Cylindrical superlens by a coordinate transformation

Cylinder-shaped perfect lens deduced from the coordinate transformation method is proposed. The previously reported perfect slab lens is noticed to be a limiting form of the cylindrical lens when the inner radius approaches infinity with respect to the lens thickness. Connaturality between a cylindrical lens and a slab lens is affirmed by comparing their eigenfield transfer functions. We numerically confirm the subwavelength focusing capability of such a cylindrical lens with consideration of material imperfection. Compared to a slab lens, a cylindrical lens has several advantages, including finiteness in cross section and ability in lensing with magnification or demagnification. Immediate applications of such a cylindrical lens can be in high-resolution imaging and lithography technologies. In addition, its invisibility property suggests that it may be valuable for noninvasive electromagnetic probing.

Figure 1

(Color online) Schematic illustration of the principle for idealized perfect cylindrical lensing. (a) Spatial mapping function. (b) Imaging of a line current source by a perfect cylindrical lens. Color map shows the real part of the $E_z$ field.

Figure 2

(Color online) Comparison of transfer functions for a cylindrical lens and a slab lens of the same thickness. Three different object positions are shown: for cylindrical lens $r_{o,1}=\frac{5ab-a^2}{8b-4a}$, $r_{o,2}=\frac{3ab-a^2}{4b-2a}$, $r_{o,3}=\frac{7ab-3a^2}{8b-4a}$; and for slab lens $x_{o,1}=-\frac{5d}{4}$, $x_{o,2}=-d$, $x_{o,3}=-\frac{3d}{4}$. Two sets of object positions are equivalent when $a$ and $b$ are much larger than $d$. For the cylindrical lens the corresponding angular momentum number is marked on the secondary $x$ axis.

Figure 3

(Color online) Effect of $a$ on cylindrical wave transfer function. Inset gives the zoom-in view of the transfer function with $a=1.5\lambda$ (inward).

Figure 4

(Color online) Effect of $\delta$ on cylindrical wave transfer function. Azimuthal momentum number is marked on the secondary $x$ axis.

Figure 5

(Color online) Radial $\left|{\text{E}}_z\right|$ field distributions with increasing material loss. The vertical solid black lines denote lens boundaries; dashed red lines denote object and image positions.

Figure 6

(Color online) Inward imaging of a line current source. (a) $\delta =0.0001–0.00001i$. Overvalued fields are shown in saturated color as dark red (positive) or dark blue (negative), and contour lines at the levels of $\pm 1000$, $\pm 2000$, $\pm 3000$, $\pm 4000$, $\pm 5000$, and 6000 are marked. (b) $\delta =0.0001-0.001i$. Contour lines at the levels of $\pm 1000$, $\pm 1500$, 2000, 2500, 3000, and 3500 are marked. Yellow crosses are object and image positions. Domain size: $2.5\lambda \times 2.5\lambda$.

Figure 7

(Color online) Imaging of two parallel line current sources. Panels in the first row are for the lens with $\delta =0.0001–0.00001i$, and those in the second row are for the lens with $\delta =0.0001-0.001i$. For panels in both rows, from left to right, $\Delta L_{\theta }$ values are, respectively, $0.16\lambda$, $0.18\lambda$, $0.20\lambda$, and $0.22\lambda$. Notice that the image has been stretched by a factor of $\left(2b-a\right)/a$ due to the higher permittivity inside the lens. Red (gray) lines: with lens; blue (dark gray) lines: without lens. Thick lines: intensity at image plane; thin lines: intensity at object plane.