# Enhancement of the Purcell Effect by the Wire Metamaterials Formed by the Hexagonal Unit Cells
## 2 Methods  
Figure 1 displays the boundary separating the structure under investigation, based on the nanowire composite [16] and the dielectric. It is worthwhile mentioning that each wire only interacts with another as a macroscopic source. The wires characterized by the permittivity \(\varepsilon_{m}\) were implanted in a host media with the permittivity \(\varepsilon_{d}\). It is worthwhile mentioning that the unit cell of the metamaterial possessed a hexagonal shape. The former arrangement was possible through utilizing the DNA origami nanostructures to guide the higher-order arrangement of the nanowires in a controlled and programmable manner [17]. In doing so, the structure under consideration stands as a promising tool for further applications. The dielectric properties of the nanowire metamaterial, in different directions, were assessed by means of the Maxwell-Garnett effective medium approximation [1], and were derived by starting from [PERSON]'s wave equation:  
\[\varepsilon_{\bot}=\varepsilon_{d}\Bigg{[}\frac{\varepsilon_{m}(1+\rho)+ \varepsilon_{d}(1-\rho)}{\varepsilon_{m}(1-\rho)+\varepsilon_{d}(1+\rho)} \Bigg{]} \tag{1}\]  
\[\varepsilon_{\|}=\varepsilon_{m}\rho+\varepsilon_{d}(1-\rho) \tag{2}\]  
Here, \(\rho\) is the metal filling fraction factor, calculated as follows:  
\[\rho=\frac{nanowire\;area}{unit\;cell\;area} \tag{3}\]  
The Drude model was adopted, with the aim of having a deeper insight into the particularities of the surface waves, and to describe the metal (i.e., silver). In doing so, permittivity was expressed as \(\varepsilon_{m}(\omega)=\varepsilon_{\infty}-\frac{\omega_{p}^{2}}{\omega^{2 }+i\delta\omega}\). The properties were found by matching this permittivity function to a particular frequency range in the bulk material [18]. It was concluded [19] that for the silver, the values of \(\varepsilon_{\infty}=5\), \(\omega_{p}=9.5 eV\), \(\delta=0.0987 eV\) gave us a reasonable fit. By making the assumption that the metamaterial unit cell had a hexagonal form, we determined the metal filling fraction (\(\rho\)). These calculations were affected by the estimates of the wires' diameters (\(d\)), and the spacing (\(S\)), as follows [20]:  
\[\rho=\frac{\pi d^{2}}{2\sqrt{3}S^{2}} \tag{4}\]  
Based on the presented assumption, it was worthwhile deriving a dispersion relation, with the aim of having a deeper insight into the mode patterns propagating at the interface, between the anisotropic media and dielectric. To have a deeper insight into the physical effects, it was particularly important to  
Figure 1: A view of the boundary, splitting up the nanowire composite and dielectric medium (**a**); a hexagonal unit cell (**b**). obtain a single surface mode by estimating the tangential components of the electric and magnetic fields at the boundary [21]:  
\[\beta=k\left(\frac{\left(\varepsilon_{d}-\varepsilon_{\parallel}\right) \varepsilon_{d}\varepsilon_{\perp}}{\varepsilon_{d}^{2}-\varepsilon_{\perp} \varepsilon_{\parallel}}\right)^{1/2} \tag{5}\]  
where \(k\) is the wavenumber being the absolute value of the wavevector in the vacuum, and \(\beta\) is the component of the wavevector parallel to the interface. By substituting Equations (1), (2), and (5), we arrived at:  
\[\beta=k\left(-\frac{\varepsilon_{d}\chi(\varepsilon_{d}-\varepsilon_{m}\rho+ \varepsilon_{d}(\rho-1))}{(\varepsilon_{d}(\rho+1)-\varepsilon_{m}(\rho-1)) \Big{(}\varepsilon_{d}^{2}+\frac{(\varepsilon_{m}\rho-\varepsilon_{d}(\rho-1 ))\chi}{\varepsilon_{d}(\rho+1)-\varepsilon_{m}(\rho-1)}\Big{)}}\right)^{1/2} \tag{6}\]  
\[\chi=(\rho-1)\varepsilon_{d}^{2}-\varepsilon_{m}(\rho+1)\varepsilon_{d}\]  
We noted that Equations (5) and (6) are valid only under the condition of the surface confinement, which can be presented in the following form:  
\[\left\{ \begin{array}{l}\Big{(}k^{2}-\beta^{2}/\varepsilon_{\parallel}\Big{)} \varepsilon_{\perp}<0\\ \Big{(}k^{2}-\beta^{2}/\varepsilon_{\parallel}\Big{)}\varepsilon_{\perp}<0 \end{array} \right.