Conditions of Perfect Imaging in Negative Refraction Materials with Gain

Light propagation is analyzed in a negative refraction material (NRM) with gain achieved by pumping. An inherent spatial “walkoff” 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.


Introduction
Negative refraction is known to offer a wide range of potential applications [1][2][3][4].However, losses, which are an inherent feature of the negative refraction, present a major impediment to the performance of NRMs [5][6][7][8][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][15][16][17][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 ε r (ω) = ε + iε and permeability μ r (ω) = μ + iμ , were studied in detail [8].The complex effective refractive index is then defined as n 2 (ω) = ε r (ω)μ r (ω) or n(ω) = n + in .In free space, the incident wave vector k is real, while in the lossy NRM, the wave vector q = q x x + (q z + iq z ) z is complex.At a given optical frequency ω, this implies that q To analyze light propagation in the NRM, the phase and group velocities are expressed as /dω are determined by the NRM dispersion.The energy propagation is approximately determined by the group velocity under the assumption of low losses [19][20][21].The Poynting vector can also be used to define the energy propagation.
For complex vectors v p and v g , the direction of the phase propagation and energy transfer in the wave packet are determined by their real parts [22]: ( In an ideal NRM, where the refractive index is negative without losses, the phase velocity and group velocity are strictly antiparallel [1,19].However, (1) and (2) show that the group velocity is no longer antiparallel because of the contribution of the last term in (2).This spatial "walk-off," that is, the noncollinearity between the phase propagation and the energy transfer, becomes obvious in a homogenous, isotropic, lossy NRM.The angle between the phase velocity and the group velocity is with the "walk-off " angle defined as 180 • − δ.
The propagation behavior is discussed here for both propagating and evanescent waves.The dispersive curve is described as the Lorentz model, with ε r host (ω 1(a) for the real and imaginary parts of the refractive index.For a typical propagating wave with |q x | < |q|, the size of the "walk-off " is numerically simulated as shown in Figure 1(b).The analysis of the "walk-off " can be extended to evanescent waves with |q x | > |q|, where it is found that the "walk-off " dramatically increases with |q x |, also shown in Figure 1(b).
For perfect focusing, the "walk-off " appearing at different |q x | should be suppressed.It was shown that this goal can be achieved by a pumping gain scheme [18].Here, we show, in a pumped four-level model of signal amplification, that the realization of perfect imaging is possible only for a specific wavelength under strict pumping condition.

NRMs with Pumping Gain
Four-level systems represent conventional gain media.The intensity of light in a chosen spectral interval can be amplified in NRMs by introducing an extra term in the electrical field susceptibility [14,17,23].It is assumed here that the gain medium is pumped in the linear regime, and no gain saturation arises.Accordingly, the population of the ground level N 1 (which can be considered as the population of the gain medium) is much larger than in the other three levels as per the usual pumping condition.With the definition of polarization P = χ ex ε 0 E, the permittivity, with the extra term in the susceptibility, is given by As shown below, the last term of ( 4) is crucial to the performance of the gain-compensated NRMs.The components of the effective refractive index, n and n , are shown in Figure 2.
Whereas the optical losses in the NRM can be effectively compensated by pumping, as shown in Figure 2(b), an amplification of the input signal is achievable; the elimination of the "walk-off " depends on both the real and imaginary parts of the refractive index.The ideal case is the one with n = −1 and n = 0, giving rise to perfect imaging [1,6].However, as shown in Figure 2, this condition holds only at the optimal frequency where ω = 121.28× 10 12 s −1 under appropriate pumping conditions.
Optical imaging with resolution above or below the diffraction limit depends on the system's ability to recover the wavevector's component for either propagating or evanescent waves.Figure 3(b) shows the size of the "walk-off " for different |q x | with the appropriate pumping rate of 0.21 × 10 9 s −1 .The propagating waves correspond to |q x | < |q|, while the evanescent ones correspond to |q x | > |q|.After introducing the pumping gain, a red shift is observed in the n curve.At the optimal frequency (ω = 121.28× 10 12 s −1 ) where n = −1, the angles between the group and phase velocities are strictly antiparallel for all |q x |.Because of the antiparallel directions of the energy transfer and phase propagation, the spatial "walk-off " is suppressed, so that the ability of directional transmission (for the propagating waves) and perfect focusing (for evanescent waves in the near-field) will be preserved.Thus, the pumping can effectively cancel the losses only in a limited spectral region, under appropriate pumping conditions.This conclusion is in agreement with those reported in [11,12].
By contrast, the propagation in an active NRM under excessive pumping exhibits a peculiar behavior.Figure 3(c) shows the "walk-off " angles at different |q x | for excessive pumping rate (here Γ pump = 0.48 × 10 9 s −1 ) at the frequency where n = −1 (here ω = 121.44 × 10 12 s −1 ).The δ angles are then larger than 180 • , indicating that the "walk-off " reappears, with the respective angles 180 • + δ.The "walkoff " becomes more significant at larger |q x |.It also shows that δ increases dramatically with the increase of |q x | for the evanescent wave with |q x | > |q|.Hence, the perfect focus for the near-field component is impossible under excessive pumping.Notice that because of the red shift in n (ω), the "walk-off " is suppressed at the frequency of 121.20×10 12 s −1 , where n (ω) ≈ −1.3 (the red arrow in Figure 3(c)).However, perfect lensing requires n (ω) = −1 [1,6], hence the perfect focus cannot be obtained under excessive pumping.
In order to achieve perfect focusing, the pumping rates should be reduced and the pumping central frequencies should be blue-shifted, as shown in Figure 3(c).Light with all values of |q x | can then perfectly focus through the slab.

Conclusions
To conclude, we have analyzed the effect of gain on the negative refraction in NRMs.In a lossy NRM, even though it is isotropic and homogeneous, the group and phase velocities    are not strictly antiparallel, yielding a spatial "walk-off ", which may restrict the applications of NRMs in a variety of fields.By introducing gain, losses can be effectively reduced, and light amplification can be realized within a narrow spectral range.Appropriately setting the gain to strictly cancel the losses, the "walk-off " for both propagating and evanescent waves can be effectively eliminated for all values of |q x |, leading to an ideal NRM.However, for excessively pumped NRMs, the spatial "walk-off " reappears.Thus, the use of optical pumping to realize perfect imaging is restricted to a very narrow spectral region, under precisely defined pumping conditions.An alternative method of overcoming NRM losses without signal distortion may involve selfinduced transparency (SIT) solitons, which were predicted in metamaterials [24], in analogy with SIT in other resonantly absorbing structures [25,26].

Figure 2 :
Figure2: The parameters for the four-level system are chosen as σ = 10 −4 C 2 /kg (C stands for Coulomb), which is the strength of the coupling between the gain material and the host NRM, γ ex = 0.76 × 10 12 s −1 is related to the linewidth of the gain medium, ω x = 121.6 × 10 12 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 Figure1(a)), and the decay time of the gain level 2 is τ 32 = 5 × 10 −12 s.The value of the occupation density is set to be N 1 = 5 × 10 23 m −3 .The pumping rates Γ pump are assumed to be 0.12 × 10 9 s −1 for insufficient pumping, 0.21 × 10 9 s −1 for the appropriate pumping (to be discussed below), and 0.48 × 10 9 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.

Figure 3 :
Figure 3: The "walk-off " angles as function of frequency ω in an active (pumped) NRM.The dashed line in all the plots shows the frequencies at which n (ω) = −1.(a) The blue line is the spatial "walk-off " angle with the pumping rate of Γ pump = 0.21 × 10 9 s −1 at |q x | = 0.01|q|.Both losses and the "walk-off " within the gain window are precisely canceled.(b) The δ angles obtained at the pumping rate of Γ pump = 0.21 × 10 9 s −1 .The group velocities and phase velocities are strictly antiparallel in this case (δ = 180 • ) at different |q x | when n (ω) = −1 (at ω = 121.28× 10 12 s −1 ).(c) The δ angles obtained at the pumping rate of Γ pump = 0.48 × 10 9 s −1 .Even though at this pumping rate the NRM is turned into a light-amplifying medium, the spatial "walk-off " reappears and depends on the value of |q x | when n (ω) = −1, which may lead to degradation of the optical performance of an NRM.