# A Distributed Terahertz Metasurface with Cold-Electron Bolometers for Cosmology Missions
## 2 Numerical Model  
The effective surface impedance \(Z_{FSS}\) (Ohm per square) of the periodic array of CEBs should be matched to the electrodynamic environment. Unlike conventional frequency selective surfaces, which do not contain active elements, a layout of the unit cell of the bolometric array must contain DC-lines for biasing and readout. It turns out that the way how DC-biasing issue is resolved determines the polarization of the incoming signal the array is sensitive to. The equivalent diagram of the array sitting on the back of a silicon lens without a back-reflector can be represented (see Figure 1) by the two impedances, vacuum impedance (\(Z_{V\!AC}\)) and silicon impedance (\(Z_{SI}\)) connected in parallel to \(Z_{FSS}\). They are frequency independent and correspond to the characteristic impedances of the vacuum and the silicon half-spaces. If \(Z_{FSS}=90\)\(\Omega\) then _Pabs_, the fraction of the RF-power coupled from \(Z_{SI}\) to \(Z_{FSS}\), will be about 80% [14]. Figure 2 shows the model of the unit cell with dimensions 64 \(\upmu\)m \(\times\) 64 \(\upmu\)m in XY-plane and its SEM image. The array is designed for receiving linear Y-polarized signal corresponding to the Floquet mode #1 and coming from the silicon half-space. E-field of the mode#1 is parallel to the DC-biasing lines that provide \(Z_{FSS}\) close to the optimal value 90 \(\Omega\) in a wide frequency range. Gaps between the neighboring rings result in capacitive \(Z_{FSS}\) and therefore poor matching with \(Z_{SI}\). The orthogonal mode #2 is sensitive to the gaps between the neighboring rings and has poor matching because of capacitive \(Z_{FSS}\). The unit cell has periodic boundaries in X and Y directions, while the two Floquet ports are defined at the boundaries corresponding to the maximum and the minimum of z-coordinates. These parts are touching vacuum and silicon spaces correspondingly. The reference planes of both ports are touching the array of model by a lumped element with a series connection of resistance \(R\) and capacitance \(C\). For a real CEB, \(R\) is determined by the DC resistance of the normal absorber, while \(C\) is the capacitance of the two SIN-junctions connected in series to the absorber. The optimal capacitance \(C=20\,f\!F\) and the resistance \(R=100\)\(\Omega\) for each CEB. Small unit cell size allows for absorption by the bolometric array in a wide range of both frequency and incidence angle \(\varphi\). The dependence of _Pabs_ as a function of frequency and \(\varphi\) computed for the two orthogonal modes is shown in Figure 3. In these simulations \(\varphi\) is defined in YZ-plane. The focus of a silicon lens is a waist of the Gaussian beam, where the condition of the planar normally incident wave is nearly satisfied. For normal incidence the array absorbs nearly 80% of mode#1 power in the range from 100 GHz to 800 GHz. However, Figure 3a demonstrates moderate _Pabs_ for \(\varphi\) values up to 40 degrees. This fact makes the array promising for integration into a waveguide or a back-to-back horn, where the k-vector of the incident wave has tangential component to the plane of the absorbing array. Figure 1: Equivalent diagram. With a new modification of the unit cell (see numerical model shown in Figure 4a) the array of CEBs absorbs an acceptable amount of power from two orthogonal modes. The equivalent network of a single ring with the four bolometers can be represented by that of a single bolometer and the inductance of the ring \(Lr\) excited by a plane wave (see Figure 4b). Capacitive coupling between the adjacent rings in the array is associated with capacitance \(Cr\), which creates negative contribution to the imaginary part of \(Z_{FSS}\). Biasing wires between the rings can be represented by small resistance \(r\) connected in series with inductance \(Lw\).