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Dynamic and steady-state aspects of wave propagation are deeply connected in lossless open systems ‎in which the scattering matrix is unitary. There is then an equivalence among the energy excited within ‎the medium through all channels, the Wigner time delay, which is the sum of dwell times in all ‎channels coupled to the medium, and the density of states. But these equivalences fall away in the ‎presence of material loss or gain. In this paper, we use microwave measurements, numerical ‎simulations, and theoretical analysis to discover the changing relationships among fundamental wave ‎properties with loss and gain, and their dependence upon dimensionality and spectral overlap. We ‎begin with the demonstrations that the transmission time in random 1D media is equal to the density ‎of states even in the presence of ultrastrong absorption and that its ensemble average is independent ‎of the strengths of scattering and absorption. In contrast, the Wigner time becomes imaginary in the ‎presence of loss, with real and imaginary parts that fall with absorption. In multichannel media, the ‎transmission time remains equal to the density of states and is independent of the scattering strength ‎in unitary systems but falls with absorption to a degree that increases with the strengths of absorption ‎and scattering, and the number of channels coupled to the medium. We show that the relationships ‎between key propagation variables in non-Hermitian systems can be understood in terms of the ‎singularities of the phase of the determinant of the transmission matrix. The poles of the transmission ‎matrix are the same as those of the scattering matrix, but the transmission zeros are fundamentally ‎different. Whereas the zeros of the scattering matrix are the complex conjugates of the poles, the ‎transmission zeros are topological: in unitary systems they occur only singly on the real axis or as ‎conjugate pairs. We follow the evolution and statistics of zeros in the complex plane as random ‎samples are deformed. The sensitivity of the spacing of zeros in the complex plane with deformation ‎of the sample has a square-root singularity at a zero point at which two single zeros and a complex ‎pair interconvert. The transmission time is a sum of Lorentzian functions associated with poles and ‎zeros. The sum over poles is the density of states with an average that is independent of scattering ‎and dissipation. But the sum over zeros changes with loss, gain, scattering strength and the number of ‎channels in ways that make it possible to control ultranarrow spectral features in transmission and ‎transmission time. We show that the field, including the contribution of the still coherent incident ‎wave, is a sum over modal partial fractions with amplitudes that are independent of loss and gain. The ‎energy excited may be expressed in terms of the resonances of the medium and is equal to the dwell ‎time even in the presence of loss or gain.‎ 

We show in random matrix theory, microwave measurements, and computer simulations that the mean free ‎path of a random medium and the strength and position of an embedded reflector can be determined from ‎radiation scattered by the system. The mean free path and strength of the reflector are determined from the ‎statistics of transmission. The statistics of transmission are independent of the position of the reflector. The ‎reflector's position can be found, however, from the average dwell time for waves incident from one side of ‎the sample.‎ 

Understanding vanishing transmission in Fano resonances in quantum systems and metamaterials and perfect and ultralow transmission in disordered media has advanced the knowledge and applications of wave interactions. Here, we use analytic theory and numerical simulations to understand and control the transmission and transmission time in complex systems by deforming a medium and adjusting the level of gain or loss. Unlike the zeros of the scattering matrix, the position and motion of the zeros of the determinant of the transmission matrix (TM) in the complex plane of frequency and field decay rate have robust topological properties. In systems without loss or gain, the transmission zeros appear either singly on the real axis or as conjugate pairs in the complex plane. As the structure is modified, two single zeros and a complex conjugate pair of zeros may interconvert when they meet at a square root singularity in the rate of change of the distance between the transmission zeros in the complex plane with sample deformation. The transmission time is the spectral derivative of the argument of the determinant of the TM. It is a sum over Lorentzian functions associated with the resonances of the medium, which is the density of states, and with the zeros of the TM. Transmission vanishes, and the transmission time diverges as zeros are brought near the real axis. Monitoring the transmission and transmission time when two zeros are close may open new possibilities for ultrasensitive detection.