Invisibility cloaks, originally existing in scientific fiction and movies, have attracted people’s interest for centuries. For military applications, they can be used as stealth coating to conceal tanks, aircrafts, submarines, and so on. For commercial applications, they can be used to prevent unwanted scattering from defects in antennas, lasers, electronic circuits, and so on. The recent development of transformation optics and metamaterials greatly promotes the experimental realization of cloaking devices [1–10]. As theoretically demonstrated by Pendry and coworkers in 2006, ideal invisibility cloaks in free space generally require near-zero permittivity/permeability components. Although several types of such cloaks have been experimentally implemented in the microwave regime [11,12], the resonant elements involved (to realize near-zero permittivity or permeability) greatly limit their bandwidths. To tackle this problem, Li and Pendry proposed the concept of a “carpet” cloak, which, instead of transparentizing the hidden object, makes it reflect light as a flat surface [13]. Since carpet cloaks no longer require near-zero value in permittivity or permeability, they have the intrinsic advantages of lower dissipation loss and broader working bandwidth as compared with free-space cloaks [13]. Under optical surface transformation, the optic-null medium helps to build a wide-illumination range and low-loss invisibility cloak, while it only works under TM polarization [14].

In general, carpet cloaks can be designed using either quasi-conformal mapping or bilinear transformation [15]. Earlier experimental realizations of carpet cloaks are generally based on quasi-conformal mappings [16–19]. This type of carpet cloaks operates on inhomogeneous refractive index and works for unpolarized light as demonstrated at both microwave [20] and optical frequencies [21]. However, the approximation made in quasi-conformal mapping leads to a lateral shift of the reflected light beam, whose value is comparable to the dimension of the hidden object, and hence makes the object detectable [22]. On the other hand, carpet cloaks designed with bilinear transformation make use of several birefringent crystals judiciously glued together [23,24]. This type of cloak eliminates the lateral shift, but it only works for a specific light polarization. Recently, with the advent of metasurfaces, an ultrathin carpet cloak design was proposed. By judiciously designing the meta-atoms to achieve the required phase shift and restore the phase front, the full-polarization metasurface-based skin cloak has been realized [25]. In addition, the tunable metasurface illusion device can be constructed by applying programmable voltage source to adjust the varactor configurations on the metasurface [26]. However, these metasurface cloak designs have limited working bandwidths. Up to now, the design and implementation of a broadband carpet cloak that is polarization insensitive while addressing the problem of lateral shift are still challenging.

In this paper, we attempt to tackle the two problems mentioned above simultaneously. We target at mid-infrared (IR) frequency range, owing to its wide application in military and industry, such as semiconductor processing, spectroscopy, chemical and biomolecular sensing, and security. Many mid-IR devices have been proposed and implemented, including lasers [27,28], sensors [29,30], and frequency combs [31]. However, to the best of our knowledge, an invisibility cloak has never been realized in this frequency range [32–40]. Our approach makes use of a judiciously designed conformal mapping, and the resultant carpet cloak relies only on an isotropic nonresonant medium. Moreover, our carpet cloak design is based on commercially available silicon-on-insulator (SOI) wafers and is suitable for mass production of on-chip cloaking devices.

According to the theory of transformation optics, light reflected from the bump surface covered with the cloak will propagate in exactly the same way as the reflected beam from a planar surface. The transformation parameters a, b, and t control the shape of the bumps (e.g., height of bumps h) as well as the range of refractive indices of the cloak (e.g., minimum and maximum values of refractive indices) according to h=ln{[be2x0−(2ab+1)e−x0+a2b+a](e2x0+2a+a2)[be2x0+(2ab+1)e−x0+a2b+a](e2x0−2a+a2)},(4)nmin=n0(e2x0−2aex0+a2)[b2e2x0−2(1+ab)bex0+(1+ab)2]|ex0|,(5)nmax=n0(e2x0+2aex0+a2)[b2e2x0+2(1+ab)bex0+(1+ab)2]|ex0|,(6)where x0=12ln(a2+ab)−t.(7)

In principle, a, b, and t can be arbitrary positive values but must be judiciously designed in order to make the implementation of the cloak experimentally feasible. In our design, these parameters are chosen as a=0.005, b=0.005, and t=3.7 to hide 0.41λ0 high bumps in a background with a refractive index n0=2.7. The corresponding refractive indices of the cloak range from 2.1562 to 3.2427. As depicted by the contour plot in Fig. 1, the minimum and the maximum refractive indices occur at the valleys and the peaks of the bumps, respectively. Away from the bumps, the cloak index gradually changes to n0 (i.e., he background index), fulfilling the condition of impedance match.

We analytically retrieve and compare the light trajectories in the systems of a cloak and a virtual slab as shown in Fig. 1. In the cloak system, the ray position vector is obtained by solving a differential eikonal equation [42] dds[n(r)drds]=∇n(r),(8)where r is the ray position vector, n(r) is the index of refraction, and s is an incremental path distance. For arbitrary n(r), [Eq. (8)] can be solved numerically, e.g., through a graphic iterative method [43]. In our case, the refractive index of the cloak takes the form of Eq. (3), and the analytical solution of the ray position vector r can be obtained.

In the system of a virtual slab, the reflecting plane denoted by the blue dashed line is located at the bottom of the cloak. The thickness of this virtual slab equals the height of the cloak indicated by H. The ray diagram presented in Fig. 1 clearly shows the lateral shift δ between the reflected light ray from the cloaked bumps and that from a flat perfect electric conductor (PEC) plane [44–46]. The lateral shift δ results from the nonzero value of H−t, i.e., height difference between the equivalent reflection plane of the cloak (yellow dashed plane) and the virtual reflection plane of the slab (blue dashed plane). Mathematically, δ takes the following form: δ=2(H−t)tanθ,(9)where θ is the incident angle.

To test the cloak performance for TM polarization, we plot the magnetic field distributions in Figs. 3(e)–3(h) when the polarization of the incident Gaussian beam is switched to TM. As expected, the scattering produced by the bumped surface in Figs. 3(e) and 3(g) is eliminated by the carpet cloak in Figs. 3(f) and 3(h), respectively. Considering the generality of our conformal transformation approach, this minimized lateral shift polarization-robust carpet cloak can be implemented in any frequency range with a proper choice of the material.

The required refractive index distribution of the cloak can be realized by drilling subwavelength holes with different radii into the silicon device layer at a fixed periodicity. As mentioned before, the refractive index of the background medium in our design is set as n=2.7, and the refractive index distributions of the cloak span from 2.1562 to 3.2427 as shown in Fig. 1. We use the following equation to deduce the hole radius distribution for the whole device: R(w)=p(n′n0)2−neff2π(neff2−1),(12)where R(w) is the hole radius, p is the periodicity of the holes, and n′ is the refractive index distribution of the cloak. When the periodicity of the holes is fixed at p=1.5μm, such a refractive index distribution can be realized by varying the hole radius from 75 to 1300 nm.

The commercial software CST Microwave Studio is used to examine the performance of the cloak implemented on the SOI wafer. The refractive indices of silicon and silica are taken from Ref. [47] and Ref. [48], respectively. Figures 4(b) and 4(c) plot the magnetic field intensities at the top plane of the silicon device layer when a Gaussian beam is incident upon (1) a bare bump and (2) a bump surface covered by the cloak. Through the comparison, we can observe remarkable scattering from the bare bump, whereas such scattering is dramatically suppressed by the presence of the cloak, giving rise to a single reflection profile. In other words, the cloak has successfully concealed the curved bump, making it reflect light as a flat PEC plane. Moreover, the suppressed scattering is observed for arbitrary incident angles, verifying the excellent performance of the cloak implemented on the SOI wafer. Since light will never penetrate into the PEC bump, any object could be hidden behind the bump and appear invisible to the observers outside.

In conclusion, we propose a mid-IR carpet cloak based on a judiciously designed conformal transformation. This cloak operates on isotropic refractive index, which can be easily implemented with nonresonant structures. Analytical calculations and numerical simulations show that through properly choosing the periodicity p and increasing the cloak height H, the lateral shift can be effectively minimized with the reflection controlled. The polarization-robust performance over a broadband frequency range is also demonstrated. The practical implementation of this cloak can be realized on the platform of commercially available SOI wafers by drilling spatially gradient holes with judiciously designed radii into the silicon layer. Our conformal transformation approach not only provides an alternative solution for cloak design but also offers a general platform for the design of other optical devices such as lenses, waveguides, and optical cavities.