Recently published methods for the quasi-static limit of the Helmholtz equation is extended to consider near cloaking for the full Maxwell equations. Effective near cloaking structures are described for the electromagnetic scattering problem at a fixed frequency. These structures are, prior to using the transformation optics, layered structures designed so that their first scattering coefficients vanish. As a result, any target inside the cloaking region has near-zero scattering cross section

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... r a band of frequencies. Analytical results show that this construction significantly enhances the cloaking effect for the full Maxwell equations. [17], this regularization point of view is adopted for the Helmholtz equation. See also [21, 28] . More recently, Bao and Liu [8] considered near cloaking for the full Maxwell equations. They derived sharp estimates for the boundary effect due to a small inclusion with arbitrary material parameters enclosed by a thin high-conducting layer. Their results show that the near cloaking scheme can be applied to cloak targets from electromagnetic boundary measurements. In the recently published papers [5, 6] , it is shown that near cloaking, from measurements of the Dirichlet-to-Neumann map for the conductivity equation and of the scattering cross section for the Helmholtz equation, can be drastically enhanced by using multilayered structures. The structures are designed so that their generalized polarization tensors (GPTs) or scattering coefficients vanish (up to a certain order). GPTs are building blocks of the far-field behavior of solutions in the quasi-static limits (conductivity equations) [4] and the scattering coefficients are "Fourier coefficients" of the scattering amplitude [6, 7] . The multilayered structures combined with the usual change of variables (transformation optics) greatly reduce the visibility of an object. This fact is also confirmed by numerical experiments [3] . The purpose of this paper is to extend the results of [5, 6] to full Maxwell's equations. It shows that near cloaking from cross section scattering measurements at a fixed frequency can be enhanced by using layered structures together with transformation optics. Of particular importance is the notion of scattering coefficients of an inclusion, which is extended in this paper to full Maxwell equations. As for the Helmholtz equation, the layered structures, prior to using the transformation optics, are designed so that their first scattering coefficients vanish. It is also shown that inside the cloaking region, any target has a near-zero scattering cross section for a band of frequencies. Analytical results prove that this construction significantly enhances the near cloaking effect for the full Maxwell equations. It is worth mentioning that even if the basic scheme of this work is parallel to that of [6], the analysis is much more complicated due to the vectorial nature of the Maxwell equations. The paper is organized as follows. In section 2, some fundamental results on the scattering problem for the full Maxwell equations are recalled. Section 3 introduces the scattering coefficients of an electromagnetic inclusion and proves that the scattering coefficients are basically the spherical harmonic expansion coefficients of the far-field pattern. Section 4 is devoted to the construction of layered structures with vanishing scattering coefficients. Numerical examples of the scattering coefficient vanishing structures are presented. Section 5 shows that the near cloaking is enhanced if a scattering coefficient vanishing structure is used. Multipole solutions to the Maxwell equations. In this section, a few fundamental results related to electromagnetic scattering, which will be essential in what follows, are recalled. Consider the time-dependent Maxwell equations where μ is the magnetic permeability and is the electric permittivity. In the time-harmonic regime, one looks for the electromagnetic fields of the form where ω is the frequency. The couple (E, H) is a solution to the harmonic Maxwell Downloaded 03/12/14 to 143.248.118.122. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php