Backward-propagating modes

Here, the longitudinal evolution of the modes is described by the functions () and () instead of a simple exponential term as in Eq. (4) since we have to take into account both forward and backward propagation modes. Substituting Eq. (11) into Maxwell equations and taking into account the relations described by Eq. (5) we find that the functions Vm (C) and have to satisfy the following very simple set of first-... 

Here, A(AC) is the transfer matrix describing the translation by A within the section s. The mode field amplitudes p(C) and q(C) can be alternatively expressed in terms of the amplitudes of forward and backward propagating modes and b. 

The calculation of field distribution in the whole waveguide structure using the immittance matrix starts from the output section. As there are no backward propagating modes in the output section, = 0, it follows from Eq. (14) that + p =, from which we get using Eq. (15)... 

Relationships between the fields of forward- and backward-propagating modes will be used frequently. We obtain these relationships by first decomposing the fields of Eq. (11-3) into transverse and longitudinal components, denoted by subscripts t and z, so that... 

The modulus ensures that Nj is positive for all modes. In the case of backward-propagating modes, Eqs. (11-7) and (11-12) give = Nj. If we combine the orthogonality conditions, Eq. (11-10), with normalization, then for two forward-propagating modes we obtain [1]... 

In this case the dipole is oriented parallel to the fiber axis, as shown in Fig. 21-2(b). On substituting Eq. (21-7) into Eq. (21-6) and referring to Table 14-1, page 304, we find that both X- and y-polarized fundamental modes are excited, with powers Pq nd Poy, respectively, in the forward- and backward-propagating modes, where... 

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]. 

We claimed in Section 22-2 that there is negligible coupling between the two polarizations of the fundamental mode due to slight nonuniformities on a weakly guiding fiber. Thus, in the first approximation the incident even, or x-polarized, mode excites power in only the forward- and backward-propagating modes with the same polarization. Hence Eq. (22-21) is replaced by... 

The transition losses discussed in Section 23-10 do not account for power reflected into backward-propagating modes from junctions or changes in bending radius. In Section 20-2, we showed that negligible power is reflected from the endface of a weakly guiding fiber, and consequently, we are justified in ignoring reflection losses. 

The total power radiated by the dipole is found by substituting Eq. (25-30) and the normalizations of Table 25-2 into Eq. (25-9). Allowing for forward- and backward-propagating modes... 

Coupling between a forward- and a backward-propagating mode is described by Eq. (27-11) with 62 replaced by f> 2 and the sign of its derivative reversed. In the special case when the backward-propagating mode has propagation constant — Pi, the coupled equations become... 

Coupling, in the special case of resonance between a forward-propagating mode and the corresponding backward-propagating mode, is described by Eq. (27-12). We substitute the factorized coupling coefficient of Eq. (27-15) and make analogous transformations to Eq. (27-16) defined by... 

The unbarred and barred fields are those of the forward- and backward-propagating modes with propagation constant /Sj. Thus we set... 

Here we determine the amplitudes of bound and radiation modes excited by a prescribed distribution of currents J, which occupy a volume y between the planes z = Zi and z = Z2 of a waveguide, as shown in Fig. 31-1. The total fields everywhere in the waveguide are expanded in terms of the complete set of forward- and backward-propagating modes... 

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