM) fiber and much smaller than the geometrical diameter in a multimode (MM) fiber. The optimum (6.1.7) MFD is given by the following formula [7]: Wg = 0.65 + 1.619.V-3/2+2.87.V-6 , (6.1.11) where a is the core radius. Equation (6.1.11) is valid for wavelengths between 0.8 Aco and 2 Aco. A = IT 4(l-1)2. - If the radial distribution for higher order modes is needed, it is necessary to use the Bessel functions [3]. In Fig. 6.1-6 the radial intensity distribution is shown for The graphs in Fig. 6.1-4 were generated using Eqs (6.1.6) five LP modes in a fiber with V = 8. Recommended and (6.1.7). The normalized propagation constant b specifications for a single-mode fiber are summarized in can vary only between 0 and 1 for guided modes; this Table 6.1-2. corresponds to k<B<nkk (6.1.8) Table 6.1-1 Cutoff frequencies of various LP modes in a step index fiber The wavelength for which b is zero is called the cutoff wavelength, i.e., l = 0 modes