Coalescence of Anderson-localized modes at an exceptional point in 2D random media

Description

In non-hermitian systems, the particular position at which two eigenstates coalesce under a variation of a parameter in the complex plane is called an exceptional point. A non-perturbative theory is proposed which describes the evolution of modes in 2D open dielectric systems when permittivity distribution is modified. We successfully test this theory in a 2D disordered system to predict the position in the parameter space of the exceptional point between two Anderson-localized states. We observe that the accuracy of the prediction depends on the number of localized states accounted for. Such an exceptional point is experimentally accessible in practically relevant disordered photonic systems. Losses are inherent to most physical systems, either because of dissipation or as a result of openness. These systems are described mathematically by a non-hermitian Hamiltonian, where eigenvalues are complex and eigen-states form a nonorthogonal set. In such systems, interaction between pairs of eigenstates when a set of external parameters is varied is essentially driven by the existence of exceptional points (EP). At an EP, eigenstates coa-lesce: Complex eigenvalues degenerate and spatial distributions become collinear. In its vicinity, eigenvalues display a singular topology [1] and encircling the EP in the parameter space leads to a residual geometrical phase [2, 3]. Since their introduction by Kato in 1966 [4], EPs have turned to be involved in a rich variety of physical effects: Level repulsion [5], mode hybridization [6], quantum phase transition [7], lasing mode switching [8], PT symmetry breaking [9, 10] or even strong coupling [11]. They have been observed experimentally in different systems such as microwave billiards [12], chaotic optical mi-crocavities [13] or two level atoms in high-Q cavities [11]. Open random media are a particular class of non-hermitian systems. Here, modal confinement may be solely driven by the degree of scattering. For sufficiently strong scattering, the spatial extension of the modes becomes smaller than the system size, resulting in transport inhibition and Anderson localization [14]. Disordered-induced localized states have raised increasing interest. They provide with natural optical cavities in random lasers [15, 16]. They recently appeared to be good candidate for cavity QED [17, 18], with the main advantage of being inherently disorder-robust. These modes can be manipulated by a local change of the disorder and can be coupled to form necklace states [19-21], which open channels in a nominally localized system [22, 23]. These necklace states are foreseen as a key mechanism in the transition from localization to diffusive regime [24]. PT symmetry has been studied in the context of disordered media and Anderson localization [25-27] but so far EPs between localized modes have not been investigated. In this letter, coalescence at an EP between two Anderson-localized optical modes is demonstrated in a two dimensional (2D) dielectric random system. To bring the system in the vicinity of an EP, the dielectric permit-tivity is varied at two different locations in the random system. We first propose a general theory to follow the spectral and spatial evolution of modes in 2D dielectric open media. This theory is applied to the specific case of Anderson-localized modes to identify the position of an EP in the parameter space. This prediction is confirmed by Finite Element Method (FEM) simulations. We show that this is a highly complex problem of multiple mode interaction where a large number of modes are involved. We believe that our theory opens the way to a controlled local manipulation of the permittivity and the possibility to engineer the modes. Furthermore, we think this approach can be easily extended to others kinds of networks e.g. coupled arrays of cavities [28, 29]. We first consider the general case of a finite-size dielec-tric medium in 2D space, with inhomogeneous dielectric constant distribution, ǫ(r). In the frequency domain, the electromagnetic field follows the Helmholtz equation: ∆E(r, ω) + ǫ(r)ω 2 E(r, ω) = 0 (1) where E(r, ω) stands for the electrical field and the speed of light, c = 1. Eigensolutions of eq. (1), define the modes or eigenstates of the problem: (Ω i , |Ψ i) i∈N | ∆|Ψ i + ǫ(r)Ω 2 i |Ψ i = 0 (2) Because of its openness, the system has inherent losses, thus is described by a non-hermitian Hamiltonian. For non-hermitian systems, modes are a priori non-orthogonal, complex and their completeness is not ensured. Here, we consider open systems with finite range permittivity ǫ(r) and where a discontinuity in the permit-tivity provides a natural demarcation of the problem. For

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