In this work we analytically and numerically study the impact of dye infiltration in multilayer hyperbolic meta- materials. The dyes we use in our model are four-level dyes with an absorption and an emission line. Starting from the Maxwell-Bloch equations we derive a semi-analytical model for the coupling of an oscillator in a one- dimensional periodic structure. We compare our model with exact results obtained using the transfer-matrix method. In the weak coupling regime, both absorption and emission lines perturb the optical mode (in a finite multilayer) or the band (in an infinite multilayer) in similar ways, introducing positive and negative imaginary parts to the dispersion of Bloch vector within the bands without altering their shapes. In contrast, in the strong-coupling regime, the two lines are responsible for different phenomena: while coupling to the absorption line causes a severe distortion of the band due to classical Rabi splitting, coupling to the emission line causes fork-shaped bifurcations reminiscent of PT-symmetry breaking scenarios, with apparition of exceptional points at the loss-gain compensation frequencies.
Hyperbolic metamaterials are examined for many applications thanks to the large density of states and extreme confinement of light they provide. For classical hyperbolic metal/dielectric multilayer structures, it was demon- strated that the properties originate from a specific coupling of the surface plasmon polaritons between the metal/dielectric interfaces. We show a similar analysis for 2D hyperbolic arrays of square (or rectangular) silver nanorods in a TiO2 host. In this case the properties derive from a specific coupling of the plasmons carried by the corners of the nanorods. The dispersion can be seen as the coupling of single rods for a through-metal connection of the corners, as the coupling of structures made of four semi-infinite metallic blocks separated by dielectric for a through-dielectric connection, or as the coupling of two semi-infinite rods for a through-metal and through-dielectric situation. For arrays of small square nanorods the elementary structure that explains the dispersion of the array is the single rod, and for arrays of large square nanorods it is four metallic corners. The medium size square nanorod case is more complicated, because the elementary structure can be one of the three basic designs, depending on the frequency and symmetry of the modes. Finally, we show that for arrays of rectangular nanorods the dispersion is explained by coupling of the two coupled rod structure. This work opens the way for a better understanding of a wide class of metamaterials via their elementary excitations.
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