Transforming Space with Non-Hermitian Dielectrics

Coordinate transformations are a versatile tool to mold the flow of light, enabling a host of astonishing phenomena such as optical cloaking with metamaterials. Moving away from the usual restriction that links isotropic materials with conformal transformations, we show how nonconformal distortions of optical space are intimately connected to the complex refractive index distribution of an isotropic non-Hermitian medium. Remarkably, this insight can be used to circumvent the material requirement of working with refractive indices below unity, which limits the applications of transformation optics. We apply our approach to design a broadband unidirectional dielectric cloak, which relies on nonconformal coordinate transformations to tailor the non-Hermitian refractive index profile around a cloaked object. Our insights bridge the fields of two-dimensional transformation optics and non-Hermitian photonics.

Figure 1

Transforming space with non-Hermitian media. (a) The virtual coordinates, with the branch cuts of the conformal (light green) and nonconformal (dark green) Zhukovsky maps. The area of each square in the grid is $0.4\times 0.4$. After the mappings, the green lines form circles, see (b) and (c), inside of which is the cloaked region (gray shaded area). The local orthogonality of the coordinate lines, a signature of conformal mapping, is present only in (b) but not in (c) (in both plots only the upper Riemann sheet is depicted). (d) Visualization of light propagation in the proposed non-Hermitian cloak profile, with the blue and red lines being the equivalent of rays in a non-Hermitian landscape (see the SM [42]). The yellow surface indicates the imaginary part of the dielectric function, that causes the blue and red lines to rise or fall in the direction orthogonal to the $x\text{−}y$ plane, indicating the local creation [$\nabla ·\mathbf{S}(x,y)>0$] or destruction [$\nabla ·\mathbf{S}(x,y)<0$] of energy flux. The projection of the lines onto the 2D plane coincides with the inversely transformed coordinates, plotted in the grid below, with the cloaked region shaded in gray.

Figure 2

Refractive index distributions for the conformal and nonconformal Zhukovsky cloaks. (a) Real-valued refractive index distribution of the conformal Zhukovsky cloak, reaching values of $n=0$ at $x=\pm 2.4\lambda$, and requiring the use of ENZ materials. Here, $\lambda ={\lambda }_0/n_0$ and $n_0=3.1$. (b) Real (left) and imaginary (right) parts of the refractive index spatial profile of the nonconformal Zhukovsky map. The refractive index based on Eq. (4) is calculated using the transformation $w(z,z^{*})$ with $\eta (z,z^{*})=1/[1+e^{\beta (|z|-R_1)}]-1/[1+e^{\beta (|z|-R_0)}]$ and $R_0=0.25$, $R_1=1$, $\beta =5.75$. The values for $n_R(x,y)$ lie between 1.013 and 3.594, whereas those of $n_I(x,y)$ lie between $-1.013$ and 1.013. The black filled circle in all panels indicates the cloaked region.

Figure 3

Non-Hermitian cloaking for an incoming plane wave at the central design spatial frequency $k=n_0{k}_0$. (a) Electric field solution for the case when cloaking is present. The refractive index distribution is that of Fig. 2, with the dashed white box denoting the area plotted there. The black ring represents the annulus of the cloak, where the Neumann boundary condition of vanishing normal electric field derivative was used. Inside the annulus there is a highly reflective material, taken here as aluminum ($n_{\mathrm{Al}}=1.52+9.26i$ at $1\mu \mathrm{m}$ [60]), with a circular cross section. (b) The electric field solution for the same incoming field but with only the highly reflective material present. The cloak’s outer radius is $2.86\lambda$, while its inner radius is $2.62\lambda$. The fields are normalized to the incoming wave amplitude. The black arrow marks the propagation direction of the incoming beam.

Figure 4

Demonstration of cloaking under pulsed illumination, for the optical media as in Fig. 3. (a) A spatially narrow pulse incident onto the cloaking region (left) is nearly perfectly transmitted (right) when the cloak is present. (b) When the cloak is absent, there is a distortion of the pulse shape, clearly revealing the presence of the reflecting material ($n_{\mathrm{Al}}=1.52+9.26i$) to the outside observer. The thin black dashed line denotes the separation of the plotting regions for the incoming (left) and outgoing (right) part of the pulse evolution; all other markings are as in Fig. 3. The time evolution was calculated by an inverse Fourier transform, using a Gaussian envelope with a width of ${\sigma }_k=0.055k$ in $k$ space.