Tunneling leaky modes

Shaded regions denote the core and dotted regions denote evanescent fields in the cladding. A tunneling leaky mode is depicted in (a) and (bX a refracting leaky mode in (c) and a bound mode in (d). The table classifies modes according to the value of... 

We derive expressions in Section 36-13 for the number of leaky modes which can propagate on an arbitrary multimode fiber with 1. The numbers of tunneling leaky modes M, and refracting leaky modes M n on a... 

The second set of curves, in Fig. 24-3(b), give U and — IP for the HEi modes (/ = 0) with w = 2, 3 and 4, including the fundamental mode (m = 1) for comparison. Below cutoff each mode becomes a refracting leaky mode there is no tunneling leaky-mode region as in Fig. 24-3(a). These curves exhibit a feature that is peculiar to the HE, modes immediately below cutoff, there is no leaky mode solution of the eigenvalue equation [19], as indicated by the breaks in the curves for U . This discontinuity is associated with the discontinuity in for these modes, which changes from = oo... 

The fraction of leaky-mode power propagating within the fiber core is given by Eq. (24-19). Using this definition and taking the real part of the quotient of Hankel functions, we plot i/ as a function of V in Fig. 24-5 for the modes of Fig. 24-3(a). The solid curves denote bound modes and the dashed curves denote tunneling leaky modes. All refracting leaky modes have / = 1 since = p and = y4 . 

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... 

It is clear from the spacing of the cutoff values of Fin Fig. 12-2 that this result holds for all values of F. Tunneling leaky modes do not propagate on planar waveguides, and the number, of refracting leaky modes is the difference between the total number of modes and the number of bound modes... 

The number of tunneling and refracting leaky modes, M, , is therefore... 

If denotes the number of tunneling leaky modes, then we deduce from Eqs. (36-40)... 

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