What is a Hollow Core Fiber? What is the difference between Hollow core Fiber and a normal optical fiber? How does light propagate in a hollow core fiber, what is the mechanism? I want to write a program to simulate the propagation of an ultrashort pulse in the hollow core fiber.
 A: To understand why hollow core fibers are particular, you need to restart from ordinary optical fibers. In an optical fiber, the core has a refractive index that is higher than the one of the cladding in order to achieve guidance. This mechanism is called index-guiding. 
On the contrary, in an hollow core fiber the core region has a refractive index that is lower than the one of the cladding. A possibility, for example, is to have a core filled with air and, around it, a cladding made of a photonic crystal. In this case, the light propagation is related to a bandgap of the photonic band structure of the photonic crystal fiber considered. 
Another possible alternative is to use, for example, a noble gas in the core region, as in hollow core pulse compression, since it can sustain higher powers with respect to solid state materials, allowing a further compression of high energy ultrashort laser pulses.
To describe, from a theoretical point of view, the propagation of light in a two-dimensional photonic crystal (as hollow core fibers), the goal is the determination of the photonic band structure of the geometry considered. At this point, we can identify the light line, that gives the dispersion relation associated to a propagation in air. It is possible to demonstrate that, for given geometries, some band gaps in the the band structure of the material can open. In these band gaps, if they are above the light line it can be possible to have modes propagating in the air core, while if they are below the air line we can have surface states of the air core (bounded to the surface between the air core and the cladding). A full description of these devices is really complicated. 
For further readings:


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*Hollow core fibers at RP Photonics

*General introduction on Wikipedia to Photonic Crystal Fibers

*See "Computing Photonic Band Structure" and links therein for a reference to computational methods
