Power drainage and energy dissipation in lossy but perfect lenses

It has recently been shown that passive lens designs can retain perfect lensing despite intrinsic loss in the comprising left-handed materials. Here we show that energy conservation is not at odds with the operation of such lossy perfect lenses: The irreversible transfer of electromagnetic power to the lossy material dictates that a smaller portion of the power emitted by the source arrives at the image, but image fidelity remains absolute. Moreover, the power that does arrive at the perfect image must drain from the system there. This is shown to apply to any layered perfect lens based on left-handed media, including the well-known Veselago lens. Combined with what is known about Maxwell's fisheye lens, our results suggest that power drainage is universal to all perfect lenses and is intrinsic to any perfect image. Properly addressing this unique power dynamics is therefore essential to any successful realization of practical perfect lenses with existing metamaterials.

Figure 1

Field solution [Eq. (9)] for a stationary current line source $(x=-h,z=0)$ in front of a matched (${\mu }_{\mathrm{DNG}}={\varepsilon }_{\mathrm{DNG}}=-1, {\mu }_{\mathrm{DPS}}={\varepsilon }_{\mathrm{DPS}}=1$) left-handed material slab $(0<x<d)$ for (a) $h=3.3\lambda >d$, (b) $h=2.8\lambda =d$, and (c) $h=1.6\lambda <d(\lambda =2\pi /k_0)$. Red arrows denote the direction of the time-averaged Poynting vector.

Figure 2

Field solution [Eq. (10)] for a stationary current line source ($x_0=-1.6\lambda , z=0$) in front of a specific matched (${\mu }_{\mathrm{DNG}}={\varepsilon }_{\mathrm{DNG}}=-1, {\mu }_{\mathrm{DPS}}={\varepsilon }_{\mathrm{DPS}}=1$) DNG-DPS multilayer (interfaces at $l_i/\lambda =0,2.8,4,6,7,9$): Its singular image points (green dots, $x_i=2l_i-x_{i-1}$) can be real [$x_1, x_{2,3}$ (coinciding), $x_6$] or virtual ($x_4, x_5$). Red arrows denote the direction of the time-averaged Poynting vector.

Figure 3

Contour line illustration of the singular [solid blue line, Eq. (12b)] and nonsingular [dashed light blue line, Eq. (12c)] field contributions for a stationary current line source in front of a mismatched left-handed material (gray). Red arrows denote the direction of the time-averaged Poynting vector related to the singular field contribution around the source and image points [green dots, Eq. (19)].

Figure 4

The TE single-interface lens point-to-point power efficiency $P_{\mathrm{out}}/P_{\mathrm{in}}$ [Eq. (20b), negative sign dropped], between a singular source and its singular image, as a function of mismatch in μ (${\mu }_{\mathrm{DNG}}=-{\mu }_{\mathrm{DPS}}$ marked by dashed blue lines). The black curve denotes the $\mathrm{Re}\left\{{\mu }_{\mathrm{DNG}}\right\}$ versus $\mathrm{Im}\left\{{\mu }_{\mathrm{DNG}}\right\}$ relation for best power efficiency [Eq. (21)]. The TM power efficiency has an identical dependence on mismatch in ε.