Introduction

Bragg Fibers and Dielectric Coaxial Fibers

5.1 Introduction

Low Index Cladding /

High Index Cladding (a) Bragg Fiber

Low Index Cladding

I

High Index Cladding

(b) Dielectric Coaxial Fiber (c) Metallic Coaxial Cable

Figure 5.1: Schematic of (a) a Bragg fiber, (b) a dielectric coaxial fiber, and (c) a metallic coaxial cable.

can be completely eliminated [105].

The possibility of guiding light using Bragg confinement was first pointed out by Yeh et al. [99], where the concept of Bragg fibers was proposed. The experimen- tal fabrication of Bragg fibers has been recently reported [102]. Fig. (5.la) is the schematic of a Bragg fiber, which consists of a low index dielectric core surrounded by cladding layers with alternating high and low refractive indices. A new approach of using Bragg reflection to transmit optical signals was suggested in Ref. [105]. In this design, Ibanescu et al. proposed to use an all-dielectric coaxial fiber to overcome problems of polarization rotation and pulse broadening in high data rate telecommu- nication. The coaxial fiber is essentially a Bragg fiber with an extra high index core, as shown in Fig. (5.lb). The cladding of the coaxial fiber is a cylindrical omnidirectional mirror, which can be designed such that there is a frequency range within which light incident from the low index medium is completely reflected back irrespective of the incident angle and polarization [108, 109, 110]. Thus analogy can be drawn between dielectric coaxial fibers and metallic coaxial cables [see Fig. (5.lb) and Fig. (5.lc)].

Based on this analogy, Ibanescu et al. predicted small dispersion for dielectric coaxial fibers.

In both Bragg fibers and coaxial fibers, we use lD Bragg reflection to achieve photon confinement. It is also possible to surround the fiber core region (silica glass or air) with silica glass patterned with two-dimensional arrays of air holes [101, 103, 104]. Such fibers are generally referred to as photonic crystal fibers, and shall not be considered here. However, we point out that as confinement mechanism, there

is no fundamental difference between lD Bragg reflection and 2D Bragg reflection.

Thus properties of photonic crystal fibers should qualitatively resemble those of Bragg fibers or coaxial fibers.

Due to the z translational symmetry of the aforementioned fibers, we can use the 2D FDTD algorithm in Sec. 2.1.1 and Sec. 2.3.2 to find dispersion and field distribution of the guided modes. This approach has the obvious advantage of being able to analyze fibers with complicated dielectric distribution. The drawback is that the numerical approach tends to be time consuming and physically less transparent.

In this chapter, we shall develop an efficient analytical method for Bragg fibers and coaxial fibers by taking advantage of their cylindrical symmetry and radial periodicity of the cladding layers.

In the original matrix formalism [99], Yeh et al. used four independent parameters to describe the solution of Maxwell equations in each layer of the Bragg fiber, and the parameters in neighbor dielectric layers were related via a 4 x 4 matrix (see also Sec.

5.2.1). Unlike the case of conventional fibers, in this approach the confined modes in a Bragg fiber were treated as quasimodes whose propagation constant and field distribution were found by minimizing the radiation loss [99]. The extra complexity associated with this matrix approach is due to the difficulty in finding the eigenmode in fiber cladding layers. For a planar air core Bragg waveguide, the eigen solution that decays in the cladding structure can be easily found according to the Bloch theorem [9]. For a cylindrically symmetric geometry, which is strictly speaking not periodic and the Bloch theorem does not apply, we cannot single out an eigen solution that decays in the fiber cladding layers. As a result, it is no longer feasible to find an exact analytical equation that determines mode dispersion by matching the cladding solution and core solution at the waveguide core-cladding interface, as in the case of conventional optical fibers [69] or planar Bragg waveguides [9].

We observe that in the asymptotic limit, the exact solutions of Maxwell equations, which take the form of Bessel functions, can be approximated as exp( ikr) /

Jr

and exp( -ikr) /

Jr

[68]. In this form, the solutions in cylindrical Bragg cladding resemble those in planar Bragg stacks and eigen solutions in the fiber claddings can be similarly

z

Core Region ,

N pl

p'l

Pco =

^{cl cl}

Cladding Region

P

cl t'cl ^{2 n'2}

nth cl~dding pair,

pll p'n

cl cl

r

Figure 5.2: Schematic of the r - z cross-section of a fiber with Bragg cladding. The dielectric layers of the Bragg fiber are classified into two regions: the core region and the cladding region, which are separated by the dash line in the figure.

found [106]. In Ref. [107], we treat the first several dielectric layers exactly and approximate the rest of the dielectric cladding structures in the asymptotic limit. We can use this method to find the dispersion relation of Bragg fibers within any desired precision simply by increasing the number of inner layers that are treated exactly.

The accuracy of the asymptotic approximation can also be estimated by comparing results obtained from treating different number of inner layers exactly, as will be demonstrated shortly hereafter.