# Absorption Enhancement in Hyperbolic Metamaterials by Means of Magnetic Plasma
## 2 Materials and Methods  
The schematic illustrations of the metamaterial samples chosen for the considerations are demonstrated in Figure 1. An obliquely incident EM wave of TM polarization was assumed. It is worthwhile mentioning that we are dealing with the propagation of waves at the interface separating the magnetized plasma layer and the metamaterial layer. A nanostructured metamaterial composed of exchanging sheets of metal and dielectric results in the required extreme anisotropy [8] (Figure 1a). Seeking for the valid homogenization approach, the thicknesses of the sheet should be smaller than the size of the operating wavelength. Metallic nanowires embedded in the dielectric material might be employed as an alternative approach seeking a hyperbolic behavior [9] (Figure 1b). Silver and gold are usually chosen as the possible metals that can be embedded in a nanoporous alumina template. Aiming to have a deep insight into the main characteristics of surface waves, we employed a Drude model for the metal (i.e., silver) characterization. In this relation, permittivity is calculated as \(\varepsilon_{m}(\omega)=\varepsilon_{\infty}-\frac{\omega_{p}^{2}}{\omega^{2 }+i\delta\omega}\). The properties are found by fitting this permittivity function to a particular frequency range of the bulk material [10]. It is concluded [11] that for silver, the values of \(\varepsilon_{\infty}=5\), \(\omega_{p}=9.5\) eV, and \(\delta=0.0987\) eV provide a reasonable fit. Figure 1: Engineering hyperbolic metamaterials. (**a**) Nanostructured metamaterial comprising oscillating metallic and dielectric layers resulting in a metal–dielectric nanostructure. (**b**) Nanowire metamaterial composed of metallic nanorods implanted in a dielectric material. (**c**) Enlarged view of the nanowire metamaterial unit cell. In both (**a**,**b**), the constituent components are subwavelength, permitting application of effective medium theory. Herein, \(d\) is the diameter of nanowires. (**d**) Structure under consideration with the SPPs propagating along the \(z\) axis. The dielectric function of magnetic plasma is a permittivity tensor and is presented as follows:  
\[\varepsilon_{p}=\left[ \begin{array}{ccc}\varepsilon_{1}&0&-i\alpha\\ 0&\gamma&0\\ i\alpha&0&\varepsilon_{1}\end{array} \right], \tag{1}\]  
where \(\varepsilon_{1}\), \(\gamma\), and \(\alpha\) are the dielectric components perpendicular and parallel to the magnetization, and the magneto-optical component, respectively. Moreover, the magnetized plasma is expressed by the effective dielectric function as follows: \(\varepsilon_{v}=\frac{\varepsilon_{1}^{2}-\alpha^{2}}{\varepsilon_{1}}\), where \(\varepsilon_{1}=1-\frac{\omega_{p}^{2}(\omega+i\
u)}{\omega\left[(\omega+i \
u)^{2}-\omega_{c}^{2}\right]}\), \(\alpha=\frac{\omega_{p}^{2}\omega_{c}}{\omega\left[(\omega+i\
u)^{2}-\omega_{ c}^{2}\right]}\), where \(\
u\) is the collision frequency in plasma, \(\omega_{p}\) is the bulk plasma frequency, and \(\omega_{c}=eB_{0}/m\) is the cyclotron frequency [12]. Herein, \(B_{0}\) is the amplitude of the external magnetic field, \(e\) is the absolute charge of the electron, and \(m_{e}\) is the mass of the electron. One may obtain a dispersion expression for the surface plasmons propagating at the boundary between two anisotropic media. It is worthwhile obtaining a single surface mode with the propagation constant [13] by calculating the tangential components of the electric and magnetic fields at the interface:  
\[\beta=k\left(\frac{\left(\varepsilon_{||}-\varepsilon_{v}\right)\varepsilon_ {\bot}\varepsilon_{v}}{\varepsilon_{\bot}\varepsilon_{||}-\varepsilon_{v}^{2 }}\right)^{1/2}, \tag{2}\]  
where \(\beta\) is the propagation constant, \(k\) is the wave number in vacuum, and \(\varepsilon_{||}\), \(\varepsilon_{\bot}\) are the frequency-dependent permittivities of the metamaterial, being a highly anisotropic medium, in the parallel and perpendicular directions to the wave propagation.