# Electromagnetic Induction Imaging with Atomic Magnetometers: Progress and Perspectives
## 2 Principles of Electromagnetic Induction Imaging  
Electromagnetic induction imaging relies on the active probing of the conductivity \(\sigma\), permittivity \(\varepsilon\) and permeability \(\mu\) of a target object by a weak oscillating magnetic field.  
As sketched in Figure 1, in its simplest arrangement, an EMI device uses a primary magnetic field (\(B_{1}\)), generated by an induction coil and oscillating at a desired frequency, \(\omega\). The primary fieldinduces eddy currents (\(J\)) in the sample of interest. An important parameter for EMI applications is the skin depth \(\delta\),  
\[\delta=\sqrt{\frac{2}{\sigma\omega\mu}}\sqrt{\sqrt{1+\left(\frac{\omega e}{\sigma }\right)^{2}+\frac{\omega e}{\sigma}}}\, \tag{1}\]  
which defines the depth below the surface of the conductor at which the current density has fallen to \(1/\mathrm{e}\), and, as it will be discussed in the following, is a key parameter to characterise the EMI penetrating power.  
The eddy currents induced in the material by the primary field produce in turn a secondary field (\(B_{2}\)), oscillating at \(\omega\). The properties of the secondary field, in particular its amplitude and phase lag with respect with the primary field, depend on the specimen's \(\sigma\), \(\varepsilon\), and \(\mu\). Therefore, by measuring \(B_{2}\) at different positions, the dielectric characteristics of the sample can be mapped, and its image can be created. For simple planar geometries, a conductivity map can be constructed simply with a point-by-point derivation of the conductivity from individual measurements. For more complex 3D structures, the formation of a tomographic image involves the use of backprojection algoritms [2] or the numerical solution of the inverse problem of electromagnetism [3].  
In general, the response of the sample is a complicated function of the parameters of the geometrical arrangements. We refer here to a simple arrangement which leads to a transparent analytic expression for the sample response, useful to capture the underlying principles of EMI. We consider a uniform disc of material placed coaxially with, and midway between, an exciter coil and a sensor coil, with the separation of the coils assumed to be large compared with their diameters. Under the condition that the penetration depth of the exciter magnetic field in the sample is much larger than the sample thickness, it is possible to show that the the relative change \(\Delta B/B\) in the receiver coil is given by [4; 5]:  
\[\frac{\Delta B}{B}=Q\omega\mu_{0}\left[\omega e_{0}\left(\epsilon_{r}-1\right) -i\sigma\right]+R\left(\mu_{r}-1\right)\, \tag{2}\]  
where \(\epsilon_{0}\) is the vacuum permittivity, \(\epsilon_{r}\) the relative permittivity of the sample material, and \(\mu_{r}\) its relative magnetic permeability. The quantities \(Q\) and \(R\) are geometrical factors.  
Equation (2) highlights the different contributions of the sample response to the variation in magnetic field recorded by the receiver coil. We note that the three different components, determined by the material permittivity, conductivity and pearmeability, can be separated by using their different dependence on the driving frequency and different phase relation. In most practical applications, the response is usually dominated by the conductivity contribution.  
Figure 1: Principles of electromagnetic induction imaging.