# Hollow-Core Optical Fibers for Telecommunications and Data Transmission
## 3 Bragg Fibers  
An optical hollow slab waveguide lined with a Bragg reflector was proposed in 1976 by [PERSON] and [PERSON] [6]. In 1978, [PERSON], [PERSON], and [PERSON] analyzed how to design a cylindrical optical fiber with a Bragg reflector around a central cavity [7], finding that \"_it is possible, in principle, to fabricate optical fibers with a multiannulus cladding whose indices may be higher than that of a core. Such fibers are strongly mode selective and can operate single mode even with large core diameter._  
The next step was made at MIT, where [PERSON] et al. made and tested a Bragg fiber [14], utilizing experience gained during earlier work for the U.S. Air Force on dielectric \"perfect mirror\" coatings, making aircraft stealthy at infrared wavelengths. As shown in Figure 1 (left), the fiber's central cavity (core) is surrounded by cladding made of alternating layers of transparent, dielectric materials with very different refractive indices, (n\({}_{1}>\) n\({}_{2}\)) constituting a Bragg reflector, which prevents the escape of radiation propagating in this cavity. High index layers are thinner, roughly in proportion to the ratio of indices. Each pair of layers has a combined thickness of half a wavelength:  
\[\mathrm{d}_{1}\mathrm{n}_{1}+\mathrm{d}_{2}\mathrm{n}_{2}=\mathrm{N}\lambda/2, \tag{1}\]  
where d\({}_{1}\) and d\({}_{2}\) are the thicknesses of each layer, n\({}_{1}\) and n\({}_{2}\) denote their refractive indices, \(\lambda\) is the operating wavelength in a vacuum, and N is an integer number (1, 2, 3,...). The higher the n\({}_{1}\)/n\({}_{2}\) ratio, the smaller is the number of layer pairs for a given reflectivity. The design of a Bragg reflector for optical fiber is complicated by the requirement that reflection must be ensured for the full range of the fiber's operating wavelengths [7; 14] by varying thickness and reflection wavelength for different pairs of layers. The reflector may be a set of concentric cylinders made of fused silica and separated by air gaps. However, to keep the cylinders in place with sub-\(\upmu\)m tolerances, each layer of air must be subdivided into narrow cells by struts providing support--in this way, we obtain a hollow-core photonic bandgap fiber, presented in Section 4. In theory, a hollow-core Bragg fiber (solid-core Bragg fibers also exist) has unique advantages:  
1. The material of the fiber's outer tube (jacket) does not interact with guided radiation; it must only provide mechanical strength and hermeticity,
2. Bandwidth and attenuation depend solely on the design of the Bragg reflector and the properties of the materials used: refractive index, absorbance, and scattering. The fiber's bandwidth is adjusted by changing the thickness and number of layers of the Bragg reflector structure, while the absorbance of the materials used can be fairly high due to the limited penetration of guided radiation into them [15]. However, there are severe difficulties in the design and manufacture of Bragg fibers:  
* For almost 100% reflection with a low number of layers, usually 8-20 (Figure 3), the difference (\"contrast\") of the refractive indices must be high, and materials of very different chemical compositions are required. Examples include poly(etherimide) (PEI) or poly(ether sulfone) (PES) as low index materials and arsenic triselenide (As\({}_{2}\)Se\({}_{3}\)) as a high index material, with \(n\) = 1.62 (PEI) and \(n\) = 2.73 (As\({}_{2}\)Se\({}_{3}\)) at 2000 nm, respectively [15; 16]. The latter material is also poisonous. * The materials of the Bragg reflector and cladding (jacket) shall have similar melting temperatures and melt viscosities to allow fiber drawing. The thickness of each reflector layer must be precisely controlled, but the preform must include no small voids prone to shrinking and collapsing due to surface tension.