Invisibility on demand based on a generalized Hilbert transform

We propose a feasible invisibility approach to suppress the scattering of waves from or to given directions and for particular frequencies, i.e., invisibility on demand. We derive a generalized Hilbert transform for a specific invisibility arrangement relating the two quadratures of the complex permittivity of an object. The scheme allows either designing objects to be invisible, or alternatively modifying the complex susceptibility of a given object to render invisibility. The theoretical proposal is further confirmed by finite-difference time-domain numerical calculations. Moreover, following an iterative chain of generalized Hilbert transforms, we propose the invisibility on demand with additional constraints, i.e., restricting the required modification of the complex refractive index within practical limits by avoiding gain areas. The proposed concept not only opens a venue of real or complex valued index landscapes, but also bridges the gap between Hermitian and non-Hermitian optical systems by unifying these systems under one single generalized design theory.

Figure 1

Illustration of unidirectional invisibility and invisibility on demand. (a) In full unidirectional invisibility, in order to prevent left reflection of every right-propagating wave, (b) all modulation components on the left half-plane, $k_x<0$, in $\mathbf{k}=\left(k_x,k_y\right)$ space must be set to zero. (c) For invisibility on demand, to prevent backscattering in a particular angular range (d) the modulation components to be set to zero are just the ones within a limited invisibility area in $\mathbf{k}=\left(k_x,k_y\right)$ space. In particular, the invisibility case (c) requires uncoupling the scattering of incident waves around $\mathbf{k}=\left(k_0,0\right)$ into back reflections, i.e., into waves in the vicinity of $\mathbf{k}=\left(-k_0,0\right)$; then, the invisibility function has to be centered at $\mathbf{k}=\left(-2k_0,0\right)$.

Figure 2

Invisibility on demand procedure workflow in the spatial and wave-vector domain. Top: Initial object: (a) electric susceptibility, and the (b) scattering potential. Center: Invisibility on demand by a Gaussian invisibility area, in the form of $exp[-{(k_x+2k_0)}^2/\left(2{\sigma }_x^2\right)-{(k_y+2k_0)}^2/\left(2{\sigma }_y^2\right)]$ centered at $-2k_0=\left(k_x,k_y\right)=\left(0,20\right)\mu {\mathrm{m}}^{-1}$ with standard deviations of ${\sigma }_x=7.0\mu {\mathrm{m}}^{-1}$ and ${\sigma }_y=7.0\mu {\mathrm{m}}^{-1}$ in the $k_x$- and $k_y$-directions, respectively, in (c) kernel in k space, (d) modified scattering potential, (e) real part, and the (f) imaginary part of the susceptibility profile after 14 iterations. Bottom: Invisibility by an eclipse-shaped invisibility area: (g) invisibility area in k space, (h) modified scattering potential, (i) real part, and the (j) imaginary part of the susceptibility profile after 14 iterations.

Figure 3

Invisibility on demand performance. FDTD simulations for the $\mathsf{S}$-shaped object [given in Fig. 2], probed by a monochromatic $({\lambda }_0=1.1\mu \mathrm{m})$ point source at different angular positions, from left to right: ${208}^{\circ }$, ${180}^{\circ }$, and ${32}^{\circ }$ (with respect to a vertical axis, in clockwise direction), when invisibility is (a)–(c) not activated, and (d)–(i) activated with the same invisibility function as in Fig. 2 [for (d)–(f)] and Fig. 2 [for (g)–(i)] after 14 iterations. The scatterer is outlined by white solid lines. The white dashed lines show the angular range of invisibility on demand. For a full angular analysis, see Movie 1 in the Supplemental Material [35]. Angle-wavelength scattering spectra obtained with a plane wave source, for (j) invisibility not activated, and (k) activated. (l) Relative scattering (ratio between scattering spectra for inactivated and activated invisibility) in the angle-wavelength domain.

Figure 4

Invisibility areas in Fourier space and corresponding kernels. Invisibility areas with (a) elliptical shape (first column), (d) rectangular shape (second column), both with widths $w_x=0.5k_0$ and $w_y=1.0k_0$—where $k_0$ is the input wave number of light (which corresponds to ${30}^{\circ }$); (g) complete circle shape (third column) and (j) solar eclipse shaped (fourth column) for these last two $w=0.1{\lambda }_0$. Corresponding associated kernels of the generalized Hilbert transform, (b), (e), (h), (k) real parts, and (c), (f), (i), (l) imaginary parts.

Figure 5

Numerical results for the case of a solar eclipse–shaped kernel with high refractive index contrasts. (a)–(i) FDTD simulations of waves coming from point sources at different positions towards the scatterer with an initial refractive index of (a), (d), (g) 2.0, (b), (e), (h) 3.0, and (c), (f), (i) 4.0. The upper row corresponds to the cases where the invisibility is not activated, the middle row corresponds to invisibility activated cases, and the lower row corresponds to cases where the invisibility is activated but where the source is located outside the invisibility range. The white dashed lines denote the incident angle where invisibility is activated. The operational wavelength is equal to $1.0\mu \mathrm{m}$.

Figure 6

Invisibility on demand procedure applied on a complex-shaped object. (a)–(c) Complex susceptibility profiles of an Einstein face–shaped object modified with a Gaussian invisibility area located at $\left(k_x,k_y\right)=\left(0,-20.0\right)\mu {\mathrm{m}}^{-1}$ with a standard deviation of $7.0\mu {\mathrm{m}}^{-1}$ in both the $k_x$ and $k_y$ directions. (a) The real part of the initial object and the (b) real and (c) imaginary parts of the modified complex susceptibility after nine iterations.

Figure 7

Numerical results of a complex-shaped scatterer. (a)–(f) FDTD simulations of waves coming from point sources at different positions. Scattering object is an Einstein face–shaped object given in Fig. 6. The upper and lower rows correspond to cases where the invisibility is not activated and activated, respectively. The white dashed lines denote the angle range where invisibility is activated. For a full angular analysis, see Movie 2 in the Supplemental Material [35].

Figure 8

Invisibility on demand procedure applied with two symmetrically shaped kernels. (a)–(c) Complex susceptibility profiles of an $\mathsf{S}$-shaped object modified with two symmetrically shaped invisibility areas. (a) The real part of the initial object and the (b) real and (c) imaginary parts of the modified complex susceptibility. The invisibility areas are located at $\left(k_x,k_y\right)=\left(0,\pm 11.4\right)\mu {\mathrm{m}}^{-1}$ with a standard deviation of $6.0\mu {\mathrm{m}}^{-1}$ in both the $k_x$ and $k_y$ directions.

Figure 9

Reflection test for two symmetrically placed kernels. (a)–(f) FDTD simulations of waves coming from point sources at different positions for the scatterer given in Fig. 8. The upper and lower rows correspond to cases where the invisibility is not activated and activated, respectively. Excitation angles correspond to (a), (d) ${24}^{\circ }$, (b), (e) ${190}^{\circ }$, and (c), (f) ${270}^{\circ }$. The white dashed lines denote the angular range where invisibility is activated. The operational wavelength corresponds to 1.1 μm.