Bound modes power

The total bound-mode power for a weakly guiding fiber of arbitrary cross-section... 

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. 

The total bound-mode power at each position along the perturbed fiber is found by summing the power in each bound mode using the expressions in Table 13-2, page 292, with a replaced by b j (z). Hence the power flowing in the positive z-direction is given by... 

Under these conditions, maximum bound-ray power results when satisfies Eq. (4-33), i.e. 9 S rtcolf o) c> nd all the light emitted from the source falls on the core endface when f9 p, where 9 is the maximum angle of emission relative to the fiber axis. By combining these two relations, all source power excites bound modes provided (P/ d)( co/ o)9c- Thus the maximum bound-ray power increases by a factor of [rjpy- compared to placing the source directly against the end of the fiber. 

The presence of a lens in front of the endface of a fiber can increase the amount of power entering the bound modes. Consider the situation in Fig. 20-5 where a... 

In the previous chapter we examined the excitation of modes of a fiber by illumination of the endface with beams and diffuse sources, i.e. by sources external to the fiber. Here we investigate the power of bound modes and the power radiated due to current sources distributed within the fiber, as shown in Fig. 21-1. Our interest in such problems is mainly motivated by the following chapter, where we show that fiber nonuniformities can be modelled by current sources radiating within the uniform fiber. Thus, isolated nonuniformities radiate like current dipoles and surface roughness, which occurs at the core-cladding interface, can be modelled by a tubular current source. 

The current sources in Fig. 21-1 launch power into bound modes, and thus specify the modal amplitudes. Expressions for these amplitudes are derived from Maxwell s equations in Chapter 31. We find from Eqs. (31-35) and... 

Eq. (21-11) that the on-axis dipole excites zero power, while the dipole at = r excites maximum power. Furthermore, the power of the bound mode due to the z-directed... 

The tubular current source was described in Section 21-6, where we showed that it is ineffective in exciting bound modes unless either of the resonance conditions of Eq. (21-15) is satisfied. A similar conclusion holds for the radiation fields. If the tube length 2L is large compared to the spatial period 2n/Sl, where 2 is the frequency in Eq. (21-13), it is intuitive that power will be radiated essentially at a fixed angle to the fiber axis. This is also a consequence of Floquets theorem [7]. However, unlike the current dipole, radiation now depends on the orientation of the currents on the tube. 

When light propagates along a fiber and impinges on nonuniformities due to imperfections in the fiber, some of its power is scattered, as shown schematically in Fig. 22-1 (a). Part of the scattered power is distributed into forward-and backward-propagating modes, while the remainder is radiated. For multimode fibers, the distribution of scattered power is best treated by the ray methods of Chapter 5. Here we are primarily interested in fibers that propagate only one or a few modes. We treat the nonuniformities of the perturbed fiber as induced current sources within the unperturbed fiber. The results of the previous chapter can then be used to describe excitation of bound modes and the radiation field [1-3]. 

Fig. 22-1 (a) Darkly shaded regions denote nonuniformities which scatter incident light power into bound modes and radiation. The profile of the perturbed fiber is n x,y,z). (b) Darkly shaded regions denote corresponding induced currents with distribution J(x,y,z) within the unperturbed fiber of profile n(x,y). 

In addition to scattering into radiated power, nonunifomiities generally redistribute power amongst all bound modes, both forward- and backward-propagating, including the incident mode. Here we use the induced current representation to calculate the power in each mode. 

We now examine the scattering of power into bound modes when an x-polarized HEj j mode is incident on an isolated nonuniformity within a single-mode, step-profile fiber. Within the weak-guidance approximation, we deduce from Eq. (22-26) and Table 14-3, page 313, that... 

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. 

In the case of weakly guiding, step-profile fibers, the power of a bound mode is given by P = al iV, where the normalization is defined in Table 14-6, page 319. Setting W= —iQ and applying the transformation of Eq. (37-71) yields... 

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 . 

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