# Inductive Imaging of the Concealed Defects with Radio-Frequency Atomic Magnetometers
## 4 Optimum Coupling between rf Primary Field and the Object
### Initial Tests--Single Edge of the Plate  
We begin the exploration of the topic of optimum coupling between an rf field and the object with a series of elementary measurements, recording the change in signal observed over the edge of the aluminium plate covered by thin aluminium sheets. This initial test represents the most basic structure, i.e., the signal produced by a single isolated sharp inhomogeneity under a conductive barrier with a controlled thickness. There is no other inhomogeneity in the vicinity that might affect the recorded signal, as in the case of the recess. The lift-off (distance between the primary rf field coil and the plate surface) is kept constant throughout the measurement. The measurement is performed in the self-compensated configuration. The data shown in Figure 3 represent the measured signatures of the edge of a plate as a function of the rf primary field frequency. The measurements have been made with the edge of a 6 mm thick aluminium plate covered by zero (blue points), one (green diamonds), two (yellow triangles) and three (red squares) aluminium sheets with a thickness of 0.5 mm. There are different ways of characterising the results by considering the measured amplitude and the amplitude/phase contrast of the edge signature. The data in Figure 3a represent the frequency dependence of the amplitude of the signature of the plate edge (\(R\)), defined as \(R_{\text{Max}}-R_{\text{Min}}\), where \(R_{\text{Max}}\) and \(R_{\text{Min}}\) are the signal amplitude measured at the opposite sides of the plate edge. For the uncovered edge, the amplitude of the signature increases with frequency, which is a simple consequence of the growth in induction with the frequency of the rf field. For the edge covered by aluminium sheets, the presence of a broad optimum in the amplitude [Figure 3a] is indicated by the shift in the slope of the frequency dependence with the number of the sheets. The presence of an optimum frequency of operation, i.e., frequency for which there is a maximum change in the signature of the edge (i.e., amplitude, contrast), is easily seen in Figure 3b,c which show the amplitude, \(C_{\text{R}}\), and phase, \(C_{\text{GE}}\), contrasts. The amplitude contrast, \(C_{\text{R}}\), is defined in a similar way to [24], \(C_{\text{R}}=(R_{\text{Max}}-R_{\text{Min}})/(R_{\text{Max}}+R_{\text{Min}})\), and analogously, \(C_{\text{GE}}=\phi_{\text{Max}}-\phi_{\text{Min}}\), with \(\phi_{\text{Max}},\phi_{\text{Min}}\) reflecting the signal phase at the opposite sides of the plate edge. The signal recorded in the immediate vicinity of the edge of the plate is defined by the secondary field produced by the induced eddy currents and the background (i.e., the residual primary field due to imperfect compensation that results from misalignment of the axes). The background value does not show any frequency dependence [30]. For low frequencies, the signal within the plate is dominated by the secondary field, while, outside the plate, it is dominated by the background. At higher frequencies, the increased induction means that the signal from the secondary field is much greater than the background. Hence, the signal inside (before the edge and over the plate), and outside (after the edge and over free space), the plate is dominated by the secondary field. This leads to the saturation of the amplitude contrast \(C_{\text{R}}\) that is visible in the blue points in Figure 3b. The dominance of the secondary field in the signal recorded in the vicinity of the edge, both inside and outside of the plate, could also be seen in \(C_{\text{GE}}\). The primary rf field produced by the coil along an axis within its cross-section is in the opposite direction to the field created along an axis that lies outside its cross-section, e.g., the opposite direction of field created at either side of a wire. This direction change leads to a 180\({}^{\circ}\) change in the phase of the secondary field as the edge of the plate is scanned through the area that is within, to outside, the cross-section of the coil.