# Inductive Imaging of the Concealed Defects with Radio-Frequency Atomic Magnetometers
## 4 Optimum Coupling between rf Primary Field and the Object
### Signature of the Recess in the Plate  
The results of the tests from the previous section show general properties of the signature of the inhomogeneity but hardly mimics real-life defects. Our next approximation of a defect consists of  
Figure 3: The change in amplitude (**a–c**) recorded scanning over an edge of the aluminum plate as a function of the rf primary frequency. The dependence is recorded with the edge covered by zero (blue dots), one (green diamonds), two (yellow triangles) and three (red squares) 0.5 mm-thick aluminium sheets. a 6 mm-thick aluminium plate containing a 50 mm-diameter recess that is 2.4 mm-deep covered by 0-3 aluminium sheets with 0.5 mm-thickness. Thinning of the plate represents a broad class of defects, ranging from cracks to corrosion. The relatively large recess diameter has been chosen to reduce the effect of the signals produced by the opposite edges of the recess overlapping. As mentioned before, the amplitude image recorded in the self-compensated configuration represents the measurement of the amplitude components of the secondary field parallel to the surface of the object (plate). For the ideal case, these components have non-zero values only in the immediate vicinity of the edge. Consequently, the amplitude image of the circular recess has a form of a ring profile, Figure 4a, while the phase of the rf signal shows a vortex centred on the recess, Figure 4b [24]. The red dashed line marks the position of the recess edge in Figure 4a,b. As mentioned before, we approximately know the position of the defect, and it has been marked for guides for the eye. The main focus of this work is on the relationship between the defect signature and operating frequency, not on the accurate mapping of the detect. This vortex in the phase image is visible across the whole range of studied frequencies and the following analysis will concentrate only on changes in the amplitude images. We characterise the recess signature similarly to that of the edge. The amplitude of the recess signature is defined as \(R=R_{\mathrm{Max}}-R_{\mathrm{Min}}\), with \(R_{\mathrm{Max}},R_{\mathrm{Min}}\) reflecting signal amplitudes at the top of the profile and in the background of the image. In order to take into account the imperfect compensation of the rf field that introduces inhomogeneities in the measured profile and background amplitudes, we define \(R_{\mathrm{Max}}\) as an average of the profile amplitude along the circumference of the ring. The value of \(R_{\mathrm{Min}}\) was calculated as an average of the image background, i.e., average of 10 columns and rows each that make up the edge of the image. Figure 4 shows the amplitude (c) and amplitude contrast (d) of the profiles representing the recess in the aluminium plate covered by; zero (blue dots), one (green diamonds), and three (red squares) 0.5 mm-thick aluminium sheets; as a function operating frequency. The lift-off between the primary rf coil and the plate with a recess has been kept constant. The frequency dependence of the amplitude, R, of the signatures can be divided into two parts. For the frequencies above 1.5 kHz, the amplitude shows different behaviour depending on the thickness of the barrier. Below this frequency, the behaviour within the three sets of data are similar, which indicates that the dependence is defined by the coupling between the rf field and the plate. As in the previous section, the measured amplitude signature of the uncovered recess increases with frequency, which is a straightforward consequence of the induction law. Consequently, the contrast of the image (\(C_{\mathrm{R}}\)) with no barriers shows saturation. This saturation occurs at high frequencies (above 4 kHz) since both the signal and the background become defined by the secondary field generated by the surface of the object. The amplitudes and contrasts of the profiles recorded in the presence of the barrier shows a broad maximum.