# Inductive Imaging of the Concealed Defects with Radio-Frequency Atomic Magnetometers
## 4 Optimum Coupling between rf Primary Field and the Object
### Defect on Inner Wall  
Figure 6 shows the amplitude (a)-(d) and phase (e)-(h) images of the aluminium plate containing a recess with variable depth. The plate is flipped such that the recess is on the surface which faces away from the rf coil, i.e., the inner surface [depicted in Figure 2d]. The area of this recess is highlighted by the dashed red square. The signature of this measurement is related to the thickness of the plate, as opposed to the depth of the recess. Within the highlighted area, the thickness of the plate gradually increases from 1 mm at \(x=23\) mm to 6 mm at \(x=64\) mm; and has a constant thickness along \(y=0\) mm to \(y=45\) mm. At the recess edge (\(y=45\) mm), there is a sharp change in thickness to 6 mm. The images are recorded for four values of the operating frequency: 0.6 kHz [(a) and (e)], 1.35 kHz [(b) and (f)], 2.26 kHz [(c) and (g)], and 10 kHz [(d) and (h)]. In contrast to the profiles recorded in the amplitude image where the recess is on the outer surface [shown in Figure 2a], the measured amplitudes produced by the edge and the tilted area of the recess for the inner surface (Figure 6a-d) have a similar value. Figure 5: The width (FWHM) of the profiles in the amplitude images representing the recess as a function the rf primary field frequency. The dependence is recorded with the recess covered by a single 0.5 mm aluminium sheet. The change in operating frequency results in a difference in the penetration depth of the primary rf field into the plate. As the frequency is increased, there is a reduction of the extent of the area where the defect signature can be observed in both the amplitude, and the phase images. While the primary rf field penetrates the full depth of the object at low frequencies, which is shown in Figure 6a by the large spatial extent of the recess signature, the contrast of the images recorded at these frequencies is relatively low. This is particularly evident in the degradation of the phase contrast in Figure 6e. The change of the operating frequency modifies the ratio of the signal components produced by the background and the secondary field. While the amplitude of the background can be assumed to be constant, the amplitude of the secondary field increases with increasing operating frequency. The actual functional frequency dependence of the total secondary field amplitude is a consequence of the fact that the sensor monitors the integrated signal, i.e., the signal produced by different layers of the object. Consequently, the expected linear increase of the secondary field resulting from the induction law is partially compensated by the reduction of the penetration depth of the rf field. The change in the ratio between the components of the signal manifests itself in both the amplitude and phase images. In the amplitude images, the zero amplitude (black) area visible for the shallow recess depth [Figure 6: (b) \(x\approx 70\,\)mm, (c) \(x\approx 60\,\)mm, and (d) \(x\approx 30\,\)mm], marks the area where the component of the residual primary field that contributes to the background signal and the total secondary field (produced by all layers of the plate) compensate each other. It is worth distinguishing the effects of the secondary field at the outer surface, which is flat, and the inner surface, in the vicinity of the recess. At the outer surface of the plate, the secondary field mostly mirrors the vertical direction of the primary field. A residual component of this is measured since the surface normal of the plate is not exactly parallel with the insensitive axis of the rf magnetometer. The space in the interior of the plate in the immediate vicinity of the recess produces horizontal components of the secondary fields. As the frequency changes, the depth at which the secondary field produced by the interior of the plate compensates the residual primary field will also change due to the change in the optimum coupling between the rf field and the plate. Therefore, the position of this black area occurs at smaller depths for increasing frequency.