Semianalytical Model for Design and Analysis of Grating-Assisted Radiation Emission of Quantum Emitters in Hyperbolic Metamaterials

We propose a semianalytical method to model, in both two and three dimensions (2D and 3D, respectively), the radiation emission of quantum emitters (QEs) interacting with nanopatterned structures. We then investigate the emission from QEs near a hyperbolic metamaterial (HMM) with a metallic cylindrical grating on its top and a poly­(methyl methacrylate) substrate embedded with QEs on its bottom. The optimization of the cylindrical grating is carried out first using a 2D model (due to its low computational cost), followed by a performance study based on a 3D model. We show that an appropriate choice of grating parameters (period, height, and fill factor) allows not only the control of the QE emission direction but also the increase of both the Purcell factor and the total power coupled from the HMM into free space. In addition, the proposed method provides a detailed mapping of both the Purcell factor and the radiated power as a function of position, enabling us to understand how the QE location affects its behavior. Furthermore, we demonstrate that the QEs with the highest Purcell factor (viz., perpendicularly polarized ones) contribute more to the power radiated into the far field than previously expected. We also show that, in addition to a high Purcell factor of about 145, perpendicularly polarized QEs radiate up to 2 times more power if placed 10 nm from the HMM as they would in free space.