Self‐consistent Purcell factor and spontaneous topological transition in hyperbolic metamaterials

Abstract

IntroductionThe Purcell effect is the ratio of the spontaneous emission rate of a dipole placed inside the media in comparison to the rate in vacuum [1]. This ratio can be tuned by engineering the environment of the source and the possibility of controllable change of radiative lifetime has been already demonstrated in a variety of different set-ups [2][3][4][5][6][7]. One of the most promising systems for this purpose are hyperbolic metamaterials, highly anisotropic uniaxial media with dielectric permittivity tensor principal components of different signs [8]. This defines the hyperboid shape of the isofrequency contours for the extra-ordinary waves, which leads to a number of unique properties and as consequence to the variety of applications [9]. In particular, hyperbolic isofrequency surface leads to a broadband singularity in photonic density of states and as a consequence to the dramatic increase of the Purcell factor [10,11]. It has been later shown that there are different mechanisms to regularize the density of states singularity: finite emitter size [12], finite size of the metamaterial unit cell [13], and nonlocal response [14]. However, experimental studies of the Purcell effect in hyperbolic metamaterials [7,[15][16][17] still demonstrate sufficient discrepancies with the theoretical predictions. In this Letter, we show that there is another source of the distinction between the theoretical predictions and measurements results originating of the strong frequency dispersion of the metamaterial effective parameters. Namely, we