Decomposition of the time reversal operator: Detection and selective focusing on two scatterers

The decomposition of the time reversal operator (DORT) method is a selective detection and focusing technique using an array of transmit–receive transducers. It relies on the theory of iterative time reversal mirrors which was presented by Prada et al. C. Prada, J. L. Thomas, and M. Fink, J. Acoust. Soc. Am. 97, 62–71 (1995). The time reversal operator was defined as K*(ω)K(ω), where ω is the frequency, * means complex conjugate, and K(ω) is the transfer matrix of the array of L transducers insonifying a time invariant scattering medium. It was shown that this time reversal operator can be diagonalized and that for ideally resolved scatterers of different reflectivities, each of its eigenvectors of nonzero eigenvalue provides the phase law to be applied to the transducers in order to focus on one of the scatterers. The DORT method consists in determining these eigenvectors and using them for the selective focusing. This paper presents a complete analysis of this method in the case of two scatterers. The mathematical expressions of the eigenvectors are given and several experimental results are described. In particular, the effectiveness of the method to focus selectively through an inhomogeneous medium is established.