Light propagation is analyzed in a negative refraction material (NRM) with gain achieved by pumping. An inherent spatial “walk-off” between the directions of phase propagation and energy transfer is known to exist in lossy NRMs. Here, the analysis is extended to the case where the NRM acts as an active material under various pumping conditions. It is shown that the condition for perfect imaging is only possible for specific wavelengths under special excitation conditions. Under excessive gain, the optical imaging can no longer be perfect. 1. Introduction Negative refraction is known to offer a wide range of potential applications [1–4]. However, losses, which are an inherent feature of the negative refraction, present a major impediment to the performance of NRMs [5–9]. To overcome these problems, NRMs with gain were proposed to compensate the losses, even to turn the materials into amplified systems. Nevertheless, it is often stated that the gain will destroy the negative refraction due to causality considerations [10], although the statement was disputed by a theory demonstrating that negative refraction may be preserved in a limited spectral region [11, 12]. Common methods to introduce gain in NRMs include optical parametric amplification (OPA) [13] and externally pumped gain materials [14–18]. Optical imaging needs to collect both propagating and evanescent waves. However, only within a limited range may the wave vectors receive gain from OPA because of the strict phase-matching condition, the application of OPA to achieve perfect imaging in NRMs is not possible. In this paper, we demonstrate that, under the action of the pumping gain, lossless and amplified light propagation may occur in a special spectral window of the NRM. The propagation behavior is shown to be closely related to the dispersion and pumping configuration. Propagation in NRMs is also examined in different pumping configurations. 1.1. Spatial “Walk-Off” in Lossy NRMs Light incidents from free space onto a homogeneous, isotropic, lossy NRM, of permittivity and permeability , were studied in detail [8]. The complex effective refractive index is then defined as or . In free space, the incident wave vector is real, while in the lossy NRM, the wave vector is complex. At a given optical frequency , this implies that for both the propagating wave ( ) and the evanescent one ( ). To analyze light propagation in the NRM, the phase and group velocities are expressed as and , where and are determined by the NRM dispersion. The energy propagation is approximately determined by the group
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