Leaky modes power

Leaky modes are a useful concept if we can apply them to problems as easily as bound modes. This is facilitated by the development of a sound, physical description of the power and attenuation of leaky modes [11,13]. As there is no formal definition of leaky-mode power, we present an intuitive description. 

Fig. 24-2 (a) Intuitive description of power flow on a fiber when a leaky mode is excited at z = 0 and (b) a differential section of length dz of a step-profile fiber, showing the element of leaky-mode power dP lost to radiation. 

The radiation and guided portions of leaky-mode power are presented as intuitive concepts, since the demarcation provided by is only meaningful for higher-order modes, and the definition of 9 assumes the attenuation is small. Nevertheless, as - oo and the attenuation becomes very small, we anticipate that Pg(z) will have the same functional form as the power of a bound mode with VP replaced by —iQ, apart from the factor exp( —2j8 z). In other words, just below cutoff the guided power of a leaky mode is the extrapolation of bound-mode power, as may be verified by evaluating Eq. (24-14) for a particular fiber and then taking the limit j8 -+ 0. 

We emphasize that the definition of N given by Eq. (24-27) is formally correct for leaky modes of arbitrary attenuation. However, although the power of a bound mode on a nonabsorbing fiber is directly related to normalization in Eq. (11-22), there is no corresponding expression for the power of a leaky mock-The leaky-mode power P of Eq. (24-16) is an intuitive concept for understanding leaky modes. Only for weakly leaky modes can we express power in terms of normalization using Eq. (11-22). However, if we are only concerned with the power in the core, then Eq. (11-28) applies rigorously to both bound and leaky modes. 

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 . 

We cannot determine directly radiation from the fundamental mode using Eq. (24-48), as there is no general definition of leaky-mode power in terms of oj (z). However, we can... 

The power attenuation coefficient (2oc) for lower order leaky modes is given by [4] thus. 

The fractional power in the cladding increases with mode number and capillary length. Thus, for sensor application, excitation of higher-order leaky modes leads to direct illumination of the immobilized fluorophores on the surface. 

The fundamental modes of all waveguides considered in this text are cut off when F = 0. At cutoff the phase velocity of the mode is equal to that of a z-directed plane wave in an unbounded medium of refractive index n, but the modal fields are not TEM waves except in special cases. In general, a significant fraction of a mode s power can propagate within the core at cutoff, i.e. r]j of Eq. (11-24) is nonzero, and the group velocity differs from the phase velocity. Below cutoff, these modes propagate with loss and are the leaky modes of Chapter 24. 

The expression within the curly brackets represents a wave propagating away from the fiber. The direction of propagation determines the direction of the power radiating from the leaky mode. In the cladding this direction makes angle with the fiber axis, whence, we deduce from Eqs. (24-8) and (24-4) that... 

In Section 11-8 we defined r) as the fraction of modal power propagating within the waveguide core. Although there is no formal definition of r) for leaky modes, we can provide an intuitive expression by adopting the description of the previous section and setting... 

Our intuitive description of a leaky mode in Section 24-12 ensures that total power is conserved as the mode propagates. This property emphasizes the idea that the power attenuation coefficient y represents the rate of power loss from the guided portion of the fields to the radiation portion of the same fields, namely... 

We emphasize that leaky modes do not obey the power orthogonality of Eq. (11-13). This is because x h is not an analytic function and thus cannot be continued into the complex plane as is necessary for the above generalization. 

To illustrate the above point, consider a weakly guiding, step-profile fiber, whose leaky mode fields are given by Eq. (24-6) together with Tables 14-1, page 304, and 14-6, page 319. For example, within the core, the z-directed power density is... 

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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