Interdimensional optical isospectrality inspired by graph networks

Abstract

A network picture has been applied to various physical and biological systems to understand their governing mechanisms intuitively. Utilizing discretization schemes, both electrical and optical materials can also be interpreted as abstract 'graph' networks composed of couplings (edges) between local elements (vertices), which define the correlation between material structures and wave flows. Nonetheless, the fertile structural degrees of freedom in graph theory have not been fully exploited in physics owing to the suppressed long-range interaction between far-off elements. Here, by exploiting the mathematical similarity between Hamiltonians in different dimensions, we propose the design of reduceddimensional optical structures that perfectly preserve the level statistics of disordered graph networks with significant long-range coupling. We show that the disorder-induced removal of the level degeneracy in high-degree networks allows their isospectral projection to one-dimensional structures without any disconnection. This inter-dimensional isospectrality between high-and low-degree graph-like structures enables the ultimate simplification of broadband multilevel devices, from three-to one-dimensional structures.Random scattering provides the spread of momentum and energy components of waves [1][2][3], differentiating disordered materials from periodic or quasiperiodic materials [4,5]. Due to the critical needs in light harvesting [6], robust bandgaps [7,8], and ultrafast optics [9], disordered optical materials have also been intensively studied to exploit their broadband responses. Although one-(1D) [8,10], two-(2D) [1,11,12], and three-dimensional (3D) [13] disordered structures have been considered promising candidates for broadband and omnidirectional operations, the deliberate control of disordered structures [1,3,7,8,14] has been achieved only in 1D or 2D structures, owing to the difficulty of manipulating 3D randomness.To understand the role of the 'dimension' in optics, we can employ the interdisciplinary viewpoint from network theory. For the structures composed of coupled resonances, light flows inside those can be interpreted as signal transport over graph networks [14][15][16][17]. Reciprocity constructs the undirected network [14,18], whose vertices and edges denote resonances and coupling, respectively. The strength of disorder then corresponds to the graph irregularity, and the dimension of material structures determines the 'degree' of graphs, which quantifies the number of coupling paths in space. This graphbased perspective reveals that optical structures in 3D real space cover only restricted parts of general graph theory [18], originating from the difficulty in connecting far-off vertices in real space. At most, nearestneighbor and next-nearest-neighbor coupling [19] have been considered in the light transport.Here, we focus on the real-space reproduction of the level statistics in hypothetical optical networks of arbitrary couplings, consequently deriving the 'isospectrality' between the stru...

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