# 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 frequency for which the maximum contrast is observed depends on the thickness of the barrier (2-3 kHz for 0.5 mm-thick barrier, 1.3-2 kHz for 1.5 mm-thick barrier). The rf primary field frequency defines the penetration depth of the rf field. The larger the depth, the larger the spatial extent (spread) of the rf primary field within the object is. In turn, the spread of the rf field limits the spatial resolution of the measurement. In this sense, the penetration depth is equivalent to the lift-off, i.e., the distance between the rf primary field coil and the particular layer of the object. Since the image recorded at a particular frequency integrates the contributions from different layers within the object, the total spatial resolution of the measurement is affected by the operational frequency. In particular, for high frequencies, the response from the object will be mostly defined by the contribution from the surface of the object and the resolution of the image will be defined by the size of the coil and step of the scan (about 2 mm). Figure 5 shows the full width at half maximum (FWHM) of the profile in the amplitude image representing the recess as a function of the rf primary field frequency. The profile width values were extracted from the images by fitting two Lorentzian profiles to the cross-section of the image. It is difficult to precisely evaluate the width for frequencies below 1 kHz because of the small profile amplitude. For lift-off values above 5 mm, the width of the profile is defined by the lift-off and does not vary with the rf primary field frequency. We have recorded images of recesses with three different diameters (50 mm, 25 mm, 12.5 mm) and confirmed that the maximum contrast level decreases with decreasing recess diameter [23]. Figure 4: The measured change of the amplitude (**a**) and phase (**b**) of the rf spectroscopy signal recorded in the self-compensation configuration over a 90 \(\times\) 90 mm\({}^{2}\) area of a 6 mm-thick aluminium plate, containing a 50 mm-diameter recess that is 2.4 mm-deep at 2 kHz. The red dashed line marks the position of the recess edge. The amplitude (**c**) and contrast (**d**) of the profile in the amplitude image representing the recess as a function of the rf primary field frequency. The dependence is recorded with the recess covered by zero (blue dots), one (green diamonds), and three (red squares) 0.5 mm-thick aluminium sheets. At this point, it is worth reminding that the sensitivity of the defect detection with the atomic magnetometer (e.g., \(\mathrm{C_{R}}\)) does not only depend on the sensitivity of the sensor but also on its spatial resolution [23]. The latter is defined by the dimensions of the rf primary field coil. For the coil dimensions bigger than these of the defect, the sensor detects a combined signal from opposite sides of the defect. This results in reduced amplitude of the defect signature and difference between observed and actual defect dimensions [23]. In practical implementations, the dimensions of the rf primary field coil are the trade-off between the required spatial resolution, image acquisition rate and the strength of the rf primary field.