## 2 Spherical Scattering Electrical-Field Probing Method  
The model of small hole diameter measurement with a non-contact probe can be expressed as:  
\[D=l+d+\delta_{1}+\delta_{2} \tag{1}\]  
where \(D\) is the hole diameter to be measured; \(d\) is the probing ball diameter; \(l\) is the macro displacement of the probe when it is moved from the first probing point to the second along the measurement line of the diameter; and \(\delta_{1}\) and \(\delta_{2}\) are micro probing gaps between the probing ball and the hole sidewall at the two probing points. In the proposed measurement method based on SSEP, \(d\) was calibrated beforehand, \(l\) was obtained using laser interferometry, and \(\delta_{1}\) and \(\delta_{2}\) were given by the SSEP probe. The quality of the diameter measurements depended on the ability to accurately determine \(\delta_{1}\) and \(\delta_{2}\). The principle of the SSEP method is shown in Figure 1. The probing ball and the small hole are electrically conductive. When the probing ball probes the internal sidewall of the hole, a dual conductor system is formed. By constructing certain boundary conditions--e.g., grounding the hole and keeping the potential of the probing ball constant--a spherical scattering electrical field is formed, and the electrical field between the probing ball and the hole sidewall, as well as the distribution of surface charge, is strongly affected by the micro probing gap. Based on this phenomenon, the micro gap between the probing ball and the hole sidewall is converted to a capacitive signal and detected with capacitive signal processing circuits [17]. The sensing characteristics of the spherical scattering electrical field is identical in arbitrary spatial directions. Therefore, the measurement performance is not influenced by the relative attitude between the probe and the hole to be measured, which is often a key limitation in existing non-contact high-precision probing methods of small holes with large depth-to-diameter ratios. Considering the actual demand in industry, only holes of conductor material are discussed--although the analysis shows that non-conductor material can also be probed using the SSEP method with much worse resolution. The bias electrical field was theoretically modeled using the seven-point finite difference method and non-uniform meshing in a spherical coordinate system [18]. The Laplace equation was solved for a model consisting of a \(\phi\)3-mm conductor sphere as the probing ball and a conducting plane (yoz plane) as the surface to be measured. The potential of the sphere was 1V and the plane was grounded. The probing gap was set to 0.2 \(\upmu\)m. The simulation software was programmed by the authors with MATLAB (MathWorks, Natick, Massachusetts, US). The surface charge density characteristics of the probing ball are shown in Figure 2. It is shown that most surface charge on the probing ball was concentrated in a very small area closest to the part being measured, and 43% of the surface charge was concentrated in an area comprising 1% of the whole surface area of the probing ball. The charge density of the waist line of the probing ball is shown in the top right corner of Figure 2, and the curve is characterized by a single peak. This phenomenon indicates that the SSEP probe has a point probing capability. Therefore, the SSEP probe is capable of isotropic sensing and point-probing, both of which are ideal for the measurement of small holes with large depth-to-diameter ratios. Figure 3a shows the relationship between the surface area ratio, defined as the ratio of the charge concentrating area to the whole surface area of the probing ball, and the corresponding surface charge  
Figure 1: Schematic diagram of the spherical scattering electrical-field probing (SSEP) method. Figure 2: Surface charge density characteristics of a \(\phi\)3-mm probing ball when a conductor plane (yoz plane) is probed. ratio when a \(\phi\)3-mm probing ball probes holes of different diameters with a 0.2-\(\upmu\)m probing gap.