## 3 Unconventional Fibers
### The PCF Case  
In the case of a holey fiber, we separate the whole fiber circular cross-section into a set of thin cylindrical layers, variable \(\eta\) along \(\varphi\) extending beyond the cladding to take into account the surrounding air with \(\eta=1\). Each layer's thickness is \(\delta r=r_{1}-r_{2}\). We can then approximate \(n(r,\varphi)\) as \(n(\varphi)\) for the average r \(<\!\!r\!>\) = \(r+\delta r/2\). The refractive index can be written as a Fourier series, i.e., as \(n(\varphi)^{2}=\langle n\rangle^{2}+\sum_{-\infty}^{+\infty}N_{k}\exp(jl\varphi)\). Taking into account the properties of the Fourier Transform we see that \(FT(\exp(jl\varphi)\cdot f(\varphi))=f(l+l^{\prime})\) so that the expressions in the second terms of Equation (18) spread around a spectrum of harmonics. This is also to be understood as a result of successive scatterings from the bored air holes. We can now use the natural geometry of the usual hexagonal lattice to see that for each set of holes we can have either \(6k\) harmonics. For the fundamental harmonic of \(l^{\prime}=1\), the derived harmonics passing through a layer of \(6k\) holes should be \(6k+1\). Thus, for the fundamental wave crossing the successive layers it \"sees\" a different set of periodic rectangle functions that will be shown rigorously to contribute a different number of harmonics (7, 13, 19,... ). For a common harmonic to pass through, one must then take an integer product which leads to higher and higher harmonics, thus cutting out the entire spectrum apart from the last highest frequency. We conclude that for holey optical fibers, the approximation for any of its cylindrical thin layers, \(\eta(r,l)^{2}\approx\eta^{2}(r)=\eta^{2}\) suffices for further analysis of the resulting equations. Thus, the original system (18) becomes  
\[\left\{ \begin{array}{c}\frac{jl}{r}\overline{H_{z}}-j\beta\overline{H_{ \varphi}}=jn^{2}\overline{E_{r}}\\ j\beta\overline{H_{r}}-\frac{\partial\overline{H_{z}}}{\partial\tau}=jn^{2} \overline{E_{\varphi}}\\ \frac{1}{r}\frac{\partial\left(r\overline{H_{\varphi}}\right)}{\partial\tau}- \frac{l}{r}\overline{H_{r}}=jn^{2}\overline{E_{z}}\end{array} \right. \tag{20}\]  
For the usual hexagonal pattern of holes, we may utilize elementary analytical geometry to derive the two separate regions where the refractive index alternates between the air refractive index \(n=1\) value and the higher value of the crystal material. We assume that along each separate layer a large circle corresponding to each cylindrical shell of radius \(r\) from the center of the fiber to the center of a smaller hole of radius \(r<<r_{0}\) is cut while moving clockwise along the large circle. Prescribing a set of circles of successive radii \(r\) for each of which we can find the air holes (in \(1/6\) angle of the PCF) which are cut by the particular radius each time. Each arc is computed inside its respective air hole and the total sum of them divided by \(\pi/3\) expresses the average squared refractive index. As a matter of fact, the square of the refractive index in this sum is equal to one, while the refractive index in the rest arc is the square of the silica refractive index. Hence the average refractive index can be easily calculated along \(r\). In Figures 9 and 10, we show the average refractive index and the electric field  
Figure 8: Average refractive indexes of circular thin layers of elliptic and rectangular core fibers. of a hexagonal PCF, of \(n=1.46\) with a twelve layers lattice, as functions of the reduced distance from the center of the fiber for the fundamental mode. Figure 8 was generated by a MATLAB code for air-hole diameter equal to 0.8 of the air-hole distance and the air hole diameter was 3.14 times the transmitted wave length (V = 1.607847).