Plasma, which is also called the fourth state of matter, contains physically energetic ions and chemically reactive radicals, making it essential for various applications, such as semiconductor fabrication [1,2], biomedicine [3,4], and agriculture [5]. Among the constituents of plasma, electrons play a crucial role in the generation of ions and radicals [6], making the electron density measurement a subject of significant interest in plasma diagnostics. Various techniques have been developed for this purpose, including electrical probes [7,8], optical emission/absorption methods [9], laser-scattering techniques [10], and microwave probes [11]. In particular, microwave probes have attracted much attention owing to their simple design and analysis, such as cutoff probes [12,13], multipole resonance probes [14,15], and hairpin probes [16]. Recently, non-invasive microwave probes have been developed for real-time plasma status monitoring while minimizing plasma perturbation [17–20]. Kim et al. [20] developed a planar-type cutoff probe, whereas Kim et al. [21,22] optimized antenna structures, and Yeom et al. [23] demonstrated a real-time planar cut-off probe system. Schulz et al. [18] developed a planar-type multipole resonance probe and demonstrated a real-time planar multipole resonance probe system [24]. Liang et al. [19] developed the curling probe, which is a planar-type hairpin probe. They demonstrated the feasibility of the finite-difference time-domain (FDTD) method in low-temperature plasma environments.

In particular, the curling probe has attracted significant interest owing to its simple structure and capacity for the precise measurements of electron density. The curling probe consists of two antennae: i) an excitation monopole antenna at which the microwave radiates and ii) a coupled-slot antenna on which standing wave resonance occurs. The curling probe measures the shift in the resonance frequency caused by the plasma, and can infer the electron density from the shift.

Liang et al. [19] proposed the equation of resonance frequency ($f_{\text{res}}^{\lambda /4}$) based on the quarter-wavelength resonance, which is

where L is the length of the slot antenna, c is the speed of light, and ϵ_r and µ_r are respectively the relative permittivity and relative permeability of material covering the slot antenna. Equation (1) is in agreement with the FDTD simulation results [19]. Arshadi et al. [25] conducted a detailed analysis of the resonant characteristics of a spiral-slot antenna, and derived a simple equation for inferring the electron density, which includes a geometrical factor [26]. In contrast with Liang et al. [19], they found that the resonance frequency ($f_{\text{res}}^{\lambda /2}$) of the curling probe is in agreement with the half-wavelength resonance, which is

Simulation results reported by Liang et al. [19] and Arshadi et al. [26] implied that the curling probe works in either quarter-wavelength or half-wavelength resonators. Recently, Boni et al. [27] explained the transition of the curling probe from the quarter-wavelength resonance to the half-wavelength resonance depending on the probe geometry. They established an empirical equation for the resonance frequency of a curling probe via a finite-element method-based electromagnetic simulation as follows:

where W is the width of the slot, d is the aperture diameter, B is a geometric factor $B=2\left(d/L\right)\left(1-{\left(d_{\text{ant}}/d\right)}^2\right)$, and d_ant is the diameter of the monopole antenna.

In addition, Ogawa et al. [28,29] demonstrated the use of doublecurling probes to perform in-situ simultaneous measurements of the thickness of a film deposited on a curling probe and the electron density. When a dielectric film of unknown thickness is deposited on the curling probe, a frequency shift can be induced by both the plasma and dielectric film; thus, they cannot be separated by a single curling probe. To address this issue, they proposed a double-curling probe system that has individual microwave input/output ports, and they calculated two unknown variables (film thickness and electron density) by deriving two equations from each curling probe under the assumption of the same electron density on each curling probe [29].

In addition, the curling probe concept can also be applied as a simple-and-compact sensor for plasma uniformity measurements via parallelization of the curling probes because its slot antenna is compact and it has a highly distinct peak (high Q-factor) in the reflection microwave frequency spectrum (S₁₁). In this study, we demonstrate the feasibility of parallel double-curling probes for the measurement of electron densities above each curling probe by performing threedimensional electromagnetic (3D-EM) wave simulations. Unlike the separate operation of double-curling probes proposed by Ogawa et al. [29], our double-curling probe was connected via a parallel configuration with one microwave input/output port, and their resonance peaks were merged in a single S₁₁.

The remainder of this paper is organized as follows. In section 2, we introduce the single-curling probe and demonstrate its operation for electron density measurements. In section 3, we establish a doublecurling probe and analyze S₁₁ and resonance peaks under various plasma conditions. Then, we established the parallel double-curling probe, analyzed the resonance peaks, and verified its operation for multisite electron density measurements. Finally, the conclusions are presented.

2.1. Simulation configuration

We employed a 3D-EM wave simulation software, CST Microwave Studio Suite, which is a commercial software tool that is widely used to investigate microwave probes [12,22,30]. In this section, we describe two cases to characterize individual curling probes, i.e., case A and case B. Figure 1(a) shows the appearance of the curling probes, especially for case A. It consists of four parts: a monopole antenna, the dielectric, the curling slot-antenna, and the case. The curling slot antenna has a thickness of 0.5 mm, slot width of 1.0 mm, three turns, an average curl length of 120 mm, and an aperture slot with a diameter of 6.0 mm to the center, at which the monopole antenna with a diameter of 2.0 mm is located. Figure 1(b) shows a cross-sectional image of the curling probe. The curling slot antenna was connected to a case with a diameter of 34 mm and height of 17 mm. The monopole antenna is connected to the core of a coaxial cable. The microwave input/output port was located at the end of the coaxial cable. A dielectric with a diameter of 30 mm and thickness of 0.5 mm is located beneath the curling slot antenna. For the curling probe in case B, the number of curling slot antenna turns increased to four. We summarize all the dimensions of the probe parts for cases A and B in Table I.

Table 1 . Geometric parameters for curling probes used in simulation cases: case A and case B.

Parts	Geometric parameters	Case Aa	Case Bb
Curling slot-antenna	Number of turns	3	4
	Inner length of curl (A)	102.5 mm	176.4 mm
	Outer length of curl (B)	138.7 mm	201.2 mm
	Average curl length [(A+B)/2]	120.6 mm	188.8 mm
	Curling slot thickness	0.5 mm
	Curling slot width	1.0 mm
	Antenna diameter	30 mm
	Aperture diameter	6.0 mm
Ground case	Case height	17 mm
	Case diameter	34 mm
	Case thickness	2.0 mm
Dielectric	Dielectric thickness	0.5 mm
	Relative dielectric constant	2.1
Monopole antenna	Monopole antenna diameter	2.0 mm

^a 3-turn curling slot-antenna, ^b 4-turn curling slot-antenna.

Figure 1. (a) Appearance of single curling probe apparatus. (b) A cross-sectional view of the single curling probe.

2.2. Reflection microwave frequency spectrum (S₁₁)

As previously mentioned, the curling slot antenna is a resonator for which the resonance frequency is described by Eq. (3); this resonance was clearly detected in S₁₁. The calculation procedure for S₁₁ in this simulation is as follows. Microwave power with a Gaussian pulse shape over time [P_in(t)], which includes a continuous frequency component from 0 to 3 GHz, enters the microwave input/output port located at the end of the coaxial cable, as shown in Fig. 1(b). Then, the microwave power is reflected on the boundaries and returned to the microwave input/output port. This is recorded as the reflected power over time [P_ref(t)]. The simulation was terminated when the remaining microwave power inside the simulation domain reached the set value. Subsequently, P_in(t) and P_ref(t) are transformed into the frequency domains, P_in(f) and P_ref(f), respectively, through a fast Fourier transformation, where f is the microwave frequency. Finally, S₁₁ was calculated as follows:

2.3. Plasma

In this simulation, plasma is established as the Drude model, and the relative dielectric constant (ϵ_p) is defined as

where ω is the angular microwave frequency (ω = 2πf ), ω_pe is the plasma angular frequency, and v_m is the electron-neutral (e-n) collision frequency. The plasma angular frequency is defined as follows:

where e is an elementary charge, n_e,input is the input electron density, ϵ₀ is the vacuum permittivity, and m_e is the electron mass [6]. The term v_m is defined as n_gK, where n_g is the gas density and K is the rate constant of a collision. Because the elastic collision is dominant among the e-n collisions, we only included it for calculating K, for which the form is well described in [6]. It is based on the Maxwellian distribution of electrons, and is valid within the electron temperature range of 1–7 eV. Here, we assumed an electron temperature of 3 eV, which is a common discharge condition used in previous studies of microwave probes [12,30,31]. Then, the calculated v_m is 2.26×p in MHz, where p is the gas pressure in mTorr. In this simulation, we employed a v_m value of 226 MHz, which corresponds to a chamber pressure of 100 mTorr.

2.4. Simulation results and discussion

S₁₁ without plasma

Figure 2(a) exhibits the S₁₁ value of the single-curling probe for case A, which shows two resonance peaks at 0.88 and 2.10 GHz. At 0.88 GHz, the peak belongs to the fundamental resonance because its quarter- and half-wavelengths correspond to 0.63 and 1.30 GHz, respectively, considering its average curl length of 120 mm. Based on Eq. (3), the resonant frequency lies between the quarter- and halfwavelengths. For the same reason, the peak at 2.1 GHz belongs to the second-harmonic resonance. However, the resonance frequency of the peak does not exactly correspond to the second fundamental resonance frequency. This may result from capacitive couplings (stray capacitances) between the curling slots, which occur at higher harmonic conditions and lead to a change in the resonance condition. In addition, the cavity resonances inside the case resulted in resonance peaks in S₁₁. To investigate this effect, we varied the radius and height in each case. Figures 2(b) and 2(c) represent S₁₁ values with various radii and case heights. Consequently, these parameters do not influence the resonance frequencies, which implies that the resonance frequencies result solely from the resonances on the curling slot antenna.

Figure 2. (a) Reflection microwave frequency spectrum (S₁₁) of the single curling probe. (b) and (c) S₁₁ values with various radii and heights of the case, respectively.

S₁₁ with plasma

To verify the operation of the single-curling probe established in this simulation, we inserted cylindrical plasma at a distance of 1.0 mm from the surface of the curling slot antenna, as shown in Fig. 3(a). Figure 3(b) exhibits S₁₁ values with various values of n_e,input. One can find a clear shift in the resonance frequency of the peak at 0.88 GHz without plasma with increasing n_e,input, which is marked by arrows. Furthermore, the peak at 2.1 GHz without plasma also shifted toward a higher frequency with increasing n_e,input, but it shifted slightly compared with the shift of the peak at 0.88 GHz without plasma. The resonance frequency shift stems from the change in the plasma dielectric constant, which is a function of the plasma electron density [11,19,27]. The resonance frequency (f_r) in a plasma environment is defined as [27]

Figure 3. (a) A cross-sectional view of the single curling probe and cylindrical plasma. (b) S₁₁ values of the single curling probe with various input electron densities (n_e,input) from 5.0×10⁹ to 1.0×10¹¹ cm⁻³.

where f₀ is the resonance frequency without plasma, f_pe is the plasma frequency (= ω_pe/2π), and a is a coefficient that depends on the curling slot antenna geometry. Using Eqs.(6) and (7), we obtain the relationship between the resonance frequency shift and the n_e,input as

Figure 4 represents the calculated $f_r^2-f_0^2$ for two resonance peaks (0.88 and 2.10 GHz without plasma) from Fig. 3(b) over n_e,input. The line graph in Fig. 4 is the curve-fitting one for the term $f_r^2-f_0^2$ of the first resonance peak, where R-square, which is an indicator of the linearity, is 0.985. This means that for the first resonance peak, $f_r^2-f_0^2$ exhibits good linearity with n_e,input. Hence, by measuring $f_r^2-f_0^2$ , we can infer the electron density over the curling probe using Eq. (8).

Figure 4. Difference between the squares of the resonance frequencies with plasma (f_r) and without plasma (f₀).

3.1. Simulation configuration

Figure 5(a) shows the configuration of the two curling probes connected to the coaxial cables. The 3-turn curling probe was located on the right side, and the 4-turn curling probe was located on the left side. The specifications of each curling slot antenna are listed in Table I. The microwave power entered the bottom coaxial cable and was channeled towards the curling probes. The divided microwaves were then returned from each curling probe to the microwave input/output port. It should be noted that the two curling probes share the same microwave input/output port, and a single S₁₁ can be obtained.

Figure 5. (a) Appearance of the double-curling probe apparatus and their heads: 3-turn and 4-turn curling slot-antennae. (b) S₁₁ values of the individual single curling probe (3-turn and 4-turn curling slot-antenna) and the double-curling probe.

3.2. Simulation results and discussion

Figure 5(b) shows the S₁₁ values obtained for individual curling probes (cases A and B) and the double-curling probe [Fig. 5(a)]. It should be noted that resonance peaks from individual curling probes emerge in S₁₁ of the double-curling probe; for instance, the resonance peaks of the 4-turn curling probe (case B) at 0.5 GHz and the 3-turn curling probe (case C) at 0.9 GHz appear in S₁₁ of the double-curling probe at the same frequencies. This indicates the capability of simultaneous measurement of S₁₁ from each curling probe with one microwave input/output port. Besides, there are other resonance peaks, for instance, the peaks near 1.0 and 2.5 GHz, which may result from the simultaneous resonance on both curling probes.

The simulation results indicate the possibility of using a doublecurling probe. However, this is impractical because it is connected via a coaxial cable. The use of a coaxial cable connection is undesirable because it creates a complicated structure and increases the sensor size. In the next section, we propose a parallel double-curling probe without a coaxial cable connection, and demonstrate its operation.

4.1. Simulation configuration

We established a parallel double-curling probe by slightly modifying the double-curling probe shown in Fig. 5(a). To verify its operation, we compared the resonance peaks of an individual curling probe with those of a parallel double-curling probe as follows. Figures 6(a) and 6(b) represent the 3-turn and 4-turn curling probe configurations, respectively, with a large antenna plate of diameter 100 mm. The coaxial cable was placed at the center of the case, whereas the curling slot antenna was located 25 mm from the center of the large antenna plate. The monopole antenna was curved at the middle of the cavity to be placed at the center of the curling slot antenna. Figure 6(c) shows the configuration of the parallel double-curling probe, which is a combination of a 3-turn and 4-turn curling slot antenna (case C + case D). In this configuration, the distance between the centers of each curling slot antenna was 50 mm. Two curling slot antennas were located opposite to each other at the same distance from the center of the large antenna plate, and their monopole antenna was placed at its center with the same length and diameter.

Figure 6. Cross-sectional view of the simulation configuration of (a) 3-turn curling slot-antenna (case C), (b) 4-turn curling slot-antenna (case D), (c) merged structure of the 3-turn and 4-turn curling probes (case C + case D), and (d) parallel double-curling probe (case E). (e) S₁₁ values of 3-turn curling slot-antenna (case C), 4-turn curling slot-antenna (case D), and merged structure of the 3-turn and 4- turn curling probes (case C + case D). (f) S₁₁ values of 3-turn curling slot-antenna (case C), 4-turn curling slot-antenna (case D), and the parallel double-curling probe (case E).

The configuration in Fig. 6(c) is a simplified configuration for the preliminary demonstration of the parallel double-curling probe; however, it is not practical because of the complicated monopole antenna configuration inside the case: separating the core of the coaxial cable and connecting the center of the curling slot antenna. To address this problem, we employed an activator instead of a curved monopole antenna, as shown in Fig. 6(d) (case E). The activator, which was a large thin-plate with a thickness of 0.5 mm, is located at the mid-point of the case in the vertical direction. The cylindrical rods, which act as individual monopole antennas, are located at the center of each curling slot antenna.

4.2. Simulation results and discussion

S₁₁ without plasma

Figure 6(e) exhibits S₁₁ values obtained from case C, case D, and the parallel double-curling probe (case C + case D). Similar to the results shown in Fig. 5(b), the resonance peaks of the individual curling probes were observed in the S₁₁ plots of the parallel double-curling probe (dashed arrow lines). An additional resonance peak at 2.25 GHz appears, which may stem from the simultaneous resonance of both curling slot antennas. Hence, the parallel connection operates similar to the coaxial cable connection, as shown in Fig. 5.

Figure 6(f) shows the S₁₁ values obtained from case C, case D, and case E. Four resonance peaks of individual curling probes appear in the S₁₁ results for case D (see dashed arrow lines): 0.87, 1.99, 2.10, and 2.75 GHz. The peaks of 0.87 and 2.10 GHz result from case C, and the peaks of 1.99 and 2.75 GHz result from case D. The depth of S₁₁ at 1.25 GHz shows a drastic decrease in case E compared to case D. The decrease in the depth of S₁₁ at the resonant frequency is not yet clearly understood. Nevertheless, a detailed analysis is not within the scope of this study, and will be discussed in a subsequent paper that focuses on the circuit modeling of a double-curling probe. The simulation results indicate the feasibility of a parallel double-curling probe for a single microwave input/output port.

S₁₁ with single cylindrical plasma

To verify the operation of the parallel double-curling probe (case E), we inserted a cylindrical single plasma, which had a uniform density n_e,input, at a distance of 1.0 mm from the surface for the doublecurling slot antennae, as shown in Fig. 7(a). The cylindrical single plasma covers two curling slot antennae, as well as the case. Figure 7(b) shows the S₁₁ values with various n_e,input values, as well as traces of the four resonance peaks with dashed lines from those without plasma (f₀). These resonance peaks shifted toward higher frequencies as n_e,input increased. Figure 7(c) represents the $f_r^2-f_0^2$ over the n_e,input for the four resonance peaks. Only two peaks of 0.87 and 2.75 GHz are linearly proportional to n_e,input. The peak at 0.87 GHz results from the resonance of the 3-turn curling slot antenna, and the one at 2.75 GHz results from the 4-turn curling slot antenna. Based on the fitting curve ($n_{\text{e},\text{input}}=a\left(f_R^2-f_0^2\right)+b$), a and b are the coefficients for curve fitting, as summarized in Table II. Furthermore, the slope, a, represents the geometrical factor, a, in Eq. (8) for each curling slot antenna. We summarize the calculated values of a in Table III. Hence, we can estimate the electron density (n_output) by measuring the resonance frequency (f_r) using the geometrical factor a in Table III as

Table 2 . Curve fitting parameters for the parallel double-curling probe.

Curve fitting parameters	3-turn curling slot-antenna (f0 = 0.87 GHz)	4-turn curling slot-antenna (f0 = 2.75 GHz)
a (GHz-2m-3)	4.9×10-2	0.18
b (m-3)	4.3×1015	4.7×1015

Table 3 . Calculated geometric parameter (a) of the parallel double-curling probe.

Geometric parameter	3-turn curling slot-antenna (f0 = 0.87 GHz)	4-turn curling slot-antenna (f0 = 2.75 GHz)
a	2.95	13.5

Figure 7. (a) Cross-sectional view of the parallel double-curling probe with single cylindrical plasma with distance of 1.0 mm from the antenna surface. (b) S₁₁ values of the parallel double-curling probe with various input electron densities (n_e,input). The f₀ indicates the resonance frequency without plasma and the dashed line indicates the trace of the resonance frequency (f_r). (c) The values of $f_r^2-f_0^2$ 2 over the n_e,input for four resonance frequencies of 0.87, 1.99, 2.10, and 2.75 GHz. The line graph is the curve fitting for the f₀ of 0.87 GHz (gray) and one of 2.75 GHz (magenta). (d) Output electron density (n_output) and difference in the input electron density values for two resonance frequencies of 0.87 and 2.75 GHz. Here, the discrepancy is defined as $\left|\left(n_{\text{output}}-n_{\text{input}}\right)\right|/n_{\text{input}}\times 100$ (%).

Figure 7(d) shows n_output obtained from two frequencies and discrepancies. The values of n_output were proportional to the input electron density. The discrepancies were large at low input electron densities compared to those at high input electron densities. This may have been due to the frequency resolution. A low input electron density induces a small frequency shift, and is comparable to the frequency resolution of the probe, leading to a large discrepancy in the resonance peak determination. Nevertheless, based on Fig. 7(d), the average discrepancy of two resonance peaks (0.87 and 2.75 GHz) is 18.7 and 15.8 %, respectively.

S₁₁ with double cylindrical plasmas

Based on geometrical factors, we can verify the operation of the parallel double-curling probe. Figure 8(a) shows a cross-sectional view of the parallel double-curling probe and cylindrical double plasmas, which have individual electron densities. Above the surface of the 4- turn curling slot antenna with a distance of 1.0 mm, the low-density plasma (LDP) with a diameter of 40 mm was aligned at the center of the antenna above the surface of the 4-turn curling slot antenna. In contrast, the high-density plasma (HDP) is aligned at the center of the 3-turn curling slot antenna at the same distance. The input electron density of HDP (n_HPD) was four times that of LPD (n_LPD), that is n_HPD = 4n_LPD. Figures 8(b) and 8(c) represent values of S₁₁ near 0.87 and 2.75 GHz, respectively, with various values of n_LPD and n_HPD. The resonance frequencies for each electron density condition are marked with arrows in the figures. Figure 8(d) shows the electron densities that were calculated using Eq. (9) for two values of f₀ in Figs. 8(b) and 8(c) over n_e,input. The measured electron densities from the two peaks are in agreement with the input electron densities, which means that the parallel double-curling probe can measure multi-site electron densities. Furthermore, the independent measurement of each curling probe can be clearly seen, as shown in Fig. 9, which exhibits the output electron densities calculated from the two resonance peaks with various n_LPD values at a fixed n_HPD and various n_HPD values at a fixed n_LPD.

Figure 8. (a) Cross-sectional view of the parallel double-curling probe with double cylindrical plasmas, which is LDP and HDP, at a distance of 1.0 mm from the antenna surface. The density of HDP (n_HPD) is four times that of LPD (n_LPD), i.e., n_HPD = 4n_LPD. (b) and (c) S₁₁ values of the parallel double-curling probe over input electron density for f₀ values of 0.87 and 2.75 GHz, respectively. (d) Output electron density (n_output) over the input electron densities (n_e,LPD and n_e,HPD) for two resonance frequencies of 0.87 and 2.75 GHz.

Figure 9. Output electron density (n_output) calculated from two resonance frequencies of 0.87 and 2.75 GHz over (a) the input electron density (n_e,LDP) at a fixed n_e,HDP value of 2.0 × 10¹¹ cm⁻³ and (b) the input electron density (n_e,HDP) at a fixed n_e,LDP value of 5.0 × 10⁹ cm⁻³.

By performing 3D-EM wave simulations, we verified the step-bystep operation of the parallel double-curling probe. The single-curling probe showed good linearity of its fundamental resonance peak with n_e,input. The double-curling probe represents the merged resonance peaks in S₁₁ obtained from each curling probe with one microwave input/output port. The parallel double-curling probe has a geometric parameter, a, of 0.74 and 5.4 for 3-turn and 4-turn curling slot antennae, respectively. In conclusion, a parallel double-curling probe with these geometric parameters can measure multisite electron density above its individual curling slot antenna.

Based on the simulation results, the multiplication and parallelization of curling probes for plasma uniformity sensors are feasible. Because the curling probe uses standing wave resonance on its curling slot antenna, its size determines the spatial resolution of the electron density measurement. This means that miniaturization of the curling slot antenna is crucial for enhancing spatial resolution. Furthermore, the resonance frequencies at each curling slot antenna do not overlap and should be separated with sufficient differences in the frequency domain. In addition, each curling slot antenna should exhibit a high Q factor for the resonance peak in S₁₁. The circuit model of a parallel double-curling probe is required to realize a plasma uniformity sensor. In a subsequent study, we will propose a circuit model, analyze the characterization of the curling slot antenna, and propose an optimum design of the parallel double-curling probe.

This research was supported by a research fund from the Chungnam National University, Republic of Korea.

The authors declare no conflicts of interest.