Analysis of the time-reversal operator for a small spherical scatterer in an electromagnetic field

The time-reversal operator (TRO) for a planar array of crossed dipole elements illuminating a small conducting and/or dielectric sphere is investigated in order to determine the general properties of an electromagnetic time-reversing array system. The behavior of such a system for a given frequency is analyzed by studying the eigenvalues and eigenvectors of the TRO. Each eigenvector specifies a set of complex driving currents for the array elements that produce received voltages that are proportional to the conjugates of the drive currents. The proportionality constant is equal to the square root of the associated eigenvalue and is the same for all elements. The eigenvalues and eigenvectors can be determined by performing a singular value decomposition (SVD) on the multistatic data matrix of the array. The eigenvalues of the TRO are the squares of the singular values, and the eigenvectors are identical to the singular vectors. We have shown that the maximum number of singular vectors associated with the sphere is equal to the number of orthogonal orientations of the dipole moments induced in the sphere when irradiated by the array, so there is a maximum of six for a conducting sphere but only three are significant when the conductivity is small and the sphere may be considered being just a dielectric. Numerical results are presented for linear and circular arrays to show the general behavior of the system.