# Arrays of Sub-Terahertz Cryogenic Metamaterial
## 3 Metamaterial Matrices of Microwave Filters: Design, Fabrication, and Experimental Studies
### Capillary High-Pass Filters  
High-pass filters can be fabricated similarly to thin-film band-pass filters. The thin-film metal mesh filters are available from industrial companies such as Virginia Diodes in the US and QMC in the UK. As with any thin-film mesh filters, they have some disadvantages, such as the Wood anomaly and losses at the edges due to the skin effect. The alternative solution is a waveguide filter. The properties of evanescent waveguides were studied in detail in [24]. The suppression of the long wavelength portion of radiation in this filter can be very high. Another important advantage of the evanescent waveguide as a filter is the sharpness of the transmission cutoff. In this case, the characteristics of the filters can be adjusted within wide limits by changing the diameters and lengths of the waveguides. The amplitude of the wave propagating in a waveguide is:  
\[\text{E}(\text{Z})=\mid\text{E}_{0}\mid\exp\{\text{i}\gamma Z\} \tag{1}\]  
where \(Z\) is the coordinate along the waveguide, E is the intensity of the electric field at point \(Z\), \(\text{E}_{0}\) is the intensity of the electric field at the initial point, and \(\gamma\) is the propagation constant, described as:  
\[\gamma=(2\pi/\lambda_{\text{cr}})\ \sqrt{(\lambda_{\text{cr}}/\lambda)^{2}-1} \tag{2}\]  
If \(\lambda<\lambda_{\text{cr}}\), we have an ordinary wave, but if \(\lambda>\lambda_{\text{cr}}\), the propagation constant \(\gamma\) will be imaginary and the wave exponentially attenuates inside the waveguide. Figure 4: Transmission spectra for different filters: (**a**) numerical modeling of copper filters on thick substrates (quartz 100 \(\upmu\)m and Teflon 800 \(\upmu\)m); (**b**) Au thin film filters: calculations and measurements; (**c**) comparison of calculated and measured results in narrow band for Al and Cu foil filters; (**d**) calculation and measurements of Cu filter on Mylar substrate; (**e**) calculation and measurements of Cu filter on Kapton; and (**f**) calculation and measurements of Cu filter on quartz. Frequency downshift is due to the difference of characteristics in the real and modeled dielectric substrate. A single below-cutoff circular waveguide as a high-pass filter was experimentally studied in [24], and it obtained an attenuation of 6-12 dB in the passband and an attenuation of 30-60 dB in the stopband. Such a filter requires input and output horns and adapters if using in quasi-optical systems and makes construction rather complicated, large in size, and with non-uniform spectral characteristics. An array of capillaries can operate as a high-pass filter at millimeter and submillimeter waves without additional adapters. At lower frequencies, each capillary can be viewed as a below-cutoff circular waveguide. An array of capillaries does not need any adapters, it is relatively short, and can be inserted in any quasi-optical beam guide. Capillary arrays can be an alternative to perforated plates (grids) that need to be arranged as four to six parallel layers, and this pseudo high-pass filter is a band-pass filter with a bandwidth of about one octave [25]. Industrial manufacturers of multimesh terahertz filters, such as QMC Instruments (UK) and TYDEX (Russia), do not offer the high-performance high-pass filters that are required for cryogenic radiation sources intended for the calibration of superconducting bolometers. We fabricated a series of capillary array filters with different diameters and lengths of capillaries. The fabrication process consisted of the following steps: electroplating stainless or nickel-copper capillaries with Ni, filling short pieces of 10 mm inner diameter tube with as many capillaries as possible, soldering such packages with tin-lead alloy, electro-erosion cutting pieces of 1, 2, 3, and 4 mm long, and finally cleaning in an ultrasonic bath (Figure 6a). A photo of the fabricated capillary filter is presented in Figure 6b.