## 3 Fast Electrical Modulation of Radiation Temperature of Cryogenic Source  
For measurements of SINIS detector optical response, a source of thermal blackbody radiation inside the cryostat was used in [12]. Thin nichrome film on a sapphire or silicon substrate was used as such source. At the sides of the NiCr film, gold contact pads were made to apply the heating current. The source was suspended on thin wires for thermal and electric insulation from the copper stage at a temperature of 0.5 K. The heating current was applied through the wires connected to the Au contact pads. DC current of a few mW provided heating up the BB source to 10-15 K. The temperature of the source was measured by an RuO\({}_{2}\) thick-film resistor thermometer glued to the back side of the substrate. The radiation from the thermal source illuminated the detector through the mesh filters with 350 GHz central frequency and 70 GHz bandwidth. A hemispherical sapphire lens was used to focus the radiation onto the detector. The radiation power was calculated using [PERSON]'s formula, taking into account the frequency band of filters. The relative value of power at different temperatures of the source depended only on the accuracy of temperature measurements. To investigate the dynamic characteristics of the detector, a pulse of radiation with a sharp front was used. This pulse was formed by heating the blackbody source on a sapphire substrate. The heat capacity of sapphire is \(C_{\text{sapphire}}\) = 29 \(\times\)\(T^{3}\) nJ/K. The heat capacity of nichrome film and gold contacts with thickness less than 1 \(\upmu\)m is low compared to the heat capacity of sapphire substrate. Applying an electrical pulse with energy of 1 \(\upmu\) heats the source to \(\sim\)3.5 K, and after that, the source holds this temperature up to a hundred milliseconds. The test measurements with pulse heating of sources on silicon substrates show that the response time of the RuO\({}_{2}\) thermometer is about 200 microseconds. For estimation of the temperature and power of the source on sapphire substrate, depending on the applied power under DC current, we compared the detector response with the measured response of the BB source on silicon substrate. Figure 11: Calculated dependence of film characteristics on surface roughness: (**a**) dependence of effective conductivity on frequency for film with roughness of 2.5 \(\upmu\)m; (**b**) absorbance coefficient of thin film at different frequencies depending on value of effective conductivity. For estimation of the radiation power variation over time, we modeled the process of thermal dynamics in the source before and after heating by current pulse. A two-dimensional thermal conductivity equation was solved by the finite element method. We used the value of 0.253 \(T\) mJ/g K [13] for a specific heat capacity of NiCr. The thermal conductivity of sapphire in such calculations was taken of 3.5 \(\times\) T\({}^{3}\) W/(m K). The temperature difference between the nichrome film and the sapphire substrate at \(T\) = 4 K was less than 0.2 K and changed with temperature as \(T^{-3}\). Such estimation used the Kapitza resistance at the indium-sapphire interface [14], which is larger compared to the nichrome-sapphire clean interface. Based on this, the Kapitsa resistance was not taken into account in the calculation. Examples of such calculations are shown in Figure 12.  
At the end of the pulse, the radiation power approaches maximum and then exponentially decreases with a time constant of \(\sim\)4 us. This decay corresponds to heat redistribution from the heating area to the area under the 2-3 mm-wide gold contact pads. The BB source was heated by current pulses of 5-90 V from the pulse generator. The output signal of the preamplifier was applied to one of the inputs of the 2-channel oscilloscope Tektronix TDS 1012B operating in waiting mode. The heating pulses were fed through 1:10 or 1:100 dividers to the second input of the oscilloscope.