PT -symmetric scattering systems with balanced gain and loss can undergo a symmetry-breaking transition in which the eigenvalues of the non-unitary scattering matrix change their phase shifts from real to complex values. We relate the PT -symmetry breaking points of such an unbounded scattering system to those of underlying bounded systems. In particular, we show how the PT -thresholds in the scattering matrix of the unbounded system translate into analogous transitions in the Robin boundary conditions of the corresponding bounded systems. Based on this relation, we argue and then confirm that the PT -transitions in the scattering matrix are, under very general conditions, entirely insensitive to a variable coupling strength between the bounded region and the unbounded asymptotic region, a result which can be tested experimentally and visualized using the concept of Smith charts.

We experimentally generate and characterize the eigenstates of the Wigner-Smith time-delay matrix, called principal modes, in a multimode fiber with strong mode coupling. The unique spectral and temporal properties of principal modes enable a global control of the temporal dynamics of optical pulses transmitted through the fiber, despite random mode mixing. Our analysis reveals that the well-defined delay time of the eigenstates are formed by multi-path interference, which can be effectively manipulated by the spatial degrees of freedom of the input wavefront. This study is essential to controlling the dynamics of wave scattering, paving the way for coherent control of pulse propagation through complex media.

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A fundamental insight in the theory of diffusive random walks is that the mean length of trajectories traversing a finite open system is independent of the details of the diffusion process. Instead, the mean trajectory length depends only on the system's boundary geometry and is thus unaffected by the value of the mean free path. Here we show that this result is rooted on a much deeper level than that of a random walk, which allows us to extend the reach of this universal invariance property beyond the diffusion approximation. Specifically, we demonstrate that an equivalent invariance relation also holds for the scattering of waves in resonant structures as well as in ballistic, chaotic or in Anderson localized systems. Our work unifies a number of specific observations made in quite diverse fields of science ranging from the movement of ants to nuclear scattering theory. Potential experimental realizations using light fields in disordered media are discussed.wave scattering | disordered media | random walk | diffusion | time delay I n the biological sciences it has been appreciated for some time now that the movement of certain insects (such as ants) on a planar surface can be modeled as a diffusive random walk with a given constant speed v (1-3). Using this connection, Blanco and Fournier (4) proved that the time that these insects spend on average inside a given domain of area A and with an external boundary C is independent of the parameters entering the random walk such as, for example, the transport mean free path (MFP) ℓ p . Specifically, the average time t between the moments when an insect enters the domain and when it first exits it again is given by the simple relation hti = πA=ðCvÞ. One finds that the mean length hli of the corresponding random walk trajectories inside the domain is also constant, hli = hti v = πA=C. Similar relations also hold in three dimensions, hti = 4V =ðΣvÞ and hli = 4V =Σ, where V is the volume and Σ is the external surface of a given domain. Extensions of this result exist for trajectories beginning inside the domain (5) or for the calculation of averaged residence times inside subdomains (6). As a generalization of the mean-chord-length theorem (7) for straight-line trajectories with an infinite MFP, this fundamental theorem has numerous applications, for instance in the context of food foraging (8) and for the reaction rates in chemistry (9).The surprising element of this result can be well appreciated when applied to the physical sciences and, in particular, to the transport of light or of other types of waves in scattering media. In that context it is well known that the relevant observable quantities all do depend on ℓ p : In the diffusive regime, the total transmission of a slab of thickness L scales with ℓ p