Approximations equivalent step profile

We also discuss generalizations of the Gaussian approximation to other low-order modes. Finally, we briefly describe the equivalent step-profile approximation [4], and compare it with the Gaussian approximation. 

The disp>ersion due to bound rays on step-profile fibers is given by Eq. (3-3) in the weak-guidance approximation. If we include leaky rays, then only those tunneling rays with effectively zero attenuation are included. Since transit time is independent of skewness, i.e. independent of /, this is equivalent to reducing the lower limit on from to defined by Eq. (8-24b). Thus the difference in transit times between the fastest, on-axis bound ray (p = n ) and the slowest tunneling ray (jS = follows from Table 2-1, page 40, as [7]... 

In Chapter 11 we discussed the fundamental properties of modes on optical waveguides. The vector fields of these modes are solutions of Maxwell s source-free equations or, equivalently, the homogeneous vector wave equations. However, we found in Chapter 12 that there are few known refractive-index profiles for which Maxwell s equations lead to exact solutions for the modal fields. Of these the step-profile is probably the only one of practical interest. Even for this relatively simple profile the derivation of the vector modal fields on a fiber is cumbersome. The objective of this chapter is to lay the foundations of an approximation method [1,2], which capitalizes on the small... 

In Section 24-18, we derived the power attenuation coefficient for tunneling leaky modes on a. step-profile, weakly guiding fiber. Here we show that, for higher-order modes, Eq. (24-36) is equivalent to the power attenuation coefficient of the corresponding skew tunneling rays. The argument of the Hankel functions in Eq. (24-36) is smaller than the order. Furthermore, we assume that / is sufficiently large that the order of both Hankel functions may be taken to be approximately /. Under these conditions, we can use the approximate forms of Eq. (37-90), and for simplicity we approximate x by the middle expression in Eq. (37-90b). Hence... 

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