Conformal transformations in design of the coatings with gain-loss permittivity

We extend a general conformal mapping approach to complex space transformations and apply it to design optical elements that possess loss and gain. We describe the methodology for tailoring the transformation optical media and analyze a ground-plane cloak as an example. We believe that our approach can be employed for the design of various optical and plasmonic devices and the potential impact can be significant in other fields.

Figure 1

Illustration of (a) the $E$-field pattern with the cloak, (b) the $E$-field pattern when a bare reflecting surface of the shape given by Eq. (15) is present, and (c) the corresponding profile of $\varepsilon$ ensuring ground cloaking. The beam width is 4 $\mu$m at a wavelength of 0.5 $\mu$m.

Figure 2

Scattering of the plasmon field from (a) the bump of the form (15) cloaked with the designed shell and (b) the bare bump.

Figure 3

(a) Spatial distribution of permittivity that operates as a conventional grating coupler of the form (18) and (b) magnetic-field pattern for the scattering of the plasmon from it and (c) from the equivalent grating. The dashed line in (b) corresponds to the boundary of the shell.

Figure 4

(a) Purely imaginary shift of the $x$ coordinate in the transformation defined by Eq. (19) $\zeta \left(x\right)$ (solid curve) and its derivative $d\zeta /dx$ (dashed curve) and (b) the corresponding spatial distribution of the imaginary part of permittivity; the real part of $\varepsilon$ is close to 1.

Figure 5

Distribution of the (a) real and (b) imaginary parts of permittivity that provides the absorption profile given in Fig. 4 and also the nonreflecting and nonscattering propagation of the plasmon along the deformed air-metal interface.

Figure 6

Magnetic-field structure of the plasmon scattered by the grating coupler (a) with the reflecting surface of the form given in Eq. (23), (b) with the dielectric coating that corresponds to the permittivity profile depicted in Fig. 5, and (c) with the fivefold increase of the absorption $a=5a_0$ [see Eq. (22)].