# Hollow-Core Optical Fibers for Telecommunications and Data Transmission
## 8 Theoretical Analysis and Simulations of Hollow-Core Fibers  
### Models  
Anti-resonant hollow-core fibers (AR-HCFs) are cylindrical waveguides in which the refractive index of the core is smaller than that of the surrounding cladding. Thus, the propagation mechanism of keeping the mode field in the fiber's core cannot be explained by the total internal reflection (TIR) phenomenon as in conventional step-index single-mode fibers. On the other hand, what distinguishes AR-HCFs from HC-PBGFs is that the former do not have a periodic cladding, which determines the existence of a photonic band gap that traps the propagating light in the core. Thus, considering the guiding mechanism, the AR-HCF fiber can be treated as a kind of leaky mode HCF. Over the years, much effort has been put into developing models that can explain the propagation mechanism in leaky mode HCF fibers. The most commonly used are the [PERSON] and Schmeltzer model, the anti-resonant reflecting optical waveguide (ARROW) model and the coupled mode model. In 1964, [PERSON] and [PERSON] published the pioneering analytical studies on the mode properties of a circular hollow-core dielectric waveguide. In their model, the air-filled core was surrounded by infinite homogenous absorption-free cladding of a higher refractive index. In this structure, regardless of the incident angle, only the partial reflection occurs at the boundary between the core and the cladding. Thus, the fraction of optical power radiates into the cladding, which causes the mode attenuation. The imaginary part of derived propagation constant \(\gamma\) represents the loss coefficient of the analyzed structure and is given by the following formula [90]:  
\[\alpha_{\text{nm}}=\text{Im}(\gamma)=(\text{u}_{\text{lm}}/2\pi)^{2}\cdot( \lambda^{2}/\text{r}^{3})\cdot\text{Re}(\
u_{1}) \tag{8}\]  
where \(\text{u}_{\text{lm}}\) is the m-the zero of the Bessel function \(\text{J}_{1-1}(\text{u}_{\text{lm}})\), l, and m are radial and azimuthal number of modes, \(\lambda\) is wavelength, \(n\) and r are core refractive index and radius, respectively, and \(\
u_{1}\) is the constant for different (TE, TM and EH) modes that depends on the refractive index of the cladding material. Using the Marcatilli and Schmeltzer model, it is also possible to calculate the bending loss according to the following formula [91]:  
\[\alpha_{\text{nm}}(\text{R})=(\text{r}^{3}/\lambda^{2}\text{R}^{2})\cdot\text{ Re}(\
u_{\text{lm}}) \tag{9}\]  
where R is the bending radius and \(\
u_{\text{lm}}\) is mode dependent constant. It is worth noting the limiting usefulness of the above model when real a AR-HCF fiber is analyzed. Formula (6) does not consider the multiple reflections of leaky modes on a complex cladding structure, which also affects the fiber mode attenuation. Moreover, the model does not support an analysis of the cladding modes. In order to explain the mentioned multiple reflections, an anti-resonant reflecting optical waveguide (ARROW) [91] was proposed and was successfully verified to analyze the light transmission in HCF [92]. In this model, the analytical formulas for resonant wavelengths (2) and anti-resonant wavelengths (3) are given and allow for the determination of the spectral bands where mode leakage is enhanced and where the propagating light is well confined in the core area, respectively. The multiband structure is similar to that obtained for the transmission spectrum in Fabry-Perot resonators. The ARROW model is based on the ray-optics approach, and the transmission bands can thus be accurately predicted when the t>> \(\lambda\) condition is satisfied. To consider the energy transfer between the core and the cladding modes, the coupled mode model was proposed. If the finite complex structure of the fiber cladding is assumed, then that high loss spectral regions result from the coupling between core (airy) modes and dielectric modes. This approach was used and numerically validated by [PERSON] and [PERSON] [93].