Conditions of Perfect Imaging in Negative Refraction Materials with Gain

Figure 2

The parameters for the four-level system are chosen as 𝜎=10⁻⁴ C²/kg (C stands for Coulomb), which is the strength of the coupling between the gain material and the host NRM, 𝛾_ex=0.76×10¹² s⁻¹ is related to the linewidth of the gain medium, 𝜔_𝑥=121.6×10¹² s⁻¹ is the central pumping frequency, determined by the frequency difference between state 3 and state 2, at which 𝑛′(𝜔)≈−1 is satisfied without the gain (see Figure 1(a)), and the decay time of the gain level 2 is 𝜏₃₂=5×10⁻¹² s. The value of the occupation density is set to be 𝑁₁=5×10²³m⁻³. The pumping rates Γ_pump are assumed to be 0.12×10⁹ s⁻¹ for insufficient pumping, 0.21×10⁹ s⁻¹ for the appropriate pumping (to be discussed below), and 0.48×10⁹ s⁻¹ for the excessive pumping, respectively. (a) The real part of the refractive index for different pumping amplitudes. A red shift of the frequency is observed at 𝑛^′=−1. (b) The imaginary part of the refractive index for different pumping amplitudes. Loss-free windows (intervals of amplification) are revealed by the 𝑛^′′ curves. Increasing the pumping rate may compensate the losses, or even turn the material into an active medium.

(a)

(b)