## 4 Single Pixel Optimization
### ac Bias  
TES microcalorimeters to be read-out under ac bias for the MHz-FDM-based experiments require a different optimization process than the one described above. The uniformity of the transition over a large array is less critical, due to the fact that in FDM each pixel can be biased individually to their optimal point. The performance of MHz biased TES is deteriorated by two major frequency dependent effects: (i) the ac losses from eddy currents generated in the normal metal structures and (ii) the ac Josephson effects describe in Section 2. The former induces thermal dissipations inside the TES and absorber [64; 117], which effectively decrease \(\alpha\) and the detector sensitivity, in particular at high bias frequency and for low resistance bias points where the best signal-to-noise is generally achieved. The latter is responsible for a relatively large non-linear Josephson inductance in parallel with the TES resistance. At high bias frequency, it reduces the range for optimal biasing at low TES resistance and generates step-like structures in the TES transition [64; 118; 119]. Through an experimental and theoretical study, the groups at SRON and NASA-GSFC have identified several optimal TES designs for the ac-bias read-out. They implemented changes both in the TES geometry and the fabrication process. The ac losses have been minimized to a negligible level by: (a) reducing the Au metal features, like the stripes and the edge banks, in the proximity of the TES and the leads, (b) making the non-microstriped loop area formed by the TES and the bias leads smaller, and (c) increasing the TES aspect ratio (AR = \(L\times W\)), where the width (W) is reduced relative to its length (L), to achieve higher value of \(R_{N}\). Moving towards higher and higher TES normal resistances, \(R_{N}\) has also been the strategy to minimize the Josephson effects in ac bias devices, as it follows from the predictions of the theoretical models presented in Section 2.2. In TESs with higher \(R_{N}\), operating at the same power, the gauge invariant phase difference across the Josephson weak link is maximized since \(\varphi\propto\sqrt{P_{sat}R}/f_{bias}\) and the TES is less affected by the Josephson effects. Similarly to the dc bias pixel optimization, a further strategy was to simplify the geometry of the TES by removing additional metal features such as the Au stripes that are responsible for a complex resistive transition. Both these actions are compliant with those also implemented to reduce the ac losses. The dependence of the Josephson effect on TES power \(P\), \(R_{N}\) and bias frequency \(f_{bias}\) is phenomenologically demonstrated in Figure 10.  
Figure 10: TES I–V curves for different X-ray microcalorimeter types and MHz bias frequencies. In the lower graph of each plot the ratio \(I_{Q}/I_{I}\) of the quadrature or Josephson (\(I_{Q}\)) and in-phase \(I_{I}\) current is shown. In (**a–c**), the bias frequency is, respectively, \(f_{bias}=1.4,4.0\) and \(3.8\,\mathrm{MHz}\). The detector is an NASA-GSFC Mo/Au bare TES [113; 117] with \(R_{N}\simeq 8\)–\(9\,\mathrm{m\Omega}\) and \(P_{sat}\sim 3\,\mathrm{pW}\)(**a,b**) and \(P_{sat}\sim 10\,\mathrm{pW}\), respectively. In (**d**), we consider a Ti/Au, \(130\times 140\,\mathrm{\SIUnitSymbolMicro}\), SRON TES [108] with \(R_{N}\simeq 220\,\mathrm{m\Omega}\), \(P_{sat}\sim 10\,\mathrm{pW}\), and biased at frequency \(f_{bias}=4.3\,\mathrm{MHz}\). Reprinted from [119] by permission from Springer Nature. Copyright ©2018, Springer Science Business Media. The ratio of the out-of-phase Josephson current (\(I_{Q}\)) to the in-phase current \(I_{I}\) is shown as a function of TES voltage for low (Figure 10a-c) and high (d) ohmic devices fabricated, respectively, at NASA-GSFC and SRON. In the top-left and top-right plots, the low power (\(P_{sat}\sim 3\,\)pW) and low resistance (\(R_{N}\simeq 8\,\)m\(\Omega\)) detectors have been bias at low (1.4 MHz) and high frequency (4.0 MHz).