# Arrays of Sub-Terahertz Cryogenic Metamaterial
## 3 Metamaterial Matrices of Microwave Filters: Design, Fabrication, and Experimental Studies
### Numerical Modeling and Simulation Algorithm  
A high background power load can saturate a single bolometer. For a SINIS bolometer, the typical saturation power level is 1 pW. To increase the dynamic range of ground-based receivers, single elements are integrated in arrays, and the incoming power is distributed between many elements. Unit antenna cells can be of different types and parameters depending on the requirements for selectivity, polarization, beam pattern, etc. In our case, the most suitable is an annular ring antenna (for details, see [14; 23]). An ordinary annular ring antenna resonant wavelength corresponds to its perimeter. As an example, an antenna for 345 GHz has a resonant wavelength of 0.87 mm in a vacuum and 0.26 mm on a silicon substrate. The radius of such an antenna without additional reactance elements on a Si substrate is 0.138 mm.  
A square array of 5 \(\times\) 5 elements is depicted in Figures 16 and 17 (left). A feeding port for an illuminating array was inserted in a circular waveguide 10 mm long and 3 mm in diameter. The two polarizations were marked as mode 1 (along connection lines) and mode 2 (across connection lines) for the first circular waveguide mode E01. The difference in spectral response for the two orthogonal polarizations was determined by the interconnection elements placed along the electric field direction for mode 2. The measurement results in the main 350 GHz frequency band are presented in Section 3.3. Furthermore, we also tested the signal response at the much lower frequency of 90 GHz, which demonstrated rather effective receiving. This experiment led to the conclusion to use the configuration of electrically small antennas much less in diameter compared to half wavelengths (about \(\lambda/10\)). We reduced the antenna diameter from 300 \(\upmu\)m to 54 \(\upmu\)m for the new array comprising 9x9 antennas (see Figure 18). As a result, on the smaller area, we were able to integrate more detecting elements and to increase the saturation power for the same focal plane area. An initial simplified estimation for a unit cell in an infinite matrix with periodic boundary conditions was presented in [5; 15]. It results in a rough estimation, but it does not reflect the fine spectral features of arrays of incoherent receiving elements. For the exact spectral simulation, we developed a full model array of 9 \(\times\) 9 elements with 324 ports (Figure 18, left). The spectral response for the two polarizations are presented in Figure 18 (right). A sharp decrease below 220 GHz is determined by the cutoff frequency of the feeding waveguide. This corresponds to the actual configuration of our experimental setup [29; 30]. The obtained spectral dependence is more complicated compared with the unit cell model with Floquet ports, and corresponds more closely to the experimental results. Such arrays of electrically small antennas are equivalent to a distributed absorber that absorbs radiation in a wide frequency range. For comparison, we also simulated six different shapes of annular antennas (see Tables 1 and 2 and Figure 19).  
Figure 18: Model of 9 \(\times\) 9 array of small antennas (**left**) and calculated spectral response (**right**).