# Hollow-Core Optical Fibers for Telecommunications and Data Transmission
## 5 Fibers with Anti-Resonant Guidance
### Optical Resonance and Anti-Resonance  
Resonance condition means a light wave is reflected back into the slab at its surface and forward again, matching wave which did not experience reflections, and both waves propagate out of the slab without loss. This occurs when the phase delay in the glass wall/slab of thickness t and made of glass with a refractive index n is of 2\(\pi\)m, where m is an integer number (1, 2, 3,...), indicating how many times the reflected wave travelled back and forth across the thickness of slab. Assuming that the refractive index of filler gas is 1, this corresponds to a resonant wavelength \(\lambda_{\rm R}\)[10; 19] as follows:  
\[\lambda_{\rm R}=2\mathrm{t}\sqrt{(}\mathrm{n}^{2}-1)/\mathrm{m}, \tag{2}\]  
and a resonance frequency \(f_{\rm R}\), as follows:  
\[f_{\rm R}=\mathrm{mc}/2\mathrm{t}\sqrt{(}\mathrm{n}^{2}-1). \tag{3}\]  
The required wall thickness is calculated as follows:  
\[\mathrm{t}=\mathrm{m}\lambda_{\rm R}/2\sqrt{(}\mathrm{n}^{2}-1). \tag{4}\]  
At resonance wavelengths, the wall or tube is transparent, and a waveguide surrounded by such structures experiences leakage; thus, the attenuation is very high. At anti-resonance, the phase shift is \((2\mathrm{m}-1)\pi\), the wave reflected back and forth is reversed in phase, and the interference with un-reflected wave is destructive--combined waves are weakened. At anti-resonance wavelengths, the waveguide exhibits the lowest confinement loss (CL). The anti-resonant wavelength \(\lambda_{\rm AR}\) and wall thickness t are as follows:  
\[\lambda_{\rm AR}=2\mathrm{t}\sqrt{(}\mathrm{n}^{2}-1)/(\mathrm{m}-0.5), \tag{5}\]  
\[\mathrm{t}=(\mathrm{m}-0.5)\lambda_{\rm AR}/2\sqrt{(}\mathrm{n}^{2}-1) \tag{6}\]  
For \(\lambda_{\rm AR}=1550\) nm and \(n=1.4440\) (fused silica), t = 0.582 \(\mathrm{\upmu m}\) for m = 1, and 1.746 \(\mathrm{\upmu m}\) for m = 2. Many fibers are designed with m \(\geq\) 2 for cladding robustness.  
The spectra of both wavelengths are periodic and the low-loss band of the fiber (often called \"window 1\" for m = 1, \"window 2\" for m = 2, etc.) can be tuned by choosing the thickness of walls and number m. The diameter of tubes or spacing of walls (in Kagome fibers) has no effect on both wavelengths, but the leakage loss rises with it.  
Each transmission window extending (theoretically) between the adjacent resonant frequencies \(f_{\rm Rm}\) and \(f_{\rm Rm+1}\) has the same bandwidth measured in the frequency domain, as follows:  
\[\Delta f=\mathrm{c}/2\mathrm{t}\sqrt{(}\mathrm{n}^{2}-1). \tag{7}\]  
In the wavelength domain, each subsequent window is narrower--see Figure 9.  
For m = 1 and m = 2, the ratio of resonant wavelengths is 2:1, and a wide low-loss band is possible, unlike HC-PBGFs. The inclusion of additional tubes in nested or double nested nodeless anti-resonant fibers (NANF and DNANF) enables the designers to further extend the low-loss bandwidth by shifting the resonant (transparency) wavelengths of each tube.  
Increasing m and/or \(\lambda_{\rm AR}\) results in larger wall thickness, which makes fiber cladding less prone to distortions during drawing, but proportionally reduces fiber bandwidth and increases the overlap of optical power with glass, because walls are thicker, while the spaces between them remain similar. This increases fiber loss if the bulk loss of the fiber material is high, e.g., at \(\lambda\geq 2100\) nm in the case of fused silica--see Figure 10 and Sections 6.1 and 6.2.  
A detailed analysis of light propagation in anti-resonant fibers can be found in [19]. Chromatic dispersion in NANF fibers was analyzed by [PERSON] et al. [40].