# Inductive Imaging of the Concealed Defects with Radio-Frequency Atomic Magnetometers
## 1 Introduction  
The non-destructive testing (NDT) toolkit covers a wide range of sensing technologies including thermal [1; 2], infra-red imaging [3], and ultrasound [4]. Like X-ray scans [5], inductive methods are a well established non-contact measurement technique. The generic scheme of magnetic induction tomography (MIT) includes a radio-frequency (rf) magnetic field (primary field) that creates a response (secondary field) within the object through inductive coupling with an electrically conductive and/or magnetically permeable object [6]. Specifically, in electrically conductive objects, the method relies on the generation of eddy currents by an oscillating magnetic field, i.e., the primary field, in the object of interest and on the detection of the magnetic field produced by those eddy currents, i.e., the secondary field. Asymmetries in eddy current flow caused by the presence of a defect (e.g., cracks, contamination, and corrosion) are mapped onto the amplitude and the direction of the secondary field. Hence, the amplitude and phase of the secondary field will contain a signature of any defect that is present in the object. The rf magnetic fields have a frequency dependent penetration depth that enables the detection of defects throughout its volume. The object response can be detected with a pick-up coil (measuring the change in coil inductance) or with a magnetic field sensor. Measurements based on pick-up coils [6; 7; 8; 9; 10] benefit from simple instrumentation, but suffer from a decrease in sensitivity at low frequencies since the measured signal is \(\propto\frac{\delta B}{\delta t}\). In contrast, the sensitivity of magnetic sensors is constant over a wide range of operating frequencies (1 kHz-1 GHz) [11]. In particular, it enables operation at low frequencies, where the change in the penetration depth of the rf field is most significant. This category of sensors includes giant magnetoresistance (GMR) magnetometers [12; 13; 14], superconducting quantum interference devices (SQUIDs) [15; 16] and atomic magnetometers [17; 18; 19; 20; 21; 22; 23; 24; 25; 26]. The measurement of the secondary field with an rf atomic magnetometer relies on the monitoring of the amplitude and phase of the rf resonance signal created by the rf field as an object is moved under the rf primary field coil. Implementation of an rf atomic magnetometer as a magnetic field sensor bringsa superior field sensitivity (\(\sim\) f\(\Gamma/\sqrt{Hz}\)) [11] and several novel functionalities to the NDT measurement process. In particular, the semi-vector mapping of the secondary field components generated by a defect was recently demonstrated in NDT measurements with an rf atomic magnetometer, although this sensor is typically regarded as scalar in nature [23; 24]. The operation of a magnetometer in the so-called spin maser mode (self-generated drive for rf magnetic field) addresses the issue of the sensor's narrow bandwidth [25; 26]. In this mode, spontaneous fluctuations of the atomic spins are detected, amplified, and fed back to the primary coil to create an rf primary field whose frequency self-tunes to the rf resonance frequency. Hence, the spin maser enables the operation of the sensor in a magnetically noisy environment, which is particularly important when studying magnetically permeable objects, where the object's local magnetisation shifts the rf resonance frequency during measurements. When operating with an external drive for the rf field, this problem can be partially addressed by active stabilisation of the local magnetic field in which the sensor operates. The residual changes of the resonance frequency make it necessary to record complete rf spectra, reducing the image acquisition rate [23]. Our previous studies have been focused on measuring object thinning in the form of a recess [22; 23; 24; 25; 26]. Analysis of the amplitude and phase images validated this method as a tool for monitoring defect (recess) depth. In all these studies, the recess has been located on the outer surface of the object.