# Inductive Imaging of the Concealed Defects with Radio-Frequency Atomic Magnetometers
## 6 Conclusions  
We have presented the results of a series of NDT measurements with an rf atomic magnetometer involving various types of concealed defects. In particular, we have studied the shape and frequency dependence of signatures of the defects that are represented by the edge of a plate, a recess and cavities in an aluminium plate. We have experimentally explored the optimum operational frequency, which addresses the issue of the coupling between the rf primary field and the investigated object. We have achieved consistency across different measurement types. The optimum in the frequency  
Figure 7: (**a**) amplitude and (**b**) phase change in the rf atomic magnetometer signal generated by scanning a \(165.6\times 165.6\) mm\({}^{2}\) area over a square \(140\times 140\) mm\({}^{2}\) aluminium plate with cavities that are \(2\,\mathrm{mm}\) in diameter and are drilled to a length of \(40\,\mathrm{mm}\) in the side of the plate. Plate edges are marked by dashed lines while the position of the holes is indicated by the arrows. Plots of the frequency dependence of the amplitude, \(C_{R}\), (**c**) and phase, \(C_{E}\), (**d**) contrast of the \(0.5\,\mathrm{mm}\) (blue dots), \(1\,\mathrm{mm}\) (green diamonds), \(2\,\mathrm{mm}\) (yellow triangles), and \(3\,\mathrm{mm}\) (red squares) deep cavities with changing frequency.  
dependence of the amplitude/phase contrast for defects under a 0.5 mm-1.5 mm conductive barrier was observed within the 1 kHz-2 kHz frequency range. It is worth pointing out that these values are significantly higher than those reported in [32] in the context of measurements with pick up coils, where the maximum in the defect signature has been observed between 5 Hz and 10 Hz. The optimum frequency for a given penetration depth is defined by the coupling between the rf field and the object. The optimum is a balance between the induced field strength and the skin depth, which are respectively proportional and inversely proportional to the operating frequency for the rf field. This balance implicitly includes the performance of the sensor, which for increased sensitivities will be able to detect a lower signal produced by a defect over a higher frequency range. In our opinion, this explains the shift of the optimum frequency range presented in this paper with respect to that described in [32]. Further improvements in the sensitivity of the rf atomic magnetometer should allow operation of a concealed defect detection system at even higher frequencies, which would be beneficial for operation in industrial environments where there are low-frequency environmental noises.  
Conceptualization, W.C. and R.G.; methodology, W.C., R.G., and P.B.; software, R.G. and P.B.; validation, R.G. and P.B.; formal analysis, R.G. and P.B.; writing--original draft preparation, W.C.; writing--review and editing, W.C., R.G., and P.B.; supervision, W.C.; project administration, W.C.; funding acquisition, W.C. All authors have read and agreed to the published version of the manuscript.  
The work was funded by the UK Department for Business, Energy and Industrial Strategy. P.B. was supported by the Engineering and Physical Sciences Research Council (EPSRC) (No. EP/P51066X/1).  
The authors declare no conflict of interest.