## 2 Development of Thin Film Blackbody Radiation Source
The results of modeling in CST with numerical calculation from [6] for film with a sheet resistance of 550 \(\Omega/\) on sapphire substrate (340 \(\upmu\)m thick) (Figure 1b) was compared to check the method. The modeling of films with different resistance values (\(\Omega/\) ) is presented in Figure 2. The largest absorbance was observed for films with resistance of 188 \(\Omega/\). The bare dielectric substrate with refractive index \(n\) at both sides should have a reflection coefficient \(R=[(n-1)/(n+1)]^{2}\), so for silicon with \(n=3.4\), \(R=0.3\). Antireflection coating with electrical thickness of one-quarter wavelength and refractive index \(n_{\text{a}}=n^{0.5}\) can be used to decrease reflections in the relatively narrow band. The modeled absorption coefficients of film with resistance of 188 \(\Omega/\) on different 340 \(\upmu\)m substrates (sapphire, quartz, silicon) and on substrate with additional antireflective coating are presented in Figure 3.  
Figure 1: Methods of modeling absorption coefficient: (**a**) schematic image of modeled project (cross-section); (**b**) results of modeling for film with resistance of 550 \(\Omega/\) on 340 \(\upmu\)m sapphire substrate and comparison with calculation of same film from [6] (Reprinted with permission. Copyright 2021 Lemzyakov, S.A.). Figure 2: Modeled absorption coefficient for films with different resistance values (\(\Omega/\) ): (**a**) irradiation from film side; (**b**) irradiation from substrate side. Figure 3: Modeled absorption coefficient of film with resistance of 188 \(\Omega\)/ on different substrates: (**a**) sapphire, silicon and quartz 340 \(\upmu\)m substrates, irradiation of film side (**left**) and substrate side (**right**); (**b**) quartz 340 \(\upmu\)m substrate with antireflection coating made of 65 \(\upmu\)m-thick Teflon, irradiation from film side (**left**) and substrate side (**right**); (**c**) comparison of results of modeling absorption coefficient of thin film on quartz substrate without and with antireflection coating and with additional backshort behind antireflection coating. When using a substrate of finite thickness, the absorption spectrum becomes more complex because the periodic spectrum of the Fabry-Perot dielectric resonator is superimposed on the smooth dependence. This dependency is shown for the case of a resistive thin film with resistance per square of 30-300 \(\Omega/\), which is too thin compared to the skin depth. So, for NiCr film with resistivity \(\rho=10^{-6}\)\(\Omega\)m, the resistance per square is 180 \(\Omega/\)  if the film thickness is 6 nm, but the skin depth of such film at \(f=100\) GHz is \(\Delta=503(\rho/f)^{0.5}=1.5\)\(\upmu\)m. The calculated dependence of the skin depth on frequency is shown in Figure 4. This means that more than 90% of power is transmitted through the substrate (see Figure 5). To increase the emissivity of a flat BB absorber, it is favorable to use relatively thick free-standing foil instead of an absorber comprising thin film on a dielectric substrate. For constantan with resistivity of 0.5 \(\upmu\Omega\) m at a signal frequency of 350 GHz, the skin depth is 0.6 \(\upmu\)m, so 50 \(\upmu\)m-thick foil is definitely opaque. In [7], 50 \(\upmu\)m constantan foil was mounted (glued by black thermally conductive STYCAST epoxy on copper foil to equalize the temperature) at a temperature stage of 0.4 K. On the reverse side, two resistors were glued to the copper carrier, one to measure the temperature and one for heating, and a miniature permanent magnet was mounted on this rotatable holder. The reflectance coefficient of the foil at a frequency of 350 GHz was measured by a BWO spectrometer/reflectometer;  
Figure 4: Dependence of skin depth on frequency for film with resistance of 180 \(\Omega/\). \[P_{CW}(f)=\frac{hf}{2}+\frac{hf}{exp\left(\frac{hf}{kT}\right)-1} \tag{2}\]0 Calculated absorption coefficient of films with different thickness. Irradiation from (a) film side and (b) substrate side.