Local modes power

The ratio P /P(0) is the fraction of power radiated from the entire loop in Fig. 23-2(a), and ignores attenuation of local-mode power along the antenna. 

Starting from the normal mode approximation, one can introduce anharmonicity in different ways. Anharmonic perturbation theory [206] and local mode models [204] may be useful in some cases, where anharmonic effects are small or mostly diagonal. Vibrational self-consistent-field and configuration-interaction treatments [207, 208] can also be powerful and offer a hierarchy of approximation levels. Even more rigorous multidimensional treatments include variational calculations [209], diffusion quantum Monte Carlo, and time-dependent Hartree approaches [210]. 

We can presuppose that transport properties due to fractal nature of percolation change physical laws of dynamics. For a sufficiently randomly diluted system, even the localized modes occur for larger frequencies, which can be introduced on basis of bizarrely called fractons [462]. Their density of states then shows an anomalous frequency behavior and, again, the power laws can characterize the dynamic properties. On fractal conductors, for example, the... 

Although the fields expressed by Eq. (19-1) vary as the profile varies from section to section, the power of a local mode must be conserved along the nonuniform fiber. This requirement is automatically satisfied if we use the orthonormal forms of Eq. (11-15) for the fields in each section, i.e. replace e and hy by e-and hy, respectively, in Eq. (19-1). 

Our derivation of the local-mode fields is sometimes called the adiabatic approximation, since it assumes all changes in profile occur over such large distances that there is a negligible change in the power of the local mode [2]. Thus, although a local mode is an excellent approximation for a slowly varying fiber, it is not an exact solution. The small correction to the local-mode fields is determined by the methods of coupled local modes in Chapter 28 or by the induced current method of Section 22-10. 

While these criteria are only qualitative, they provide insight into the requisite slowness in variation along a fiber for a local mode to propagate without losing significant power. Later, in Chapter 28, we derive exact expressions for the slowness criterion from Maxwell s equations. 

Consider a clad fiber whose core radius p (z) changes along its length, such as the taper in Fig. 19-1 (a). For simplicity we assume the fiber is weakly guiding and has a step profile. As a particular local mode propagates, the fraction of its power within the core, rj z at each position z decreases as p (z) decreases. This is evident from Fig. 14-3(c) since the local fiber parameter V is proportional to p (z). 

The local-mode concept also applies to slowly varying composite waveguides, such as the two identical fibers in Fig. 19-3(a) and the pairs of nonidentical fibers in Fig. 19-4, and is therefore a powerful method for studying the properties of nonuniform couplers. 

One immediate consequence of using local modes for pairs of identical, slowly varying fibers is a simple description of power transfer due to cross-talk between fibers. If fiber 1 in Fig. 19-3(a) is initially illuminated with unit power and fiber 2 with zero power, the distribution of power along the composite waveguide is given by a simple modification to the corresponding problem for cylindrically symmetric fibers in Section 18-13. We... 

The simplicity of the local-mode description of propagation on couplers is evident in the analysis of cross-talk between the constituent fibers. If only fiber 1 is illuminated at z —/in Fig. 19-4(d), it is clear from the previous section that the P+ modeofEq. (19-11) is excited and no power enters the P. mode. Consequently propagation is described entirely by the characteristics of the 4 + mode, and thus all of its power is carried by fiber 2 for z /. In other words, there is essentially a 100% transfer of power from one fiber to the other on a tapered coupler, provided only that the slowness criterion below is satisfied. 

Induced-current representation for local modes 22-11 Redistribution of guided power... 

The philosophy for determining power loss is analogous to the construction of the local-mode fields in Section 19-1. We determine the loss from each section in terms of J, sum the loss from each section and then approach a continuum limit. 

We are primarily interested in radiation from the fundamental modes of bent, single-mode fibers. Within the weak-guidance approximation, the power radiated is insensitive to polarization, since p. Thus we can conveniently assume that the transverse electric field is parallel to the Z-axis in Fig. 23-2(a), i.e. orthogonal to the plane of the bend. Close to and within the core, the magnitude of the electric field on the bend is given by aj Fo (R) exp (ifiz), using the local-mode approxinution, where is the modal amplitude, Fq (R) is the... 

In addition to the radiation loss associated with bending of a fiber, there is a transition loss due to abrupt changes in curvature, as occur at the cross-sectional plane AA of the fibers in Fig. 23-4. As we show below, there is a mismatch between the fields on AA, and, consequently, the incident field on one side excites both the local modes and the radiation field on the other side. The power in the radiation field accounts for the transition loss [4-6]. 

However, before we begin, we emphasize that coupled local-mode theory is particularly useful in situations when there is a large transfer of power between local modes. Apart from such exceptional situations, power transfer is slight and-the induced-current methods of Sections 22-10 and 22-11 are sufficient. 

In Section 27-1 we showed that in applying the coupled mode equations there is an intrinsic restriction to weakly guiding waveguides. This restriction does not occur in the application of the coupled local-mode equations. A general solution of the coupled local-mode equations is derived for weak power transfer in the following section, and the radiation modes are discussed later in the chapter. 

The local-mode fields are orthonormal, and consequently the total guided power propagating in the positive z-direction follows from Eqs. (11-16), (11-27) and (28-1) as... 

The coupled local-mode equations can be solved approximately when only a small fraction of the total power of the perturbed waveguide is transferred between modes. We show that the first-order solution is identical to the induced-current solution of Chapter 22. For convenience we assume that only the Ith forward-propagating local mode is excited at z = 0. To lowest order we ignore coupling to all other modes. The solution of Eq. (28-2) is then... 

An important application of the solution of the coupled local-mode equations for weak power transfer determines how slowly a waveguide must vary along its length in order that an individual local mode can propagate with negligible variation in its power. If we assume the /th local mode alone is initially excited with unit power, i.e. h((0) = 1, then the fraction of power excited in the jth... 

The power P (z) in the E mode is given by Eq. (28-10) with F = C- +/5) , where (5/3 = 2C and C denotes the mean value of C (z) over the taper length. Since the integrals in C + and C decreases exponentially with increasing separation, it is clear that P- (z) will be small only if the taper angle is minute. In other words, for a given taper angle the local-mode description is acutely sensitive to separation, as anticipated in Section 19-9. 

The solution of Eq. (29-30) must satisfy the requirement that the total power Pi (z) -I- P2 (z) of the local modes is the same everywhere along the system. On substituting Eq. (29-30) into Eq. (29-9), we deduce that the functional forms of p+ and p are given by... 

In the central region of the coupler where the fibers are virtually identical, it follows that F = a+ =1, and each fiber carries half the total power. At the end of the taper, z = L, the dissimilarity between vhe fibers is the reverse of that at z = 0, and thus 2 la this case f = 0 and a+ p 1, so that Eq. (29-37) reduces to Pj (L) = 0, PiiL) = 1. In other words, all of the power initially in the first fiber has tran erred to the second fiber, which is the conclusion reached in Section 19-7 using the composite local modes of the coupler. 

In Chapter 19 we introduced local modes to describe the fields of waveguides with large nonuniformities that vary slowly along their length. As an individual local mode is only an approximation to the exact fields, it couples power with other local modes as it propagates. Our purpose here is to derive the set of coupled equations which determines the ampUtude of each mode [10]. First, however, we require the relationships satisfied by the fields of such waveguides. 

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