## 1 Introduction  
Consisting of charged particles (electrons and ions) and neutral particles (atoms, molecules, radicals, excited and metastable species), plasma is controllable via electrostatic and electromagnetic fields [1]. Its application covers various fields such as material fabrication, nuclear fusion, and medical, environmental, and aerospace industries [1, 2]. In particular, plasma is one of the key factors in semiconductor and display fabrication, since both its physical (energetic ions) and chemical (reactive radicals) properties can be exploited. Advanced process control (APC) refers to the fine-tuning of plasma processing based on real-time feedback signals from various plasma process monitoring devices. In fabrication fields, recent demandsfor sub-nanometer patterning and high aspect ratio- and atomic scale-etching and deposition [3; 4; 5] have greatly increased process difficulty, leading to wide interest in APC and real-time plasma process monitoring. The latter has contributed to productivity improvements through providing feedback signals to process control units; the feedback signals are produced via the gathering and post-processing of myriad monitoring parameters, such as plasma emission light, voltage and current of the electrode, antenna, electrostatic chuck, chamber pressure, gas flow rate, and plasma parameters (electron density, electron temperature, etc.) [4]. Among these monitoring parameters, electron density is one of the crucial factors because it is directly related to processing time and quality [1; 2; 6]. Many diagnostic techniques including electrical, laser, optical, and microwave methods have been developed to measure electron density. Examples include the Langmuir probe, laser Thomson scattering diagnostics, optical emission and absorption spectroscopy, and microwave probes. Most of these approaches, though, are not suitable for plasma process monitoring; the Langmuir probe cannot operate under conditions with probe tip contamination (especially in the deposition process), laser Thomson scattering diagnostics is highly sensitive to the environment and requires quite a large space, and the optical emission and absorption method operates only within a narrow window [7]. On the other hand, the microwave method has attracted great attention for application to plasma process monitoring since microwave probes are not affected by probe tip contamination, afford high measurement accuracy [8; 9], and further, the required microwave power is small enough to not disturb the plasma. One drawback, though, is that microwave probes provide few parameters, namely electron density and temperature. Nowadays, research is focused on the development of a planar and compact microwave probe for real-time plasma process monitoring, since a small and planar probe can be embedded into a wafer chuck or chamber wall, and is therefore non-invasive. Variations such as the curling probe (CP), the planar multipole resonance probe (pMRP), and the planar cutoff probe have been developed and are currently under improvement via commercial three-dimensional (3D) electromagnetic simulation (CST Microwave Studio [10]), as well as experimental validation. Here, the computer simulation approach is quite simple and economical for the optimization and analysis of microwave probes. [PERSON] et al. developed the CP to measure the shift of standing wave (SW) resonance frequency caused by plasma in a reflection microwave frequency spectrum (S\({}_{11}\)) [11]. The authors optimized the CP via computer simulation [11], as well as physical modeling [12; 13]. They have recently demonstrated the in situ simultaneous measurement of the thickness of a deposition film and electron density using two CPs [14; 15]. Developed by [PERSON] et al., the pMRP uses the resonance characteristics of the probe itself and can measure muti-resonance frequencies in S\({}_{11}\)[16]. The antenna of the pMRP consists of two semi-circle planar plates with a dielectric cover.