Leaky modes attenuation

These properties of leaky-mode attenuation coefficients are consistent with the attenuation of the family of leaky rays in Chapter 7, which comprise the mode. This is discussed in Section 36-11. 

While this longitudinal loss is detrimental for communications or applications involving transport of energy over long distances, this property is potentially very beneficial for sensors utilizing capillaries. Most of the leaky modes will directly excite molecules immobilized on the inner surface of the capillary. The effective attenuation for each of the leaky modes is found to be inversely proportional to the diameter of the capillary and exhibits unacceptable values for all modes with the exception of a few lower order modes, corresponding to almost normal incidence at the proximal end of the capillary, i.e., Oq < 5", ... 

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

Ramskov Hansen, J. J., Ankiewicz, A. and Adams, M. J. (1980) Attenuation of leaky modes in graded noncircular multimode fibers. Electron Lett 16, 94-6. 

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. 

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. 

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... 

The attenuation of leaky rays on multimode fibers is discussed in Section 7-1. For higher-order modes, it is intuitive that there should be good agreement between the leaky-mode and corresponding leaky-ray attenuation coefficients. We discuss this agreement both qualitatively and quantitatively in Section 36-11 for step-profile planar waveguides and fibers. 

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 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... 

In Section 22-5 we determined the attenuation of the fundamental mode on a weakly guiding, step-profile fiber due to radiation from a sinusoidal perturbation of the interface, using free-space antenna methods and correction factors. Here we consider the situation when the radiation field is well approximated by a single leaky mode, which can be realized by having an on-axis sinusoidal nonuniformity of the form of Eq. (22-14). The azimuthal symmetry ensures that only HEi leaky modes are excited. Further, the direction of radiation should coincide with the direction of the leaky-mode radiation [23]. If we represent the nonuniformity and the incident fundamental-mode fields by the induced current method, as in Section 22-5, the direction condition is satisfied by setting C = in Eq. (24-43), whence... 

When the 7th leaky mode is just below its cutoff value V = the attenuation coefficient of Eq. (24-53) agrees with the corresponding expression derived by the free-space approximation and correction factor for the fiber profile. For the on-axis source, it is clear by setting ro = 0 in Eqs. (22-15) and (21-38) that the attenuation coefficient 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... 

If we substitute for T from Eq. (7-19b) and for Zp from Table 2-1, page 40, and note that p = n in the weak-guidance approximation, it is readily verified that the mode and ray attenuation coefficients are identical [7]. A similar, analytical comparison can be made for refracting leaky modes and rays on the step-profile planar waveguide [8]. 

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