## 3 Experimental Results and Discussion
In the so called Leontovich approximation, see formula A3 from [PERSON] paper [27] and other details therein, we have:  
\[\frac{1}{\lambda}=\frac{1}{2\pi d_{m}n_{m}}\text{arctan}\frac{2n_{m}n_{eff}^{2 }(\rho^{2}-n_{eff}^{2})^{1/2}}{n_{eff}^{4}+n_{m}^{2}n_{eff}^{2}-n_{m}^{2}\rho^ {2}} \tag{3}\]  
As a function of, this is simple biquadratic equation  
\[Q^{2}n_{m}^{4}\rho^{4}-2\rho^{2}(Q^{2}n_{m}^{2}(n_{eff}^{4}+n_{m}^{2}n_{eff}^{ 2})+2n_{m}^{2}n_{eff}^{4})+Q^{2}(n_{eff}^{4}+n_{m}^{2}n_{eff}^{2})^{2}+4n_{m}^ {2}n_{eff}^{6}=0 \tag{4}\]  
where \(Q^{2}=\tan^{2}\frac{2\pi d_{m}n_{m}}{\lambda}\), and its solution gives us \(\rho=1.0033+6\cdot 10^{-6}i\) corresponding to the theoretical propagation length of \(\cong\frac{\lambda}{2\pi\text{Im}\rho}=17\) mm, which is two orders of magnitude larger than the experimentally measured value. Similarly, for gold at 710 nm and for the thickness of 5 nm, cf. [8], we have \(\text{Im}\rho\cong 10^{-6}\), for palladium at 739 nm with \(d\) = 8 nm and \(n_{Pd}\) = 1.9 + 4.8\(i\)[10,27] we have \(\text{Im}\rho\cong 10^{-5}\), and so forth. Certainly, for the case of nanofilms, additional damping mechanisms exist and should be taken into account. The most well-known and fundamental one is the collision-induced scattering of conducting electrons at the walls of the nanofilm (\"anomalous skin- and size-effects\") which has been profoundly discussed starting from the end of the forties (see e.g., [30,31] and references therein; see also [27]). For our purposes, it is sufficient to present here only the final formula pertinent for the imaginary part of the dielectric constant for a continuous thin film at a frequency \(\omega\)[30]:  
\[\varepsilon^{\prime\prime}=\varepsilon^{\prime\prime}_{bulk}+\frac{3\omega_{p }^{3}}{8\omega^{3}}\frac{v_{F}}{c}\frac{1+\cosh^{2}(\omega_{p}d/c)}{\sinh( \omega_{p}d/c)\cosh(\omega_{p}d/c)+\omega_{p}d/c}(1-p) \tag{5}\]where \(d\) is nanofilm thickness, \(\omega_{p}\) is plasma frequency, \(v_{F}\)--Fermi velocity and \(p\)--the so called \"scattering parameter\" defined as a fraction of electrons specularly reflected on the walls. (Real part of the dielectric constant is practically non-affected by this same scattering). For a completely diffuse scattering, \(p=0\), Formula (5) gives the following limit for small thicknesses:  
\[\varepsilon^{\prime\prime}\cong\varepsilon^{\prime\prime}_{bulk}+\frac{3\omega _{p}^{3}}{8\omega^{3}}\frac{v_{F}}{d} \tag{6}\]  
Taking for estimations for our case of silver nanofilm \(v_{F}=1.4\cdot 10^{6}\) m/s [32], \(\hbar\omega_{p}=9\) eV [33], \(d=12.5\) nm and the frequency corresponding to the He-Ne laser used, we have \(\frac{3\omega_{p}^{3}}{8\omega^{3}}\frac{v_{F}}{d}\cong 0.4\). This is non-negligible but still not such a significant change given that \(\varepsilon^{\prime\prime}_{bulk}=n_{Ag}^{2}=-15.87+1.07i\), and it results in the new value of an effective refractive index of the \"gold in the nanofilm\"\(n_{film}=\sqrt{\varepsilon^{\prime\prime}}=0.1846+3.99i\). Substitution of this value into formula (4) instead of \(n_{Ag}=0.1345+3.986i\) leads only to a minor change in the dissipation giving Im\(\rho\cong 8\cdot 10^{-6}\) The same conclusion can be drawn applying formula (6) for other cases, such as gold and palladium. This same aforementioned noticeable difference between the experimentally observed and theoretically possible propagation lengths of LRSPP supported by Photonic Crystals should not be considered as surprising, because the ideal matching of the design of the PC and a real structure parameters, as well as ideal quality of all interfaces and materials, are needed (not to mention that in reality we certainly have a finite PC only). In any case, obtained values are already quite impressive and suitable for practical applications. Nevertheless, we should bear in mind that apparently there is still significant room for improvements in this field.