Leaky modes eigenvalue equation

So far we have presented all the properties necessary for understanding the physical attributes of leaky modes, and for determining their excitation. All these properties, though, require a knowledge of the complex propagation constant for each leaky mode. The solution of the eigenvalue equation for values of F below cutoff must, in general, be performed numerically. Only for weakly leaky modes when V is close to cutoff are analytical solutions available [13]. We now consider examples. 

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 leaky mode eigenvalue equation of Eq. (24-30) and its solutions U U are independent of A, but the corresponding attenuation coefficient is very sensitive to the value of A. If we equate real an imaginary components of 17 using the definition at the back of the book, eliminate and substitute the solution of the resulting quadratic equation for (P f into Eq. (24-20), then... 

The eigenvalue equations for leaky TE and TM modes on the step-profile planar waveguide in the weak-guidance approximation are obtained by setting IF = — t Q in the TE mode eigenvalue equations of Table 12-2, page 243. Hence... 

U and Q are defined in terms of 6 by Eq. (21-37). Consequently, as 0 Varies, the peak values of Cf(6) occur at the minima of G([/). Now G(U) vanishes only at a leaky-mode solution of the eigenvalue equation of Eq. (24-30) in these cases U is not real. Since Eq. (21-37) requires U to be real, we deduce that G([/) has small but finite minima when U is approximately equal to the real part of the leaky-mode solution. Thus the peaks in Cj(6) correspond to the finite minima of G U ) and, therefore, to the excitation of leaky modes, i.e. to resonances in the fiber cross-section, as illustrated in Fig. 21-6(0). 

The above decomposition complements our intuitive description of leaky modes in Chapter 24, and shows formally that the fields of leaky modes have the same analyticalforms as bound-modefields. Each leaky mode is associated with a transverse resonance in the fiber cross-section, specified by the eigenvalue equation Wi(U,Q) = 0. Leaky modes only contribute to the radiationfield within an angular sector of the (r,z)-plane of Fig. 26-2. Consider the qith leaky-mode pole with coordinates (steepest descent path sd and the path p in Fig. 26-1 (b). In this case (J, and f/, satisfy the inequalities in Eq. (26-10). As r and z vary, the angle 6 defined by Eq. (26-10) will vary and the steepest descent path defined by Eq. (26-9) will be continuously modified. Clearly, there is a maximum angle 0, for which Eq. (26-10) is only just satisfied, when the leaky-mode pole lies on the steepest descent path. In this case. 

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