## 2 Simulation Details  
The high-frequency time domain solver in CST Microwave Suite was adopted in this study. This solver used the finite-difference time-domain method to solve [PERSON]'s equations in 3D space, typically in the microwave range. Although this software did not solve the basic plasma equations self-consistently, compared to fluid plasma simulation [21] or particle-in-cell simulation [22], it was a useful tool to study the characteristics of the microwave probes mentioned in Section 1. Since electron density was a controllable input parameter in this simulation, the software allowed us to easily establish the ideal measurement accuracy by defining it as the deviation of the output electron density from the input electron density. Besides, it should be noted that this simulation did not need to define the plasma source, such as a capacitively or inductively coupled plasma source, since plasma is regarded as a dispersive dielectric material. Such a material can be represented by the Drude model, in which the plasma dielectric constant (\(\epsilon_{\mathrm{P}}\)) is given by \(\epsilon_{\mathrm{P}}=e_{0}\Big{(}1-\big{(}2\pi f_{\mathrm{P}}\big{)}^{2}/ \{\omega(\omega-\dot{\
u}_{\mathrm{m}})\}\Big{)}\), where \(e_{0}\) is the vacuum dielectric constant, \(f_{\mathrm{P}}\big{(}=8980\sqrt{n_{\mathrm{e,input}}}\big{)}\) is the plasma oscillation frequency, \(\omega\) is microwave frequency, \(\
u_{\mathrm{m}}\) is electron-neutral collision frequency, and \(n_{\mathrm{e,input}}\) is the input electron density. Here, the \(\
u_{\mathrm{m}}\) term only includes the momentum transfer collision between electrons and Argon atoms at an electron temperature of 2.0 eV, with Maxwellian distribution for simplicity. The \(\epsilon_{\mathrm{P}}\) is related to the complex wavenumber (\(k\)) from the dispersion relation for electromagnetic waves (\(k=\omega/c\sqrt{\epsilon_{\mathrm{P}}}\)) [5]. The real and imaginary parts of \(k\) are related to the refractive index and attenuation constant, respectively. Figure 1a,b show schematic diagrams of the top and cross-sections of the RCP and BCP, respectively, which are embedded in a cylindrical holder. Both probes consisted of radiating and detecting antennae; the RCP had a point-type radiating antenna and ring-type detecting antenna, while the BCP had bar-type radiating and detecting antennae. All antennae had 1.0 mm height, were insulated by a dielectric of height 2.0 mm, and were connected with a 50 \(\Omega\) coaxial line. The plasma was a cylinder of 330 mm diameter and 40 mm height, and was positioned with an interlayer (sheath) distance of 5.0 mm from the RCP and BCP, as shown in Figure 1c; here, the simulation considers the sheath as a vacuum dielectric material (\(\epsilon_{\mathrm{sheath}}=\epsilon_{0}\)). Figure 1: Schematics of the (**a**) ring-type cutoff probe (RCP) and (**b**) bar-type cutoff probe (BCP), with various geometrical parameters: \(a\)—antenna diameter; \(b\)\(h\)—antenna distances; \(c_{\mathrm{S}}\)—dielectric thickness; \(e\)—antenna length; \(df\)—antenna widths. (**c**) Geometry of the plasma and sheath. Input and output ports were located at the ends of the 50 \(\Omega\) coaxial lines. First, a Gaussian voltage pulse (\(V_{\text{in}}(t)\)) with 1 V maximum, including frequencies from zero to 10 GHz, was applied to the input port and radiated by the radiating antenna. Then, a fraction of the pulses (\(V_{\text{out}}(t)\)) entered the output port via the detecting antenna. The S\({}_{21}\) is defined as S\({}_{21}=20\log_{10}(V_{\text{in}}(f)/V_{\text{out}}(f))\) (dB), where V\({}_{\text{in}}(f)\) and V\({}_{\text{out}}(f)\) are the processed data through a fast Fourier transform of \(V_{\text{in}}(t)\) and \(V_{\text{out}}(t)\). All simulations were conducted in the open boundary condition except for the \(z=0\) plane where the input and output ports are located. On this plane, the ground boundary condition (zero tangential electric field) was applied.