We present a computational scheme allowing for a self-consistent treatment of a dispersive metallic photonic metamaterial coupled to a gain material incorporated into the nanostructure. The gain is described by a generic four-level system. A critical pumping rate exists for compensating the loss of the metamaterial. Nonlinearities arise due to gain depletion beyond a certain critical strength of a test field. Transmission, reflection, and absorption data as well as the retrieved effective parameters are presented for a lattice of resonant square cylinders embedded in layers of gain material and split ring resonators with gain material embedded into the gaps. 78.20.Ci, 41.20.Jb The field of metamaterials 1,2 is driven by fascinating and far-reaching theoretical visions such as, e.g., perfect lenses, 3 invisibility cloaking, 4,5 and enhanced optical nonlinearities. 6 This emerging field has seen spectacular experimental progress in recent years.1,2 Yet, losses are orders of magnitude too large for the envisioned applications. Achieving such reduction by further design optimization appears to be out of reach. Thus, incorporation of active media (gain) might come as a cure. The dream would be to simply inject an electrical current into the active medium, leading to gain and hence to compensation of the losses. However, experiments on such intricate active nanostructures do need guidance by theory via self-consistent calculations (using the semi-classical theory of lasing) for realistic gain materials that can be incorporated into or close to dispersive media to reduce the losses at THz or optical frequencies. The need for self-consistent calculations stems from the fact that increasing the gain in the metamaterial, the metamaterial properties change, in turn changes the coupling to the gain medium until a steady-state is reached. A specific geometry to overcome the severe loss problem of optical metamaterials and to enable bulk metamaterials with negative magnetic and electric response and controllable dispersion at optical frequencies is to interleave active optically pumped gain material layers with the passive metamaterial lattice.For reference, the best fabricated negative-index material operating at around 1.4 µm wavelength 7 has shown a figure of merit (FOM) = −Re(n)/Im(n) ≈ 3, where n is the effective refractive index. This experimental result is equivalent to an absolute absorption coefficient of α = 3 × 10 4 cm −1 , which is even larger than the absorption of typical direct-gap semiconductors such as, e.g., GaAs (where α = 10 4 cm −1 ). So it looks difficult to compensate the losses with this simple type of analysis, which assumes that the bulk gain coefficient is needed. However, the effective gain coefficient, derived from self-consistent microscopic calculations, is a more appropriate measure of the combined system of metamaterial and gain. Due to pronounced local-field enhancement effects in the spatial vicinity of the dispersive metamaterial, the effective gain coefficient can be substantially larg...
We design three-dimensional (3D) metallic nanowire media with different structures and numerically demonstrate that they can be homogeneous effective indefinite anisotropic media by showing that their dispersion relations are hyperbolic. For a finite slab, a nice fitting procedure is exploited to obtain the dispersion relations from which we retrieve the effective permittivities. The pseudo focusing for the real 3D wire medium agrees very well with the homogeneous medium having the effective permittivity tensor of the wire medium. Studies also show that in the long-wavelength limit, the hyperbolic dispersion relation of the 3D wire medium can be valid even for evanescent modes.Comment: 7 pages, 9 figure
Abstract. A self-consistent computational scheme is presented for one dimensional (1D) and two dimensional (2D) metamaterial systems with gain incorporated into the nanostructures. The gain is described by a generic four-level system. The loss compensation and the lasing behavior of the metamaterial system with gain are studied. A critical pumping rate exists for compensating the losses of the metamaterial. There exists a wide range of input signals where the composite system behaves linearly. Nonlinearities arise for stronger signals due to gain depletion. The retrieved effective parameters are presented for one layer of gain embedded in two layers of Lorentz dielectric rods and split ring resonators with two different gain inclusions: (1) gain is embedded in the gaps only and (2) gain is surrounding the SRR. When the pumping rate increases, there is a critical pumping rate that the metamaterial system starts lasing.
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