An Explicit Formula for the Magnetic Polarizability Tensor for Object Characterization

The magnetic polarizability tensor (MPT) has attracted considerable interest due to the possibility it offers for characterizing conducting objects and assisting with the identification and location of hidden targets in metal detection. An explicit formula for its calculation for arbitrary-shaped objects is missing in the electrical engineering literature. Furthermore, the circumstances for the validity of the magnetic dipole approximation of the perturbed field, induced by the presence of the object, are not fully understood. On the other hand, in the applied mathematics community, an asymptotic expansion of the perturbed magnetic field has been derived for small objects and a rigorous formula for the calculation of the MPT has been obtained. The purpose of this paper is to relate the results of the two communities, to provide a rigorous justification for the MPT, and to explain the situations in which the approximation is valid

Efficient computation of the magnetic polarizabiltiy tensor spectral signature using proper orthogonal decomposition

The identification of hidden conducting permeable objects from measurements of the perturbed magnetic field taken over a range of low frequencies is important in metal detection. Applications include identifying threat items in security screening at transport hubs, location of unexploded ordnance, and antipersonnel landmines in areas of former conflict, searching for items of archeological significance and recycling of valuable metals. The solution of the inverse problem, or more generally locating and classifying objects, has attracted considerable attention recently using polarizability tensors. The magnetic polarizability tensor (MPT) provides a characterization of a conducting permeable object using a small number of coefficients, has an explicit formula for the calculation of their coefficients, and a well understood frequency behavior, which we call its spectral signature. However, to compute such signatures, and build a library of them for object classification, requires the repeated solution of a transmission problem, which is typically accomplished approximately using a finite element discretization. To reduce the computational cost, we propose an efficient reduced order model (ROM) that further reduces the problem using a proper orthogonal decomposition for the rapid computation of MPT spectral signatures. Our ROM benefits from a posteriori error estimates of the accuracy of the predicted MPT coefficients with respect to those obtained with finite element solutions. These estimates can be computed cheaply during the online stage of the ROM allowing the ROM prediction to be certified. To further increase the efficiency of the computation of the MPT spectral signature, we provide scaling results, which enable an immediate calculation of the signature under changes in the object size or conductivity. We illustrate our approach by application to a range of homogenous and inhomogeneous conducting permeable objects

Identification of a Spheroid based on the First Order Polarization Tensor

Polarization tensor (PT) has a lot of useful and important practical applications. In this case, it must be firstly determined by some appropriate method. Besides, understanding some properties of the PT might also be very useful in order to apply it. In this study, we investigate the first order PT for ellipsoid and use it to describe the first order PT for spheroid as well as identify the spheroid. Numerical examples are also given to further justify our results

Understanding the magnetic polarizability tensor

The aim of this paper is to provide new insights into the properties of the rank 2 polarizability tensor M̆ proposed by Ledger and Lionheart for describing the perturbation in the magnetic field caused by the presence of a conducting object in the eddy-current regime. In particular, we explore its connection with the magnetic polarizability tensor and the Pólya-Szegö tensor and how, by introducing new splittings of M̆, they form a family of rank 2 tensors for describing the response from different categories of conducting (permeable) objects. We include new bounds on the invariants of the Pólya-Szegö tensor and expressions for the low-frequency and high-conductivity limiting coefficients of M̆. We show, for the high-conductivity case (and for frequencies at the limit of the quasi-static approximation), that it is important to consider whether the object is simply or multiply connected but, for the low-frequency case, the coefficients are independent of the connectedness of the object. Furthermore, we explore the frequency response of the coefficients of M̆ for a range of simply and multiply connected objects

In-Situ Dual-Port Polarization Contrast Imaging of Faraday Rotation in a High Optical Depth Ultracold 87Rb Atomic Ensemble

We study the effects of high optical depth and density on the performance of a light-atom quantum interface. An in-situ imaging method, a dual-port polarization contrast technique, is presented. This technique is able to compensate for image distortions due to refraction. We propose our imaging method as a tool to characterize atomic ensembles for high capacity spatial multimode quantum memories. Ultracold dense inhomogeneous Rubidium samples are imaged and we find a resonant optical depth as high as 680 on the D1 line. The measurements are compared with light-atom interaction models based on Maxwell-Bloch equations. We find that an independent atom assumption is insufficient to explain our data and present corrections due to resonant dipole-dipole interactions

The depolarization factors for ellipsoids and some of their properties

The terminology depolarization factors was firstly highlighted in the study of problems involving magnetic, where, it was initially used to describe magnetic properties of material. Recently, this terminology was investigated to describe composites, improve imaging techniques, and other field of researches related to potential theory in mathematics and physics. Due to our interest in electrical imaging using polarization tensor (PT) and since PT is actually related to the depolarization factors, in this paper, some properties of the depolarization factors are investigated for future applications. The values of these depolarization factors are firstly proven to be non-negative. Based on the previous studies which consider the incomplete elliptic integrals of the first and second kind with some suitable identities, the summation of the depolarization factors are shown to be equal to one. By using these two properties, the value for each depolarization factor for ellipsoid is then explained to be between zero and one. It is also shown in this paper that the depolarization factors can be characterized based on the values of the semi principal axes of the ellipsoid. Reversely, the semi principal axes of the ellipsoid can be classified based on the values of the depolarization factors. All properties presented in this paper could be useful and important in the future especially to use the depolarization factors in any related applications

Hidden security threat identification: A reduced order model for the rapid computation of object characterisations

This work presents computational results of a reduced order model for the rapid calculation of conducting object characterisations as a function of frequency in metal detection. Such characterisations are called their spectral signature. We present a brief description of the eddy-current model and the magnetic polarizability tensor (MPT) used for our object characterisations. The transmission problem required for the computation of the MPT and its discretisation is then described followed by a summary of the reduced order model. As an illustration of the capabilities of the approach for characterising realistic objects, we show MPT spectral signatures of a British £1 coin