# Arrays of Sub-Terahertz Cryogenic Metamaterial
## 3 Metamaterial Matrices of Microwave Filters: Design, Fabrication, and Experimental Studies
### Methods of FSS Modeling  
Alternative methods for FSS modeling can be applied in CST STUDIO SUITE, HFSS, and Mathcad projects. One approach is using [PERSON]'s theorem with the unit cell in the infinite matrix, assuming the electromagnetic field in each cell can differ only by a complex multiple. Another approach is modeling the whole array with all complex interactions between elements. A Floquet cell is a space element limited on the sides by periodic boundary conditions, with the radiation source presented by a plane wave emerging at coordinate Zmax (schematic model in Figure 15). Electric field E (and similar for magnetic field H) for a Floquet port can be presented, similar to [28], as:  
\[\text{E(x,y,z)}=\sum_{n}\sum_{m}\text{a}_{n,m}\cdot\exp\{-\text{i}\text{k}_{n }\text{x}-\text{i}\beta_{m}\text{y}-\gamma_{n,m}\text{z}\} \tag{3}\]  
where \(\text{a}_{n,m}\) are the amplitudes of n, m harmonics; k is the wave number for free space; \(\text{k}_{n}=\text{k}_{0}+2\pi\text{m}/\text{L}_{\text{x}}\); \(\beta_{m}=\beta_{0}+2\pi\text{m}/\text{L}_{\text{y}}\); \(\gamma_{n,m}=\sqrt{(\text{k}^{2}-\text{k}_{n}{}^{2}-\beta_{m}{}^{2})}\); k\({}_{0}\), \(\beta_{0}\) are determined by the applied field, which is presented by a plane wave; and \(\text{L}_{\text{x,y}}\) are the repetition periods for axes \(OX\) and \(OY\). Incident field is plane wave \(\text{k}_{0}=\text{k}\cdot\text{sin}\theta\cdot\text{cos}\phi\), \(\theta\) is inclination angle, and \(\phi\) is azimuth angle. Figure 14: (**a**) Blackbody radiation source, tunable filter at 3 K stage, and bolometer under test at 273 mK temperature stage; and (**b**) response measured with blackbody source (red) and 232 GHz IMPATT source (black). Reduced and broadened spectrum for black body source is due to broad illumination angle for such wide aperture and corresponding different resonant distances for different angles of incidence. Such a method allows us to quickly obtain rough estimations of any periodic structure, and is widely used for the simulation of periodic quasi-optical filters. For the simulation of detector arrays, such a method can provide only approximate results, because the dissipation makes phases of reflected and re-illuminated signals uncertain. For nondissipative case the lateral sides of the unit cell were allocated a Floquet periodicity conditions with a wave vector resulting from the calculated values of the wave vector within one cell. Absorption in each unit cell is independent and phases of wave vectors are indefinite, the mutual influence of the elements with actual phases is not accounted for. In Figure 15, the single-cell approach according to [PERSON]'s theorem is presented. For comparison, the results of the modeling of the whole array of conventional half-wave antennas with SINIS detectors are presented in Figure 16 (the results of modeling from [29]). Here, we should mention that such a technique is suitable for the modeling of phased antenna arrays. By scaling the dimensions of an FSS, it is easy to obtain arrays for different spectral bands. The modeling and scaling of circular antennas for the central frequencies of 90, 150, 230, 350, and 480 GHz, and photos of fabricated samples are presented in [14] and in Figure 17, with experimental results at 95 and 350 GHz. Figure 15: Modeling of half-wave antenna array using [PERSON]’s theorem in CST STUDIO SUITE. (**a**) Boundary conditions; (**b**) unit cell of the investigated structure antenna is made of gold with four integrated SINIS detectors on the Si substrate. On the back side of the substrate, a 1 μm film of an Au counter-reflector is deposited (Zmin–electrical boundary conditions). The source is a plane wave emerging at the boundary Zmax (**c**). Figure 16: Results of modeling of the whole array of 25 annular antennas as in Figure 2: (**a**) schematic image of modeled structure with description; and (**b**) optical image of fabricated array (left) and results of modeling for two polarization modes, along connection lines (mode 1) and across (mode 2), as in [22].