1206 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 39, NO. 6, JUNE 2001 On the Wideband EMI Response of a Rotationally Symmetric Permeable and Conducting Target Lawrence Carin, Fellow, IEEE, Haitao Yu, Yacine Dalichaouch, Alexander R. Perry, Senior Member, IEEE, Peter V. Czipott, and Carl E. Baum, Fellow, IEEE Abstract—A simple and accurate model is presented for compu- tation of the electromagnetic induction (EMI) resonant frequen- cies of canonical conducting and ferrous targets, in particular, fi- nite-length cylinders and rings. The imaginary resonant frequen- cies correspond to the well known exponential decay constants of interest for time-domain EMI interaction with conducting and fer- rous targets. The results of the simple model are compared to data computed numerically, via method-of-moments (MoM) and finite- element models. Moreover, the simple model is used to fit measured wideband EMI data from ferrous cylindrical targets (in terms of a small number of parameters). It is also demonstrated that the general model for the magnetic-dipole magnetization, in terms of a frequency-dependent diagonal dyadic, is applicable to general ro- tationally symmetric targets (not just cylinders and rings). Index Terms—Induction, magnetization, resonance, sensing. I. INTRODUCTION E LECTROMAGNETIC induction (EMI) is widely used for the detection and discrimination of conducting and fer- rous targets. In the context of subsurface sensing, EMI is used to sense buried metal mines as well as unexploded ordnance (UXO) [1], [2]. Since the conductivity of such targets is typi- cally many orders of magnitude larger than the conductivity of soil, the target can usually be analyzed in free space. We con- sider such an analysis here. Most land mines and UXO exhibit rotational symmetry, with this symmetry yielding a simplifica- tion of the EMI phenomenology. In particular, as discussed fur- ther in Section II, the magnetization tensor of such targets be- comes diagonal (with regard to a coordinate system with an axis along the target axis of symmetry). Moreover, we demonstrate that each element of the diagonal corresponds to a magnetic dipole moment, oriented either parallel to or orthogonal to the target axis. The wideband frequency-domain EMI response of a con- ducting and ferrous target is characterized by EMI resonant frequencies, which exist along the imaginary axis of the Manuscript received June 18, 2000; revised October 27, 2000. This work was supported in part by Strategic Environmental Research and Development Pro- gram Contract DACA72-99-C-0012-CU-1123, a U.S. Army Research Office Demining MURI, and by the Air Force Office of Scientific Research. L. Carin and H. Yu are with the Department of Electrical and Computer En- gineering, Duke University, Durham, NC 27708-0291 USA. Y. Dalichaouch, A. R. Perry, andP. V. Czipott are with Quantum Magnetics, San Diego, CA 92192 USA. C. E. Baum is with the Air Force Research Laboratory, Directed Energy Di- rectorate, Kirtland Air Force Base, Albuquerque, NM 87108 USA. Publisher Item Identifier S 0196-2892(01)04840-9. complex-frequency plane [3]. Note that here we use the plane for complex frequencies, instead of the plane as in the two-sided Laplace transform [4]. Assuming an time dependence, a purely imaginary resonant frequency corresponds to the exponential decay characteristic of time-domain EMI interaction with conducting and ferrous targets. As is discussed in Section II, the imaginary resonant frequencies corresponding to magnetic-dipole modes play an important role in characterizing the frequency and time-domain properties of the magnetization tensor. The magnetization tensor and EMI resonances have been discussed previously [4]. The principal contribution of this paper is a recognition that the EMI resonant modes of high-con- ductivity targets, characterized by purely imaginary resonant frequencies, are analogous to the resonant modes of high-per- mittivity, low-loss microwave resonators [5], which resonate at nearly purely real resonant frequencies. Once this duality is understood, all of the many techniques developed in the microwave community for characterization of high-permittivity dielectric resonators [6]–[8] (operating at nearly real frequen- cies) can be directly transitioned to the EMI high-conductivity, high-permeability problem (with resonant frequencies at imaginary frequencies). Moreover, this understanding yields an explicit interpretation of the EMI resonant modes in terms of dipole and higher order magnetic moments, from which the physical significance of particular EMI modes can be accrued. Many metal land mines and UXO are composed of parts that resemble finite-length cylinders and rings and therefore, the EMI resonant modes of such targets are of interest. Such targets, as well as more-general shapes, can be analyzed via numerical algorithms, such as the method of moments (MoM) [3]. Unfortunately, such calculations are very expensive computationally, since the size of the MoM basis functions typically must be small relative to the skin depth [3]. This requirement significantly limits the size of the targets that can be considered, and results in significant computer run times, and computer memory requirements. However, for simple but important shapes, such as finite-length cylinders and rings, one can make use of the duality between EMI resonances and microwave resonances to yield very simple but highly accurate approximate models. In particular, as detailed in Section II, we have employed these simple models to calculate the (imaginary) resonant frequencies of conducting cylinders and rings, with the computed data compared favorably to data simulated numerically. In those examples we restrict ourselves to relatively small targets, for which the numerical analysis is tractable. The approximate model is applicable to cylinders and 0196–2892/01$10.00 © 2001 IEEE