**6. Analytical approach for omniguiding fibers**

Various attempts have been made to analyze the modes in omniguiding Bragg fibers. The most common approach remains as the use of transfer matrix theory that can be applied to any cylindrically symmetric fiber structure surrounded with periodically stacked Bragg cladding [23]. In this kind of formalism, the exact treatment of arbitrary number of inner dielectric layers is taken into account, and the structure of the outermost clad is approximated in the asymptotic limit. The exploitation of transfer matrix theory can yield the confined modes in Bragg fibers by minimizing the radiation loss in the radial direction. Apart from this technique, the asymptotic analysis and finite difference time domain (FDTD) method may also be used [3].

The method of asymptotic analysis of omniguiding fibers involves dividing the bulk of periodic multi-layered cylindrical dielectric mirrors into two groups, namely the inner and outer ones. The former one is in close proximity of the core section, whereas the latter group is assumed to be at relatively larger distance from the center of fiber. In the analyses, however, both the kinds of groups involve several dielectric mirrors. Further, the field in the inner group is represented by Bessel functions, whereas that in the outer group is treated asymptotically using the plane wave approximation. It has been found that the results obtained in this formalism match very well with those achieved by implementing the FDTD technique and/or the transfer matrix method [5].

In the analyses of omniguiding Bragg fibers, the propagation of Bloch waves is of extreme importance to determine the nature of propagation. Within the context, Bloch wave constant remains vital to evaluate as a complex value of it shows the forbidden bands of the periodic structure, whereas a real-valued Bloch constant indicates the propagation of waves. In the former case, however, the fields are evanescent.

Bragg fibers support modes that lie above the light line. These modes have the wave vector that corresponds to a frequency situated at the band-gap of multilayered dielectric mirror. The imaginary part of wave vector indicates the radiative loss of modes that decreases exponentially with the increasing number of layers.

As stated before, the light wave propagation in Bragg fibers can be investigated in the analogy of electron flow in periodic lattice structures. This can be utilized in order to determine the working principle of omniguiding optical fibers. As such, the allowed and forbidden regions of these guides may be obtained by exploiting the quantum theory of electrons in solids. This has been justified that the use of simple Bloch formulation in omniguiding fibers exhibits continuous electric fields and power in dielectric boundaries [24, 25]. Furthermore, it has been found that, in the case of omniguiding fibers, the number of allowed and forbidden bands increases with the increase in the difference between the values of refractive index of different dielectric layers. Furthermore, the widths of the allowed band remain larger in the case of fibers having stacked layers of larger thickness values [26]. In the dispersion characteristics of omniguiding fibers as well, it has been found that the width of allowed range decrease with the increase in *k*/*k*0, *k* and *k*0 being the wave vector in the medium and that in the free-space, respectively. This has been demonstrated through obtaining thick curves [26] that represent the existence of allowed and forbidden bands of wavelengths, instead of simple lined curves shown by conventional optical fibers.
