Overcoming Losses with Gain in a Negative Refractive Index Metamaterial

On the basis of a full-vectorial three-dimensional Maxwell-Bloch approach we investigate the possibility of using gain to overcome losses in a negative refractive index fishnet metamaterial. We show that appropriate placing of optically pumped laser dyes (gain) into the metamaterial structure results in a frequency band where the nonbianisotropic metamaterial becomes amplifying. In that region both the real and the imaginary part of the effective refractive index become simultaneously negative and the figure of merit diverges at two distinct frequency points.

Figure 1

Illustration of the double-fishnet structure with a square unit cell of side-length $p$ highlighted. The two perforated silver films are embedded in a dielectric host material which holds the dye molecules (translucent). Dimensions are given in the text. Pump (red dashed line) and probe (blue solid line) pulses illustrate the pump-probe configuration with the electric field polarized along the $x$ direction.

Figure 2

(a) Snapshot of the occupation inversion $\Delta N_{\mathrm{e}}$ in a plane 5 nm below the upper silver fishnet film just before probing and (b) the electric field enhancement at 710 nm in the same plane; both for a pump-field amplitude of $2.0\mathrm{kV}/\mathrm{cm}$. The silver film perforation is indicated by a white rectangle. (c) and (d) show the same physical quantities in a plane given by the propagation direction and the long side of the perforation. The dashed black lines in (a) and (c) highlight the positions of the intersection with the other plane, respectively.

Figure 3

(a) Real and imaginary part of the retrieved effective refractive indices of the double-fishnet structure for different pump amplitudes. The peak electric field amplitude of the pump increases in steps of $0.5\mathrm{kV}/\mathrm{cm}$ from no pumping (cyan line, lightest) to a maximum of $2.0\mathrm{kV}/\mathrm{cm}$ (black line, darkest). The inset shows the real and imaginary part of the effective permeability (black and red line, respectively) and the result of the Kramers-Kronig relation (black and red dotted lines) for the highest peak electric field amplitude of $2.0\mathrm{kV}/\mathrm{cm}$. (b) The figures-of-merit (FOM) for the same pumping amplitudes.

Figure 4

Detailed view of (a) the imaginary part of the retrieved effective refractive indices $n^{\prime \prime }$, (b) the absorption, and (c) the figures of merit (FOM) for peak pump-field amplitudes close to and above compensation between 1.5 and $2.1\mathrm{kV}/\mathrm{cm}$ in steps of $0.1\mathrm{kV}/\mathrm{cm}$.